Stratigraphy and Structure of Part of the Western Sierra Nevada Metamorphic Belt, California GEOLOGICAL SURVEY PROFESSIONAL PAPER 410 Prepared in cooperation with the State of California, Department of Conservation, Division cf Mines and GeologyStratigraphy and Structure of Part of the Western Sierra Nevada Metamorphic Belt, California By LORIN D. CLARK GEOLOGICAL SURVEY PROFESSIONAL PAPER 410 Prepared in cooperation with the State of California, Department of Conservation, Division of Mines and Geology UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director The U.S. Geological Survey Library has cataloged this publication as follows: Clark, Lorin Delbert, 1918- Stratigraphy and structure of part of the western Sierra Nevada metamorphic belt, California. Washington, U.S. Govt. Print. Off., 1964. iv, 70 p. illus., maps (7 col.) diagrs. (1 col.) tables. 30 cm. (U.S. Geological Survey. Professional paper 410) Part of illustrative matter fold, in pocket. Prepared in cooperation with the State of California Dept, of Natural Resources, Division of Mines and Geology. Bibliography: p. 58-59. (Continued on next card) Clark, Lorin Delbert, 1918- Stratigraphy and structure of part of the western Sierra Nevada metamor-phic belt, California. 1964. (Card 2) 1. Geology, Stratigraphic. 2. Geology—California—Sierra Nevada Mountains. I. California. Division of Mines and Geology. II. Title. III. Title: Sierra Nevada metamorphic belt, California. (Series) For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C., 20402CONTENTS Page Abstract_______________________________________________ 1 Introduction............................................. 2 Location, culture, and accessibility___________________ 2 Physical features______________________________________ 3 Previous work________________________________________ 3 Purpose and scope of investigation_____________________ 4 Field work and acknowledgments_________________________ 5 Terminology__________________________________________ 5 Synopsis of geology_________________________________________ 6 Stages of the Middle and Late Jurassic_________________ 7 Stratigraphy______________________________________________ 8 Calaveras formation__________________________________ 8 Distribution and character________________________ 8 Eastern belt_______________________________________ 8 Clastic member_______________________________ 9 Volcam.,”'ember________________________________ 9 Argillaceous member___________________________ 9 Chert member__________________________________ 10 Western belt______________________________________ 12 Cosumnes River exposures______________________ 12 Mokelumne River and Jackson Creek exposures__________________________________ 13 Fossils and age.__________________________________ 13 Rocks of Jurassic age_____________:_______________ 15 Rocks exposed in the central block and Melones fault zone_____________________________________ 16 Rocks of Middle(?) Jurassic age_______________ 16 Amador group__________________________________ 17 Cosumnes formation_______________________ 17 Lithology___________________________ 17 Fossils and age______________________ 18 Logtown Ridge formation.................. 18 Lithology___________________'____ 19 Fossils and age______________________ 21 Penon Blanco volcanics________________________ 22 Lithology________________________________ 23 Age__________________________________ 23 Mariposa formation____________________________ 23 Lithology________________________________ 24 Fossils and age__________________________ 25 Strata of uncertain stratigraphic position____ 26 Volcanic strata______________________________ 26 Epiclastic strata_____________________________ 26 Rocks exposed in the western block and Bear Mountains fault zone____________________________ 27 Gopher Ridge volcanics_______________________ 27 Lithology______________________________ 28 Age------------------------------------ 29 Salt Spring slate_____________________________ 29 Lithology________________________________ 29 Fossils and age_________________________ 29 Merced Falls slate____________________________ 30 Lithology________________________________ 30 Age.................................. 30 6>£ 7^ Pk v.y-io-Li\\ (L■e-jp, L^- Page Stratigraphy— Continued Rocks of Jurassic age—Continued Bear Mountains fault zones—Continued Copper Hill volcanics________________________ 30 Lithology________________________________ 30 Age______________________________________ 31 Peaslee Creek volcanics______________________ 31 Lithology________________________________ 31 Age______________________________________ 31 Stratigraphic relations between the formations of the western block___________________ 31 Stratigraphic relations between the Salt Spring slate and Gopher Ridge volcanics___________________________________ 31 Stratigraphic relations between the Salt Spring slate and Copper Hill volcanics___________________________________ 33 Stratigraphic relations of the Merced Falls slate to the Peaslee Creek volcanics and Gopher Ridge volcanics.. 33 Rock and mineral fragments in rocks of Jurassic age_____________________________________________ 33 Composition and shape of fragments___________ 35 Origin of limestone blocks___________________ 37 Source of fragments__________________________ 38 Bedding features and conditions of deposition. . 39 Green schist__________________________________________ 41 Plutonic rocks_____________________________________________ 41 Ultramafic rocks______________________________________ 42 Age_____________________________________________ 42 Structural relations______________________________ 42 Gabbroic to granitic rocks____________________________ 42 Age_______________________________________________ 43 Hypabyssal rocks___________________________________________ 43 Structural geology_________________________________________ 44 Top determinations____________________________________ 45 Faults________________________________________________ 46 Melones fault zone________________________________ 47 Description of localities____________________ 49 Bear Mountains fault zone_________________________ 50 Direction and amount of movement------------------ 51 Mother Lode fissure system________________________ 51 Minor faults______________________________________ 52 Age of faulting___________________________________ 53 Structure of the eastern block______________________ 53 Structure of the central block________________________ 54 Structure of the western block________________________ 54 Small-scale structures________________________________ 55 Cleavage and schistosity__________________________ 55 Minor folds_______________________________________ 56 Lineations________________________________________ 56 References cited___________________________________________ 58 Description of map units___________________________________ 59 Index______________________________________________________ 69 in 467IV STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT ILLUSTRATIONS [Plates 1-11 In map case] Plate 1. Geologic map of the western Sierra Nevada between the Merced and Cosumnes Rivers, Calif. 2. Expanded block diagram showing geology of the western Sierra Nevada between the Merced and Cosumnes Rivers, Calif. 3-8. Geologic maps and composite cross sections along: 3. Merced River. 4. Tuolumne River. 5. Stanislaus River. 6. Calaveras River. 7. Mokelumne River. 8. Cosumnes River. 9. Stratigraphic sections within the central fault block. 10. Geologic map showing type section of Amador group. 11. Geologic map and cross section of epiclastic rocks east of Merced Falls, Merced County, Calif. Page Figure 1. Index map showing location of area studied. 2 2. Canyon of Stanislaus River___________________ 3 3. Block diagram showing intertonguing_________ 16 4. Photomicrograph of mudflow conglomerate. 5. Photograph of volcanic breccia of probable 18 mudflow origin_____________________________ 20 Page Figure 6. Photograph showing scattered large clasts in bedded pyroclastic groundmass__________ 21 7. Photograph of thin-bedded to laminated tuff. 21 8. Photograph of thin-bedded tuff showing intra- formational breccia bed below dime and small-scale crossbedding under breccia_ 22 9. Photograph of clastic sills of medium-grained tuff_______________________________________ 22 10. Geologic map and cross section showing struc- ture of slate on Tuolumne River east of La Grange, Stanislaus County, California__ 34 11. Photomicrograph of very coarse graywacke from Cosumnes formation____________________ 35 12. Photomicrograph of very coarse graywacke from Mariposa formation____________________ 36 13. Photomicrograph of graded tuff bed__________ 45 14. Photomicrograph showing relation of cleav- age to graded bed of fine graywacke in Salt Spring slate_______________________________ 46 15. Photograph of small asymmetric anticline showing axial plane cleavage_______________ 47 16. Geologic map and cross section showing struc- ture of bedded pyroclastic rocks west of Don Pedro Dam______________________________ 48 TABLES Tables 1-6. Description of map units: 1. Merced River_______ 2. Tuolumne River_____ 3. Stanislaus River___ 4. Calaveras River____ Page 59 61 62 64 Tables 1-6. Description of map units—Continued 5. Mokelumne River___________ 6. Jackson Creek______M------ 7. Cosumnes River____________ Page 64 66 66STRATIGRAPHY AND STRUCTURE OF PART OF THE WESTERN SIERRA NEVADA METAMORPHIC BELT, CALIFORNIA By Lorin D. Clark ABSTRACT The area described in this repoft lies on the lower western slopes of the Sierra Nevada due east of San Francisco Bay and northwest of Yosemite National Park. It is about 90 miles long and 30 miles wide and is underlain by steeply dipping meta-morphic rocks of Paleozoic and Mesozoic age lying between the Sierra Nevada batholith on the east and overlapping Tertiary deposits of the Central Valley of California on the west. The historically famous Mother Lode gold belt extends northwestward through the center of the area. Distribution of the different units of metamorphic rocks is profoundly influenced by two nearly vertical major fault zones that divide the region into three structural blocks. The orientation and amount of net slip along the fault zones have not been definitely established but the horizontal component is probably much greater than the vertical component; the apparent vertical component of movement amounts to many thousands of feet Rocks of Paleozoic age constitute the eastern fault block, which lies east of the Mother Lode. All the Paleozoic rocks have previously been assigned to the Calaveras formation, and this usage is retained. The upper part of the Calaveras formation is of Permian age, but the formation is perhaps tens of thousands of feet thick, and the age of lower parts is unknown. Five lithologic members have been distinguished within the Calaveras formation, but no revision of the nomenclature is proposed. The Calaveras formation consists largely of originally argillaceous rocks and chert and is notable for a paucity of detrital material as coarse as sand, except in the lowermost part. Volcanic rocks form lenses in most of the formation and thick, extensive sequences in some places. Limestone also forms widespread lenses, but is prominent only in the east-central . part of the area. Mesozoic rocks constitute the western fault block and most of the central block. Paleontologically dated Mesozoic rocks are of Late Jurassic age. They are divisible in a general way into sequences of epiclastic rocks, largely slate, graywacke and conglomerate, and volcanic rocks, although epiclastic and volcanic rocks are commonly interbedded and in some places intertongue. The graywacke and conglomerate of all the formations are similar in composition. The most abundant clastic fragments are of volcanic rocks, slate, and chert, but fragments of metamorphic rocks, plutonic rocks, and quartzite are widespread. The various volcanic formations also have many features in common; most are composed largely of tuff and volcanic breccia, but lavas, in part having pillow structure, form thick sequences locally. Felsic volcanic rocks are common in one of the volcanic formations, but lacking in the others. Type localities of previously named formations are within the central fault block, and because of the difficulty of correlating between the two fault blocks new names are proposed for formations of the western fault block. Formations exposed in the central block are conformable and include the Cosumnes formation, of probable Late Jurassic age, consisting largely of epiclastic rocks; the Logtown Ridge formation, ranging from Callovian to late Oxfordian or early Kimmeridgian in age, and consisting of volcanic, chiefly pyroclastic rocks; the Penon Blanco volcanics, of Late Jurassic age, consisting of lava and pyroclastic rocks; and the Mariposa formation, of late Oxfordian or early Kimmeridgian age or both, consisting chiefly of epiclastic rocks. Conglomerate is very abundant in the Cosumnes formation and part of the Mariposa formation and contains many pebbles and boulders of limestone derived from the Calaveras formation. Coarse, poorly sorted conglomerates, previously thought to be tillite, are probably of mudflow origin. The section is everywhere truncated by faults or erosion, but the total thickness of the Jurassic strata is in excess of 15,000 feet. In the western fault block, a lower formation of volcanic rocks, a middle formation of epiclastic rocks, and an upper formation of volcanic rocks can be distinguished. Names here proposed for these formations are, respectively, the Gopher Ridge volcanics, the Salt Spring slate, and the Copper Hill volcanics. These units form a conformable and, in part, inter-tonguing, sequence. Fossils have been described from only one locality; these are from the Salt Spring slate and indicate late Oxfordian or early Kimmeridgian age. The Salt Spring slate is therefore about equivalent to the Mariposa formation, and the Gopher Ridge volcanics are presumably about equivalent to the Logtown Ridge formation. .The youngest unit, or Copper Hill volcanics, may be equivalent to the upper exposed part of the Mariposa formation but may be younger. Two other formations, here named the Peaslee Creek volcanics and the Merced Falls slate, of probable Late Jurassic age, have been distinguished in part of the block. The metamorphic rocks are on the west limb of a faulted synclinorium, the axial part of which is occupied by the Sierra Nevada batholith. Large folds having essentially horizontal axes were formed during development of the synclinorium but wide homoclinal belts also occur. Axial planes of the folds dip steeply eastward or are vertical, and! most beds dip eastward more steeply than 60°. The folds and homoclinal sections are truncated by the two major fault zones, which strike nearly parallel to bedding and are parts of a fault system that extends throughout and beyond the western Sierra Nevada metamorphic belt. The fault zones are characterized by map-pable belts of cataclastieally deformed and in part recrystallized rocks and by thin elongate pods of serpentine. The general sequence of strata within each fault block is from older on the west to younger on the east. This sequence was reversed by movements on the major fault zones, so that the youngest exposed strata are in the western fault block and the oldest in the eastern fault block. Pervasive shearing that accompanied the faulting resulted in the development of steeply plunging 12 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT minor folds and 6-lineations, particularly in the fault zones and the eastern fault block. Whether extensive deformation accompanied emplacement of the Sierra Nevada batholith during middle Cretaceous time has been neither established nor disproved. Most of the deformation apparently took place during part of Late Jurassic time as major folds involving upper Jurassic strata are truncated by the major faults, some of which are in turn cut by a Late Jurassic pluton. INTRODUCTION LOCATION, CULTURE, AND ACCESSIBILITY The area covered by this report lies on the lower western slope of the Sierra Nevada in California (fig. 1). It extends from a short distance south of the Merced River to a short distance north of the Cosumnes River, or from north latitude 37° 30' to about north latitude 38°35'. The area is bisected by the historically famous Mother Lode gold mining belt and the towns and villages of the region were established as centers of mining activity during the 19th Century. The largest town is Sonora, with 2,500 people. The population of the region has decreased markedly since the days of active mining. The road and highway network is still much the same as shown on topographic maps published in 1900 and before, but grades and surfaces of the highways have been modernized and most Figure 1.—Index map showing location of area studied.INTRODUCTION 3 Figure 2.—Canyon of the Stanislaus River about two miles northeast of Parrotts Ferry bridge, looking northeast. The canyon is cut into a rolling upland surface underlain by the Calaveras formation of Paleozoic age. The cliffs are formed in limestone. Flat-topped ridge in left background is part of Table Mountain, which is composed of river-channel deposits of Tertiary age. These deposits consist of alluvial gravels and rhyolite tuffs capped by latite lava. of the secondary roads also have hard surfaces. Houses are sparse between towns. At the lower altitudes near the western edge of the area are extensive treeless semiarid grasslands. With increasing altitude, evergreen and deciduous oaks occur, and near the Mother Lode belt open oak forests are interspersed with grasslands dotted by scattered trees and by dense thickets of manzanita and chamise covering areas of poor, thin soil. At altitudes of about 2,000 feet, oaks give way to a conifer forest in which pines predominate. Little of the land is tilled, and most of the nontimbered land is used for grazing. Precipitation is directly proportional to elevation, so that the annual rainfall increases from about 20 inches at the west edge of the area, to 30 inches or more on the Mother Lode, and to 40 inches farther east. It occurs mostly during the fall, winter, and spring months— rain is rare between about the first of June and the middle of September. Snow seldom whitens the ground west of the Mother Lode. PHYSICAL FEATURES The western slope of the Sierra Nevada in this region is for the most part a gently rolling upland with occasional scattered higher hills. This surface largely was developed prior to Eocene time, and is deeply weath- ered. It is buried in some places by younger deposits, which, in the western part of the area, project as buttes and cuestas. The upland surface is dissected by the rivers and their larger tributaries, which have cut deep, youthful canyons. In many of these the average slope of the walls from bottom to top is more than 40°, and the bottoms are little wider than the stream itself at intermediate water levels. The Calaveras River is an exception, as it flows in a broader valley inherited from an earlier erosion cycle, except for a few miles above the point where it leaves the bedrock. Part of the upland surface and canyon of the Stanislaus River are shown in figure 2. The western edge of the area stands about 300 feet above sea level, but near the eastern edge the upland surface stands 5,000 to 6,000 feet above sea level. In the western part of the area long ridges carved from resistant parts of the bedrock extend parallel to the range front and project as much as 1,000 feet above the surrounding country. These are probably monad-nocks on the upland surface (Eric, Stromquist, and Swinney, 1955, p. 5). PREVIOUS WORK Numerous publications on the geology of the western part of the Sierra Nevada have appeared, beginning4 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT shortly after the Gold Rush of 1849-50; the earlier of these were listed and summarized by J. P. Smith (1894). In the present account only those which bear on the subject of this investigation will be mentioned, but some others will be referred to later in the report. The first comprehensive investigations were those of H. W. Turner, Waldemar Lindgren, and F. L. Ransome. The results were published between 1893 and 1900, in a series of geologic folios covering most of the western Sierra Nevada, and in separate papers dealing with specific areas or problems. The geologic maps of the folio series are shown by later work to have distinguished the various lithologic units with much accuracy. Besides the positions of intrusive rocks, these maps show that the western part of the area is underlain by north-west-striking alternating belts of volcanic rocks (diabase and porphyrite of the folio maps) and sedimentary rocks, and that the eastern part of the area is underlain largely by sedimentary rocks, among which only limestone is differentiated. Structural interpretations in the folios are diagrammatic, and bedded volcanic rocks are shown as massive igneous bodies, without stratigraphic significance. Adolph Knopf (1929), in studying many of the gold mines of the Mother Lode region, found that the quartz veins with which the gold ore bodies are associated are in reverse faults. He also recognized that many of the volcanic rocks were bedded and that tuff was intimately interbedded with some of the sedimentary sequences. The results of N. L. Taliaferro’s extensive investigations in the region are presented in a series of abstracts, papers, and a quadrangle geologic map published between 1942 and 1949. Taliaferro (1942, p. 96) treated the volcanic units as formations and has presented a concise summary of the sequence of major geologic events in the western Sierra Nevada. He has also summarized the regional aspects of the structure and stratigraphy (Taliaferro, 1942, p. 77-81, 89-105; 1943b, p. 280-286, 302-323), but his interpretations are supported by little specific locality data. During World War II, a number of copper mines in the region were mapped by G. R. Heyl, M. W. Cox, and J. H. Eric (in Jenkins, ed., 1948). About 1946, the U.S. Geological Survey began a program of geologic mapping for the purpose of gathering the field data necessary to develop an understanding of the regional structure and stratigraphy. Results of this program appeared in the geologic maps and descriptions of the Sonora and Angels Camp (Eric, Stromquist and Swinney, 1955) and Calaveritas (Clark, 1954) 7y2 minute quadrangles. Although these maps contributed to the desired end, it became apparent that to approach the larger scale problems through standard mapping procedures would be very costly in time and manpower. At the same time the feasibility of a different approach to the regional geology was demonstrated. PURPOSE AND SCOPE OF INVESTIGATION The investigation described in this report is designed to supplement previous geologic mapping with new data for interpreting major features of the regional structure and stratigraphy of the metamorphic rocks (pis. 1 and 2). Plutonic rocks were not studied nor were the nearly flat lying Tertiary strata that locally cover the ancient surface. Field work consisted of detailed study of the geology along the major streams, which in general cross the regional structure at about right angles. The folio maps indicate the continuity of formations between the streams. This approach is reasonable not only because of the existing map coverage, but because of the physiographic and geologic features of the region. Exposures, particularly of sedimentary rocks, are generally poor on the upland surface, and contacts between sedimentary rocks in the valleys and volcanic rocks in the ridges are commonly concealed beneath extensive surface debris. In contrast, the major stream canyons provide exposures that are continuous for considerable distances. Fossils are so sparsely and unevenly distributed in the region that stratigraphic sequences must be worked out by systematic observation of structures showing the direction of tops of beds; such structures are abundant in the region. They are readily observed on stream-polished surfaces but in many places where exposures are weathered and covered with lichens they can be discovered only after tedious search. Similar field methods have been used for many years in areas of Precambrian rocks, particularly in the Lake Superior region, but have been little used in California. In the field, the metamorphic rocks were mapped and described in terms of the numbered map units shown on plates 3 through 8, and the basic rock descriptions in tables 1 through 7 (p. 59 to 66) are independent of structural and stratigraphic interpretations. During compilation each unit was assigned a number in sequence from west to east and the units were grouped into formations. The rock descriptions do not represent columnar sections, for each outcrop belt resulting from repetition of a unit by a fold or fault is described separately. This investigation is limited to the metamorphic rocks of the region—rocks of Tertiary age and the plu-tonic rocks are not considered. This is a preliminary report on a continuing study, and some of the interpre-INTRODUCTION 5 tations may be changed as a result of further work. At present, only some of the sedimentary rocks have been studied petrographically, and these not fully. Petrography of the volcanic rocks probably has less bearing on the stratigraphy and has not yet been studied. FIELD WORK AND ACKNOWLEDGMENTS Field work covered a total period of 5i/> months during 1953 and 1954. Field mapping was done on aerial photographs and topographic maps at a scale of 1/24,000. The data were compiled at 1/48,000 for publication at 1/62,500 scale. In addition, short sections have been mapped by plane table at a scale of one inch to 500 feet. The investigation was performed in cooperation with the California Division of Mines and Geology. In compiling the map and cross section along the Calaveras River free use was made of an unpublished map and report on the San Andreas northwest quadrangle prepared by A. A. Stromquist and C. M. Swinney. A detailed map of the type section of the Amador group in the Cosumnes River area prepared by G. R. Heyl, J. H. Eric and M. W. Cox is incorporated in this report. Richard Pack assisted in the field work during the early part of the 1953 field season, and E. H. Pampeyan later in that season. C. D. Rinehart participated in mapping the shores of the reservoirs in the area. The East Bay Municipal Water District granted permission to enter the closed area around the Pardee Reservoir. TERMINOLOGY Discussion of some of the terminology used in this report is necessary for clarity. All the sedimentary and volcanic rocks in the region are metamorphosed, but original structures and textures are commonly well preserved, and the emphasis of this study is on their structure and stratigraphy. For this reason the rocks are generally designated by such names as chert and tuff rather than metachert and metatuff. Lack of consistency will be apparent, however, for originally argillaceous rocks are referred to as slate, phyllite, and schist, and some volcanic rocks in which original textures are destroyed are distinguished by the term “metavolcanic.” This nomenclature has proved convenient and understandable to geologists of my acquaintance. In this region, original textures and structures of pyroclastic rocks forming much of the volcanic formations are indistinguishable from those of rocks of similar grain size composing the “sedimentary” formations. This similarity indicates that both sandstone and tuff, for example, were introduced to the area of deposition and were deposited in the same manner—they are both sedimentary rocks. The significant difference between 724-275 0-65-2 them is the origin and manner of comminution of the fragments. In recognition of this, shale, sandstone, conglomerate, and the like will be designated “epi-clastic” rocks to indicate that the component debris was formed by weathering of pre-existing rocks. The term “graywacke” designates poorly sorted sandstone with a matrix of silt-size and originally clay-size material (Pettijohn, 1954; 1957, p. 290-292). Sand grains are mostly chert or other lithic fragments, but various amounts are mineral grains, most commonly quartz. Rocks in which the sand-size fraction consists entirely of volcanic rock and mineral detritus are called tuff, but those containing even widely scattered grains of non volcanic material are called graywackes. No fundamentally important aspect of the interpretation of geologic history of the region is masked by this usage, as sequences of definite volcanic material alternate with sequences of definite epiclastic material on both the regional scale and the scale of a single formation. The term “tuffaceous sandstone” is not appropriate unless the rock contains both grains formed by epiclastic processes and first-cycle grains formed by pyroclastic processes. In the absence of a purely descriptive term, the name “chert” is applied to microcrystalline, granoblastic rocks composed of quartz. In this area, these rocks have been referred to by most previous geologists as either phtan-ite or quartzite. Phtanite has fallen into disuse in this country, and although quartzite is often used as a descriptive term, it is more properly applicable to a rock formed from quartz sand. Most of the chert is seri-citic, and rock containing so much sericite so that it can be scratched with a hammer point is here referred to as quartzose slate. “Limestone” is used loosely to include carbonate rock ranging in composition from limestone to dolomitic limestone and possibly to dolomite, and in texture from aphanitic to coarsely crystalline. Three terms—cleavage, schistosity, and slip cleavage—are used to distinguish different kinds of parting surfaces of metamorphic origin. “Cleavage,” without a modifier, designates the sort of planar structure characteristic of slate, the slaty cleavage of many authors. In this region development of cleavage is widespread. It has formed in fine-grained pyroclastic rocks and sandstones as well as slate. It commonly crosses bedding, forms a simple regional pattern, and in most places is about parallel to the axial planes of folds. “Schistosity” designates a planar structure formed by the parallel arrangement of tabular mineral grains and clastic fragments in rocks sufficiently coarse grained to be termed schist. In strongly sheared schist, slip surfaces parallel to grains and fragments accentuate the6 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT cleavage. “Slip cleavage” consists of closely spaced crinkles and micro-faults (White, 1949) cutting earlier schistosity and cleavage. Previously formed mica flakes are commonly bent to a direction parallel to the micro-faults. SYNOPSIS OF GEOLOGY Rocks of the western Sierra Nevada are divisible into two groups which have long been known as the bedrock complex and the superjacent series; the latter, consisting of nearly flat-lying alluvial and volcanic deposits, is not considered here. The surficial rocks have been described by Storms (1894); Piper, Gale, Thomas, and Robinson (1939); Clark (1954, p. 9-11); and Eric, Stromquist and Swinney (1955, p. 16-19). The bedrock complex consists of metamorphosed sedimentary and volcanic rocks of Paleozoic and Mesozoic ages, and plutonic rocks of Mesozoic age. The distribution of the metamorphic rocks and the interpretation of their stratigraphy is profoundly influenced by two northwesterly-trending fault zones that divide the metamorphic rocks into three structural blocks (pi. 2). For convenience these are referred to as the eastern, central, and western blocks. The fault zones are not obscure features drawn to explain anomalous distribution of formations, but are mappable belts of cata-clastically deformed and recrystallized rock ranging from a few hundred feet to a few miles wide. Most of the metamorphic rocks of Paleozoic age are in the eastern block and all the metamorphic rocks of Mesozoic age are in the western and central blocks and within the fault zones. The oldest rocks in the region are those of the Calaveras formation, which has been defined to include all of the metamorphic rocks of Paleozoic age in the area (Turner, 1893a, p. 309; 1893b, p. 425). This usage is followed in this report, although five lithologically distinct members within the formation have been distinguished. Four of these members are in the eastern block and one is exposed in a narrow belt in the central block. All the metamorphic rocks of the eastern structural block are included in the Calaveras formation. The most complete section of the formation is exposed along the Merced River, where four members have been recognized. Two members are separable from one another near the Merced River, but not elsewhere. Strata of the Calaveras formation are apparently progressively younger eastward along the Merced River. Diagnostic fossils have been found at only one locality in the eastern block, near the top of the Merced River section of the Calaveras formation, and suggest a Permian age. For convenience, the four members of the Calaveras formation in the southern part of the area have been designated from west to east, the clastic, volcanic, argillaceous, and chert members. These terms do not adequately describe the various sequences of beds, but emphasize the distinguishing features of each. The clastic member, exposed between Bagby and the North Fork of the Merced River, consists in its lower part of thinly interbedded slate and graywacke. In its upper part it consists largely of slate, with interbedded pyroclastic rocks, and a thin-bedded chert unit several hundred feet thick. The volcanic member consists mostly of volcanic breccia and tuff, but contain some pillow lava. The argillaceous member is composed largely of black carbonaceous slate and siltstone, but contains appreciable amounts of bedded chert. Except in deep road cuts along the Merced River, the chert is preferentially exposed and for this reason the argillaceous member is not readily distinguishable from the chert member in most places. The chert member consists dominantly of interbedded black carbonaceous quartzose slate, phyllite, schist, and chert, but limestone lenses are larger in the lower part of the chert member than elsewhere in the formation and north of Sonora limestone forms a prominent part of the section. The chert member underlies a much greater area than the other three members combined. A fifth member of the Calaveras formation is exposed in a small part of the central fault block, where it underlies the Cosumnes formation, of probable Late Jurassic age. This member has long been referred to as the western belt of the Calaveras formation. It includes lenticular clastic limestone which at several localities has yielded fossils of Permian age, and probably includes quartzose slate and minor pyroclastic rocks. Other rocks may be part of the Calaveras formation here, but the central block is structurally complex, and it is uncertain whether some of the rocks in this vicinity belong to the Calaveras formation or to younger formations. The member in the central block is unlike any of those exposed in the eastern block. The western and central fault blocks are characterized by northwesterly trending alternating belts of epiclastic and volcanic rocks (pi. 1). Many of these belts are several miles wide and continuous for great distances, but the stratigraphy is complex. Rocks were correlated by earlier geologists on the basis of lithologic similarity, although fossils are sparse and have not been found in some belts. To avoid unjustified correlation and possible oversimplification of the geologic history of the region, previously proposed formation names are used only in belts which contain their type localities. New names are proposed for the formations of other belts. Mesozoic stratified rocks in the central block are conformable and consist of the Amador group, comprisingSYNOPSIS OF GEOLOGY the Cosumnes and Logtown Eidge formations, the Penon Blanco volcanics, the Mariposa formation, and epiclastic and volcanic rocks of uncertain stratigraphic position. No fossils have been found in the Penon Blanco volcanics. But fossils indicate that the age of the upper part of the Cosumnes formation is within the interval extending from middle Bajocian to late Callovian, that the Logtown Eidge is of Callovian to late Oxfordian or early Kimmeridgian age, and the Mariposa is of late Oxfordian and early Kimmeridgian age. STAGES OF THE MIDDLE AND LATE JURASSIC After Arkell (1956, p. 7-8) except that the Callovian is here included in the Late Jurassic. Late Jurassic Tithonian Kimmeridgian Oxfordian Callovian Middle Jurassic Bathonian Bajocian The Cosumnes formation consists of dark-gray slate, abundant conglomerate, graywacke, and tuff. Much of the conglomerate contains limestone fragments derived from the Calaveras formation. The Mariposa formation is similar to the Cosumnes formation except that conglomerate is much less abundant in most places and the Mariposa contains a thick volcanic member. The Logtown Eidge formation consists largely of coarse porphyritic volcanic breccia, but contains some tuff and minor pillow lava. The Penon Blanco volcanics consist largely of volcanic breccia and tuff, but a thick mass of lava, some showing pillow structure, forms the lower part of the section. Stratified rocks of Mesozoic age in the western block are also conformable. They consist in most of the area of three formations, here named the Gopher Eidge volcanics, Salt Spring slate, and Copper Hill volcanics. Two other formations, the Peaslee Creek volcanics and Merced Falls slate, are distinguished in the southern part of the area. The Salt Spring slate is about equivalent to the Mariposa formation and of late Oxfordian or early Kimmeridgian age, and the Gopher Eidge volcanics are presumably about equivalent to the Logtown Eidge formation. The Copper Hill volcanics may be of Kimmeridgian age, or may be younger. The only area where continuity of these formations across the strike can be observed is in the north where the Salt Spring slate thins and intertongues eastward with the Gopher Eidge and Copper Hill volcanics. This suggests that the Salt Spring slate was not laterally con- 7 tinuous with the Mariposa formation, at least not in all places. Graded bedding, widely and abundantly developed in the tuff and graywacke of Mesozoic age, indicates that these beds accumulated in quiet and probably deep water. Textures and structures indicative of deposition in agitated water are rare or absent. Plutonic rocks in the region include ultramafic rocks which are mostly altered to serpentine, hornblende gab-bro, diorite, and granodiorite. All the plutonic rocks are younger than the metamorphic rocks. Ultramafic rocks form narrow, elongate bodies in major fault zones, and less commonly form somewhat thicker bodies that extend beyond the limits of the fault zones. The proximity of ultramafic rocks to fault zones, including possible sill-like bodies, and the general absence of such rocks elsewhere suggests that emplacement of the ultramafic rocks was controlled largely by the fault zones. As cataclastic deformation associated with these fault zones is much more extensive, both along and across the strike, than are the ultramafic rocks, it is not likely that the serpentine controlled the locus of faulting. Only a few individual plutons in the region have been dated, but most of the granitic rocks appear to fall into two separate series, those of the western and central' blocks apparently being appreciably older than those of the eastern. The easternmost of the two major fault zones is the Melones fault zone and the western one, the Bear Mountains fault zone. The Melones fault zone coincides throughout much of its length with the Mother Lode gold belt. Schistosity within the fault zones suggests that they dip eastward, at angles between 65° and 90°. The two fault zones form part of the Foothills fault system (Clark, 1960a), which extends from south of the area studied to the northern limit of exposures of metamorphic rock in the western Sierra Nevada, a distance of about 200 miles, and perhaps farther. Steeply plunging lineation and minor fold axes suggest that the dominant component of movement along the Melones and Bear Mountains fault zones was horizontal. Some faults of the system are cut by granitic plutons of the older series. Previously mapped reverse faults that control the quartz veins of the Mother Lode are within the Melones fault zone in most, but not all, of the area. These faults also dip eastward but less steeply than the Melones fault zone, and in the northern part of the area are exposed as much as one mile west of the Melones fault zone. They are probably younger than the Melones fault zone, and may have formed concurrently with emplacement of the Sierra Nevada batholith.8 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT Although large folds repeat the outcrop of formations in places, older beds are generally exposed in the western part and younger beds in the eastern part of each of the three fault blocks. This arrangement suggests that the region lies on the west side of a large synclinorium, but is reversed by the Melones and Bear Mountains fault zones, so that the youngest exposed metamorphic rocks of the region are in the western block, the oldest in the eastern block, and rocks of intermediate age in the central block. Only general features of the structure of the eastern block have been worked out because of the monotonous lithology of a great part of the Calaveras formation and because bedding has been destroyed by shearing in much of the region. Much of the Calaveras formation is schistose. Near the Merced River, where it is less recrystallized, the structure of the eastern block is apparently homoclinal with tops east. Where bedding is preserved, it nearly everywhere dips steeply. Large folds may occur near the Merced River, but none are large enough to repeat any of the four members distinguished in the Calaveras formation. In the southern part of the area, the northwest regional trend of belts of rock in the Calaveras formation is truncated at an acute angle by the Melones fault zone. Near San Andreas, a large easterly plunging complex anticline is truncated at a large angle by it. The amount of deformation in the central and western blocks is about equal. Isoclinal folds occur in the central block near the middle of the area, but the beds are homoclinal with tops east farther north and south. In the north half of the western block are large open folds overturned to the west. In the southern part of the western block, folds also occur, but their form and position are obscure because they occur in thick sequences of sheared volcanic rocks with little bedding and no graded bedding. Intersections of bedding and cleavage, the plunges of minor fold axes, and the areal patterns of outcrop show that the major folds plunge at low angles, generally less than 25°. Cleavage occurs widely and schistosity locally in rocks of the central and western blocks. Most rocks of the eastern block are schistose. Most of the deformation within all three fault blocks preceded the major faulting and probably took place during an orogeny that appreciably antedates intrusion of the Sierra Nevada batholith. STRATIGRAPHY CALAVERAS FORMATION The term Calaveras formation was used for all the metasedimentary and metavolcanic rocks of Paleozoic age in the Sierra Nevada, except the Silurian and Upper Carboniferous rocks of the Taylorsville region (Turner, 1893a, p. 309; 1893b, p. 425). This broad usage is followed here. The formation was named for beds in the central block about 2 miles east of the crest of the Bear Mountains in Calaveras County that in this report are included in the Mariposa formation. DISTRIBUTION AND CHARACTER Within the report area, most of the Calaveras formation lies between the Melones fault zone on the west and the Sierra Nevada batholith on the east. A much smaller belt of Calaveras formation in the central fault block extends from south of the Mokelumne River to north of the Cosumnes River. Other small, isolated blocks derived from the Calaveras formation, surrounded by younger rocks, occur in major fault zones at widely separated localities, and in conglomerate of the overlying Cosumnes and Mariposa formations. The Calaveras formation within the area can be divided into five members. One of these is in the central block and forms the western belt of the formation; the other four are in the eastern block. The Calaveras formation of the western belt consists largely of cherty phyllite and thin-bedded chert, with minor amounts of calcarenite. The eastern block comprises, from older to younger, (1) a member composed of slate, graywacke, chert, and tuff, here designated the clastic member; (2) an upper volcanic member; (3) an argillaceous member, recognized only on the Merced River; and (4) a member composed of thin-bedded chert, black carbonaceous slate, and limestone, here designated the chert member. The four members of the eastern block are exposed along the Merced River. Proposal of formal stratigraphic names for these units is deferred until named subdivisions of the Calaveras formation exposed north of the area covered by this report have been studied. EASTERN BELT The most complete section of the eastern belt of the Calaveras formation is in the southern part of the area, where it is divisible into four members. The section is apparently conformable, and contacts between members are drawn to group beds of similar lithology. Graded bedding and attitudes of beds suggest that successively younger members are exposed from west to east along the Merced River. Northwest of the Merced River the lower members of the Calaveras formation are successively truncated by the Melones fault zone; the argillaceous and chert members that form the upper part of the formation are more extensively exposed. The northeastern part of the eastern belt includes undifferentiated rocks of the Shoo Fly formation of probable Silurian age (Clark and others, 1962, p. B15-B18).STRATIGRAPHY 9 CLASTIC MEMBER The clastic member is exposed in the Merced River between Bagby and the mouth of the North Fork of the Merced River. It is bounded on the west by serpentine and on the east by the volcanic member. It is truncated along the Melones fault zone a few miles north of the Merced River (pi. 1). It is inferred to extend south of the Merced River to a point near the boundary of the map area where it is truncated by a granitic pluton. Fine-grained detrital rocks, chiefly black slate and siltstone, make up most of this member, but pyroclastic rocks and bedded chert constitute important parts of it. Abundant thin beds of graywacke are interbedded with the black slate in the western part of the section. Small lenses of black limestone occur at several horizons and crinoid plates are abundant in one. Poorly sorted conglomerate or breccia, consisting of pebbles and boulders of feldspathic graywacke in a matrix of sandy argillite, forms units 31 and 33 (pi. 3). These are probably mudflows, as suggested by the poor sorting, argillaceous matrix, and absence of internal bedding. The total thickness of the clastic member in the Merced River is more than 10,000 feet. The black slate (unit 29, pi. 4) west of the bridge on State Highway 49 across the Tuolumne River is arbitrarily included with the clastic member, following Turner (1897), but may be of Mesozoic age. The slate contains two small limestone masses. VOLCANIC MEMBER The pyroclastic rocks exposed in the Merced River east of the mouth of the North Fork of the Merced are continuous to the northwest with a belt of metavolcanic rocks mapped previously (Turner, 1897) as amphibolite, green schist, and, locally, as gabbro. The belt extends northwestward as far as State Highway 108 (pi. 1); between State Highway 108 and San Andreas metavolcanic rocks that form separated masses are included. Although the stratigraphic position of most of those rocks has not been established, their distribution suggests that they are near the same stratigraphic level. The metavolcanic rocks may be infolded with or inter-tongue with the surrounding rocks. The mass east of San Andreas forms the core of an anticline and is over-lain by interbedded black carbonaceous schist and chert similar to that forming the argillaceous and chert members in the Merced River (Clark, 1954, p. 8-9). It also includes thin, probably interbedded layers of chert and black schist. Small masses of volcanic rock north of San Andreas (pis. 1 and 2) are included, but may be interbedded with the argillaceous or chert member of this report. The volcanic member extends southeastward from the Merced River beyond latitude 37° 30' north. It is about 5,000 feet thick east of the mouth of the North Fork, assuming a uniform dip of 80°. In the Merced River, the volcanic member consists almost entirely of coarse volcanic breccia, similar to some breccias of Jurassic age in color, grain size, and the presence of pyroxene phenocrysts as much as 1 cm in diameter. Two layers of black slate, each about 50 feet thick, with interbedded volcanic conglomerate and tuff lie in the breccia near its western side. Fragments of black, carbonaceous slate are common in the eastern part of the volcanic breccia. The volcanic rocks exposed on the North Fork of the Merced River, less than a mile from the section exposed in the main stream, are more varied. Besides volcanic breccia they include green phyllite, volcanic conglomerate, and tuff. Differences in the two sections suggest very rapid interlensing of different rocks, or small folds that cannot be demonstrated by the observed bedding attitudes. Gradually waning volcanism and conformable relations between the volcanic member and the overlying argillaceous member are suggested by the presence of interbedded epiclastic and pyroclastic rocks in the southwestern part of the section included with the argillaceous member. In secs. 25 and 26, T. 3 S., R. 17 E., on the North Fork of the Merced River, pillow lava and layers of volcanic breccia and tuff too thin to map individually are interbedded with argillaceous slate and silt-stone. On the west side of the Merced River near the south end of McCabe Flat, the thick porphyritic breccia of the volcanic member is succeeded on the north by conglomerate less than 20 feet thick consisting of pebbles of volcanic rock in a black slate matrix. This is succeeded by another volcanic breccia layer less than 100 feet thick and this by black slate containing some chert. Another volcanic breccia layer is exposed at the north end of McCabe Flat. Farther east along the Merced River is fine tuff and a green phyllite that was probably a very fine tuff. ARGILLACEOUS MEMBER A member composed dominantly of bedded, black argillaceous rock which locally contains bedded chert occurs on the Merced River east of the volcanic member. It differs from the clastic member to the west, as it contains almost no graywacke and less variety of lithologic types, and it differs from the chert member in containing much less chert. Nevertheless, the argillaceous member cannot be distinguished from the chert member along the North Fork of the Merced River, perhaps because of increase in abundance of chert northwestward. It may, however, be due to the nature of the exposures, as the extensive fresh highway cuts along the Merced River are not comparable to the stream-polished expo-10 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT sures examined elsewhere. The resistant chert and quartzose slate may be preferentially exposed in some streams, giving an erroneous impression of the amount of chert. Existence of a distinct member of argillaceous rocks is suggested also by the normally weathered rock in road cuts immediately north and south of Briceburg, and about 3 miles northeast of Coulterville, which consist largely of gray- and brown-weathering slate, at least superficially similar to weathered exposures of the Mariposa formation. Rocks of this sort do not occur in other parts of the Calaveras formation. The total thickness of the argillaceous member is uncertain, because at several places in the Merced River bedding has been destroyed by shearing for several hundred feet across the regional strike. The widest unbroken block with apparently simple structure is between Briceburg and McCabe Flat. In this interval the beds are more than 5,000 feet thick. This represents only about one-fourth of the width of outcrop of the argillaceous member, and may represent a comparable fraction of its total thickness in the Merced River. Volcanic rock to the west and the belt of limestone to the east converge northward, suggesting that the argillaceous member thins to the north. Rocks of the argillaceous member are chiefly slate and siltstone, the latter partly massive, partly thin-bedded. Seams of thin-bedded chert are present locally. Tuff and aphanitic green phyllite that may be derived from tuff are common on the Merced River west of Briceburg, but are lacking east of Briceburg. Along the Tuolumne River, on the other hand, pyroclastic rocks are interlayered throughout the interval between the volcanic member and the limestone that is near the southwest side of the chert member. Limestone was not observed in the argillaceous member along the Merced River, but H. W. Turner (unpublished map of the Yosemite quadrangle) mapped limestone on the hill slope about half a mile east of Briceburg. Thin beds of limestone are exposed in the comparable section on the Tuolumne River. Conglomerate, graywacke, and sandy siltstone are interbedded with black slate and siltstone in the Merced River northeast of the mouth of Sweetwater Creek (unit 41, pi. 3). The conglomerate has a matrix of slate or siltstone and some of the graywacke beds are graded. Some of these rocks are tightly folded and the total thickness of the part of the section that contains them is unknown. Although these coarse epiclastic rocks are near the contact between two of the members they do not indicate emergence, or even shallowing of the water. The pebbles in the conglomerate are volcanic rocks, lime- stone, chert, quartzose slate, and carbonaceous slate which could have been derived from underlying parts of the Calaveras formation; no crystalline, metamorphic, or other rocks foreign to the Calaveras formation occur. A local source is suggested by contrasting composition of adjacent conglomerate beds; in some the pebbles are almost all volcanic rocks, and in others these are rare or absent. The argillaceous matrix of the conglomerates, the extremely poor sorting and lack of internal bedding in some layers, and the grading of associated sandstone beds suggest deposition by mudflows and turbidity currents. Very coarse slump breccia composed of sandstone blocks more than 15 feet long in a slate matrix is exposed along the South Fork of the Merced River about 1 mile east of the mouth of Devil Gulch (pl.l). CHERT MEMBER The easternmost member of the Calaveras formation in this area is characterized by abundant chert interbedded with black carbonaceous slate and siltstone. A belt of lenticular limestone masses lies near the western boundary of the member, and it also includes the large mass of limestone northeast of Sonora. Some quartz-rich siltstone resembling the chert also occurs. Sandstone beds exposed near El Portal on the Merced River and along the South Fork of the Merced are here included with the chert member although their stratigraphic relations are uncertain. Lenses of volcanic rocks are widespread. The chert member is the most widely distributed subdivision of the Calaveras formation in the area, and extends from the west side of the belt of limestone lenses eastward to the batholith. It is exposed on all streams mapped and good exposures are readily accessible on all except the Calaveras and Co-sumnes Rivers. It is the only subdivision of the Calaveras formation recognized on the Calaveras, Moke-lumne, and Cosumnes Rivers, although perhaps some of the beds on these three streams should be included with the argillaceous member. The chert and quartz-rich siltstone form layers 0.04 to 8 inches thick, but mostly less than iy2 inches thick, in the black carbonaceous slate, phyllite, and schist. Relative proportions of chert and quartz siltstone are unknown, but most of the textures shown in thin sections of fine-grained quartzose rocks studied do not indicate a clastic origin. The chert is aphanitic and is gray to black on fresh surfaces; the luster is chal-cedonic to vitreous. Weathered chert is white and has a line sugary appearance. Rocks of various compositions ranging from chert to phyllite are common. Rocks composed of elliptical fragments of chert in a matrix of black carbonaceous slate or phyllite areSTRATIGRAPHY 11 common in the chert member. The short diameter of the fragments is mostly less than an inch. These aggregates might be intraformational conglomerates, but they are not bedded and are probably breccias, formed tectonically from interbedded chert and carbonaceous slate (Clark, 1954, p. 6-7). Thin section study of cherts of the Calaveras formation of the eastern block indicates that they are composed chiefly of microcrystalline, equigranular quartz generally with less than 5 percent of biotite and sericite. Many cherts contain pyrite and a few contain epidote and zoisite. A carbonate mineral forms widely dispersed, small, irregular microcrystalline aggregates in most specimens. The cherts have a granoblastic texture; boundaries between quartz grains interlock but are not dentate or sutured. The micas are interstitial to the quartz, and in the coarser grained specimens penetrate boundaries of quartz grains and form inclusions in the quartz. Biotite, with X, colorless; Y, red-brown; and Z, red-brown; and absorption Xertella sp. aff. S. kingi Dunbar and Skinner; Schuber-tella(l) ; and three species of Parafusulina (or possibly Pseudo-fusulina in part) are barely recognizable. Permian age is definitely indicated. Fusulinids of the apparent grade of evolution represented by these specimens characterize the late Hueco and Leonard Faunas of West Texas. I doubt that they are as old as the Wolf camp at the type locality, but may be as young as the early Guadalupian. Fossils from a third locality in the western belt of the Calaveras formation, the Allen marble quarry in sec. 13, T. 6 N., R. 10 E., are reported by Mr. Henbest and Miss Helen Duncan (written communication) to be of Permian age. Fossils other than crinoid debris have been found at only one place in the eastern belt of Calaveras formation in the area covered by this report. These were Foraminifera discovered by Turner (1893a, p. 309) near Hite Cove, in the El Portal quadrangle, which were identified as Fusulina cylindrica and regarded as of Carboniferous age. Turner’s collection has apparently been lost and by modern standards “Fusulina cylindrica” only means Pennsylvanian or Permian age. Regarding a collection from NW^kW1^ sec. 2, T. 4 S., R. 19 E., near Hite Cove, Mariposa County, by L. D. Clark and N. K. Huber, Mr. Henbest wrote that the fossils are altered beyond recognition. In the Sonora quadrangle fossils were found by the writer in a block of limestone more than 30 feet wide associated with (strongly sheared conglomerate that probably is part of the Mariposa formation. The locality is in Mormon Creek, 1.1 miles S. 38° IV. of Jackass Hill and is within the Melones fault zone. The conglomerate contains pebbles of limestone, chert, and black carbonaceous slate. These, as well as the fossiliferous limestone, resemble the lithologic assemblage near Hite Cove rather than that of the Calaveras formation of the western belt. Among the fossils, Mr. Henbest (written communication, 1955) identified: ?Neofusulinella mantis Thompson, 1946 fParafusulina turgida Thompson, 1946 Crinoid columnals and says: The fusulinids and other fossil remains are attenuated by •plastic flow of the rock. Permian age is definitely indicated. Upper McCloud age is strongly suggested, but the exact age in the Permian and the generic and specific determinations listed above must be held in question because of the obscurity of the fossils. Fragmental megafossils were found in the SE14NE14 sec. 22, T. 4 N., R. 11 E. about 0.6 mile southeast of the Calaveras River in a limestone mass that probably slumped into beds of the Mariposa formation during their deposition. The limestone is exposed in an area about 50 feet long and 20 feet wide. The relation of the limestone to surrounding rocks is not evident; rocks west of the limestone are not exposed, but east of the limestone are poorly exposed sandstone and conglomerate containing fragments of chert, quartz, and slate. The sandstone and conglomerate are in strike with similar rocks which form unit 26 of the Calaveras River section (pi. 6). The limestone contains abundant fragmental organic material. The following have been identified by Jack E. Smedley (written communication, 1955) of the TLS. Geological Survey: Crinoid columnals Spiriferina sp. undet. Rhynchonellid brachiopod Terebratulid brachiopod Trigonid pelecypod Pectenid pelecypod Murchisonid gastropod Spirorbis sp.STRATIGRAPHY 15 According to Smedley: The scraps are too incomplete and poorly preserved and the forms represented are not sufficiently diagnostic to permit accurate dating of the beds from which the fossils were collected. Moreover the forms identified belong to groups in which it is often difficult to distinguish between the Permian and Triassic genera and species even with good material. The presence of Spiriferina, rhynchonellids, and terebratulids in fair numbers suggests that this is a Triassic fauna. The Spiriferina has certain characters which relate it more to Triassic than to Permian species. The possibility that this is a Permian fauna cannot, however, be ruled out completely. As the fossils do not permit accurate dating, and as no geologic evidence has been found to indicate that Triassic rocks occur in the region, this limestone may be of Permian age and derived from the Calaveras formation. Although the Permian age of part of the Calaveras formation is well established, the very size of the area east of the Melones fault zone that is underlain by steeply dipping rocks included with the Calaveras formation suggests that rocks of other than Permian age might be present. In addition, available structural data suggest that the rocks included with the Calaveras formation on the Merced River constitute a generally homoclinal section with the younger beds on the east. None of the four members distinguished on plate 3 is repeated by major folds. Allowing for considerable thickening by smaller folds and faults, the section is still apparently tens of thousands of feet thick. This section can be conveniently divided by the belt of lenticular limestone masses that extend northwestward from the Merced River west of Incline which Turner (1897) has previously considered to be a stratigraphic horizon. The rocks northeast of this belt and inter-bedded with the limestone are the same in all sections examined during this investigation and in the Cala-veritas quadrangle (Clark, 1954), except for the distinctive quartz-rich sandstone exposed near El Portal on the Merced River. These rocks apparently represent a single episode of deposition in which no significant break is known. The rocks northeast of the limestone may all belong to the same period as the limestone near Hite Cove, or they might be in part of Triassic age. Lithologic similarity of the two lower members of the Calaveras formation to Mesozoic formations in the region suggests the possibility that these members may be of Mesozoic age. This interpretation cannot be eliminated, but is not supported by more substantial evidence. On the contrary, abundant crinoid plates in a limestone pod (unit 29, pi. 3) in the clastic member suggest Paleozoic rather than Mesozoic age and structural data indicate that the section from Bagby east- ward to Hite Cove is essentially homoclinal with tops eastward; no major fault zone was found within this interval. ROCKS OF JURASSIC AGE Steeply dipping strata of Jurassic age underlie most of the area of bedrock exposures west of the Melones fault zone but are known east of that zone only near the south edge of the area covered by this report. These Jurassic rocks are divided into two main blocks by the Bear Mountains fault zone, along which the minimum stratigraphic separation, at least in the northern part of the area, must be many thousands of feet. The Jurassic strata consist largely of volcanic rocks, probably mostly of andesitic and basaltic composition but including some rhyolite or dacite; and of epiclastic rocks, mostly slate and partly graywacke and conglomerate. All are apparently marine deposits. The stratigraphic nomenclature used in this report for the rocks of Jurassic and probable Jurassic age is consistent with that of previous geologists in that it separates sequences composed dominant^ or entirely of epiclastic material from those consisting dominantly or entirely of volcanic material. The formations so defined are several hundred to nearly 15,000 feet thick. Even with a gross separation of this kind the stratigraphy is complicated (pi. 9) and it is impossible to work out from the fragmentary record on the present surface a complete three-dimensional picture showing the relations in time and space of all stratigraphic units to one another. Fossils are sparse and have yet to be found in some formations. Lithologic variations within any one formation, either volcanic or epiclastic, may be greater than that between two formations of either class. The stratigraphic succession can be deciphered in any cross section normal to the strike where adequate structural information is available, but use of individual formation names can be extended with confidence along the strike only so far as the component rocks can be traced by surface exposures. If a formation name be used to include areas of exposure that are not continuous on the surface with the type locality, significant geologic relations might be masked by an oversimplified concept of the stratigraphy. Difficulties in arriving at a complete understanding of the stratigraphy, although augmented by later folding and faulting, result primarily from the original distribution of strata constituting the Jurassic section. The volcanic rocks were derived from a number of vents distributed in an unknown pattern about the area in which material of epiclastic origin was accumulating. Activity of individual vents was sporadic and certainly not all volcanic centers were active at the same time. Nevertheless,16 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT the distribution of volcanic rocks indicates that in some places deposits from two or more centers overlapped or intertongued. Because the lower, middle, or upper part of a volcanic pile related to one center can conceivably intertongue with any part of other piles, as well as with the epiclastic rocks, the possible number of complex relationships is considerable. In this report the use of previously accepted formation names is restricted along strike to rocks that form belts of exposure continuous with type areas. In a direction across the regional strike, formation names are extended to include outcrop areas repeated by folding. New names are proposed for other outcrop areas of sufficient size to warrant naming. Type sections are designated to fulfill the requirements of stratigraphic nomenclature, but cannot be used as rigid standards of the sequence and character of deposits within each formation over an area, say, equivalent to a 15-minute quadrangle. A minor problem of stratigraphic nomenclature involves small elliptical outcrop areas of volcanic rocks within belts of epiclastic rocks, or the reverse. Are these lenticular members of the epiclastic formation or are they tongues of the overlying or underlying volcanic formation? In some places intertonguing relationships can be shown by mapping, but nowhere is it possible to establish rigorously that a particular map unit is a lens. This results largely from the fact that dips are steep and many sections are homoclinal so that only in a few places can stratigraphic sections be studied in a direction normal to the strike. Figure 3 illustrates relationships that may be encountered in the western Sierra Nevada; the east and west faces of the block can be regarded as faults. Unit 2 (fig. 3) is readily recognizable as a tongue of volcanic formation C because in part of the area shown it can be seen on the surface to form an unbroken sequence with the rest of the formation. Exposures of unit 1, on the other hand, suggest that it is an isolated lens of volcanic rock in an epiclastic formation. As illustrated, however, it is connected beneath the surface with volcanic formation A and is therefore a tongue of that formation. Unit 3 is well within volcanic formation C and is not connected at the surface or at depth with other epiclastic rocks. It is reasonable to believe, but impossible to prove by surface mapping, that unit 3 is a member of formation C—it may be the end of a tongue formerly connected above the erosion surface with an epiclastic formation. In making the interpretations shown on the accompanying maps and cross sections, it has been necessary to decide arbitrarily whether some map units or groups of units would be shown as tongues or lenses. These decisions affect the stratigraphic nomenclature, but do Figure 3.—Block diagram of homoclinal section showing intertonguing and possible lenticular relationships. not notably influence interpretation of the geologic history—whether unit 1, for example, is a tongue or a lens, it indicates that unconsolidated volcanic material was supplied at that particular stratigraphic position. They do, however, bear on problems relating to the limits and physiography of the basin of deposition and directions of transport. Descriptions of the stratigraphic units of Jurassic and probable Jurassic age in the following pages are brief. In many respects, the composition, texture, and original structures of the rocks are similar throughout the section and can be more conveniently discussed in terms of the area as a whole. ROCKS EXPOSED IN THE CENTRAL BLOCK AND MELONES FAULT ZONE HOCKS OF MIDDLE (1) JURASSIC AGE A small isolated body of volcanic and epiclastic rocks exposed about 6 miles southwest of Sonora contains fossils determined by Imlay (in Eric, Stromquist, and Swinney, 1955, p. 12) to resemble closely species in the Mormon sandstone of Middle Jurassic age. The fossils were found in the SE1/^ sec. 20, T. 1 N., R. 14 E., about 5% miles southwest of Sonora (pi. 1). Fossils of Middle Jurassic age have not been found elsewhere in the area between the Merced and Cosumnes Rivers. The fossils from southwest of Sonora occur in tuff, about 65 feet from the contact of the tuff with conglomerate. Eric, Stromquist, and Swinney (1955, p. 12 and pi. 2) correlated the tuff with the Logtown Ridge formation and the conglomerate with the Mari-STRATIGRAPHY 17 posa formation, both of Late Jurassic age, and hesitate to accept the interpretation that these rocks are of Middle Jurassic age. They do not, however, discuss the geologic evidence for their correlation and it seems entirely possible that these rocks are, in fact, of Middle Jurassic age. The fossiliferous tuff forms part of a block that is nearly surrounded by gabbro and serpentine and is separated from the main belts of the Log-town Ridge and Mariposa formations by a major fault (Eric, Stromquist, and Swinney, 1955, pi. 2). The block has been displaced and rotated, for strikes within the block are northeasterly, nearly normal to the consistent northwesterly strike of rocks outside of the Me-lones fault zone in this vicinity. AMADOR GROUP The Amador group was named by Taliaferro (1943b, p. 282-284) who designated two type sections, one on the Cosumnes River and the other on the Merced River. On the Cosumnes River, he divided the Amador group into two formations, the Cosumnes formation below, and the Logtown Ridge formation above. A generalized cross section along the Cosumnes River at a scale of about 2 miles equal 1 inch (Taliaferro, 1943b, fig. 2) indicates roughly where the contacts of the formations are to be placed. Taliaferro did not recognize the existence of the Bear Mountains fault zone at this latitude and included the rocks of both the western and central blocks with the Amador group, nor did he designate which part of the section in the Cosumnes River is meant to be the type section. Presumably, however, the type section extends westward from Huse Bridge on Highway 49, for the bridge is at the east side of the volcanic sequence that underlies Logtown Ridge. Along the Merced River, Taliaferro (1943b, p. 283) divided the Amador group into five formations, but provided only very brief descriptions of these units and did not indicate where the contacts are to be placed. The formation names proposed by Taliaferro for the Cosumnes River section are useful, but more complete descriptions of the formations are necessary. In redescribing the formations, it will be convenient simultaneously to modify Taliaferro’s usage, and propose definitions that will be more readily applicable in geologic mapping. Only one type section should be designated for the Amador group. The type section is on the Cosumnes River southwest of Huse Bridge and is shown on plate 10. The name Logtown Ridge formation is applied to the volcanic sequence, and the name Cosumnes formation is applied to the underlying epiclastic sequence. The lower contact of the Cosumnes formation is tentatively placed about 2.1 miles southwest of Huse Bridge at the west unit 42, pi. 8. The contact is not exposed on the river banks. The contact between the Logtown Ridge and Cosumnes formations, drawn at the top of the slate layer exposed on the north side of the river about 7,400 feet southwest of Huse Bridge (pi. 10), is parallel to bedding in both formations. At Huse Bridge, the Logtown Ridge formation is in depositional contact with an overlying epiclastic sequence previously included with the Mariposa formation (Lindgren and Turner, 1894; Taliaferro, 1943b, fig. 2). The upper contact of the Logtown Ridge formation is at the base of the lowermost epiclastic bed of the overlying sequence. COSUMNES FORMATION Exposures of the Cosumnes formation extend southward beyond Sutter Creek and probably extend northward beyond the boundary of plate 1. The most abundant rock in the Cosumnes formation is black to gray slate and siltstone. Conglomerate in which the pebbles and boulders are composed of a large variety of rock types is common in the type section. Pyroclastic rocks and graywacke are less common than the other rocks. The Cosumnes formation is about 3,600 feet thick near the Cosumnes River. Lithology In the type section of the Cosumnes formation, conglomerate is dominant in the lower part of the formation, interbedded graywacke and slate in the middle part, and interbedded mafic tuff and black slate in the upper part, but most of these rocks occur in varying amounts throughout the section. The proportion of conglomerate is greater in the type section of the Cosumnes formation than elsewhere in this formation or any of the others, but the conglomerates are apparently lenticular and the type section may not be representative of the whole formation. Both the conglomerates and graywackes in the Cosumnes formation resemble those in the other formations. Conglomerates of the Cosumnes formation can be divided on the basis of sorting into two types, one resulting from current transport and the other from mudflow. The two types are interbedded with slate, graywacke, and each other. Layers of the better sorted conglomerate contain internal bedding and, at least superficially, a normal gradation of grain size from the largest fragments to the smallest—space between the larger pebbles is largely filled with smaller pebbles and sand. These are features of current-deposited conglomerate. Layers of poorly sorted conglomerate are not internally bedded and the boulders and pebbles are dispersed in a matrix of slate and siltstone (fig. 4). Exposures of poorly sorted conglomerates contain widely scattered boulders18 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT Figube 4.—Photomicrograph of mudflow conglomerate from Mariposa formation, NE14NE14 sec. 2, T. 3 N., R. 11 E. Only a small fraction of the total size range of clastic fragments is shown. Felsic volcanic rock with quartz (white) phenocrysts, fv; mafic volcanic rocks, mv, in part with plagioclase and pyroxene phenocrysts; quartz, q; granophyre composed of intergrown K-feldspar and quartz with minor biotite, a; volcanic rock partly replaced by pyrite, v; mica-quartz phyllite, p; chert-carbonate rock, c; clastic limestone, 1. which are several times as large as the other boulders in the layer. These conglomerates are probably of mudflow origin. Both kinds of conglomerate occur in the Cosumnes formation on the Cosumnes River and are well exposed in a small canyon cut into rocks of the Mariposa formation in the SW1^ sec. 1, T. 3 N., R. 11 E., about 4 miles southwest of San Andreas. Fossils and age Taliaferro (1943b, p. 284) ascribed to the Amador group a late Middle to early Late Jurassic age, but does not give a specific age for the Cosumnes formation. He mentions several fossil localities but does not describe the localities or faunas in detail. Later collections indicate that the age of the upper part of the Cosumnes formation is within the interval extending from middle Bajocian to late Callovian. Eric, Stromquist, and Swinney (1955, p. 10, 11) report finding an ammonite in the Cosumnes River section about 500 feet below the top as the formation is described in this report (Meso- zoic locality 22175, U.S. Geol. Survey collections). This specimen was compared by Imlay (1952, p. 975) with the Callovian genus Grossouvria, but he now considers (written communication, January 1960) that: its coiling is too involute for that genus and that it agrees better in coiling and rib pattern with Pseudocadoceras which occurs in considerable numbers in the lower part of the over-lying Logtown Ridge formation.1 The speciman suggests, therefore, that the upper part of the Cosumnes formation is not older than Callovian. Furthermore, considering that the Logtown Ridge formation grades downward into the Cosumnes formation, the presence of Pseudocadoceras in the basal beds of the Log-town Ridge formation implies that the Cosumnes formation is at least in part of Callovian age * * *. LOGTOWN RIDGE FORMATION The name Logtown Ridge formation is applied in this report to the sequence of volcanic rocks that overlies the Cosumnes formation in the Cosumnes River 1 While this report was being prepared for publication, a report describing late Jurassic fossils from the western Sierra Nevada in more detail was published by Imlay (1961).STRATIGRAPHY 19 southwest of Huse Bridge. It extends from north of the area studied to a point between Drytown and the Mokelumne River where it is overlapped by the Brower Creek volcanic member of the Mariposa formation. The southern boundary of the Logtown Ridge formation is drawn arbitrarily on the regional geologic map and block diagram (pis. 1 and 2), for detailed mapping that might show the position of the contact between the Logtown Ridge formation and the younger volcanic sequence is lacking. The Logtown Ridge formation has been examined only in the type section, where it is about 4,300 feet thick and consists largely of volcanic breccia and tuff. Two thin layers of pillow lava are present in the type section. The name Logtown Ridge formation was used by Eric and others (1955, p. 11-12, pis. 1, 2) to include volcanic rocks extending as far south as Chinese Camp, and by Taliaferro and Solari (1949) to include volcanic rocks in the vicinity of Copperopolis. These sequences will be given new names because some occupy a different stratigraphic position from the Logtown Ridge formation and the position of others relative to the Logtown Ridge formation is unknown. Lithology In the Cosumnes River, tuff and lapilli tuff dominate the lower part of the Logtown Ridge formation and volcanic breccia dominates the upper part. Nearly all the volcanic fragments are dark green, with an apha-nitic texture. Pyroxene phenocrysts in both clasts and matrix of some beds range from less than one millimeter to more than one centimeter in diameter. A few breccia beds contain scattered clasts of medium- to coarse-grained pyroxenite and coarse-grained rocks composed of plagioclase and pyroxene in various proportions. No megascopic features that indicate differences in chemical composition between the various breccia beds were observed, but the amount of por-phyritic or amygdaloidal fragments differs from bed to bed. Beds may also differ in the size of the phenocrysts and in the proportion of clasts that contain phenocrysts or amygdules. Original textures and structures of the pyroclastic rocks are well preserved along the Cosumnes River and are probably representative of features in other volcanic formations of the region that were studied in less detail. Tire Cosumnes River exposures show that the detritus composing various beds of pyroclastic rock was emplaced in several different ways. Much of the volcanic breccia of the Logtown Ridge formation was apparently emplaced by subaqueous mudflows. Mudflow layers have poor sorting, lack internal bedding, and some contain boulders several feet long. Packing of fragments larger than one inch in diameter ranges widely. Clasts are angular to subrounded or well-rounded (fig. 5). The mudflow deposits range widely in composition—some contain only one or two textural varieties of lava and others contain many textural and perhaps even compositional varieties. In some layers part of the fragments are amygdaloidal but scoriaceous clasts are infrequent. Rare layers contain fragments of pyroxenite and plagio-clase-pyroxene rocks. Although clasts larger than two inches in diameter are abundant in most of the probable volcanic mudflow breccias, a layer of finer volcanic breccia about 0.8 miles southwest of Huse Bridge contains widely scattered chunks as much as several feet long of somewhat contorted, thin-bedded, coarse to very fine tuff. These chunks are consistent with the notion that this layer is also a mudflow. In stream polished exposures the matrix of most of the volcanic breccias wastes away at about the same rate as the fragments (fig. 5). In one breccia layer about 100 feet thick, however, the matrix weathers much more rapidly than the fragments, leaving them in bold relief; it is exposed about 300 feet west of Huse Bridge on the Cosumnes River. Clasts are angular to subrounded and range from sand size to boulders more than 4 feet long and 2 feet thick. The large boulders are elongated parallel to the bedding in this vicinity, but no layering suggestive of bedding was observed. The clasts have little lithologic variety as nearly all contain pyroxene phenocrysts more than 2 mm in diameter—about a third of them are amygdaloidal (quartz and calcite amygdules). The matrix is composed of sand size grains of lava and pyroxene cemented by calcite. Although this layer is apparently the product of a mass movement process, “mudflow” seems hardly an appropriate term for a mass lacking in clay-size material. This layer is separated from an underlying pillow lava by about 10 feet of rubble atop the lava, and about 4 feet of bedded tuff immediately under the breccia. It is overlain by a thin amygdaloidal lava. The mudflow deposits represent material that has moved to a permanent site after temporary accumulation in another place. A different kind of volcanic breccia, probably the result of direct deposition from explosive vents, is represented by bedded tuff and lapilli tuff layers containing scattered boulders of volcanic rock (fig. 6). Flatfish blocks generally are parallel to the bedding. The shape of these fragments suggests that they are not volcanic bombs, but rather are blocks of lava. Their well-defined regular bedding indicates that the poor sorting is not the result of mass flow; and the absence of graded beds, as well as extremely wide range in the size of the clasts indicates that they were20 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT Figure 5.—Photograph of volcanic breccia of probable mudflow origin. Closely packed fragments. Dark spots in clasts are pyroxene phenocrysts. Fragment of plagioclase-pyroxene rock (light colored with scattered black crystals) in lower left corner between two pyroxenite (black) fragments. Part of block more than two feet long shown in upper left corner. Logtown Ridge formation, Cosumnes River, about 2,000 feet southwest of Huse Bridge. not formed by turbidity currents. Such deposits probably record the sporadic activity of the source vent. Load casts (Kuenen, 1957, p. 246-248, fig. 13), shown in figure 7, and more wildly contorted structures resulting from contemporaneous deformation are common in the lower part of the Logtown Ridge formation and have been noted elsewhere. Cut-and-fill structures measurable in inches are widespread in volcanic rocks of the western Sierra Nevada, but one erosion channel noted in the lower part of the Logtown Ridge formation in the Cosumnes River is about two feet deep and six feet wide at the top. The channel is cut into thin-bed- ded, medium- to very fine-grained tuff and filled with the same kind of material. Many beds in the lower part of the Logtown Ridge formation contain angular fragments of very fine black tuff, like that in place in associated beds (fig. 8). In some beds the fragments are widely scattered, suggesting that they were well stirred into the enclosing coarser tuff and may have been carried a considerable distance by mudflows. In other beds the fragments are closely packed and have moved only a short distance from their source, apparently as result of restricted subaqueous slumping that involved a single layer or possibly twoSTRATIGRAPHY 21 Figure 6.—Photograph of volcanic breccia showing scattered large clasts in bedded pyroclastic groundmass. Tops of beds are toward the left. Logtown Ridge formation, Oosumnes River, about 1,000 feet west of Huse Bridge. or three adjacent layers of the very fine tuff that constitutes the fragments. Small clastic sills of medium-grained tuff that intruded very fine tuff (fig. 9) can be observed in the lower part of the Logtown Eidge formation along the Co-sumnes Eiver. Beds in the photograph are of subaqueous origin and graded beds are abundant stratigraphi-cally above and below. That the medium-grained tuff forms sills, rather than lenticular beds, is shown by the associated cross-cutting dikes and by the separation and breaking of septa of very fine dark-colored tuff visible below the coin in the photograph (fig. 9). Intrusion of the sills apparently took place after deposition of the beds but before they were indurated, for although the very fine tuff was sufficiently coherent to break into angular fragments the coarser tuff was able to flow. Clastic dikes more than a few inches long are rare and both these and the sill-like bodies are the results of processes that apparently operate only locally. Fossils and age The Logtown Eidge formation apparently ranges in age from Callovian to upper Oxfordian or lower Kimmeridgian. Ammonites collected by Clark and E. W. Imlay (Mesozoic localities 24710 and 27397, respectively, in the U.S. Geological Survey collections) from above an abandoned ditch on the hill slope north of the Cosumnes Eiver at the abrupt southward bend of the river about 4,550 feet southwest of Huse Bridge and about 1,000 feet above the base of the Logtown Eidge formation (figs. 4 and 28) are described as fol- 724-276 0-65-4 lows by E. W. Imlay (written communication, 1959) : The ammonites belong to species of Pseudocadoceras that have been found in Alaska, British Columbia, and the Taylorsville area * * *. The genus Pseudocadoceras in Europe ranges from * * * [lower to upper Callovian], Imlay (1952, p. 975-976) reported the occurrence of an ammonite in volcanic breccia of the Logtown Eidge formation near Huse Bridge, apparently in the upper part of the formation. He then described this ammonite as being “similar to Divisosphinctes from the upper Oxfordian and to some of the finer ribbed species of Pachysphinctes from the Kimmeridgian” but later (Imlay, written communication, January 1960) stated that “This ammonite is now identified with the genus Idoceras of late Oxfordian to early Kimmeridgian age.” Because the Logtown Eidge formation was previously assumed to be overlain by the Mariposa formation, of Kimmeridgian age, Imlay concluded (1952, p. 976) that this specimen is of upper Oxfordian age. The slates that Figure 7.—Photograph of thin-bedded to laminated tuff showing graded bed and load casts. Dime shows scale. Elongate hard areas are concretions. Logtown Ridge formation, Cosumnes River, 1% miles southwest of Huse Bridge.22 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT tuff with some interbedded chert or silicified ash and the unnamed epiclastic rocks is exposed on the margins of Lake McClure in the eastern part of sec. 7, T. 4 S., R. 16 E. The Penon Blanco volcanics were mapped as porphyrite and amphibolite by Turner (1897) except that he differentiated a unit of thin-bedded tuff and chert which is here included in the Penon Blanco volcanics. The Penon Blanco volcanics of this report were placed by Taliaferro (1943b, p. 283) in the Amador group of his southern type section although he was unable at that time to establish complete correlation with the Cosumnes River section. Taliaferro (1933, p. 149-150; 1943b, p. 283) divided the Amador group in the Figure 8.—Photograph of thin-bedded tuff showing intraforma-tional breccia bed below dim and small-scale crossbedding under breccia. overlie the Logtown Ridge formation, however, have not been demonstrated to be of Kimmeridgian age, so it is possible that the Logtown Ridge formation extends into the Kimmeridgian. PENON BLANCO VOLCANICS The name Penon Blanco volcanics is here applied to the sequence of volcanic rocks that underlies the Mariposa formation from the south end of the area shown on plate 1 to a point west of the town of Melones. The formation underlies and is named for Penon Blanco Ridge between the Merced and Tuolumne Rivers. The type section is along the Merced River. Of the three sections studied, this is the only complete one; in the Stanislaus and Tuolumne Rivers the Penon Blanco volcanics are in fault or intrusive contact with serpentine and part of the formation is missing. The Penon Blanco volcanics consist almost entirely of dark green mafic volcanic rocks. In the type section, where the formation is nearly 15,000 feet thick, tuff, lapilli tuff, and volcanic breccia constitute the upper part of the formation, and lava, mostly massive but some with pillow structure, constitutes the lower part. The areal pattern south of the Merced River suggests that the Penon Blanco volcanics thin rapidly southward, beyond the edge of the map area. The top of the Penon Blanco volcanics is well exposed on the banks of the Merced River near the northeast corner of sec. 27, T. 3 S., R. 16 E. (fig. 23). Bedding in the Mariposa formation is parallel to that of the Penon Blanco volcanics near the contact. The contact of the Penon Blanco volcanics with an underlying, unnamed sequence of epiclastic rocks is in sec. 15, T. 3 S., R. 15 E., but is not exposed at the abandoned railroad grade on the south side of the Merced River where the traverse was made. A conformable contact between Figure 9.—Photograph of clastic sills of medium-grained tuff (light gray) in very fine tuff (dark gray). Dime shows scale. Lowest light-gray band may be a bed. Note wedge shape of various light-gray layers and brecciation of dark-gray material.STRATIGRAPHY 23 southern area into five formations but he did not define his formations and they are not used here. The Penon Blanco volcanics of this report were included in the Logtown Ridge formation by Heyl, Stromquist, Swin-ney, and Wiese (pi. 2, in Eric, Stromquist, and Swinney, 1955). The Penon Blanco formation is here distinguished from the Logtown Ridge formation because the two are not continuous on the surface nor are the epi-clastic rocks that overlie the Logtown Ridge formation continuous with the Mariposa formation. LITHOLOGY Most of the Penon Blanco volcanics are medium- to dark-green mafic pyroclastic rocks, except along the Tuolumne River where massive lavas form most of the incomplete section. Some of the pyroclastic rocks, notably those near the Stanislaus River, contain pyroxene phenocrysts. Felsic crystal tuff with interbedded chert or silicified ash, apparently in the basal part of the Penon Blanco volcanics, is exposed in the western part of sec. 8, T. 4 S., R. 16 E. Original textures are well preserved, except on the Stanislaus River where original textures and structures have been almost obliterated by recrystallization, and in the central part of the section along the Merced River, which has been strongly epidotized. Massive aphanitic lavas, in part amygdaloidal, occur in all sections studied but form only a small part of the incomplete section along the Stanislaus River. Pillow lava forms three units near the top of the lava sequence on the Merced River; pillow structure is possible at places in lower parts of the lava sequence, but very poorly developed. Two thin epiclastic units (units 30 and 33, pi. 5) are exposed near the Stanislaus River and are interpreted to be interbedded with the Penon Blanco volcanics. Unit 33 is black slate; unit 30 is slate with interbedded tuff, conglomerate, and graywacke. Most of the pebbles in the conglomerate are composed of slate and volcanic rocks, but some are limestone. With the possible exception of microcrystalline quartzose layers interbedded with the felsic crystal tuff in the western part of sec. 8, T. 4 S., R. 16 E., no chert was found in exposures along the Merced River and Lake McClure. Unit 17 (pi. 3) is on the projection of the strike of a chert unit mapped by Turner (1897) south of the Merced River and referred to by Taliaferro (1943b, p. 283) as the Hunter Valley cherts, but near the river unit 17 consists of tuff with interbedded silicified ash. AGE No fossils have been found in the Penon Blanco volcanics. The upper part of the formation is not younger than Kimmeridgian for the overlying Mariposa forma- tion is of late Oxfordian or early Kimmeridgian age. The age of the lower part of the Penon Blanco volcanics is unknown. Part of the formation is probably of the same age as the Logtown Ridge formation. MARIPOSA FORMATION The Mariposa formation was named for exposures on the former Mariposa estate, southeast of Bagby (Becker, 1885, p. 18-19). As used by Turner (1894a, 1897), the Mariposa slate included all the dominantly epiclastic rocks in the various sequences west of the Melones fault zone, except those which he assigned to the western belt of the Calaveras formation. Taliaferro (for example, in Taliaferro and Solari, 1949) gave the formation a similar scope, except that he assigned some of the epiclastic rocks to the Cosumnes formation. Eric, Stromquist, and Swinney (1955, p. 12) substituted the term Mariposa formation for Mariposa slate because other rock types form important parts of the unit. Here, the name Mariposa formation is restricted to rocks in the central fault block and the Melones fault zone that are continuous on the surface with the type area, or are repeated across the strike by adequately documented structures. All previously described strata attributed to the Mariposa formation that have yielded diagnostic fossils and are within the area of figure 21 are retained in the Mariposa formation, but some rocks in the central block that were previously correlated with the Mariposa formation are excluded and will be discussed later. New names are proposed for rocks in the western block that were previously correlated with the Mariposa formation. Exclusive of the Brower Creek volcanic member, the Mariposa formation is chiefly black slate and silty slate, but tuff and graywacke form important parts of some sections. Conglomerate is mostly scarce but is locally abundant in the lower part southwest of San Andreas. The name Brower Creek volcanic member is here applied to a sequence of volcanic rocks, chiefly volcanic breccia, that forms part of the Mariposa formation north of the Stanislaus River. This member extends northwestward beyond the Mokelumne River, but probably not as far northwestward as the Cosumnes River. It is named for Brower Creek, whose headwaters are in the vicinity of Fowler Lookout. The Brower Creek member is thickest near Fowler Lookout, where it apparently represents an accumulation in the vicinity of a vent. Northwest of the lookout, the member divides into two tongues that are interlayered with the epiclastic rocks of the Mariposa formation. The Brower Creek member was mapped as diabase and porphyrite by Turner (1894a) and was included with the Logtown Ridge formation by Eric, Stromquist, and Swinney (1955, pi. 1). The Brower Creek volcanic member is24 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT surrounded on three sides by epiclastic rocks continuous on the surface with the type area of the Mariposa formation. Exposures of the member near Fowler Lookout and to the northwest along the strike are on the western flank of a syncline, and the beds dip steeply. Accordingly, the surface pattern portrays the true stratigraphic relations of the Brower Creek member to other parts of the Mariposa formation. The top of the Mariposa formation is not preserved, as it is everywhere truncated by faults or erosion. North of the Stanislaus River, the Mariposa formation lies in a syncline whose eastern and western limbs are truncated by the Melones and Bear Mountains fault zones. South of about the latitude of Chinese Camp, the eastern limb of the syncline is absent and the Mariposa formation forms part of an eastward-dipping homocline. The partial section of the Mariposa formation exposed near the Merced River is about 2,700 feet thick. Near the Stanislaus River the formation seems to be much thicker, but the structure is not well enough known in the eastern part of its outcrop to justify more than a crude estimate here. Near the Mokelumne River, the Mariposa formation is about 4,000 feet thick, of which nearly all is the Brower Creek volcanic member. The Mariposa formation is identified on all streams studied except the Cosumnes River, but the most readily accessible good exposures are on the Merced River, northwest of Bagby and in readouts along State Highway 49 south of Bagby. The lower part of the formation is well exposed in the Tuolumne River west of Jacksonville, and higher parts of the formation are exposed in State Highway 49 cuts northwest of Jacksonville. The railroad grade on the north side of the Calaveras River about 2i£ miles west of San Andreas provides large exposures of both the epiclastic rocks and the Brower Creek volcanic member. A departure from the usual practice of this investigation has been made in tentatively correlating with the Mariposa formation a belt of epiclastic and volcanic rocks extending from west of Altaville to south of the Stanislaus River. These rocks lie east of the main belt underlain by the Mariposa formation and are separated from it by a fault of the Melones fault zone; nevertheless, correlation of rocks in this area seems more certain than that of rocks in other outlying areas. LITHOLOGY The Mariposa formation, exclusive of the Brower Creek member, consists largely of slate, tuff, graywacke and conglomerate in the same manner as the Cosumnes formation. In most places, thin beds of tuff and graywacke are interbedded with the slate throughout the formation, but proportions of these rocks to slate varies considerably from place to place. In most of the sections studied, a unit of tuff about 300 feet thick is interbedded in the lower part and a unit of graywTacke about 500 feet thick farther up. Eric, Stremquist, and Swinney (1955) have differentiated the slate, tuff, and graywacke of the Mariposa formation on their maps of the Angels Camp and Sonora quadrangles. The conglomerates of the Mariposa formation mostly resemble those of the Cosumnes formation, but one type is peculiar to the Mariposa formation. This type is exposed at the south end of the area underlain by the Brower Creek volcanic member, and west of the north end of the north-trending arm of Melones Reservoir. Pebbles and cobbles of the conglomerate, as much as 5 inches in diameter in some beds, are well rounded and consist largely of resistant rocks, such as white and dark-gray vein quartz, black chert, orthoquartzite, and quartz conglomerate. Less resistant and less abundant pebbles consist of gray chert and slate, medium-grained granitic rocks, and fine-grained gneissic granitic rocks. Pebbles of volcanic rocks are rare or absent. The conglomerate and associated sandstones contain little silt-or clay-size material. The Mariposa formation includes poorly exposed sandstone and fine conglomerate rich in quartz and chert and with little detrital matrix near the Calaveras River (unit 26, pi. 6). The quartz grains are angular to well rounded and the chert grains are angular to subangular; the sandstone also contains fragments of slate and probable quartzite made up of quartz grains with sutured contacts. The associated conglomerate has a larger proportion of slate fragments than the sandstone. The quartz and chert sandstone surrounds an exposure of limestone in the NWA^EiA sec. 22, T. 4 N., R. 11 E., about 700 feet southeast of an abandoned farmhouse. The limestone is black, thin-bedded cal-carenite, some beds of which contain finely comminuted organic detritus and scattered quartz grains. The shape and attitude of the limestone and its contained quartz grains suggest that the limestone may be interbedded with the sandstone. These rocks differ from others of the Mariposa formation in being better sorted and containing a more restricted range of pebble types. However, they do not contain fragments of rocks foreign to the graywacke and other conglomerates of the Mariposa formation. Thin beds of megascopically similar sandstone are interbedded with limestone pebble-bearing conglomerate in rocks of uncertain stratigraphic position near the Cosumnes River (unit 31, pi. 8). Although the sandstone is not known to be interbedded with limestone pebble-bearing conglomerate near the CalaverasSTRATIGRAPHY 25 River, such conglomerates are exposed in strike with the sandstone about 3 miles to the southeast (A. A. Stromquist, written communication 1952). No chert has been observed in the Mariposa except in the lower part west and southwest of San Andreas, where it forms parts of units 23, 25, and 28 of the Calaveras River section. Here, it is thin bedded and light to dark gray and is interbedded with slate and quartzose slate of the same colors. Some is sericitic, with varied composition, and probably there is a complete gradation between chert and sericite slate. In thin section, much of the sericitic chert can be seen to contain small elliptical bodies composed of quartz more coarsely crystalline than the surrounding material; perhaps they formed from Radiolaria, as they differ from detrital quartz grains in their shape and polycrystalline character. The Brower Creek volcanic member of the Mariposa formation consists largely of coarse dark-green volcanic breccia, in places containing blocks 2 or more feet in diameter. Rarely, clasts of pyroxenite and limestone occur in the breccia of the upper tongue. Pyroxene phenoorysts as large as 1 centimeter in diameter are common in fragments and matrix of many breccia layers. The upper tongue of the Brower Creek member contains some tuff and lapilli tuff and one small exposure of pillow lava. The only epiclastic detritus in the Brower Creek member, other than the rare limestone fragments, occurs on the margins of Pardee Reservoir on the Mokelumne River. Here, thin beds of slate are interlayered with the volcanic breccia of unit 31/ (pi. 7) on the west limb of a syncline, and small conglomerate lenses with a few pebbles of epiclastic origin form part of unit 35 on the east limb of the syncline. J. H. Eric (written communication, 1950) reports similar conglomerates north of the reservoir about on the strike of unit 35. FOSSILS AND AGE According to R. W. Imlay (written communication, 1959) : The Mariposa formation is definitely of late Oxfordian to early Kimmeridgian age throughout most of its extent. The most common fossil is Buchia concentrica (Sowerby) which in northern Eurasia and in Alaska ranges from upper Oxfordian into middle Kimmeridgian (Imlay, 1959, * * * p. 157, 158) * * *. The Oxfordian age assignment is based on the occurrence of the ammonites Dichotomosphinctes and Discosphinctes. Of these Dichotomosphinctes ranges through the Oxfordian, but is particularly characteristic of the upper Oxfordian. Discosphinctes occurs in the upper Oxfordian and questionably in the lower Kimmeridgian * * *. The early Kimmeridgian age assignment of part of the Mariposa formation is based on the ammonite Amoebites, a subgenus of Amoeboceras [see locality 719 below]. * * * As a result of subsequent investigations, Imlay (written communication, January 1960) believes that faunas at localities 902-903 below are probably also early Kimmeridgian. Faunal localities in the Mariposa formation from a compilation by Imlay, are listed below. Numbers designate Mesozoic locality numbers in collections of the U.S. Geological Survey. References to current (1959) maps have been added in parentheses to old locality descriptions. 243 Near Pine Tree Mines, Mariposa Estate, Mariposa County. (See topographic map of the Coulterville 15-minute quadrangle.) Buchia concentrica (Sowerby) Perisphinctesl sp. 253 Left bank of Merced River 14 mile below Bentons Mills, Mariposa County. (Bentons Mills known as Bagby in 1959; see pi. 1.) Buchia concentrica (Fischer) 719 Texas Ranch, Calaveras County. (Near Texas Charlie Gulch, pi. 5.) Amoeboceras (Amoebites) dubium (Hyatt) Buchia concentrica (Sowerby) 901 Trail on south side of Stanislaus River, opposite Bosticks Bar, near Reynolds Ferry, NW!4 sec. 3, T. 2 N., R. 13 E., Tuolumne County, flooded by Melones Reservoir (see pi. 5 or topographic map of Copperopolis 15-minute quadrangle, edition of 1916). Perisphinctes (Discosphinctes) virgulatiformis Hyatt P. (Dichotomosphinctes) cf. P. (D.) miihlbachi Hyatt Taramellicerasl denticulatum (Hyatt) “Belemnites” paciflcus Gabb Turbot sp. Cerithiuml sp. Aviculal sp. (Hyatt, 1894, p. 429) 902 South bank of Tuolumne River at Moffit Bridge site, Tuolumne County (near Jacksonville, pi. 4) and east of contact between Penon Blanco volcanics and Mariposa formation. Buchia concentrica (Sowerby) Subdichotomocerasl aff. 8. filiplex (Quenstedt) 903 Stanislaus River near canyon opposite mouth of Bear Creek, near center of sec. 11, T. 1 N., R. 13 E., Calaveras County (pi. 5). Buchia concentrica (Sowerby) “Belemnites" pacificus Gabb Subdichotomocerasl aff. 8. filiplex (Quenstedt) 904 Six miles from Copperopolis on road to Sonora and on grade to Angels Creek, Calaveras County. (Probably in sec. 33, T. 2 N„ R. 13 E., pi. 5.) Subdichotomoceras aff. S. filiplex (Quenstedt) Buchia concentrica (Sowerby) “Belemnites” pacificus Gabb Amusium aurarium Meek 1982 Hell Hollow, north of Bear Valley, Mariposa County (pl. 3). Perisphinctesl sp. “Belemnites” pacificus Gabb\ STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT 26 1983 5 miles southeast of Princeton, Mariposa County. (Princeton is shown as Mount Bullion on the topographic map of the Coulterville 15-minute quadrangle.) Buchia sp. Amusinm aurarium Meek 20554 Woods Creek Canyon 3 miles northwest of Jacksonville, Tuolumne County. (PI. 1 and topographic map of Sonora 15-minute quadrangle.) Buchia concentrica (Sowerby) 22176 Left bank of Mokelumne River, 400 feet west of the east edge of sec. 17, T. 5 N., R. 11 E., Sutter Creek 15-minute quadrangle, Calaveras County (pi. 7). Ammonite fragment 24317 Loose boulder in bed of Cherokee Creek, sec. 22, T. 3 N., R. 12 E., San Andreas 15-minute quadrangle, Calaveras County. Matrix is identifiable with beds in the Mariposa formation west of Cherokee Creek. Perisphinctes (Dichotomosphinctest) sp. 27311 Moffit bridge site. Railroad cut on south side of Tuolumne River 200 to 225 feet east of contact between Penon Blanco volcanics and Mariposa formation. Near Jacksonville, Tuolumne County, plate 4. Buchia concentrica (Sowerby) 27312 North bank of Merced River, NW corner, sec. 26, T. 3 S., R. 16 E., Mariposa County, plate 3 Buchia concentrica (Sowerby) 27398 Crest of Woods Creek Ridge 1 mile northwest of Jacksonville, Tuolumne County. Coll, by George R. Heyl, 1946. (See topographic map of Sonora 15-minute quadrangle.) Buchia concentrica (Sowerby) Lima ? sp. 27459 Woods Creek, Tuolumne County. Received from Becker. (See topographic map of Sonora 15-minute quadrangle.) Buchia sp. Lima1! sp. 27460 From Mount Bullion, Mariposa County, collection by W. D. McLeam, Coulterville, California. Received May 3, 1922. Perisphinctes (Dichotomosphinctes) cf. P. (D.) miihVbachi Hyatt STRATA OR UNCERTAIN STRATIGRAPHIC POSITION VOLCANIC STRATA The Jurassic of the central part of the region includes various bodies of volcanic rock that have not been correlated with the standard formations because they are separated from them by faults, or have been insufficiently appraised during this investigation. A body that underlies Mount Bullion constitutes the only known occurrence of Jurassic rocks east of the Melones fault zone within the area covered by this report. Its true extent is unknown, but volcanic rocks extending from near the Merced River to the south edge of plate 1 have been included arbitrarily. Poor exposures on Mount Bullion suggest that the unit consists largely of coarse mafic volcanic breccia, in part containing pyroxene phenocrysts, but it also includes some mafic tuff and pillow lava. The internal structure of the unit and the kind of contact with the Calaveras formation are also unknown, owing to a lack of detailed mapping in this part of the region. The age of the unit is established by an ammonite recovered from a ground-sluice mine on top of a narrow ridge 600 feet southeast of the northwest comer of sec. 25, T. 4 S., R. 17 E., Bear Valley 7%-minute quadrangle, California, The mine was cut into bedrock and the dumps contained no material foreign to the immediate area. The ammonite, identified as Perisphinctes (Dichotomosphinctes) cf. P. (D.) miihlbachi Hyatt, is of late Oxfordian age (Imlay, written communication, 1959). Another large body extends from north of the Cosumnes River to southwest of Mokelume Hill immediately west of the Melones fault zone. Near the Cosumnes River, this body is in fault contact with the epiclastic rocks to the west and with sheared grano-diorite on the east. In this section the western part of the volcanic body is composed of dark-green mafic volcanic breccia with pyroxene phenocrysts. A thin unit of interbedded tuff and black slate occurs in the eastern part. Volcanic rocks of this narrow belt might belong to the Logtown Ridge formation, the Brower Creek volcanic member of the Mariposa formation, or to neither. An elongate area in the eastern part of the Bear Mountains fault zone near Jackson Creek is shown on plate 1 as volcanic rocks of uncertain stratigraphic position but includes some bands of epiclastic rocks to small to differentiate. Near Jackson Creek, the rocks of the area are tuff, with some interbedded slate and porphy-ritic volcanic breccia. These rocks are faulted against serpentine on the west and to the east lie with undetermined relations against the quartzose phyllite assigned to the Calaveras formation. The elongate fault-bounded block west of Fowler Lookout, consists mostly of mafic pyroclastic rocks but includes an amygdaloidal mafic lava flow, a thin-bedded chert unit, a thin unit of slate with some interbedded conglomerate, and small lenticular bodies of very finegrained massive limestone. The geographic position of this block suggests that it is part of the Copper Hill volcanics or the Penon Blanco volcanics. EPICLASTIC STRATA The epiclastic unit exposed along the Cosumnes River east of Huse Bridge consists largely of black slate and siltstone but contains some tuff, graywacke, and fine conglomerate. The epiclastic belt conformably overlies the Logtown Ridge formation for several miles north and south of the Cosumnes River. Absence of a break in deposition at the top of the Logtown Ridge formation is shown by top determinations along the Cosumnes River, and parallelism of beds in both units.STRATIGRAPHY 27 The unit was previously mapped as Mariposa formation (Lindgren and Turner, 1894; Taliaferro, 1943b, fig. 2), but detailed mapping in the area between the Mokelumne and Cosumnes Rivers will be necessary to determine whether this correlation is valid; for the present, therefore, the unit is not designated by a formation name. Existing maps (Lindgren and Turner, 1894; Lindgren, 1900) show the epiclastic rocks east of Huse Bridge to be part of a belt that extends continuously more than 40 miles north of the Cosumnes River. There may, however, be structural complications in part of the belt, and Imlay (1952, p. 975) believes that strata near its north end are of Callovian age, or more nearly equivalent in age to the Amador group than to the Mariposa formation. Buchia concentrica, Entolium'l. sp., and Nucula? sp., were identified by Imlay (written communication, 1959) in a collection from a narrow projection of this belt exposed in a small gully % mile north-northwest of Plymouth. In the northern part of the region another belt of epiclastic rocks lies along the western side of the central fault block from the north end of the area shown on plate 1 to south of the Mokelumne River. These rocks are black slate, graywacke, conglomerate, and mafic volcanic rocks. Pebbles in the conglomerates are composed chiefly of volcanic rocks, slate, chert, and vein quartz; some beds contain limestone pebbles. Sandstone consisting largely of well-rounded quartz grains is interbedded with the conglomerate of unit 31 (pi. 8) along the Cosumnes River. These rocks were mapped as Calaveras formation by Turner (1894a, and Lindgren and Turner, 1894), and Taliaferro (1943b, fig. 2) included them in the Amador group. They resemble both the Cosumnes and Mariposa formations and it cannot be determined whether they belong to one or the other. A narrow belt of conglomerate (unit 11, Jackson Creek section, pi. 7) exposed along Jackson Creek probably belongs to the Mariposa formation but may be part of the Cosumnes formation. It was included by Turner (1894a) in the Calaveras formation. The conglomerate, like those of the Cosumnes and Mariposa formations, contains varied pebbles, including limestone. It apparently lies conformably below the Brower Creek volcanic member of the Mariposa formation. The thin-bedded black chert and slate of unit 10 on Jackson Creek, exposed immediately west of the conglomerate, is here included with the Calaveras formation, but may be part of the same sequence as the conglomerate. The epiclastic sequence exposed along the northeast side of Lake McClure (unit 13, pi. 3) consisting of interbedded black slate and tuff, underlies the Penon Blanco volcanics. It was included by Turner (1897) with the Mariposa formation, but its stratigraphic position is incompatible with this assignment. No new name is proposed for this sequence because it extends south of the area shown on plate 1 and the exposures studied constitute but a small part of the formation. BOCKS EXPOSED IN THE WESTERN BLOCK AND BEAR MOUNTAINS FAULT ZONE The western block includes the area lying between the Bear Mountains fault zone and the western limit of bedrock exposures. In the northern part of this area, the rocks consist of a lower volcanic formation, a middle epiclastic formation, and an upper volcanic formation. These formations extend to the southern part of the area studied, where additional igneous and epiclastic rock formations may occur. Relations between the three most extensively exposed formations of the western block are best displayed in the Cosumnes River (pi. 8). Volcanic rocks of the western block resemble those of the central block in that they are dominantly of pyroclastic origin and are mostly of probable andesitic or basaltic composition. Rhyolite or dacite, however, is much more common in the westernmost belt of volcanic rocks than in others. The epiclastic rocks of the western block apparently contain less graywacke and conglomerate than the Cosumnes and Mariposa formations of the central block, but are otherwise similar. The stratigraphic sequence of the western block has not previously been distinguished from that in the central block. Structural data collected during the current investigation, however, indicate that some of the rocks of the western blocks may be younger than any exposed in the central block and that correlation of other units is complicated by the difficulty of establishing lateral continuity between the two blocks. New names are proposed for subdivisions of the western block. GOPHER RIDGE VOLCANICS The name Gopher Ridge volcanics is here used for the westernmost exposed volcanic formation in the part of the area shown on plate 1 that lies north of the Stanislaus River. The formation extends northwestward, and probably southeastward beyond the area studied, but the stratigraphic relations in the southern part of the area studied are less clear. The formation underlies and is named for Gopher Ridge, a prominent topographic feature near the west side of the exposed belt of metamorphic rocks between the Stanislaus and Calaveras Rivers. The base of the Gopher Ridge volcanics is probably not exposed. The formation is overlain by, and intertongues with, the Salt Spring slate. The section along the Calaveras River, at the northern end of Gopher Ridge, is designated as the type section.28 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT The Gopher Ridge volcanics were mapped as diabase, porphyrite, and amphibolite schist by Turner (1894— 1897). They were included with the Amador group by Taliaferro (1943b, fig. 2), Heyl (1948c, p. 114) and Heyl, Cox, and Eric (1948, p. 67), and with the Log-town Ridge formation of the Amador group by Taliaferro and Solari (1949, pi. 1). The Gopher Ridge volcanics consist largely of pyroclastic rocks but they contain lavas with and without pillows. Rocks of basaltic, andesitic, and rhyolitic composition have been identified by Heyl, Eric, and Cox (in Jenkins and others, 1948) in detailed studies of several mine areas in the belt of Gopher Ridge volcanics. Rocks of rhyolitic or dacitic composition are locally prominent, particularly between the Stanislaus and Calaveras Rivers, but the rocks of mafic or intermediate composition are much more abundant. Some of the felsic rocks have pyroclastic textures and are in-terbedded with the more mafic pyroclastics, but some are massive and are probably flows, or shallow intrusives penecontemporaneous with the bedded deposits. In mapping, it has not been possible to distinguish in most places between the flows and intrusives. Only partial thicknesses of the Gopher Ridge volcanics are known because their base is not exposed. The exposed part of the Gopher Ridge volcanics below the lowest bed of Salt Spring slate is about 4,000 feet in the Mokelumne River, 7,000 feet in the Calaveras River, and 12,000 feet in the Stanislaus River. Tongues of the Gopher Ridge in the Salt Spring slate give additional thicknesses of 8,000 feet in the Mokelumne River, 1,500 feet in the Calaveras River, and 3,000 feet in the Stanislaus River. The greatest total exposed thickness, 15,-000 feet, is in the Stanislaus River. Probably the formation is at least as thick in the southern part of the area as it is in the northern but the structure of the southern part is too poorly known to permit making a meaningful estimate. LITHOLOGY Both pyroclastic rocks, ranging from very fine tuff to coarse volcanic breccia, and lavas constitute the formation throughout the length of the area studied. Pyroclastic rocks are far more abundant than lavas in most of the area, but lavas form thick units near the Calaveras River. Massive rocks that may be lavas form much of the section exposed on the sides of La Grange Reservoir on the Tuolumne River. Near the Merced River, volcanic breccia is more abundant than tuff. Most of the tuff is either lithic or vitric, but recrystallization resulting from metamorphism of the rocks makes it difficult to determine in thin section whether the fragments were originally glass or microcrystalline lava. Shards have not been identified in thin section, but polished surfaces of some very fine tuff beds exposed about 500 feet west of the gaging station on the Tuolumne River (fig. 16) show thin arcuate structures that may be shards. Most of the very fine tuff is yellow green and siliceous and, where fresh, has a porcellaneous luster; where weathered, this material is easily mistaken for drab slate. Another very fine grained black rock interbedded with coarse to very fine tuff resembles the slate of the epiclastic sequences. It commonly forms beds one inch or less thick, with a gradational lower contact with the tuff and a sharp upper contact. The rock is very well indurated, probably siliceous, and breaks with a con-choidal fracture. It is probably also very fine tuff or an argillaceous material formed by submarine weathering of volcanic detritus. Most of the rocks in the formation referred to as lavas are not amygdaloidal, but amygdaloidal lavas occur in unit 12 on the Tuolumne River (pi. 4), and small masses of amygdaloidal rocks are dispersed in non-amygdaloidal material in unit 7 on the Calaveras River (pi. 6). Near the Calaveras River, on the west limb of the large anticline, gross layered structure suggests superimposed flows. Near the Tuolumne River are several masses of silicified massive volcanics that are possibly flows. Some of these so-called lavas may be hypabyssal intrusives, but their metamorphism appears identical to that of the pyroclastic rocks, so that they are not significantly younger. Pillow lavas are nowhere prominent, but form part of unit 1 on the Mokelumne River (pi. 7) and unit 6 on the Calaveras River (pi. 6). Felsites containing quartz phenocrysts, probably rhyolite or dacite or both, occur on the Cosumnes and Merced Rivers, but are more common on the Stanislaus and Mokelumne Rivers. They include both massive and pyroclastic varieties. The pyroclastic rocks are tuff, siliceous ash, and volcanic breccia and are interbedded with the more mafic volcanics (for example, unit 11, pi. 7). Part of the massive felsite is intrusive, but a part may be extrusive, for strip mapping along the streams did not provide data to determine whether most individual masses were one or the other. Intrusive felsite is probably about contemporaneous with the Gopher Ridge volcanism for it is not known to intrude the Salt Spring slate, which overlies the Gopher Ridge volcanics, and close spacial association of massive, possibly intrusive, felsite with bedded pyroclastic felsite in unit 21 on the Cosumnes River (pi. 8) and units 12 and 13 on the Calaveras River (pi. 7) suggests that the two types are essentially contemporaneous. On the other hand, the boundary of fresh-appearing amygdaloidal felsite cuts across schistositySTRATIGRAPHY 29 of the greenstone on the south side of the Stanislaus River near the west quarter-comer of sec. 21, T. 1 S., R. 12 E. (pi. 5). Downstream from section 21, the relations between massive felsite and mafic or intermediate volcanics are too complex to show at the scale of this mapping. Metamorphism has produced abundant epidote, al-bite, and tremolite and some chlorite in most of the rocks of the Gopher Ridge volcanics, as it has in the rocks of the central block. Nevertheless, where the rocks are not schistose, primary sedimentary textures are readily visible in outcrops, although less so in thin section. Secondary planar structure (schistosity) has in places obscured or destroyed the primary textures. In most schistose volcanics the texture of originally coarse pyroclastics is recognizable, although the fragments are elongated and flattened, but the texture of fine-grained pyroclastics is lost. Schistosity is prominent in the Bear Mountains fault zone and in the Merced River section. On the Calaveras River, some of the massive rocks, presumably lavas, have been brec-ciated tectonically and the cracks strongly epidotized. AGE No fossils have been found in the Gopher Ridge volcanics. They are probably of Late Jurassic age, although the lower part of the formation may be somewhat older. The upper part of the formation is not younger than Kimmeridgian in the northern part of the area, as fossils of late Oxfordian or early Kimmeridgian age have been found in the Salt Spring slate near the Cosumnes River, and this overlies and intertongues with the Gopher Ridge volcanics. At least part of the Gopher Ridge volcanics is probably of the same age as the Logtown Ridge formation (Callovian to late Oxfordian). SALT SPRING SLATE The name Salt Spring slate is here used for the dominantly epiclastic rocks that overlie and intertongue with the Gopher Ridge volcanics. The formation is named for Salt Spring Valley, which lies immediately east of Gopher Ridge (pi. 1). Exposures of the formation are poor in Salt Spring Valley and the section exposed in the Cosumnes River near the Michigan Bar Bridge (pi. 8) is designated as the type section. Black sericite slate dominates in the formation, but gray-wacke and tuff are widespread and thin conglomerate layers occur in some places. The Salt Spring slate is of about the same age as the Mariposa formation, but the two formations may have been deposited in separate basins, as suggested by Turner (1894b, p. 457). The Salt Spring slate is exposed in a continuous belt, throughout the length of the area mapped, according 724-276 0-65-5 to existing maps. North of the Stanislaus River the slate forms a single formation that overlies the Gopher Ridge volcanics. Near the Tuolumne River, however, the available data suggest that the relations between part of this slate belt and the belt of volcanics to the west may be different than they are north of the river, so that correlation of the slate exposed in Don Pedro Reservoir and to the south is tentative. The Salt Spring slate was mapped by previous geologists as Mariposa formation (Turner, 1894a, 1897; Taliaferro, 1943b, fig. 2; Taliaferro and Solari, 1949; Heyl, Cox, and Eric, 1948, p. 66). LITHOLOGY The Salt Spring slate is lithologically much like the epiclastic parts of the Mariposa and Cosumnes formations except that it contains little conglomerate. Slates and graywackes in all three formations are megascopi-cally indistinguishable. Most of the conglomerate in the Salt Spring slate is associated with graywacke and both are exposed on all streams studied. The conglomerates are mostly fine grained, have well rounded pebbles, and form bedded layers. None was found to contain limestone fragments. Dark-gray limestone in lenses one to two feet thick and a few feet long, and black quartzose slate are in-terbedded with the black sericite slate at the west side of Lake McClure, near a plesiosaur locality at the high-water line at the south boundary of sec. 12, T. 4 S., R. 15 E. The limestone is a medium-grained calcarenite and in thin section is seen to consist of rounded to angular calcite grains. The black color results from abundant very finely disseminated carbonaceous material which forms inclusions in most of the grains. The quartzose slate is thin bedded and has a subvitreous luster on fresh surfaces. FOSSILS AND AGE The Salt Spring slate is of Late Jurassic age. Fossils are rare in the Salt Spring slate and the few found previously are not diagnostic. During the present investigation, fossils were found at four localities, but fossils from only one of these have been dated. In collections by Clark and R. W. Imlay (Mesozoic localities 25638 and 27318, respectively in U.S. Geological Survey collections) from the south side of the Cosumnes River, 0.1 mile east of the Michigan Bar Bridge, in the SE14 sec. 36, T. 8 N., R. 8 E., Imlay (written communication, 1959) identified fBuchia concentrica (Sow-erby), “Belerrmites” sp., and Phylloceras sp. He considered them diagnostic of late Oxfordian to early Kimmeridgian age. Belemnite molds were found in the Salt Spring slate near Don Pedro reservoir on the Tuolumne River, and30 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT part of a plesiosaur skeleton was found on Lake McClure, on the Merced River. The belemnite molds were found in slate in the south part of sec 35, T. 2 S., R. 14 E., below the high-water line on Don Pedro Reservoir. The plesiosaur skeleton was found at the high-water mark of Lake McClure on the south line of sec. 1, T. 4 S., R. 15 E., enclosed in coarse-grained black clastic limestone that forms a lens about 1 foot thick in the black slate. The skeleton has not yet been studied. MERCED FALLS SLATE The name Merced Falls slate is here used for the epi-clastic rocks shown by Turner (1897) in the belt extending southeastward from La Grange to beyond the southern boundary of the area (pi. 1). The type section is along the Merced River east of the town of Merced Falls (see pi. 11). The formation is well exposed along the Tuolumne River and weathered exposures are plentiful on the north side of the Merced River. The top of the formation is not preserved but about 5,000 feet of strata form the type section. The Merced Falls slate overlies and intertongues with the Gopher Ridge volcanics. It apparently underlies and intertongues with the Peaslee Creek volcanics. LITHOLOGY Near the Tuolumne River the formation is almost entirely thin-bedded dark gray slate and siltstone, and only one thin bed of graywacke was observed. Along the Merced River, however, it includes considerable graywacke and some felsic tuff. Graywacke is inter-bedded throughout the Merced River section, but is coarser grained and forms thicker beds in the eastern upper part of the formation, where tuff is also common, than in the western lower part. Rare thin beds of conglomerate occur south of the Merced River. AGE L. E. Mannion of Stauffer Chemical Co. (oral communication, 1954) found Buchia in the Merced Falls slate about 1.5 miles southeast of La Grange in the south part of sec. 21, T. 3 S., R. 14 E. The present writer found none, but this genus indicates that the beds containing it are not older than late Oxfordian, as Buchia is unknown in older beds (Ralph W. Imlay, written communication, 1957). Belemnite molds, not adequate for dating, were found in black slate about 20 feet west of the contact between the slate and the Gopher Ridge volcanics, near the top of the penstocks of the La Grange power plant (fig. 10). COPPER HILL VOLCANICS The name Copper Hill volcanics is here used for the sequence of volcanic rocks that overlies and intertongues with the Salt Spring slate. It is named for the inactive Copper Hill mine, in the NEi,4 sec. 34, T. 8 N., R. 9 E., in Amador County. The type section of the formation is on the Cosumnes River (pi. 8), in Amador and El Dorado Counties. The Copper Hill volcanics are mainly pyroclastic rocks, probably mostly andesitic. The formation extends from north of the Cosumnes River to south of the Stanislaus River within and west of the Bear Mountains fault zone; volcanic rocks southward as far as the Merced River are included, but this correlation is less adequately supported. The Copper Hill volcanics are truncated on the east by faults and the top of the formation is probably not preserved in the area. The formation is exposed along the Stanislaus River and all rivers to the north. The best exposed and most complete section is along the Cosumnes River, but exposures are also excellent along the Moke-lumne River. The Copper Hill volcanics were mapped as diabase and amphibolite by Turner (1894a) and by Lindgren and Turner (1894). They were placed in the Logtown Ridge formation by Taliaferro (1943b, fig. 2), and by Taliaferro and Solari (1949). The part near Copper-opolis was tentatively assigned to the Amador group by Heyl (1948b, p. 99). Parts of the Copper Hill volcanics near the Newton mine, about 6 miles west of Jackson, were named the Mountain Spring volcanics and Newton Mine volcanics by Heyl and Eric (1948, p. 52-53), and were tentatively correlated with the Amador group (1948, p. 51). Partial sections show that the volcanic rocks of this unit are more than 7,000 feet thick on the Cosumnes River, and perhaps equally thick on the Stanislaus River, although here the rocks are more strongly foliated so that the thickness is less certain. The formation is more than 3,000 feet thick on the Mokelumne River. In the Calaveras River, the formation is strongly schistose and the thickness cannot be measured. LITHOLOGY The Copper Hill volcanics are moderately deformed in the northern part of the area, but much of the formation is strongly sheared near the Calaveras River and to the south. In some of the strongly sheared rocks the primary textures and structures are obscure or completely destroyed. In the northern part of the area, pyroclastic rocks form most of the formation, and tuff is more abundant than the coarser pyroclastics. Preserved textures suggest that similar proportions exist farther south. Minor pillow lava is interbedded near the Cosumnes and Stanislaus Rivers. Most of the Copper Hill volcanics are dark to medium green in color, without quartz phenocrysts, and are probably andesitic of basaltic. Porphyritic rhyoliteSTRATIGRAPHY 31 or dacite with quartz phenocrysts occurs on the Cosum-nes and Mokelumne Rivers; some of it is massive and either of hypabyssal or volcanic origin. Other parts on the Cosumnes River (unit 21, pi. 8), are bedded and show pyroclastic textures and are certainly extrusive. A distinctive porphyritic lava containing green saus-suritized plagioclase phenocrysts as much as 8 mm long in an aphanitic groundmass is interlayered with the more common volcanic rock types in the Cosumnes River (unit 25, pi. 8). Some of this forms pillow lava, with pillows as much as 5 feet in diameter. The direction of tops indicated by the pillows is consistent with that indicated by adjacent graded beds. This porphyry is extrusive, but similar rocks form sills in the Cosumnes and Logtown Ridge formations, and in the type section of the Amador group along the Cosumnes River (pi. 10). West of Amador City a sheet of similar rock, apparently a sill (Knopf, 1929, p. 17), is interlayered in greenstone of either the Log-town Ridge formation or the Brower Creek member of the Mariposa formation. age • No fossils have been found in the Copper Hill vol-canics. The lower part of the formation intertongues with the Salt Spring slate and may therefore be as old as late Oxfordian. Higher parts of the formation are probably of Kimmeridgian age or younger. It is apparently the youngest formation of the bedrock complex in the area studied, although the volcanic rocks south of La Grange or the slate exposed east of Merced Falls may conceivably be younger. PEASLEE CREEK VOLCANICS The name Peaslee Creek volcanics is given here to an isolated mass of volcanic and possible hypabyssal rocks, at the western edge of the area of exposed bedrock south of La Grange. The type area is about 2 miles south of La Grange. These volcanics were mapped as porphyrite by Turner (1897). The base of the formation, lying on the Merced Falls slate, is exposed immediately south of the Tuolumne River (fig. 10) but its top is not preserved. The thickness of the formation as exposed is uncertain, but is probably more than 3,000 feet. Its structure is shown diagrammati-cally on plate 4. LITHOLOGY The Peaslee Creek volcanics are not exposed in the Tuolumne River and relatively poor exposures about 1 mile south of La Grange were examined to complete the Tuolumne River section. Here, bedded pyroclastic rocks form the northeastern and lower part of the unit, and consist of tuff and volcanic breccia of probable mafic or intermediate composition, similar to those of the other volcanic sequences. The southwestern higher part of the unit consists of massive rocks without amygdules but with small feldspar phenocrysts and hardly with quartz phenocrysts. Most of the massive rocks are fresh-appearing and may be of hypabyssal origin, perhaps appreciably younger than the bedded pyroclastic rocks in the northeastern part of this belt. AGE No fossils have been found in the Peaslee Creek volcanics, but its stratigraphic relations to the Merced Falls slate indicate that it is of Late Jurassic age, and probably Oxfordian or younger. STRATIGRAPHIC RELATIONS BETWEEN THE FORMATIONS OF THE WESTERN BLOCK STRATIGRAPHIC RELATIONS BETWEEN THE SALT SPRING SLATE AND GOPHER RIDGE VOLCANICS Stratigraphic relations between the Salt Spring slate and the Gopher Ridge volcanics are shown best near the Cosumnes River. Moreover, this section is the only one where facies changes normal to the strike can be examined. Less complete data in the rivers as far south as the Stanislaus supplement those in the Cosumnes River, and show that the Gopher Ridge volcanics underlie and intertongue with the Salt Spring slate. Near the Tuolumne River, however, some of the epi-clastic rocks previously interpreted as continuous with those in Salt Spring Valley apparently underlie the Gopher Ridge volcanics. In the Merced River, the contact between the main belts of these formations is faulted. The more significant sections are described below. Cosumnes River exposures.—The Gopher Ridge volcanics and Salt Spring slate are exposed near the western end of the Cosumnes River section and are repeated by folding east of the Michigan Bar Bridge (pi. 8). Bedrock is concealed near the contact between the two formations east and west of the Michigan Bar Bridge. Enough observations can be made of tops of beds, bedding attitudes, and kinds of rock to show that the epi-clastic rocks are conformable with the volcanic rocks and that there is little if any interbedding. No slate body of comparable thickness occurs on the east limb of the anticline east of Michigan Bar Bridge, and the approximate stratigraphic equivalent of the slate at Michigan Bar Bridge is interbedded slate and pyroclastic rocks. Gross aspects of the interbedding are shown on plate 8, but the two rocks are also much more thinly interbedded, and exposed contacts are common. Here bedding dips consistently eastward and graded beds are abundant. Although precise correlations cannot be made between the two limbs of the anticline, the stratigraphic interval corresponding to the epi-32 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT clastic sequence at Michigan Bar Bridge almost certainly lies within the more than 12,000 feet of strata on the east limb of the anticline. The contrast between the lithology in the syncline at Michigan Bar Bridge and that on the east limb of the anticline indicates that the Salt Spring slate intertongues eastward with volcanic rocks. In the Co-sumnes River area, the volcanics interbedded with slate on the east limb of the anticline have been divided arbitrarily between the Gopher Ridge volcanics and the Copper Hill volcanics. The outcrops suggest that the Salt Spring slate may be absent and the two volcanic formations in contact with one another at depth at the latitude of the Cosumnes River. Mokelum/ne River exposures.—Near the Mokelumne River, the Salt Spring slate forms three belts in a part of the section that is interpreted to be homoclinal with younger beds on the east. Each belt of slate must therefore be a tongue in the volcanic rocks. The volcanic rocks at Pardee Dam have been included arbitrarily with the Gopher Ridge volcanics. The volcanics at the dam are not continuous either northward or southward according to Turner (1894a), and might be a volcanic member within the Salt Spring slate, or a tongue of the Copper Hill volcanics. Thin layers of tuff interbedded in dominant slate units and thin layers of slate interbedded in dominant pyroclastic units are well exposed in the spillway and river canyon immediately west of Pardee Dam. Again, only gross relations could be mapped. The eastern belt of Salt Spring slate is poorly exposed around the margins of Pardee Reservoir and yields little stratigraphic information. Calaveras River exposures.—Exposures of the Salt Spring slate in the Calaveras River are generally poor but top determinations within both the slate and the Gopher Ridge volcanics near the western contact of the slate indicate tops towards the east so that the slate overlies the Gopher Ridge volcanics. Near Hogan Dam the structure has arbitrarily been interpreted as homoclinal with the volcanic rocks forming a tongue of the Gopher Ridge volcanics, but definitive structural data are lacking. Stanislaus River exposures.—The belt underlain by the Gopher Ridge volcanics is wide, and the belt underlain by the Salt Spring slate is narrow, south of the Stanislaus River. Near the river the map pattern (pi. 5; Taliaferro and Solari, 1949) indicates that the two formations intertongue or are infolded. Graded beds are scarce and exposures do not certainly show the relations between the two formations. Taliaferro and Solari (1949) interpreted the pattern as being the result of tight folding, but details within single exposures indicate that pyroclastic rocks are interbedded, rather than infolded, with epiclastic rocks. Small folds, such as are common where rocks of differing competency are tightly folded are rare or absent. The tuff and epiclastic rocks alternate with each other in layers ranging down to less than one inch in thickness so that the units shown on plate 5 are generalized. Tuolmrvne River exposures.—Exposures near the Tuolumne and Merced Rivers contribute little to an understanding of the relations between the Gopher Ridge volcanics and the Salt Spring slate to the east. Near the margins of Don Pedro Reservoir on the Tuolumne River (pi. 4), epiclastic rocks on the trend of the Salt Spring slate are cut out by a plutonic body. Those between the pluton and Don Pedro Dam, tentatively included in the Salt Spring slate, project southwestward into the belt of the Gopher Ridge volcanics and apparently underlie at least part of them. An apparently depositional contact between the two is exposed in the NE14 sec. 2, T. 3 S., R. 14 E., where the rocks are folded on a small scale but are not much sheared. The part of the volcanic section that appears to be structurally conformable with the epiclastic rocks immediately northeast of the dam extends southwestward from the dam for nearly a mile and is at least 700 feet thick. Volcanic rocks farther west are separated from this part of the section by shear zones and possible faults and their relations to the rocks nearer Don Pedro Dam are less certain. As suggested on plate 4, the epiclastic rocks northeast of Don Pedro Dam may form the core of an anticline and be a tongue of the Salt Spring slate overlain by a tongue of the Gopher Ridge volcanics that includes units 13,16,17, and 19. The tongue of epiclastic rocks would correspond approximately to the lower part of the slate in Salt Spring Valley, and the tongue of pyroclastic rocks to the upper part of the Gopher Ridge volcanics southeast of the Stanislaus River. However, if the fault extending toward the northwest corner of T. 3 S., R. 15 E. (pi. 1) continues into the approximate axis of the inferred anticline, epiclastic rocks southwest of the fault might be appreciably older than the Salt Spring slate. Merced River exposures.—Near the Merced River epiclastic rocks tentatively placed in the Salt Spring slate are exposed on the shores of the northwest-trending arm of Lake McClure and west of Exchequer Dam. Of these, only those west of the dam are known to be in depositional contact with the Gopher Ridge volcanics. The epiclastic rocks on the shores of Lake McClure and in railroad cuts on the east side of the lake are on the extension of the belt of epiclastic rocks that underlies Salt Spring Valley. No long uninterrupted sections across the strike are available in this vicinity andSTRATIGRAPHY 33 scattered bedding attitudes and top determinations indicate that the rocks in this area are folded. At least part of the contact between these epiclastic rocks and the Gopher Ridge volcanics to the west is faulted as well. The epiclastic rocks west of Exchequer Dam (units 6-8, pi. 3) form a syncline in the Gopher Ridge volcanics and are probably separated by a fault from the epiclastic rocks that underlie the western part of Lake McClure. The western contact of the epiclastic rocks is not exposed, but about one foot of the eastern contact is exposed immediately below the road. Here, undisturbed slate is in contact with a layer of massive, amygdaloidal, nonporphyritic lava about 25 feet thick. Graded beds in the tuff immediately east of (below) the lava indicate that tops are to the west and that the slate overlies the lava. Graded graywacke beds in the western part of the slate belt indicate that the graywacke and associated slate overlie the volcanics to the west. STRATIGRAPHIC RELATIONS BETWEEN THE SALT SPRING SLATE AND COPPER HILL VOLCANICS Stratigraphic relations between the Salt Spring slate and Copper Hill volcanics are shown best on the Cosumnes River. The contact between the two formations on the Mokelumne and Calaveras Rivers is poorly exposed and top determinations have not been made nearby. The Salt Spring slate and Gopher Ridge volcanics are exposed near, but not at, the contact on the Stanislaus River and top determinations have not been made near the contact. On the Tuolumne River, the formations are separated by intrusive rocks and near the Merced River they are separated by a fault. On the Cosumnes River, the volcanics interbedded with epiclastic rocks on the east limb of the anticline east of Michigan Bar Bridge have been divided arbitrarily between the Gopher Ridge volcanics and the Copper Hill volcanics. The stratigraphically lowest unit included with the Copper Hill volcanics is unit 18, plate 8. Within this unit and the epiclastic units on the two sides graded beds are common and exposures are excellent, showing an interbedded, conformable relation. Graded beds were not found for several hundred feet stratigraphically above the uppermost epiclastic unit (unit 19, plate 8). STRATIGRAPHIC RELATIONS OF THE MERCED FALLS SLATE TO THE PEASLEE CREEK VOLCANICS AND GOPHER RIDGE VOLCANICS No graded beds were found in the Merced Falls slate along the Tuolumne River, but minor folds near La Grange suggest that the slate is on the western limb of an anticline (fig. 10). The Merced Falls slate thus probably overlies the Gopher Ridge volcanics and underlies the Peaslee Creek volcanics. The western contact of the slate is not exposed near the river, but the eastern contact is well exposed at the La Grange powerhouse (fig. 10). Neither this exposure nor a smaller one in the canal on the north side of the river indicate the stratigraphic relations although they suggest that the two formations are conformable. Below the headgates of the penstocks, the massive volcanic rocks east of the slate form a vertical face, and the slate is exposed in a gully at the base of the face. The volcanics are not bedded, but bedding in the slate parallels the surface of the volcanic rocks. Within a few feet of the contact, fragments of volcanic rock are present in the slate—they might be interpreted as either the first or the last small episode of volcanism related to the accumulation of volcanic rocks east of the slate. In the bank of the canal north of the river a similar relation is shown. Here, a layer of amygdaloidal volcanic rock about two feet thick is interlayered with pumice-bearing slate near the contact of the slate with the volcanic sequence. Abundant graded bedding between tightly folded areas in the slate east of Merced Falls shows that tops are consistently westward in noncrumpled parts of the belt (plate 11). This is consistent with the gently plunging minor folds in the westernmost slate belt exposed in the Tuolumne River. At the eastern side of the belt underlain by the Merced Falls slate, the rocks are strongly crumpled, and the volcanic rocks with which they are in contact are schistose, suggesting that the two formations may be in fault contact. The slate belt becomes narrower, and the Gopher Ridge volcanic belt wider, northeast of Merced Falls. This probably results from intertonguing of the slate with the volcanics, for the most southerly projection of volcanic rock extends nearly to the Merced River, and tops are to the west in the epiclastic rocks on both sides of this projection. In addition, thin tuff beds are commonly interbedded with the slate and graywacke near the tongue. ROCK AND MINERAL, FRAGMENTS IN ROCKS OF JURASSIC AGE Field examination of conglomerate pebbles was made at about 50 localities and laboratory study was made of 34 thin sections of graywacke and fine conglomerate. More information is available from the Mariposa and Cosumnes formations than from other units which contain fewer graywacke and conglomerate beds.EXPLANATION LA GRANGE DAM Headgate lfl»fSi Powerhouse// SEA LEVEL 50//- ,.o> yr wm TURLOCK Geology and base mapped by L. D. Clark and Richard Pack, 1953 APPROXIMATE MEAN DECLINATION, 1962 500'- Approximate altitude of river surface SEA LEVEL -500'- Peaslee Creek volcanics s 'f Merced Falls slate jg Gopher Ridge volcanics U tf) to < cr D Contact Dashed where approximately located Fault Anticline, showing crest line Syncline, showing position of trough -a-+ Bearing and plunge of minor anticline Bearing and plunge of minor syncline -^*35 Bearing and plunge of minor folds, showing trace of bedding 70X _ Strike and dip of beds Overturned beds on section are not distinguished as such on map became evidence (minor folds) is indirect 90^ Strike of vertical beds BX Strike and dip of cleavage SO/-* Strike of vertical cleavage Bearing and plunge of intersection of bedding and cleavage Figure 10.—Geologic map and cross section showing structure of slate along Tuolumne River east of La Grange, Stanislaus County, California. STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELTSTRATIGRAPHY 35 COMPOSITION AND SHAPE OF FRAGMENTS The pebbles are similar in all formations studied and consist largely of volcanic rocks, slate, and chert, but those of quartzite and high-grade metamorphic rock are ubiquitous also (figs. 11, 12). Pebbles and boulders of felsic plutonic rocks have been found in the Cosumnes and Mariposa formations, and some sand grains in the Salt Spring slate may have been derived from the same rocks. Although the various formations are alike in composition of fragments, proportions between various rock types in adjacent beds of single formations vary strikingly, even in beds of similar grain size. Moreover, slate is less abundant and chert and quartz are more abundant in graywacke than in conglomerate. Most of the Jurassic rocks are in the greenschist or higher metamorphic facies, and most fragments derived from terranes of lower metamorphic grade have now been altered to the same metamorphic grade as the enclosing rock. Pebbles in the conglomerate are well rounded to angular. In some beds nearly all are angular, in others nearly all well rounded, but they range widely in degree of roundness. Degree of rounding is not everywhere directly related to hardness; some beds contain both well-rounded pebbles of vein quartz, and angular pebbles of chert, quartzose phyllite, and volcanic rocks. Carbonate rock fragments occur only in the Cosumnes and Mariposa formations. They are chiefly calcarenite containing much rounded organic detritus, including abundant crinoid fragments. Two fragments of oolitic limestone were found in coarse breccia consisting mostly of volcanic rocks in the western part of unit 28 on the Calaveras Kiver (pi. 6). A large boulder of fusulinid limestone of Permian age was mentioned previously (p. 14). The metamorphic rocks include quartz-mica schist, nonmicaceous aggregates of strongly strained quartz, and crinkled sericite slate. Granitic detritus is rare in all the conglomerates. In addition to fragments of coarse-grained rocks composed of quartz and potassium feldspar, separate grains of untwinned feldspar and one grain showing a micrographic intergrowth of quartz in feldspar were observed. Figure 11.—Photomicrograph of very coarse graywacke from Cosumnes formation, Cosumnes River, NWM sec. 22, T. 8 N., R. 10 E., showing mixture of detritus of volcanic, metamorphic, and sedimentary rocks. Chert with carbonate spicule(?), Ci; schistose quartz sandstone partly replaced by carbonate, Q; chert with scattered carbonate, C2; granular limestone with scattered embayed quartz grains, L; scoriaceous mafic volcanic rock, av; pyroxene, p; chlorite, ch; mafic volcanic rock, mv. Plain light.36 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT Figure 12.—Photomicrograph of very coarse graywaeke from Mariposa formation, SE14 sec. 16, T. 3 N., K. 12 E., showing association of detritus from diverse terranes. Chert with carbonate rhombohedrons, C; quartzite(?), Q; pyroxene, P; porphyritic mafic volcanic rock, mv ; slate with quartz veinlet, S; devitrified volcanic glass (?), G; chlorite pseudomorph, ch; microcrystalline epidote rock, E. Plain light. Pebbles and boulders of granitic rocks are rare but occur as fragments in the conglomerates of the Cosum-nes and Mariposa formations; they have not been found in the Salt Spring slate or Merced Falls slate, where conglomerates are less abundant. Most of these are fine-grained aplite or slightly coarser. In the Mariposa formation pebbles of medium-grained granitic gneiss occur in conglomerate on the south flank of the ridge topped by Fowler Lookout, and Knopf (1929, p. 13) reports a pebble of coarse diorite near the contact of the Mariposa formation and the Penon Blanco volcanics “one-half mile west of Kittridge” in the canyon of the Merced River. Untwinned feldspar, in part perthitic, is common in graywackes of the three epiclastic formations of Mesozoic age, and granophyre clasts are widespread. They are probably derived from felsic volcanic or hypabyssal rocks as they are most abundant in exposures of the Merced Falls slate east of Merced Falls. Here, felsic tuff of the Peaslee Creek volcanics intertongues with the Merced Falls slate. Graywackes of the Salt Spring slate also contain more undeformed quartz than those of other formations. The Salt Spring slate overlies and intertongues with the Gopher Ridge volcanics, which contain more felsic volcanic and hypabyssal rock than the other volcanic formations. Pebbles of coarse quartz-mica schist occur in the Cosumnes and Mariposa formations, and books of muscovite, probably derived from areas of high-grade meta-morphic rocks, are widespread in all the epiclastic formations. Fragments of fine-grained quartz schist containing very little mica is sparsely present in the Mariposa formation and Salt Spring slate. Fragments of chert and quartzose slate are abundant in the graywackes and conglomerates. The cherts range in color from nearly white to nearly black as well as several shades of green, and very rarely red. The chert varies in purity and texture. The silica in nearly all the chert is microcrystalline quartz, but chert consisting of opal or chalcedony occurs in graywaeke of the Mariposa formation west of Angels Camp and near the Tuolumne River. Chert, mostly phyllitic, withSTRATIGRAPHY 37 abundant finely disseminated carbonaceous material was found in all formations, but in a larger proportion of thin sections of the Salt Spring slate. Chert with scattered carbonate rhombs was observed in several thin sections of the Mariposa formation and Salt Spring slate. Chert, in part phyllitic, with ovoid structures (Radiolaria?) occurs in all formations. Nearly all of the clast-forming slate and siltstone resembles at least superficially the rocks with which the graywackes and conglomerates are interbedded. Some slate and siltstone clasts, however, contain abundant finely disseminated black material, probably carbonaceous, which is absent from most of the Jurassic rocks of the region. Fragments of volcanic rocks and minerals are abundant in most of the graywackes and conglomerates (fig. 11) and in some form more than 90 percent of those megascopically visible. Most of these fragments are microcrystalline or porphyritic with a microcrystalline groundmass. Their present texture, except for the phenocrysts, results largely from the regional metamorphism which has affected the Jurassic rocks, and they may have been derived either from volcanic glass, lava, or both. The phenocrysts are pyroxene and sericitized plagioclase, and these minerals also form discrete grains in some of the graywackes. Quartz is abundant in most of the graywackes and conglomerates, but adds little data about source ter-ranes. Both white and gray vein quartz have been distinguished in the conglomerates of the Cosumnes and Mariposa formations, but vein quartz has not been distinguished from quartz of other origin in thin sections of graywacke. Many of the quartz grains seen in thin section are strongly deformed aggregates that may have been derived from quartzite, veins, or granitic rocks. Nearly all the quartz shows strain shadows and vacuoles that are commonly arranged in planes. Sandstone fragments (figs. 11 and 12) occur in all three formations but especially in the Cosumnes formation. They are of three sorts: orthoquartzite, or quartz sandstone with quartz cement (Pettijohn, 1957, p. 295-296), graywacke, and sandstone having a carbonate matrix. The orthoquartzite ranges from fine sandstone to fine conglomerate, and consists of well-rounded quartz grains, many of which show overgrowths, but are not markedly sutured. Most of the component quartz grains show undulatory extinction and lines of vacuoles. The graywacke consists of angular grains of quartz or quartz and feldspar in a sericite matrix which commonly contains a minor amount of carbonate. The proportion of quartz to feldspar ranges widely, half of the grains in some clasts being feldspar. Potassium feldspar is more abundant than plagioclase. Sandstones with a carbonate martix show similar variations between individual grains in proportion of quartz to feldspar. The carbonate of the matrix is coarse grained and generally twinned. Recrystallization has destroyed many of its original features. ORIGIN OF LIMESTONE BLOCKS Isolated small masses of limestone, surrounded by other rocks, are widely distributed in the region studied. Some are lenses formed in place in the enclosing rocks and others are probably horses in faults. However, other blocks were probably emplaced into younger beds by sliding or slumping. Many of these lie in conglomerates of the Mariposa formation; some lie in slate of the Mariposa formation. Contacts of the blocks are rarely exposed but their shape and character is such as to indicate that they neither developed simultaneously with surrounding rocks nor were emplaced by plastic flow; available evidence suggests that they were emplaced by slumping during deposition of the surrounding beds. Modem subaerial mudflows containing boulders 40 to 50 feet long have been described by Blackwelder (1928, p. 471) and much larger blocks emplaced by slumping in other sediments have been described by King (1937, p. 66-68, 89-92), by Renz, Lakeman, and Van Der Meulen (1955), and others. Limestone masses are commonly grouped. Two groups of limestone blocks have been mapped by A. A. Stromquist (written communication, 1952) on the San Andreas northwest quadrangle, in the western belt of the Mariposa formation near the Calaveras River. One group (in strike with unit 23, plate 6) is about 14 mile northwest of the river in sec. 15, T. 4 N., R. 11 E. and as much as V2 mile southeast of the river in sec. 22, T. 4 N., R. 11 E. Another group (in strike with unit 28, figure 26) occurs in the SW1^ sec. 14, T. 4 N., R. 11 E. within )4 mile both northwest and southeast of the river. In unit 28, plate 6, contact relations of smaller limestone blocks, about 5 feet and less in diameter, are exposed in two places. One block is in the previously mentioned 14 railroad cut in sec. 10, T. 4 N., R. 11 E., where fusulinid limestone of Permian age is surrounded by slate and conglomerate of the Mariposa formation. A narrow zone of gouge surrounds the limestone and can be traced from the bottom to the top of the cut. The rocks on the two sides of the zone are similar, however, and the displacement along the fault marked by the gouge is probably minor. On the north bank of the Calaveras River (plate 6), near the west side of unit 28, is a very coarse sheared breccia consisting largely of porphyritic andesite like that of the Brower Creek volcanic member to the east but con-38 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT taining scattered blocks 6 feet and less in length of limestone and bedded chert. Another group of limestone blocks, within the Me-lones fault zone in the NW14 sec- 22, T. 3 N., R. 12 E., about 7 miles slightly east of south of San Andreas, has been mapped by D. B. Tatlock (written communication, 1952). These are surrounded by strongly sheared conglomerate in which the less resistant rock fragments are much elongated. The fragmental rock is probably not a tectonic breccia, as the fragments include a considerable variety of rocks including widely scattered granitic rocks. A group of limestone blocks exposed near the Stanislaus River in sec. 26, T. 2 N., R. 13 E. (pi. 5) was mapped by Eric and Stromquist (in Eric, Stromquist, and Swinney, 1955, pi. 1). A block composed of limestone on the west and carbonaceous quartz-mica schist on the east is distinguished as unit 46 on plate 5. The contact of the schist with conglomerate on the east is poorly exposed. On the west, the limestone is in fault contact with pyroclastic rocks of the Brower Creek volcanic member. The limestone and schist may be a tiny remnant of the basement on which the Mariposa formation was deposited, fortuitously exposed on the present land surface. Relations of other limestone masses in the area are not clear. The limestone blocks at the localities mentioned are associated with conglomerate. Of the group west of San Andreas a limestone block has been found to be in contact with conglomerate only at the railroad cut exposure, but lenticular conglomerates are in strike with the beds surrounding the limestone blocks in the two groups west of San Andreas. Conglomerate surrounds the limestone southwest of San Andreas and overlies the exposure in the Stanislaus River. The shape and size of the limestone blocks, together with their internal texture and structure and their relations to surrounding rocks, are not consistent with the hypotheses that they were formed in place or were emplaced by faulting. The limestone blocks range in shape from very elongate to nearly equidimensional. They apparently range in size from the smallest pebbles of the conglomerate to blocks more than 100 feet long. Bedding is not apparent in most of the blocks. Breccia structure is common, but whether it is of epiclastic or cataclastic origin wTas not determined. No conglomerate or other rock is interbedded with the limestone of the individual blocks, nor are there tongues of limestone extending into surrounding rocks. Although some limestone blocks that are apparently horses in faults are coarsely crystalline and have planar structures suggestive of great deformation and plastic flow, these features have not been found in the blocks at the localities described above. Several of the blocks contain fossils but the visible original textures indicate that the limestones are bioclastic deposits rather than reefs. SOURCE OF FRAGMENTS Some of the fragments in the Cosumnes and Mariposa formations and the Salt Spring slate were apparently derived from older formations exposed in the Sierra Nevada, and others from nearly contemporaneous beds in or near the Upper Jurassic basin of deposition of the western Sierra Nevada. Derivation from sources west and northwest of the Sierra Nevada has been neither established nor disproved. No evidence has been found by the writer to support Taliaferro’s (1942, p. 89) contention that sediments of Jurassic age become coarser westward. Furthermore, when considering deposits transported by turbidity currents and mudflows, the site of deposition of coarse material provides no reliable evidence of its source, as coarse detritus carried down submarine canyons may bypass finer deposits laid down on the continental shelf on either side of the canyon. Other than the region of the Sierra Nevada, conceivable sources include the sites of the present Klamath Mountains, California Coast Ranges, and Central Valley. The eastern part of the Klamath Mountains, about 200 miles northwest of the Cosumnes River, may have been exposed during Late Jurassic time. It is about on the projection of the strike of the Upper Jurassic rocks under discussion. The strike of the Upper Jurassic rocks may be parallel to the axis of the basin in which they were deposited but some debrisladen currents probably move parallel to the axis of a basin of deposition as well as across it. Common rocks exposed in the eastern Klamath Mountains include mafic, intermediate and felsic volcanic rocks, chert, shale, graywacke, and intrusive rocks ranging in composition from ultramafic to granitic. Chert and quartz-mica schist are not abundant. Taliaferro (1942, p. 103) believes that the site of the California Coast Ranges was a land mass during deposition of the Amador group and the Mariposa formation. In that region, sedimentary rocks shown to be older than the Mariposa formation are rarely exposed except in the Santa Lucia Range, about 100 miles southwest of the Merced River. These rocks, the Sur series, consist of “quartzose schists and gneisses, quartz-biotite schists, marbles, and plagioclase amphibolites” (Reiche, 1937, p. 118) and quartzite (Trask, 1926, p. 127). They may have been more widely exposed in the Coast Ranges during Late Jurassic time. On the other hand, Hill and Dibblee (1953, p. 449), suggest that displacement of pre-Cretaceous rocks by the San Andreas faultSTRATIGRAPHY 39 may total 350 miles since Jurassic time, so that these rocks may not then have been in their present position. Well-borings in the Central Valley provide some information on the basement rocks but are too widely spaced in most places to provide complete information. The well borings have penetrated slate, amphibolite, schist, granitic rocks, serpentine, and quartzite, suggesting that the basement rocks are similar to those exposed in the western Sierra Nevada (Taliaferro, 1943b, p. 129). Evaluation of the possible sources is hindered by factors other than the extensive cover of post-Mariposa strata. There is insufficient knowledge of the petrography and composition of both the transported fragments and possible source rocks in place and the possibility that some rocks exposed during Late Jurassic time may have since been either removed by erosion or emplacement of plutonic rocks, or metamorphosed beyond recognition. Large parts of the Klamath Mountains and Sierra Nevada have not been mapped in detail and possibly not all the distinctive lithologic varieties in these areas are known. Some geologists, for example (Taliaferro, 1943b, p. 187) have suggested that areas now buried beneath the sea, or younger deposits are more attractive sources of epiclastic rocks in northern California than exposed areas of older rocks, when these areas apparently do not include the lithologic types found in the clasts. Nevertheless, erosion has removed much material from the exposed areas and all the kinds of rocks exposed there during Late Jurassic time are not necessarily still represented. Moreover, the Paleozoic rocks now exposed in the Sierra Nevada, are almost certainly of higher metamorphic grade than the Paleozoic rocks exposed during Late Jurassic time, as the batholith was since then emplaced and eroded to considerable depth. The metamorphism would probably affect the texture of much of the chert. The volcanic rocks are characteristically lenticular, and distinctive varieties originally present in modest volume might have since disappeared, owing to erosion and emplacement of plutonic rocks, or their appearance may have changed owing to extensive recrystallization. Other rocks less notably lenticular might also have been removed or altered beyond easy recognition. Fragments of sandstones, and carbonaceous slate, siltstone, and chert were probably derived from formations exposed in the Sierra Nevada, and limestone clasts in the Cosumnes and Mariposa formations were derived from the part of the Calaveras formation that is represented by the western belt of exposures. Nearly all of the limestone in both the clasts and the masses believed to be in place in the western belt is coarse cal- carenite with abundant organic debris. In addition, Permian fossils have been identified in one limestone boulder in the Mariposa formation (p. 14). Clasts of black carbonaceous slate, phyllite, and chert strongly resemble rock types that are abundant in the chert unit of the eastern belt of the Calaveras formation and may well have been derived from this unit. The granules and pebbles of orthoquartzite, gray-wacke, and quartz-feldspar-carbonate sandstones may have been derived from the Blue Canyon formation, exposed in the Sierra Nevada in a wide belt extending beyond the north and south limits of the Colfax 30-minute quadrangle (Lindgren, 1900). All these rocks are in the Blue Canyon formation, including the unusual sandstone consisting of angular quartz grains in a seri-cite matrix. Of the rocks already discussed, exposures of all but the calcarenite of the Calaveras formation are east of the exposures of the Upper Jurassic formations. However, these rocks may also have been exposed west of this belt during Late Jurassic time. The abundance of calcarenite boulders and blocks, however, indicates that the source of the calcarenite was close to the site of deposition. Fragments of slate, noncarbonaceous chert and volcanic rocks may have been derived from penecontempo-raneous or older Mesozoic beds in the western Sierra Nevada. These rocks might also have been derived in part from the Calaveras and other formations of Paleozoic age in the western Sierra Nevada, the Central Valley area, or the Klamath Mountains. Many of the fragments of volcanic rocks might be the result of contemporaneous volcanism. The source of the granitic rocks, schist, and gneiss is unclear. Little can be added to Knopf’s (1929, p. 19) thesis that the granitic pebbles were derived from masses exposed within the area of this investigation. Other possible sources for these materials are the previously mentioned regions northwest and west of the Merced-Cosumnes River area. BEDDING FEATURES AND CONDITIONS OF DEPOSITION The Cosumnes and Mariposa formations and the Salt Spring slate have similar bedding features which indicate that all three were deposited in like environments. The three volcanic formations are likewise similar to one another in their primary structural and textural features. Besides, structures of some of the pyroclastic rocks are almost identical with those of the epiclastic graywackes and conglomerates and indicate that both kinds of rock were carried to the site of deposition in a similar manner. Fossils in the Logtown Ridge and Mariposa formations and Salt Spring slate indicate40 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT that the deposits arte marine; the Cosumnes formation, Copper Hill volcanics, and Gopher Ridge volcanics differ from them in no way that would suggest a different origin. Structures in the sands and finer sediments indicate deposition in quiet water, but structures in coarser sediments are not diagnostic. Wide distribution of graded beds in both epiclastic and pyroclastic rocks is indicated by the maps of individual traverses. Current-ripple bedding (small-scale cross-beds), in places associated with minor cut-and-fill structures, is widespread but not common. Load casts and crumples generally associated with contemporaneous deformation are also widespread but are only locally common. They are strikingly exposed in the lower part of the Log-town Ridge formation and the upper part of the Cosumnes formation in the Cosumnes River (pi. 8). Flutes and grooves have not been observed. Most of the graywacke forms beds less than 3 feet thick but tuff is generally thicker bedded and some tuff and lapilli tuff layers several tens of feet thick show no internal discontinuities in grain size. At least some volcanic breccias have no textural discontinuities through stratigraphic intervals of several hundred feet. Some beds of slate and very fine tuff are only a few inches or millimeters thick. Graded beds have been described elsewhere (Petti-john, 1943, p. 949; Kuenen and Migliorini, 1950; and Shrock, 1948, p. 78-84) and have been noted previously in the western Sierra Nevada by Heyl, Cox, and Eric (1948, p. 68), but they are fundamental to the structural and stratigraphic interpretations of the region and are therefore described here in some detail. Graded units are marked by progressive changes in color as well as grain size. In graywacke-slate sequences, the coarser material is of lighter color than the fine, but among the pyroclastic rocks either the coarser or the finer material can be lighter, depending on the mineral content. Some beds appear megascopically to grade uniformly from the coarsest to the finest material, but in others the grading is apparent in only part of a bed and is obscure or absent in other parts. In these, size-grading may be in the lower, middle, or upper part of the bed. There are no single isolated graded beds; instead, they form sequences commonly many tens of feet thick in which more than half of the beds are graded. Graded graywacke2 layers are common both in dominant graywacke and dominant slate. In both, bottoms of the coarse layers precisely mark one boundary of a 2 The terminology used in this paragraph to describe phenomena related to size sorting is that of the epiclastic rocks. Although these features are equally common in the pyroclastic rocks the nomenclature for comparable size grades is different and verbiage can be reduced by describing a single genetic type. bed. The upper boundary of some coarse layers is well marked by the base of the overlying coarse layer or by an abrupt change to finer grain size. In other beds the upper boundary of the coarse layer is ill-defined because changes in grain size from sand to clay take place over appreciable parts of the beds. The sediments constituting graded beds range from fine conglomerate to shale, but the complete range between these size grades seldom occurs in a single layer or unit. Pebble conglomerate forms the lower parts of rare graded layers. Most of the well-graded beds are ^4 inch to 6 inches thick, but some wTell-graded beds are as thin as 2 mm or as thick as 30 inches. The bottoms of most graded beds are smooth, but some are irregular from minor channeling of the underlying finer material, formation of load casts, and from post-depositional crumpling. Compositional sorting is lacking in graded graywacke layers but is pronounced in some graded tuff layers. Thin sections indicate that such beds range from crystal tuff to lithic or vitric tuff. Rock fragments are microcrystalline, in part at least, because of metamorphism, and it is difficult at best to distinguish originally glassy from originally lithic material. Crystal fragments are much more abundant in the coarser lower parts of these beds and lithic fragments are equally or more abundant in the finer upper parts. The original composition of the very fine tuff is uncertain, as it consists of microcrystalline mixtures of quartz, zoisite(?), white mica, and chlorite, with scattered plagioclase and pyroxene crystal fragments. In thin sections studied, proportion between rock and crystal fragments does not always vary directly with the grain size. In some, contiguous layers of similar grain size have different proportions of rock and crystal fragments, and in others an upward increase in proportion of fairly coarse rock fragments reverses the size grading in small thicknesses of the bed. Current-ripple bedding is generally associated with small cut-and-fill structures and is common in sequences having many graded beds. Nearly all cross-laminated units are less than 2 inches thick and the thickness of cross laminae is measurable in millimeters. Grain sizes in most cross-laminated layers range from very fine sand to clay. Features suggestive of deposition in agitated water are rare or ambiguous. Distorted ripple marks are exposed on many surfaces of thin-bedded tuff near the gaging station about 1 mile west of Don Pedro Dam on the Tuolumne River (fig. 16) but have not been found elsewhere. Graded beds are absent or rare near the ripple marks. No diagnostic bedding features were noted in the quartz-rich sandstone included with the Mariposa formation near the Calaveras River (unit 26,PLUTONIC ROCKS 41 pi. 6), but the scarcity of clay and silt in this rock suggests deposition in agitated waters. In contrast with other sandstones in the Jurassic formations, there is little fine detrital material in the matrix of this rock, and labile constituents are scarce. If these isolated beds do result from a shallow water environment, they were probably formed in places where the depositional basins temporarily filled to wave base. Structures and textures of the graded sandstones and tuffs and the clean, bedded conglomerates suggest that they are turbidity current deposits. The mud-matrix conglomerates were probably emplaced by submarine slides. Similar deposits of Pliocene age in the Ventura Basin have been ascribed a similar origin (Natland and Kuenen, 1951, p. 102-105). Lawson (1933, p. 10) held that unsorted conglomerate in rocks near Colfax equivalent to the Mariposa formation is tillite, and reported a glaciated surface beneath them. The alleged glaciated surface may, however, be a fault surface, as faults are common in the region. The thickness of the prism of Jurassic deposits suggests that the basement surface upon which they were laid formed a basin, as the thickness of the stratigraphic section is as great or greater than the depth of the modem ocean except in the deeps. The aggregate thickness of the Gopher Ridge volcanics exposed in the Mokelumne River and the Copper Hill volcanics exposed in the Cosumnes River is about 19,000 feet. The total apparent thickness of the Penon Blanco volcanics and Mariposa formation in the Merced River is nearly 20,000 feet. The bedding of some of the deposits, especially of volcanic rocks, may have been inclined, so these values may be greater than the depth of the floor below sea level. Nevertheless, the figures are minimum values because the complete stratigraphic section is nowhere exposed and some allowance must be made for the water column above the uppermost deposits, which must have been laid down at considerable depth. The deposits were probably bathyal or abyssal, but the absolute depth probably cannot be ascertained in the absence of diagnostic fossils (Kuenen, 1950, p. 203). The wide distribution of graded beds, and the general absence of sorted elastics of other than clay size indicate not only transportation by turbidity currents and mudflow, but also that the bottom water was not sufficiently agitated by currents or wave action to destroy the characteristic structures. The depth of the basement beneath the mass of deposits is only indirectly related to the depth of water over the accumulating deposits, for the two depend on the rate and time of downwarp and the rate and time of accumulation of the deposits. At one extreme, down warping might have preceded significant accumulation of detrital material, and at the other subsidence might have coincided exactly with filling. If the first, each succeeding bed would be laid down in shallower water whereas in the second all would be laid down at equal depth. The history of most filled basins is probably between these two extremes. Moreover, lava and coarse pyroclastic rocks characteristically are built up as mounds or ridges on the sea floor so that volcanic material in a given basin might be deposited at shallower depths than contemporaneous epiclastic sediments. GREEN SCHIST Green schist, derived from volcanic rocks, has been mapped within the Melones fault zone near San Andreas and in the Bear Mountains fault zone at the north end of the area studied. Because of their occurrence in fault slices these rocks are of unknown age. The mass near San Andreas probably includes rocks of both Paleozoic and Jurassic ages because in this vicinity volcanic rocks of both ages lie adjacent to the Melones fault zone. Most of the rock included with this unit near San Andreas is markedly schistose. In most places, the primary textures and structures are obliterated, but volcanic breccia texture is preserved locally. The mass at the north end of the map area as exposed in the Cosumnes River is locally schistose, but consists largely of massive fine- to medium-grained amphibolite. PLUTONIC ROCKS Plutonic bodies exposed in the region consist of ultramafic rocks and rocks ranging in composition from gabbro to granite. The ultramafic rocks form linear belts and are most abundant west of the central part of the area. The less mafic plutonic rocks form the Sierra Nevada batholith in the eastern part of the area and small isolated plutons elsewhere. Most of the plutonic rocks east of the Melones fault zone are granodioritic to granitic whereas most of those west of the fault zone, except for the ultramafic rocks, are gab-broic to quartz dioritic, although all types occur in varying amounts on both sides of the fault zone. The composition, origin, and mode of emplacement of the plutonic rocks were not studied in detail for this project. General descriptions of the plutonic rocks are given by Turner (1894a, 1897, 1899) Turner and Ran-some (1897) and Knopf (1929, p. 18-21). Calkins (1930) discusses the petrology and sequence of emplacement of some plutons of the western part of the Sierra Nevada batholith near the Merced River. Several of the ultramafic bodies have been mapped and described by Cater (1948a, 1948b).42 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT ULTRAMAITC ROCKS Ultramafic rocks include serpentine, talc-antigorite-ankerite schist, and less commonly, peridotite, dunite, and saxonite. Serpentine is much more abundant than the other rocks in most places, but talc-antigorite-ankerite schist is more abundant east of the Melones fault zone near the latitude of San Andreas. Peridotite, dunite, and saxonite occur in some of the serpentine masses. The ultramafic rocks in the region form elongate masses from less than a foot wide and several feet long to nearly four miles wide and more than 15 miles long. Most of the serpentine is blocky, with each surface strongly slickensided, but part is strongly foliated. AGE The ultramafic rocks of the area are probably of Late Jurassic age. The faults which apparently controlled the emplacement of the serpentine cut nearly all the Jurassic and Paleozoic stratified rocks of the region. The youngest of these, the Copper Hill volcanics, over-lie beds of Kimmeridgian age. Serpentine is cut in turn by felsic plutonic rocks north of the area of this report. A felsic pluton northeast of Folsom (see Lindgren, 1894) which cuts serpentine has been dated by Curtis, Evernden, and Lipson (1958, p. 6) by the potassium-argon method as 131 million years and quartz diorite from the same pluton as 142.9 million years old. Serpentine is also cut by another, undated, pluton near Placerville (see Lindgren and Turner, 1894). STBUCTUEAL RELATIONS In plan, ultramafic bodies are elongate parallel to the regional strike of bedding and fault zones. Where adjacent to schistose rocks, the contact of the ultramafic rocks is parallel to the schistosity. Surface exposures suggest that the bodies dip steeply, but whether more or less steeply than adjacent bedded rocks is unknown. The extent and magnitude of the fault zones has not been recognized previously and the orientation of the ultramafic bodies has been cited as evidence that they were intruded as sills (Taliaferro, 1943b, fig. 2). Ferguson and Gannett (1932, p. 21-22), on the other hand, recognized as a major fault a large part of the structure here referred to as the Melones fault zone and suggested that it determined the position of serpentine emplacement. Distribution of the ultramafic rocks supports this, as many of the smaller masses are confined to belts of cataclastically deformed rocks and even the larger masses in the area that intrude layered rocks are adjacent to fault zones. The strongly sheared condition of the serpentine, however, indicates that it was emplaced before fault movement ceased. Ultramafic bodies that extend beyond the margins of known fault zones are the two large masses east of Latrobe, the mass exposed in the eastern part of Pardee Reservoir, the mass about 3 miles west of Fowler Lookout, the large mass crossed by the Stanislaus and Tuolumne Rivers, and the mass near Bagby. These have irregular contacts which locally cross the structure of layered rocks. Although these masses are largely serpentine like the rest, they differ from most of the smaller masses in containing some preserved peridotite, dunite, and saxonite, (Cater, Rynearson, and Dow, 1951, p. 120-125; and Cater, 1948a, 1948b). In contrast to these large masses the smaller ones are characterized by straight contacts and are confined to zones of sheared rocks. On the regional map (pi. 1) several of these are indicated west of the Bear Mountains, but many more are shown on individual river sections. Small sheets of serpentine, some less than a foot thick, are widely distributed in the shear zones. The serpentine bodies with straight contacts are probably horses or masses that were injected plastically into their present positions. The close areal relation of ultramafic rocks and the shear zones shown on the regional maps (pis. 1 and 2) suggests that the serpentine and fault zones are somehow related. By far the greater part of the ultramafic rocks of the region are within the area encompassed by the Melones and Bear Mountains fault zones and many of even the sill-like bodies are bounded on at least one side by faults. Ultramafic rocks are scarce east of the Melones fault zone and are absent west of the Bear Mountains fault zone. It is more likely that the major faults controlled the position of the ultramafic rocks than that the ultramafic rocks controlled the position of the faulting (Clark, 1960a, p. 488). It is true that the ultramafic rocks are strongly sheared, probably owing to later movements on the fault zones, but they have not localized the shearing, as the belts of sheared rocks marking major fault zones are more persistent than the ultramafic rocks. The large mass of serpentine exposed on the Calavaras River is thus flanked on the west by a wide belt of schistose rocks, and the serpentine southeast of Bagby is flanked on the west by a belt of sheared slate about 200 feet wide exposed in cuts of State Highway 49 about iy2 miles southeast of Bagby. The Melones fault zone is believed to extend the length of the metamorphic belt, but in much of the segment covered here it contains little serpentine. GABBROIC TO GRANITIC ROCKS Hornblende gabbro and diorite are abundant in the isolated plutons of the western Sierra Nevada. They form nearly all the mass exposed in the northeast partHYPABYSSAL ROCKS 43 of Don Pedro Reservoir, the part of the mass southeast of Jackson that is cut by the Mokelumne River, the mass 5 miles southwest of Sonora that is associated with serpentine, and many smaller masses. These rocks also form the marginal parts of some less mafic plutons such as the one east of San Andreas (Clark, 1954, p. 11), and the whole of many smaller bodies (for example, Eric, Stromquist, and Swinney, 1955, pi 1). The hornblende gabbro and diorite range in texture from medium to fine grained. Masses of gabbro and diorite are generally adjacent to or are surrounded by meta-volcanic rocks, and in places grade into them texturally, suggesting that they may have been formed in part by metamorphism of the volcanics. Eric, Stromquist, and Swinney (1955, p. 21) suggest that they are at least partly differentiates from the same magma as that of the serpentines, because of gradational contacts. Dur-rell (1940, p. 74) found evidence of similar origin of the gabbro in the western Sierra Nevada about 100 miles south of Sonora. Compton (1955, p. 18-20) on the other hand found that some of the gabbroic rocks in the western Sierra Nevada, north of the area covered, are products of replacement or recrystallization in place and that others are intrusive. Granitic rocks form most of the larger isolated plutons of the area, the Sierra Nevada batholith, and the cores of some smaller masses that are mostly mafic. The granitic plutons, except near their borders, are much more uniform in texture and composition than the mafic bodies. Contacts of the isolated plutons with metasedimentary rocks are sharp, and in many places the metasediments at the contact are but slightly coarser grained than elsewhere. Contact metamorphism is more pronounced at the border of the batholith. Contacts of isolated plutons with metavolcanic rocks are less sharp and the grain size of the meta volcanics increases markedly toward the contact. Granitic dikes commonly cut the wall rocks near the plutons. AGE Although most geologists previously considered all the granitic rocks of the area covered here to be of about the same age, Knopf (1929, p. 18-19) suggested two sets of intrusions, one of late Carboniferous age and another of post-Mariposa age. Curtis, Evemden, and Lipson (1958, p. 6-9) more recently showed that two distinct sequences of emplacement of granite rocks can be distinguished by potassium-argon dating. Two bodies of the older series near the map area have been dated. Granodiorite from Rocklin, dated as 131 million years old, and quartz diorite from Horseshoe Bar, dated as 142.9 million years old, form parts of a pluton north of the area that extends to within 4 miles of the northern map boundary (for location, see Curtis, Evemden, and Lipson, 1958, fig. 1). Quartz monzonite from the Guadaloupe intrusive about 5 miles south of the southern boundary of plate 1 and immediately east of the projection of the Bear Mountains fault zone was originally dated as 142.9 million years old but later recalculated as 136 million years (Best, 1963, p. 113). A granitic body west of Columbia has been correlated by Baird (1962, p. 18) with the older series on the basis of structural evidence. The younger series composes the Sierra Nevada batholith in the Yosemite region east of El Portal and ranges in age from about 77 to about 95 million years. Evidence of Carboniferous age of some granitic intrusions is indirect and inconclusive, but has not been disproved. Curtis, Evemden, and Lipson (1958, p. 10-11) pointed out that according to the Holmes A time scale (in Zeuner, 1952) the older series is of Early Cretaceous age and according to the Holmes B time scale it is of Early Jurassic age. They cite stratigraphic evidence to show that the older series is of Late Jurassic age and that the Holmes scales need revision. The intrusions assigned by Knopf (1929, p. 18-19) to the Carboniferous are those east of Plymouth, at Mokelumne Hill, and southeast of San Andreas. They intrude the Calaveras formation and the plutons east of Plymouth and at Mokelumne Hill are in fault contact with rocks of Late Jurassic age. Knopf’s age assignment was based on (a) the occurrence of granitic pebbles in the Mariposa formation, (b) cataclastic deformation of rocks in the plutons, (c) higher meta-morphic grade of the Calaveras formation as compared with the Mariposa formation, and (d) exceptionally strong pleochroic haloes in the micas of these plutons. This evidence is suggestive rather than conclusive, and the present author believes that all intrusives in the area are of Late Jurassic to middle Cretaceous age, but the occurrence of Carboniferous intmsives has not been disproved. The plutons east of Plymouth and at Mokelumne Hill are truncated by the Melones fault zone and are therefore apparently older than the dated pluton that truncates faults of the Foothills system north of the mapped area. On the other hand, emplacement of granitic rocks certainly, and fault movements probably, occupied finite parts of geologic time, and it is unlikely that granites of a single series of intrusions would everywhere show the same relations to the faults. HYPABYSSAL ROCKS Massive rocks of possible hypabyssal origin, because of their uncertain relations, are grouped with the formations with which they are associated and are not discussed here. Most of the rocks known to be of44 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT hypabyssal origin form dikes. Swarms of fine-grained leucocratic dikes cut the metamorphic rocks west of the gaging station downstream from Don Pedro Dam on the Tuolumne River (fig. 16) and along the Merced River between Bagby and the mouth of the North Fork of the Merced River (pi. 3). A porphyritic leucocratic rock with fine-grained groundmass, probably forming a dike, is poorly exposed in the south bank of the Calaveras River at the western side of bedrock exposures (pi. 6). Knopf (1929, p. 21-22) has described several albitite dikes from other localities in the region. Some mafic sills consisting of saussuritized plagio-clase phenocrysts in a fine-grained groundmass intrude the lower part of the Amador group near the Cosumnes River (see pi. 10). A similar rock is intruded at the base of the upper tongue of the Brower Creek volcanic member of the Mariposa formation south of the Calaveras River. The dike rock is not exposed along the river, and is represented only by boulders. A vertical dike of porphyritic rhyolite or dacite intrudes unit 2 of the Calaveras River section (pi. 6). The unit shows horizontal columnar jointing. Dikes and sills intruding the metamorphic rocks cannot be precisely dated. The upper age limit is defined by the land surface which truncates the dikes. The surface apparently developed largely during the Late Cretaceous. The oldest Mesozoic rocks cut by dikes are probably not older than Middle Jurassic and the youngest are of Late Jurassic 'age. The sills in the lower part of the Amador group post-date the consolidation of enclosing beds, for dike walls are straight for considerable distances, bedding is not more distorted at dike contacts than elsewhere, and some dike walls show sharply angular offsets. Some dike rocks in the eastern block may be of Paleozoic age. The vertical rhyolite or dacite dike exposed near the Calaveras River is tentatively interpreted as of Tertiary age on the basis of its fresh appearance, jointing and attitude. STRUCTURAL GEOLOGY Metamorphic rocks of the western Sierra Nevada are on the west limb of a faulted synclinorium, the axial part of which is occupied by the Sierra Nevada batho-lith (Bateman and others, 1963). The strike of bedding and major fault zones, about parallel to the trend of the axis of the synclinorium, is generally northwest except in the northern part of the map area where bedding and the fault zones turn northward. Beds in most places dip eastward at angles of more than 60° or are vertical. Isoclinal folds occur in some places and more open folds elsewhere, but there are wide areas with homoclinal dip. The major folds and homoclinal sequences are truncated by the Bear Mountains and Me- lones fault zones, which are elements of the Foothills fault system (Clark, 1960a). In most homoclinal sequences, tops of beds are eastward, and in large parts of each of the three blocks separated by the Bear Mountains and Melones fault zones older beds are on the west and the younger beds on the east. This distribution is reversed by the major fault zones, as the youngest bedrock unit of the region, the Copper Hill volcanics, is in the western block and the oldest, the Calaveras formation, is in the eastern block. Metamorphic rocks exposed in the central block are of intermediate age. Axes of major and minor folds related to development of the snyclinorium are essentially horizontal; they plunge northwest and southeast at angles of 35° and less. The folding was followed by a second stage of deformation that involved the major faulting and development of slip cleavage and steeply plunging minor folds and 5-lineations, particularly east of the Melones fault zone (Clark, 1960b). Most of the minor structures in the region are identifiable with these two stages, but minor structures in some small parts of the map area have different attitudes than those so identified. Baird (1962, p. 35-44), for example, has described a prominent slip cleavage and lineation direction in limestone northwest of Columbia. He found these structures to be younger than the first regional deformation but older than the second. The structure of the western and central blocks is similar, but the eastern block is deformed more severely and differently than the others. These differences have led some geologists to infer that the Calaveras formation was strongly folded before deposition of the Cosumnes formation, but this is not assured. The greater age of the metamorphic rocks in the eastern block relative to those farther west indicates that the eastern block has been relatively uplifted by faulting, hence that the exposed metamorphic rocks of the eastern block were deformed under greater pressure at lower levels of the earth’s crust than metamorphic rocks exposed farther west. Most of the deformation of the region is older than the Sierra Nevada batholith, and is part of a zone of deformation that coincides with a belt of eugeosynclinal deposits and granitic rocks of Jurassic age that extends with some interruptions from the western Sierra Nevada through the western part of the Klamath Mountains into southwestern Oregon. The Jurassic rocks of the western Sierra Nevada were folded and the folds were truncated by major steep northwest-striking faults before emplacement of the Upper Jurassic plutons. Much of the deformation of the eastern block was also earlier, for regional structures in the block are truncated by the Melones fault zone, and slip cleavage relatedSTRUCTURAL GEOLOGY to the faulting locally offsets bedding and previously formed schistosity in the eastern block. Little evidence for large-scale deformation accompanying emplacement of the Sierra Nevada batholith has been found. Faults, probably thrusts, in and near the Melones fault zone are later than the strike-slip deformation and may be related to emplacement of the batholith. Perhaps spacial adjustments of considerable magnitude were accommodated by previously formed structures without leaving a record that is readily distinguishable. Structures available for such adjustments include folds, wide shear zones in the Melones and Bear Mountains fault zones, and an infinitude of slip surfaces in schists of the Calaveras formation. TOP DETERMINATIONS Structural interpretations presented here, as well as those of stratigraphic sequence, depend greatly on determinations of tops of beds, based on graded bedding, crossbedding, pillow lavas, minor folds, and relations between cleavage and bedding (Shrock, 1948). More determinations have been based upon graded bedding than on all other features. As tuff is more abundant than graywacke, most of the determinations are on tuff beds (fig. 13). No plotted determination has been based on less than three graded beds, and some represent scores of them. Exposures showing obscure, or spurious, grading and apparently conflicting top directions within stratigraphic intervals of a few feet have been ignored. Curved cleavage planes, convex strati-graphically upward, distinguish some steeply dipping graded beds showing no megascopically apparent size gradation, as well as many beds with such gradation (fig. 14). In these beds, the angle between cleavage and bedding is a function of grain size so that cleavage planes make a large acute angle with bedding in the coarsest part of the bed, and approach parallelism to the bedding as the material becomes finer. This phenomenon is more common in graywacke and slate than in tuff. Throughout the region relation of cleavage to bedding indicates that cleavage is parallel or nearly parallel to the axial planes of the folds (fig. 15). In much of the area of Jurassic rocks, this relation should provide reliable top determinations. However, locally developed cleavage which is at least superficially similar to the pervasive cleavage is probably related to a stage, or several stages, of deformation later than that which formed the regional folds and may have a random relation to axes of these folds. Because of these uncertainties only two top determinations based on cleavage were recorded. Figure 13.—Photomicrograph of graded tuff bed in Copper Hill volcanics, NE%NE% sec. 28, T. 2 N., R. 12 E. Light-colored grains are pyroxene and sodic plagioclase crystal fragments. Dark grains are chlorite after lithic or glassy fragments. Dark layer in upper part of photograph—originally dust-tuff— is also chlorite. Most of the coarse layer is about of the same grain size, but this layer grades into the fine material through an interval that is appreciable relative to the thickness of the bed. Note sharp contact of succeeding coarse layer at extreme top of photograph. Gray band in uppermost layer is an epidote veinlet. Plain light. Minor folds were used to interpret structures and stratigraphic sequence in areas on the Tuolomne River near Don Pedro Dam and La Grange. Those in the Gopher Ridge volcanics near Don Pedro Dam (fig. 16) are consistent for a mile west of the slate contact near the dam and suggest that this area is on the west limb of an anticline. The folds near La Grange (fig. 10) are much less extensive. At both places the minor folds plunge as gently as the major folds of the region and46 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT Figure 14.—Photomicrograph showing relation of cleavage to graded bed in fine graywacke of Salt Spring slate, NEtiNEti sec. 14, T. 4 S., R. 15 E. Cleavage direction, influenced by grain size, is about 60° from bedding in coarse part but changes progressively to about 25° from bedding in finer part. Such curved cleavage, convex stratigraphically upward, commonly indicates the existence of graded bedding where the gradation is not otherwise recognizable in the field. Most of the clastic grains are quartz and the matrix is sericite. Original texture is well preserved although cleavage is well developed. Plain light. are probably related to the deformation that produced them. FAULTS Most major faults are components of the Melones and Bear Mountains fault zones which in turn are part of the Foothills fault system (Clark, 1960a). The Melones and Bear Mountains fault zones are the dominant structural features of the region. Faults that control the quartz veins and gold ore deposits of the Mother Lode belt are younger than the Foothills faults and are much less important as structural features. Faults with only a few feet or a few tens of feet of displacement do not notably affect rock distribution and few have been mapped. The Foothills fault system extends more than 100 miles northwest and an unknown distance southeast of the area shown on plate 1. The width of the fault system within the latitudes considered here is unknown. Additional fault zones parallel to the exposed Melones and Bear Mountains fault zones may underlie the younger deposits of the Central Valley, and others may have once been present in the area now occupied by the Sierra Nevada batholith. The Melones and Bear Mountains fault zones have components of reverse movement, but the orientation of their net slip and that of the Foothills fault system as a whole is not established. Both faults and shear zones shown on the accompanying maps express loci of deformation that disrupted structural or stratigraphic continuity, or produced cataclastic deformation, crumpling, or recrystallization. Most zones of dislocation of less than mappable width are indicated as faults whereas dislocation belts that are of mappable width are indicated as shear zones; some structures shown as shear zones on the detailed maps are indicated as faults on the regional map and diagram. Other faults are drawn arbitrarily at contacts of sheared ultramafic rocks where they occupy the position of major faults and at stratigraphic discontinuities where the shearing extends for many hun-STRUCTURAL GEOLOGY 47 Figure 15.—Photograph of small assymetric anticline showing axial plane cleavage. Fold is in bedded tuff of the Gopher Ridge volcanics on the north side of the Cosumnes River in the west part of sec. 31, T. 8 N., R. 9 E. Cleavage is most pronounced in axial region and gently dipping east limb of the fold. Fold axis plunges north about 15° (away from observer). dreds of feet in both directions, as east of Angels Camp (pi. 1; Eric, Stromquist, and Swinney, 1955, pi. 1). Shear zones were mapped on the basis of thin platy structure; of schistose belts, some markedly more crystalline than rocks on the two sides; of zones of crumpling; and of cataclastic deformation; or of a combination of these. Rocks in the shear zones—except the metavolcanic and granitic rocks—are generally too much fractured for collecting hand specimens; some phyllitic rocks in shear zones separate into paper-thin layers. Schistosity in the shear zones results from parallel arrangement of tabular minerals and rock fragments, flat mineral pods, and closely-spaced slip surfaces. Plates have feather edges due to acute intersection of slip surfaces, and are distinct from cleavage plates of slate. Although similar schistosity is widely developed in rocks of the Calaveras formation east of the Melones fault zone, it is not there restricted to such well-defined belts. In some phyllite and fine-grained nonschistose serpentine, platy structure is caused by the slip surfaces alone. Original textures and structures are generally obscure or absent in the schists in the fault zones. The schists and phyllites are commonly lineated. Amplitudes of crumples and minor folds in the shear zones are generally measurable in millimeters or inches, and limbs of many are truncated by slip surfaces. In some places rock fragments are rotated and mineral grains broken. MELONES FAULT ZONE The Melones fault zone trends about N. 30° W. through most of the area studied, but near Plymouth the strike changes to northerly. From south of the area studied to about the Mokelumne River, it coincides with the area known as the Mother Lode belt. Within the area covered by the regional map (pi. 1) the Melones fault zone has been mapped in detail for a total distance of about 30 miles, through the Sonora, Angels Camp, and San Andreas NW 71^-minute quadrangles (Eric, Stromquist, and Swinney, 1955, pis. 1, 2; A. A. Stromquist, written communication, 1952). The Melones fault zone is cut about 3y2 miles south of the map area by a granitic pluton that crystallized after the last movement on the fault zone (Cloos, 1932a, p. 303), but may cut metamorphic rocks exposed south of this pluton. Existence of the Melones fault zone is indicated by both a break in stratigraphic and structural continuity, and by mechanical deformation and recrystallization. Through an 80-mile segment of the report area, the fault zone separates plutonic rocks and rocks of Paleozoic age on the east from rocks of Jurassic age on the west. Southeast of San Andreas, the regional trend of the volcanic rocks and limestone lenses in the Calaveras formation is truncated at an acute angle by the fault zone, and successively younger strata of the Calaveras formation are cut by the fault zone to the northwest. The plutonic bodies north of Mokelumne Hill are terminated on the west by the fault zone. The Mariposa formation bounds the western side of the fault zone except in the north part of the report area. The most striking truncation of structure of the eastern block is southeast of San Andreas, where an easterly trending anticline having a core formed by volcanic rocks of the Calaveras formation is terminated abruptly on the west by the fault zone. The Melones fault zone is more nearly parallel to structures of the rocks to the west, but it cuts at a small angle a syncline formed in the Mariposa formation. The axis of this syncline is about two miles west of the fault zone near the Mokelumne River, and intersects the fault east of00 EXPLANATION Gopher Ridge volcanics Mostly tuff and fine volcanic breccia. Coarse volcanic breccia distinguished by symbol br and by pattern in vertical projection d// Dike rock of Cretaceous or Jurassic age (J U) if) ■< tr D Quartz vein of Cretaceous or Jurassic age Contact Dashed where projected Fault Dashed where projected ----------!------ Anticline, showing crest line Dashed where approximately located Syncline, showing position of trough Dashed where approximately located Base map from aerial photograph and U. S. Geological Survey Merced Falls quadrangle, California Geology mapped by L. D Clark, and Richard Pack, 1953 canyon wall r 1000' - 500' L SEA LEVEL Strike and dip of beds Strike and dip of cleavage Strike of vertical cleavage Horizontal lineation Intersection of bedding and cleavage Trace of bedding Trace of beds on steep north canyon wall mapped by intersection from points of plane table traverse along abandoned ditch on south canyon wall Figure 16.—Geologic map and vertical projection showing structure of bedded pyroclastic rocks of the Gopher Ridge volcanics of Late Jurassic age west of Don Pedro Dam, Tuolumne River, Mariposa County, California. STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELTSTRUCTURAL GEOLOGY 49 Chinese Camp. Farther southeast, the trough and east limb of the syncline are missing. The west side of the Melones fault zone is marked by an abrupt transition from moderately to strongly cleaved rocks that coincides with a break in the stratigraphic sequence, whereas on the east side of the zone, slip cleavage or schistosity parallel to the fault zone commonly persists thousands of feet northeastward beyond the stratigraphic discontinuity. The greatest width of the fault zone in the map area, about 4 miles, was measured between the major breaks in stratigraphic and structural continuity northwest of Alta-ville, that is, between the northeast side of the Mariposa formation and the contact between the green schist of the fault zone and Calaveras formation. Across most of this interval, the rocks are greatly sheared. In addition, slip cleavage parallel to the schistosity in the fault zone can be recognized as much as 6 miles east of the fault zone in the southern part of the Calaveritas quadrangle (Clark, 1954, pi. 1). The slip cleavage is most pronounced near the fault zone, and less prominent eastward. In the northeast part of the Calaveritas quadrangle and other places where the schistosity of the Calaveras formation is parallel to that of the Melones fault zone, movements related to the faulting cannot be distinguished. The fault zone is much narrower near the Merced River. At Bagby, near the river, exposures are inadequate to determine the width, but the shear zone is certainly less than 200 feet wide. At the Pine Tree mine, about 1% miles south of the river, the shear zone west of the serpentine is about 100 feet wide. The serpentine, more than 600 feet wide near the Pine Tree mine, may have displaced sheared rocks and has no doubt itself absorbed some of the movement. Dip of the Melones fault zone is indicated by internal planar structures which in most places are vertical or dip to the east at an angle of 70° or more. Lineation within the fault zone is in the planes of schistosity and marked by hornblende crystals in sheared granitic rocks, widespread elongate fragments in pyroclastic rock and conglomerate, and axes of small folds. With rare exceptions it bears eastward and is nearly normal to the strike of the schistosity; the plunge is 60° to vertical, and in most places about 80°. The Melones fault zone is not uniformly sheared. It includes large blocks in which original structures are well preserved and the cleavage is no more strongly developed than in the Jurassic rocks west of the fault zone. In some places most of the movement was near the western side of the zone, in some places near the eastern side, but in others the locus of greatest displacement cannot be established. DESCRIPTION OF LOCALITIES The Melones fault zone is crossed by the Cosumnes River east of Huse Bridge (pi. 8), where there are three zones of shearing, separated by non-sheared rocks. The west sides of these zones are about 1,400 feet, 2,700 feet, and 3,600 feet east of the mouth of the North Fork of the Cosumnes River. The westernmost shear zone, about 200 feet wide, contains carbonaceous sericite schist quite unlike the rocks on either side, which retain their original textures. The central shear zone is marked by green schist that is probably formed by deformation and recrystallization of part of the volcanic breccia immediately west of the shear zone. A narrow block of slate with interbedded tuff having well-preserved original textures and structures (too small to map separately) separates the chlorite schist from metamorphosed granitic rock of the eastern shear zone. The granitic rock, exposed for about 2,500 feet on the river, is schistose throughout and contains a pronounced lineation resulting from parallel arrangement of amphibole crystals. In thin section, the feldspars are seen to be completely saussuritized and although textures suggest that some of the altered feldspar crystals have been broken, this has not been established. Quartz forms microcrystalline lenticles with grano-blastic texture; undulatory extinction is weak or absent. Epidote also forms lenticles. Cataclastic deformation is shown by broken amphibole crystals. The schistose granitic rock was included by Lindgren and Turner (1894) and by Turner (1894a) with a much larger mass lying east of the Melones fault zone. Exposures examined during reconnaissance on roads east of the fault zone suggests that the granitic rocks become progressively less sheared to the east—the granitic rocks north of Fiddletown are apparently undeformed. The granitic rock abutting the Melones fault zone on the Mokelumne River also is schistose on the western side, but the feldspars are megascopically little altered, and no lineation is apparent. Epidote is abundant and suggests metamorphism of the granitic rocks. The fault zone extends more than 3,000 feet westward from the granitic rock. Immediately west of the granitic rock is a narrow, poorly exposed belt of chlorite schist, but west of this is a belt several hundred feet wide underlain by black Jurassic quartzose slate with interbedded tuff. Another chlorite schist belt extends westward from the interbedded slate and tuff nearly to Middle Bar Bridge, but two narrow belts of black carbonaceous quartzose mica schist, very similar to parts of the Calaveras formation, occur within the chlorite schist 1,200 feet and 2,000 feet, respectively, northeast of the bridge.50 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT On the Calaveras River, the fault zone is marked by a belt of green schist and serpentine with a total width of about 4,500 feet. Schistosity in the green schist dips eastward about 70°. Within the Calaveras formation east of the fault zone as delineated on the accompanying maps, strong schistosity parallel to the Melones fault zone extends an unknown distance eastward. This schistosity may be at least in part related to the fault. On the Stanislaus River (pi. 5) the fault zone is divisible into two parts separated by a body of Jurassic rocks that are no more strongly deformed than those outside the fault zone. The westernmost fault of the Melones zone truncates the easternmost belt of the Brower Creek volcanic member of the Mariposa formation. The fault is marked in the river 'by a narrow shear zone immediately east of the Brower Creek member, and a narrow block of Calaveras formation lies against the west side of the fault. The top of the Brower Creek member west of the fault is westward, but tops in the Jurassic strata to the east are eastward, indicating that displacement along this fault is at least great enough to eliminate the crest of an anticline. Rocks of the Mariposa formation are not notably deformed and graded beds are well preserved. The main part of the Melones fault zone is between the highway bridge at Melones and the mouth of Coyote Creek because west of the bridge are characteristic Mesozoic rocks and east of the creek mouth are characteristic Calaveras rocks. Exposures in the banks of the river are poor in this interval. Exposures of the Melones fault zone in the Tuolumne River add little to the information available elsewhere but excellent exposures of the strongly sheared rock characteristic of the fault zone are readily accessible near the State Highway 49 bridge. Only the part of the fault zone composed of serpentine is exposed on the Merced River. BEAR MOUNTAINS FAULT ZONE The Bear Mountains fault zone is parallel to the Melones fault zone, and like the Melones fault zone, extends beyond the limits of the area shown on plate 1. Schistose rocks along the projection of the Bear Mountains fault zone have been described by R. J. P. Lyon3 from near Miles Creek, about 8 miles south of the southern boundary of plate 1, suggesting that the fault zone extends at least this far south. Near the Cosumnes River, the fault zone divides into two parts. The fault zone is widest between the Stanislaus and 8 Lyon, R. J. P., 1954, Studies in the geology of the western Sierra Nevada: (I) Tectonic analysis of the Miles Creek area, Mariposa County, California: (II) Mineralogy of the ores of the southern foothill copper belt in California : California Univ., Berkeley, Ph. D. thesis. Calaveras Rivers where it is marked by a belt of greatly sheared volcanic rocks as well as by the individually mapped faults. It is narrowest in the southern part of the area. Rocks in the Bear Mountains fault zone are not as well exposed as those in the Melones fault zone, because four reservoirs extend across it. When the reservoirs are filled the original exposures on the former banks of the river are submerged and at low water they are covered with slime. The Bear Mountains fault zone includes the Bostick Mountain fault zone, the Copper-opolis fault, and the Hodson fault of Taliaferro and Solari (1949), the first-named being near the east side and the last-named near the west side. Schistosity has not been measured in the individual shear zones in the Cosumnes and Mokelumne Rivers, but farther south it dips eastward about 70° to 80°, although both steeper and more gentle dips occur. On the banks of Lake McClure the schistosity in shear zones is vertical. In general, the schistosity of the shear zones is as steep as, or steeper than, the dip of the rocks, on either side. Near the Cosumnes and Mokelumne Rivers, the fault zone consists of two main parts—a belt of serpentine and metavolcanic rocks on the west, and a belt of narrow shear zones causing repetition of strata on the east. The metavolcanic rocks associated with the serpentine on the Cosumnes River are fine grained. On the Cosumnes River (pi. 8) the position of the main part of the Bear Mountains fault zone is marked by two bodies of serpentine, separated by schistose and massive metavolcanic rock of unknown age. The shear zone cutting the Mesozoic rocks east of the serpentine and the shear zone separating this block of Mesozoic strata from the rocks tentatively included with the Calaveras formation are interpreted to be less important parts of the Bear Mountains fault zone. Near the Mokelumne River (pi. 7) the locus of greatest displacement is again marked by the serpentine. Whether it is at the largest body or the smaller mass farther east depends on correlation of the volcanic rocks (unit 22, pi. 7) east of the largest serpentine here arbitrarily included with the Copper Hill volcanics. Displacement on lesser faults of the eastern part of the zone has produced horst and graben structure. The fault zone as exposed near the Calaveras River differs from that farther north in that the volcanic rocks west of the serpentine are strongly sheared for iy2 miles west of the serpentine. The largest stratigraphic break is apparently about 1 mile east of the serpentine, at the east side of exposures of the Copper Hill volcanics. Near the Stanislaus River, the volcanic rocks west of the serpentine are also sheared, but much less so, nearly as far westward as O’Bymes Ferry.STRUCTURAL GEOLOGY 51 Taliaferro and Solari (1949) have mapped blocks of Calaveras formation within the serpentine immediately south of the Stanislaus River. Near Don Pedro Reservoir on the Tuolumne River, sheared rocks are limited to a belt about 500 feet wide near the northeast corner of sec. 24, T. 2 S., R. 14 E., and within a volcanic sequence arbitrarily included with the Penon Blanco volcanics. West of the shear zone is a granitic pluton which may have displaced other extensively sheared rocks. A shear zone along the strike of the one exposed in the Tuolumne River is exposed in a quarry where the road between La Grange and Coul-terville crosses Piney Creek. The shear zone exposed in Piney Creek, presumably controlled the straight course of Piney Creek, and short segments of faults along this trend are found on the banks of the northwesterly trending arm of Lake McClure (pi. 3). Another shear zone, about 1,000 feet wide, is exposed in cuts alongside the La Grange-Coulterville road about 2 miles south of the quarry at the position indicated by a fault on plate 1. No corresponding shear zone was found in Don Pedro Reservoir at a period of low water. DIRECTION AND AMOUNT OF MOVEMENT Age relations of the rocks exposed in the three main fault blocks indicate that the vertical component of movement in the Bear Mountains and Melones fault zones was in the reverse direction, but direct determination of the orientation and amount of net slip along the fault zones is impossible, as corresponding points or combinations of planes on opposite sides have not been identified. The structure here referred to as the Melones fault zone has been described as a reverse fault by Ferguson and Gannett (1932, p. 21) and Knopf (1929, p. 45, 46). Ferguson and Gannett used the term “reverse fault” strictly in a descriptive sense, but Knopf believed the dominant component of net slip to be vertical. Taliaferro (1942, p. 90), states that “The Mother Lode represents a great Upper Jurassic thrust along which the Paleozoic was thrust westward over the Mesozoic.” Cloos (1932b, p. 392-394; 1935, p. 234) discussed chiefly faults that controlled the quartz veins of the western Sierra Nevada and compares these faults to marginal thrusts related to batholiths. Cloos (1935, p. 238, 247) also, however, identified a long continuous fault (apparently the Melones fault zone of this report) as the “Mother Lode proper” and said that “It seems possible that movement along this fault began long before intrusion of the larger batholithic masses.” Clark (1960a, p. 491-492) suggested the possibility that the dominant component of movement along major fault zones is horizontal (strike-slip). Lineations within the fault zone plunge steeply, are parallel to axes of most minor folds measured within the fault zones, and parallel to the b tectonic axis. The steeply plunging fold axes and lineations suggest an important strike-slip component in fault movement, but many more fold axes must be measured to provide a reliable basis for interpretation. Limbs of most of the minor folds so far observed are sheared off, so these folds do not provide evidence of sense of movement. Evidence for strike-slip deformation elsewhere in the western Sierra Nevada has been reviewed previously (Clark, 1960a, p. 492). The apparent vertical component of movement on the Melones and Bear Mountains fault zones is large. Because rocks of Jurassic age west of the Melones fault zone are juxtaposed against rocks of Paleozoic age east of the fault zone for a distance of more than 90 miles, the apparent vertical component of movement is equivalent to, or greater than, the thickness of the Jurassic section west of the fault zone. Partial sections of these rocks range from about 3,000 to almost 15,000 feet thick. Strata along both sides of the Bear Mountains fault zone are of Jurassic age, but those adjacent to the east side of the zone are at least 10,000 feet strati-graphically lower than those adjacent to the west side. MOTHER LODE FISSURE SYSTEM The term “Mother Lode fissure system” is used here for the system of eastward dipping faults with which the quartz veins and gold ore bodies of the Mother Lode belt are associated. The controlling faults have usually been described as reverse (Knopf, 1929, p. 45) or thrust faults, although evidence for direction of movement is scarce (Eric, Stromquist, and Swinney, 1955, p. 27). During the present investigation little study was made of the Mother Lode fault system and most of the following discussion is based largely on previous descriptions. The resulting picture may be biased, for published descriptions are, appropriately, of the large mines where veins and ore zones have been explored for distances of several thousand feet along the dip or strike, or both. Veins and ore zones within the Melones fault zone that meet this qualification post-date significant movement along the major fault zone; otherwise the veins or their controlling structures would have been broken by fault zone movements into trains of horses or boudins extremely difficult to trace successfully in mining. If small remnants of ore bodies dating from an earlier stage do exist, they would receive slight consideration in any but the most exhaustive investigation of gold ore deposits of the Mother Lode belt. Knopf (1929) gave excellent descriptions of the faults exposed in many of the mines, and the areal pattern of the quartz veins and gold mines is shown by Ransome52 STRATIGRAPHY AND STRUCTURE, (1900) and Eric, Stromquist, and Swinney (1955, pis. 1 and 2). Most of the Mother Lode fissures are in or near the Melones fault zone, but eastward-dipping faults that may be part of the same system have been recognized in the Penn mine (Heyl, Cox, and Eric, 1948, p. 75) and in the adjacent Grayhouse area (Heyl, 1948a, p. 89-90), near Campo Seco. The distribution of mapped eastward dipping faults closely parallels that of the gold and copper mines, suggesting that their recognition may depend on detailed study of well-exposed areas, and that they may be more extensive than known. From a point about half way between the Merced and Tuolumne Rivers northward to the Mokelumne River, most of the Mother Lode fissures are within the Melones fault zone, but near the southern end of the belt and from the Mokelumne River to the north end of the Mother Lode belt, northwest of Placerville, most of the veins crop out one mile or more west of the Melones fault zone. Although the strike of some veins is moderately sinuous, the strike of most veins throughout the Mother Lode belt is about parallel to that of the Melones fault zone. Veins generally dip less steeply than cleavage and bedding west of the fault zone and schistosity within the fault zone. The veins dip eastward, mostly at angles between 50° and 70°, although one dips as gently as 20° for a considerable distance (Knopf, 1929, p. 24). Among the veins whose attitude is given by Knopf (1929) or Ransome (1900), no significant difference was noted between the dips of veins within the Melones fault zone and those west of the fault zone. Veins west of the fault zone cut and offset bedding in the country rock at acute angles. Some veins and zones of replacement ore within the Melones fault zone bear similar relations to schistosity and tabular bodies of rock drawn out parallel to the schistosity of the fault zone but others are parallel to schistosity and contacts within the fault zone. Of the veins that crop out west of the fault zone, none have been described as having been traced down dip into the fault zone. The movement picture of the Mother Lode fissure system evidently is not simple. Knopf (1929, p. 45) believed as a result of finding older rocks on the hanging wall side of many veins that most of the veins occupied dip-slip reverse faults. He points out that flutings on post-mineral gouge indicate movement parallel to the dip in the Plymouth mine (Knopf, 1929, p. 52), but faulting that has produced the main fissure of the central Eureka mine had horizontal displacement of 120 feet (p. 62), and a strong lateral component of movement is suggested by structural relations in the Argonaut mine (p. 67). The greatest measurable displacement found by Knopf was an apparent dip-slip SIERRA NEVADA METAMORPHIC BELT displacement of 375 feet in the Gover shaft of the Fremont mine (Knopf, 1929, p. 25, 54). Cloos (1932b, p. 393-394) believed the fissures to be related to emplacement of the Sierra Nevada batholith and comparable to marginal thrusts. The mine maps and sections prepared by Knopf provide evidence that some of the Mother Lode veins and gold ore bodies post-date most of the movement along the Melones fault zone. In several of the mines within the Melones fault zone the principle ore bodies replace schist. Schistosity is much stronger in the Melones fault zone than in most of the bordering rocks and is apparently related to fault movements. Replacement of schistose rock by gold ore accordingly suggests that the ore is later than fault movement. Ore bodies of this kind are found in the Eagle Shawmut and Clio mines which are in the Melones fault zone near the Tuolumne River. In these mines Knopf interprets the sequence of events as follows: (a) reverse faulting, (b) emplacement of peridotite or similar rock, (c) serpentinization of the peridote and renewed fault movement, (d) deposition of ore. The first faulting was no doubt part of the Melones fault zone movement—the later movement noted by Knopf may be related to the same tectonic episode or to a later episode. In contrast with ore bodies at the Eagle Shawmut and Clio mines, some of those at Carson Hill, in the Melones fault zone about one mile north of the Stanislaus River, are associated with smaller faults that cross and offset structures characteristic of the chief movement along the major fault zone. Here, gold-quartz ore bodies were found at the intersection of gently dipping veins with a steeply dipping vein. Along at least one of the gently dipping veins, contacts of the wall rocks, including serpentine and schist, have been offset in a reverse direction (Knopf, 1929, p. 76-77, fig. 23). Mapping by Eric and Stromquist (pi. 1 in Eric, Stromquist, and Swinney, 1955) shows that in the vicinity of Carson Hill serpentine and schist form narrow bodies elongate parallel to schistosity in the Melones fault zone as they do elsewhere in the zone. MINOR FAULTS The minor faults discussed here are those found during this investigation which are apparently not related to the fault systems discussed previously. Their displacements probably do not exceed a few hundreds of feet at the most. The westward-dipping faults exposed in the north wall of the Tuolumne River gorge below Don Pedro Dam (see fig. 16) are apparently thrust faults related to the folding. The apparent offset of the westernmost fault is shown by the coarse volcanic breccia bed. IfSTRUCTURAL GEOLOGY the volcanic breccia exposed near Don Pedro Dam is the same layer, the displacement of the easternmost fault is much greater. The faults were observed only from the south side of the river, and the true dip and direction of movement were not measured. In the cut immediately north of Huse Bridge steeply dipping slickensided surfaces separating several blocks of volcanic breccia suggest a minor fault. Most of the slickensides plunge less than 30 °, both north and south, but on one surface the plunge is steep. No comparable slickensides were found immediately south of the bridge but here about an inch of gouge separates the Logtown Ridge formation from the overlying epiclastic rocks. No evidence was found to suggest that this fault has more than a few feet of movement. The steeply dipping fault exposed in the Cosumnes River near the SW corner of sec. 31, T. 8 N., R. 9 E., on the west flank of the large anticline, is near where Taliaferro (1943b, fig. 2) showed a major reverse fault. The fault is prominent in stream-bed exposures as it separates moderately folded but generally gently dipping beds on the east from nearly vertical beds on the west. The fault is marked by a fracture less than an inch wide, and the beds on either side are apparently not crumpled. This fault is in such striking contrast to those known to be major faults in the region that it is here interpreted as a minor structure. AGE OF FAULTING Movement on faults of the Foothills system began during Late Jurassic time and may have continued into Early Cretaceous time. The youngest rocks cut by the Melones fault zone include early Kimmeridgian strata of the Mariposa formation, and the youngest rocks cut by the Bear Mountains fault zone conformably overlie strata of late Oxfordian or early Kimmeridgian age but are probably pre-Tithonian. Branches of the Bear Mountains fault zone are truncated north of the report area by the Rocklin granodiorite and Horseshoe Bar quartz diorite which have yielded potassium-argon dates of 131 and 142.9 million years, respectively, and are believed by Curtis, Evemden, and Lipson (1958, p. 6, 10-12) to be of Late Jurassic age. The latest significant movement along the Melones fault zone is less closely dated: the zone is truncated south of the map area by a pluton that is presumably a lobe of the Sierra Nevada batholith (Cloos, 1932a) and of middle Cretaceous age. Faults of the Mother Lode fissure system may be of about the same age as the Sierra Nevada batholith. They are younger than the Melones fault zone, but older than the unconformably overlying auriferous gravels of the region, generally considered to be of 53 Eocene age (see discussion by Eric, Stromquist, and Swinney, 1955, p. 16). STRUCTURE OF THE EASTERN BLOCK Only some of the larger aspects of the geologic structure of the eastern block can be described, as most of the metamorphic rocks of this area lack distinctive marker units and in large areas bedding has been destroyed by shearing. Bedding is preserved in some isolated small blocks within sheared areas, but in such places it is almost invariably parallel to the regional schistosity, leading to the suspicion that it is preserved only because of this orientation. Attitudes of bedding in these isolated blocks have not been recorded. The structure of the eastern block is particularly obscure in the part that lies north of the latitude of San Andreas, except near the Mokelumne River where a northerly trend is suggested by the distribution of limestone. Regional trends can be distinguished south of this latitude. In the southeast half of the eastern block, the volcanic member of the Calaveras formation and the belt of limestone lenses both trend northwest at an acute angle to the Melones fault zone. This trend is further substantiated by the fact that the lowermost unit of the Calaveras formation in this region, exposed east of Bagby, is missing north of the Tuolumne River, where it is truncated by the Melones fault zone. West and northwest of Sonora, distribution of volcanic rocks in the Calaveras suggests that they are repeated in folds, or that they form large lenses at various stratigraphic positions instead of a single continuous sequence. The eastward-trending mass of volcanic rock southeast of San Andreas forms the core of a large anticline (Clark, 1954). The volcanic rocks and limestone lenses converge north of the Merced River suggesting that one unit or the other, or both, thickens at the expense of intervening rocks. This is further suggested by the fact that limestone is in contact with volcanic rocks north of the Stanislaus River. More data on the structure of the eastern block were obtained near the Merced River than elsewhere. Here, top determinations, absence of areas of general low dip, and lack of repetition of lithologic members of the Calaveras formation, suggest that the structure of the block is essentially homoclinal with younger beds on the east. Nevertheless, folds with amplitudes of a few hundred or even a few thousand feet may be present in parts of the area lacking distinctive marker units. The data are most complete between Bagby and Brice-burg, and the possibility of undetected large folds is greatest between Briceburg and the abandoned cement plant where the argillaceous unit yielded few top determinations or lithologically distinctive horizons. Also,54 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT exposures on canyon walls are poor in this interval, so that the trace of bedding is not readily observed. East of the abandoned cement plant, long limestone lenses and abundant chert horizons can be traced readily on the canyon walls, and these show regular, near vertical dips. An area of tight, complex minor folding well shown in riverbank exposures in the northeast comer of sec. 19, T. 3 S., R. 19 E., may indicate the axis of a larger fold, but the limestone exposed west of this area is not repeated to the east. Structure of the meta-morphic rocks east of Indian Flat Guard Station is obscure due to poor exposures and extensive crumpling. STRUCTURE OF THE CENTRAL BLOCK In the northern and southern parts of the area the structure of the central block is homoclinal with tops east. Near the Cosumnes River, beds are vertical and near the Merced River they mostly dip eastward 50° to 75°. In the central part of the block are isoclinal folds, the largest of which is the syncline whose core is formed by the Mariposa formation. It has been traced down the Mokelumne River to the Stanislaus River. The trough of the fold is at an acute angle to the Melones fault zone, lying 2 miles west of the fault zone on the Mokelumne River, and intersecting the fault zone between the Stanislaus and Tuolumne Rivers. The areal pattern and generally very low plunge of the lineation formed by intersection of bedding and cleavage indicates that the fold axis is nearly horizontal. The fold has been verified by repetition of beds on the Mokelumne, Calaveras, and Stanislaus Rivers, and also by top determinations in the first two areas. Most of the bedding on the limbs of the folds is nearly vertical. An isoclinal anticline is exposed east of this syncline near the Calaveras River, where it has been established by top determinations and attitudes of bedding. The anticline is shown as far north as the Mokelume River on the accompanying maps, but here the structural interpretation, is not well established. This anticline is truncated by the Melones fault zone south of San Andreas. The structure of the Penon Blanco volcanics west of Melones Dam is uncertain, as only one top determination was made and bedding is obscure in much of the area between the dam and the serpentine to the west. STRUCTURE OF THE WESTERN BLOCK The northern part of the western block is characterized by open folds overturned to the west. Large folds are also formed in the south part of the block, but the shapes of folds are unknown because the Gopher Ridge volcanics, which there constitute much of the block, show little bedding. In the western block, structure is best documented near the Cosumnes River (pi. 8), where the syncline at Michigan Bar Bridge, the anticline to the east, and the wide homoclinal section on the east flank of the anticline are adequately controlled by top determinations and attitudes of beds. On plate 8, the structure of parts of the anticline is generalized, as the transition from gently dipping to steeply dipping beds on both sides of the crestal region is marked by small tight folds. Axes of subsidiary folds near the crest of the anticline indicate that it plunges northwest at an angle of about 25°. The nearly horizontal attitude of beds at the west end of the Cosumnes River section is inferred from exposures in the banks of the river near Bridge House. Although no bedding was found here, lava is exposed at the low-water level for a distance of about 200 feet, whereas coarse volcanic breccia is exposed higher on the banks in the same area, suggesting a horizontal contact. On plate 7 the epiclastic and volcanic rocks west of Campo Seco are interpreted to be a homoclinal section with tops east. This implies that the volcanic rocks in the core of the anticline at the Cosumnes River plunge beneath the bedrock surface near lone, and that the volcanic rocks near Campo Seco are on strike with the westernmost belt of volcanic rocks exposed on the Cosumnes River. However, if it be assumed that the volcanic rocks near Campo Seco (pi. 7) are a continuation of the belt that is overlapped by Tertiary deposits near lone, the distribution of epiclastic and volcanic rock units suggests, instead, that the large folds found on the Cosumnes River extend southward beyond the Mokelumne River. Heyl, Cox and Eric (1948, p. 67) state that tops are westward at the Penn mine (pi. 7) which may support this interpretation, although they do not indicate the size of the area covered by their observation. If this indicates a major fold, the epiclastic rocks west of the mine should be in a syncline. Nevertheless, these epiclastic rocks dip consistently eastward at an angle of about 35°, and no trace of a trough is apparent. Search for graded beds on the surface in the Penn mine area during this investigation revealed only one small group consisting of about 5 dubiously graded beds. These are on the north bank of the Mokelumne River near the west side of unit 7. The west parts of the beds were consistently lighter colored and apparently finer grained than the east parts but the light-colored parts apparently are fine-grained aggregates resulting from alteration of feldspar crystals that originally constituted the coarser parts of the beds. If so, tops of the beds would be toward the east.STRUCTURAL GEOLOGY 55 Structural data are meager along the Calaveras River within the interval between A and B on the section (pi. 6) and near Hogan Dam. Between unit 8 and the western part of unit 15 the structure is indicated by top determinations and attitudes. In the SW14 sec. 14, T. 3 N., R. 10 E., beds dip westward at angles between 65° and 90° or are vertical, and the relation of cleavage to bedding suggests tops wTest. In the eastern parts of secs. 22 and 27, gross layered structure, suggesting superimposed lava flows, strikes northeast, parallel to the river, and dips northwestward about 50°. The structure near Hogan Dam is arbitrarily interpreted to be homo-clinal. Near the Tuolumne and Merced Rivers, tops are westerly near the western side of the area of bedrock exposures and beds are folded near the eastern side of the block. Structure of much of the intervening area is obscure because the Gopher Ridge volcanics consist largely of massive volcanic and pyroclastic rocks in which bedding can be distinguished only locally. In the slate east of La Grange, on the Tuolumne River, no graded beds occur, but minor folds in the western part of the slate belt (fig. 10) suggest that these beds are on the west limb of an anticline. The epiclastic rocks east of Merced Falls, on the Merced River, are tightly folded in places, but many graded beds show tops west wherever the structure straightens out (pi. 11). Drag folds exposed for nearly a mile west of Don Pedro Dam on the Tuolumne River (fig. 16) indicate that the eastern part of the Gopher Ridge volcanics lie on the west limb of an anticline. The axis of the anticline is apparently along the northwest-trending part of Don Pedro Reservoir, for tops are toward the east immediately east of this arm. However, the crest of the anticline may be cut out by a continuation of a fault extending from Lake McClure toward the northwesttrending arm of Lake McClure. Because tops are westward on the west side of the main belt of Gopher Ridge volcanics on the Merced River, and eastward on the east side, the Gopher Ridge volcanics probably form an anticline, but part of the anticline may be faulted out as suggested by strong schistosity in sec. 2, T. 5 S., R. 15 E. A syncline with a core of slate is suggested by graded bedding west of Exchequer Dam on the Merced River (pi. 3), and folds of moderate size near the western shores of Lake McClure are indicated by the differing directions of tops of beds at various localities. SMALL-SCALE STRUCTURES CLEAVAGE AND SCHISTOSITY Cleavage is widely developed in rocks of the western and central blocks, and schistosity is well developed in metamorphic rocks of the shear zones and the eastern block. Slip cleavage (White, 1949) pervades large parts of the eastern block, and occurs locally in rocks of the western and central blocks. Nearly all the slate, and much of the graywacke, conglomerate, and tuff of the western block are cleaved. Much of the lava and coarse volcanic breccia seems massive in fresh exposures but cleavage becomes apparent in weathered exposures. Regionally, and in single exposures (fig. 15), cleavage is about parallel to the axial plane of folds. In the eastern block, argillaceous rocks east of Bagby on the Merced River show only cleavage, but most of the rocks east of the volcanic member of the Calaveras formation are phyllitic or schistose. In all rocks here described as schistose, a large proportion of the tabular minerals are parallel to the parting surfaces. In some, however, the parting is attributable solely to parallel mineral grains and in others it results in part from parallel rock fragments of epiclastic or cataclastic origin. In most places, the schistosity results from slip parallel to the parting planes as well as recrystallization. Within major fault zones slip is indicated in conglomerate and sandstone by microfaults and mortar structure with trains of fragments and in phyllite by small drag folds with sheared-off limbs. In much of the eastern block slip parallel to the schistosity is indicated by fragmentation of beds, the development of spindle-shaped fragments, and minor folds with sheared-off limbs. In large areas bedding in the chert and argillaceous sequences of the Calaveras formation has been completely destroyed in the development of strongly lineated fragmental schist. In other parts of the eastern block, especially in some places where schistosity is parallel to bedding, the significance of slip in the development of schistosity is not apparent. Slip cleavage that is younger than schistosity can be distinguished in some places east of the Melones fault zone. It is most apparent southeast of San Andreas where it is parallel to the schistosity in the Melones fault zone, but cuts eastward trending schistosity and related structures in the Calaveras formation (Clark, 1954, p. 12). There schistosity is commonly parallel to bedding and both are deformed along the slip cleavage planes. The slip cleavage consists of microfaults and crinkles. In thin section mica is seen to be sharply bent into the slip cleavage direction. The slip cleavage becomes less prominent eastward, where the trend of the earlier schistosity is more nearly parallel to the Melones fault zone. "Where slip cleavage surfaces are closely spaced, they can be distinguished only with difficulty from schistosity. Slip cleavage is shown by the same symbol as the schistosity on the map56 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT of the Angels Camp quadrangle (Eric, Stromquist, and Swinney, 1955, pi. 1, p. 29) although it is described separately in the text as shear cleavage. In some places the only remaining foliation in the rocks of the eastern block may be related to the slip cleavage. MINOR FOLDS Minor folds have amplitudes of a few inches to a few tens of feet. Minor folds resulting from both stages of regional deformation are found throughout the meta-morphic rocks of the map area, but most of the minor folds west of the Melones fault zone plunge gently and are related to the first stage, whereas most of those east of the Melones fault zone plunge steeply and are related to the second stage. Axes of most minor folds here interpreted to be related to the first stage of deformation plunge at angles of less than 20°, but axes of some plunge as much as 35°. Gently plunging minor folds can be readily observed on the western limb of the anticline east of Michigan Bar Bridge on the Cosumnes River, and near both Don Pedro Dam and La Grange on the Tuolumne River. Gently plunging minor folds are rare east of the Melones fault zone, but are associated with gently plunging lineations on the north side of the Mokelumne River in sec. 32, T. 6 N., R. 12 E., and along the Tuolumne River in sec. 1, T. 1 S., R. 15 E. Axes of minor folds interpreted to be related to the second stage of deformation plunge more steeply than 60°. The steeply plunging minor folds occur in all three fault blocks and the fault zones but are scarce in the central and western blocks. Too few steeply plunging folds were mapped to provide independently a regional movement picture, but the folds served to indicate the tectonic orientation of lineations which are much more abundant. Axial planes of steeply plunging minor folds are parallel to slip cleavage and schistosity. Further work will be necessary to establish the significance of minor folds having different orientations than folds characteristic of the two stages of regional deformation. An example of such folds is in the Merced Falls slate along the Merced River. Here, minor fold axes are of diverse orientation except near the east side of the slate belt where the plunge is consistently steep and to the southeast (pi. 11). The diversity of orientation possibly results from a local modification of regional deformation patterns by a buttress of volcanic rocks that ends north of the river (pi. 1), or from superimposed folding. LINEATIONS Lineations discussed in the following paragraphs are in the more strongly deformed and sheared rocks that are chiefly in the eastern block and the two fault zones. Most lineation in the less deformed Mesozoic rocks that constitute the central and western blocks results from intersections of bedding and cleavage; a pencil structure occurs where bedding and cleavage are nearly at right angles near the crests or troughs of moderately to steeply plunging minor folds in epiclastic rocks of Mesozoic age. Pencil structure in the slate east of Merced Falls provided a useful guide in locating axes of minor folds. Lineations in the greatly sheared rocks are of several kinds: in schists some lineations are marked by flat triaxial ellipsoids derived from fragments that are of pyroclastic, cataclastic, and epiclastic origin, others by elongate flat pods of chlorite or mica or by parallel amphibole crystals. Lineations marked by minerals are commonly associated with those marked by rock fragments, but are not restricted to fragmental schists. Each kind has been found to be parallel to minor fold axes and hence parallel to b. The most common lineations east of the Melones fault zone consist of elongate fragments of chert in a matrix of carbonaceous and commonly quartz-rich mica schist. Associated with them in many places are thin elongate mica pods. The lineations are very similar to some of the mullion structures in the Moine series of Scotland described by Gilbert Wilson (1953.) The chert fragments (granoblastic microcrystalline quartz) take many forms depending upon the relations of cleavage and bedding and the amount of deformation. Axial ratios range from the order of 1:1:200 or 1:5:200 in the more elongate fragments to about 1:1:5 in spindle-shaped fragments. In cross section, the fragments are circular in some places and angular in others, but more commonly show streamlined shapes. Lineations lie in the plane of schistosity where schistosity can be identified, but in rare places where the rock consists of closely packed cylinders of chert no prevailing direction of schistosity was found. Although chert fragment lineation is of the same kind throughout areas underlain by the chert-bearing rocks east of the Melones fault zone, they are not necessarily everywhere related to the same stage of deformation. The chert fragments were formed by cataclastic processes; they are not elongated epiclastic pebbles (Clark, 1954, p. 6-7). The abundant and widespread lineation in the Calaveras formation probably results from the great extent of rocks consisting of alternating thin beds of chert and relatively incompetent carbonaceous quartz-mica schist. In exposures and specimens showing the relation of cleavage to bedding, lineation is parallel to axes of minor folds and is formed in two ways; both result in lineation parallel to the bSTRUCTURAL GEOLOGY tectonic axis. Where bedding crosses the schistosity, the chert beds are broken into long prisms by offset along schistosity surfaces, but where bedding is parallel to the schistosity elongate fragments are formed by boudinage. Where offset is slight, the fragments of chert beds are angular. With further movement they assume streamlined or roundish forms in cross section. A third process depends upon the tendency, common in rocks consisting of alternate layers of competent rock and slate, for cleavage which is nearly parallel to the bedding in the slate to be refracted at a large angle to the bedding in the competent layer. Such refracted cleavage breaks the competent layer into prisms elongate parallel to b. In much of the area deformation is so severe that bedding is completely destroyed, and the origin and tectonic orientation of the chert fragment lineation is obscure ; no pattern that is consistent throughout the eastern block is apparent. Nevertheless, the attitude of lineation is reasonably consistent over large areas within which it is possible to show that the lineation is parallel to axes of small folds. Not all small folds, however, were necessarily formed during the same stage of deformation. Lineations with gentle to moderate dip are, at least in places, related to large folds apparently formed during the first stage of severe deformation of the Calaveras formation. Steeply plunging lineations, also characteristic of large areas, seem to be best explained as the result of a second stage of deformation acting on previously folded beds. Relation of moderately plunging lineations to major folds is best shown east of San Andreas where the lineations generally plunge ESE. 30° to 50°. The pattern of volcanic rocks southeast of San Andreas shows that the anticline in which they are exposed plunges eastward, but the angle of plunge is not known directly. Lineations in the Mokelumne River also plunge at moderate angles in a direction slightly more southerly than most of the lineations east of San Andreas and are probably related to the same fold system. The few measurements on the Tuolumne River show both gentle and steeply plunging lineations and form a pattern that is not readily interpreted. Those on the Merced River and the North Fork of the Merced River fall into two groups, both lying in a vertical plane that strikes N. 70° W. and is close to the average for schistosity in this region. Lineations in the argillaceous sequence nearest the volcanic sequence plunge northwest at angles of 50° to 80° whereas those farther east plunge southeast at similar angles. The steepness of plunge of these lineations and the associated minor folds suggest that they are related to a stage of deformation 57 that followed rotation of the beds into a near vertical position. Intersection of slip cleavage that is parallel to schistosity of the Melones fault zone with earlier schistosity and bedding forms a prominent set of lineations consisting of axes of minor shear and flexure folds, crinkles, and crenulations on schistosity surfaces. These structures are developed southeast of San Andreas, where the eastward-trending anticline in the Calaveras formation is at a large angle to the Melones fault zone. Although the slip cleavage and associated lineations are prominent, earlier schistosity and bedding can be readily traced in most exposures. The strike of the slip cleavage ranges as much as 25° and attitudes of the bedding and schistosity range much more widely. Consequently, the plunge of lineations related to slip cleavage is apparently at random, even in areas of less than a square mile. Lineations of this sort can be expected in the Calaveras formation within a few miles of the Melones fault zone wherever slip cleavage is developed. Linear structures related to slip cleavage can be seen conveniently in the floor of the Calaveritas Hill Consolidated hydraulic pit, about 4 miles southeast of San Andreas (see Clark, 1954, pi. 1). Lineations within the Melones fault zone lie within the planes of schistosity and shearing, and plunge eastward and southeastward at angles of 60° to 80°. The bearing of the lineation is nearly normal to the strike of the schistosity. Abundant parallel amphibole crystals form a well-marked lineation in the sheared western part of the granitic pluton north of Plymouth. The lineation is developed throughout the width of the western tongue of the pluton near the Cosumnes River, a distance of more than half of a mile, but no lineation is visible in unsheared granodiorite of the same pluton several miles east of the Melones fault zone. Between San Andreas and Angels Camp, lineation is marked by elongated fragments of volcanic breccia and elongate blebs of chlorite in the metavolcanic rocks and by elongated pebbles and boulders in the mixed-pebble conglomerate of the Mariposa formation. 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W., Jr., 1953, San Andreas, Garlock, and Big Pine faults, California—a study of the character, history and tectonic significance of their displacements: Geol. Soc. America Bull., v. 64, p. 443-458. Hyatt, Alpheus, 1894, Trias and Jura in the Western States: Geol. Soc. America Bull., v. 5, p. 395-434. Imlay, R. W., 1952, Correlation of the Jurassic formations of North America, exclusive of Canada: Geol. Soc. America Bull., v. 63, p. 953-992. ------ 1959, Succession and speciation of the pelecypod Auoella: U.S. Geol. Survey Prof. Paper 314-G, p. 155-169. ------1961, Late Jurassic ammonites from the western Sierra Nevada, California: U.S. Geol. Survey Prof. Paper 374-D, p. D1-D30. Jenkins, O. P., ed., and others, 1948, Copper in California: California Div. Mines Bull. 144, 429 p. King, P. B., 1937, Geology of the Marathon region, Texas: U.S. Geol. Survey Prof. Paper 187, 148 p. Knopf, Adolph, 1929, The Mother Lode system of California: U.S. Geol. Survey Prof. Paper 157, 88 p. Kuenen, P. H., 1950, Marine geology: New York, John Wiley and Sons, 568 p. ------ 1957, Sole markings of graded graywacke beds: Jour. Geology, v. 65, p. 231-258. Kuenen, P. H., and Migliorini, C. I., 1950, Turbidity currents as a cause of graded bedding: Jour. Geology, v. 58, p. 91-127. Lawson, A. C., 1933, The geology of middle California : Internat. Geol. Cong., 16th, Washington 1933, Guidebook 16, p. 1-12. Lindgren, Waldemar, 1894, Sacramento, Calif.: U.S. Geol. Survey Geol. Atlas, Folio 5, 3 p. ------1900, Colfax, Calif.: U.S. Geol. Survey Geol. Atlas, Folio 66,10 p. Lindgren, Waldemar, and Turner, H. W., 1894, Placerville, Calif.: U.S. Geol. Survey Geol. Atlas, Folio 3, 3 p. Natland, M. L., and Kuenen, P. H., 1951, Sedimentary history of the Ventura Basin, California, and the action of turbidity currents, in Soc. Econ. Paleontologists and Mineralogists, Turbidity Currents and the transportation of coarse sediments to deep water—a symposium : Soc. Econ. Paleontologists and Mineralogists, Spec. Pub. no. 2, p. 76-107. Pettijohn, F. J., 1943, Archean sedimentation: Geol. Soc. America Bull., v. 54, p. 925-972. ------1954, Classification of sandstones: Jour. Geology, v. 62, p. 360-365. ------1957, Sedimentary rocks: New York, Harper and Brothers, 718 p.DESCRIPTION OF MAP UNITS 59 rfi "V" ' . ; Piper, A. M., Ga)e, H. S., Thomas, H. E., and Robinson, T. W., 1939, Geologyand ground-water hydrology of the Mokelumne Hill area, Califfenia: U.S. Geol. Survey Water-Supply Paper 780, Ransome, F. L., 1900, Mother Lode district: U.S. Geol. Survey Geol. Atlas, Folio 63, 11 p. Reiche, Parry, 1937, Geology of the Lucia quadrangle, California : California Univ. Dept. Geol. Sci. Bull., v. 24, no. 7, p. 115-168. Renz, O., Lakeman, R., and Van Der Meulen, E., 1955, Submarine sliding in western Venezuela: Am. Assoc. Petroleum Geologists Bull., v. 39, no. 10, p. 2053-2067. Shrock, R. R., 1948, Sequence in layered rocks: New York, McGraw-Hill, 507 p. Smith, J. P., 1894, Age of the auriferous slates of the Sierra Nevada: Geol. Soc. America Bull., v. 5, p. 243-258. Storms, W. H., 1894, Ancient channel system of Calaveras County: California State Mining Bureau, Rept. 12, p. 482-492. Taliaferro, N. L., 1933, Bedrock complex of the Sierra Nevada, west of the southern end of the Mother Lode [abs.] : Geol. Soc. America Bull., v. 44, no. 1, p. 149-150. ------1942, Geologic history and correlation of the Jurassic of southwestern Oregon and California: Geol, Soc. American Bull., v. 53, no. 1, p. 71-112. ------1943a, Franciscan-Knoxville problem: Am. Assoc. Petroleum Geologists Bull., v. 27, no. 2, p. 109-219. ------1943b, Manganese deposits of the Sierra Nevada, their genesis and metamorphism: California Div. Mines Bull. 125, p. 277-332. Taliaferro, N. L., and Solari, A. J., 1949, Geology of the Copper-opolis quadrangle, California: California Div. Mines Bull. 145 (map only). Trask, P. D., 1926, Geology of the Point Sur quadrangle, California: California Univ. Dept. Geol. Sci. Bull., v. 16, p. 119-186. Turner, H. W., 1893a, Some recent contributions to the geology of California : Am. Geologist, v. 11, p. 307-324. ------1893b, Mesozoic granite in Plumas County, California, and the Calaveras formation: Am. Geologist, v. 11, p. 425-426. ------1894a, Jackson, Calif.: U.S. Geol. Survey Geol. Atlas, Folio 11, 6 p. ------1894b, The rocks of the Sierra Nevada : U.S. Geol. Survey, 14th Ann. Rept., pt. 2, p. 435-495. ------ 1894c, Geological notes on the Sierra Nevada: Am. Geologist, v. 13, p. 228-249, 297-316. ------1895, The age and succession of the igneous rocks of Sierra Nevada : Jour. Geology, v. 3, p. 385-414, map. Turner, H. W., and Ransome, F. L., 1897, Sonora, Calif.: U.S. Geol. Survey Geol. Atlas, Folio 41. ------1897, Description of the Big Trees quadrangle: U.S. Geol. Survey Geol. Atlas, Folio 51. ------ 1899, The granitic rocks of the Sierra Nevada: Jour. Geology, v. 7, p. 141-162. White, W. S., 1949, Cleavage in east-central Vermont: Am. Geophys. Union Trans., v. 30, p. 587-594. Wilmarth, M. G., 1938, Lexicon of geologic names of the United States: U.S. Geol. Survey Bull. 896, pt. 1. Wilson, Gilbert, 1953, Mullion and rodding structures in the Moine series of Scotland: Geologists’ Assoc., London, Pr. v. 64, pt. 2, p. 118-151. Zeuner, F. E., 1952, Dating the past; an introduction to geochronology: London, Methuen and Co., Ltd., 495 p. DESCRIPTION OF MAP UNITS Table 1.—Description of map units, Merced River (pi. 3) Map unit Approximate thickness (feet) 5___ 3000+ .._ 6 Lithology 1 __ 2000_______ Top not exposed. Black slate, locally pyrite bearing, with interbedded gray-wacke and siltstone. Graywacke beds commonly less than one-half in. thick below Merced Falls dam, several inches thick elsewhere, locally thick bedded. Interbedded rhyolite(?) tuff eastern part. Graywacke beds commonly graded throughout the section. 2 __ ___________ Coarse rhyolite tuff. 3 ___ 1000+ Like unit 1. 4 ___ 1000+ Schistose volcanic rocks with some rhyo- lite^) tuff on west side; mafic and intermediate flows, volcanic breccia and minor tuff elsewhere. No graded tuff beds found. Mafic or intermediate thick-bedded pyroclastic rocks. 500_____ Laminated black slate with interbedded thin graywacke beds. Includes thick-bedded graywacke about 100 ft thick on west side. Rare graded beds in thick-bedded graywacke. Interpreted to be equivalent to unit 8. 7 __ 200________ Bedded fine conglomerate with interbedded slate. Pebbles angular to well rounded, as much as 1 in. long in some beds, but smaller in most beds. Slate matrix. Rock fragments include slate, dark-and light-gray chert, white and gray vein quartz. 8 __ 500________ Laminated black slate with minor inter- bedded graywacke on west side. 9 __ 400________ Thin- to thick-bedded rhyolite tuff, lapilli tuff and silicified ash. Lapilli tuff contains fragments of chert or silicified volcanic glass or ash. Large proportion of beds are graded. 10— 1500+—_ Locally faulted on east side. Silicified ash and volcanic breccia with quartz pheno-crysts; in part, at least, rhyolitic(?); dark-gray massive rhyolite(?) porphyry; and minor black chert. 11— 500+----- Structure uncertain; faulted on east side. Laminated black slate with interbedded thin beds of graywacke or tuff in some places. Contains interbedded conglomerate near south quarter-corner of sec. 12, T. 4 S., R. 15 E. Slate is locally quartzose, with sub vitreous luster. Near south quarter corner of sec. 1, T. 4 S., R. 15 E., contains pods about 1 ft thick of dark-gray calcarenite. One such pod contains Plesiosaur bones.60 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT Table 1.—Description of map units, Merced River (pi. 3)—Con. Approximate Map thickness unit (feet) Lithology 12 _ 1000+—- Faulted on west side. Mafic or intermediate volcanic breccia and tuff; some tuff beds graded. Includes chlorite schist of the fault zone. 13 _ 1000+—_ Thin-bedded black slate and tuff. 14 _ 300+_______ Thin-bedded rhyolite(?), crystal tuff, and silicified ash or chert. Top not mapped. 15 _ 1000....... Coarse dark-green volcanic breccia. 16 _ 5000_______ Massive aphanitic dark-green mafic lavas. Three bands of pillow lava present in upper 2000 ft; rare suggestions of pillow structure in lower part. 17 _ 1500_______ Thin-bedded tuff and silicified ash; graded bedding common. Hunter Valley cherts of Taliaferro (1933). 18 _ 3000_______ Recrystallized and epidotized mafic or intermediate pyroclastic rocks; top poorly exposed. 19 _ 2000+Coarse massive dark-green volcanic breccia with large augite phenocrysts. 20 _ 2500_______ Thick-bedded coarse tuff and volcanic breccia. A few thin-bedded zones with graded bedding are present. 21 _ 600________ Black laminated slate with rare Aucella. 22 _ 300________ Coarse dark-green volcanic breccia. 23 _ 500________ Interbedded thin-bedded graywacke and black slate. 24 _ 200-500. Thin-bedded tuff and silicified volcanic ash. Graded bedding common. Unit thickens to the southeast. 25 _ 500+_______ Interbedded black slate and graywacke with thick-bedded graywacke horizon at base. Graywacke contains fragmental plant remains in Highway 49 roadcut on south side of Merced River at Bagby. Fault contact with serpentine on east. 26___ 2000_______ Faulted at base—in contact with serpentine. Laminated black silty slate in lower part, containing lens of black massive aphanitic limestone. One exposure of volcanic breccia with pyroxene phenocrysts as much as one-quarter inch in diameter at Bagby. Upper part is interbedded black silty slate and graded graywacke. Beds are about one-half inch thick. Parts of some graywacke beds are cross-laminated. 27— 50-400-. Interbedded thin-bedded green tuff, silicified volcanic ash, and black slate. Volcanic breccia occurring as float on northeast side contains pyroxene phenocrysts as much as one-quarter inch in diameter. 28— 500__________ Dark-green quartzose slate. Bedding not visible. Cleavage fairly well developed. 29— 300------ Black slate, with lenticular dark-gray lime- stone containing abundant crinoid debris. 30— 1000--------- Interbedded tuff and black slate. Includes thin unit of interbedded limestone (in part calcarenite), black slate, and silty sandstone exposed at sand bar in NEJ+ SWtf sec. 10, T. 4 S„ R. 17 E. Table 1.—Description of map units, Merced River (pi. 3)—Con. Approximate Map thickness unit (feet) Lithology 31____ 0-300_____ Poorly sorted schistose conglomerate con- taining angular to well-rounded fragments of graywacke as much as 18 in. in diameter in a mottled yellow-green and black sandy argillite matrix. Not bedded. 32 _ 1100_______ Interbedded dark-gray chert and black slate. Most chert and slate beds are about 1 in. thick. 33 _ 400________ Poorly sorted conglomerate, like unit 31. 34 _ 600________ Black slate with minor interbedded tuff and volcanic breccia near center of unit. 35 _ 400________ Massive dark-green recrystallized tuff. 36 _ 4000_______ Interbedded laminated black slate, massive black siltstone, and dark-green slate. Individual rock types form subunits 50 to 100 ft thick. 37 _ 5000_______ Coarse dark-green volcanic breccia, volcanic conglomerate, and very fine grained metatuff with well-developed slaty cleavage Volcanic breccia contains fragments with pyroxene crystals as much as one-half inch in diameter, and locally contains abundant black slate fragments Two interbedded units about 50 ft thick near base consist chiefly of black slate with interbedded tuff and conglomerate containing volcanic rock fragments in black silty slate matrix. 38— 900_________ Fine-grained green phyllite and phyllitic tuff with some interbedded black phyllite. 39— 4000________ Interbedded black slate, massive black siltstone, green slate, and fine-grained tuff. 40— 5000 +—. Thin-bedded black slate and phyllite, gra- phitic in part. Contains thinly interbedded metachert locally. 41— 1000____ Conglomerate, argillaceous sandstone, and massive black sandy siltstone. Conglomerate contains fragments of mafic volcanic rocks, some of which are amyg-daloidal, limestone, chert, and calcareous orthoquartzite. Fragments in some layers are nearly all volcanic rocks, in others, nearly all chert. 42— 1500______ Thinly interbedded chert and black car- bonaceous slate. Contains massive black quartz-mica phyllite. Contains knots of sericite, probably after andalusite, near small plutonic body southwest of abandoned cement plant. 43___ 90-200... Black aphanitic thin-bedded limestone. Locally crossbedded. 44— 1000_____ Thinly interbedded chert and black car- bonaceous phyllite. 45— 1500_____ Fine-grained hornblende-feldspar schist. Probably a metavolcanic rock. 46— ------------ Like unit 44. 47— 35_______ Dark-gray aphanitic limestone.DESCRIPTION OF MAP UNITS 61 Table 1.—Description of map units, Merced River (pi. 3)—Con. Table 2.—Description of map units, Tuolumne River {pi. 4)—Con. Map unit 48___ 49___ 50___ 51.1. Approximate thickness (feet) Lithology 4000 + ___ Thin-bedded chert interbedded with black carbonaceous phyllite and schist and massive black quartzose argillite. Chert layers range in thickness from about to 10 in. May include some quartz siltstone. In easternmost part, massive dark-gray sandstone with widely dispersed roundish quartz grains in very fine grained quartzose matrix. __________ Massive microcrystalline black quartz horn- fels in western part; thick-to thin-bedded quarz-mica hornfels with scattered round quartz grains, probably sandstone, in eastern part. __________ Massive argillite, thinly interbedded chert, and black phyllite or schist. Some feldspathic quartzite near west side. _________ Lile unit 50 but contains limestone, in part silicated or altered to tactite at eastern margin. Table 2.—Description of map units, Tuolumne River (pi. 4) Map unit 1_____ 2____ 3_____ 4 ___ 5 ___ 6____ 7____ 8____ 9____ Approximate thickness (feet) 500+... 4000. 1000+ — 500+.. . 100 Lithology Massive black aphanitic porphyritic rhyolite (?) with feldspar phenocrysts about 1 mm long. Possibly intrusive. West boundary not exposed. Volcanic and metavolcanic rocks. From west to east these are: fine-grained, massive greenstone; massive aphanitic freshlooking rhyolite or dacite porphyry with quartz phenocrysts 0.5 mm in diameter and feldspar phenocrysts 1 to 3 mm long; medium-bedded tuff and silicified ash; interlayered light- and dark-green vol-canics including some pyroclastics and massive rhyolite porphyry. Thin- to medium-bedded black slate and siltstones, rare graywacke. No graded beds found. Contains belemnites east of LaGrange powerhouse. Medium-green massive silicified aphanitic volcanic rock. Shows vertical columnar jointing locally. Schistose, light-green metatuff and volcanic breccia; contains fragments of silicified ash and volcanic rocks. Fault on east side. Medium-bedded silicified volcanic ash and tuff, lapilli tuff, and volcanic breccia. Schistose, silicified epidotized volcanic rocks, with some volcanic breccia, lapilli tuff, chert or silicified ash, and massive rhyolite!?) porphyry with feldspar phenocrysts. Sheared agglomerate or pillow lava consisting of ellipsoidal bodies of amygdaloidal aphanitic lava as much as 6 in. thick and 3 ft long. Contains rare fragments of medium-bedded tuff. Dark-green sheared pillow lava. Map unit 10--- 11--- 12.-. 13- -- 14- _. 15... 16— 17— 18— 19— 20___ 21___ 22___ 23 — 24... 25— 26— 27— 28... 29... 30... 31 — 32___ Approximate thickness (feet) 500 +___ 1500 +___ 1000+ — 300_____ 500 ±--- 300 ±--- 300 ±--- 2000 + — 3000+ „. 700 300 600 1000 + — 500 + ... Lithology Phyllitic light-colored volcanic rock; includes some tuff and slate or very fine tuff. Massive light-colored volcanic rock. Includes some tuff and volcanic breccia. Schistose on east. Thick-bedded pyroclastic rocks, massive light-gray silicified volcanic rocks and minor amygdaloidal lava. Thin- to thick-bedded light-green tuff and lapilli tuff with minor volcanic breccia. Thickness uncertain because of folding and faults. Thin-bedded black slate with interbedded tuff near west side. Interbedded graywacke with abundant volcanic rock fragments; graded beds on east side. Interbedded tuff and black slate. Massive dark-green amygdaloidal lava. Interbedded tuff and volcanic breccia. Black slate. Dark-green volcanic breccia, cut by grano-diorite sills and porphyritic dikes with feldspar phenocrysts. Schistose greenstone. Porphyritic rhyolite, locally sheared. Probable fault on west side. Massive dark-green metavolcanic rock. Massive dark-green metavolcanic rock containing small masses of hornblende gabbro. Dark-green massive porphyritic volcanic rock with feldspar phenocrysts 1 to 2 mm long. In part silicified. Eastern part amygdaloidal. Medium-green volcanic breccia. Amygdaloidal fragments abundant in places. Black slate with interbedded graywacke beds 1 in. to 1 ft thick. Graywacke beds are commonly graded. Conglomerate at base of formation consists of pebbles of volcanic rocks in black slate matrix. Conglomerate lenses about 45 ft above the base of the unit contain pebbles of vein quartz. Invertebrate fossils abundant 200 to 225 ft above the base of the unit. Thick-bedded coarse graywacke. Graded bedding not common. Only the basal contact exposed on line of traverse—upper contact projected from mapping by G. R. Heyl (written communication). Black slate; very poorly exposed along line of traverse. Contains medium-bedded interlayered graywacke and slate in sec. 20, T. 1 S., R. 15 E. Many graded graywacke beds. Thin- to medium-bedded black slate. Contains two small limestone lenses. Faulted on east and west sides. Schistose metavolcanic rock. Black carbonaceous quartzose mica schist. Schistose metavolcanic rock. Fault on east side.62 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT Table 2.—Description of map units, Tuolumne River (pi. 4)— Continued Map unit Approximate thickness (feet) 33... 2000 ± — 34... 200 35... 1500 36... 200 37... 1500 38... 1500+.. 39.. . 40.. . 0-500... 41... 200 42... 1100 43... 1800 44... 300 45... 200 46... 300 47... 1000 48.. . 1500 49... 3000 50... Lithology Black thin- to medium-bedded interlayered metachert and carbonaceous mica schist. Volcanic conglomerate and schistose meta-volcanic rock. Black quartzose mica schist; bedding obscure in most places. Interbedded limestone and black quartzose mica schist. Medium to very thin bedded interlayered chert and black carbonaceous schist with minor interbedded volcanic breccia and massive metavolcanic rock. Black thin- to medium-bedded interlayered chert and carbonaceous schist. Includes some medium-bedded quartz siltstone. Locally includes pods of limestone and calcareous quartz-mica schist. Limestone with interbedded black, carbonaceous quartzose mica schist. Includes limestone breccia at the mouth of the North Fork of the Tuolumne River. Dark-green tuff and volcanic breccia. Thins rapidly west of the mouth of the North Fork of the Tuolumne River. Includes thin zone of black quartzose carbonaceous schist at contact. Thin-bedded limestone. Thin- to medium-bedded interlayered black metachert and carbonaceous schist. Contains zone of limestone breccia in middle part of unit. Thick-bedded limestone—probably includes some dolomite. Massive very fine grained black metavolcanic rock. Medium-bedded limestone. Thin- to medium-bedded vitreous metachert and black, carbonaceous schist. Limestone and dolomite with interbedded metachert and black, carbonaceous schist. Thin- to medium-bedded black, carbonaceous quartzose mica schist. Coarse crystalline limestone and mediumgrained dolomite with interbedded quartzose mica schist. Black, carbonaceous quartzose mica schist and gneiss with rare pods of coarse crystalline limestone. Cut by granitic rock and pegmatite dikes. Table 3.—Description of map units, Stanislaus River (pi. 5) Approximate Map thickness unit (feet) Lithology 1____ ____________ Includes volcanic rocks of mafic or inter- mediate composition and rhyolite (?) porphyry. Most mafic and intermediate rocks are schistose, whereas most of the rhyolite is fresh looking and massive. Contacts of rhyolite cross schistosity of more mafic rocks at least locally. Table 3.—Description of map units, Stanislaus River (pi. 5)— Continued Approximate Map thickness unit (.feet) Lithology 2 ___ __________ Light-green rhyolite or dacite porphyry with aphanitic groundmass. Quartz pheno-crysts rare. Locally sheared. 3 ... 200_ - Dark-green massive metavolcanic rock, probably lava. 4 ___ 400_____Light-green volcanic breccia with abundant fragments of white chert or silicified ash. Feldspar crystals 1 to 2 mm long are abundant in groundmass. 5 ... ......... Light-green thick-bedded lithic tuff and volcanic breccia. 6 ___ __________ Coarse volcanic breccia with interbedded tuff. 7 ___ 2000______ Medium to very thick bedded medium- to dark-green coarse tuff. 8 ___ 200______ Dark-green volcanic breccia. 9 ___ 200_______ Massive amygdaloidal volcanic rock. 10 __ 3000______ Dark-green tuff and volcanic breccia. Medium to very thick bedded. Some tuff beds graded. Schistose except in eastern part. 11 __ 300_______ Medium-bedded silicified ash and tuff. Some graded beds. 12 __ 500+______ Black slate. 13— —........ Volcanic breccia and coarse tuff. 14 __ 500_______ Quartz-bearing rhyolite(?) tuff. 15 __ 200_______ Black slate. Includes a medium-bedded tuff and silicified ash sequence about 10 ft thick. 16 __ 300_______ Very fine grained tuff. 17 __ 1000______ Black slate with interbedded tuff layers 1 mm to about 1 m thick. 18 __ 400+______ Quartzose slate. 19 __ 1000______ Medium to very thick bedded volcanic breccia, tuff, and silicified ash with minor amygdaloidal lava. Contains abundant feldspar phenocrysts as much as 5 mm long. 20 __ 400_______ Interbedded black slate, volcanic breccia, and tuff. 21 __ 1400______ Black slate with interbedded graywacke. 22 __ 1000______ Coarse volcanic breccia; contains fragments of white chert or silicified ash on east side. Minor tuff. 23 __ 300______- Medium-bedded tuff and silicified ash. Graded beds common. 24 __ 1100______ Black slate with interbedded tuffaceous gray- wacke, graywacke, and minor conglomerate. Conglomerate has angular to well-rounded fragments of vein quartz, chert, and silicified ash(?), and angular fragments of black slate. 25 ___ __________ Coarse dark-green tuff on west; dark-green volcanic breccia in eastern part. 26 __ 300-------- Medium- to thick-bedded tuff and very fine tuff. 27 __ 5000 + __ Coarse yellow-green tuff with scattered thin layers of volcanic breccia. Bedding obscure. Contains zone about 25 ft thick of pillow lava in the NEJ4 sec. 32, T. 1 N., R. 13 E.DESCRIPTION OF MAP UNITS 63 Table 3.—Description of map units, Stanislaus River (pi. 5) — Continued Table 3.—Description of map units, Stanislaus River (pi. 5)— Continued Map unit 28... 29- -. 30- .- 31.-- 32.. . 33-.. 34.. . 35. .. 36 __ 37 __ 38 __ 39.. . 40.. . 41__. 42.. . 43.. . 44.. . 45.. . 46.. . 47.. . Approximate thickness (feet) 200 100.... 1000+.. 200.... 200+.. . 1000+.. 400_____ 500+ ... 500_____ 300_____ 500_____ Lithology Includes massive fine- to medium-grained metavolcanic rock; interbedded chlorite schist and black phyllite with rare limestone beds about 4 in. thick; foliated medium-bedded interlayered black chert, dark-gray phyllite, and meta volcanic rock; and dark-gray quartzose phyllite. Lithology and structure complex. Correlation uncertain; structure shown diagram-matically. Volcanic breccia and tuff, faulted against serpentine on west side. Medium-bedded black slate with interbedded tuff. Minor interbedded graywacke and conglomerate. Most pebbles are black slate and volcanic rocks; contains some chert and rare limestone pebbles. Volcanic rocks, chiefly pyroclastics. Bedding obscure. Zone of massive amyg-daloidal volcanic rock on west side. Pyroclastic rocks. Bedding and texture obscure. Black medium-bedded slate. Limestone float on surface. Coarse dark-green volcanic breccia. Locally contains augite phenocrysts as much as one-quarter inch in diameter. Coarse dark-green volcanic breccia. Black slate. Coarse dark-green tuff. Eastern contact concealed. Black slate with interlayered thick to very thick bedded graywacke. Some graywacke beds graded. Slate contains abundant plant remains in the SW% sec. 33, T. 2 N., R. 13 E , and invertebrate fossils in addition to plant remains in the NWJ4 sec. 3, T. 1 N., R. 13 E. Medium- to thick-bedded graywacke; graded bedding common. Black slate. Folded in NE^ sec. 3, T. I N., R. 13 E. Massive dark-green tuff. Black slate with interbedded tuff. Tuff. Interbedded tuff and black slate, poorly exposed at high-water mark. Contains tuff units as much as 300 ft thick, but in general slate and tuff zones are much thinner. Medium- to thick-bedded tuff with augite crystals. Coarse crystalline limestone and black carbonaceous quartzose schist. Conglomerate, containing pebbles of volcanic rocks, slate, chert, vein quartz, and limestone. Map unit 48- -. 49- .. 50.. . 51.-- 52.. . 53--. 54.-. % 55— 56... 57— 58— 59— 60 — 61 — 62— 63 — 64... 65 — Approximate thickness (feet) Lithology Black slate with minor interbedded tuff and chert. Conglomerate, like unit 47. Medium- to thick-bedded tuff with interbedded black slate. Slate most abundant in eastern part. Tuff beds commonly graded. Volcanic conglomerate. Contains scattered pebbles of slate. Melones fault zone—mostly green schist containing blocks of black carbonaceous quartzose schist, and volcanic conglomerate. Thin- to medium-bedded chert and black carbonaceous very fine grained schist. Mostly derived from medium to very thick bedded mafic volcanic rocks with some volcanic breccia, but western part is interbedded black carbonaceous schist and green schist; and central part is quartzose sericite schist. In part fine-grained and schistose, and in part massive with medium-grained gabbroic texture. Eastern part is massive amphibolite, becoming coarser grained to the east. Thin zone of black carbonaceous schist in SWJ4NEJ4 sec. 17, T. 2 N., R. 14 E. Brecciated and cut by granitic dikes near contact with granodiorite. Limestone. Limestone. Thin- to medium-bedded coarse- to mediumgrained limestone. Aphanitic dark-green metavolcanic rock. Massive limestone, in part coarse-crystalline, and massive dolomite or dolomitic limestone. Medium- to thick-bedded dolomite and limestone. Contains interbedded black carbonaceous quartzose schist on west side. Contains thin zone of quartz-mica-magnetite schist, probably metarhyolite, in the NEJ4NWJ4 sec. 34, T. 3 N., R. 14 E. Massive limestone. Includes thin zone of coarse calcareous quartz-biotite schist on east side. Alphanitic dark-green metavolcanic rock. Coarse crystalline limestone. Thin- to medium-bedded chert and black, carbonaceous quartz-mica schist. Chert is metamorphosed and locally vitreous. Includes some limestone. Bedding destroyed in most places. Lineation extensively developed. Like unit 64.64 STRATIGRAPHY AND STRUCTURE, SIERRA NEVADA METAMORPHIC BELT Table 4.—Description of map units, Calaveras River (pi. 6) Table 4.—Description of map units, Calaveras River (pi. 6)—Con. Map unit 2 3 ___ 4 ___ 5 ___ 6 ___ 8____ 9____ 10... 11... 12... 13.. . 14.. . 15.. . 16.. . 17.. . 18.. . Approximate thickness (feet) 3000+.. 200.... 1000+.. 1000+.. 800..... 1000_____ 1500____ 400 150 300 1800+.. 1500____ 800..... 200..... 500..... Lithology Massive quartz-feldspar porphyry, containing subhedral quartz phenocrysts as much as 2 mm in diameter, euhedral plagioclase phenocrysts as much as 1.5 mm in diameter, and ferromagnesian minerals in a white very fine grained groundmass. Very fine grained dark-green massive por-phyritic lava with abundant feldspar phenocrysts averaging 1 mm in diameter. Feldspar partly replaced by epidote. Includes vertical rhyolite(?) dike about 15 ft thick with horizontal columnar joints. Dike rock contains quartz phenocrysts 3 mm in diameter and plagioclase phenocrysts 2 mm long in aphanitic dark-gray groundmass. Aphanitic light-green volcanic rock. Massive fine-grained drark-green volcanic rock. Dark-green coarse metamorphosed volcanic breccia. Massive dark-green porphyritic metavol-canic rock, in part brecciated. Phenocrysts are of augite or amphibole after augite and are 1 to 2 mm in diameter. Includes zones of volcanic breccia and pillow lava. Massive dark-green metavolcanic rock, mostly brecciated. Some breccia contains amygdaloidal fragments widely scattered among nonamygdaloidal fragments. Breccia fragments are separated by epi-dote-rich seams. Some nonbrecciated parts that are not amygdaloidal contain amygdaloidal bodies 1 to 2 ft long. Contacts with unit 8 are poorly defined. Massive dark-green metavolcanic rock like unit 6, [but] contains zones of volcanic breccia and pillow lava. Volcanic breccia containing some coarse tuff zones with some graded beds. Massive dark-gray rhyolite(?) porphyry containing quartz and feldspar phenocrysts about 1 mm in diameter. Dark-green coarse volcanic breccia. Includes some massive lava. Coarse porphyritic rhyolitic(?) volcanic breccia with abundant feldspar phenocrysts and widely scattered quartz phenocrysts. Massive dark-gray porphyritic rhyolite(?). Contains common plagioclase phenocrysts 1 to 2 mm in diameter and widely scattered quartz phenocrysts. Light-green volcanic breccia, tuff, lapilli tuff, and silicified ash. Black slate with some interbedded gray-wacke and minor interbedded tuff. Pyroclastic rocks. Black slate. Pyroclastic rocks. Map unit 19.. . 20.. . 21... 22... 23... 24.. . 25.. . 26... 27... 28... 29... 30... 31— 32... 33— 34.. . 35.. . Approximate thickness (feet) 2000+.. 3000___ 500+-.- 500 ±--- 400+---- 1000____ 2000+ — 1000 + — 2000+— Lithology Black slate with some interbedded gray-wacke—poorly exposed. Schistose metavolcanic rocks. Includes some metatuff and feldspar porphyry on east side. Poorly exposed. Metamorphosed volcanic breccia containing phenocrysts probably composed of amphibole after augite. Tuff, with some interbedded volcanic breccia. Slate, in part quartzose. Weathers gray. Minor interbedded very coarse gray-wacke. Tuff. Not exposed in river. Probably dark-gray quartzose slate and sericitic chert as exposed south of river. Well-indurated dark-gray coarse sandstone and fine conglomerate composed chiefly of fairly well-rounded fragments of vein quartz and chert. Slate fragments common. Coarse dark-green volcanic breccia containing augite phenocrysts more than 2 mm in diameter. Dark-gray quartzose slate and sericitic chert. Includes scattered zones of fine conglomerate containing fragments of chert, slate, and volcanic rocks. Conglomerate is possibly infolded. Dark-green massive porphyritic dike rock with tabular saussuritized plagioclase phenocrysts. Not exposed on river banks. Coarse dark-green volcanic breccia containing augite phenocrysts more than 2 mm in diameter. Thin- to medium-bedded black slate with interbedded tuff. Thin- to medium-bedded black slate with interbedded tuff. Includes lenticular conglomerates and possibly some chert. Green schist. Green schist. Black medium-bedded interlayered chert and carbonaceous schist. Some interbedded limestone. Table 5.—Description of map units, Mokelumne River (pi. 7) Approximate Map thickness unit (feet) 1_________ 1500+... 2____ 700 Lithology Massive dark-green metavolcanic rocks; includes lavas and volcanic breccia. Pillow lava present in SWl/4sec. 5, T. 4 N, R. 10 E. Some lavas and volcanic breccia in eastern part contain feldspar phenocrysts. Poorly exposed. Massive feldspar (rhyolite?) porphyry with very dark gray aphanitic groundmass. Quartz phenocrysts 2 mm and less in diameter.DESCRIPTION OF MAP UNITS 65 Table 5.—Description of map units, Mokelumne River (pi. 7)— Continued Table 5.—Description of map units, Mokelumne River (pi. 7)— Continued Map unit 3____ 4____ 7____ 8____ 9____ 10... 11... 12... 13.. . 14.. . 15.. . 16.. . 17.. . 18.. . 19... 20... 21... 22... 23— 24... Approximate thickness (feet) 1500_______ 200 600.... 600+.. . 800..... 500+____ 200..... 500+____ 1000 ± — 500..... 600..... 1500 + ... 1000 + ... 2000____ 1500.... 700+.. . 1500+___ 200_____ 2500+.. 500 + ... 200 + .._ Lithology Thin- to very thick bedded medium-green rhyolite (?) tuff, lapilli tuff, and light-green silicified ash. Tuff contains abundant feldspar phenocrysts and scattered quartz grains. Graded beds common. Medium-green volcanic breccia. Feldspar crystals and crystal fragments common and quartz grains scattered in groundmass. Tuff and lapilli tuff, like unit 3. Thin- to medium-bedded black silty slate with some interbedded graywacke. Contains a tuff horizon about 15 ft thick near the east side. Foliated medium- to thick-bedded metaandesite and metadacite pyroclastics. Foliated metamorphosed volcanic breccia. Includes minor intrusive rhyolite porphyry. Medium- to dark-green basaltic pillow lavas. Dark-green foliated pyroclastic rocks. Interbedded rhyolite and mafic or intermediate pyroclastic rocks. Contains many vertical quartz veins. Porphyritic rhyolite or dacite. Mafic or intermediate volcanics. Porphyritic rhyolite, foliated in eastern part. Quartz phenocrysts are as much as 2 mm in diameter. Vein quartz, sericite schist, and much-altered strongly foliated volcanic rocks in eastern part. Light-green foliated fine- to medium-grained tuff. Interbedded medium- to thick-bedded tuff and black slate. Predominantly slate. Tuff beds commonly graded. Medium- to very thick-bedded fine tuff, lapilli tuff, and volcanic breccia. Black slate. Poorly exposed around margins of reservoir. Tuff with minor interbedded slate. Includes chlorite schist and sericite schist in the NE^SWK sec. 23, T. 5 N., R. 10 E. Thickness measured in sec. 15, T. 5 N., R. 10 E. Porphyritic foliated metarhyolite or metadacite. Dark-green tuff and volcanic breccia in western part; massive metavolcanic rocks with some interbedded metachert and metatuff in eastern part. On northwest side of reservoir; from west to east: coarse massive dark-green metavolcanic rock, massive amygdaloidal lava. Exposed on southeast side of reservoir, light-gray thin- to medium-bedded very fine grained limestone about 30 ft thick, and bedded tuff. On southeast side of reservoir, undifferentiated volcanic rocks. Tuff with interbedded conglomerate. Quartzose phyllite, weathers light gray. Map unu 25- - 26— 27— 28— 29... 30— 31_ 32— 33— 34— 35— 36— 37— 38— 39— . 40— . 41 _ 42 _ 43 _ 44... 45.. . 46— 47.. . Approximate thickness (feet) 500 + ... 100 + -.- 1500+.. 500_____ 500_____ 1500____ 100 + -.-1300____ 500+-.- Lithology Interbedded black slate, conglomerate, graywacke, and tuff. Limestone. Interbedded black slate and tuff on west. Coarse dark-green volcanic breccia with augite phenocrysts as much as 5 mm in diameter on east. Not exposed on northwest side of reservoir. Quartzose phyllite, weathers light gray to light brown. Contains light-green foliated amygdaloidal lava about 6 ft thick and limestone about 20 ft thick in the SE)+ NE)i sec. 18, T. 5 N., R. 11 E. Limestone is calcarenite and breccia with limestone and dolomite fragments. No non-calcareous detrital grains noted. Coarse dark-green volcanic breccia containing augite crystals more than 5 mm in diameter. Tuff, poorly exposed. Medium- to thick-bedded tuff with inter-layered black slate. Medium- to very thick-bedded tuff and lapilli tuff. Black slate with interbedded tuff. Medium- to very thick-bedded tuff and lapilli tuff. Tuff with interbedded conglomerate or breccia. Conglomerate consists chiefly of fragments of volcanic rock in a tuffaceous matrix, but also contains scattered fragments of black slate and vein quartz. Most volcanic rock fragments are angular. Black slate with interbedded tuff, very coarse graywacke, and fine conglomerate. Medium-bedded coarse to fine tuff. Black slate with interbedded tuff. Tuff with interbedded slate. Poorly exposed. Chlorite schist. Fault zones contain thin units of black carbonaceous quartzose schist, probably derived from the Calaveras formation. Black carbonaceous quartzose phyllite with lamellae of metatuff. Interbedded tuff and black phyllite. Includes chlorite schist on east side. Chlorite schist, coarse crystalline limestone, and black quartzose schist. Intruded by gabbro dikes. Medium-bedded metachert and black carbonaceous quartz-mica schist. Possibly includes quartzose siltstone. Includes chlorite schist and limestone pods on east side. _________ Interbedded black carbonaceous schist, metachert, and possibly quartzose siltstone. _________ South part like unit 45, north part coarse crystalline limestone. _________ Like unit 45.66 STRATIGRAPHY AND STRUCTURE SIERRA NEVADA METAMORPHIC BELT Table 5.—Description of map units, Mokelumne River (pi. 7)— Continued Table 7.—Description of map units, Cosumnes River (pi. 8)— Continued Map unit 48— 49— 50— Approximate ihicknett (feet) Lithology __________ Coarse crystalline limestone with interbedded quartz-mica schist. __________ Like unit 45. __________ Coarse crystalline limestone. 51___ __________ Like unit 45. Table 6.—Description of map units, Jackson Creek Unit No. Thkknete Lithology 1 ________________ Dark-green metavoleanie rock. 2 ________________ Coarse- to fine-bedded tuff. 3 ________________ Interbedded black slate and graywacke. 4 ________________ Interbedded black slate and tuff. 5 ________________ Tuff and porphyritic volcanic breccia with pyroxene phenocrysts. 6 ________________Light-green buff-weathering quartzose phyl- lite. 7 ________________ Gray-green fine-grained volcanic breccia with feldspar phenocrysts less than 2 mm long. 8 ________________Light-green quartzose phyllite with thin in- terbedded conglomerate. Conglomerate contains pebbles of chert, slate, gray-gray wacke, and volcanic rocks. 9________________ Interbedded thin-bedded gray quartzose phyllite and black chert. Includes limestone lens, probably bioclastic, with Foraminifera locally abundant, in eastern part. 10 ______________Thin-bedded black chert, with phyllite partings. Limestone lens on east side. 11 ______________Interbedded coarse and fine conglomerate, containing pebbles of volcanic rocks, chert, vein quartz, quartz schist, limetone, and rare granitic (?) rocks. 12 ______________Volcanic breccia. Table 7.—Description of maps units, Cosumnes River (pi. 8) Approximate Map thickneet unit (feet) Lithology 1____ ___________ Coarse dark-green volcanic breccia contain- ing augite phenocrysts as much as 5 mm in diameter. Probably equivalent to unit 3. Exposed in upper part of river banks; not shown on map. 2 ___ __________ Massive dark-green very fine grained lava. Base not exposed. 3 ___ 1100______ Coarse dark-green volcanic breccia. Frag- ments have quartz amygdules as much as .1 in. in diameter, and chalcedony amygdules as much as l}i in. in diameter. 4 ___ 400_______ Massive dark-green very fine grained lava. Contains some epidotized breccia. Map unit 5_____ 6___ 7 __ 8 __ 9___ lO- 11— 12— 13— . 14— 15— 16— 17... 18— 19— 20— 21 — 22— 23— 24- 25— 26— 27- Approximate thickness (feet) Lithology 500+_____ Medium-green rhyolite volcanic breccia and coarse tuff. Contains abundant roundish quartz grains mostly less than 1 mm in diameter. 500+_____ Fine to coarse medium- to very thick-bedded tuff and lapilli tuff. 1000_____ Black slate with interbedded graywacke. Includes fine-grained metatuff on east side. 200 +____ Graywacke. 200+_____ Black slate with interbedded graywacke. Includes thin crystal tuff unit on east side. 1000_____ Thick- to thin-bedded graywacke, minor slate. Graded beds common. Coarse to fine tuff about 100 ft thick at western side. 2000_____ Black slate with minor interbedded gray- wacke. Eastern part fossiliferous. 1000+ - - Medium- to thick-bedded tuff and lapilli tuff with minor silicified ash. Graded beds common. 200______ Black slate with interbedded tuff. 500______ Volcanic breccia and tuff. 300______ Black slate with interbedded tuff. 1000_____ Medium-bedded tuff. 700______ Medium-bedded interlayered graywacke and black slate. 600______ Tuff and volcanic breccia. 300______ Black slate with interbedded tuff. 500______ Mafic or intermediate pyroclastic rocks. 400_____- Porphyritic rhyolite (?), including massive lava and pyroclastic rocks. 1200_____ Mafic or intermediate pyroclastic rocks with minor pillow lava. 700+_____ Massive porphyritic rhyolite (?) with some interlayered mafic or intermediate volcanic rocks. 3000+-- Massive fine- to medium-grained tuff with some volcanic breccia, lapilli tuff, and pillow lava. Zone of lapilli tuff and volcanic breccia 80 ft thick about 1600 ft north of the south quarter-corner of sect 27, T. 8 N., R. 9 E., contains feldspar crystals as much as 1 in. in diameter and abundant scoria fragments. Pillow lava is porphyritic, with saussuritized euhedral feldspar phenocrysts about 5 mm long. Zone of black carbonaceous slate about 10 ft thick occurs about 1500 ft west of the east quarter-corner of sec. 27, T. 8 N., R. 9 E. 1100_____ Porphyritic lava containing saussuritized euhedral feldspar phenocrysts and clusters of phenocrysts as much as 8 mm long in aphanitic matrix. Much of this lava has pillow structure. 1000+ - - Mafic or intermediate tuff and volcanic breccia. Some graded beds and cross-laminated beds. 1000+ - - Tuff in western part, green schist in eastern part.DESCRIPTION OF MAP UNITS 67 Table 7.—Description of map units, Cosumnes River (pi. 8)— Approximate Map thickness Continued Map Approximate thickness Continued unit (.feet) Lithology unit (feet) Lithology 28.-- Metavolcanic rocks, including massive greenstone, green schist, and medium-bedded metatuff. 42.-. 300 Interbedded black slate, volcanic conglomerate, and tuff. 29--. Green schist. 43--. 200 Fine conglomerate. 30.-- Medium-bedded fine- to coarse-grained tuff. 44... 500 Coarse conglomerate. 31--. 700+--. Thick- to very thick-bedded conglomerate with interbedded sandstone on east side. 45.-. 600 Interbedded black slate, silt-stone, with subordinate 32... 1000+-- 33.-. 1000+.. 34.. . 35.. . 36.. . 37.. . 38.. 600. 300. 39. 40. 1000+.. 41... 400- Conglomerate consists mostly of fragments of volcanic rocks but also contains vein quartz, black and gray chert, and black slate, in a tuffaceous matrix. Fragments of volcanic rocks and slate are angular to well rounded; chert and quartz fragments are well rounded. Some beds contain pebbles of fine-grained gray limestone. The sandstone is well indurated and consists mostly of well-rounded quartz grains, but contains some less resistant grains. Thin- to medium-bedded interlayered fine to coarse graywacke and black slate. Gray-wacke beds commonly graded. Contains rare limestone beds less than 1 ft thick. Near west side, includes massive porphy-ritic lava or hypabyssal intrusive rock with feldspar phenocrysts. Eastern part includes volcanic rocks about 200 ft thick. Interbedded graywacke, massive gray slate, and fine-grained tuff or tuffaceous sandstone. Tuff and volcanic breccia. Volcanic rocks. Black slate with interbedded pyroclastic rocks. Well-indurated dark-gray sandstone consisting chiefly of quartz grains. Chert and black slate grains are common. Grains are angular to well rounded. Contains some fine conglomerate. Black medium-bedded interlayered. chert and carbonaceous phyllite. Some quartz schist. Quartz sandstone, like unit 37. Black slate. Contains beds of black argillaceous very fine grained limestone 1 in. to 1 ft. thick on southeast side of river near the north quarter-corner of sec. 28, T. 8 N., R. 10 E. Contains poorly exposed fine conglomerate with interbedded arenaceous limestone (calcarenite) on the southwest side of the river near the center of sec. 21, T. 8 N., R. 10 E. Conglomerate contains well-rounded granules of volcanic rocks, vein quartz, chert, and quartzite. Very fine-grained dark-green massive meta-volcanic rock. 46— 300. 47... 800. 48-.. 1000____ 49--- 1500... O 03 fl 03 .5 A 03 « 5 aT g M ® o _ o cj 3 O' T3 G o3 ® s £ « 9 SX) s a 1 a .a T3 _ o W fig < Ld olq: 1y I o o N o z Ld U o ► U) U) < CL => (J o N r o LD Ld 2 Jp Penon Blanco volcanics Upper part is dark-green mafic volcanic breccia and tuff, in part porphyritic (pyroxene phenocrysts); lower part is dark-green mafic lava and some pillow lava U O J N ^ O CO Ld Eastern fault block Undifferentiated argillaceous member and chert member Pcc, argillaceous rocks and siltstone and some thin-bedded chert; includes minor lenticular limestone in southwestern part; thin-bedded chert and black carbonaceous slate containing minor lenticular mafic pyroclastics elsewhere Peel, limestone Z < CL Ld a Ficv Upper volcanic member Mafic pyroclastic rocks, in part porphyritic (augite phenocrysts). Minor pillow lava and black slate Ld < 0. u o N V o Ld _l < a. ftcc Clastic member Thin-bedded dark-gray slate and graywacke, thin-bedded chert, and tuff; minor limestone lenses GEOLOGIC MAP OF THE WESTERN SIERRA NEVADA BETWEEN THE MERCED AND COSUMNES RIVERS, CALIFORNIA SCALE 1:316 800 5 10 15 20 MILES 5 0 5 10 15 20 25 KILOMETERSUNITED STATES DEPARTMENT OF THE INTERIOR GEOLOGICAL SURVEY PROFESSIONAL PAPER 410 PLATE 3 EXPLANATION LITHOLOGY Various symbols are combined to show interbedded lithologic types SEA LEVEL - 5000' - 10,000' —1 10,000' GEOLOGIC MAP AND COMPOSITE SECTION ALONG THE MERCED RIVER, CALIFORNIA & SCALE 1:62 500 1 3 MILES 1 4 KILOMETERS I—I H H H R~ D CONTOUR INTERVAL 50 AND 100 FEET DATUM IS MEAN SEA LEVELUNITED STATES DEPARTMENT OF THE INTERIOR GEOLOGICAL SURVEY PROFESSIONAL PAPER 410 PLATE 4 EXPLANATION g I QT Volcanic rocks and alluvial gravels > £ < LITHOLOGY Various symbols are combined to show interbedded lithologic types F- Z UJ ^ CE < i_ y < o KJp ~N Western fault block _________A________ Plutonic rocks Ju U ^ w vi < a. D Jpc Jch Peaslee Creek volcanics Copper Hill volcanics Ultramafic rocks Central fault block __________A_________ Eastern fault block __________A_________ Jmf Js Jm Merced Falls slate Salt Spring slate Mariposa formation Jg Jp Gopher Ridge volcanics Penon Blanco volcanics Gw\\w\\\ v,\\W\w Tuff * * 7 ^ 0 ■*/ * t ■ V Slate f» Schistose metavolcanic rock \\Vx\x\ 'G\nsXY\ Contact Dashed where approximately located; short dashed where projected in plane of observation; dotted where inferred Volcanic breccia MFTt Sandstone I Schistose metasedimentary rock, generally showing marked lineation HP }'| I fill .) h v \\ l S ) h n \\ I H h h s, x O U) a. D ~i Lava Thin-bedded chert and slate Gneiss vr V- /'TFj\ Win1 a Fault Dashed where approximately located; queried where doubtful; short dashed where projected in plane of observation Shear zone Strike and dip of beds Strike of vertical beds Massive volcanics, massive and pyroclastic types not differentiated Limestone Massive metavolcanic rock Map Top of beds Section o*«»o « 0 0 0 « O O V 0 ° 0 % o O \ ° ° Vo°o • °.» O 0 Q o O Massive porphyritic volcanic or hypabyssal rock Conglomerate Determination based on graded bedding. Plotted along strike with point of observation so Strike and dip of joints 50 Strike and dip of schistosity Strike of vertical schistosity Strike and dip of parallel bedding and schistosity Strike of parallel vertical bedding and schistosity Bearing and plunge of minor fold axis --------------------*■ 20 Bearing and plunge of lineation 60 7^5 Strike of vertical schistosity, showing bearing and plunge of lineation Strike of vertical schistosity, showing bearing and plunge of lineation Geology mapped b L Q V Ciark 1953-54 __________ir----1 SU' X-—* s' Base from U. S. Geological Survey maps of the Merced Falls, Coulterville, Sonora, and Tuolumne 15' quad-rangles, California Note: Numbers refer to units described in Table 2 __ Approximate altitude of river surface SEA LEVEL - 5000' u 10,000' GEOLOGIC MAP AND COMPOSITE SECTION ALONG THE TUOLUMNE RIVER, CALIFORNIA SCALE 1:62 500 1 3 MILES .5 0 1 4 KILOMETERS CONTOUR INTERVALS 50 AND 100 FEET DATUM IS MEAN SEA LEVELUNITED STATES DEPARTMENT OF THE INTERIOR GEOLOGICAL SURVEY PROFESSIONAL PAPER 410 PLATE 5 QT Volcanic rocks and alluvial gravels KJh Porphyritic dike KJp •„Hs .a. o Western fault block _________A._______ Plutonic rocks >- a. | < Q t cr < uj < I- 3 o in 2 O < ~z < EXPLANATION LITHOLOGY Various symbols are combined to show interbedded lithologic types • • • • —5“ • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . , . L\VXVY-T f \\\\ % ^ n\\v \\W v 'W«w Tuff Slate Schistose metavolcanic rock b '<1 .« 4~ ■ a • ° 1 Volcanic breccia i ii u 1 U wl Sandstone Schistose metasedimentary rock, generally showing marked lineation Ju Jch Copper Hill volcanics - Ultramafic rocks Central fault block and Melones fault zone ___________________a___________________ Eastern fault block __________A_________ Js Jm Jmb Salt Spring slate Mariposa formation Jm, tujf, black slate ami some graywacke Jmb, Brower Creek volcanic member , Jgy Jgf Jg<5: Jp Gopher Ridge volcanics Jg, mafic volcanic rocks; Jgf, felsic volcanic rocks; Jgu, undifferentiated mafic and felsic volcanic rocks Pefion Blanco volcanics gs V*V\ V Pif Lava Thin-bedded chert and slate Massive metavolcanic rock x * n X x X x X A X x A „ X X * U in vc. D ~> Massive volcanics, massive and pyroclastic types not differentiated Limestone o' 0 00 o p 0n Massive porphyritic volcanic or hypabyssal rock Conglomerate Contact Dashed where approximately located; short dashed where projected in plane of obser-iration: dotted where inferred Fault Dashed where approximately located; queried where doubtful; short dashed where projected in plane of observation Shear zone 60^ Strike and dip of beds -P- 85 Strike and dip of overturned beds 90 Strike of vertical beds Map Section Top of beds Determination based on graded bedding. Plotted along strike with point of observation 70 Strike and dip of schistosity Strike of vertical schistosty Bearing and plunge of lineation <6 65 Strike and dip of schistosity, showing bearing and plunge of lineation Strike of vertical schistosity, showing bearing and plunge of lineation Bearing and plunge of minor folds showing trace of bedding Strike and dip of parallel bedding and schistosity Pillow lava Green schist Pc Pcc Peel Calaveras formation Interbedded chert and pliyllite, minor limestone Pcc, interbedded black micaceous schist and chert; Peel, limestone MytfA ‘ tTpi •■■'.'•■•■',7? Xy-'V.j>f,: / wzmSm ftev Mafic volcanic rocks v . ^ vip mm „ ..y f jijfTv, \ yY/ii T.m:X u ' ,>\n tel U 5 \ CrZ ■ cY \ ,.V§Sfwimmi wAlimm j n —1 s JM> (lL A mm 1 i i fypCi k/JJSL KKJt' r 39 Jm 43.yi8: feliliulLIL v, /Jmb Jm us Peel jif H , ’*QT»Tii I'Efi jej ' >3 \ 11 i r i -'f i 24' / 'jwj 47 Jch :J f ,,, - Jg i (! i I : OH i 8 ,«/ Js / t7s * ^ 4. •% r 1' ' Is 1 '"PP"'.--^-7—.........rh: - r./---» ■ni- 33 M JpdO £---1/ i «* V Base from U, S. Geological Survey maps of the Copperopolis. San Andreas, Sciora, and Columbia 15' quadrangles. California 'Tm. ^ y11~ .) Hi-,-'- -y 2 , \ f 'f //A X ■ .■Fi.xt i ! • . r. T 4 Y: "-; ■ojeo ' / Pcc Geology mapped by L. 1953 '■-.'•y- ,.0/ ^ vi rA'■ "W . Bear Mountains fault zone Melones fault zone Note: Numbers refer to units described in Table 3 Approximate altitude of river surface - SEA LEVEL - 5000' L 10.000' GEOLOGIC MAP AND COMPOSITE SECTION ALONG THE STANISLAUS RIVER, CALIFORNIA SCALE 1:62 500 1 y2 o J________________2_______________3 MILES F^C I—I I—I I—I I—II— ' ---1 I I 1 .5 0 1 2 3 4 KILOMETERS hhhhh --------I -1 1-------' CONTOUR INTERVALS 25, 50, 80 AND 100 FEET DATUM IS MEAN SEA LEVELUNITED STATES DEPARTMENT OF THE INTERIOR GEOLOGICAL SURVEY PROFESSIONAL PAPER 410 PLATE 5 QT Volcanic rocks and alluvial gravels KJh Porphyritic dike KJp •„Hs .a. o Western fault block _________A._______ Plutonic rocks >- a. | < Q t cr < uj < I- 3 o in 2 O < ~z < EXPLANATION LITHOLOGY Various symbols are combined to show interbedded lithologic types • • • • —5“ • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • . , . L\VXVY-T f \\\\ % ^ n\\v \\W v 'W«w Tuff Slate Schistose metavolcanic rock b '<1 .« 4~ ■ a • ° 1 Volcanic breccia i ii u 1 U wl Sandstone Schistose metasedimentary rock, generally showing marked lineation Ju Jch Copper Hill volcanics - Ultramafic rocks Central fault block and Melones fault zone ___________________a___________________ Eastern fault block __________A_________ Js Jm Jmb Salt Spring slate Mariposa formation Jm, tujf, black slate ami some graywacke Jmb, Brower Creek volcanic member , Jgy Jgf Jg<5: Jp Gopher Ridge volcanics Jg, mafic volcanic rocks; Jgf, felsic volcanic rocks; Jgu, undifferentiated mafic and felsic volcanic rocks Pefion Blanco volcanics gs V*V\ V Pif Lava Thin-bedded chert and slate Massive metavolcanic rock x * n X x X x X A X x A „ X X * U in vc. D ~> Massive volcanics, massive and pyroclastic types not differentiated Limestone o' 0 00 o p 0n Massive porphyritic volcanic or hypabyssal rock Conglomerate Contact Dashed where approximately located; short dashed where projected in plane of obser-iration: dotted where inferred Fault Dashed where approximately located; queried where doubtful; short dashed where projected in plane of observation Shear zone 60^ Strike and dip of beds -P- 85 Strike and dip of overturned beds 90 Strike of vertical beds Map Section Top of beds Determination based on graded bedding. Plotted along strike with point of observation 70 Strike and dip of schistosity Strike of vertical schistosty Bearing and plunge of lineation <6 65 Strike and dip of schistosity, showing bearing and plunge of lineation Strike of vertical schistosity, showing bearing and plunge of lineation Bearing and plunge of minor folds showing trace of bedding Strike and dip of parallel bedding and schistosity Pillow lava Green schist Pc Pcc Peel Calaveras formation Interbedded chert and pliyllite, minor limestone Pcc, interbedded black micaceous schist and chert; Peel, limestone MytfA ‘ tTpi •■■'.'•■•■',7? Xy-'V.j>f,: / wzmSm ftev Mafic volcanic rocks v . ^ vip mm „ ..y f jijfTv, \ yY/ii T.m:X u ' ,>\n tel U 5 \ CrZ ■ cY \ ,.V§Sfwimmi wAlimm j n —1 s JM> (lL A mm 1 i i fypCi k/JJSL KKJt' r 39 Jm 43.yi8: feliliulLIL v, /Jmb Jm us Peel jif H , ’*QT»Tii I'Efi jej ' >3 \ 11 i r i -'f i 24' / 'jwj 47 Jch :J f ,,, - Jg i (! i I : OH i 8 ,«/ Js / t7s * ^ 4. •% r 1' ' Is 1 '"PP"'.--^-7—.........rh: - r./---» ■ni- 33 M JpdO £---1/ i «* V Base from U, S. Geological Survey maps of the Copperopolis. San Andreas, Sciora, and Columbia 15' quadrangles. California 'Tm. ^ y11~ .) Hi-,-'- -y 2 , \ f 'f //A X ■ .■Fi.xt i ! • . r. T 4 Y: "-; ■ojeo ' / Pcc Geology mapped by L. 1953 '■-.'•y- ,.0/ ^ vi rA'■ "W . Bear Mountains fault zone Melones fault zone Note: Numbers refer to units described in Table 3 Approximate altitude of river surface - SEA LEVEL - 5000' L 10.000' GEOLOGIC MAP AND COMPOSITE SECTION ALONG THE STANISLAUS RIVER, CALIFORNIA SCALE 1:62 500 1 y2 o J________________2_______________3 MILES F^C I—I I—I I—I I—II— ' ---1 I I 1 .5 0 1 2 3 4 KILOMETERS hhhhh --------I -1 1-------' CONTOUR INTERVALS 25, 50, 80 AND 100 FEET DATUM IS MEAN SEA LEVELUpper Jurassic UNITED STATES DEPARTMENT OF THE INTERIOR GEOLOGICAL SURVEY PROFESSIONAL PAPER 410 PLATE 6 EXPLANATION QT - ------------------------------- Volcanic rocks and alluvial gravels LITHOLOGY Various symbols are combined to show interbedded lithologic types Tt ■m Tuff Slate Schistose metavolcanic rock Western fault Block and Bear Mountains fault zone < ! A ' Qiiartz-feldspat* porphyry Porphyritic felsic dike rock JQP i ■■xi KJpd mM w.V'V.' Volcanic breccia Sandstone Feldspar porphyry dike Schistose metasedimentary rock, generally showing marked lineation Ju Jch i Copper Hill volcanics Ultramafic rocks Central fault block and Melones fault zone ____________A___________ Js Salt Spring'slate Jm Jml Jeu Mariposa formation Jm, black slate containing interbedded tuff; Jmb, Brower Creek volcanic member; Jml, limestone Epiclastic rocks of uncertain stratigraphic position Jg, Gopher Ridge Volcanics mafic volcanic nfA'* Jgf, felsic volcanic rocks VWX x \i ^ a n B 4 0 * Lava Thin-bedded chert and slate Feldspar porphyry x v a x „ X x x , * y. X v >< v * x * x X xx o if) . in X < tr 3 “1 Massive volcanics, massive and pyroclastic types not differentiated Limestone = s\ «' , * '' ^ V ^ // - n _ w ❖ # " * 0°0°.° ^°o°0 00oo0 °0°co 0 o o °n ° O O ” Massive porphyritic volcanic or hypabyssal rock Conglomerate Pillow lava V & -1 ' I • • - •< * •• 4-i K \y |>V V X r ~ %,• ?•:- ■,, gs Green schist Pc Eastern fault block _________a_________ Calaveras formation Pc, epiclastic rocks; Pci, limestone ftcc ftccl Z < Of LU 0. Calaveras formation Undifferentiated argillaceous member and chert member; ftcc, interbedded black micaceous schist and chert; Ricci, limestone y o N ^lil _J < CL , ^ i Contact Dashed where approximately located; short dashed where projected in plane of observation; dotted where inferred Fault Dashed where approximately located; queried where doubtful; short dashed where projected in plane of observation ---------i------- Syncline, showing trace of axial plane so Strike and dip of beds 60 Strike and dip of overturned beds _t_ _i_ O fa Map Section Tops of beds Determination based on: a, graded bedding; b, relation between bedding and cleavage. Plotted along strike with point of observation 25 pu Generalized strike and dip of folded beds 80 Strike and dip of cleavage 60 s' Strike and dip of schistosity Strike of vertical schistosty 77 Strike of vertical schistosity, showing bearing and plunge of lineation Base from U. S, Geological Survey maps of the Valley Springs and San Andreas 15' quadrangles, California Geology mapped by L, D, Clark, 1953, and by A. A. Stromquist, 1950-52 Bear Mountains fault zone ____________A------------ Melones fault zone Note: Numbers refer to units described in Table 4 GEOLOGIC MAP AND COMPOSITE SECTION ALONG THE CALAVERAS RIVER, CALIFORNIA % SCALE 1:62 500 1 3 MILES ~F=r 1 .5 1 4 KILOMETERS CONTOUR INTERVALS 25 AND 50 FEET DATUM IS MEAN SEA LEVELUNITED STATES DEPARTMENT OF THE INTERIOR GEOLOGICAL SURVEY PROFESSIONAL PAPER 410 PLATE 7 EXPLANATION CO CO C 5*. § QT Volcanic rocks and alluvial gravels LITHOLOGY Various symbols are combined to show interbedded lithologic types \ \ \ *. • \ A \ \ N\ \ " \\ in KJp X y u w in a. 5 > c/> o 0 *. \ % *• ’• • LLV.L f < _ ui rr Q n Tuff Slate Schist.n.c Contact Dashed where approximately located; short dashed where projected in plane of observation; dotted where inferred -Q- 50 Western fault block _________a_________ Plutonic rocks 3 2 < —7 Jet jci Jen----- Ultramafic rocks Central fault block and fault zones Copper Hill volcanics Jch, mafic volcanic rocks; Jet, felsic volcanic rocks; Jcl, limestone Eastern fault block _________a_________ A < \- A • V tu>' n • ■(V T v.o . ■ ■£ 0 O - y- <, Ju Volcanic breccia Sandstone Schistose mptaspHimp Fault Dashed where approximately located; queried where doubtful; dotted where projected in plane of observation Small fault showing dip Strike and dip of overturned beds Map Section Top of beds generally showing marked lineation Js Jeu Jvu Jm Jmb Salt Spring slate Epiclastic rocks of uncertain stratigraphic position U U) a D Lava Thin-bedded chert and slate Shear zone Anticline, showing trace of axial plane Determination based on graded bedding. Plotted along strike with point of observation Strike and dip of cleavage Strike and dip of schistosity Volcanic rocks of uncertain ■ stratigraphic position Jgu Jg Mariposa formation Jm, black slate containing minor interbedded tuff; Jmb, Brower Creek volcanic member Gopher Ridge volcanics Jg, mafic volcanic rocks; Jgt.felsic volcanic rocks; Jgu, undifferentiated mafic and felsic volcanic rocks v-v =” 7" Massive porphyritic volcanic or hypabyssal rock Limestone 000« ’ll J 0 o o 0r 0 O o O o ° ° ’ 0oo o 0 o 0 ° o o 0 o Syncline, showing trace of axial plane Strike and dip of beds Bearing and plunge of lineation Strike and dip of schistosity, showing bearing and plunge of lineation Strike of vertical schistosity, showin bearing and plunge of lineation Pillow lava Conglomerate 10,000' -1 L 10,000' GEOLOGIC MAP AND COMPOSITE SECTION ALONG THE MOKELUMNE RIVER, CALIFORNIA SCALE 1:62 500 1 3 MILES 1 .5_______0 4 KILOMETERS CONTOUR INTERVAL 25, 50 AND 80 FEET DATUM IS MEAN SEA LEVELUpper Jurassic UNITED STATES DEPARTMENT OF THE INTERIOR GEOLOGICAL SURVEY PROFESSIONAL PAPER 410 PLATE 8 Western fault block _________A_________ Jch Copper Hill volcanics Jch. mafic vulcanic rocks; Jcf.fetsiC volcanic rocks Js Salt Spring slate Jg Gopher Ridge volcanics Jg, mafic volcanic rocks; Jgs, black slate I fW! * i/t QT m Volcanic rocks and alluvial gravels i ' -1K KJp Plutonic rocks Ju Ultramafic rocks A; ;■? X k * 4k Jvu Volcanic rocks of uncertain stratigraphic position I 94 Central fault block and fault zones Jeu Epiclastic rocks of uncertain stratigraphic position a 3 O & Logtown Ridge formation Jc Cosumnes formation gs Green schist Calaveras formation Pc, chiefly phyllite and pyroclastic rocks, some sandstone, limestone and chert; Pci, limestone EXPLANATION Eastern fault block ---------A--------V Pci Pcc Pc Calaveras formation < >-cr 5 cr ■ cc Q tr u z u t- < H U ij in UoO < u DC Q U D Z < m < H U 0) CO < o: D 1 cr LiJ n Tuff ,0 .» < • r ■ a ‘P «►' »; ,f.0_ 9. v.' 0* . Volcanic breccia \ \\\X V Lava * «■ " \ II ^ \\ * Massive porphyritic volcanic or hypabyssal rock Pillow lava LITHOLOGY Various symbols are combined to show interbedded lithologic types l'Y\\V>'W| " ' v Slate Sandstone Thin-bedded chert and slate Limestone „ o « 0 'o o 0 oOo„0 oc » Q o o o ® ° O ° 9 o o O Conglomerate Schistose metavolcanic rock 'W \ ' \u T\av l \ \ V ^ N Schistose metasedimentary rock, generally showing marked lineation b'n Schistose plutonic rock /' / IN , nI - L I N 1 Is /N V- N Massive metavolcanic rock Contact Dashed where approximately located; short dashed where projected in plane of observation; dotted where inferred Fault Dashed where approximately located; queried where doubtful; short dashed inhere projected in plant of observat ion Small fault showing dip Shear zone ->♦■25 Bearing and plunge of minor folds showing trace of bedding 60 Strike and dip of beds sfP Strike and dip of overturned beds j!90 Strike of vertical beds ,4r A a b —► Map Section Tops of beds Determination based on: a, graded bedding; fa, pillow lava. Plotted along strike with point of observation 10 Generalized strike and dip of folded beds 60 Strike and dip of schistosity tTeo 80 Strike and dip of schistosity, showing bearing and plunge of lineation 38°30' Base from U. S. Geological Survey maps of the Carbon-dale 7Vi quadrangle and the Folsom and Placerville 15 quadrangles, California Geology mapped by L. D. Clark, 1953 and compiled from G. R. Heyl, J, H, Eric and M. W. Cox (plate 10, this report) N*OoN ■ ’*6, Bear Mountains fault zone Melones fault zone SEA LEVEL- 5000' - 10,000'—1 Note: Numbers refer to units described in Table 6 _ Approximate altitude of river surface ■SEA LEVEL INTERIOR—GEOLOGICAL SURVEY. WASHINGTON. D. C. —1964—G61165 ■ 5000' ■- 10,000' GEOLOGIC MAP AND COMPOSITE SECTION ALONG THE COSUMNES RIVER, CALIFORNIA SCALE 1:62 500 1 3 MILES H H H H~H~ 4 KILOMETERS CONTOUR INTERVALS 10, 25 AND 80 FEET DATUM IS MEAN SEA LEVELGEOLOGIC MAP AND DIAGRAMMATIC SECTION OF EPICLASTIC ROCKS EAST OF MERCED FALLS, MERCED COUNTY, CALIFORNIA ■.k | iv PROFESSIONAL PAPER 410 PLATE 1 1 EXPLANATION Jmf Merced Falls slate . Gopher Ridge volcanics , y (/) l/) < o: D Contact Dashed where approximately located; dotted where concealed -3 Plunge of minor anticline -e-*™ Plunge of minor syncline W° Plunge of minor folds, showing trace of bedding 6 Oy'' Strike and dip of beds e0*f Strike and dip of overturned beds 90^ Strike of vertical beds -*~70 Generalized strike of folded beds showing plunge of fold axis 85^ Strike and dip of schistosity 55 ^5 Strike and dip of schistosity showing bearing and plunge of lineation *y Strike and dip of cleavage Strike of vertical cleavage eoy Strike and dip of parallel bedding and cleavage «— Section Map Direction of tops of beds determined by graded bedding Structure lines in section are spaced to represent stratigraphic intervals of 20 feet. Their spacing depends on the angle between bedding and the plane of the section. No allowance was made for thickening and thinning that resulted from folding 724-276 0 - 65 (In pocket)UNITED STATES DEPARTMENT OF THE INTERIOR GEOLOGICAL SURVEY PROFESSIONAL PAPER 410 PLATE 2 ooo' \:50oo' I-Sea level :—5000' F--ioooo’ KJp Gopher Ridge volcanics Jg, mafic pyroclastic rocks including much lava and some pillow lava near the Calaveras River Jgf, felsic porphyritic massive and pyroclastic rocks Jgu, undifferentiated mafic and felsic pyroclastic and massive rocks KJp Pcc KJp KJp INTERIOR GEOLOGICAL SURVEY WASHINGTON. D C — 1 964 —G6 I 1 65 L Jeu Western fault block Jch Cjjchf l~s V Peaslee Creek volcanics Light- to dark-green bedded pyroclastic rocks; includes undifferentiated massive dark-gray porphyritic felsic lavas or hypabyssal intrusives (plagioclase and quartz phenocrysts) Jmf Merced Falls slate Dark-gray slate and siltstone; contains interbedded graywacke and minor conglomerate near Merced River Copper Hill volcanics Jch, mafic pyroclastic rocks, lava, and pillow lava Jchf, felsic porphyritic massive and pyroclastic rocks Salt Spring slate Mostly dark-gray clay slate; but includes subordinate tuff graywacke, and conglomerate CONTOUR INTERVAL 500 FEET DATUM IS MEAN SEA LEVEL Sect ions by L. D. Clark EXPLANATION QTu Volcanic rocks and alluvial gravels undivided Shown only at west side of map area KJp Plutonic rocks Chiefly granodiorite, quartz monzonite, and granite, but includes some hornblende gabbro and rocks of intermediate composition Ju Ultramafic rocks Chiefly serpentine in and west of the Melones fault zone and talc-ankerite schist east of the Melones fault zone. Periodotite and dunite rare Central fault block and Melones fault zone _____________________A_____________________ Jvu Jeu Volcanic rocks of uncertain Epiclastic rocks of uncertain stratigraphic position stratigraphic position a Similar in lithology to Logtown Ridge Lithologically similar to Cosumnes o formation, Pehon Blanco volcanics, and Mariposa formations So and Brower Creek volcanic member ol of Mariposa formation e Mariposa formation Jm, mostly dark-gray clay slate; but includes subordinate tuff, graywacke and conglomerate Jmp, Brower Creek volcanic member, dark-green mafic volcanic breccia; includes so me tuff and rare pillow lava Jl _________ Logtown Ridge formation Coarse mafic volcanic breccia, in part porphyritic (augite phenocrysts). Subordinate tuff and lapilli tuff, minor pillow lava Cosumnes formation Dark-gray clay slate, pyroclastic rocks, coarse conglomerate, and some graywacke Mafic volcanic rocks, conglomerate, and slate gs Green schist Metavolcanic rocks of unknown age distinguished in the Melones fault zone and at the north end of the map area. Mostly schistose, but includes some massive rocks Pc Calaveras formation Thin-bedded slate, chert, and siliceous slate containing calcarenite lenses. Includes some volcanic rocks Z < cr LlI CL !>■ < < L cr vc. ^ <• LlI cr r < D 0 U O N W u 5 in <0 o w >d) ^ tr & 2 u o in in r< cr D o o N ro in in 2 Jp Penon Blanco volcanics Upper part is dark-green mafic volcanic breccia and tuff in part porphyritic (pyroxene phenocrysts); lower part is dark-green mafic lava and some pillow lava O O N o in in Eastern fault block ________a_________ Undifferentiated argillaceous member and chert member Pcc, argillaceous rocks and siltstone and some thin-bedded chert; includes minor lenticular limestone in southwestern part; thin-bedded chert and black, carbonaceous slate containing minor lenticular mafic pyroclastics elsewhere Peel, limestone ftev Upper volcanic member Mafic pyroclastic rocks, in part porphyritic (augite phenocrysts). Minor pillow lava and black slate Z < cr LlI CL LJ _l < CL u o N 111 _l < 0- Rzcc Clastic member Thin-bedded dark-gray slate and graywacke, thin-bedded chert, and tuff; minor limestone lenses Contact Fault EXPANDED BLOCK DIAGRAM SHOWING GEOLOGY OF THE WESTERN SIERRA NEVADA BETWEEN THE MERCED AND COSUMNES RIVERS, CALIFORNIAA Solution of the Differential Equation of Longitudinal Dispersion in Porous Media GEOLOGICAL SURVEY 'PROFESSIONAL PAPER 411-A SEP 5 1961 library UNIVERSITY Of CALIFORNIA. J&S.S.RA Solution of the Differential Equation of Longitudinal Dispersion in Porous Media By AKIO OGATA and R. B. BANKS FLUID MOVEMENT IN EARTH MATERIALS GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-A UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1961UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. -Price 15 cents (paper cover)CONTENTS Page Abstract______________________________________________________________________ A—1 Introduction________________________________________________________________________ 1 Basic equation and solution_________________________________________________________ 1 Evaluation of the integral solution_________________________________________________ 3 Consideration of error introduced in neglecting the second term of equation 8------- 4 Conclusion__________________________________________________________________________ 7 References__________________________________________________________________________ 7 ILLUSTRATIONS Page Figure 1. Plot of equation 13______________________________________________ A-5 2. Comparison of theoretical and experimental results-------------- 5 3. Plot of equation 18--------------------------------------------- 6 mi ’ . ....FLUID MOVEMENT IN EARTH MATERIALS A SOLUTION OF THE DIFFERENTIAL EQUATION OF LONGITUDINAL DISPERSION IN POROUS MEDIA By Akio Ogata and R. B. Banks ABSTRACT Published papers indicate that most investigators use the coordinate transformation (x—ut) in order to solve the equation for dispersion of a moving fluid in porous media. Further, the boundary conditions C= 0 at x=co and C=C0 at x= — <*> for f>0 are used, which results in a symmetrical concentration distribution. This paper presents a solution of the differential equation that avoids this transformation, thus giving rise to an asymmetrical concentration distribution. It is then shown that this solution approaches that given by symmetrical boundary conditions, provided the dispersion coefficient D is small and the region near the source is not considered. INTRODUCTION In recent years considerable interest and attention have been directed to dispersion phenomena in flow through porous media. Scheidegger (1954), deJong (1958), and Day (1956) have presented statistical means to establish the concentration distribution and the dispersion coefficient. A more direct method is presented here for solving the differential equation governing the process of dispersion. It is assumed that the porous medium is homogeneous and isotropic and that no mass transfer occurs between the solid and liquid phases. It is assumed also that the solute transport, across any fixed plane, due to microscopic velocity variations in the flow tubes, may be quantitatively expressed as the product of a dispersion coefficient and the concentration gradient. The flow in the medium is assumed to be unidirectional and the average velocity is taken to be constant throughout the length of the flow field. BASIC EQUATION AND SOLUTION Because mass is conserved, the governing differential equation is determined to be n _ b(J. bC Dv2C—u -5-—F-5-7 bx bt v bx^by^bz2 (1) where D—dispersion coefficient C— concentration of solute in the fluid u=average velocity of fluid or superficial velocity/ porosity of medium x=coordinate parallel to flow y,2= coordinates normal to flow i=time. In the event that mass transfer takes place between the liquid and solid phases, the differential equation becomes Dv‘lC~u bCbCbF bx bt bt (2) where F is the concentration of the solute in the solid phase. The specific problem considered is that of a semiinfinite medium having a plane source at x=0. Hence equation 1 becomes b2C <>C=dC' dx2 U bx bt (3) Initially, saturated flow of fluid of concentration, (7=0, takes place in the medium. At t—0, the concentration of the plane source is instantaneously changed to C—C0. Thus, the appropriate boundary conditions are C(0,t) = C0; t>0 G(x,0)=0; x>0 (7(<»,f)=0; *>0. The problem then is to characterize the concentration as a function of x and t. To reduce equation 1 to a more familiar form, let C(x,t) = T(x,t) exp (4) 585211—61----2 A-lA-2 FLUID MOVEMENT IN EARTH MATERIALS Substituting equation 4 into equation 1 gives where i>T_n d2r dt dx2 The boundary conditions transform to T(0,<) = Co exP ('u2t/4J)); t> 0 r(x,0) = 0; 2>0 (5) r(»,0=0; <>0. It is thus required that equation 5 be solved for a time-dependent influx of fluid at 2=0. The solution of equation 5 may be obtained readily by use of Duhamel’s theorem (Carslaw and Jaeger, 1947, p. 19): If C=F(x,y,z,t) is the solution of the diffusion equation for semi-infinite media in which the initial concentration is zero and its surface is maintained at concentration unity, then the solution of the problem in which the surface is maintained at temperature (t) is _d_ F(x,y,z,t—\)dk. q= tJp/d. The boundary condition as x—>&> requires that B—0 and boundary condition at 2=0 requires that A—l/p, thus the particular solution of the Laplace transformed equation is r=- e'~tx. V The inversion of the above function is given in any table of Laplace transforms (for example, Carslaw and Jaeger, p. 380). The result is r=i-erf(i/!*■'*■ (7) 2,fDt Utilizing Duhamel’s theorem, the solution of the problem with initial concentration zero and the time-dependent surface condition at 2=0 is r”J.'*(T)s[iL=s=‘"’v,]o r(2,0) = 0; 2>0 r(co,t) = 0; <>0. Since e~”2 is a continuous function, it is possible to differentiate under the integral, which gives JLA s: e ,2dr;= 2V-D((—r) 2 VttD(<-t)3/2 0-z*/4DU-t) Thus Letting r 2 *73 e-x* HD(t-r) dr (t — r)3/2' \=x/2^D(t—T) This problem is readily solved by application of the Laplace transform which is defined as n oo r(2,p)=J e~p‘T(x,t)dt. Hence, if equation 5 is multiplied by e~pt and integrated term by term it is reduced to an ordinary differential equation dx2 D The solution of equation 6 is (6) r=Ae-tx+BeQX the solution may be written (8) 2tiDt Since +«->.J%xp[-(X-i)’]rfx}. (10) Since the method of reducing integral to a tabulated function is the same for both integrals in the right side of equation 10, only the first term is considered. Let z=e/\ and adding and subtracting the integral may be expressed 7l-d>[-(x+0> =~e£(1-?)e£-(i+2)’],fe +‘“/.".exp[“(;+sfk Further, let 0 in the first term of the above equation, then /1=-e*J“e e-f>2dp+e*' J” exp [~(€ + 2)2] dz- A-3 Similar evaluation of the second integral of equation 10 gives /,=e"£“p[-(\-zJ]dz Again substituting —0=1 — z into the first term, the result is /!“2""X-„ «-'■**-*"£,exp [-(H)1]*• a Noting that X".mpK2+0’+2']*=X*.mp[_(j-2)!_2*]* substitution into equation 10 gives I=e~il e~^dfi—eit e~p2d0. --a —j-c* a a Thus, equation 9 may be expressed vM i [.-f e~fi2d0—e2t f e—"V - (11) --a —\-a ' a a However, by definition, e J* e fi2d0=~ e2e erfc a+- also, 2 u l ( --a a Writing equation 11 in terms of the error functions [e2e erfc (^x+^)+e~2< erfc Thus, substituting into equation 4 the solution is [erfc («-i)+e“ erfc («+i)} T (x,t)=^e“> (12)A-4 Resubstituting for e and a gives FLUID MOVEMENT IN EARTH MATERIALS which may be written in terms of dimensionless parameters, zB [erfo (^)+exp G)erfc (i^)] (13) where £—ut/x and t]=D/ux. Where boundaries are symmetrical the solution of the problem is given by the first term of equation 13. This symmetrical system was considered by Dankwerts (1953) and Day (1956), utilizing different analytical methods. The second term in equation 13 is thus due to the asymmetric boundary imposed in the more general problem. However, it should be noted also that if a point a great distance away from the source is considered, then it is possible to approximate the boundary condition by C(— °° ,f) = C0, which leads to a symmetrical solution. A plot on logarithmic probability graph of the above solution is given in figure 1 for various values of the dimensionless group i)=Dlux. The figure shows that as t] becomes small the concentration distribution becomes nearly symmetrical about the value £=1. However, for large values of tj asymmetrical concentration distributions become noticeable. This indicates that for large values of D or small values of distance x the contribution of the second term in equation 13 becomes significant as £ approaches unity. Experimental results present further evidence (for example, Orlob, 1958; Ogata, Dispersion in Porous Media, doctoral dissertation, Northwestern Univ., 1958) that the distribution is symmetrical for values of x chosen some distance from the source. An example of experimental break-through curves obtained for dispersion in a cylindrical vertical column is shown as figure 2. The theoretical curve was obtained by neglecting the second term of equation 13. CONSIDERATION OF ERROR INTRODUCED IN NEGLECTING THE SECOND TERM OF EQUATION 8 Experimental data obtained give strong indication that in the region of flow that is of particular interest it is necessary to consider only the first term of equation 13. Owing to complexity of the overall problem of determining the error, it would facilitate analysis to determine the value of £ at which the function elln erfc /l+£\ is a maximum. This then will enable the deter- V*lh) mination of the value of r) at which equation 13 may be reduced to without introducing errors in excess of experimental errors. The necessary condition that the function /(r?,£) is a stationary point is given by (15) To determine whether the function is either maximum or minimum at a given point the sufficient conditions are given by d2/ (b) Maxima, ^ <(0, consequently ^<0 d2/ d2/ (c) Minima, ^ >0, consequently ^ >0 (16) (Irving and Mullineux, 1959, pp. 183-187) Further, if 16(a) is greater than zero, the stationary point is called a saddle point. Let erfc Differentiating the function _____L_ d£ 2 -yfinj [(£-l)r3/2]e-'* where e2=(l—£)2/4£jj. From the above expression it can be seen that £=1 and £=® are the stationary points of the function. The second differentials can readily be obtained by direct methods. The results are and e1,v erfc also + 1 2V?r£ -£)V7'2- d2/_.l_ d£d?7 2 V^r (£_l)r3/V3/2« Glass beads Theoretical curve ______I______I______I -0.5 Figure 2.—Comparison of theoretical and experimental results.A-6 FLUID MOVEMENT IN EARTH MATERIALS At point {=1, the following expressions are obtained: &2/ £>2/ dr,2 Vf_ d&v {-1 =— (4irr?) 1/2 i=^-3+i v-4^ eVv erfc (v~1/2)~^ V 6/2 =0. (17) {-i <0 It can be shown that 5Z>mX102 the second term of equation 13 becomes negligible. Orlob and Radhak-rishna (1958) obtained values of Dm ranging from 0.09 cm to 2.79 cm. Using these values, measurements must be obtained at values of x greater than 45 cm or 1395 cm. However, if an error of 5 percent is permitted, the above values are reduced by a factor of 4, thus x must be greater than 10 cm or 350 cm. CONCLUSION Consideration of the governing differential equation for dispersion in flow through porous media gives rise to a solution that is not symmetrical about x=wt for large values of ij. Experimental evidence, however, reveals that D is small. This indicates that, unless the region close to the source is considered, the concentra- tion distribution is approximately symmetrical. Theo- O 1 retically, 77only as tj—>0; however, only errors of Oo ^ the order of magnitude of experimental errors are introduced in the ordinary experiments if a symmetrical solution is assumed instead of the actual asymmetrical one. REFERENCES Carslaw, H. S., and Jaeger, J. C., 1947, Conduction of heat in solids: Oxford Univ. Press, 386 p. Dankwerts, P. V., 1953, Continuous flow system distribution of residence times: Chem. Eng. Sci., v. 2, p. 1-13. Day, P. R., 1956, Dispersion of a moving salt-water boundary advancing through saturated sand: Am. Geophys. Union Trans., v. 37, p. 595-601. deJong, G. de J., 1958, Longitudinal and transverse diffusion in granular deposits: Am. Geophys. Union Trans., v. 39, p. 67-74. Horenstein, W., 1945, On certain integrals in the theory of heat conduction: Appl. Math. Quart., v. 3, p. 183-187. Irving, J., and Mullineux, N., 1959, Mathematics in physics and engineering: Academic Press, 883 p. Orlob, G. T., and Radhakrishna, G. N., 1958, The effects of entrapped gases on the hydraulic characteristics of porous media: Am. Geophys. Union Trans., v. 39, p. 648-659. Pierce, B. O., and Foster, R. M., 1956, A short table of integrals: Boston, Mass., Ginn and Co., 189 p. Scheidegger, A. E., 1954, Statistical hydrodynamics in porous media: Appl. Phys. Jour., v. 25, p. 994-1001. oTransverse Diffusion in Saturated Isotropic Granular Media < GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-B DOCUMtNrS DEPARTMENT SEP 5 1961 library UNIVERSITY OF CALIFORNIA Transverse Diffusion in Saturated Isotropic Granular Media By AKIO OGATA FLUID MOVEMENT IN EARTH MATERIALS GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-B UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1961UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C. - Price 15 cents (paper cover)CONTENTS Page Abstract___________________________________________________________________________ B-l Introduction_______________________________________________________________________ 1 Acknowledgments____________________________________________________________________ 1 Two-dimensional dispersion___________________________________________________________ 1 Solution of the differential equation______________________________________________ 2 Solution of special cases of r equals 0 and r equals a_______________________________ 4 Some properties of the hypergeometric series_________________________________________ 5 General solution for computation of concentration distribution_______________________ 5 Steady-state solution________________________________________________________________ 6 Conclusion___________________________________________________________________________ 6 References__________________________________________________________________________ 8 ILLUSTRATIONS Figube 1. Mathematical model__________________________________J_______________ B-2 2. Plot of solution for various values of r/a_________________________ 7 3. Plot of solution for various values of a2/4D<______________________ 7 m#FLUID MOVEMENT IN EARTH MATERIALS TRANSVERSE DIFFUSION IN SATURATED ISOTROPIC GRANULAR MEDIA By Akio Ogata ABSTRACT An analytical method of determining the ionic diffusion transverse to the direction of flow in granular media is presented. The basic assumption that the frontal zone created by longitudinal dispersion is fully developed and stationary before the transverse diffusion can cause spreading is made to simplify the mathematical analysis. Solutions were obtained in terms of both a series of Bessel’s functions and a hypergeometric series. Numerical computations were made and results are presented in graphical form. INTRODUCTION The necessity for disposal of radioactive and other wastes and for more quantitative knowledge of microscopic flow in porous media has focused interest on dispersion phenomena occurring in saturated flow through porous media. Various laboratory techniques have been developed for determining the dispersion coefficient appearing in the analytical solutions. Emphasis, however, has been placed on studies of longitudinal dispersion—for example, those of Beran (1957) and Day (1956) and by the writer (Dispersion in porous media, doctoral dissertation, Northwestern Univ., 1958). The data available on transverse diffusion are still extremely meager, owing to the difficulty in modeling transverse diffusion and measuring the concentration distribution once a model is constructed. The Geological Survey in its ground-water research office at Phoenix, Ariz., has developed methods for measuring transverse diffusion, by means of radioactive isotopes and commercially available counters (Skibitzke and others, 1960). However, the studies to date have been directed toward feasibility of the methods, and quantitative data are not available. This paper will consider the mathematical aspects of an approximate description of the phenomena by dealing with a simplified mathematical model. ACKNOWLEDGMENTS The problem treated in this report was brought to the attention of the writer by H. E. Skibitzke, mathema- tician in charge of the Phoenix ground-water research office. The writer is greatly indebted to Mr. Skibitzke, whose initial development and ideas greatly facilitated the preparation of this report. The author would like to express appreciation also to Prof. Richard Skalak, of Columbia University, for his thorough discussion and review of the paper. Professor Skalak points out that Goldstein, in his paper “Some Two-Dimensional Diffusion Problems with Circular Symmetry” (1932), obtained equation 27. However, his paper was not available to the writer at the time of writing. Professor Skalak points out also that there is a shorter method of derivation of equation 9 by use of the Hankel transform. The author utilized the Laplace transformation because this method has a wider range of applicability and is a more familiar method of approach. TWO-DIMENSIONAL DISPERSION The differential equation that describes the two-dimensional dispersion of a contaminant in saturated granular media, where radial symmetry exists, is „ 1 d / dC\ . n d2<7 dC r r dr \ dr) z dx2 U dx dC dt (1) where C= concentration of contaminants in lb/ft3 r=radial distance, in feet, measured from center of contaminant filament x=longitudinal distance, in feet, measured in the direction of fluid flow i=time, in seconds Dr=radial diffusion coefficient, in ft2/sec D*=longitudinal dispersion coefficient, in ft2/ sec u= average fluid velocity, in ft/sec Equation 1 is readily obtained from the law of conservation of mass. Owing to difficulty in solving the general case, it is necessary to make some simplifying assumptions that are physically realistic. B-l 584498—61----2B-2 FLUID MOVEMENT IN EARTH MATERIALS The physical nature of the dispersion process is shown in figure 1. The system may be divided into two regions, the frontal zone and the cylindrical filament of contaminant upstream from the frontal zone. In the frontal zone the dispersion, or spreading, of the contaminant is due to longitudinal plus transverse dispersion. However, in the region upstream from this frontal zone the spreading is due principally to transverse diffusion and convection because of the virtual nonexistence of a concentration gradient in the direction of flow. Thus, if the frontal zone is not considered, d2<7 Dx-^—0 and equation (1) reduces to Ad/ dC\_ , dC_dC r dr \r dr ) U dx dt (2) same as the one-dimensional system. But, if it is assumed that the establishment of this front takes place at a relatively rapid rate in comparison with transverse diffusion, then the time rate of change of concentration depends only on the radial dispersion. Thus, the differential equation describing the process is Ad / dC\=dC' r dr \ dr / dr Substituting £=x—ut, t=r in equation 2 and resubstituting t for r gives Equation 3 is the fundamental differential equation describing the dispersion process throughout the region for long times. This is the differential equation found in various radial-diffusion or heat-flow problems, whose solutions are known for various boundary conditions. The boundary conditions for the case to be considered are (1) (=0, C=Cq, ra (3) r=0, g=0 (4) It is evident that an initial-value problem needs to be solved. Because the longitudinal-dispersion coefficient, Dx, no longer appears, the subscript r will be dropped from the radial-diffusion coefficient. In discussing the frontal zone, consider a moving coordinate system—that is, let %=x—ut and r=t. Substituting this transformation into equation (1) gives Ad/ dC\_ n d2C dC r dr \ dr) x d£2 dr Physically, the above equation indicates that the observer is moving at a velocity, u. Experimental and analytical development of the dispersion in a one-dimensional system indicates that the length of the frontal zone remains constant, once it is fully developed. Von Rosenberg (1956) indicates that the zone in which the concentration varies by a certain percentage depends on the velocity and the dispersion coefficient. In the system considered here, in which no mass exchange occurs between the liquid and the solid phase, this front will progress through the medium with the velocity of the fluid. Because of the existence of another interface in the radial direction, the two-dimensional system is not the SOLUTION OF THE DIFFERENTIAL EQUATION Solution of equation 3 may be obtained by employing the Laplace transform; that is, assume that there exists a function C{r,p)—^ e~ptC(r, t)dt If equation 3 is multiplied by e~vt and integrated with respect to t within the limits 0 to °°, equation 3 may be written 7 3?(,'f)+c"-J>c~0;r<“ <5a) (5b) _ n It should be noted that by setting 6’=X—~ in 5a the two equations become equivalent.TRANSVERSE DIFFUSION IN SATURATED ISOTROPIC GRANULAR MEDIA B-3 Hence, equation 5 may also be written d?\ .1 d\ 2r „ / i?+-, rr5 x=0; '<“ PC, 1 iC^ ifi+r A r>a, Above is Bessel’s equation, whose solution is as follows (Hildebrand, 1954, p. 167) C=AI0 (qr)+BK0 (qr) where 70 (a) = modified Bessel function of the first kind of order zero K0(a) = modified Bessel function of the second kind of order zero A,B= arbitrary constants Since K0 remains finite as r—>oo, this solution is valid; that is, A—0 for r)>a. In the region r0, and also since (£L must be zero, I0 is the solution, or B=0. Hence, C1=^+AI0(qr) Jr C2=BK0(qr) At r=a C\=C\; thus Co ra (6) BKo(qa) =— +AI0(qa) In addition, the mass rate of transfer across r—a is given by giving (dtP).A° - BKi (aq)=AB (qa) dC, ZrJT-_ From the above two equations the coefficients A and B may be determined; that is, _A=C*_____________Kiiqa)_________ p I0(qa)Ki(qa)+h(qa)K0(qa) _ nCn_____________/i(ga)_________ p K0(qa)B(qa) +Ki(qa)I0(q) Substituting values for A and B into equation 6 gives the Laplace solution of equation 5, Ki(qa)I0(qr) I0(qa)Ki(qa)+K0(qa)I1(qa)J a—a 1 V \ Ii(qa)K,(qr) 1 [K0(qg)B(qa)+Ki(qa)Io(qg)j To obtain the solution of equation 3 for O' as a function of r and t it is necessary to apply the Bromwich inversion theorem, C{r ■Ail C+i<*> C+f co ep‘C(r,p)dp The inversion of equation 7, however, has been evaluated by Carslaw and Jaeger (1948, p. 285); the result for this specific case is r (r f “ J1 (««) [J 0 ( (a) ]da OHM;- _ | e aW(a)) * Jo a (r t)=^ r {r,t) A Jo e-BaH Jo(ar)Ji(aa)da ^ where «V(«)] 4>(a) =J i (act) Y0 (act) —J o (gct)Yi(ga) Further (Watson, 1948, page 77), (z)Y+l (z) -J,+l(z) Yr(z)= 7TZ Thus, because v=0, 4>(°d—— irga Substituting the value for (a), the result is „-DaH O1=O2=O=O0a J™e~Da2t Ji (ota)Jo(ar)da (9) where Ji(a) and t70 (a) = Bessel function of the first kind of order one and zero, respectively a=radius of contaminant filament By direct substitution, the above can be shown to satisfy the original differential equation. It is now desired that equation 9 be shown to satisfy the boundary conditions. For t=0 the solution reduces to the following (Watson, 1948, p. ^06): C C “ 7f—g I Ji(ag)J0(ar)da Jo f 0 for r<(g =J ± r=g (10) (. 1/a r>a The condition at r—g is due to an averaging process such that the concentration curve becomes a continuous function. In any numerical integration process difficulties are inherent; thus, if possible, equation 9 should be written in terms of tabulated functions to facilitate computation. However, before going into the general case itB-4 FLUID MOVEMENT IN EARTH MATERIALS may be advantageous to consider special cases for r=0 and r=q. For these two values of r, equation 9 may be readily integrated, thereby simplifying the analysis considerably. The diffusion coefficient may be readily obtained from experimental data by using these two expressions. SOLUTION OF SPECIAL CASES OF r=0 AND r=a Consider first the concentration variation along r=0. Because J0(x)-*l as 2—>0 the equation above becomes simply 9.=a fVB'“2 «7i(ao)0 and for x= — 1 only if p— ax — a2 + l>0 (Hildebrand, 1954, p. 179). GENERAL SOLUTION FOR COMPUTATION OF CONCENTRATION DISTRIBUTION As stated previously, although equation 9 is the required solution of the problem, the numerical evaluation of the integral is extremely difficult. Thus, it would be advantageous to rewrite it in such a way as to facilitate computation. Equation 9 may be written in a series of Bessel functions or hypergeometric functions. Because tables of Bessel functions were readily available, this series was used. However, both of the series will be presented. The product of Bessel functions generally may be written co J„(aa)JJar) = AnJn(aa) Jn(ar) n=0 by use of the recurrence formula Explicitly, Jp-i(/3x) + J7+i(/3x) e/Afa) (19) Jn(ar)^(aa)—y', (¥)(“)' To n+1 Jn(aa)Jp(ar) .(■ ua\ a 2 )\r -S ■ • , Jn(aa)Jn(ocr) 71=2 n—i (20) Substituting equation 19 into equation 9 and using the known integral J0 e~»2‘2 J.(at)J,{bt)tdt=~ exp (-—)/» (21) Equation 9 may be readily integrated. The result of the integration is C a 1=1 exp r-^1 / ? T* Ml / ro 2expL ADt J \^offl+l ’ (?) (if)’ 2 m- wliere £= M and »?=-• s 2 Dt r 1-2 A (22) By use of the recurrence relationship for the modified Bessel function, /,_i(/3a:) — lu+i(px)=j^l,(px) (23) The second term in equation 22 may be written 771 = 2 Wl J- 771 = 0^+1 771 = 1 This on substituting into the original equation gives 5=Mp(-w)a’’/-(o <24> The above result was obtained by Goldstein (1953) in the study of exchange processes in fixed columns. Equation 24 may be written in another form,however: Equation 25 also may be written in its alternate form (Goldstein, 1953) g_1_exp[_k=ll%]s_L6-i/,({) (26)B-6 FLUID MOVEMENT IN EARTH MATERIALS Note that for thus equation 26 con- verges more rapidly than 25 in the region ^>1. The converse is true for equation 25; that is, it would be advantageous to use equation 25 for computation in the region -1, and J0(ar)Maa)=~ 2 2 m=o m\m\ 2Fi(—m,— m-, 2; i?2) for„2=^-,i/2(»+n 2 p the expressions obtained are C_ Go m=0 2Fi(-m-l-m; l;—2 S (-1)”------------------- (X)B+1; (27) and C_ C0= = S (— l)m(arm)2- 771 = 0 Wi(—m,—m; 2; ??2) m! Vm2/ ’ where X a2 4.Dt „2<1 (28) Hypergeometric functions appearing as coefficients of the power series are polynomials of m. Although both equations 27 and 28 and equations 25 and 26 may be readily computed, equations 25 and 26 were used because of their rapid convergence. A plot , G & j, • 1 p Qj • * ■ of -~r versus -pyrr lor various values ol ij=- is given in Oq 4jl/6 T Cy CL figure 2, and a plot of yy versus - for various values of Co T is given in figure 3. STEADY-STATE SOLUTION Owing to the nature of the experiments conducted (Skibitzke and others, 1960), it would be advantageous to assume that a steady state is reached. Accordingly, dC consider-—in equation 2, which gives The boundary conditions become C(r,a)=C0 C (r,0)=0 0a K>0 The conditions given are the same as those given for the X unsteady-state problem. Further, by letting r=—, u equations 29 and 3 are identical. Thus, the solution for X steady-state condition may be obtained by letting t=- u in equation 25, 26, 27, or 28. CONCLUSION In attacking the mathematical problem of transverse diffusion, an important assumption was made: that the front due to the longitudinal dispersion is established rapidly as compared to the transverse diffusion. If the diffusion transverse to the direction of flow includes mechanical dispersion, this assumption would be erroneous. However, since a homogeneous medium is assumed, the assumption that diffusion is wholly due to molecular agitation seems to be valid. Although there have been no quantitative studies, qualitative study using dye tracers in various types of models seems to bear this out. Up to this time, investigation has been directed primarily toward evaluating experimental and analytical methods. The approximate method presented leads to what is believed to be a realistic solution that can be computed. When laboratory experiments are completed the resulting data, in conjunction with the solution presented, will furnish a means of determining the magnitude of the coefficient of transverse diffusion.TRANSVERSE DIFFUSION IN SATURATED ISOTROPIC GRANULAR MEDIA B~7 4 Dt Figure 2.—Plot of solution for various values of r/a. Figure 3.—Plot of solution for various values of a2/4Dt.B-8 FLUID MOVEMENT IN EARTH MATERIALS REFERENCES Bateman Manuscript Project, 1953a, Higher transcendental functions, v. 1: New York, McGraw-Hill, 302 p. ----— 1953b, Higher transcendental functions, v. 2: New York, McGraw-Hill, 396 p. Beran, M. J., 1957, Dispersion of soluble matter in flow through granular media: Jour. Chem. Physics, v. 27, no. 1, p. 270-274. Carslaw, H. S., and Jaeger, J. C., 1948, Conduction of heat in solids: London, Oxford Univ. Press, 386 p. Day, P. R., 1956, Dispersion of moving salt-water boundary advancing through saturated sand: Am. Geophys. Union Trans., v. 37, no. 5, p. 595-601. Goldstein, S., 1932, Some two-dimensional diffusion problems with circular symmetry: London Math. Soc. Proc., Ser. 2, 34, p. 51-88. Goldstein, S., 1953, On the mathematics of exchange processes in fixed columns: Royal Soc. (London) Proc., v. 219, p. 151-185. Hildebrand, F. B., 1954, Advanced calculus for engineers: New York, Prentice-Hall, 594 p. Skibitzke, H. E., Chapman, H. T., Robinson, G. M., and McCullough, R. A., 1961, Radio-tracer techniques for the study of flow in saturated porous material: Internat. Jour. Applied Radiation and Isotopes v. 10, no. 1, p. 38-46. Von Rosenberg, D. U., 1956, Mechanics of steady-state singlephase fluid displacements from porous media: Am. Inst. Chem. Eng. Jour., v. 2, p. 55. Watson, G. N., 1948, A treatise on the theory of Bessel functions. Cambridge Univ. Press, 804 p. oucuiugi^ai ouivc) nuii'Hsiuiiai raper ±11 Transverse Dispersion in Liquid Flow Through Porous Media ^(GEOLOGICAL SURVEY H * * 3 3 3 3 3 3 3 3 3 H ) ? 5 i » 5 i i \ ; > iTransverse Dispersion in Liquid Flow Through Porous Media By EUGENE S. SIMPSON FLUID MOVEMENT IN EARTH MATERIALS GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-C UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1962UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C.CONTENTS Page Symbols and glossary___________________________________ iv Abstract______________________________________________ C-l Introduction______________________________________________ 1 Acknowledgements_________________________________________ 2 Theoretical analysis___________________________________ 2 Description of apparatus and experimental procedure___ 7 Page Experimental results__________________________________ C-13 Comparison between theory and experiment________________ 23 Discussion_____________________________________________ 26 References____________________________________________ 28 Appendix________________________________________________ 29 ILLUSTRATIONS Page Figure 1. Diagram of porous medium showing line source and coordinate axes__________________ C-3 2. Diagram illustrating concepts of tortuosity and circuity__________________________________ 4 3. Diagram illustrating set of idealized flow paths_________________________________________ 4 4. Diagram illustrating motion of tracer ele- ments between adjacent flow paths_____________ 4 5. Diagram of idealized porous medium_____________ 5 6. Diagram of idealized pore space________________ 6 7. Graph of probability q plotted against 1/X__ 8 8. Graph of a/a VdL plotted against 1/X___________ 8 9. Flow diagram of dispersometer_i________________ 9 10. Diagram of experimental porous block______ 10 11. Flow diagram of apparatus to supply de- aired water__________________________________ 11 12. Photograph of dispersometer, front view___ 12 13. Photospectrograph of tracer solution__________ 12 14. Mechanical analysis of sand used in porous block________________________________________ 13 15. Concentration distribution, run 1, suites A, B, and C_____________________________________ 14 16. Concentration distribution, run 2, suites A, B, and C_____________________________________ 14 17. Concentration distribution, run 3, suites A, B, and C_____________________________________ 15 Page 18. Concentration distribution, run 4, suites A and B___________________________________ C-15 19. Concentration distribution, run 5, suites A and B_____________________________________ 16 20. Concentration distribution, run 6, suites A and B_____________________________________ 16 21. Concentration distribution, run 7, suites A and B_____________________________________ 17 22. Concentration distribution, run 8, suites A, B, and C_______________________________ 17 23. Concentration distribution, run 9, suites A and B_____________________________________ 18 24. Concentration distribution, run 10, suites A and B_____________________________________ 18 25. Concentration distribution, run 11, suites A, B, and C_______________________________ 19 26. Concentration distribution, run 12, suites A, B, and C_________________________________ 19 27. Concentration distribution, run 13, suites A and B___________________________________ 20 28. Graph of dispersion versus distance in hypothetical aquifer______________________ 24 29. Graph of a* plotted against H*, theoretical and experimental—_________________________ 25 TABLE Table 1. Summary of experimental results:. C-21 mSYMBOLS a half-diameter of idealized pore space X spatial flow coordinate parallel to direction of C concentration of tracer elements bulk flow Co initial concentration of tracer elements y spatial flow coordinate transverse to direction d characteristic grain diameter of bulk flow, and direction in which dispersion D coefficient of bulk dispersion of tracer elements was measured Dm coefficient of molecular diffusivity of tracer z spatial flow coordinate transverse to direction elements in host fluid of bulk flow, and direction wherein dC/dz=0 f lithologic factor accounting for the effect of a and dvb/dz=0 porous medium on the diffusion of tracer ai dimensionless constant related to distance elements in a fluid that saturates the medium. between idealized flow paths H dimensionless number, H=vvdlDM ot dimensionless constant related to distance H* dimensionless number, H*=(a/2)2H between idealized pore spaces L length of porous medium a dimensionless constant; a=a1/f a2 P permeability of porous medium, or porosity of X dimensionless number; \=DMt/a2 porous medium M mean or expected position of tracer element on V probability that a tracer element will move from y axis one flow path to an adjacent flow path in an V kinematic viscosity of water idealized pore space a standard deviation of dispersion of tracer i q=l-p elements along y axis t time <7* dimensionless measure of dispersion along y Vi bulk velocity of host fluid axis; 0. C, Tortuosity>0; circuity=0. D, Tortuosity >0; circuity>0. diffusivity of tracer elements, when combined with the tortuosity of fluid stream paths, provides a mechanism for transverse dispersion. The fluid paths coalesce and diverge repeatedly. In a pore space where two paths coalesce, a very small motion of a tracer element, which is due to molecular diffusivity, will carry the element from one path into the other. The paths then diverge and coalesce again with other paths and the same opportunity for transfer occurs. In this way the relatively small total distance traveled by a tracer element because of molecular diffusivity c | Arbitrary flow path y Average position of flow path Figure 3.—Diagram illustrating set of idealized flow paths. is amplified by the nature of fluid movement in the medium through which, it is being carried. The analysis that follows seeks to put this line of reasoning in mathematical terms. It is necessary to idealize flow conditions in a manner that will permit mathematical treatment and at the same time be related to measurable flow and media variables. The first idealization will be to assume that all possible positions of the tracer elements are restricted to an arbitrary set of flow paths. The flow-path set is arranged in an orderly manner as shown on figure 3. The spacing between successive paths in the y direction is determined by a characteristic grain diameter, d, multiplied by a dimensionless coefficient, <*i, which depends on grain shape and packing and on porosity—in other words, on the geometry of the medium. The spacing between paths in the z direction is of no consequence because there is no change in concentration in that direction. It is assumed also that the motion of each tracer element, which is due to molecular diffusivity, is independent of the motion of every other tracer element and that for each unit distance a2d (fig. 4) each tracer element will have a probability, p, of moving to an adjoining flow path, and a probability, g, of staying in the same flow path, where a2 is a dimensionless constant also depending on the geometry of the medium, and where p + g = 1. In effect, each tracer element will experience a succession 1*- -*l f- V -j V J o l / 1 Figure 4.—Diagram illustrating motion of tracer elements between adjacent flow paths. A, Relation between flow paths and grains of idealized medium. B, In bulk-flow distance aid each tracer element has probability p of moving to an adjacent flow path and probability g of staying in same path.TRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C-5 of Bernoulli trials (Feller 1950, pp. 104-110), with the probability p for “success” and q for “failure” in each trial. In order to preserve symmetry, it will be necessary to modify the idealized flow pattern shown on figures 3, 4, and 5. According to the figures, if the first trial of the tracer element permits a transfer to the left, then the next trial permits only a transfer to the right, and the next left, and so on, and two different distributions would result, depending on which way the first trial went. If, however, it is assumed that the direction at each trial is uncertain and is independent of the preceding trial, then symmetry is preserved and only one solution is possible. Since in a real three-dimensional porous medium the orientation of the grains and the tortuosity of the flow paths are random and not regular, this departure from the idealized model probably is toward real conditions. Under this assumption, at each trial there is a probability of %p to move to the right (positive y direction), a probability of %p to move to the left (negative y direction), and a probability of q to remain in the original path. If the zero point of the y axis (the intersection of the x and y axes) is defined by the flow path initially occcu- pied by the tracer element, then the y ordinate of the mean, or expected, or “average,” position of the element for each trial is (Feller, 1950, p. 180): Mi=| [(aid)2>-t-(—ad)p]=0, (la) and in n trials Mn=SMi= 0. (lb) i=l The variance, a2, of each trial is 2C W~Dm w where C= concentration of tracer elements, and Dm=coefficient of molecular diffusivity. (5) Particular solutions of equation 5 depend on choice of boundary conditions. For the present purpose it is sufficient to assume the simplest possible. The boundary is defined as a rectangle of width aid and length a2d; at time f=0 the tracer elements are assumed to occupy one-half this space in concentration C0 (fig. 6) the time available for molecular diffusion of the elements is assumed to equal the length of the pore space divided by average pore velocity. It will be noted that, by this method of getting an estimate of p, the tracer elements are not restricted to the arbitrary flow paths, but by virtue of molecular diffusivity they may occupy any position within the idealized pore rectangle. This is a departure from the original idealization but is closer to real conditions. However, the elements cannot move out of the pore X Figure 6.—Diagram of idealized pore space. A uniform concentration Co of tracer elements is assumed to occupy the region to the left of centerline at time=fo. rectangle except by motion of the host fluid. The limit of validity of this latter assumption is considered under “Discussion.” With the aforementioned boundary conditions, two equivalent solutions of equation 5 may be obtained: n—1 (—1) 2 nirv „ , ----- cos exp 2 tt n= w... n 2a \ /mr' \2a J DMt (6) and C(y,t)=Q £ L n— — o feri<.(l+4^-i'+erf«(l-4«)+.y\. \. 2 \ 1 t 2 \ / )\; t j (7) Equation 6 was obtained by the method of separation of variables (see, for example, Miller, 1953, p. 93-108) and equation 7 was taken from Crank (1956, p. 15). By definition p= 1—g, where q is both (1) the probability that a given tracer element remains in the same flow path in going through a pore rectangle and (2) the fraction of tracer elements that remain in the same semipore space after the diffusion of tracer elements occurs. By the latter definition By substituting equations 6 and 7 in 8 and integrating each with respect to y over the distance y=a, we obtain two equivalent solutions for q, where q will be a function of the semipore diameter, a, the time, t, and the coefficient of diffusivity of tracer elements, Dm. As will be seen, one solution is convenient to use for small values of t and the other for large. First substituting equation 6 in 8, and integrating term by term, we get (see app., p. C-29): 2 2+tt2 £ n=1,3,5 .. n2 D\ft where 14 "1 =4+;5 S zr2 exp—iX Z 7T n= 1,3,5 . . . ^ (9a) or i 4 « 1 P=S n2 exP-/:X Z 7T 71 = 1,3,5 . . . 71 (9b) Equation 9b converges satisfactorily for moderate to large values of t. To obtain a solution that will con-TRANSVERSE DISPERSION IN LIQUID PLOW THROUGH POROUS MEDIA C-7 verge satisfactorily for small values of t, we use 7 in conjunction with 8 and obtain the following (see app., p. C-29). +ertc(|)~<1^2") orfc0vr)} (10a) where as before. a? To a very close approximation, when XiSO.Ol, equation 10a simplifies to q= 1—^=1-0.5642VX> (10b) and 05642VX. (10c) By figure 5 Standard7 Run Suite Date Start of run Suite sampled Temperature* (°C) (cm Vb per sec; vp Gradient * Diffusivity Dm 5 (cm* per sec) deviation (cm) 1 1A 1/22/58 ? 4:00P 26 0. 0141 0. 0381 0. 0655 0. 215 0. 19 0. 63X 10-5 290 1. 04 IB do. ? 6:00P 26 . 0139 . 0376 . 0655 . 212 . 18 . 63 290 1. 06 1C 1/24/58 ? 3:40P 27 . 0139 . 0376 . 0645 . 216 . 18 . 63 290 1. 09 2 2A 1/30/58 12:15P 3:30P 28 . 0245 . 0663 . 114 . 215 . 18 . 64 500 1. 34 2B do. do. 5:00P 28 . 0244 . 0660 . 113 . 216 . 18 . 64 490 1. 28 2C 1/31/58 11:30A 2:05P 27 . 0238 . 0644 . 112 . 212 . 18 . 63 490 1. 33 3 3A 2/3/58 12:30P 3:25P 25 . 0184 . 0497 . 0900 . 204 . 18 . 63 380 1. 20 3B do. do. 4:50P 25 . 0181 . 0489 . 0885 . 204 . 18 . 63 370 1. 21 3C do. do. 7:40P 25 . 0176 . 0476 . 0880 . 200 . 18 . 63 360 1. 19 4 4A 2/10/58 11:30A 3:35P 24 . 00811 . 0219 . 0427 . 190 . 17 . 63 170 1. 14 4B do. do. 7:10P 24 . 00731 . 0198 . 0378 . 193 . 18 . 63 150 1. 28 5 5A 2/11/58 10:45A 1:10P 24 . 0320 . 0865 . 160 . 200 . 18 . 63 660 1. 16 5B do. do. 2:30P 24 . 0312 . 0843 . 157 . 199 . 18 . 63 640 1. 14 6 6A 2/14/58 1:30P 4:10P 24 . 01365 . 0369 . 0700 . 195 . 18 . 63 280 1. 11 6B do. do. 6:05P 24 . 0128 . 0346 . 0662 . 193 . 18 . 63 260 1. 11 7 7 A 2/19/58 11:40A 2:40P 23 . 0223 . 0603 . 1170 . 191 . 18 . 63 460 1. 14 7B do. do. 4:15P 23 . 0220 . 0595 . 1153 . 191 . 18 . 63 450 1. 13 8 8A 2/26/58 12:00N 2:40P 28 . 0183 . 0495 . 0858 . 213 . 18 . 64 370 1. 12 8B do. do. 4:25P 28 . 0174 . 0470 . 0812 . 214 . 18 . 64 350 1. 14 8C 8/27/58 1:00P 3:25P 27 . 0164 . 0443 . 0785 . 209 . 18 . 63 340 1. 23 9 9A 3/12/58 12:00N 2:00P 29 . 0605 . 164 . 291 . 208 . 17 . 64 1230 1. 01 9B do. do. 3:25P 29 . 0594 . 160 . 286 . 208 . 17 . 64 1200 1. 01 10 10A 3/19/58 12:10P 2:30P 28 . 0324 . 0875 . 162 . 200 . 17 . 64 660 1. 06 10B do. do. 4:05P 28 . 0316 . 0855 . 157 . 201 . 17 . 64 640 1. 03 11 11A 3/28/58 11:00A 3:40P 29 . 00383 . 01035 . 0185 . 207 . 17 . 64 78 1. 31 11B do. do. 6:25P 29 . 00306 . 00828 . 0146 . 209 . 17 . 64 62 1. 42 11C do. do. 7:15P 29 . 00293 . 00792 . 0140 . 209 . 17 . 64 59 1. 57 12 12A 4/3/58 11:15A 3:35P 28 . 00387 . 01045 . 0202 . 192 . 16 . 64 78 1. 24 12B do. do. 5:00P 28 . 00327 . 00884 . 0172 . 190 . 16 . 64 66 1. 31 12C do. do. 6:00P 28 . 00322 . 00870 . 0168 . 192 . 16 . 64 65 1. 35 13 13A 4/30/58 2:30P 3:55P 28 . 0859 . 232 . 3940 . 218 . 18 . 64 1740 . 89 13B do. do. 5:18P 28 . 0837 . 226 . 3835 . 218 . 18 . 64 1700 . 89 5 Coefficient of diffusivity adjusted for temperature change by T formula DM=7frD' M where, T is absolute temperature of water I 20 72o = 293°K ( = 20°C) Z)'m=coefficient of molecular diffusivity at 293°K = 0.62X 10~5 cm1 2 3 4/sec. 6 H=~— is dimensionless parameter discussed in text as equa- JJm tion 12c; d = 0.048 cm. 7 Figure shown is standard deviation taken from graphs, figs. 15-27. 1 Temperature shown is that of influent water; effluent water never more than a few tenths of a degree different. Db V h 2 Bulk velocity vb =Q/A. Pore velocity vp=------- porosity 0.37 where Q is discharge in cm3/sec. A is area of cross section= 100 cm2. 3 Figure shown is average of gradient measured by manometers on each side of porous block. 4 Permeability computed by relationship P=Q/AI. P<=Permeability at temperature of experiment. Pm=Permeability at temperature 20° C __p Kinematic viscosity of water at temperature t 1 Kinematic viscosity of water at 20° CC-22 FLUID MOVEMENT IN EARTH MATERIALS At the outset, it should be pointed out that most uncontrolled variables, such as the small change with time in the permeability of the medium and in the hydraulic gradient, or undetected nonuniformities3 of the medium, or adsorption of tracer on the medium, either will have no effect on dispersion or will act to increase the dispersion over that which would cocur in an ideal experiment. In other words, the measured dispersion may be thought of as the sum of two parts: (1) the dispersion that would occur under ideal experimental conditions, and (2) the dispersion caused by experimental conditions that departed from ideal and that may not be identifiable or, if identifiable, may not be measurable. The question of uniformity of the medium was perhaps the most perplexing, and in attempting to assess it, an experiment was performed as follows: (1) The block was completely saturated with clear water; (2) a dye solution was introduced simultaneously on both sides of the centerline along the entire base of the block; (3) the progress of the dye front through the block was then observed. If the block were absolutely uniform, all parts of the dye front would travel at the same average velocity and the front would appear as a horizontal line moving upward through the block. In point of fact, a bulge developed at one side and the fluid in that portion of the block moved about 50 percent faster than the overall average. In a repetition of the experiment, at a different velocity, the bulge developed in exactly the same place, showing that the flow was stable, but that the block was not completely uniform. Also, as was previously pointed out, the block was built up in a series of layers so that a number of “bedding planes” existed perpendicular to the flow direction. According to the theory developed herein, transverse dispersion is velocity-dependent, but a velocity difference of 50 percent in the range studied is not enough to produce a measurable change in dispersion. However, if the nonuniformity produced a circuity greater than zero (contrary to the original premise), then the dispersion could be affected measurably, but independently of velocity. Therefore, the nonuniformity of the medium either produced no measurable effect on dispersion or it affected dispersion measurements at all 3 Uniform is here used in the sense that the properties, such as porosity, grain size, grain orientation, and permeability, of any small region (containing, say, 10 to 100 grains) have values nearly identical to those of any other small region, so that (1) the standard deviation of any property of a large number of small regions, taken along any line through the medium, will be close to zero, and (2) the average of any property taken along a line will approach the same value as the average of the same property taken along any other parallel line, as the length of the lines increases. A medium is also isotropic if the average of any property will approach th^ same value regardless of direction. Since absolute uniformity is a mathematical convenience having no real counterpart in nature, the term nonuniform used herein merely implies a gross or readily measurable departure from the conditions described above. velocities to the same degree. Furthermore, if the effect of the nonuniformity on one side of the block was substantially different from that on the other side, a skewed concentration distribution should result. This did not occur. The 24 spouts through which the fluid was discharged were individually adjustable. At relatively high flow velocity, it was possible to adjust the spouts so that individual departures from average rate of discharge did not exceed 15 percent. At low velocity, however, individual departures from average were as much as 100 percent. Since the forces producing these variations were associated with the spouts, the effect on flow through the medium would be confined to the immediate vicinity of the upper flow-separation plates. At most, this type of error would cause a scattering of the points on the concentration-distribution curve. As is seen, except for the extreme values, all points fit the curves rather closely. The migration of the interface during the course of a run would tend to skew the dispersion. In run 13, where velocity of flow was maximum of all runs, the rate of sideways migration at the spouts was about 0.02 mm per minute; in run 11, where velocity was minimum, the rate of migration was about 0.10 mm per minute. The reason for the difference is that, when flow velocity is low, small changes in entrance resistance (which caused the migration) are large compared to total gradient; when velocity and gradient are high, then the changes in entrance resistance are almost negligible by comparison. In run 11, the average time required for fluid to move through the porous block was about 220 minutes. In this interval the sideways migration amounted to about 2 cm. In run 13 the average time for fluid movement through the block was 8.6 minutes; during this interval sideways migration was about 0.02 cm. Hence, it is concluded that (1) at high-flow velocity the maximum possible effect of interface migration is less than the error of measurement of the standard deviation and may be neglected; (2) at low-flow velocity the effect may not be neglected on this basis but may be treated as follows: To begin with, if the effect is significant, the distribution should be skewed; that is, the dispersion should be greater on the side away from which the interface migrated. An examination of the plotted points on suites 11B and 11C (fig. 25) show this to be the case. On the other hand, the dispersion on the side toward which the interface is migrating should be compressed. This is because the flow lines, which started at the base of the block by occupying half the width of the base, end by occupying less than half. Therefore, if only the latter points are considered, the standard deviation based on those points should give a value that is tooTRANSVERSE DISPERSION IN LIQUID FLOW THROUGH POROUS MEDIA C~23 low. In order for one to be on the conservative side, insofar as this analysis is concerned, the latter points are taken as correct. Also, it should be pointed out that the interface migrated by moving parallel to itself except in the region immediately above the flow-separation plate at the base of the block. In other words, the flow adjustment required by the changing head relationship was accomplished close to the base of the block. Also, no tendency to “smear” could be seen; that is, near the base the interface was always clear and sharp in spite of the slow migration. Farther along, of course, the interface became fuzzy independently of migration, as dispersion progressed. It is concluded that, in spite of the various departures from ideal experimental conditions mentioned above, the measured values of dispersion for various flow velocities are valid. To provide some notion of the meaning of these data in terms of dilution and dispersion over long distances within an aquifer, the following idealized analogy is proposed. Assume an aquifer of finite thickness but of infinite area, whose physical characteristics are similar to those of the experimental medium; assume an injection well that penetrates the full thickness of the aquifer; and assume that a solution containing tracer elements in concentration C0 is injected into the well. As a function of the distance downgradient from the injection well, compute the width of the band of tracer elements between points where relative concentration C/C0=0.05, and compute the change in maximum concentration of the tracer elements, under the following conditions: (1) ground water velocity is 22.5 ft per day (equivalent to pore velocity of run 11C), and ground-water velocity is 658 ft per day (equivalent to pore velocity of run 13A), and (2) initial width within the aquifer of injected band of tracer elements is 1 foot, initial width is 2 feet, and initial width is 4 feet. The coefficient of bulk dispersion D is calculated for each case from values of standard deviation obtained in runs 11C and 13A, according to the relationship D=a2/2t. The resulting computations are shown graphically on figure 28. It is seen, for example, that 10 miles down-gradient maximum concentration, which would occur along the centerline of the band, ranges from 44 percent of initial concentration for the faster velocity and 4-foot initial band width to 6.6 percent for the slower velocity and 1-foot initial width. The width of the band containing tracer elements at relative concentration of 5 percent or higher, at 10 miles downgradient, ranges from 8.8 feet for the faster velocity and 1-foot initial width, to 22 feet for the slower velocity and 4-foot initial width. It is also seen that the change in maximum (centerline) concentration is sensitive to initial band width, and that the wider the band the longer the distance of downstream travel before the maximum concentration begins to change. In addition to the cases shown on figure 28, it may be calculated that where ground-water velocity is 658 feet per day, a band of 24 feet initial width would travel 10 miles before its centerline concentration would be affected; where ground-water velocity is 22.5 feet per day, a band of 42 feet initial width would travel 10 miles before its centerline concentration would be affected. In this connection it should be noted that in practical cases of fluid injection into permeable formations, the head on the injected fluid is apt to be considerably higher than that of the ground water in the formation. Consequently, the initial width of band leaving an injection well is apt to be many times the diameter of the injection well. COMPARISON BETWEEN THEORY AND EXPERIMENT In a previous section (see equation 4c) the following was derived For a given porous medium the denominator of the term on the right side is constant, and the term as a whole is dimensionless. This term may be used to transform the measured standard deviation of dispersion, tr, into dimensionless form. Or, where a* replaces yp for convenience in terminology. The length L is obtained by direct measurement of the porous block. The characteristic grain diameter d is, in general, not well defined, but for artificial media of nearly constant grain size it may be taken as the average grain size (fig. 14). A first approximation of the dimensionless constant a can be made by assuming values that seem intuitively reasonable. Referring to figures 4 and 5 and recalling that a=a^a2, one assumes that a^d (average distance be tween=0.5r/; adjacent flow paths) that is, «i = 0.5; a2d (average distance between=0.7c?; successive pore spaces along a flow path) that is, <*2=0.7.C-24 FLUID MOVEMENT IN EARTH MATERIALS DISTANCE DOWNGRADIENT, IN MILES 0.01 0.1 1.0 10. Figure 28.—Dispersion plotted against distance in imagined aquifer. Computations are based on formula: dr -I/erf W/2~V I erf w/2+?A 6/C°-2\ert 2jDt +ePi 2jDi J where, w=initial width of band <=time required to reach downgradient point indicated y — transverse distance from centerline of band vv = ground-water velocity /) = coefficient of bulk dispersion= 0.04 for the porous block used in the experiments. It can be shown by use of equation 18a (below) that, when R > 0.04, the standard deviation of the dispersion is less than 0.003 cm in the average time required for fluid to move through a pore 0.01 cm long. Hence, the bulk of tracer elements would be carried out of a pore space by fluid flow before they could move out by molecular diffusion. “Unrestricted” molecular diffusion considers that tracer elements may move both within and out of pore spaces by molecular diffusion. In the first idealization listed above, that of dispersion caused by unrestricted molecular diffusion with no amplification by the medium, diffusion would occur indpendently of fluid flow. Consider a porous medium (such as that used in the experients) half of which is occupied by fluid containing an initially uniform distribution of tracer elements flowing parallel to and in contact with fluid in the other half containing no tracer elements. Dispersion of tracer elements across the interface will occur as a function of time and of the coefficient of molecular diffusivity. It can be shown that in a semi-infinite region the standard deviation of the diffusion across the interface is a—V2 Drat. (18a) In the case of a porous medium it is necessary to take the “lithology” into account, and <7=V 2 DmJ t—■%] 2DMj — (18b) 1 vp The lithologic factor, /, accounts for the effect of the solid boundaries of the medium which inhibit the diffusion of the tracer elements. For media of nearconstant grain size its value is always close to 0.71 (R. P. Rhodes, oral communication, 1959). From table 1, Dm=0.64X 10-5 cm2/sec, and Z=length of porous block=119 cm. We substitute these values in equation 18b V»P (18c) and by equation 16b o-*=0.77 (/?>0.05) the mechanism of molecular diffusion without amplification by the medium has a small effect on dispersion, and cannot account for the experimental results. The second idealized mechanism listed is that of unrestricted molecular diffusion with amplification by the medium, and with zero circuity. By examination of figure 29, it would appear that this mechanism is of special importance in the range H*=0.7 to H*=7 (R=0.005 to I?=0.05). This is an intermediate range where the assumptions leading to the construction of both the solid-line curve and the dashed-line curve break down. Hence, it is concluded that in this range a theoretical curve should be drawn that is higher than both the curves just mentioned. The labor of performing a mathematical analysis of this case appears to be formidable and, for the present purpose, unnecessary. A curve of “reasonable fit” (dotted line) is drawn by inspection on figure 29. The experimental data lie almost wholly outside the range covered by this case and thus are not affected. The third mechanism postulated is that of zero molecular diffusion and nonzero circuity. By definition, when circuity is nonzero, any two originally contiguous fluid paths will tend to separate farther and farther apart as flow proceeds downgradient, independently of the molecular diffusion of the tracer elements. (It is well to note, however, that in practice fluid paths always are identified by the motion of tracer elements.) Because of the requirements of continuity, if any two contiguous paths separate, other paths must enter to fill the intervening space. The process is the sum of a large number of independent random events, and, by the central limit theorem, it should be expected to produce a normal distribution of tracer elements across an originally planar interface (such as in the experimental arrangement). In addition to being independent of molecular diffusion, the fluid paths are fixed (within the range of validity of Darcy’s law) and do not change with change in velocity. Circuity, then, is a characteristic of the medium and is a constant. It is a matter of observation (H.E. Skibitzke, oral communication, 1959) that flow through grossly non-uniform media is circuitous, within the meaning of circuitous as previously defined. Fluid paths are deflected away from the direction of bulk-flow gradient in response to spatial changes in media permeability, and paths (identified by dye streams) are seen to be separated, twisted, and inverted. Whether or not dispersion resulting from aquifer nonuniformity would be normally distributed probably depends on the magnitude and shape of the nonuniformity. Probably, certain kinds of nonuniformity could yield a normally distributed dispersion. The significant point to be noted is dispersion in grossly nonuniform media may be orders of magnitude greater than the dispersion either measured or calculated for uniform media. A question may now be raised that appears to lead to an explanation of the twofold difference in magnitude between prediction and experiment indicated on figure 29. Can any randomly packed granular medium be exactly uniform? The two words “randomness” and “uniformity” are mutually exclusive except in a statistical sense. Any randomly packed “uniform” medium, will consist of a multitude of small regions, the size of each being, say, on the order of a few grain diameters, and each having a permeability slightly different from all the others. The medium is called uniform when their sum-total (that is, their statistically expected) effect on flow is the same in all parallel sections of the medium. The medium is also isotropic when their sum-total effect is the same in every direction. Therefore, if circuity is a nonzero function of medium nonuniformity, and if, by the argument just given, all randomly packed media are at least slightly nonuniform, it follows that the circuity of the experimental porous medium was in fact nonzero. A nonzero circuity could account for the nearly constant difference in magnitude between prediction and experiment, but could not account for the observed decrease in dispersion with increase in fluid velocity. The writer concludes that the observed dispersion may best be explained by assuming that a velocity-dependent dispersion resulting from tracer-element diffusion amplified by the medium is superposed on a constant dispersion which results from circuity of the medium. The fourth case listed, that of unrestricted molecular diffusion with amplification by the medium and withC-28 FLUID MOVEMENT IN EARTH MATERIALS nonzero circuity, includes all postulated mechanisms. For reasons already given, the effect of medium amplification becomes unimportant at very low values of the H number; at high values of the II number the effect of molecular diffusion beyond individual pore spaces becomes unimportant. At intermediate values of this number the two mechanisms are of comparable magnitude. Medium circuity is assumed constant at all values up to the limit of validity of Darcy’s law. The portions of the theoretical curves given on figure 29 appropriate to each of the various ranges, and for the particular medium studied, were previously indicated. The effect of nonzero circuity may be included by increasing the magnitude of all theoretical curve values by some constant amount, say 0.5 in this case. The fifth case listed, that of restricted molecular diffusion with amplification by the medium and with zero circuity, is the model analytically developed in this report. It is not a comprehensive description of transverse dispersion, but it probably is the most complex combination of mechanisms that it is practical to treat analytically. The further development of analytical procedures probably will require the accumulation of systematic experimental data; in particular, data on the effects of increasing nonuniformity of the medium will be required. REFERENCES Crank, J., 1956, The mathematics of diffusion: London, Oxford Univ. Press, 347 p. Feller, William, 1950, An introduction to probability theory and its application: New York, John Wiley & Sons, 419 p. Kitagawa, K., 1934, Sur le dispersement et lYcart moyen de lYcoulement des eaux souterraines: Kyoto (Japan) Imper. Univ., Coll. Science Mem., ser. A, v. 17, p. 37-42. McBain, J. W., and Liu, T. H., 1931, Diffusion of electrolytes, non-electrolytes, and colloidal electrolytes: Am. Chem. Soc. Jour., v. 53, p. 59-74. Miller, K. S., 1953, Partial differential equations in engineering problems: New York, Prentice-Hall, 254 p. Pai, Shih-I, 1956, Viscous flow theory, I-Laminar flow: New York, Van Nostrand Co., 384 p. Rifai, M. N. E., Kaufman, W. J., and Todd, D. K., 1956, Dispersion phenomena in laminar flow through porous media: California Univ., San. Eng. Research Lab., Prog. Rept. 2, 157 p. Scheidegger, A. E., 1958, Typical solutions of the differential equations of statistical theories of flow through porous media: Am. Geophys. Union Trans., v. 39, no. 5, p. 929-932. Taylor, Geoffrey, 1953, Dispersion of soluble matter in solvent flowing slowly through a tube: Royal Soc. (London) Proc. A, v. 219, p. 186-203.APPENDIX Derivation of equation 9a of text: C(y,t)=%+^ ± { 1)2 2 T n=1,3,5 ... n COS fki 6XP~(la)2 Dut (20a) Integrating both sides of 20a with respect to y over interval y=0 to y=a n— 1 (-i)~ P Cdy=^f^-^ ± Jo 2 7T n n=1,3,5. . . n 2 1 7T By equation 8 sin y exp—(|^) DMt (20b) (70a , 4C'0a ^ 1 /nif\2 „ , . .S..V“p-(2i) ^ (20C) i ra 2(aI0=^oJo Cdy 1,4 - 1 =o+Z2 X 172 exp- T n—1,3,5 . . . n (£j» 'Mt (20d) Note: When t= <», q= 1 When <=0, 2=^+4 X “2=4+H=l 2 2T n=l,3,5,.. n A A Derivation of equation 10a of text: The preceding solution for q(a,t) converges satisfactorily for moderate and large t (low fluid velocity), but not for small t (high fluid velocity). It turns out that an equation that will converge satisfactorily for small t can be developed from the error-function solution of the original differential equation. This solution of the differential equation as given by Crank (1956, p. 15), letting h=a, and L=2a is The error function is itself an integral, and a method integrating it with respect to y over the distance a is not apparent. However, the integral is easily differentiated with respect to y. We may take advantage of this fact by developing the following relationship: Start with the differential equation for diffusion in one dimension bC_n d2C bt Um by2 (23) Integrating with respect to y over the distance a (24a) Intrgrating left side of above, and interchanging order of integration and differentiation of right side, we get rfLKJ>» <24b> Because the fluid region is bounded at y=0 D, bC by hence, (24c) Referring to 22 for any a= constant, q is a function of time only. Hence, we may differentiate 22 with respect to time, substitute it in 24c, and get dq____1_ j-. bC dt C0a M by a (25a) We now integrate both sides with respect to time =A* r be 2 CoaJo by a dt -\~A (25b) C(y, t)=~ X erf ^ 71 = — oo .(1+4g_~y+erf 2 fDMt q(l—4n)+y (2l) We desire q(a, t), where where A=constant of integration when f=0, q=l=A Thus, (22) 4(.,0 = l+g dt (25c) C-29C-30 FLUID MOVEMENT IN EARTH MATERIALS Substituting 21 into 25c g(„,rnuz CaCl^ . 9 ± [erf (F,)+erf (F2)]')| \dt oy\ & n=—co /|aj (26) where, Ft= and, F2 g(l+4n)—y 2-jDMt a{\ — 4ra)-fy Interchanging order of summation and integration, 2(M) = 1+^ {|)[erf (FO+erf (F,)]L (27) where X= Dtft For X<0.01 equation 28, to a very close approximation simplifies to q= 1-^=1—0.564 VX (29) Relation between the Reynolds and H numbers: dvv jT=dvv v _ R DM Dm Dm Each term within brackets on the right side of equation 27 is first differentiated with respect to y, then evaluated at y=a, and then integrated with respect to t. This gives -XL* {{2n) erfc (I)-'1-2”)erfc (X5)} (28) r=Dmh=Du(4l_\h* v v \or/ for, a=0.6 (see text, eq. 17b) 7?ii/=0.64X10~5cm2 per sec r=0.01 cm2 per sec R=0.71X10~2 H* or, H*=140R. U S. GOVERNMENT PRINTING OFFICE : 1962 0—618550Review of Some Elements of Soil-Moisture Theory DOCUMENTS DEPftKIMENT OCT 3 119GZ ubrary UNIVERSITY Of CALIFORNIAReview of Some Elements of Soil-Moisture Theory By IRWIN REMSON and J. R. RANDOLPH FLUID MOVEMENT IN EARTH MATERIALS GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-D UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1962UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington 25, D.C.CONTENTS Page List of symbols____________________________________________ iv Abstract____________________________________________________ D1 Introduction_________________________________________________ 1 Zone of aeration_____________________________________________ 2 Background physics___________________________________________ 2 Surface tension_________________________________________ 2 Molecular causes of surface tension_________________ 2 Definition of surface tension_______________________ 3 Pressure differential across a liquid-gas interface- 3 Angle of contact__________________________________ 5 Rise of water in a capillary tube___________________ 5 Water vapor in unsaturated soils________________________ 6 Molecular causes of vapor pressure__________________ 6 Vapor pressure and temperature______________________ 6 Vapor pressure and amount of dissolved solute. 7 Vapor pressure and hydrostatic pressure in the liquid____________________________________________ 7 Barometric equation_________________________________ 7 Kelvin equation_____________________________________ 8 Derivation_____________________________________ 8 Significance__________________________________ 10 Diffusion______________________________________________ 12 Osmosis.___________________________________________ 13 Force, work, energy, potential, and gradient___________ 14 Potentials in soil water_________________________________ 17 Gravitational potential________________________________ 17 Hydrostatic-pressure potential_________________________ 19 Page Potentials in soil water—Continued Osmotic-pressure potential___________________________ D20 Adsorption potential__________________________________ 21 Thermal potential_____________________________________ 21 Chemical potential____________________________________ 22 Total potential and its measurement___________________ 22 Equilibrium distribution of potential and water in soil.. 24 Distribution in the zone of saturation---------------- 24 Distribution in the capillary fringe__________________ 24 Distribution in pinching and swelling tubes without horizontal connections__________________ 25 Distribution in tightly packed spheres____________ 27 Distribution above the capillary fringe--------------- 28 Distribution in tightly packed spheres____________ 28 Distribution in randomly distributed pores____ 29 Movement of soil moisture__________________________-— 29 Physical picture______________________________________ 29 Pore underlain by a cavity________________— 30 Pore underlain by pores of similar size----------- 31 Pore underlain by pores of dissimilar size---- 31 Randomly distributed pores________________________ 32 Differential equation for unsaturated flow------------ 33 Historical background.---------------------------- 33 Equation of continuity____________________________ 34 Darcy’s law__________,________________________ 36 Flow equation_________________________________ 36 References_________________________________________________ 37 ILLUSTRATIONS Page Figure 1. Intermolecular force as a function of molecule separation_________________________________________________________ D2 2. Bubble of air blown at the end of a capillary tube_________________________________________________________________ 4 3. Contact angle of a liquid-vapor surface near a solid_______________________________________________________________ 5 4. Rise of water in a capillary tube----------------------------------------------------------------------------- 5 5. Column of vapor in a gravitational field___________________________________________________________________________ 8 6. A curved liquid-vapor surface in contact with a free flat liquid surface___________________________________________ 6 7. Position of the meniscus in a capillary tube and adjacent vapor pressures_________________________________________ 12 8. Position of the meniscus in a wedge-shaped pore and adjacent vapor pressures______________________________________ 12 9. Ground-water gradient_____________________________________________________________________________________________ 16 10. Approximate flow net in the vicinity of an irrigation pond________________________________________________________ 16 11. Moisture characteristic of a fine sand____________________________________________________________________________ 23 12. Plot of the capillary-rise equation for water_____________________________________________________________________ 25 13. Hypothetical variation of pore radius with height above the water table___________________________________________ 26 14. Plot Of the capillary-rise equation on the same scale used in figure 13___________________________________________ 26 15. Unit rhombohedron of a mass of spheres in the most compact packing________________________________________— 28 16. Unit element of pore space in a mass of spheres packed in the most compact manner_________________________________ 28 17. Dewatering of a main pore and the reentrants______________________________________________________________________ 28 18. Pore underlain by a cavity_______________________________________________________________________________________ 30 19. Pore underlain by a pore of similar size__________________________________________________________________________ 31 20. Pore underlain by a smaller pore______________________________________________________________________________ 31 21. Pore underlain by a larger pore___________________________________________________________________________________ 32 22. Unit rectangular parallelepiped________________________________________________________________________________ 34 inLIST OF SYMBOLS a acceleration A area of a liquid surface b mole fraction of dissolved material C a constant d a digit or group of digits dS area of an infinitesimal element of capillary surface D density of a liquid or gas I)(9.) diffusivity as a function of concentration of ions or molecules e the exponential base, 2.71828 . . . j a digit F force or resultant force Fa force of adhesion Fc force of cohesion F„ force of gravity g acceleration of gravity G gravitational potential Gm gravitational potential in ergs per gram of water G, gravitational potential in ergs per cubic centimeter of water h water-table elevation Ah difference in water-table elevation i, j, k a system of unit direction vectors K coefficient of capillary conductivity l length of a capillary tube m mass M molecular weight o a digit p (a) pressure; (b) pressure on the concave side of a curved liquid surface; (c) pressure of the vapor in a capillary opening; (d) pressure of the vapor a distance above a free flat water surface; or (e) vapor pressure of a solution p0 (a) pressure of the vapor immediately above a flat water surface; (b) pressure on the convex side of a curved liquid surface; or (c) vapor pressure of a free pure solvent p' pressure of the liquid on the convex side of a meniscus pa pressure just over an element of capillary surface p'a pressure just under an element of capillary surface P hydrostatic pressure Q discharge or flow rate r radius of a capillary tube r0 critical distance of separation between the molecules of a liquid R radius of curvature of a hemispherical liquid surface R\ and R2 principal radii of curvature of a liquid surface IR the gas constant dS area of a small element of capillary surface T absolute temperature t time u a scalar point function such as potential v volume of a mass of water Vi specific volume of a liquid vv specific volume of a vapor V mass flow of moisture (a vector) Vx, Vv, Vz components of the mass flow of moisture in the x, y, z directions Vv volume flow of moisture (a vector) W work or energy x, y, z distances in a rectangular coordinate system z distance above or below a datum a angle between a force and direction of movement d moisture content on a dry-weight basis ^ specific moisture capacity 7 density of a gas or vapor A operator indicating change in value of a quantity t osmotic-pressure potential r? coefficient of viscosity 0 (1) angle of contact; or (2) angle between the incli- nation of the water table and the horizontal k diffusivity X heat of vaporization at constant temperature £ adsorption potential it 3.14159 . . . p density of liquid water p„ bulk density of a soil a surface tension 2 capillary potential total potential \p hydrostatic-pressure potential 12 concentration of ions or molecules V the vector operator del, or nabla IVFLUID MOVEMENT IN EARTH MATERIALS REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY By Irwin Remson and J. R. Randolph ABSTRACT This review was assembled from the existing literature to make available a convenient introduction to this subject, which is of interest to workers in many diverse fields of hydrology. Surface tension at the liquid-vapor interfaces largely controls the occurrence and movement of moisture in unsaturated soils. Surface tension is defined as the amount of work or energy required to produce a unit increase in the area of a liquid surface. The pressure on the concave side of a liquid-vapor surface in a round capillary tube exceeds the pressure on the convex side by an amount equal to twice the surface tension divided by the tube radius. The rise of liquid in a round capillary tube, if the angle of contact is assumed to be zero, is predicted by the capillary- 2cr rise equation pgz = —; where p is the density of the liquid, g is the acceleration of gravity, z is the height of capillary rise, a is surface tension, and r is the radius of the tube. Important soil-moisture phenomena result from the transfer of water molecules across liquid-vapor surfaces and from the diffusion of water vapor within the system. Vapor pressure decreases with increasing height above a free water surface according to the barometric equation. The vapor pressure over a water meniscus is less than the vapor pressure over a flat water surface by an amount that increases with surface curvature according to the Kelvin equation. When the barometric and Kelvin equations are compared, the capillary-rise equation shown above is obtained. This equation holds at equilibrium throughout the zone of aeration. The total potential of soil water is the minimum energy per gram needed to transport a test body of water from a water-table datum to any point within the liquid phase of a soil-water system at equilibrium. It is the scalar sum of six main component potentials: (a) gravitational potential, which increases with height above the water table; (b) hydrostatic-pressure potential, which is associated with surface tension at liquid-vapor interfaces; it becomes increasingly negative with increasing height above the water table at equilibrium, and the force associated with its negative gradient exerts a suction on any external body of water with which it is in contact; (c) osmotic-pressure potential, which is present in soil that shrinks upon drying; it is produced by osmotic pressures resulting from differences in ion concentrations at surfaces of soil particles having electric (Gouy) double layers; the force associated with its negative gradient also exerts a suction on externally applied bodies of water; (d) adsorption or adhesion forces, which strongly retain water near the surfaces of soil particles; adsorption potential is important only in soils that are drier than those normally encountered in nature; (e) gradients of thermal poten- tial, which occur when temperature differences occur in the system; they are relatively unimportant in causing moisture movement except when the temperature gradients are large; (f) finally, chemical potential, which is due to the osmotic energy of free ions in the aqueous solution; it is usually unimportant in nonsaline soils because solutes generally move more readily than the water with respect to the soil. In a hypothetical distribution of vertically swelling and pinching horizontally disconnected capillary tubes, there are positions of both minimum and maximum capillary rise. These are the most common positions of the top of the capillary fringe, which is generally at the maximum or minimum position according to previous fluctuations of the water table, whether rising or falling. A hypothetical model of rhombohedrally packed spheres differs from the vertically pinching and swelling tubes mainly in that isolated wedges of water remain above the capillary fringe as the water table falls. Water is either added to or taken away from the water wedges by vapor diffusion so that the capillary-rise equation is satisfied at equilibrium. Water may move in the zone of aeration as a liquid, a vapor, or both a liquid and a vapor. Vapor transfer occurs where a vapor continuum fills the pores. Where only the pore necks are filled with water, movement occurs by vapor transfer in the pores and by liquid transfer in the necks. Where the pore necks and parts of the pores are filled with liquid water, movement may occur by discontinuous jumps of liquid water from neck to neck. Finally, where a moisture continuum fills the pores, water may move completely as a liquid. The hydraulic conductivity of a wet soil decreases very rapidly as the moisture content decreases from its saturation value. Field-capacity and wetting-front phenomena are largely a result of this decrease of hydraulic conductivity with moisture content. The differential equation for unsaturated flow is derived from the equation of continuity and Darcy’s law. INTRODUCTION This report is not an exhaustive treatise on soil moisture. It is not meant to display a full understanding of the physics of the basic phenomena. Indeed, the writers do not possess such a thorough knowledge of this adjacent but alien field. Rather, it is intended to provide for the hydrologist a concise and easily mastered picture of some of the important phenomena and practical manifestations of soil-moisture theory. DlD2 FLUID MOVEMENT IN EARTH MATERIALS The Seabrook investigation of the U.S. Geological Survey, of which this report is a product, was made under the supervision of Allen Sinnott, district geologist, U.S. Geological Survey, Trenton, N.J. The cooperation of the Seabrook Farms Co. in all phases of the investigation is acknowledged with gratitude. ZONE OF AERATION The zone of aeration is between the water table and the land surface. It is “the zone in which the interstices of the functional permeable rocks are not filled (except temporarily) with water. The water is under pressure less than atmospheric” (Am. Geol. Inst., 1957, p. 325; after Meinzer, 1923b, p. 31). Thus, three physical phases exist in an unsaturated soil in the zone of aeration: the solid soil matrix, the liquid water, and the soil air which includes the vapor of the liquid. Because of the presence of these phases and the interfaces between them, the occurrence and movement of water in the zone of aeration are more complex than in saturated flow. The zone of aeration (vadose zone), according to Meinzer (1923a, p. 81; 1923b, p. 26), has three divisions: the capillary fringe, the intermediate belt, and the belt of soil water. The lowermost or capillary fringe is “a zone, in which the pressure is less than atmospheric, overlying the zone of saturation and containing capillary interstices some or all of which are filled with water that is continuous with the water in the zone of saturation but is held above that zone by capillarity acting against gravity” (Am. Geol. Inst., 1957, p. 44; after Meinzer, 1923b, p. 26). The intermediate belt is “that part of a zone of aeration that lies between the belt of soil water and the capillary fringe. It contains intermediate vadose water” (Am. Geol. Inst., 1957, p. 151; after Meinzer, 1923b, p. 26). The uppermost belt, or belt of soil water, is “that part of the lithosphere, immediately below the surface, from which water is discharged into the atmosphere in perceptible quantities by the action of plants or by soil evaporation” (Am. Geol. Inst., 1957, p. 29; after Meinzer, 1923b, p. 23). This report reviews some of the elements of soil-moisture theory, which deals largely with the unsaturated occurrence and movement of water in the zone of aeration. It reviews also some of the background physics needed for an understanding of this theory. BACKGROUND PHYSICS SURFACE TENSION Surface-tension phenomena associated with the liquid-vapor interfaces within the soil matrix have important effects upon the occurrence and movement of soil water. Part of the following description of these surface-tension phenomena is based upon Sears (1950, p. 319-330). MOLECULAR CAUSES OP SURFACE TENSION Whereas the average distance between the molecules of a gas at atmospheric pressure is about 10 times the size of each molecule, the molecules of a liquid are almost touching. The forces between the individual liquid molecules are largely electrical. As shown in figure 1, Figure 1.—Intermolecular force as a function of molecule separation. (After Sears, F. W., 1950, Mechanics, heat, and sound; 2d ed., Addison-Wesley Pub. Co., Inc., Reading, Mass., p. 321.) an attractive force exists between two molecules that are separated by a distance greater than some value, r0. (See “List of symbols.”) As the separation distance increases from r0, the attractive force first increases to a maximum and then gradually decreases. When the separation between two molecules of a liquid is less than r0, there is a large repulsive force. This repulsive force is responsible for the high pressures that are needed to compress a liquid. When the separation distance is r0, the repulsive and attractive forces equate to zero, and the molecules are in a state of equilibrium. A molecule in the interior of a liquid is surrounded by similar molecules. The molecule has thermal energy that tends to make it move (to become displaced from the equilibrium separation, r0). However, when displaced from r0, it is attracted by the adjacent molecules on one side and repelled by the adjacent molecules on the other side in accordance with the relation shown on figure 1. The combination of attractive and repulsive forces tends to restore the molecule to the equilibrium separation. As a consequence of the displacing effect, due to the thermal energy, and the restoring effect, due to the intermolecular forces, the moleculesREVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D3 in the interior of a liquid tend to oscillate about their equilibrium separation, r0. A molecule at the surface of a liquid generally has a component of vibration normal to that surface. When the molecule is moving outward from the surface, it is attracted back to the surface by the molecules below. However, there are no molecules above the liquid surface to give the repulsive force shown in figure 1. Therefore, the molecule may move out a greater distance than molecules within the body of the liquid. If the surface molecule has sufficient thermal energy it may escape (evaporate) from the liquid. The molecule performs a series of excursions out to a distance slightly greater than that of normal separation, and spends most of its time in a region where an inward force of attraction is exerted on it in accordance with the relation shown in figure 1. “The fact that the environment of those molecules in or very near the surface differs from that of the molecules in the interior gives rise to the surface effects we are npw considering” (Sears, 1950, p. 322). DEFINITION OF SURFACE TENSION A definite amount of work per unit area is required to increase the area of a liquid surface. The work can be recovered when the area decreases, so that the surface layer appears capable of storing potential energy. The enlarged liquid surface is not stretched like a balloon. Instead, additional molecules move into the surface layer and the final molecular spacing is unchanged. Unlike a balloon or rubber membrane, the work done, or energy of the surface, is directly proportional to the increase in the surface area of the liquid. The proportionality constant, or the work per unit area, is known as the surface tension, = the pressure, in dynes per square centimeter, at a distance z above the reference point v=specific volume of the gas T= absolute temperature R=gas constant, in ergs per degree gramD8 FLUID MOVEMENT IN EARTH MATERIALS Figure 5.—Column of vapor in a gravitational field, po is the vapor pressure at the free water surface and p, vapor pressure at a height, z above the free water surface. (After Edlefsen and Anderson, 1943, p. 129.) IR is equal to the universal gas constant, 83,140,000 ergs per degree grain mole divided by the weight of a gram mole. The molecular weight, or the weight of a gram mole of water vapor, is 18.02 grams. For water vapor, R would therefore be about 4,620,000 ergs per degree gram. Rewriting equation 25, and remembering that specific volume is the inverse of density, (26) where y is the density of the gas, in grams per cubic centimeter. The vapor pressure of the gas in the column in figure 5 changes by the amount — dp in going from A to B. This decrease in pressure is equal to the weight of the gas or vapor in the length of column represented by dz. The weight is ygdz. Thus, —dp=ygdz (27) Eliminating y by means of equation 26 and rearranging, JL p FT (28) Integrating from z= 0 at the free water surface where p—Po, to z=z where p=p, [lnp]£0=—j^.[z]o (29) and 1 Po FT (30) (Edlefsen and Anderson, 1943, p. 130). Equation 30 can be put into the final form p=p0e RT (31) Thus, if p0 is the vapor pressure at the free water surface, where 2=0, equation 31 determines the pressure p at a height 2. It is interesting that the rate of change of vapor pressure with height outside a capillary tube is much smaller as predicted by equation 31 than the rate of change of hydrostatic pressure with height within the tube, pgz. This is as predicted by equation 23. KELVIN EQUATION DERIVATION The vapor pressure over a concave water meniscus is less than the vapor pressure over a free flat water surface. Thus, a vapor-pressure deficiency exists over a meniscus in a capillary tube when compared with the vapor pressure over a free flat water surface. The relation between the vapor pressure and the radius of curvature of the meniscus is described by the Kelvin equation (Thompson, 1871). Because of the importance of this equation, it is now derived for a capillary surface connected to a flat surface following the derivation of W. O. Smith (1936, p. 228-230). Let the system in figure 6 be isolated so that there is no air pressure. Then p0 is the water-vapor pressure over the flat surface. The hydrostatic pressure in the liquid just under the flat surface also is equal to po and is due to the weight of the overlying water vapor. Let pa be the water-vapor pressure just over an element, dS, of capillary surface, and let p„' be the hydrostatic pressure just under the element, dS, of capillary surface. The hydrostatic pressure, p'a, under the element of curved surface equals the vapor pressure, pa, minus the pressure difference across the curved surface due to surface tension. If the system in figure 6 were open to the air, all the pressures would be increased by the atmospheric pressure. Furthermore, the change in atmospheric pressure with height would have to be considered, as well as the change in water-vapor pressure with height. Therefore when suitable adjustments are made theREVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D9 Figure 6—A curved liquid-vapor surface/ n contact with a free flat liquid surface. Po is the water-vapor pressure over the flat surface, pa, the water-vapor pressure just over and p'a, the hydrostatic pressure just under an element, dS, of capillary surface at a height, *, above the flat surface. (After Smith, W. O., 1936, p. 229). following derivation is applicable regardless of the type of gas in the system: The pressure in the water in the capillary tube at the same level as the flat surface is p0=Pa+pgz (32) Similarly, the pressure in the vapor outside the capillary tube just over the flat surface is Po=Pa+Jo ygdz (33) where y is the density of the vapor. Equating equations 32 and 33, Pa+J^ ygdz=Pa + pgz (34) This can be rearranged to Pa—Pa = pgz— f ygdz (35) Jo The value for the difference of pressure across the curved surface can now be substituted from equation 10: v{k+^)=pgz-^i/gdz (36) taking the radii of curvature as positive. The two terms on the right side of equation 36 give the pressure differences between the flat and curved water surfaces in the water and vapor columns, respectively. To make them more usable, everything will be put in terms of pressure. This can be done by starting with the simple relationship between pressure and height in the vapor, equation 27. and ' J>0 (p—Po)=—Jo ygdz z (37) p, ygdz (38) ygdz (39) Equation 39 may be substituted for the second term on the right side of equation 36, giving (^+]^)=PSf2+(p—Po) (40) To put the first term on the right side of equation 40 into terms of pressure, equation 37 is rewritten as dz = dp yg Setting up the integral between 0, p0 and z, p, (41) (42) Integrate the left side, and z= (43) We cannot integrate the right side of equation 43 because y, the density of the vapor, varies with pressure. To eliminate 7 from equation 43, consider Boyle’s law: or y_=P. To Po (44) (45) Putting this value of 7 into equation 43, it becomes Integrating, _ Po rp dp gy0 Substituting for p0 in equation 49 and remembering that p, the density of water, is the inverse of vh the specific volume of water, we arrive at the following forms of the Kelvin equation: r <» and _2a Vi p=p„e r RT (53) The minus is in equation 53 because the values of Ru R2, and r were assumed as positive if the meniscus is concave to the va-por. If positive values of r are substituted into equation 53, the pressure over the meniscus comes out less than that over a flat surface. Using the form shown in 53, «the radius of curvature is negative when the meniscus is convex to the vapor, as in a raindrop. If a negative value of r is inserted in the equation, the vapor pressure over a raindrop comes out greater than that over a flat surface. Equation 53 applies where the capillary surface is in liquid continuity with the flat water surface. The equation may be derived, from thermodynamic considerations (Edlefsen and Anderson, 1943, p. 142-146), for a curved liquid-vapor interface that is separated from the flat liquid surface. For the discontinuous-liquid system, we again take the radius of curvature of the interface to be positive for a liquid surface concave to the vapor and negative for a surface convex to the vapor. From the equation, the smaller the radius of curvature of a capillary meniscus, or the greater the curvature of the meniscus, the greater is the vapor-pressure deficiency above the meniscus. SIGNIFICANCE Equating the exponential terms in equation 53, the Kelvin equation, and equation 31, the barometric equation, rearranging terms, and substituting density for the inverse of the specific volume of the water, we arrive back at equation 18 Equation 54 does not contain any of the properties of the gas. Thus, the height of capillary rise of a given liquid in a tube or interstice of given radius is the same whether the surrounding atmosphere consists of water vapor alone or water vapor and air. The Kelvin equation may be looked at in several ways. If suction is applied to a wet sand, the vapor pressure is decreased. Therefore, the radii of curvature of the equilibrium minisci are greatly reduced, and the menisci are drawn deeper into the narrowing necks of the soil pores. The hydrostatic pressure in the water on the convex sides of the menisci also is decreased because of the pressure differentials across such curved surfaces. Adam (1941, p. 13-14) gives a physical explanation of these phenomena: The vapor pressure over a convex surface is greater than that over a plane; and over a concave surface it is less. The difference depends on the fact that condensation of vapor on a small convex drop of a liquid increases its surface area, so that the surface tension tends to oppose the condensation and to increase the vapor pressure. On a plane surface condensation does not alter the surface area, and on a concave surface the surface area is diminished by condensation of more vapor, so that the surface tension aids condensation in this case. Adam (1930, p. 22) gives solutions of the Kelvin equation for the increase of vapor pressure, p/po, over a drop of water of radius r, at 20° C, with p0 equal to 17.5 mm of mercury: r (cm) Pi Po -10-4 1.001 -10-5 1.011 -10-' 1.114 -10-7 2.95 Thus, a small change in vapor pressure greatly changes the radius of the equilibrium meniscus. To demonstrate the use of these equations, assume that a suction equivalent to 15 atmospheres is applied to a soil. This value of negative hydrostatic pressure is commonly taken for the permanent wilting percentage. Let us compute the radius of the circular pore in which the menisci will be lodged at equilibrium.REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY Dll For this case, Vi= 1 cm3 per gram ^=57,800 cm3 per gram dP= 15 atmospheres= 15,194,000 dynes per cm2 Therefore, the vapor-pressure deficiency can be computed from equation 23. It is dp=263 dynes per cm2. Thus, even at the permanent wilting percentage, the vapor-pressure deficiency in the soil is very small. (See also p. D7.) To determine the equilibrium meniscus, or the pore that has a sufficiently small radius to support the equilibrium meniscus, we refer to the Kelvin equation. For a temperature of 20°C, p0, the vapor pressure over a flat water surface, may be taken as 17.5 mm of mercury. p0=17.5 mm of mercury=23,300 dynes per cm2 where Q=the concentration of ions or molecules f=time z = distance D(Q) is the diffusivity which is a function of the concentration, 0. This equation states that the change dQ in concentration with time, ^ > is a function of the rate of change with distance, of the product of the diffusivity, D{U), and the concentration gradient,-^-OSMOSIS Suppose a vessel containing a sucrose solution is placed in contact with one containing plain water. There will be a diffusion-pressure gradient for sucroseD14 FLUID MOVEMENT IN EARTH MATERIALS from the solution toward the water and a diffusion-pressure gradient for water from the water toward the solution. According to the principle of independent diffusion, water molecules and sucrose molecules will diffuse in opposite directions until the solute concentration and the diffusion pressure become uniform. Suppose that the two vessels are connected through a differentially permeable or semipermeable membrane that is permeable to the solvent (water) but impermeable to the solute (sucrose). Water will diffuse into the sucrose solution but the sucrose will not be able to penetrate the membrane and diffuse into the pure water. This is called osmosis. As a result of the net movement of solvent into the solution, the total pressure in the solution becomes greater than that in the pure solvent at equilibrium. We say net movement because solvent actually moves in both directions across the membrane. However, at equilibrium, the movements in both directions are equal. Thus, osmosis may be considered as the building up of a differential total pressure in one part of a system through the restricting of diffusion in one direction and the allowing of diffusion in the other direction by means of a differentially permeable membrane. “Osmotic pressure may be defined as the maximum pressure which can be developed in a solution when separated from pure water by a rigid membrane permeable only to water” (Meyer and Anderson, 1939, p. 94). The osmotic pressure of a solution can be determined by enclosing it in a vessel formed of a rigid membrane permeable only to water, immersing this vessel in pure water, and exerting just enough pressure on the solution to prevent any increase in its volume due to the entrance of water. The osmotic pressure of the solution is quantitatively equal to the imposed hydrostatic pressure (Meyer and Anderson, 1939, p. 94-95). It is common practice to speak of solutions possessing osmotic pressures whether or not they are under such conditions that a hydrostatic pressure can develop within them. In other words, the term “osmotic pressure” is commonly used to denote the potential maximum hydrostatic pressure which would develop in a solution were it placed under the necessary conditions. It will be used in this way in this report. Osmotic pressure is used sometimes to designate also the actual hydrostatic pressures or turgor pressures developed as a result of osmosis. Actual hydrostatic (turgor) pressures developed during osmosis seldom equal the osmotic, or potential maximum, pressure. For example, assume that a solution having an osmotic, or potential maximum, pressure of 12 atmospheres is enclosed in a stoppered vessel formed of a membrane permeable only to water. Assume further that the vessel in turn is immersed in a solution having an osmotic pressure of 8 atmospheres. Water will diffuse inward until, at equilibrium, the actual hydrostatic (turgor) pressure developed in the internal solution will be 4 atmospheres. Its osmotic, or potential maximum, pressure will still be nearly 12 atmospheres. Even if the external liquid is pure water, the actual hydrostatic (turgor) pressure developed in the internal solution will not be equal to its original osmotic pressure unless the membrane is completely inelastic (Meyer and Anderson, 1939, p. 94-95). Because osmotic pressures are defined in terms of final hydrostatic (turgor) pressures developed by osmosis, flow occurs in the direction of greater osmotic pressure. For example, flow occurs from pure water (lower osmotic pressure) into a solution (higher osmotic pressure). However, flow is always from the region of greater to the region of lesser partial pressure of the diffusing substance. This is because the region of greater osmotic pressure is the region of lesser partial pressure of the diffusing substance. When a substance is dissolved in water, the diffusion pressure of the water in the resulting solution is decreased and its osmotic pressure increased as compared with that of pure water at the same temperature and pressure. The diminution of diffusion pressure is proportional, within a wide range of solution concentrations, to the number of solute particles in a given volume of the solvent. The osmotic pressure of a solution is a measure of the diffusion-pressure deficit of the water in that solution. Suppose that solution A has an osmotic pressure of 20 atmospheres and solution B an osmotic pressure of 12 atmospheres. Then the diffusion-pressure deficit of A is 20 atmospheres, and that of B is 12 atmospheres. Water moves toward solution A where the diffusion-pressure deficit, or the osmotic pressure, is greater. After the water movement, solution A will exert an actual hydrostatic (turgor) pressure of 8 atmospheres at equilibrium, and the membrane will exert a wall pressure of 8 atmospheres to contain it. Because the actual pressure is greater in A at equilibrium, the diffusion pressure of water in A will increase, and the diffusion-pressure deficit of the water in A will no longer be 20 atmospheres, but will be 12 atmospheres. Thus at equilibrium the increase in actual hydrostatic, or turgor, pressure inside the solution raises the diffusion pressure of the water in the solution to equal that in the outside water (Meyer and Anderson, 1939, p. 97). FORCE, WORK, ENERGY, POTENTIAL, AND GRADIENT The following sections of this report are devoted largely to discussions of potentials and forces in soil water. Therefore, it is advisable to discuss first the meaning of force, work, energy, potential, and gradient.REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D15 Force is commonly defined as a push or pull (Sears, 1950, p. 1). It may be thought of as the cause of acceleration or the cause of a change in the state of rest or state of uniform motion of a body. The relation between the force on a body and its mass and acceleration is described by Newton’s second law of motion (Sears, 1950, p. 78) F—ma (56) where m=mass, in grams a=acceleration, in centimeters per second squared F= force, in gram centimeters per second squared =dynes When a force acts on a body the product of the force and the distance the body moves in the direction of the force is the work performed by the force. Mathematically, W=Fx cos a (57) where F=the force causing the motion, in dynes a;=the distance that the body is moved by the force, in centimeters a=the angle between the direction that the force is acting and the direction that the body is moving H7=the work done, in dyne centimeters, or ergs Energy is the capacity for doing work. A moving body possesses kinetic energy equal to the work it can do before it is brought to rest. By definition it is equal to the product of one-half the mass and the square of the velocity of the body. Potential energy is the capacity that a body or system of bodies has for doing work by virtue of its position or configuration (Smith, A. W., 1948, p. 45). For example, a spring expanded beyond its unstretched length has potential energy because it can do work when it contracts. Similarly, a body at some elevation above a gravitational datum has potential energy with respect to that datum because it can do work as it, falls from the higher elevation to the datum. The amount of potential energy it has is proportional to the height above the datum. In other words, the higher it is, the more work it can do upon falling to the datum. Because energy is the capacity to do work, it has the same dimensional units as work, ergs. A scalar “is any quantity which although having magnitude does not involve direction. For example, mass, density, temperature, energy, quantity of heat, electric charge, potential, ocean depths, rainfall, numerical statistics such as birth rates, mortality or population, are all scalar quantities” (Coffin, 1911, p. 1). Any quantity which involves both magnitude and direction is a vector quantity. “Any vector quantity may be represented graphically by an arrow” (Coffin, 1911, p. 1). According to d’Abro (1951, p. 71), the name" field” is given to the continuous distribution of some “condition” prevailing throughout a continuum. When the condition is adequately described at each point of space by a scalar, it is known as a scalar field. Temperature is such a condition and the temperature distribution throughout a volume is a physical illustration of a scalar field. In many cases the condition at each point of space has a direction as well as a magnitude, and the field is a vector field. A field of force or the field defined by the instantaneous velocities of the different points of fluid in motion are illustrations of vector fields. Consider a number of points in space. Suppose that with each of those points there is an associated value of potential. Potential is the work required to take a unit mass or volume from a datum to the point in question against the field forces. The potential might be due to gravitational, mechanical, or electrical causes. The potential at a point may be thought of as the potential energy of a unit body at the point. Because energy has no direction, the potential at the point is a scalar property or a scalar point function. Its value depends only upon the values of the coordinates of the point. Suppose further that the scalar point functions together with their first space derivatives are continuous and singly valued functions. Then the ensemble of points in the given region together with the corresponding values of the scalar point function “constitute what is called the Field of the Scalar Point Function” (Wills, 1931, p. 75). Examples of such fields are gravitational or electrical fields. A level or equipotential surface of a scalar point function is a surface for all points of which the function has the same value (Wills, 1931, p. 76). For example, all points at the same elevation have the same gravitational potential and can be connected by such an equipotential surface. For another example, the equipotential surfaces around an isolated electrostatic point charge would consist of concentric shells. Consider next an ensemble of points each of which is associated with a vector function the value of which depends only upon the coordinates of the point. Examples would be the gravitational force field associated with a field of gravitational potential, the velocity of a moving fluid associated with a field of pressure potential, or an electric force field due to a distribution of electric charges. The ensemble of points in the given region, together with the corresponding values of the vectorD16 FLUID MOVEMENT IN EARTH MATERIALS point function, constitute “the Field of the Vector Point Function in the region” (Wills, 1931, p. 75). The potential gradient at a point is the rate of change of potential with distance measured in the direction in which this variation is a maximum (Gray, 1958, p. 219). As shown in figure 9, the ground-water gradient would water table at two wells a horizontal distance Ax cos 8 apart, where 8 is the angle between the inclination of the water table and the horizontal. “The symbol V, read ‘del,’ defined by writing: (58) is an operator which, acting upon the scalar point function u, produces the gradient of this function” (Wills, 1931, p. 77). x, y, and z are the rectangular coordinates of the system in which u is defined, i, j, and k are three mutually perpendicular unit vectors, which are present in the operator because the gradient of the scalar function is a vector. A bar over a term indicates that the term is a vector quantity. Therefore, the gradient of u is _ du , du , T du v”=’ 5+’ aj?51 (59) where u is a scalar point function such as potential. “Thus, the gradient of a scalar field is a vector field, the vector at any point having a magnitude equal to the most rapid rate of increase of * * * the scalar point-function * * * at the point and in the direction of this fastest rate of increase, i.e., perpendicular to the level surface at the point” (Hague, 1939, p. 35). The total change of the value of u in a distance dr, is du. This is the scalar product of the gradient and the distance, or du=dr ■ Vu (60) where the dot indicates the scalar product (Wills, 1931, p. 77). The gradient of a scalar field is a vector field as shown by equations 58 and 59. Furthermore, because the direction of maximum change is perpendicular to the equipotential surfaces, the gradient of a scalar field denotes a vector field such that the vectors are normal to the equipotential surfaces. Such fields, where the equipotential surface and the vectors are perpendicular, are said to be orthogonal. In effect, for hydraulics this simply expresses the fact that water flowing down a hill follows the steepest grade. It is often valuable to map fields of associated scalar and vector point functions. An example is given in figure 10. In studies of gravi- Figure 10—Approximate flow net in the vicinity of an irrigation pond. The dashed lines are equipotential lines, or lines of equal hydraulic head, in feet. The solid lines are flow lines which enclose areas of equal ground-water flow. tation, the lines or surfaces of equal potential are lines or surfaces of equal gravitational potential. The orthogonal vectors are lines of equal gravitational force. In hydraulics, a map of scalar and vector point functions is called a flow net (fig. 10). The scalar field consists of distributions of values of pressure or head. Equipotential surfaces are surfaces of constant pressure or head. The lines orthogonal to the scalar field map the vector field and are streamlines which indicate directions of flow.REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D17 In hydraulics, a velocity potential is often defined as a scalar function of space and time such that its space rate of change with respect to a given direction, or its gradient in that direction, is the fluid velocity in that direction. It is analogous to the force potential, whose gradient in a given direction is the force intensity in that direction. In such a flow net, as in figure 10, flow lines, or streamlines or flow tubes, include elements of equal flow. Therefore they are closer together in areas of greater velocity. Rewriting equation 57, and changing the notation so that F stands for force, which is always a vector, in Gouy double layers associated with the solid surfaces, (d) adhesion of water to the solid surfaces, (e) temperature gradients, and (f) gradients of chemical potential due to changes in solute concentration. If the last two of these are considered to he negligible, the total potential at any point is the scalar sum of the potentials due to the first four effects (Childs and Collis-George, 1948, p. 78). Thermodynamic considerations and the use of the free-energy concept are necessary to include the thermal and chemical potentials (Edlefsen and Anderson, 1943). GRAVITATIONAL POTENTIAL W—Fx COS a (61) Because potential energy is the ability a body has to do work by virtue of its position or configuration, IT is the potential energy of the body after it is moved a distance from a datum. The negative of the work done by a conservative field of force is the potential u, and according to Joos (1934, p. 81) dW=Fdx——du (62) The minus sign is necessary because the potential decreases in the direction of the force or flow. Suppose we want the force. It is simply obtained from F— — vu (63) Thus, “every conservative force may be represented as the negative of the gradient of a scalar point function, the potential” (Joos, 1934, p. 81). POTENTIALS IN SOIL WATER In soil moisture, the total potential is defined as the minimum energy per gram of water which must be expended in order to transport an infinitesimal test body of water from a specified reference state to any point within the liquid phase of a soil-water system which is in a state of rest. The reference state is commonly taken to be a pool of pure water at the same temperature with a flat surface exposed to atmospheric pressure at a known elevation. Under conditions in which the soil will spontaneously adsorb water from the reference state, the potential is a negative quantity. In any soil-water system in a state of rest, the total potential does not vary from point to point in the system (Bolt and Miller, 1958, p. 918). The total potential depends upon the individual potential fields and upon individual force fields which affect the test body of water as the transfer is made. In unsaturated soils these potentials are presumed to be due to (a) gravitational attraction, (b) hydrostatic-pressure difference, or gradient of pressure potential, (c) osmotic-pressure differences due to differences of content both of soluble salts and of anions and cations All terrestrial water lies within the earth’s gravitational field. Gravitational potential is the energy required to move an infinitesimal mass or volume of water from a datum where gravitational potential is arbitrarily taken as zero, to a given position in the field against the attraction of gravity. According to this definition of gravitational potential, G=Fgz (64) where (?= the gravitational potential, or gravitational energy, at an elevation z above the datum, where the gravitational potential is arbitrarily taken as zero. It is the work done, in ergs, in moving an infinitesimal mass or volume of water from the datum to the height z, in centimeters. A„=the force of the gravitational field against which the water is moved, in dynes. The relation between G and z is positive because the gravitational potential increases as the given mass or volume of water is moved in the direction of increasing z, or vertically upward. According to Newton’s second law of motion, F„=ma=mg (65) where w=the mass of water moved, in grams. a=g= tli0 acceleration of gravity, in centimeters per second squared. The acceleration is assumed to be constant within the relatively small range of values of 2 normally encountered in soil-moisture work. Combining equations 64 and 65, the energy, in ergs, to move any mass of water, m, from the datum to the height 2 centimeters is the scalar quantity G=mgz (66)D18 FLUID MOVEMENT IN EARTH MATERIALS To find the gravitational potential in ergs per gram of water, Gm, divide equation 66 by m. Then Gm=^=gz (67) To find G„, the gravitational potential, in ergs per cubic centimeter of water, divide equation 66 by v, the volume of the test mass of water, and where p=density of the liquid water. The gravitational potential, in ergs per gram of water, or in ergs per cubic centimeter of water, depends only upon the height within the field above the datum. Furthermore, it increases at a uniform rate with increasing z, assuming g to be constant. The gravitational potential is not affected by the contents of the field, whether liquid, vapor, or solid. It depends only upon the energy required to move vertically a unit test mass or volume from a datum to any position in the field. To determine the gravitational force on a mass rn of water at elevation z in the field one applies the operator V, equation 58, and differentiates the potential. For equation 66 operated upon by 5.8, F,—>(69) The x and y terms present in equation 58 do not exist because the gravitational force operates only in the vertical, or z, direction. The minus sign shows that the force is directed downward while height, z, increases upward and the potential increases with z. In other words, the force is directed in a direction opposite to that of increasing potential. Differentiating, the gravitational force on the mass of water, m, is Fg=—kmg (70) where F, is in dynes. Changing the notation to represent the magnitudes of the quantities only, we arrive back at equation 65, F„=mg (71) Similarly, if F„ is force per gram of water, F0=g (72) Thus, the gravitational force, in dynes per gram of water, is numerically equal to the acceleration of gravity. Finally, if Fg is the gravitational force per unit volume of water, Fs=Pg (73) The force on a unit mass or volume at any point in the field is constant because of the assumption that g is constant. This independence of the gravitational potential of the contents of the field explains why equation 54 is equally applicable whether or not there is a liquid continuum. It explains why equation 54 applies equally to the continuous liquid phase below the capillary fringe and to the discontinuous liquid phase above the capillary fringe. The acceleration of gravity, g, may be taken as 980 cm per sec2. Therefore, according to equation 67, the change in gravitational potential per gram equals 980 ergs per gram per centimeter of change in height. The density of water, p, is 1 gram per cm3. Therefore, according to equation 68, the change in gravitational potential per cubic centimeter of water is 980 ergs per cm3 of water per centimeter of change in height. The datum for gravitational potential is usually taken as the water table. At this point 2=0 and G= 0. Therefore 2 is the height above the water table and — 2 is the depth below the water table. Below the water table the gravitational potential is negative by an amount equal to 980 ergs per gram of water per centimeter depth below the water table, or 980 ergs per cm3 of water per centimeter depth below the water table, times the depth below the water table. Above the water table, it is equal to 980 ergs per gram of water per centimeter height above the water table, or 980 ergs per cm3 of water per centimeter height above the water table, times the height above the water table, regardless of whether the liquid phase is continuous or discontinuous. According to equations 72 and 73, the force on the water is equal to 980 dynes per gram of water, or 980 dynes per cm3 of water throughout the field. Although the gravitational potential and force at any point in the gravitational field are independent of the material filling the field, this is not true for the pressure within the field. The variation of pressure with height is directly proportional to the density of the phase filling the field. In equation 73 the force on each cubic centimeter of water is constant. However, when the water is in a liquid state, there are more of these cubic centimeters of water per unit change in height than it the water were in a vapor state. Therefore, the forces add up more quickly in the liquid phase, and the changes in pressure are greater than in the vapor phase. Looked at in another way, a column of liquid water weighs more than a column of water vapor, and the pressureREVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D19 is a function of the weight of the overlying column of water. According to elementary hydrostatics, A p—Dgz (74) where Ap = change in pressure between two points a vertical distance z apart 19=density of the liquid or vapor The soil is saturated for some distance above the water table. For a somewhat greater distance above the water table, some interconnecting systems of pores are in liquid continuity with the water table. Because the density of water is 1 gram per cm3, the pressure in the water above the water table changes at the rate of 980 dynes per cm2 per centimeter of change in height, according to equation 74. In the unsaturated soil outside the interconnecting systems of liquid-filled pores, the pores are partly filled with water vapor. Taking the density of saturated water vapor, at a temperature of 20 °C and at atmospheric pressure, as 0.000017 grams per cm3, equation 74 shows that the change in the pressure in a vapor-filled continuum is about 0.02 dyne per cm2 per centimeter of change in height. For greater accuracy, the compressibility of water vapor should be considered by using the barometric equation. HYDROSTATIC-PRESSURE POTENTIAL According to Childs (1957, p. 14), a suction must be applied in order to withdraw water from a soil or to prevent the soil from imbibing water. The greater the applied suction, the more water is withdrawn, and the lower will be the moisture content when the soil has reached equilibrium at the applied suction. If water is withdrawn from a soil that does not shrink upon drying, air must enter the pore space and air-water interfaces must be present in the pore space. Such curved interfaces can be maintained only by capillary forces. Hence, surface tension acting in the interfaces provides a mechanism of soil-water retention against externally applied suctions (Childs, 1957, p. 14). This is one of several types of force that retain water against externally applied suctions. Furthermore, this is the mechanism whereby a negative hydrostatic pressure and pressure potential are developed within the soil. As in the case of gravitational potential, hydrostatic-pressure potential may be taken as zero at the water table. Of course the absolute pressure at the water table is atmospheric pressure, equivalent (at sea level) to that of a column of mercury 76 cm high, or 1,013,000 dynes per cm2. Because this pressure is taken as zero, all pressures are relative to atmospheric pressure or are gage pressures. Below the water table at equilibrium hydrostatic pressures^ are positive because the hydrostatic-pressure potential increases with increasing depth. Above the water table at equilibrium the water is under a negative hydrostatic pressure, or tension, because the hydrostatic-pressure potential becomes increasingly negative with increasing height above the water table (Baver, 1948, p. 204). At the water table, then, the gravitational potential and the hydrostatic-pressure potential are arbitrarily taken as zero. In the absence of other forces the total potential at the water table is therefore zero. Under equilibrium conditions the total potential must also be equal to zero at all points above and below the water table. Thus, at equilibrium and where other forces are not present, the gravitational potential and the hydrostatic-pressure potential must everywhere be numerically equal (Baver, 1948, p. 204; Childs and Collis-George, 1950b, p. 243). According to equation 67, the gravitational potential per gram of water decreases at the rate gz with increasing depth below the water table. Therefore at equilibrium the hydrostatic-pressure potential per gram of water must increase at the same rate with increasing depth below the water table, or * *=-gz (75) where ^=the hydrostatic-pressure potential. It is the energy, in ergs per gram of water, to move a gram of water from the water table to a depth — 2 below the water table against the hydrostatic-pressure forces. The minus sign is present because the pressure potential increases downward, whereas 2 is measured upward. In similar fashion from equation 68, t= — pgz (76) \p is now the hydrostatic-pressure potential, in ergs per cubic centimeter of water. It must be remembered that below the water table 2 is negative, and according to equations 75 and 76 the hydrostatic-pressure potential, i, is therefore positive. A liquid continuum extends to a height above the water table that is determined by the pore sizes and by equation 54, as will be discussed later. This is the same capillary rise that occurs when a capillary tube is immersed in water. This region, where some or all of the pores are filled with water that is continuous with water in the zone of saturation, is the capillary fringe. At equilibrium, and in the absence of other potentials, the hydrostatic-pressure potential equals the gravitational potential in the capillary fringe, and its value is given by equations 75 and 76. In this case z is positive,D20 FLUID MOVEMENT IN EARTH MATERIALS and the hydrostatic-pressure potential is negative. This situation corresponds to the fact that in the zone of aeration the soil exhibits a negative pressure, or hydrostatic-pressure deficiency, or suction, or tension. The liquid-water system is discontinuous above the capillary fringe. It consists of isolated capillary interstices, or groups of such interstices, that are filled with water and separated from similarly filled interstices by others that are filled with soil air and water vapor. In this part of the zone of aeration, equations 75 and 76 also hold at equilibrium and give negative values of hydrostatic-pressure potential for positive values of z. Curved liquid-vapor interfaces are present in many of the pores and equation 54 also applies. The actual pressures in the water vapor in this belt can be determined from the barometric equation 31. The pressures in the water on the convex sides of the menisci can be determined from equations 9 or 10. Above the water table the hydrostatic pressure in the liquid is negative relative to atmospheric pressure. The greater the height above the water table, the greater will be the curvature of the equilibrium liquid-vapor interfaces. The interfaces are drawn deeper into the smaller interstices, and the larger interstices are emptied of water, giving larger hydrostatic-pressure deficiencies in the pore water. However at a sufficiently great height above the water table, or at sufficiently great values of externally applied suction, only very small pores remain full of water. Other forces then become more important than surface tension as discussed below. Thus extremely high negative hydrostatic pressures, or tensions, do not exist in the soil water. In fact, as shown by Edlefsen and Anderson (1943, p. 205), much of the water is actually under compression, even though it is in equilibrium with an extremely large applied suction. OSMOTIC-PRESSURE POTENTIAL A third type of potential found in soil water is osmotic-pressure potential. It is due to differences in content of both soluble salts and anions and cations in Gouy double layers associated with the solid surfaces of soil particles (Childs and Collis-George, 1948). According to the concept of Gouy (Adam, 1941, p. 342), cations, or positively charged particles, frequently become dissociated from soil particles of clay size (Childs, 1957, p. 7-8). The surface of the soil particle then has a negative charge. Such particles are called micelles. Particles having similar electrical charges repel each other and particles having opposite electrical charges attract each other. Therefore, in the vicinity of negatively charged soil-particle surfaces, there is an increased concentration of cations (positively charged ions) and a decreased concentration of anions (negatively charged ions), whereas cation and anion concentrations are equal in parts of the solution that are more remote from the solid-liquid interface. The arrangement of charges at the surface of the micelles is known as an electric, or Gouy, double layer. Because the total ion concentration increases as the solid-liquid interface is approached, the osmotic pressure also increases. Because the negatively charged soil micelles attract cations from the soil solution, the net concentration of ions in the space between any two given micelles is greater than that in the solution remote from the micelles. Because of this difference in concentration, an osmotic force tends to force water into the space between the micelles, thereby tending to force them apart. Thus the hydrostatic pressure between the micelles is in excess of the hydrostatic pressure in the remote solution. If suction is applied to a soil that exhibits this phenomenon, removal of water as a result of application of the external suction relieves the hydrostatic pressure on the space between the micelles. Therefore, the soil contracts upon drying (Childs, 1957, p. 23-28). The ultimate ionic concentration at equilibrium in the neighborhood of a Gouy double layer at the surface of a clay-size particle represents a balance between the segregating forces and the diffusing forces. The segregating, or ion-concentrating, forces are those due to the mutual electrical attraction between the micelle and oppositely charged ions and to adsorptive forces. The diffusing, or ion-diluting, forces include the normal ionic diffusion that would occur from a region of higher ion concentration to one of lower ion concentration and the diffusion of water in the direction of the higher concentration. Corresponding to this equilibrium, there is an osmotic-pressure gradient which is a function of distance from the solid surface. The osmotic pressure at any point corresponds to a negative hydraulic-pressure potential. It must be regarded as a hydrostatic-pressure deficiency because it tries to force water in the direction of greater ion concentration—that is, toward the particle and into the electric double layers. This contribution to total potential might be appreciable within as much as 50 A (A = 1 Angstrom unit = 10~8cm) from the dissociating surface (Childs and Collis-George, 1948, p. 79). Osmotic-pressure potential due to differences in ion concentration is important in soils that shrink upon drying. Such soils contain clay minerals and humus. Childs (1957, p. 23) refers to water retention in such soils as the mechanism of “water retention by particle repulsion.” It is the main source of the negative hydrostatic pressure that is exerted against an externally applied suction applied to a soil that shrinks upon drying. It could be of importance also in extremely dryREVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D21 sandy soil, where the remaining water is retained in very small wedges or close to the particle surfaces. ADSORPTION POTENTIAL “Attractive (adsorptive) forces between solids and water can be divided in two categories, short-range forces and long-range forces” (Bolt and Miller, 1958, p. 918). The short-range forces are effective in a range less than 100 A from a solid surface, whereas the long-range forces may extend beyond 100 A. Of the two types of short-range forces, one consists of chemical forces, which are “caused by localized interactions between the electron clouds of surface atoms and water molecules” (Bolt and Miller, 1958, p. 918-919). These forces are localized close to the surfaces of the particles and have little effect on adsorption. The second type of short-range force consists of Van der Waals forces. They have their origin in the electrostatic attraction of the nucleus of one molecule for the electrons of another. Because of the geometry of the molecule this attraction is largely but not completely compensated for by the repulsion of electrons by electrons and of nuclei by nuclei (Pauling, 1958, p. 321). These short-range forces, although not as localized as the chemical forces, are in the case of interaction between water and a solid surface still of short range as a result of the destructive interference with fields from other atoms as the distance between the atoms under consideration increases. Although such forces account for the cohesive strength of the water and the phenomenon of surface tension, it seems unlikely that the effective range with regard to the adsorption of water on solid surfaces extends beyond the first few molecular layers adjacent to the interface (Bolt and Miller, 1958, p. 919). In general, the short-range forces may be neglected except at low water contents usually encountered only in the laboratory. It must be noted, however, that other workers ascribe the cohesive strength of water and the phenomenon of surface tension to the molecular hydrogen bonding, or dipole nature of the molecule (Hendricks, 1955, p. 11-12). To understand the long-range forces, consider the nature of the water molecule. “* * * The water molecule may be supposed to have a localized distribution of electrical charge in its outer parts. The concentrations of charge are located at the apexes of a tetrahedron, two being positively charged with two negatively. The molecule is thus a dipole; i.e., it is, as a whole, electrically neutral but the ‘center of gravity’ of the negative charge is separated from that of the positive” (Childs, 1957, p. 9). The long-range adhesive forces are due to the attraction between the positive ends of the water dipoles and the electrostatic field emanating from the negatively charged soil particles. The relatively small separation between the oppositely charged ends of the dipole becomes less important the farther away the dipole is from the soil particle. At some distance the dipole acts as a neutral particle because, for all practical purposes, the two opposite charges can be considered to coincide. Therefore, even these “long-range” adhesive forces must be of rather short range even though they may attain large values near the soil particle. This attractive or negative force may extend to a maximum distance from the soil particle of more than 100 A (Bolt and Miller, 1958, p. 919). The adsorptive forces strongly retain water near the solid surfaces, and large forces may be required to remove this water. However, these adsorptive forces become prominent only when soils are dried far beyond the state of dryness that is normally found in nature. THERMAL POTENTIAL According to Edlefsen and Anderson (1943), the free energy of soil moisture, or any other fluid, increases with the temperature. Furthermore, temperature and other thermodynamic variables of state help determine phase changes in the soil fluids. Therefore, thermal potential must be accounted for in determining the total potential of soil water. Unfortunately, “There can be little doubt that the most complex and least understood area in the field of soil-water relationships is that of the effect of temperature gradients' applied to moist soils” (Winterkorn, 1958b, p. 113). Moisture movement by vapor diffusion or by a combination of vapor diffusion and capillarity can be appreciable when thermal gradients are present (Smith, W. O., 1943; Gurr and others, 1952; Taylor and Cavazza, 1954). However, thermal transfer of water occurs in great quantity only when relatively large thermal gradients are present in the soil water. Furthermore, evidence suggests that moisture flow in response to temperature gradients occurs mainly or completely in the vapor phase (Hutcheon, 1958, p. 114; Kuzmak and Sereda, 1958, p. 146). It is most important near the land surface where the largest thermal gradients are found. At depth, thermal gradients are generally small (Baver, 1948). The moisture transfer due to these small thermal gradients at depth may be quantitatively substantial over a period of time and for a given area. However, it is relatively small when compared with the moisture transfer due to the other gradients. As stated by Richards and Richards (1957, p. 53), “There is evidence that, except very near the soil surface and then mainly for longtime effects involving weeks or months, vapor transfer of water in soil in the root zone of growing crops is not very significant agriculturally.” Therefore,D22 FLUID MOVEMENT IN EARTH MATERIALS it will be assumed that thermal gradients are negligible and that moisture movement occurs isothermally. CHEMICAL POTENTIAL By chemical potential is meant the potential due to the osmotic energy of ions free in the aqueous solution. This is distinct from the potential due to the osmotic energy of ions held in the electrical double layers on the solid particle surfaces (Winterkorn, 1958a, p. 330). If free salts are absent or are uniformly distributed, chemical potential may be ignored (Low, 1958, p. 56). In addition, it “is usually omitted in considering water movement because the solutes generally move more readily than the water with respect to the soil” (Gardner, 1958, p. 78). TOTAL POTENTIAL AND ITS MEASUREMENT The advantages of using potentials in the soil water rather than the forces on the soil water now become apparent. Assuming isothermal conditions and uniform solute concentration, the total force on the soil water would be the sum of the forces due to gravity, hydrostatic pressure, osmosis, and adsorption. Because forces are vectors and have direction, a vectorial addition would be required at every point. On the other hand, potential is a scalar property, and the total potential at any point in an unsaturated soil is the scalar sum of the four component potentials discussed above: *=/#*+«—«-*) (77) where = total potential, in ergs per cubic centimeter pgz—gravitational potential, in ergs per cubic centimeter \f/=hydrostatic-pressure potential, in ergs per cubic centimeter e=osmotic-pressure potential, in ergs per cubic centimeter £= adsorption potential, in ergs per cubic centimeter (Childs and Collis-George, 1948, p. 79.) The total force can be determined from the gradient of the total potential. The potential of soil water can be measured by bringing the soil water into equilibrium with a manometer. Because all unsaturated soils have some continuous air-filled pore space, a manometer would read zero, or atmospheric pressure, in contact with an unsaturated soil. Therefore, a porous membrane made of sintered glass spheres, unglazed ceramic, cellophane, porvic, or sausage casing is interposed between the soil on one side and an external body of water which is in contact with the manometer. The membrane to be used should have pores sufficiently small to remain full of water at all values of negative pressure to be encountered in an experiment. Because these pores remain full of water, air cannot enter them and put the manometer out of operation. On the other hand, the water in the soil is continuous with the water in the membrane, with the body of water on the manometer side of the membrane, and with the manometer. Thus, pressures are transmitted from the soil water to the manometer. In unsaturated soils a negative pressure or suction must be maintained on the manometer side of the porous membrane for equilibrium to be achieved with the soil water. One commonly speaks of measuring soil-water suction because one measures “the suction prevailing in an external body of water which is in equilibrium with the soil. In some cases, as in sand, we shall see that we may safely infer the internal soil water suction from such measurements, while in others, as in clays, we may not; but in the latter case the true internal suction is almost meaningless because [of its] varying from point to point within wide limits, while the equilibrium suction of an external water body is definable and significant” (Childs, 1957, p. 16). The “internal soil water suction” refers to the actual tension or hydrostatic-pressure deficiency in the water. When a sand or other soil that does not shrink upon drying is brought into equilibrium with such a manometer, the suction in the external water body comes to equilibrium with the resultant of the forces due to gravity and to hydrostatic-pressure deficiency from surface-tension effects. In these nonshrinking soils the osmotic effect is unimportant. The adsorptive effects are also unimportant at normal moisture contents. In this case, the external suction is a measure of the hydrostatic-pressure deficiency resulting from the surface tension at the curved air-water interfaces. When a clay or other soil that does shrink upon drying is brought into equilibrium with such a manometer, the suction in the external water probably comes to equilibrium with the resultant of the forces due to gravity, to hydrostatic-pressure deficiency from surface-tension effects, and to osmotic pressure (or pressure involved in water retention by particle repulsion). Where the shrinkage effect is completely dominant, the necessary external suction required to bring about equilibrium is equal to the osmotic repulsive pressure (Childs, 1957, p. 27). The relation between the external suction applied to a soil and the amount of water that the soil retains against that dewatering suction (the moisture content) is the moisture characteristic. A typical moistureREVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D23 Figure 11—Moisture characteristic of a fine sand. (Courtesy of N. A. Willits, assistant professor of soils, Rutgers University.) characteristic is shown in figure 11. The moisture characteristic of a nonshrinking soil is related to the withdrawal of moisture from pores against the forces of surface tension. Therefore it is an indication of the pore-size distribution of the soil. For soils that do shrink upon drying, it is related to the osmotic pressure and to the separation of the clay micelles (which depends upon the amount of water between the micelles). Thus, when shrinkage experiments show that water loss is not accompanied fully by the entry of air into the pore space, the soil suction may be interpreted either as repulsive pressure between the micelles due to osmosis or as hydrostatic-pressure deficiencies due to surface tension. In nonshrinking soils the contribution of osmotic potential, «, and adsorption potential, $, to total potential, = Pfi'S+S (78) where 2= (*-«-*) (79) and 2 is the capillary potential. Its interpretation for shrinking and for nonshrinking soils is outlined above. In most situations, as discussed above, it equals the suction potential. The gradient of the sum of the gravitational and capillary potentials gives the water-moving forces in unsaturated soils. EQUILIBRIUM DISTRIBUTION OF POTENTIAL AND WATER IN SOIL The distribution of component potentials within an unsaturated soil is discussed by Bolt and Miller (1958). The distribution of water and potentials at equilibrium in a nonshrinking sand is discussed below. In a nonshrinking sand the capillary potential, 2, is equal to the hydrostatic-pressure potential, \f/, because osmotic and adsorption potentials are negligible. At equilibrium the total potential, , is everywhere equal to zero because there are no gradients of total potential at equilibrium, and the total potential is equal to zero at the water table. Therefore, letting 4> equal zero in equations 77 and 78, the gravitational potential, pgz, is everywhere equal to the negative value of the capillary potential (—2), or to the negative value of the hydrostatic-pressure potential Furthermore, equation 54 also applies; pgz in 54 may be interpreted as the gravitational potential, in ergs per cubic centimeter, and 2 is zero and 2 must equal the gravitational potential. The required value of negative capillary potential, or hydrostatic-pressure potential, can develop only in a pore 0.0015 cm in radius. If there is no pore that small, sufficient retaining capillary or hydrostatic-pressure force cannot be developed to balance the dewatering gravitational force. Equilibrium is not possible, and the interstice is dewatered by gravity flow. At 100 cm above a rising water table, water vapor would diffuse to any meniscus in a pore less than 0.0015 cm in radius. The vapor would condense on the meniscus, reducing its curvature and energy of water retention until the equilibrium meniscus size of 0.0015 cm was reached. DISTRIBUTION IN RANDOMLY DISTRIBUTED PORES Consider next the hypothetical extreme case where there is a random distribution of pore sizes. Furthermore, assume that the soil is of infinite horizontal extent. As a final simplification, assume circular pores so that equation 81 may be used. Somewhere in the system, a pore having a radius greater than that required by equation 81 will be located close above the water table. At that point, there will be no capillary fringe and no capillary rise. Elsewhere, it will be possible to find a continuum of small pores extending to a great height above the water table. Furthermore, each pore radius will be equal to or smaller than the radius required by equation 81 at that height above the water table. There will be a saturated continuum filling these pores and thus a very large height of capillary rise. Finally, all heights of capillary rise between these two extremes will occur, and the top of the capillary fringe will be a very uneven surface. Of course the exact height and shape of the capillary fringe will depend upon whether the soil is in a dewatering or rewatering phase. Above a falling water table, for example, very large pores can remain full of water provided that the required meniscus is maintained in an overlying pore of sufficiently small radius. In this random distribution of pore sizes, occasional isolated water-filled pores will be surrounded by vapor-filled pores. Similarly, occasional isolated vapor-filled pores will be surrounded by water-filled pores. In general, however, at a given level there will be different combinations or groups of water-filled pores whose menisci are sufficiently small to satisfy equation 81. Here again hysteresis is important. For example, consider a region containing a group of relatively large pores completely surrounded by pores small enough to retain water at a particular elevation and to satisfy equation 81. At a given stage in a rewatering phase, water might diffuse toward and condense in the smaller pores until they were filled with water. However the larger internal pores might not be filled at that stage of the rewatering phase, and menisci might face inward toward the larger pores. In dewatering from a state of saturation, on the other hand, the situation would be similar to that at the top of the capillary fringe for a falling water table. The surrounding smaller pores might reach equilibrium according to equation 81 and retain the water in their own pores and in the larger interior pores as well. Therefore the entire region might remain saturated even though the larger internal pores would normally be dewatered at that elevation. An analogous hysteresis effect would prevail in a region consisting of small pores surrounded by larger pores. Two extremes have been presented. One consists of uniform interstices having a capillary fringe of virtually uniform height and individual reentrant wedges retaining water above the capillary fringe. The other consists of a random distribution of pores having a capillary fringe the top of which is extremely uneven. Water would be retained erratically above the capillary fringe in all combinations from single filled pores to very large webs of water filling great numbers of pores. The actual situation at equilibrium in the field lies between these extremes. Most soils have some sorting and horizontal stratification. Therefore, in most cases, the water distribution at equilibrium should be closer to the first case than to the second. In extremely variable material the actual water distribution at equilibrium might approximate more closely the random situation. MOVEMENT OF SOIL MOISTURE PHYSICAL PICTURE Soil moisture can exist and move in three ways—as liquid water, as water vapor, and as adsorbed water. “It seems probable that, in soils, moisture transfer inD30 FLUID MOVEMENT IN EARTH MATERIALS the adsorbed phase can be significant only under rather special conditions, such as where the soil is very dry and possesses a large specific surface” (Philip, 1958, p. 157). Thus in the following discussion it is assumed that movement of adsorbed water is negligible and that the water moves chiefly in the liquid and vapor phases. Soil moisture moves in response to gravitational, hydrostatic-pressure, adsorption, osmotic, temperature, and chemical potential gradients. As discussed previously, temperature and solute concentration are taken as constant throughout the system. In addition, the discussion is restricted to a sand, where the osmotic and adsorption potentials are negligible. Moisture movement is then governed by the gradient of the total potential, which according to equation 78 is =pgz+2 (88) Here, 2 is the capillary potential of a nonshrinking soil. It is virtually equivalent to \f/. PORE UNDERLAIN BY A CAVITY Assume that a given pore has the general shape shown in figure 18 and is underlain by a cavity. Figure 18A shows the pore neck partly filled with water and at A B Figure 18.—Pore underlain by a cavity. equilibrium. Therefore the upward and downward forces on the water are equal. The downward forces are the pull of the lower meniscus and the weight of the water in the pore, pgz, where z is now the height of the water in the neck. If the pore is circular, this pull is denoted by equation 9, which is (2>-Po) = y (89) The upward force is the pull of the upper meniscus, also denoted by equation 9. The upward pull by the upper meniscus must exceed the downward pull by the lower meniscus by an amount equal to pgz. Consequently the radius of the upper meniscus must be smaller according to equation 9, and it will be lodged in a smaller part of the neck than will the lower meniscus. Suppose water is added to the system until the soil overlying the cavity becomes saturated. Eventually the upward meniscus will not be able to develop a small enough radius to support the growing weight of water in the pores in addition to the pull of the lower meniscus. Therefore the lower meniscus will move to the bottom of the pore and become convex toward the cavity, as shown in figure 185. The pressure in the cavity is atmospheric pressure. At equilibrium, the pressure on the upper side of the lower meniscus (concave to the liquid) is greater than atmospheric pressure according to equation 9. Thus the upper and lower menisci act in the same direction— to support the weight of water in the pore. Positive hydrostatic pressure due to the weight of the overlying liquid is required to overcome the pressure differentials across the menisci before water can flow into the cavity. Thus the soil above a cavity must be saturated and under sufficient head to overcome the pressure differences across the interfaces before water can move into the cavity. If the pore radius is large, it will take very little positive pressure to overcome the pressure difference, according to equation 89. If the pore is small, it might take considerable hydrostatic pressure—a considerable zone of saturation over the cavity—to overcome the larger interfacial pressure differences associated with pores of smaller radius. Assume that the pore is in the moisture condition shown in either A or B of figure 18 and that the cavity is underlain by pores having a lower moisture content. No liquid movement can occur across the cavity as discussed above. However, equilibrium does not exist because the pores are drier below the cavity than above it. The water menisci below the cavity are drawn farther into the pore necks and their radii of curvature are smaller than above the cavity. According to the Kelvin equation, vapor pressures are therefore smaller below the cavity than above it. A vapor-pressure gradient would be established across the cavity, and water would evaporate from the upper menisci, diffuse across the cavity, and condense on the lower menisci. Movement would cease at equilibrium when all the moisture contents and menisci are such that equation 80 is satisfied everywhere in the system (assuming the validity of the simplifying assumptions). Once the upper pore has sufficient positive hydrostatic head for liquid flow to occur, it occurs very quickly. On the other hand, the diffusion of water vapor is very slow. Thus water moves at greatly different rates under these different conditions.REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D31 PORE UNDERLAIN BY PORES OF SIMILAR SIZE Figure 19 shows two similar necks separated by a pore. There is less water in the lower neck than in the upper. The balance of forces in each neck is similar to that described for the case of the cavity. In each neck the capillary pull of the lower meniscus plus the weight of the water in the neck equals the capillary pull of the upper meniscus; the upper meniscus exerts a greater pull than does the lower meniscus and must be lodged in a narrower part of the neck. As in the case of the cavity, water cannot flow as a liquid across the pore, and moisture equilibration occurs by slow evaporation at the upper neck, diffusion to the lower neck, and condensation at the lower neck. Assume that entrapped air can be bled from the pores by lateral connections. As water is added to the upper neck in figure 19A, meniscus b moves into the upper pore and meniscus a moves into the lower pore. Meniscus b is always lodged in a narrower neck section than meniscus a, however, because it must balance the weight of the water in the neck as well as the pull of meniscus a. If insufficient water is added to move meniscus a to the midpoint of the lower pore, liquid flow does not occur. Instead, relatively slow downward movement of water occurs across the pore by evaporation, diffusion, and condensation. Because the radius of curvature of meniscus a in figure 19 B is larger than in figure 19 A, vapor pressure at this meniscus is greater and a diffusion occurs more rapidly than in figure 19 A. If sufficient water is added to move meniscus a past the midpoint of the pore, water will flow rapidly to the lower part of the pore. At equilibrium the amount of water in the lower neck will be slightly greater than in the upper neck in order to satisfy equation 80. If sufficient water is added to fill all the pores and entrapped air is removed, surface tension ceases to be a factor. Water movement through the pores then follows the rules of saturated flow. PORE UNDERLAIN BY PORES OF DISSIMILAR SIZE The upper pore in figure 20 is underlain by a smaller one. Meniscus a will reach the midppint of the lower pore while meniscus b is still in the neck of the larger pore. After meniscus a reaches the midpoint of the smaller pore, water will flow rapidly to the lower part of the small pore. Thus liquid flow will occur at a moisture content which is lower than that required for liquid flow where the upper pore is underlain by a pore of equal or larger size. In fact, if the lower pore is sufficiently smaller than the upper, it can fill with water while the upper pore remains almost empty. If the lower pore is larger than the upper, the reverse situation is present. In figure 21, for example, it would be necessary for the upper pore to be fullD32 FLUID MOVEMENT IN EARTH MATERIALS Figure 21—Pore underlain by a larger pore. and to have sufficient hydrostatic head before the water could move as a liquid into the larger pore below. Vapor diffusion would occur across the underlying large pore as long as there was any difference in the vapor pressures above and below. This is similar to the case of the pore underlain by a cavity. RANDOMLY DISTRIBUTED PORES Several conclusions are possible from the considerations discussed above. For a given pore size and distribution, there is a moisture content below which liquid movement through the pores does not occur. Movement occurs exclusively by vapor transfer or else by vapor transfer from neck to neck and by liquid transfer through the necks. At higher moisture contents, the liquid movement occurs as a series of discontinuous jumps from neck to neck as discussed above. At sufficiently high moisture contents, a series of pores might be filled so that the flow through them follows the laws of saturated flow. For some types of soil geometry there might be a continuous vapor bypassing liquid-filled wedges. Similarly there might be liquid-filled passages bypassing vapor-filled pore centers. The following description pertains to a soil containing a random distribution of pores. Some passageways are filled only with water vapor, and water moves through them slowly only as a vapor. Some passageways contain vapor-filled pores and liquid-filled necks. The movement through these channels occurs by vapor transfer through the pores and liquid transfer through the necks. Some passageways are in a similar condition except that there is enough water for the menisci to jump across the pores. Water movement then occurs relatively rapidly by the mechanism of liquid jumps. Finally, some passageways are completely full of water and movement occurs rapidly by saturated flow. A given arrangement of pores at a given moisture content transmits water at a given velocity under a given head difference. The ability to transmit water under a unit head difference is known as the capillary conductivity, or unsaturated conductivity, of the soil at the given moisture content. Capillary conductivity describes the sum of the movement by the different mechanisms described above. It includes movement by vapor flow, by vapor flow in the pores and liquid flow in the necks, by liquid jumps, and by liquid flow. As moisture content increases, more systems of pores contain more water. Therefore more systems of pores transmit water by the liquid-flow mechanisms, and capillary conductivity increases. Abundant experimental data show that capillary conductivity increases with increasing moisture content and with decreasing suction. Poiseuille’s law Q=^fPl~Pi) (90) gives the discharge rate through a capillary tube under conditions of viscous flow, where Q=the discharge rate r=the radius of the tube ,7=the absolute or dynamic coefficient of viscosity f=the length of the tube Pi and P2 = the pressures at the ends of the tube The largest pores empty first as soil moisture decreases. Because of the fourth-power relationship between Q and r in equation 90, saturated flow through a soil decreases very rapidly as the radii of the saturated pores diminish in size. In addition, as moisture content decreases “ * * * the chance of water occurring in pores or wedges isolated from the general three-dimensional network of water films and channels increases. Once continuity fails, there can be no flow in the liquid phase, apart from flow through liquid ‘islands’ in series-parallel with the vapor system * * *” (Philip, 1958, p. 153). Most of the conductivity of a wet soil stems from pores in material of silt size and larger. When a soil is dry the only pores that are filled and capable of liquid flow are the smaller pores. Liquid movementREVIEW OP SOME ELEMENTS OF SOIL-MOISTURE THEORY D33 through these pores is very small. The larger pores, substantially dewatered, transmit water only by the slow mechanisms of (1) vapor transfer, and (2) vapor transfer through the pores and liquid flow through the necks. At such low moisture contents the conductivity is very small. In field soils, drainage and water transmission occur rapidly at high moisture contents. At a critical range of moisture contents, where pores of silt size and larger are being virtually dewatered, the rate of drainage becomes very small. For some soils this critical moisture content corresponds to field capacity. Field capacity, or field moisture capacity, is the “amount of water remaining in a well-drained soil when the velocity of downward flow into unsaturated soil has become small” (Soil Sci. Soc. America, 1956, p. 433). Applied water distributes itself uniformly throughout a soil only if it is sufficient in quantity to bring the entire soil to a certain critical moisture content or range of moisture contents. These critical moisture contents correspond to those at which values of capillary conductivity become small during drainage. The applied water percolates downward as a belt, whose moisture content is between the critical one and saturation. It leaves the overlying soil near the critical moisture content. After the applied water moistens a certain depth to this moisture content, movement practically ceases. More water must be applied to obtain further rapid penetration as a liquid (Veihmeyer, 1939, p. 544). According to Bodman and Colman (1944, p. 117), downward moving water advances behind a “wetting-front,” or sharp differential in moisture content (Remson and others, 1960, p. 153). If only enough water is applied to wet a certain depth of soil to the critical moisture content, movement of the wetting front practically ceases. Further moisture movement occurs only by vapor transfer and other slow processes discussed above. More water must be applied to obtain further penetration as a liquid. Of course, all movement finally stops at static equilibrium, when only enough moisture has been left behind so that the capillary forces equal the gravitational forces. This behavior results because a certain degree of saturation is necessary before any given distribution of pore sizes can begin to transmit water rapidly as a liquid. At lower moisture contents, water movement through the larger of these pores depends upon slow vapor diffusion. Water movement through the smaller of these pores is negligible even when the pores are filled. Therefore rapid water transmission can occur only at moisture contents where series of larger pores are sufficiently wet for water to move through them as a liquid. Below these moisture contents, water movement as a vapor through the larger pores and as a liquid through the smaller pores is very slow, and the wetting front remains practically stable. DIFFERENTIAL EQUATION FOR UNSATURATED FLOW HISTORICAL BACKGROUND In order to understand the derivation and meaning of the unsaturated-flow equation, it is helpful to consider first the derivation and meaning of the similar equation for saturated flow. This is because “the simplest type of porous flow problem deals with ‘saturated’ media in which all of the pores are completely filled with one homogeneous liquid” (Miller and Miller, 1956, p. 324). Historically, the equation was first derived for the saturated case and later adapted to the unsaturated case. Henry Darcy discovered in 1856 an empirical proportionality between macroscopic flow rate and driving force (Darcy, 1856). By adding a conservation-of-mat-ter condition for steady flow to Darcy’s law written in differential form, Slichter in 1899 obtained an equation identical in form with the Laplace equation (Slichter, 1899, p. 330). Finally, by adding a conservation-of-matter condition for unsteady flow to Darcy’s law written in differential form, Theis obtained an equation identical in form with the heat-flow equation (Theis, 1935). “This has since formed the basis for the successful development of saturated flow technology” (Miller and Miller, 1956, p. 324). This type of equation is known as the equation of heat conduction in studies of heat conduction, as the diffusion equation in studies of chemical diffusion, and as the nonequilibrium equation in studies of ground water. Several difficulties delayed the development of the analogous equation for unsaturated flow. The first difficulty stems from the nature of the potentials and the driving forces in unsaturated systems. In saturated systems the potentials are relatively simple, involving only position and pressure, and are easily measured by means of water wells and piezometers. The potentials in unsaturated flow systems are much more complicated and involve components of gravitational, hydrostatic-pressure, osmotic-pressure, adsorption, temperature, and chemical potentials. Unsaturated-flow potentials also vary in a complicated way with moisture content, and hysteresis enters into the relationship between them. Furthermore, measurements of potential and moisture content in soils are difficult and tedious to make. An additional difficulty not encountered in saturated flow is due to the fact that the transmission constant, or capillary conductivity, is dependent upon the moisture content and thus is variable.D34 FLUID MOVEMENT IN EARTH MATERIALS Buckingham (1907) recognized the analogy between potentials in unsaturated-flow systems and in other flow systems and introduced the concept of capillary potential. Richards (1931, p. 323-324) was able to adapt the heat-flow or diffusion type of equation to unsaturated flow by writing the moisture content and capillary conductivity as unspecified independent functions of the capillary potential or the suction. In this way, he arrived at the type of equation that forms the basis for unsaturated-flow studies. Richards assumed that Darcy’s law holds for unsaturated systems and subsequent work has shown the validity of this assumption (Philip, 1958, p. 153). As has been discussed, unsaturated flow involves several different mechanisms. The particular mechanism that dominates varies with the moisture content. It may seem surprising that a single equation can describe flow under these different conditions, but it can be done because of the fortunate circumstance that the equation for saturated flow is of the same form as the equation governing the diffusion of water vapor. Thus by proper specification of the transmission coefficient as a variable to cover the sum of the different mechanisms operating at different moisture contents a single equation can be used (Philip, 1958, p. 158). It is recognized by some investigators (Edlefsen and Anderson, 1943) that the thermodynamic specification of the liquid and vapor phases is incomplete in current use of the equation. However, no solution to this difficulty is currently available. The best answer available for the hysteresis problem has been to confine the solution of the equation to periods of either soil drying or soil wetting so that moisture content and capillary conductivity can be treated as single-valued functions of the capillary potential. Because of the importance of the unsaturated-flow equation, its derivation is presented. It is derived from the equation of continuity and the Darcy equation, which are discussed first. EQUATION OF CONTINUITY The equation of continuity is chiefly a statement of the law of conservation of matter. It “states that the fluid mass in any closed system can be neither created nor destroyed” (Muskat, 1937, p. 121). It can be derived from the fact that the change of mass in a small unit rectangular parallelepiped equals the difference between the mass entering and the mass leaving. Figure 22 shows a unit rectangular parallelepiped with center at x, y, z. The mass flow of moisture through the parallelepiped is to be computed first. This is the net flow of moisture out of the volume element per unit time, in grams of water per square centimeter per second. Let Vx equal the mass flow of moisture per unit time in the x direction, through a unit cross-sectional area at a distance x from the yz plane. Vx is measured in grams per square centimeter per second. Then, the expression dV, dx equals the change in the mass flow of moisture per unit time per unit cross-sectional area in the x direction, with distance from the yz plane. The mass flow through an elemental plane of the parallelepiped parallel to the yz plane and cutting the point x, y, z is Vidydz, in grams per second. The mass dx flow into the side dydz at a distance x—from the yz plane is V4ydz-^ ^ dydz (91) or [V--Tidi]d«d‘ <92> Similarly, the flow out of the side dydz at a distance x-\-dx from the yz plane is 2 (*» The net mass flow in the x direction out of the unit rectangular parallelepiped per unit time is the flow out,REVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D35 equation 93, minus the flow in, equation 92. Making the subtraction, it is dxdydz (94) Similarly, let F„=the mass flow of moisture per unit time in the y direction, through a unit cross-sectional area at a distance y from the xz plane. F2=the mass flow of moisture per unit time in the z direction, through a unit cross-sectional area at a distance z from the xy plane. Then the net mass flow in the y direction out of the unit parallelepiped per unit time is dxdydz (95) The net mass flow in the z direction out of the unit parallelepiped per unit time is dF dxdydz (96) The total mass flow out of the unit parallelepiped per unit time, in grams per second, is the sum of equations 94, 95, and 96. Adding and simplifying, it is dF* (ix + iiVy dy (97) The loss of mass of water in the unit parallelepiped per unit time is to be computed next. Let per=the bulk density of the medium or soil, in grams per cubic centimeter. /3=the moisture content on a dry-weight basis. This is a decimal fraction obtained by dividing the mass of water in grams by the mass of dried soil in grams. The mass of water per volume of soil, in grams per cubic centimeter, is P'P (98) The mass of water in the unit parallelepiped of volume dxdydz is pafidxdydz (99) The loss of mass of water in the unit parallelepiped per unit time, in grams per second, is dxdydz (100) According to the law of conservation of matter, the mass loss of water in the parallelepiped equation 100 must equal the flow out of the parallelepiped equation 97, or (101) Dividing by dxdydz, d(pe0)_dVx . 5F, dV2 dt dx + dy + dz (1U2j which is the equation of continuity. Let F be a vector giving the mass flow of moisture, in grams per second per square centimeter, in the direction of the line of flow. Then, V=iVx+jVy+kV, (103) Using the vector operator defined in equation 58 on equation 103, V-F=f~-(JVX) +j ~ ■ QVy) ^ • (kV,) (104) But, Therefore, i ■ i=j ■ j=k ■ k= 1 ox oy oz Comparing equations 102 and 106 d(P„j3) df -=V • F (105) (106) (107) V is the vector operator known as del or nabla. It will be recalled also that the product of this operator and a scalar such as potential is a vector known as the gradient. The scalar product of this operator and a vector, such as velocity above, is known as the divergence (Brand, 1947, p. 183). The divergence operator applied to a vector function gives at each point the rate per unit volume at which the physical entity is issuing from that point. If the divergence is positive, as in equation 107, there must be a source of water located at the point, or else water must be leaving the point. If water is leaving the point the storage of water at that point must be decreasing as shown by the negative time derivative on the left side of equation 107. If the divergence is zero and there are no sources or sinks, equation 107 shows that the rate of change of storage also is zero and that steady-state conditions pertain. where <=time, in seconds.D36 FLUID MOVEMENT IN EARTH MATERIALS DARCY’S LAW As stated by Richards (1931, p. 323), “Darcy, working with mediums under saturated conditions, found that the flow of water through a column of soil is directly porportional to the pressure difference and inversely proportional to the length of the column. For low pressure gradients it has been found by numerous investigators (Stearns, 1927; King, 1899) that this law is in exact agreement with experiment and it is entirely analogous to the well-known law of Poiseuille for the flow of liquids through capillary tubes. However, both of these laws fail to hold for high pressure gradients. The limits within which they are true and the modifications which a second approximation requires can be determined only by exhaustive experiments on a wide range of materials. In view of the experimental data now available it is assumed that Darcy’s law holds for the low velocities and pressure gradients dealt with in this paper.” Darcy’s law may be written as V,= -Kv4> (108) where F„=a vector giving the volume flow of moisture, in cubic centimeters per square centimeter per second K = the coefficient of capillary conductivity —V=the negative gradient of the total potential If is in ergs per gram, K is in seconds. FLOW EQUATION The mass flux is the product of the fluid density and the volume flux, or V = PV„ (109) where p=the density of the water, in grams per cubic centimeter. Therefore, equation 107 can be changed to d(pgfl) dt V-IpVA (110) where Vv is the vector giving the volume flow of moisture. The value of V„, the volume flow, can be replaced by means of Darcy’s law, equation 108. Then the equation of continuity, equation 110, becomes dt V-{pKv) (111) This is the general equation for unsaturated flow. For a system in which flow is occurring in the vertical direction only, and for which p„ is constant, equation 111 becomes (112> Equation 78 may be rewritten as =gz+ 2 (113) where 0=the total potential, in ergs per gram 2= the vertical height above the datum <72=the gravitational potential, in ergs per grain 2=the capillary potential, in ergs per gram Substituting this into equation 112, K Lp. dp dzy P'9 dz k=P-K^ p« (116) (117) the equation for vertical flow is dp_ d_ / dP\ , _p dK dt dz \ dz) p, ® dz (118) Similarly, for unidirectional horizontal flow, the flow equation is dp=d_{ dp\ dt dx V dx) (119) k is known as the diffusivity. The reciprocal of the term ^ is analogous to specific heat in the theory of heat flow, and Klute (1952a, p. 106) proposes the name specific moisture capacity for it. Under conditions of steady-state flow the mass loss of water, or the change in moisture content with time, is zero. Therefore, the left sides of equations 102, 107, 111, 118, and 119 are equal to zero. This was discussed previously in regard to equation 107, the continuity equation. It was pointed out that if theREVIEW OF SOME ELEMENTS OF SOIL-MOISTURE THEORY D37 divergence is zero and there are no sources or sinks, the rate of change of storage of water also is zero and steady-state conditions pertain. Under steady-state conditions the general equation for unsaturated flow, equation 111, becomes 0=v-(PKv) (120) When the density of water, p, and the coefficient of capillary conductivity, K, are constant, equation 120 can be rewritten as vV=0 (121) 5V dV dV dx?'dy2~^dz2 (122) This is the well-known Laplace equation. Similarly, for steady-state conditions, the equation for vertical flow of soil moisture, equation 118, becomes 5 z (123) Finally, for steady-state conditions the equation for unidirectional horizontal flow, equation 119, becomes dx (124) These equations are difficult to use because the capillary conductivity, K, the specific moisture capacity, 5/3 the total potential, , the diflusivity, k, and the capillary potential all depend on the moisture content. To solve the equations, the functional dependence of capillary conductivity, specific moisture capacity, potential, and diflusivity on moisture content must be determined by empirical or other means. Exact solutions of equation 123 under given boundary conditions are available from Richards (1931, p. 329), Remson and Fox (1955, p. 308), Wind (1955), and Gardner (1958a). In the past few years, numerical, or iterative, methods have been devised for the solution of the flow equation for the nonequilibrium or transient case under given boundary conditions (Klute, 1952a, Klute, 1952b; Luthin and Day, 1955; Day and Luthin, 1956, p. 445; Philip, 1957; Youngs, 1957). It is hoped that a large number of solutions applicable to soil-moisture flow under different conditions will be available within a few years. REFERENCES Adam, N. K., 1930, The physics and chemistry of surfaces: 1st ed., Oxford, The Clarendon Press, 332 p. ------1941, The physics and chemistry of surfaces: 3d ed., Oxford, Oxford Univ. Press, 436 p. American Geological Institute, 1957, Glossary of geology and related sciences: Washington, Natl. Acad. Sci., Natl. Research Council, 325 p. Baver, L. D., 1948, Soil" physics: 2d ed., New York, John Wiley & Sons, 398 p. Bodman, G. B., and Colman, E. A., 1944, Moisture and energy conditions during downward entry of water into soils: Soil Sci. Soc. 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Fermi, Enrico, 1937, Thermodynamics: New York, Prentice-Hall, 160 p. Gardner, W. R., 1958a, Some steady-state solutions of the unsaturated moisture flow equation wtih application to evaporation from a water table: Soil Sci., v. 85, no. 4, p. 228-232. ------- 1958b, Mathematics of isothermal water conduction in unsaturated soils, in Winterkorn, H. F., ed., Water and its conduction in soils: Highway Research Board Spec. Rept. 40, Washington, Natl. Acad. Sci., Natl. Research Council, Pub. 629, p. 78-87. Gray, H. J., ed., 1958, Dictionary of physics: London, Longmans, Green & Co., 544 p. Gurr, C. G., Marshall, T. J., and Hutton, J. T., 1952, Movement of water in soil due to a temperature gradient: Soil Sci., v. 74, no. 5, p. 335-345. Hague, B., 1939, An introduction to vector analysis: London, Methuen & Co., 118 p.D38 FLUID MOVEMENT IN EARTH MATERIALS Hendricks, S. B., 1955, Necessary, convenient, commonplace, in Yearbook of Agriculture: U.S. Dept. Agriculture, p. 9-14. Hildebrand, J. H., 1947, Principles of chemistry: New York, The Macmillan Co., 446 p. Hutcheon, W. L., 1958, Moisture flow induced by thermal gradients within unsaturated soils, in Winterkorn, H. F., ed., Water and its conduction in soils: Highway Research Board Spec. Rept. 40, Washington, Natl. Acad. Sci., Natl. Research Council, Pub. 629, p. 113-133. Joos, George, 1934, Theoretical physics: Hafner Pub. Co., 748 p. [English translation by I. M. Freeman.] King, F. H., 1899, Principles and conditions of the movements of gound water: U.S. Geol. Survey 19th Ann. Rept., pt. 2, p. 59-294. Klute, Arnold, 1952a, A numerical method for solving the flow equation for water in unsaturated materials: Soil Sci., v. 73, no. 2, p. 105-116. ------ 1952b, Some theoretical aspects of the flow of water in unsaturated soils: Soil Sci. Soc. America Proc., v. 16, no. 2, p. 144-148. Kuzmak, J. M., and Sereda, P. J., 1958, On the mechanism by which water moves through a porous material subjected to a temperature gradient, in Winterkorn, H. F., ed., Water and its conduction in soils: Highway Research Board Spec. Rept. 40, Washington, Natl. Acad. Sci., Natl. Research Council, Pub. 629, p. 134-146. Low, P. F., 1958, Movement and equilibrium of water in soil systems as affected by soil-water forces, in Winterkorn, H. F., ed., Water and its conduction in soils: Highway Research Board Spec. Rept. 40, Washington, Natl. Acad. Sci., Natl. Research Council, Pub. 629, p. 55-64. Luthin, J. N., and Day, P. R., 1955, Lateral flow above a sloping water table: Soil Sci. Soc. America Proc., v. 19, no. 4, p. 406-410. Meinzer, O. E., 1923a, The occurrence of ground water in the United States, with a discussion of principles: U.S. Geol. Survey Water-Supply Paper 489, 321 p. ------ 1923b, Outline of ground-water hydrology, with definitions: U.S. Geol. Survey Water-Supply Paper 494, 71 p. Meyer, B. S., and Anderson, D. B., 1939, Plant physiology: New York, D. Van Nostrand Co., 696 p. Miller, E. E., and Miller, R. D., 1956, Physical theory for capillary flow phenomena: Jour. Applied Physics, v. 27, no. 4, p. 324-332. Muskat, Morris, 1937, The flow of homogeneous fluids through porous media: New York, McGraw-Hill Book Co., 763 p. Partington, J. R., 1951, An advanced treatise on physical chemistry: v. 2, London, Longmans, Green & Co., 448 p. Pauling, Linus, 1958, General chemistry: 2d ed., San Francisco, W. H. Freeman & Co., 710 p. Philip, J. R., 1957, The theory of infiltration: Soil Sci., v. 83, p. 345-357, 435-448; v. 84, p. 163-178, 257-264, 329-339. ------1958, Physics of water movement in porous solids, in Winterkorn, H. F., ed., Water and its conduction in soils: Highway Research Board Spec. Rept. 40, Washington, Natl. Acad. Sci., Natl. Research Council, Pub. 629, p. 147-163. Remson, Irwin, and Fox, G. S., 1955, Capillary losses from ground water: Am. Geophys. Union Trans., v. 36, no. 2, p. 304-310. Remson, Irwin, Randolph, J. R., and Barksdale, H. C., 1960, The zone of aeration and ground-water recharge in sandy sediments at Seabrook, New Jersey: Soil Sci., v. 89, no. 3, p. 145-156. Richards, L. A., 1931, Capillary conduction of liquids through porous mediums: Physics, v. 1, no. 5, p. 318-333. Richards, L. A., and Richards, S. J., 1957, Soil moisture, in Yearbook of Agriculture: U.S. Dept. Agriculture, p. 49-60. Schofield, R. K., 1948, Discussion of Childs, E. C., and Collis-George, N., Soil geometry and soil-water equilibria: Faraday Soc. Disc., v. 3, p. 129. Sears, F. W., 1950, Mechanics, heat, and sound: 2d ed., Reading, Mass., Addison-Wesley Pub. Co., 564 p. Slichter, C. S., 1899, Theoretical investigation of the motion of ground waters: U.S. Geol. Survey 19th Ann. Rept., pt.2, p. 295-384. Smith, A. W., 1948, The elements of physics: New York, McGraw-Hill Book Co., 745 p. Smith, W. O., 1936, Sorption in an ideal soil: Soil Sci., v. 41, no. 3, p. 209-230. -------1943, Thermal transfer of moisture in soils: Am. Geophys. Union Trans., p. 511-523. Smith, W. O., Foote, P. D., and Busang, P. F., 1931, Capillary rise in sands of uniform spherical grains: Physics, v. 1, no. 1, p. 18-26. Soil Science Society of America, 1956, Report of definitions approved by the Committee on Terminology: Soil Sci. Soc. America Proc., v. 20, no. 3, p. 430-440. Stearns, N. D., 1927, Laboratory tests on physical properties of waterbearing materials: U.S. Geol. Survey Water-Supply Paper 596-F, p. 121-176. Sulman, H. L., [London] 1919, A contribution to the study of flotation: Mining Metall. Inst. Trans., v. 29, p. 44-138. Taylor, S. A., and Cavazza, Luigi, 1954, The movement of soil moisture in response to temperature gradients: Soil Sci. Soc. America Proc., v. 18, no. 4, p. 351-358. Theis, C. V., 1935, The relation between the lowering of the piezometric surface and the rate and duration of discharge of a well using ground water storage: Am. Geophys. Union Trans., pt. 2, p. 519-524. Thompson, W., 1871, On the equilibrium of vapor at a curved surface of liquid: Philosophical Mag., v. 42, p. 448-452. Veihmeyer, F. J., 1939, The validity of the assumption that it is possible to produce different moisture-percentages in field soils: Rept. Comm, on Physics of Soil Moisture, 1938-39, Am. Geophys. Union Trans., p. 543-545. Wark, I. W., and Cox, A. B., 1932, Principles of flotation—an experimental study of the effect of Xanthates on contact angles at mineral surfaces: Am. Inst. Mining Metall. Engineers Tech. Pub. 461, 48 p. Wills, A. P., 1931, Vector analysis: New York, Prentice-Hall, 285 p. Wind, G. P., 1955, A field experiment concerning capillary rise of moisture in a heavy clay soil: Netherlands Jour. Agr. Sci., v. 3, p. 60-69. Winterkorn, H. F., 1958a, Mass transport phenomena in moist porous systems as viewed from the thermodynamics of irreversible process, in Winterkorn, H. F., ed., Water and its conduction in soils—An international symposium: Highway Research Board Spec. Rept. 40, Washington, Natl. Acad. Sci., Natl. Research Council Pub. 629, p. 324-338. Winterkorn, H. F., ed., 1958b, Water and its conduction in soils—An international symposium: Highway Research Board Spec. Rept. 40, Washington, Natl. Acad. Sci., Natl. Research Council, Pub. 629, 338 p. Youngs, E. G., 1957, Moisture profiles during vertical infiltration: Soil Sci., v. 84, no. 4, p. 283-290. U. S. GOVERNMENT PRINTING OFFICE : 1962 O - 643856Multiphase Fluids in Porous Media—A Review of Theories Pertinent to Hydrologic Studies documents department '-'AN 29 1364 LIBRARY 1 UNIVERSITY OF CALIFORNIAMultiphase Fluids in Porous Media—A Review of Theories Pertinent to Hydrologic Studies By ROBERT W. STALLMAN FLUID MOVEMENT IN EARTH MATERIALS GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-E UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402CONTENTS Page Abstract________________________________________________ El Introduction_______________________________________________ 1 Acknowledgments------------------------------------------ 3 Scope------------------------------------------------------ 3 Nomenclature_______________________________________________ 4 Static fluid occurrence____________________________________ 4 Terminology___________________________________________ 4 Structural stage _________________________________ 5 Adsorbed stage____________________________________ 6 Pendular stage____________________________________ 7 Funicular stage______________:________________ 7 Capillary stage___________ _ _____-________ __ 7 Computation of fluid content__________________________ 8 Limitations imposed by characteristics of the fluid and the solid_____________________________ 8 Adsorbed liquid___________________________________ 8 Water held by capillarity_________________________ 8 Observation of the curve liquid content versus liquid pressure head______________________________________ 11 Manometric control_______________________________ 12 Use of fluids other than water___________________ 12 Centrifuge acceleration__________________________ 13 Vapor pressure control___________________________ 16 Effects of temperature on head and liquid content— 17 Effects of salts in the liquid phase on liquid pressure . head and liquid content_________________________ 22 Page Multiphase flow________________________________________ E24 General flow relations________________________________ 24 Measurement of liquid conductivity____________________ 27 Calculation of liquid conductivity____________________ 28 Differential equation for liquid flow_________________ 29 Computation of one-dimensional flow___________________ 30 Water-vapor transmission______________________________ 32 Anisothermal flow of liquid and vapor----------------- 36 General relations--------------------------------- 36 Liquid phase____________________________________ 36 Vapor phase_______________________________________ 37 Composite of anisothermal flow equations______ 38 Criteria for hydraulic models of liquid flow_______________ 39 Dimensional similitude________________________________ 40 Stationary model in earth gravity field_______________ 41 Artificially accelerated model________________________ 42 Hydrologic investigations and the unsaturated zone----- 43 Nature of hydrologic problems_________________________ 43 Problems in defining the behavior of the unsaturated zone________________________________________________ 44 Research requirements_________________________________ 45 Summary----------------------------------------------- 49 References_________________________________________________ 49 ILLUSTRATIONS Page Figure 1. Classification of liquid occurrence by stages____________________________________________________________________ E6 2. Curvature on liquid-gas interface----------------------------------------------------------------------------- 9 3. Liquid rise in a capillary tube___________________________________________________________________________________ 9 4. Liquid volume versus height above free surface in pendular ring__________________________________________________ 10 5. Moisture content versus height above free surface in sands observed by King and calculated by Smith-------------- 11 6. Liquid pressure head versus liquid content, observed on tension plate apparatus-------------------------------- 13 7. Moisture retention versus acceleration after 5 minutes of centrifuging------------------------------------------- 14 8. Liquid content versus height above the free-liquid surface----------------------------------------------------- 15 9. Diagram showing sample arrangement in centrifuge_________________________________________________________________ 15 10. Controlling liquid pressure head by regulating vapor pressure with solutes_______________________________________ 16 11. Water adsorption and liquid pressure-head control by vapor regulation in fine-grained soils---------------------- 18 12. Static head, moisture content, and temperature relations for a sand______________________________________________ 20 13. Gross changes in storage above the water table as a function of temperature and water-table position------------- 20 14. Measurements of temperature and liquid pressure-head in a muck soil______________________________________________ 21 15. Vapor phase in vicinity of liquids____________________________________________________________________________ 22 16. Tensiometer in contact with moist soil___________________________________________________________________________ 23 17. Quality, electrical conductivity, and osomtic pressure relations for soil water extracts----------------------- 24 18. Schematic relative conductivities for wetting liquids and gas phase______________________________________________ 26 19. Multifluid flow data_____________________________________________________________________________________________ 26 20. Schema of liquid conductivity measurement apparatus______________________________________________________________ 27 21. Calculated and observed conductivities versus liquid content_____________________________________________________ 28 mIV CONTENTS Page Figure 22. Moisture content and pressure-head values after horizontal flow into dry media________________________________ E31 23. Calculated moisture profiles in nondimensional form for horizontal flow in semi-infinite medium____________ 33 24. Water vapor diffusion through dry soil from wetted section in contact with sample__________________________ 34 25. Schema of a sealed sample under a thermal gradient_________________________________________________________ 34 26. Vapor flow rates versus temperature gradients______________________________________________________________ 35 27. Heat conductivity effects caused by gravity flow components________________________________________________ 35 28. Schema of combined anisothermal flow of gas and liquid through porous media________________________________ 36 29. Ratio of calculated anisothermal flow of fluid to calculated vapor diffusion and observed fluid flow as a function of liquid content_________________________________________________________________________________________ 38 30. Diagram showing steady state profiles of liquid content above water table__________________________________ 46 31. Water content and conductivity of carbon plug showing hysteresis___________________________________________ 48 TABLES Page Table 1. Water removed from clay colloids by oven drying____________________________________________________________________ E6 2. Pressure head in pore liquid versus relative humidity of gas phase_____________________________________________ 12 3. Static pressure head, moisture content, and temperature relations for King’s No. 40 sand_______________________ 19 4. Soil-water salinity and moisture content at the wilting point for tomato plants________________________________ 24FLUID MOVEMENT IN EARTH MATERIALS MULTIPHASE FLUIDS IN POROUS MEDIA—A REVIEW OF THEORIES PERTINENT TO HYDROLOGIC STUDIES By Robert W. Stallman ABSTRACT Studies of the occurrence of surface water and ground water are commonly made with little or no regard to the unsaturated zone. Yet, this zone between the land surface and the saturated earth materials has a significant influence on the distribution of both surface water and ground water as a function of time and space. Where man’s use of water results in the full development or major rearrangement of water distribution, the dynamic effects on water distribution caused by hydrologic processes in the unsaturated zone must be considered in evaluating the total water resource. Further, the unsaturated zone must be treated as a dynamic influence on water distribution. Treating the unsaturated zone as a dynamic component in the hydrologic cycle requires (1) understanding of the basic laws of multiphase fluid movement through porous media, (2) means for identifying the features of the earth materials in the unsaturated zone on which flow is dependent, and (3) techniques whereby understanding of this system might be translated into practical engineering computations which will define the effects to be expected from a postulated change in water distribution. This report summarizes the present status of the theory of occurrence of multiphase fluids in physically and chemically stable porous media and the laws of flow of the vapor and liquid phases. Review of the literature indicates that the theory of isothermal liquid flow has been sufficiently developed and tested to provide an adequate basis for use in practical engineering investigations of flow through the unsaturated zone. However, the extensive application of theory evidently must await the development of more practical means for evaluating the pertinent hydrologic and physical characteristics of earth materials in the field and for making the required calculations to define the flow regime. Theoretical descriptions of anisother-mal liquid and vapor flow through unsaturated rocks have been advanced, but have not yet been fully tested in the laboratory. The magnitude of vapor transport, even by thermal gradients, may in many places be far more significant to water availability than heretofore assumed. Thus, it appears that anisothermal flow in the unsaturated zone should be the subject of further and more intensive laboratory and field study. Details of modeling unsaturated liquid flow in the laboratory have been developed from dimensional considerations. It is apparent that many laboratory results could be made more useful to field practice if flow experiments were designed according to dimensional criteria. Specifically, moisture equivalent determinations made by centrifuging are currently made without complete regard for modeling criteria, and as a result, interpretations of reported data may bear but slight relation to the I hydraulic characteristics of the unsaturated zone. Changes in testing procedure are suggested to improve laboratory centrifuge experiments. Other laboratory test procedures reported in the literature are also discussed in relation to theory, and reasons for anomalous experimental results are advanced, particularly for anisothermal flow. Also included is a review of the concept of “specific yield” and a discussion of its usefulness for defining a system which must be considered as a dynamic flow regime. It is concluded that, for those hydrologic problems in which flow through the unsaturated zone occurs isothermally in the liquid phase, theory of flow has been sufficiently developed and tested to use with assurance for forecasting changes in water occurrence caused by changing the environment of the unsaturated zone. However, the basic laws of flow, field observation techniques, and computing methods are either inadequately developed or tested to permit treatment of the more complicated flow systems. The most critical areas of insufficient basic knowledge or technique appear to be the following: (1) The rate of movement of adsorbed water, (2) techniques for measuring thermal properties of moist soils, (3) methods of calculating anisothermal multiphase fluid flow through a region where the boundary conditions are known to be variable in both time and space, (4) multiphase fluid occurrence in fine silts and clays, and (5) field measurements of the hydraulic characteristics of porous media in the unsaturated zone. INTRODUCTION Hydrologic investigations are mainly concerned with the distribution of water on and beneath the land surface in time and space. In these investigations the physical and hydraulic characteristics of the flow systems are studied by either a direct or indirect approach. A direct approach is one in which the water distribution is defined by measurement of the physical aspects of the system such as aquifer size and shape, drainage density, basin slopes, and other factors. Indirect methods may be characterized as those procedures from which the gross flow relations are obtained without reference to the physical and hydraulic variables controlling flow. Flow-duration curves obtained from stream gaging and discharge-drawdown relations for pumping wells are examples. ElE2 FLUID MOVEMENT IN EARTH MATERIALS Direct methods, as considered here, are becoming increasingly more important in hydrologic studies. This is because the large-scale water developments of today inherently create significant changes in system conditions. Description of a hydrologic unit on the basis of indirect methods only defines those flow relations that obtain if there is no change in the system. Extrapolation of such information to predict the consequences of a large-scale water development on future water availability is unrealistic. Where large-scale water development significantly changes the water distribution from its natural state, the hydrologist must rely on knowledge of the physical and hydraulic characteristics of the system to evaluate the available resources for the future. Ground water and surface water are interconnected through the unsaturated zone which lies between the land surface and the water table (Meinzer, 1923). The function of the unsaturated zone in the hydrologic cycle is predominantly one of storage. Overland flow derived from precipitation is reduced by the rate at which water is absorbed by this zone, and the aquifer response to drainage or pumping is dependent on the manner in which water is stored by or moves from this zone to the saturated section of rocks beneath it. The functional role of the unsaturated zone in the distribution of ground water and surface water is now defined mainly through indirect methods of study. An example of the application of the indirect approach is the determination of infiltration and its application in hydrologic investigations. Hydrologists generally have adopted the indirect approach for describing in filtration except for a comparatively small amount of work paralleling Horton’s (1933). Statistical correlation is most often used as a means for relating infiltration to factors such as antecedent precipitation, time of the year, and rainfall intensity. Although such correlations are certainly useful for determining the infiltration within given probability limits, the accuracy attainable from such studies will usually be low if they are applied to estimate infiltration rates over short time intervals. Fundamentally this is because correlation of hydrologic phenomena is based on many months or years of observation. Data correlation, as applied in hydrologic analysis, generally yields flow relations representing the average or mean of conditions existing over a past period of observation. Yet, these mean values are sometimes applied to much shorter periods, say one storm interval, for estimating infiltration under conditions likely to be far different from the mean of the period of observation. If the total hydrologic environment is not near the mean during the storm interval studied, the infiltration estimate from statistical correlation cannot be correct. Infiltration is the resultant of several factors: (1) The distribution and depth of water on the land surface; (2) the underground permeability, moisture distribution, and temperature; (3) the length of time water is supplied at the land surface; and (4) the structural stability of the rock mass. All these factors influence the infiltration rate, and each of these is variable with time. Entering the variable factors in the appropriate equation of flow would permit the computation of instantaneous or total infiltration for any set of circumstances for any given time or time period. The validity of such a computation would be dependent primarily on the accuracy with which all pertinent variables could be evaluated. Estimating infiltration by this means is difficult because the variables controlling it are many. Nevertheless, if estimates are to be made realistically on a cause-effect or dynamical basis, the identity of all pertinent variables, including time, must be retained until it is demonstrated that some can be omitted from analysis or can be grouped. The hydraulic properties of aquifers are often obtained from pumping tests. Generally these tests are conducted over a comparatively short time, rarely for longer than 3 days. The aquifer characteristics obtained from analysis of such tests are often applied to forecast the occurrence of water under diverse conditions of water movement, and often over time periods either much shorter or much longer than that taken for testing. Accuracy of the forecast made depends on the accuracy with which the test and test analysis represent the hydraulic conditions at the applied extrapolation or interpolation. Attention has been focused on aquifer transmission and storage characteristics as being measures of the ability of aquifers to yield water. It is generally agreed among ground-water hydrologists that if the aquifer remains saturated its transmission characteristic changes insignificantly as a function of time, pressure, and hydraulic boundary conditions. In the saturated zone where water appears to be released almost instantaneously with a decline in pressure, storage changes are also relatively independent of all factors other than pressure changes. That is, at any given point in porous media saturated with water the hydraulic characteristics may be described by constants to a satisfactory order of approximation. In unsaturated media, however, flow is highly dependent on the degree to which interstices are filled with liquid, the pressure and pressure distribution in the fluid, temperature and temperature distribution, and the physical and chemical properties of the fluid and media. Further, all these are dependent on climatologic and hydrologic conditions which are variable with time. Thus it can be visualized that drainage attending a lowering of theMULTIPHASE FLUIDS IN POROUS MEDIA E3 water table is inherently difficult to describe exactly and completely because the interrelations between the variables of flow are many and complicated. Further, application of a complete expression for storage changes in the equation of flow for the contiguous saturated zone would present added and rather formidable difficulties. Some of this complexity may be reduced by accepting an approximate expression relating storage changes to only the most significant variables. The concept of specific yield was introduced for this purpose by Mein-zer (1923). As defined, specific yield refers to storage changes caused by gravity drainage. Other conditions affecting drainage from the unsaturated zone such as the rate of downward percolation caused by recharge, thermal and chemical potential differences, and the time-wise variations in storage between two equilibrium states are accounted for by the specific yield concept in a very general way. Although it is an approximation, the specific yield concept is a practical and useful means for describing ground-water storage for some purposes. It is limited in application partly because it is measured under physically unspecified environmental conditions. Yet, we frequently apply aquifer storage characteristics determined from pumping-test analysis, even from tests of only a few hours duration, as an adequate measure of the drainage characteristics expected over many years of dewatering. Furthermore, the test analyses are based generally on an expression which is derived from a mixture of static and dynamic viewpoints. The reporter often makes the statement or implies that near-equilibrium conditions exists above the -water table through even short tests, and therefore the specific yield observed adequately describes the aquifer storage characteristics in the system observed. Too often, in actuality, this viewpoint is but a reflection of the simplifications inherent in the concept of specified yield. At the present time, it appears impossible to assess the errors inherent in ground-water hydrology studies wherein the simplified concept of drainage from the unsaturated zone is applied. In order to clarify this point, the unsaturated zone must be studied as a dynamic flow system. In hydrologic analysis requiring the viewpoint that flow within the hydrologic cycle is a dynamic process, inclusion of physical concepts based on static or steady-state conditions yields a system or method of analysis constructed by addition of incongruous terms. The construction of analytical expressions by addition of component terms, some arbitrarily defined by statics, some by steady state, and some by nonsteady relations, is equivalent to adding terms of differing dimensional properties. The final result is dimensionally indescribable and has such a poor theoretical foundation that its validity cannot be evaluated, unless the assumed state of each component term is rigorously justified. To the time of this writing, 1960, the influence of the unsaturated zone on flow has been treated as a statical term, without adequate justification. Continuing research is being conducted by the U.S. Geological Survey and many other agencies that will lead eventually to a more comprehensive description of the effects of the unsaturated zone on ground- and surface-water movement. For example, work is being done to improve the field and laboratory measurement of specific yield, as defined, and to devise a system of observing and presenting aquifer storage data which treats the unsaturated zone as a dynamic hydrologic entity. Infiltration, evapotranspiration, base flow of streams, bank storage, and a host of other hydrologic phenomena are directly dependent on the dynamic changes of water storage within the unsaturated zone. Evaluation of these factors as functions of time and physical environment can be greatly improved through a more thorough application of the physical principles of fluid occurrence in unsaturated porous media. ACKNOWLEDGMENTS The writer is indebted to Mr. P. E. LaMoreaux, formerly chief, Ground Water Branch, and Mr. E. K. Bennett, formerly chief, Besearch Section, Ground Water Branch, whose support and encouragement made this review possible. Interest in, and understanding of, the subject was gained chiefly through several years of association with W. O. Smith, physicist, Ground Water Branch. To Smith; E. H. Brown, chief, Eesearch Section, Ground Water Branch; E. Schneider, geologist, and Akio Ogata, mathematician, Ground Water Branch; C. V. Theis, geologist, Water Eesources Division; associates in the Ground Water Branch Hydro-logic Laboratory, Denver, Colo., and many others, the writer is especially grateful for extended discussion of the subject and many helpful suggestions in review of this manuscript. SCOPE Multiphase fluid occurrence and movement is of interest to a wide variety of physical science students. Theory and applications practices pertaining to this subject have originated in studies concerned with oil production, soil mechanics, hydraulics, applied mathematics, chemical engineering, soil science, hydrology, and other fields. In hydrology, as it is with some of the other specialized studies, there is a need for a composite review of the potentially useful theory and applications found in the specialized literature of each field. Such a composite is prerequisite to a full understanding of the role played by the unsaturated zone in hydrologic activity. In this report the general aim has been to presentE4 FLUID MOVEMENT IN EARTH MATERIALS a brief description of fundamental relations among the variables known to govern the occurrence and movement of fluids in the unsaturated zone, and to indicate those areas where knowledge is deficient insofar as hydrologic studies are concerned. A few experimental data have been selected from the published literature for illustrating the significance of some of the variables involved. The structural stability of fine-grained materials is an important factor in the occurrence of fluids underground. In this report, however, only those factors pertaining to flow through rigid physically and chemically stable porous media are discussed. NOMENCLATURE Where equations are quoted from other texts the originally published notation is changed to conform with that used in this report. Notation used in this text is defined where it first appears. In addition, the following summary list of symbols used at many places in the text is presented for the convenience of the reader. Units of the cgs system are used except where, as indicated, a computation is employed for direct or implied comparison with numerical results published elsewhere. A, B, C, F, representations of fluid or solid characteristics. Df, liquid diffusivity. D0, coefficient of diffusion of water vapor at standard pressure and temperature (equals 0.239 cm2-sec-1 at 281° Kelvin and 1 atmosphere). D„ coefficient of diffusion of water vapor. H, height above free surface. H', height above free surface in model. K, specific permeability of porous medium (cm2). Kf, conductivity to liquid flow, a function of (cm-sec-1). K„ conductivity to vapor flow, (cm-sec-1). Kr{, relative conductivity to liquid Af (at — 1) Krt, relative conductivity to gas=t • Arr(at 0f—U) L, unit of length. M, unit of mass. N, rotational velocity in rpm. P, total pressure. P„, standard total pressure. R, gas constant per gram of water (4.51 X103). T, temperature, degrees Kelvin. T0, standard reference temperature. V, liquid volume contained in pendular ring. a, b, * * * d, constants. f(rc) function of characteristic radius on liquid-gas interface. g, acceleration due to gravity. \\xa:-\^-\\x.v^ ipil .VxWxWvWv ^wisi ililitf w\\ x Vv N X\\X \V.X \'x Solid'. v\x\\V^x\\x\\xx' ‘N'“ xx\NSolid^\\J m \p\d W,\xx\x\' X\\\V x\\\ W\^'- Lna L'\X\: RSolid Cx Mmm, IW' 'Solid',, W «» W' in this stage provides means for judging test feasibility and accuracy. For example, Hoseh (1937) subjected selected soil colloids to temperatures as high as 500°C and compared the water content observed from the weight reduction with the weight observed after drying at 500°C. Table 1 shows the results obtained. It may be noted that drying at 110°C still permitted a residual liquid content of about 25-35 percent of the pore space, assuming that all liquid had been driven off by drying the samples at 500°C. Heat of wetting observations made by Hoseh indicated that the samples heated above 400°C were evidently sintered. Alteration of porosity and structure in samples with high organic content are caused also by drying at high temperatures. For materials like soil colloids, clays, and organic materials, oven drying is inadequate for dry weight determinations. Drying in a vacuum over a liquid with very low vapor pressure, at room temperature, is preferred (Orchiston, 1953). Table 1.—Water removed from clay colloids by oven drying 1 Oven temperature (°C) Water removed, in percent of total removed by drying at 500 °C, for indicated soil type Altamont Yolo Vina Aiken Room _ 0. 0 0. 0 0. 0 0. 0 47 42. 07 43. 57 33. 00 45. 85 70 70. 97 72. 93 66. 51 64. 23 no 76. 25 78. 12 71. 41 66. 06 200 81. 58 81. 65 76. 76 69. 57 340 90. 17 89. 14 83. 30 81. 16 400 95. 11 97. 67 91. 43 96. 40 500 100. 0 100. 0 100. 0 100. 0 i From Hoseh (1937, table 5, p. 265). ADSORBED STAGE If a small amount of water vapor is introduced into the pore space of a dry medium, water will be adsorbed on the surface of the solid particles because of the attractions between the molecules of water and those of the solid. The amount of fluid adsorbed per unit area of surface or per unit weight of solid depends mainly on the chemical structure of both the fluid and solid, Figure 1—Classification of liquid occurrence by stages. A, Structural: Pore space completely filled by gas phase or non wetting liquid. Wetting liquid found in molecular form as part of the chemical structure of the solid; B, Adsorbed: Pore space largely filled by gas phase or nonwetting liquid. Wetting liquid contained on adsorption sites of the solid as a continuous or discontinuous film of one or more molecular layers; C, Pendular: Pore space largely filled by gas phase or nonwetting liquid. Wetting liquid exists mainly in small isolated rings around grain contacts, termed pendular rings, and on adsorption sites. Continuity of the wetting liquid is provided by an adsorbed film on the solid surface; D, Funicular: Pore space partly filled by gas phase or nonwetting liquid. Wetting liquid is continuous from one area to another within the pore space, and is relatively Independent of adsorption characteristics of the solid; E, Capillary: All pore space is completely filled by wetting fluid in the liquid phase.MULTIPHASE FLUIDS IN POROUS MEDIA E7 on the temperature and pressure in the medium, and on the amount of fluid available to the pore space. In fine-grained media, or in materials where the pores are small and porosity is high, the surface area per unit volume is large. The amount of liquid adsorbed per unit volume of the medium is approximately proportional to the surface area of solid. Consequently, in fine-grained materials such as silts and clays, water retention may be regulated predominantly by the nature of their adsorption characteristics. Adsorbed water, by virtue of the forces causing adsorption, is under a pressure greater than the pressure in the contiguous vapor or gas phase. In many generalized treatments of flow through porous media it is assumed that adsorbed water is immobile. However, recent studies (see page E48^f9) indicate that adsorbed water does in fact move along the surface in response to energy gradients. In coarsegrained materials the contribution to flow from adsorbed fluid may be negligible, whereas in fine-grained materials movement of adsorbed water may constitute a large part of the total flow, although the total flow itself is small. PENDULAR STAGE Upon increasing the liquid content in the pore space through the adsorbed stage, the distance between the solid surface and the liquid-gas interface is increased. Thus as the liquid content is increased, the force interaction between molecules of the solid and molecules of liquid in the liquid-gas interface decreases. At higher liquid contents the shape of the liquid-gas interface is primarily dependent on interaction between liquid molecules only. In porous media this becomes important first in the vicinity of grain contact areas, where doughnut-shaped rings of liquid are formed, as indicated in figure 1. Liquid occurrence is defined as being in the pendular stage between that moisture content at which the liquid intermolecular forces are a dominating influence on shape of the liquid-gas interface, and that moisture content at which the pendular rings first begin to coalesce. In that part of the liquid affected predominantly by the liquid-gas interface, the pressure is less than the pressure in the gas phase. For the water-vapor systems, curvature of the interface is convex toward the solid at all points where the liquid pressure is less than the pressure in the gas phase. Fundamentally, the surface curvature relations in porous media are the same as those applied to the study of interfaces, or menisci, in capillary-sized tubes. Considering liquid occurrence on a volumetric basis rather than by single grain contacts, the pendular stage may be visualized as that stage for which (1) there are many continuous paths, through liquid, traversing both adsorbed liquid and liquid contained in rings under re- duced presure, and (2) there is at least one path entirely through the pore spaces passing continuously through only the vapor or gas phase, or nonwetting liquid. The paths taken are assumed to begin from some point on the surface enclosing a selected volume of the medium to another point on the surface diametrically opposite the starting point. At a grain contact where water occurrence is in the pendular stage, one visualizes the liquid contained to occupy a type of ring form. Even though on a gross volumetric basis the rings may coalesce locally, if the conditions prevail as indicated in note (2) above, the selected volume would be considered in the pendular stage. For applying laws of flow, the nature of the occurrence of water must be considered on a volumetric basis rather than on a microscopic scale so as to provide a finite sample size sufficient to afford a degree of homogeneity. Without such a view, it would be necessary to consider all the intricate spatial relations in the pore system—an approach found to be impractical for the study of fluids in naturally occurring porous media. FUNICULAR STAGE Increasing the moisture content to a value above that for the upper limit of the pendular stage will cause the pendular rings io coalesce to such an extent that the gas or vapor phase locally will be entirely enclosed by liquid. In a finite but small sample of the medium, as discussed above, pendular rings might exist at a few points, but again considering the liquid system on a volumetric basis, the vapor or gas phase is discontinuous through the pore space. As a parallel to the description of liquid continuity in the pendular stage, in the funicular stage (1) there are many continuous paths through liquid under tension and (2) there are no paths entirely through the pore space passing through only the gas or vapor phase. The upper part of the capillary fringe and the lowest part of the intermediate belt (Meinzer, 1923, p. 23,26), for static conditions, probably contain water in the funicular stage. Conditions of nonsteady flow, recharge, and stratification of rock in depth may develop the funicular stage at other locations in the unsaturated zone, either permanently or temporarily. CAPILLARY STAGE The stages of liquid occurrence just described all refer to situations in which part of the pore space is occupied by non wetting fluid. As explained by Smith (1933a) the liquid pressure in places within the pore space may be less than the total pressure outside the system, yet wetting fluid may occupy all the pore space. The assemblage of pores is much like a group of irregularly shaped and interconnected tubes of small diameter. Liquid will rise in the pore spaces in much the sameE8 FLUID MOVEMENT IN EARTH MATERIALS fashion as it would rise in capillary-sized tubes of uniform cross section if the ends are immersed in a body of free liquid. The liquid head everywhere above the source of free liquid for static conditions will be negative just as it is in the capillary tube. In the process of filling the pores by capillary rise, part of the liquid flow will be at a higher velocity than other parts, because the pore space is irregular in cross section. The faster moving liquid may flow around sections containing gas and vapor, thereby trapping gas locally and forming a funicular stage of liquid occurrence. The gas or vapor phase entrapped in this way eventually establishes a stable funicular stage almost down to the free liquid surface. Any vapor or gas trapped in excess of that required for equilibrium is dissipated by either condensation or gas diffusion through the liquid. Upon dewatering by lowering the source of free water, gas may be liberated from the capillary fringe because the liquid pressure is reduced, provided the liquid is initially nearly saturated with gas. Water near the water table is generally saturated with gases peculiar to the soil zone—a mixture of normal atmospheric gases enriched with C02. Changes in the water-table altitude are, in effect, changes in position of the free-water surface at the base of the unsaturated zone. In most places the storage changes accompanying changes in the water-table position are probably dependent on the entrapment and release of gas from solution. Thus the capillary fringe saturated with water, as defined, may exist only as a rarity in the field. COMPUTATION OF FLUID CONTENT LIMITATIONS IMPOSED BY CHARACTERISTICS OF THE FLUID AND THE SOLED The solid particles composing most naturally occurring porous media are highly irregular in shape, have a micro- and macroscopically nonhomogeneous surface roughness and vary in chemical characteristics. Fluid distribution in the pore space in the unsaturated zone is dependent on all these factors, among others. Limiting study to the occurrence of water permits some simplification in defining the relations between the solid and fluid, but the nature of the pore space with respect to content of fluid can, in the final analysis, be obtained only by measurement. Thus far means for calculating the fluid content in naturally occurring media, with only the environmental fluid conditions known, have not been found. Methods for predicting fluid content have been developed, however, for ideal porous media. Idealizations include, variously, assuming that the particles are spherical, perfectly smooth, chemically homogeneous, do not combine chemically with the fluid, or that they form pore spaces that may be considered equivalent to circular capillary tubes. Idealizations of this nature have been adopted specifically to describe the solid in simple form, a form amenable to further mathematical manipulation and eventual computation of liquid content for prescribed environmental conditions. Although the relations found by study of ideal media may not be directly applicable to natural media, they do serve to indicate, in a general way, the influence environment has on fluid retention. ADSORBED LIQUID The laws governing the adsorption of water on solid surfaces are not yet clearly understood. Present status of our ability to calculate the amount of liquid adsorbed was summarized by Herbener (1958, p. 16) as follows: Progress made in the field of adsorption during the last decade, leads one to the conclusion that adsorption processes in general will be predictable by theoretical methods within the next decade. The method or theory for achieving this result may already have been stated in rough form but it needs sophistication or requires a combination of the above methods, that is, an expanded equation * * *. The results from theoretical studies of adsorption (Barrer, 1954; Bradley, 1986a, 1936b; Brunauer and others, 1938; and Harkins and Jura, 1944) have been applied for indirectly estimating the surface areas of soil colloids and clays. (See Orchiston, 1953; and Quirk, 1955, for example.) However, the validity of such applications is questionable. The two basic models used for developing theoretical relations assume (1) that water is adsorbed as a monomolecular layer only and that capillary effects predominate if the liquid content exceeds the amount required to form such a layer and (2) that water is adsorbed in multimolecular layers. Neither may be completely correct if Quirks’ (1955) belief that water molecules tend to cluster around cation sites is valid. The very foundation of adsorption theory, the molecular distribution on the solid surface, appears to be undecided. In the writer’s opinion, at present the adsorption characteristics of porous media can be obtained effectively only by direct measurement. WATER HELD BY CAPILLARITY The pendular and funicular stages are defined by a resolution of the forces on the liquid-gas interface and the solid surface. In a macroscopic sense these forces are defined by the interfacial surface tension of the fluid phases, the contact angle between the liquid-gas interface and the solid surface, and the shape and size of the opening bridged by the interface. Where the occurrence of water in the pore space can be defined by these factors, the water is said to be held by capillarity. The curvature of any element of the liquid-gas inter-MULTIPHASE FLUIDS IN POROUS MEDIA E9 face is given (Smith, 1933b; and Smith and others, 1931) as where pc is the pressure difference across the liquid-gas interface, r is the interfacial tension, and rcl and rc2 are the principal radii of curvature of the interface. As an example, the radii reference axes are indicated on the center of a pendular ring in figure 2. Equation 1 is in essence a differential statement in that it must be satisfied at every point on the interface. For static conditions pc is very nearly constant over one pore width. Thus the term (l/rci+l/ra2) is nearly a constant over the space of one pore width. In irregularly shaped pores, the relative values of rcl and rc2 are different at almost every point on the interface. To define the shape of the interface geometrically the relation between r0l and rc2 must be obtained by integration over the whole interface, satisfying the condition that minimum energy was consumed in forming the surface. The complexity of the calculations required necessitates the adoption of an idealized model of the pore space—such as the circular capillary tube. A capillary tube open to the atmosphere at one end with its other end immersed below a free liquid surface is shown in figure 3. The liquid rises a distance H above the free surface in the tube, and the liquid-gas interface assumes an angle a with the solid surface. The contact angle is approximately a constant characteristic of a Figure 3.—Liquid rise in a capillary tube. given liquid-gas-solid combination. Equation 1 applied across the tube opening yields Vc——Pt(jH 2 t cos a rct (2) where pf is the liquid density, g is the acceleration due to gravity, and rct is the radius of the capillary tube. From equation 2, evidently the liquid pressure just beneath the meniscus is less than atmospheric if a is less than 90°. Assuming that the capillary tube serves as an adequate model of the pore space, equation 2 shows that the change of pressure across the interface is directly proportional to the interfacial surface tension and cos a and is inversely proportional to the pore width. At the liquid-gas interface, water molecules continually escape from the liquid phase and enter the vapor phase. If there is a net vapor movement to or from the interface, static equilibrium in the liquid phase is not complete because this loss or gain will be compensated by movement of the liquid phase. As is evident from the assumptions leading to equation 2, only the hydrostatics of the liquid phase have been considered. Thus equation 2 alone does not fully describe static liquid occurrence, except as an approximation. Edlefsen and Anderson (1943) have described the details of vapor-phase statics, showing that the vapor phase must satisfy the following: —H=RT log PvlPo (3) where R is the gas constant per gram of the vapor phase (=4.51 X103 for water), T is the temperature in degrees Kelvin, pv is the partial pressure of the vapor over the interface, and p0 is the partial pressure of the vapor phase over the free surface. Equation 3 is the definition of static conditions in the vapor phase in the capillary tube. If completely static conditions exist in the liquid phase of figure 3, both equations 2 and 3 must be satisfied at the interface.E10 FLUID MOVEMENT IN EARTH MATERIALS The terms pc and H of equations 1, 2, and 3 or then-equivalents are variously referred to in the literature on porous media as soil moisture tension, capillary potential, and other factors. In this report pressure of a particular phase is denoted by p and the subscript f will refer to the liquid phase. Thus the pressure in the liquid phase is written pt. In terms of head, p{=P-\-pc=P-ptgH=P+Ptght (4) where P is the absolute pressure at the free liquid surface and h is the pressure head of the liquid phase which is the height to which liquid will rise above the point tapped by a manometer. In this report the pressure head Aj- will be referred to simply as liquid head. By definition of static conditions H=—h{. Thus equations 2 and 3 may be written and , 2r cos a fi{—---------- PiffTc, h;=^RT log pv!p0. (5) (6) Failure by some investigators to recognize that both equations 5 and 6 apply to the static occurrence of liquids in capillaries has led to several erroneous conclusions being drawn from the use of the capillary tube as a model, when associated with laboratory observations of fluid distribution in porous media. If porous media could be modeled as a group of cap-ilary tubes, each with a different known radius, and all ending in a body of free water, the amount of space filled with liquid at a given altitude above the free surface could be computed with the aid of equation 5. How- volume OF LIQUID IN PENDULAR RING VOLUME OF GRAIN Figure 4.—Liquid volume versus height above free surface in pendular ring. (After Smith, 1933b.) ever, the spaces in porous media occupied by fluid are greatly different in physical shape than the continuous round capillary tube, and such computations would be relatively meaningless because the model is geometrically inadequate. Smith (1933b) successfully avoided the capillary tube hypothesis in a classical contribution to the study of liquid distribution. His model consisted of spherical particles of uniform size, which were chemically inert, had smooth surfaces, and were homogeneously packed. The volume of water in the capillary, funicular, and pendular stages contained by the model were calculated by means of equations he developed from equation 1 and geometrical considerations. Figure 4 shows the relation among grain radius rg, height above the free surface H, and the volume contained in a pendular ring around the contact between two grains, V. The curve for liquid contained in the pendular stage is from Smith’s figure 3 (1933b, p. 429) calculated from his equation 10: V/2rrr*=/32[ 1 - {2/3+/32} * sin ~1 {1/(1 +/})} ] (7) where &=Tlp{gHrt assuming r/pf grg— 1. For water, t is about 72 dynes-cm-1, pf=l g-cm-3, and 3=980.7 cm-sec-2. Therefore, assuming the pendular ring is formed of wate r, r„ is about 7.3 X10'2 cm. Thus figure 4 shows the general relation between the pendular ring volume and height above the free water surface for a spherical grain diameter of about 1.5 mm. The curve emphasizes that for a given grain radius, as H increases, the volume of liquid contained in each pendular ring undergoes progressively less change per unit change in H. Because the total volume of liquid retained in the medium is simply the volume of the pendular ring multiplied by the number of grain contacts, it is recognized that the shape of the curve in figure 4 schematically represents also the general form of the curve of total liquid content versus height above the free surface for the pendular stage. Smith also derived equations defining the amount of liquid retained in a medium of spherical grains not in intimate contact. This model was developed for computing the expected equilibrium liquid content for the sand columns drained by King (1898, p. 85-95). The calculated and observed values were compared by Smith (1933b, fig. 7, p. 438) as shown here in figure 5. The calculated curves show saturation to point B, which corresponds with the top of the capillary fringe calculated from equations for the maximum height of the capillary stage (Smith and others, 1931). Other information on the extent of the various stages of liquidMULTIPHASE FLUIDS IN POROUS MEDIA Ell LIQUID CONTENT. PERCENT OF POROSITY 0f X100 A LIQUID CONTENT, PERCENT OF POROSITY 0.X 100 B LIQUID CONTENT, PERCENT OF POROSITY LIQUID CONTENT, PERCENT OF POROSITY LIQUID CONTENT, PERCENT OF POROSITY 0.X 100 0.X 100 ^.XIOO C D E Figure 5.—Moisture content versus height above free surface in sands observed by King (indicated by dashed line) and calculated by Smith (indicated by solid line). Ot is the fraction oi pore volume filled with liquid, A-B is height of maximum capillary rise, BCD is the funicular stage, DE is the pendular stage, and m marks the upper limit of microscopic hysteresis. (After Smith, 1933b, fig. 7, p. 438.) occurrence, as calculated by Smith, are indicated in figure 5. Using virtually the same model as used by Smith, Gardner and Gardner (1953) also derived expressions for the pendular ring volume. Pendular ring volume was described as a function of the ratio of grain radius to pendular ring radius. A second expression derived from equation 1 was used to show the relation among the fluid properties, H, and radii ratios. Smith’s result incorporates all these interrelations into one expression much more convenient to apply. OBSERVATION OF THE CURVE LIQUID CONTENT VERSUS LIQUID PRESSURE HEAD The general case of fluid occurrence in a liquid-solid complex is evidently not amenable to practical mathematical description. Yet, the relations among liquid content and liquid head, or liquid content and vapor pressure of the gas phase, for the porous medium must be defined if the occurrence of fluid in the unsaturated zone is to be predicted for various environmental conditions. For water held in the capillary, funicular, and pendular stages, equations 5 and 6 indicate the factors controlling liquid content. The term (2 cos a)/ptgre, of equation 5 was derived from the assumption that water occurs in smooth round capillary tubes. Assuming any other shape, surface roughness, irregularity in cross section, or other factors, would produce a different function, but it would still be comprised of the same variables. Thus, the liquid pressure bead, h{, is expected to be related in some fashion to the volume of liquid retained. Also, for static equilibrium the ratio pvlp0 provides a measure of Aj, according to equation 6. Evidently, the static liquid content canE12 FLUID MOVEMENT IN EARTH MATERIALS be controlled as a function of Af by direct manometric connections with the pore liquid or by regulation of the humidity of the vapor phase. The significance of the humidity of the vapor phase is illustrated in table 2. Numerical values of Af are given as a function of pjp0 (relative humidity, expressed as a decimal fraction), calculated from equation 6 for pure water at 298°K (25°C). Note that the liquid pressure head undergoes large changes for very small changes in humidity where the humidity is nearly one. Accurate control and measurement of humidity are difficult to attain over this region of high sensitivity. Thus control of the vapor phase as a means for regulating liquid content is usually restricted to pjpo values less than about 0.99. For Af values less than one atmosphere in magnitude direct manometric connection with the liquid phase is more convenient for regulating liquid content. Measurement and control limitations attending the simpler laboratory procedures for regulating liquid content may be circumvented partly by using analogous fluids, exerting external control on the total pressure of the vapor phase, and placing the porous medium in an accelerated field. These techniques are discussed briefly in the following sections of this report. MANOMETRIC CONTROL An example of manometric control has already been given in figure 5. King’s data shown thereon were obtained by maintaining a free surface near the bottom end of upright sand columns about 8 feet long, and by measuring the liquid contained as a function of height above the free surface after the columns had remained under this control for about 2% years. Table 2.—Pressure head in pore liquid versus relative humidity of gas phase Relative humidity (P./Po) Pressure head (h{ in cm of H2O Relative humidity (P./Po) Pressure head (h{ in cm of HjO) 0. 9999 — 1. 34X102 0. 80 -3.00x10s . 999 — 1. 34X103 . 50 -9. 3X 10s . 99 — 1. 34X104 . 10 -3. 1X106 . 90 — 1. 41X105 .01 -6. 2X106 It might be presumed that the moisture content had reached equilibrium at the end of this period, but according to Smith (1933b), equilibrium had not been attained in the finer sands studied. Vapor losses to a relatively dry laboratory atmosphere probably caused the short segment at the top of the columns to exhibit a markedly lower moisture content than found over the upper half of the columns. The effect is noticeable as a sharp change in curvature of the relation between height and moisture content near the upper ends of the samples. Samples in long columns require long periods of time to attain equilibrium with the free surface, and therefore do not appear to be practical for studying the relation between liquid head and liquid content under static conditions. Richards (1928) proposed a “tension plate” for making such measurements in less time. The tension plate is a fine-grained porous body such as a thick asbestos or ceramic plate. A manometer is connected to the plate through a flexible tube. The liquid pressure head in the plate equals the distance between the plate surface and liquid level in the manometer tube, and can be adjusted and regulated easily by raising or lowering the manometer tube as required. The Af in a thin sample placed on the tension plate will be very nearly that of the liquid in the plate for equilibrium conditions. Because there is a highly permeable connection between a relatively small sample and the manometric reference, equilibrium liquid content in the sample is reached in a comparatively short time. Richards measured equilibrium moisture contents with the tension plate apparatus for several manometer positions corresponding with prescribed values of Af. Some of his reported measurements are shown in figure 6. Instead of controlling the liquid pressure only by direct manometric connection with the liquid phase, Richards and Fireman (1943) applied pressure to the vapor phase while the liquid was restrained against complete removal to the atmosphere by a fine porous disc. This technique is the equivalent of Richards’ (1928) tension plate except for the method of controlling the value of pe applied to the sample. In the study reported, total pressure of the vapor phase was limited by mechanical design to 2 atmospheres. There appears to be no difficulty in adopting other mechanical components for extending observations to higher Af or pc values. However, observations made are applicable only for defining water content as held by capillarity. USE OF FLUIDS OTHER THAN WATER Some of the difficulties involved in controlling liquid heads that are nearly equal to or greater in magnitude than one atmosphere can be overcome by injecting a nonwetting liquid in the pores. According to equation 2, if the interface curvature of the liquid and vapor is convex toward the vapor phase, the liquid pressure will be higher than the total pressure of the vapor phase. This is because the right side of equation 2 is negative for convex curvature. By inspection, it can be seen that if a liquid is introduced in the pore space having a contact angle of (180—a) °, the pressure difference will be of opposite sign. Water occurrence is of prime in-MULTIPHASE FLUIDS IN POROUS MEDIA E13 MOISTURE, IN PERCENT OF DRY WEIGHT 0.05 0.3 1 5 20 125 500 RADIUS, IN MICRONS SIZE-DISTRIBUTION CURVE Figobe 6.—Liquid pressure head versus liquid content, observed on tension plate apparatus. (After Richards, 1928.) terest, and the angle of contact between liquid water and most clean rock surfaces is nearly 0°. Thus, mercury with a contact angle of 180° would be a suitable liquid for forming a liquid-gas interface system exactly duplicating the water-water vapor interface. The volume enclosed by mercury in the pore space will be the same as the volume occupied by gas in the solid-water-vapor system if in each case the interface curvatures are made identical by control of the pressure drop across the interfaces. The relation between applied pressures for mercury injection and equivalent water sorption at a given liquid content may be derived from equation 2. For the two systems, mercury and water, at complimentary liquid contents the term 2 cos a/rct is equal in magnitude but of opposite sign, as indicated above. Therefore (pc/r) mercury = — (pe/r) water. (8) The amount of water sorbed at any given value of pc is simply equal to the total pore volume less the volume of mercury injected. The obvious advantage of mercury injection is that much lower effective water heads can be simulated than can be attained by direct manometric 695-719—63--3 control using water. For example, a liquid head of —2,000 cm of water may be simulated by mercury injection at a head of about +985 cm of mercury, according to equation 8. To avoid entrapment of air or other gases in pendular rings under mercury injection, samples are initially evacuated before injection. It should be noted that if the porous medium is of such a nature that water retention is caused largely by adsorption processes, equation 2 does not apply. Because the mercury injection method is founded on the presumption that water is held by capillarity it applies only for the description of liquid content in the pendular, funicular, and capillary stages. Details of the mercury injection techniques are given in the literature by Burdine and others, 1950; Purcell, 1949; and Ritter and Drake, 1945. CENTRIFUGE ACCELERATION The manometric control methods outlined above depend upon the normal earth gravity field for creating pressure differences across the liquid-vapor interface. The centrifuge has been used to exert a higher accelerative force on the liquids. Hassler and Brunner (1945)E14 FLUID MOVEMENT IN EARTH MATERIALS showed that if a sample is subjected to a constant acceleration, g', h[=Pc/pt9— —H'g'/g (9) where IT is the distance, measured along the direction g’, between (1) a point in the accelerated sample where the liquid head is zero and (2) the point where pc is observed in the accelerated model; and g is the acceleration due to the earth gravity field. Marx (1956) also derived this relation from a dimensional study of the boundary conditions in both prototype and model columns. He indicated that the ratio g'/g for a centrifuged sample is given by the relation g'lg=l. 118X10-W+2r (10) where N is the velocity of the sample, in rpm, and r is the mean radius to the sample measured from the center of rotation, in centimeters. Leverett and others (1942) also obtained equation 9 from a more general dimensional analysis of model requirements for multiphase fluid flow. Details are given on pages E39^3 of this report. Application of equation 9 to centrifuge observations presumes equilibrium liquid distribution, and that both the liquid and porous medium centrifuged have the same properties as the prototype. Dimensional analysis (Leverett and others, 1942; and Marx, 1956) serves to emphasize the origin of equation 9. In essence, it has been shown that if a centrifuged model of a column of porous material contains the same fluid and porous substance as the prototype, a characteristic model length, //, is related to its analogous prototype length, Z, by high, about 33 feet, acted upon by the earth’s gravity field. Beginning with a saturated sample and draining by centrifuging is directly analogous to draining the initially saturated prototype column by lowering the free surface to its base, provided the free surface reference is actually maintained in the centrifuge system. As shown in another section of this report, dimensional analysis also provides the relation between time in the prototype and the analogous time scale of the accelerated sample. (See E42). These are related by t=(9'/g)r. (12) For the moisture equivalent tests g'lg= 1,000, andt'=40 minutes. From equation 12, i=lX106X40=4X107 1 cm long at l,000g for 40 minutes is equivalent to gravity drainage of a vertical column 33 feet long for 76 years. If the free water surface below the centrifuged sample is not permitted to contact the sample between the time the centrifuge is slowed and the time the sample is removed for weighing, the moisture retained should equal the average moisture retention in the prototype column if it had been observed at the end of a 76-year period of gravity drainage. Lebedeff (1927) made a series of observations of moisture retention in fine soils as a function of centrifuging time and acceleration. Estimating from the data given in his paper, the soil sample used was about 0.14 cm long. It was placed in a specially designed box and the bottom of the sample rested on filter paper, which in turn was supported by a fine sieve. Selected observations from his table 1 (Lebedeff, 1927, p. 554) are shown in figure 7. Centrifuging time for these data was 5 L/L'=g'/g. (11) In equation 9, h( can be taken as a characteristic length in the prototype, for static conditions as expressed by equation 4, and H' is the corresponding length in the model. Thus equation 9 can be obtained directly as a form of equation 11, considering that the negative sign in equation 9 arises by definition. Equation 11 indicates the short samples normally employed in centrifuging are equivalent to long columns in the gravity field. Briggs and McLane (1910) proposed g’/g = 1,000 for measuring the “moisture equivalent” of soils. An initially saturated soil sample, 1 cm in height, placed in a perforated cup was subjected to this acceleration for 40 minutes. The amount of water retained by the soil against the known centrifugal force, as a percentage of dry weight, was designated as the moisture equivalent. Provided the modeling requirements are all satisfied according to equation 11 the 1 cm sample is equivalent to a prototype soil column 1,000 cm Figure 7—Moisture retention versus acceleration after 5 minutes of centrifuging. (Data from Lebedeff, 1927, table 1, p. 554.).MULTIPHASE FLUIDS IN POROUS MEDIA E15 minutes. The soil column lengths and drainage times for equivalent gravity drainage are also shown at a few selected ratios of g'/g. From one viewpoint, it might be concluded from figure 7 that a column 1.4 X103 cm long, about 46 feet, would still be in the nonsteady state after drainage by gravity for nearly a thousand years. Circumstances can, however, be visualized that may account for the shape of the curves in figure 7, even for steady-state conditions. A hypothetical moisture distribution at equilibrium in a soil column is shown as a function of H in figure 8A, as might be observed in a comparatively long column. The moisture equivalent tion. His description of the centrifuge equipment indicates that no free surface was established at the base of the samples. Furthermore, he stated: “A strong air current develops from the center of the bowl to the periphery * * *” part of which probably flowed through the sample. Thus the reported equilibrium moisture contents were probably as dependent on the vapor pressure of the room atmosphere and drying air currents as on centrifuge acceleration. Quantitative comparison of evaporation losses and liquid flow from the sample presented by Lebedeff seems extremely difficult to justify. "t r A B Figure 8.—Liquid content versus height above the free-liquid surface. observed by centrifuging is simply the average of the moisture contents exhibited from top to bottom in figure 8A. If liquid retention is observed in a shorter column, as in figure 8B, the average soil moisture content observed will be materially higher than in the longer column. The differences in average moisture content effected as a result of employing different column lengths, established by the different centrifuge accelerations, are obviously closely related to the ratio between column length and height of the funicular and capillary stages above the free surface. If the top of the funicular stage were about 15-30 feet above the free surface in the prototype of the materials studied by Lebedeff, the pronounced reduction in observed average moisture content as a function of increased prototype column length shown in figure 7 could be ascribed to this effect. It is not unreasonable to expect that the funicular stage extends this far above the free surface in the fine-grained materials investigated by Lebedeff. The foregoing discussion of figures 7 and 8 aids in forming an appreciation for the relation between the moisture content in centrifuged samples and moisture content in columns drained in a 1 g field. This appreciation, however, may be highly conjectural if one must assume that Lebedeff’s centrifuged samples were under the control of a free surface at a fixed and known posi- CENTER OF ROTATION 7 7 f / » iiil Free surface ===. eSF— Water Water — — — A B Figure 9.—Diagram showing sample arrangement in centrifuge. Interpretation of centrifuge data by means of equations 9 to 12 requires that the sample be contained as shown schematically in figure 9A. The free surface reference, where h' f=0, is formed by the top of the free water body contained below the bottom of the sample. The centrifuge model of a prototype column of porous rock is valid, according to dimensional analysis, only if this free surface is maintained. By equation 10 it is obvious that the g'/g ratio is not exactly uniform over the length of the sample. However, the radius f is usually large, about 16 cm, compared with a sample length of about 1 cm. Thus, for many applications the g'/g ratio may be considered virtually constant over the sample length. The centrifuge might be used also for controlling pressure of the liquid in a finite length of sample for the purpose of constructing the characteristic curve of liquid pressure head versus liquid content. To avoid the problems associated with modeling the entire prototype sediment profile in the centrifuge, as discussed in connection with figure 8, a thin sample may be mounted a fixed distance, 7, from the center of rotation and the free water surface located at a radius of rs, near the bottom of the centrifuge cup as shown in figure 9B. Applying equation 11 to differential lengths we obtain —dH=g'/gdr (13)E16 FLUID MOVEMENT IN EARTH MATERIALS wherein the reference r—0 is taken at the center of rotation and H=Q is taken at the water table in the prototype column. Substituting the required form of equation 10 in equation 13 dH= — 1.118X 10~5N2rdr. (14) Integration between ~ and rj" yields -h{=H=5.59X10-*N2(rf2-ri) (15) where ry is the radius to the free water surface, in centimeters, and is the mean liquid pressure head in the sample. Length of the segment of the prototype column represented by the sample may be computed from the following form of equation 15: A/7 5.59X10-8.V2 (ri*-r»2) (16) where rb and rt are the radii from the center of rotation to the bottom and top of the sample respectively. The ratio g'/g for the sample is given by g'/g=1.118X10-5 N2r. (17) The arrangement indicated in figure 9B would be decidedly superior to that of figure 9A for the determination of liquid content because the funicular and capillary stages could be eliminated by the space rb—rt. The equations developed for relating liquid pressure head to centrifuge conditions are founded, however, on the postulate that there is liquid continuity between the free surface and the sample. This has been stressed by Hassler and Brunner (1945). Further investigation of the vapor phase and its effect on the liquid phase under the influence of an angular acceleration should be undertaken to clarify the validity of applying dimensional analysis to centrifuging and the meaning of data obtained. In particular, much of the soil testing by centrifuge is done with centrifuge cups specifically designed to avoid retention of the liquid phase beneath the sample. (See for example, Am. Soc. for Testing Materials, 1958.) Thus the free surface reference position required for the construction of a dimensionally correct model does not exist. Moisture retention observed under such test conditions may not reflect prototype conditions accurately because they are founded on an incomplete model control, being influenced strongly by the vapor pressure in the atmosphere of the centrifuge chamber. VAPOR PRESSURE CONTROL As previously indicated, the liquid pressure head beneath a curved liquid-gas interface may be controlled for static conditions indirectly by regulating the vapor pressure over the meniscus. This is apparent from equation 6. If a sample of a dry porous medium is left in a water-vapor environment, vapor will condense in the sample until the static head prescribed by equation 6 is reached, provided no outflow of liquid from the sample is permitted. Control of pv might be attained by placing the sample in a closed vessel a given distance H above a free pure water surface, as shown schematically in figure 10, but for practical reasons such observations are restricted to H values of less than about 10 feet. For larger H or Aj values it is more practicable to regulate the vapor pressure of the liquid free surface beneath the sample. This is done readily by introducing a solute in the control liquid which reduces the vapor pressure above the free surface to some value below that for pure water. In effect the reduction of vapor pressure caused by the solute can be used to simulate the H that might be obtained by physical movement of the sample to a given height above the free surface of pure water. The general head relations are shown schematically in figure 10. If a free surface of pure water is located at altitude A, the equilibrium liquid pressure head in the sample at altitude G is given by equation 6. If, for example, a salt is added to the pure water to form a solution, the vapor pressure directly over the free solution surface will be less than that over the surface of free pure water. The resulting decrease in vapor pressure is osmotic in nature. In terms of pure water the change in pressure head in the sample effected by the solute may be stated as h0sm—RT log ps/p0 (18) where ps is the vapor pressure over the free solution surface, and as previously defined p„ is the vapor pressure over the free pure water surface. Equation 18 is simply the equation for osmotic pressure put in terms of water head. It can be seen from figure 10 that for simulating any desired value of H in the sample, the distance between the free solution surface and the sample, As, can be regulated by controlling the solution characteristics. As can be made zero or nearly SAMPLE Free solution Surface H= Free pure Liquid surface Figure 10.—Controlling liquid pressure head by regulating vapor pressure with solutes.MULTIPHASE FLUIDS IN POROUS MEDIA E17 zero by selecting the solute and solute concentration such that hom=—H=h{ for any desired value of Af. This permits control of liquid pressure head over a wide range in comparatively small vessels. For example, data of Stokes and Robinson (1949) indicate that an Af of about —6X104 cm of water, about 60 atmospheres, can be simulated by a solution of water and only 5.54 percent NaOH, or 9.33 percent CaCl2, or 11.02 percent II2S04, the percentage referring to weight of anhydrous solute in the solution. A CaCl2 concentration of 14.95 percent of the solution weight will produce an Af of about — 12X105 cm of water, for example. These numerical examples are based on the postulate that the porous sample is located at the same altitude as the free solution surface; that is, As of figure 10 equals zero, and that the sample is isolated from environmental influences other than those of the solution vapor. Orchiston (1953) used this technique to measure the water vapor adsorption on seven New Zealand soils. The inadequacy of oven drying for determining dry weight, as previously discussed on page E6, was noted by Orchiston in preliminary observations. Orchiston therefore elected to dry his samples over a free surface of pure concentrated H2S04. The dried samples were then placed in a dessicator with a dilute solution of H2S04 and water and the entire assembly, evacuated, was placed in a constant-temperature water bath. The amount of water adsorbed by the samples at equilibrium was observed for several values of ps which were controlled by increasing the acid dilution after each observation. The two highest vapor-pressure ratios (ps/p0) reported were 1.000, at which the pore space should be completely filled, and 0.842. All other controlled values of Ps/po reported were less than 0.842. According to equation 6, the liquid pressure head for Pv/Po—0.842, T—298° K, and A?=4.5 X103, is about —2.3X105 cm H20. From the capillary tube model it is estimated that the radius of the largest pores filled at this head value may be of the order of 10-6 cm. Because the water molecule radius is about 1.5 X 10~8 cm (Dorsey, 1940, p. 43), and because the molecular forces between the liquid and solid are believed dominant if the meniscus is less than about 100 water molecule diameters in width, most of the moisture retention observed by Orchiston is likely in the adsorbed stage (fig. 1). Thus, equation 6 may not apply strictly as a measure of liquid pressure head, because it is founded on assumptions which do not account for adsorptive forces. Equation 6 was used, nevertheless, to calculate the h{ values corresponding with the pjp0 reference used by Orchiston so as to emphasize the magnitude of the effective liquid pressure head attained by control of vapor pressure. The results are given in figure 11. The water contained by the samples observed was largely under the influence of adsorption characteristics. Furthermore, assuming that the soils were comprised of spherical particles, the specific surface area of 26.9 m2/g for sample F requires that the individual particle radius be about 5 x 10-8 cm. Thus the solid particle size may have been nearly the same as the size of water molecules. With such proportional relations between particle size and size of the water molecule, definition of the physical occurrence of the pendular, funicular, and capillary stages in the pore spaces does not apply because the basic assumptions leading to equations 5 and 6 are not met. EFFECTS OF TEMPERATURE ON HEAD AND LIQUID CONTENT Equations 5 and 6 both contain temperature-dependent variables; consequently h( is temperature dependent. As implied in connection with equation 5 the liquid content is dependent on Af and now it will further be presumed to be a function of temperature. In order to proceed toward an evaluation of thermal effects on liquid pressure head in unsaturated porous media, we must first obtain a more general expression describing liquid head than is afforded by the capillary tube model and equation 5. From equation 1 and its dimensional characteristics it may be inferred that for the general case of a capillary interface (19) where / (1 /rc) is an unspecified function of a characteristic radius, rc, on the liquid-gas interface which is dependent on the amount of liquid held in the pore space. rc might be identified, for example, as the smallest interface curvature developed at a given liquid content and existing in a given finite volume of the pore space. It will be assumed here that the liquid content is a singled value function of (1 /rc). The latter assumption is generally accepted for changes in liquid content made consistently from either initially dry or initially moist conditions. With this assumption equation 19 may be rewritten as ht=wm) (20) in which 6$ is the decimal fraction of the pore volume filled with liquid, and /(0f) is an unspecified and independent function of the liquid content. Assuming the content of pore liquid is held constant as the temperature is changed and the liquid density is constant, the differential of equation 20 becomes bAj-_ 1 . dr_Af dr dT-7 M1' (21)E18 FLUID MOVEMENT IN EARTH MATERIALS Figure 11.—Water adsorption and liquid pressure-head control by vapor regulation in fine-grained soils. A, Podzol, surface area, .<4=140.6 m2 per g; D, granular clay A=105.2 m2 per g; F, Recent alluvium, A=26.9 m2 per g. (Data from Orchiston, 1953, table 1, p. 456.) For the conditions prescribed, equation 21 may be used for estimating the effect on liquid pressure head caused by small changes in temperature. From Dorsey (1940, table 225, p. 514), the interfacial tension for the water-water vapor system can be expressed by r/mf ~bT~~~bT (26) in which m, is the mass of water vapor contained in each unit pore volume and mf is the mass of liquid water contained in each unit pore volume. It is also recognized that and m,=pv( 1—0f) (27) mt=P[dt. (28) Substituting the differentials of equations 27 and 28 in equation 26, and assuming that density of the liquid phase may be considered independent of temperature, we find d0f (1—0{) bp, bT (p,-pt)bT (29) p„— is nearly equal to minus one, and, because the sample was relatively dry, (1 — 0f) is of the order of one. Therefore, d0r bp, bT~ bT (30) At a temperature of 25°C, ^=1.3X10 6 g-cm~3-°C_1. From Gardner’s observations the mean was about 10 cm-°C_1, and Af= —400 cm. These data, asE22 FLUID MOVEMENT IN EARTH MATERIALS finite-difference approximations, and equation 30 may be put in equation 25, which yields: dht in d/q dT dffr dAr 9.2 d0f~1.3XlO-6= (1.3X10-6) =7X106. 400 f 768-298 Numerically the latter result is probably more nearly correct than the figure obtained from analysis of the warming curve. It is believed that the physical basis for the two analyses are sufficiently diverse, and the numerical results are sufficiently close, that one may conclude vapor transfer had a pronounced effect on the observations recorded by Gardner. Certainly it does not appear possible to justify the observations shown in figure 14 as representing changes due predominantly to the functional relation between temperature and surface tension. A more adequate study of Gardner’s data would require a more precise estimate of the amount of vapor driven from the sample upon warming and consideration of the vapor driven into the pore space upon cooling. The latter would require knowledge of the room humidity and temperature, and the df value at some time during the observations. However, the questions arising from the uncertainty as to the state of the thermal field and its effect on the observations reported preclude the desirability of a more detailed study. It may be remarked that a more precise development of the vapor-liquid interrelations outlined above might form the basis for accurately measuring the slope of the liquid pressure head versus liquid-content curve at low moisture contents. Gravimetric methods have comparatively low accuracy where the liquid content in the sample is small. In observing a liquid pressure head versus liquid-content curve at small moisture contents in the laboratory, changes of liquid content could be effected by temperature control of the sample. Analysis of such data by means of equations like 26 to 30 may provide a more accurate observation of changes in liquid content at many points in the experiment than can be made by the more direct gravimetric methods. EFFECTS OF SALTS IN THE LIQUID PHASE ON LIQUID PRESSURE HEAD AND LIQUID CONTENT To this point, the static liquid pressure head versus liquid-content characteristics have been discussed only for pure liquids. On pages E16-17, the reduction in vapor pressure caused by dissolved salts was mentioned briefly as a means for conveniently obtaining large h{ values in laboratory observations of liquid content versus liquid pressure head. Using the information presented there one might suspect that salt concentra- tion, or differences in salt content, from place to place in the porous medium may have a pronounced effect on head or distribution of liquid. First consider the static occurrence of liquid described by equation 19, (19) Pt9 from which it is recognized that ht is proportional to the liquid-gas interfacial tension. According to Moore (1955, p. 503) aqueous solutions with ionic salts generally have a higher surface tension than pure water. The increase in surface tension caused by the ionic salts is comparatively small. For example, the surface tension of sea water with a density of 1.025 g per cm8 is only about 1 percent greater than the surface tension of pure water (Dorsey, 1940, p. 514). Therefore considering the liquid phase only in two separate columns both comprised of the same type of porous medium, the relations between static liquid head and liquid content should be found very much alike regardless of the salt concentration, provided the salt concentration were uniform in each column. However, salts may not be uniformly distributed in naturally occurring earth materials because of differences in structure, land use practices, differences in rates of leaching, and other factors. Near the water table the salt concentration is usually much smaller than at points higher in the zone of aeration. Nonhomogeneous salt concentration in the zone of aeration may play a significant role in the moisture distribution above the water table. Some of the pertinent features of the relation between a salty-liquid system and a gas-solid system not apparent from the capillary tube hypothesis can be learned from a study based on equation 6, applied to the inverted capillary-sized U tube of figure 15. The internal radii of the two "arms of the U’are identical and uniform. The lower end of tube B is inserted beneath the free surface of pure water in the container on the right. Tube A is inserted beneath the surface of an aqueous ionic salt solution in the container on the left. The free surface positions in both are held constant and Figure 15.—Vapor phase in vicinity of liquids.MULTIPHASE FLUIDS IN POROUS MEDIA E23 at the same level. From equation 19 and the knowledge that surface tension and density change very little with the salinity in A, we expect to find HA very nearly equal to Hb. It might be presumed, therefore, that this system is in a static condition. However, to satisfy static conditions the vapor phase must also be static. This may be checked by applying equation 3 to the vapor phase. Accordingly, Ha=—RT log pv/ps (31) and HB=—RT log pjp0 (32) Comparison of equations 31 and 32 shows that because p, is different from p0, the vapor phase cannot be under static conditions. Rather, there will be a continuous vapor flow through the U section from B to A causing dilution and an increase in temperature in A, and a decrease in temperature in B. If the salt content in the container on the left were held constant the system could never become static. Thus it is evident that salt distribution in the unsaturated zone will have comparatively little direct effect on liquid heads but will have a pronounced effect on the movement of the vapor phase. As a consequence of the occurrence of dissolved salts in natural waters in porous media, measurements of liquid pressure head in unsaturated rocks, commonly made with tensiometers in field studies, are of doubtful accuracy. Certainly the conditions postulated for figure 15 may arise around tensiometer cups as installed in the field. Figure 16 schematically shows a tensiometer cup Figure 16.—Tensiometer in contact with moist soil. associated with a segment of unsaturated soil. It is assumed that the tensiometer cup and the attached manometer line were filled with pure water, and the liquid solution in the porous medium at A contains ionic salts. The manometer connected to the tensiometer cup can only reflect the correct pressure head in the soil liquid if the vapor phase is static. If the vapor phase is not static, vapor movement to or from the cup assembly will continuously change the head values observed in the manometer. From equation 3, for static conditions in the vapor phase, the pressure drop across the menisci of the soil solution is given by ^=RT log pv/pt (33) Pi9 and across the menisci of the porous cup by ^=RT log pjp0. (34) It can be seen that the difference in head between the soil liquid and liquid in the tensiometer may be as much as PcaIPt9—PcbIpM- This difference is equal to the osmotic pressure of the salty solution (Edlefsen and Anderson, 1943, equations 222, 226, and 227). pa and p„ are significantly different even for comparatively dilute salt solutions. Ordinarily then the tensiometer observation, unless corrected for errors due to this difference, may indicate liquid pressure heads appreciably greater in magnitude than those which actually exist. If the soil-moisture content varies slowly, static conditions might be postulated because the amount of vapor available for transfer from the cup is restricted by the limited tensiometer storage. Such a restriction was not implied in connection with figure 15. Actually the errors in the tensiometer readings may be any value between zero and the osmotic pressure of the soil solution, depending on rate of change of 0f, the degree of liquid phase continuity between the soil and cup, and other factors. Breazeale and McGeorge (1955) showed the effects of salinity on liquid content in the wilting range by a laboratory study. Soil water was depleted to the wilting point by tomato plants. The soil salinity was controlled by first desalinizing the original field samples. Equal parts of sodium chloride and sodium sulphate were then added to produce various salinity levels in the liquid retained up to a total of 1,000 ppm salt. Presumably in the experiments the relative humidity of the soil air was controlled by the plant at wilting and was therefore relatively independent of the salt content. Thus, pv as observed in the various suites of samples was relatively independent of the salt content of the liquid phase. The free solution vapor pressure, ps, decreases as the salt content is increased. Therefore, according to equation 33, as the salt content increases, for a given pore vapor pressure, the value of pcA increases. Or, in effect, the liquid pressure head tends toward zero. In turn, this causes a higher percentage of moisture to be retained with increasing salinity, in accord with equation 19. The data shown in table 4, taken from Breazeale and McGeorge (1955), verify this. Evi-E24 FLUID MOVEMENT IN EARTH MATERIALS Table 4.—Soil-water salinity and moisture content at the wilting point for tomato plants (After Breazeale and McGeorge (1955)] Soils and solution Conductivity of saturation extract (mmho-cm-1) Tomato plant wilting percentage (by weight) Safford soil, Cajon deep silted phase: Desalinized _ _ 1. 9 15. 9 Desalinized plus 100 ppm salt 2. 8 18. 1 Desalinized plus 500 ppm salt 3. 3 18. 9 Original — - 5. 1 20. 0 Mesa soil, Laveen clay loam: Desalinized . 9 6. 7 Original _ - - 1. 3 7. 9 Desalinized plus 100 ppm salt 1. 6 7. 1 Desalinized plus 500 ppm salt . - - 2. 3 8. 5 Desalinized plus 1,000 ppm salt - 3. 0 8. 9 Vinton soil, very fine sand: Desalinized - _ _ 1. 0 3. 6 Original 2. 5 3. 9 Desalinized plus 100 ppm salt __ __ 2. 6 4. 0 Desalinized plus 500 ppm salt 3. 9 4. 8 Pima soil, very fine sandy loam: Desalinized - - - - - - 2. 1 9. 3 Desalinized plus 100 ppm salt -- 3. 4 10. 4 Desalinized plus 500 ppm salt 5. 2 12. 8 Original- 14. 5 13. 2 dently the functional relation between moisture content and salinity is dependent on the liquid pressure head versus liquid-content characteristic of the soil, the osmotic pressure of the soil-water solution, and its distribution in space. Fox (1957) suggested applying these relations to determine the amount of moisture available to plants in soils containing saline solutions. The significance of the relation of salinity to osmotic pressure is apparent through the work of Campbell and others (1948). They plotted electrical conductivity and total cations in milliequivalents per liter (meq per 1) against osmotic pressure and electrical conductivity versus milliequivalents per liter as observed in saturation extracts. They calculated osmotic pressures from observed freezing point depressions and measured electrical conductivities with special laboratory apparatus. Their results are shown in figure 17. Typical potable ground water generally falls within the range of 1-10 meq. per 1 total cations. The osmotic pressure associated with this range would be between 0.03 and 0.3 atmospheres or approximately 30 to 300 cm of water, as found by extrapolating milliequivalents per liter versus the osmotic pressure curve of figure 17. Thus the liquid pressure head versus liquid-content distribution is affected very strongly by comparatively small differences in salt distribution in the profile. For example, the sand characterized in figure 12A under static conditions in the vapor phase could remain saturated at the top of the TOTAL CATIONS, IN MILLIEQUIVALENTS PER LITER 1 3 6 10 30 60 100 300 ELECTRICAL CONDUCTIVITY, MILLIMHOS PER CENTIMETER Figure 17—Quality, electrical conductivity, and osmotic pressure relations for soil-water extracts. (After Campbell and others, 1948, fig. 1.) column if the cation content were 4 meq per 1 at the top and nearly zero at the bottom. It is, however, doubtful that static conditions could exist in a porous medium in nature where the salt concentration differs appreciably from place to place. If, in the case cited in the previous paragraph, the upper section of the sand column became nearly saturated by vapor motion from the free surface of pure liquid at its base, gravity would cause the salty solution in the liquid phase to flow downward. It is more likely that such flow would continue until the pure water became contaminated and the salt concentration above was diluted by flushing. Presumably this action would continue all along the profile until both the liquid and vapor systems reached static conditions. Evidently the salt concentration must be uniform over the profile if static conditions are to prevail in both phases simultaneously. MULTIPHASE FLOW GENERAL FLOW RELATIONS Flow through unsaturated porous media is a multiphase process that is far more complicated than flow through saturated media. The rate of flow is dependent on the configuration of the liquid and gas phases in the pore space and on energy gradients formed chiefly by spatial changes in the pressure, temperature, and chemical content of the fluids in the pores. Flow characteristics are described by the conductivity of the medium to the passage of fluid under a specified energy gradient in much the same way that the coefficient of permeability (Wenzel, 1942) has been used for defining the flow characteristics of saturated aquifer mate-MULTIPHASE FLUIDS IN POROUS MEDIA E25 rials. The term conductivity will be applied in this report to flow characteristics of unsaturated media so as to distinguish these from references to the nature of flow in saturated media. Considering the nature of multiphase flow of water, three basic types of conductivity may be distinguished, each defining the potential mobility of the liquid, gas, and vapor phases. Individually, each of these types might be subdivided further according to the type of energy gradient creating flow. Liquid may be driven through porous media by pressure, temperature, or salinity gradients for example. Thus the conductivity of a medium to liquid flow, K{, might be expressed in terms of the unit gradient for each form of energy. This holds also for the vapor and gas conductivities, Kv and Kg respectively. Evidently the conductivity values must be defined in terms of the particular energy drive involved, unless it can be shown that the various energy forms can be interrelated and their total effect defined by a tensor. As used in this report, the term “conductivity” is defined as the rate at which fluid moves through a unit cross-sectional area, taken normal to the direction of flow, under a unit gradient of the appropriate form of energy. Although the thermal and chemical effects on unsaturated flow have been studied at some length, practical computations for predicting the flow within prescribed boundaries are still limited mostly to those cases in which unsaturated flow can be treated as a pure liquid system, completely independent of heat flow. The general relations among conductivity to liquid flow, velocity of liquid movement, and pressure potential were discussed by Buckingham (1907) in connection with the movement of soil moisture. He pointed out that decreasing 0f reduces both the area of liquid-filled pore space available to flow and the number of continuous flow paths through the liquid phase. Thus the liquid conductivity decreases with decreasing Zero liquid conductivity was presumed only for zero liquid content. Richards (1931) reasoned that gas occupying part of the pore space influenced the liquid flow in basically the same manner as if the gas were a solid. Thus, liquid flow that results from head gradients applied across a short section of unsaturated media should follow Darcy’s law, at least approximately. This relation may be stated as v{=—Ks (~ + sm x) (35) where V{ is the bulk velocity of the liquid phase, l is length taken along the direction of flow, and X is the angle between the direction of flow and the horizontal. According to Scheidegger (1957, p. 154-156), at about the same time that Richards and others working with soil moisture migration forwarded this postulate, workers in the oil industry were also suggesting that Darcy’s law be applied to liquid flow through unsaturated porous media. Experimental observations dating from about 1930 in the chemical, oil, and soils disciplines gave considerable support to this hypothesis and by now it appears to have reached acceptance (Irmay, 1954; Pirson, 1958). As will be indicated later, Darcy’s law is an inadequate approximation where the flow of water and water vapor become interdependent. Even by intuitive reasoning one might certainly question the applicability of Darcy’s law for describing the complex flow of liquid and vapor. For some flow regimes over which d{ is comparatively high, omission of flow contributions in the vapor phase may be justified. It must be recognized, however, that even at large d-( values, the distribution of thermal and chemical potentials may cause more liquid motion than do the liquid-head gradients if the latter are relatively small. Therefore the degree of approximation in expressing flow and energy relations must be devised with an acute respect for all the conditions causing flow. Within the general class of naturally occurring porous media, approximate grain shape is restricted mostly to spheres or plates. For two assemblages of spherical grains, each packed identically but of different grain size, the pore shapes are congruent, and the two pore systems differ only by a factor of length. In study of static liquid occurrence, it has been noted that the liquid configuration in the pore space is closely related to the liquid content. Thus for a given value of in the pore space, we may perceive that the relative interconnections among liquid elements are nearly congruent regardless of the size of grain. At any given 0;-, then, the proportionate reduction in permeability due to decreasing the number and continuity of liquid elements should be somewhat similar for all grain sizes. Variability in this relation is contained within the limits of grain shape and packing arrangement differences as found especially in natural media. This concept is important to the study of unsaturated flow because it permits grouping of natural porous media under comparatively few forms of conductivity versus curves. At any given the conductivity may be expressed conveniently in dimensionless form as a relative conductivity. The relative conductivity is defined as the ratio of (1) the conductivity to fluid flow at df and (2) the conductivity to fluid flow at 6r=l. In this report, relative conductivity of the liquid phase is denoted by Krl and of the gas and vapor phases by Kre and Krv respectively. Schematic curves of relative conductivity, as may be obtained by laboratory measurements (Wyckoff and Botset, 1936, for example), are shown in figure 18E26 FLUID MOVEMENT IN EARTH MATERIALS Figure 18.—Schematic relative conductivities for wetting liquids and gas phase for the liquid and gas phases. The potential field driving either phase is presumed to be derived from control of the pressure gradient only, whereas the temperature and chemical gradients are presumed to be zero. Furthermore, in measuring liquid flow rates the gross movement of the gas phase was held to zero; and, conversely, in measuring the gas flow rates the gross movement of the liquid phase was held to zero. In the funicular stage the gas content is discontinuous through the pore space, and therefore the gas conductivity should be zero, by definition. The minimum liquid content in the funicular stage is thus given by the intercept Krg=0 near 0f=O.8 in figure 18, for example. Liquid continuity in some form is indicated by the Kr( curve down to a 0f of about 0.2, showing possible finite conductivity to liquid in the pendular stage. The curves of figure 18 apply for fluid flow under controlled conditions. That is, only one fluid type is permitted to move in each set of observations. If this control were not exerted, liquid movement may induce parallel movement of the gas phases or nonwetting fluids. The results of Leverett and Lewis (1941) show the area of concentration levels in which appreciable movement was noted in more than one fluid. The relative concentrations of oil, water, and gas found in a permeable sand, after fluid mixtures of constant composition flowed through the sample until equilibrium was attained, are given in figure 19A. The relative conductivity to oil for this system is shown in figure 19 A. A comparison of these data shows that flow by induction with an adjoining fluid would likely be com- paratively insignificant. Thus, each fluid or phase held in the pores may be considered as an individual flow, unaffected by adjacent fluids. This conclusion, however, may not hold for the water-water vapor and air system. For relatively volatile liquids like water, interaction between the liquid and vapor phases may preclude their separation and treatment as individual fluids in the pore space. Most studies of unsaturated flow have been made with the assumption that motion of each fluid or fluid phase is independent of the others, and comparatively little advance in the theory of flow of interacting fluids has been made. 100 percent gas A 100 percent gas Figure 19.—Multifluid flow data. (After Leverett and Lewis, 1941.) A, Composition of pore fluid and parallel fluid flow; B, Relative conductivity to oil, in percent, versus pore-fluid composition.MULTIPHASE FLUIDS IN POROUS MEDIA E27 MEASUREMENT OF LIQUID CONDUCTIVITY Conductivity measurements are made using virtually the same techniques employed for permeability measurements. In the latter, the head or pressure difference across a saturated and confined sample of the porous medium is controlled, and the resulting gross velocity of liquid, V{, is observed (Wenzel, 1942). The permeability is calculated from these observations. Conductivity measurements differ from permeability measurements in that more than one fluid must be subjected to control and measurement, and the degree of saturation by the principal flowing fluid must be determined because conductivity is dependent on the degree of saturation. Field observation of a complete relative conductivity versus 6? curve is not practical now because multiple control and measurement is required over a wide range of 0{ values. An adequate range of is not usually found at any one place over a conveniently short period of time. The inadequacy of field equipment also prohibits accurate field measurement of conductivity over an extended range of values. In most instances accurate conductivity measurement over a sufficient range of 0f values are attainable only by laboratory techniques. The basic features of equipment used for measuring liquid conductivity of core samples are shown schematically in figure 20. The sample (a) is held in a gas chamber (c) supported at each end by fine porous disks (b). The disk material is selected so that gas from (c) will not pass through either disk, and both will remain saturated if the drop in total pressure across either disk is greater than the maximum value of hi of interest. Chambers (e) are kept full of liquid during the conductivity tests and liquid continuity between (e) and the contained sample is maintained through disks (b). Either the pump shown or a gravity Differential pressure gage Figure 20— Schema of liquid conductivity measurement apparatus: (a) sample, (6) porous retaining disks, (c) gas chamber, (d) impermeable sleeve and support, and (e) liquid distribution chamber. drive may be used to produce a head difference across and hence flow through the sample. The rate of liquid flow at any given degree of saturation is monitored at the flowmeter. This information may be converted to gross liquid velocity if the cross-sectional area of the core is known. The conductivity may be calculated from equation 35, the observed velocity, and the observed head gradient. Any one of several methods might be adopted for finding the value of 0f for a particular test run. The difference between the mean liquid pressure head applied at the ends of the sample and the head in (c) equals the hi across the gas-liquid interfaces. This head may be compared with the static liquid pressure head versus liquid content curve (discussed on pages E4-22) of the sample to determine 6( for each test run, or the core may be removed from the holder at the end of each run for a more direct method of measuring 01. A continuous accounting of the inflow and outflow of liquid, and a material balance check, through all measurement runs, could be kept to indicate indirectly the residual 0i in the sample and its changes from one run to another without dismantling the test equipment. The most critical feature of the liquid-conductivity measurement equipment is the porous disk (6). Its primary function is to act as a hydraulic filter. In most testing methods the gas phase from chamber (c) is not permitted to flow through (6), whereas the liquid phase does flow from one retaining disk through the sample and out of the opposite retaining disk. Such control is obtained by selecting the material for disks (b) so that the disks will retain liquid in the capillary or funicular stages at all values of hi to be established in the sample. Special-purpose testing required by oil-reservoir engineering for evaluating oil-drive techniques has given rise to the development of many laboratory conductivity devices. Scheidegger (1957, p. 157-163) has discussed the relative merits of the more popular equipment and appended an excellent set of references to the basic literature on instrumentation and measurement of conductivity. A more detailed description of a few of the laboratory techniques is given by Pirson (1958, p. 74^87). Conductivity measurements made in the laboratory yield highly accurate results; error is usually less than about ±2 percent. However, the cost of such measurement in terms of funds and time is comparatively high. Even though the ocnductivity measurements are needed to identify the hydraulic properties of the unsaturated zane, the cost of measurement is not so readily justified for water exploration as it is for oil exploration.E28 FLUID MOVEMENT IN EARTH MATERIALS An alternative method of obtaining the conductivity versus 0f curve would be desirable. Richards and Weeks (1953) describe a system of observing flow rates and tension on a soil column about 1 foot in length, under nonsteady flow conditions, from which conductivity can be estimated. Liquid head is measured by standard tensiometer equipment at several points in the sample, and these head values are used as an indication of 0f by comparison with the liquid content versus liquid-head curve obtained from static measurements. Conductivity values may be calculated, from these measurements, taken over a period of nonsteady flow, by finite-difference approximation of the differential equation of nonsteady liquid movement. Rose (1951) also discussed measurements of this nature, with special reference to the time required by the tensiometers to reach equilibrium with the liquid phase. He emphasized the difficulty of obtaining correct head measurements if only a small amount of liquid was contained in the sample. Low conductivities in the sample, associated with small 0f, result in slow manometer response. A procedure for adjusting the manometer positions to minimize lag in response of the manometer due to storage changes in the liquid phase as the tests proceed was offered. Minimizing storage changes in the manometer reduces the time needed for attaining equilibrium between the sample and tensiometer and thereby reduces the time needed for laboratory observation. CALCULATION OF LIQUID CONDUCTIVITY A less costly approach to determining conductivity than by laboratory measurement alone might be found through making measurements for a relatively small number of 0r values and interpolating between measured values, or possibly defining the entire curve by calculation from the curve h{ versus 0f. The fundamental characteristic that best defines the liquid content of the medium is the curve of liquid content versus liquid pressure head, which is obtained from static observations. But in order to relate the static liquid occurrence to the steady-flow characteristics there must be a clear definition of the configuration of the pore liquid as a function of 0r. As indicated by Miller and Miller (1956), the interconnections between liquid filaments in the pore space are complicated, and it is difficult to obtain a completely sensible model of unsaturated liquid flow in terms of interconnected capillary tubes. However, the simplified capillary tube models remain as the chief basis for developing methods of computing liquid conductivity. Childs and George (1948) assumed that pore space may be modeled as a bundle of parallel cylindrical capillary tubes of various sizes. For the cylindrical capillary tube, equation 5 states that h{ is proportional to 1 jrct. Accepting the capillary tube model allows viewing the 0j- versus /q static relation as an index of an rct versus 0f curve. Thus the h{ versus 0f curve may be employed directly for estimating the pore-size distribution at any degree of saturation. Poiseuille’s well-known equation for flow through capillary tubes was applied to each pore size. A summation of the relative contribution of flow from each size by this means yields a figure for relative conductivity of the medium. This procedure was used by Childs and George for calculating the curve shown in figure 21. o —-------------------------------1---------- 0 0.1 0.2 0.3 0.4 WATER CONTENT, AS FRACTION OF TOTAL APPARENT VOLUME Figure 21.—Calculated and observed conductivities versus liquid content. (After Childs and George, 1948.) They compared their calculated results with a set of conductivity observations made on the sample, a sand whose grains ranged between 0.5 and 1.0 mm in diameter. The calculated curve was arbitrarily matched with the observed value of conductivity at 0f/0=O.3. Rather than the Poiseuille equation for flow in a filled cylindrical tube, Rapoport and Leas (1951) used Kozeny’s equation relating the solid-liquid interface area to permeability for weighting the relative conductivity of any given continuous liquid element. In Kozeny’s equation the total surface area was taken to include both the liquid-solid and the liquid-gas interface areas. The liquid-gas interfacial area was obtained by use of the principles of thermodynamics. The latter investigation defined maximum and minimum liquid-gas areas for any given value of 0f, which were applied through Kozeny’s equation to find a maximum and minimum value of relative conductivity. TheMULTIPHASE FLUIDS IN POROUS MEDIA E29 final expression for i£r?(maX) obtained from analysis of this model may be written as max) — 0 where = 03 (36) 2 2s /—J 1 1 +\* (MW’J -d„ do (37) s S=-** T f r hfddf (38) s ro0 I h{d6{ (39) ro0, h(dd r. (40) In equations 37 through 40, 60 is that liquid content, expressed as a fraction of the total pore volume at which the conductivity to liquid flow is zero, and h{m is the mean liquid pressure head in the range 0f to 0„. The minimum relative conductivity at any given 0f value was found to be (41) wherein the definitions given by equations 37 through 40 apply. It may be noted that all the integrals defining the terms of equations 36 and 41 simply refer to specified areas under the 0j- versus h[ curve. Thus, ifrc(maX) and Kr{(mn) are easily evaluated by graphical or numerical methods. The Kozeny equation used as a model for developing equations 36 and 41 contains a constant dependent on the tortuosity of the liquid-flow paths, which has been assumed by Rapoport and Leas to be independent of 0j-. The validity of equations 36 and 41 is in question because the relation between 0f and tortuosity does not appear to be constant (Burdine, 1953). Wyllie and Gardner (1958a, b) developed a model of porous media using somewhat the same viewpoint as Childs and George (1948), except for the variability due to tortuosity, and derived the following: pddt/hl Kr(=e^---------- (42) d0f[h\ do 0 wherein the term 02, defined by equation 37, accounts for a variable tortuosity. Equation 42 reportedly yields estimates of Kr$■ highly compatible with labora- tory test results on a wide variety of natural porous media. The integrals in equation 42 may be evaluated simply by numerical or graphical interpretation of the h{ versus 0j- data. Evidently the amount of computation needed to adequately define the Krt versus 0f curve by equation 42 is much less than is required for solving equations 36 and 41. Thus equation 42 is preferred provided it is eventually shown to yield reliable predictions. For those media in which 1 /h2( is a linear function of 0 it can be shown (Wyllie and Gardner, 1958b) that, Kr^Q. (43) However, it seems doubtful that the simplified form of Kr{ represented by equation 43 can be applied without reservations. A linear relation between 1 /h2t and 0 has not been noted except for a few samples, or over a short range of 0 values. In most of the tests over the range of 0f from 0O to 1, this relation generally has been observed to be nonlinear. The equations for Kr{ given above, and the empirical tests made to judge their validity, show that Kr( at low moisture contents is highly dependent on 0„. At the present time, however, there does not appear to be a way to measure or observe 6„ directly. For coarsegrained and well-sorted material, 0o might be estimated by inspection of the h{ versus 0f curve. It is known (Smith, 1933b) that the slope of the latter curve is much greater for the pendular stage than for the funicular stage. Thus, 60 for the h{ versus 0f curve of figure 12A might be assumed approximately equal to 0.08. However, the 0r value at which moisture distribution is entirely pendular probably extends to some much higher value, if it is correct to assume that liquid conductivity is zero over all the pendular stage (Smith, 1933b). The latter assumption, however, cannot be upheld by evidence in the literature. Thus estimation of a reliable 0„ value from static observations alone does not appear feasible. Rather, it has been suggested by many that the Kri versus 0? relation may be determined by combining calculations based on an equation like 41 with a few measured relative conductivities. The shape of the relative conductivity curve is calculated for several assumed values of 0„ by equation 41 and these are matched to the selected measured values. The computed curve providing the best fit to the measured values is then taken as the medium characteristic sought, affording also an indirect measurement of 0„. DIFFERENTIAL EQUATION FOR LIQUID FLOW In a pore space filled with water, water vapor, and gas, all three fluids are free to move with varying degrees of mobility depending on their relative concen-E30 FLUID MOVEMENT IN EARTH MATERIALS trations and the energy distribution in the flow system. At high liquid concentrations in most natural environments, it is probable that the vapor and gas motion neither retards nor augments the liquid flow appreciably. Under these circumstances, the movement of liquid water may be defined without reference to the other fluids contained in the pores. In any given infinitesimal element of such a flow field, the rate of change in mass of stored liquid equals the spatial change in mass flow. This continuity relation may be expressed as bp(v( . c>prrf dprrr dpr0f bx + by + bz 0 bt (44) where t refers to time. Provided the liquid density is constant at all points and is uniform from place to place, equation 44 may be written as bV{ . bV{ bV{_ bd{ bx + by + bz~ * bt' (45) For the type of flow assumed here the gross velocity of liquid may be expressed in terms of h{, according to equation 35. Making the indicated substitution and taking z to be the vertical axis, equation 45 becomes „ d2/h i d2/q~| bKt bh{ bKt bh( f Ldr2 by2~*~bz2 J bx bx by by , *Kt r bz Richards (1931) is generally credited as being one of the pioneers in adopting the concepts of unsaturated liquid occurrence and movement embodied in equations 35, 44, 45, and 46. These fundamental concepts have gradually gained acceptance because observed heads and liquid contents are comparable generally with those predicted from the above equations for liquid flow systems. Equation 46 may be written in any one of several combinations of variables selected for convenience of application. Dividing through by the permeability of the medium converts the conductivity to the relative conductivity. Assuming that h( and K% may be expressed as single-valued functions of 0j, equation 46 may be written in terms of only 0?, for example. The construction of equation 46 does not in itself presume limitations on the functional relations among K(, d(, and h(. However, solutions to equation 46 are easily obtained only through the further assumption that A| is a single-valued function of either 0£ or h%, and that h( is a single-valued function of 0£. The latter assumptions, in part, follow* logically from the more general conditions postulated: that all flow is in the liquid phase and that thermal and chemical effects on the flow regime are negligible. It is known the relations among 0j, and A$ are not single-valued in most porous media. On increase of the liquid content the value of Aj at a given 0$ will be less than the h( observed at the same value of 0t if the liquid content is decreased. An example is given in the adsorption-desorption curves presented in figure 31 A, p. E48. Because the curves of (such as fig. 31A for example) h( versus 0j observed over several wetting and drying cycles closely resemble the hysteresis curves noted in other stress phenomena, the multivalued relations are usually termed hysteresis effects. The geometrical shape of the pore space is generally recognized as the underlying cause of hysteresis, as described by Smith (1933a). COMPUTATION OF ONE-DIMENSIONAL FLOW As a general rule the relations among conductivity, liquid head, and liquid content are nonlinear, and K( is a function of Aj. Therefore equation 46 is a nonlinear differential equation. Thus, analysis of unsaturated flow in more than one direction is virtually impossible by straight-forward methods of calculus. One must evidently resort to some form of approximation in order to obtain a solution, either in the analytical process adopted or in approximating the relatively complicated A{ functions by simple algebraic expressions. Richards (1931) discussed a solution of equation 46 for one-dimensional steady flow in materials for which the conductivity could be expressed as K{=ah(-}-b (47) where a and b are constants. At Af=0 the medium is saturated and therefore b in equation 47 apparently equals the permeability. Equation 47 is a good approximation (Gardner and Mayhugh, 1958) over the range 1>0;OO.65. Even for the least complex boundary conditions, however, equation 47 does little toward simplifying the solution of equation 46 for nonsteady flow. Remson and Fox (1955) studied a case of one-dimensional steady flow for K{=a/h{. (48) Equation 48 specifies that the liquid conductivity is infinite where 0f=l. Where the unsaturated section above the water table is thick, the section near the water table, corresponding to the capillary fringe, normally contributes little resistance to flow compared with the drier sections higher in the profile, and perhaps equationMULTIPHASE FLUIDS IN POROUS MEDIA E31 48 may be justified as an effective average for many situations. Remson and Fox applied their final analytical expression for computing the relation between liquid pressure head and height above the free surface in an evapotranspirometer. Their computed results agreed well with observed data even though, as they point out, observations were made while the soil-water system was in the nonsteady state, the constant, a, in equation 48 was obtained from only one observation, and some of the flow-rate measurements were affected by instrumental errors. Equations 47 and 48 are symbolic of a wide variety of approximate relations which might be utilized for construction of a soluble differential equation of onedimensional steady flow. In the way of contrast, equation 47 best fits the K{ versus 0r curve at large values of 0f, and equation 48 might be applied most successfully to small values of 0f. Further search for, application of, and evaluation of simple functional relations between K{ and h{ or 0f seems desirable so as to extend the limited means now available for simply estimating liquid flow rates under steady or near-steady conditions. Perhaps because study of nonsteady flow is considered of greater practical import, more attention has been focused recently on investigating it than steady flow. Approximations like equations 47 and 48 applied to solve nonsteady-flow problems evidently produce differential equations intractable except through the use of methods like numerical analysis. Where such methods have been used it has been unnecessary to represent the conductivity function in approximate form, but the calculations required for obtaining a description of flow are generally lengthy. Thus far only one special problem of one-dimensional flow along the axis of the earth’s gravity field seems to have been treated successfully, that is, a solution has been devised which does not require an impractical amount of calculation. Equation 46 for horizontal flow is ! bx2 bx dx dt (49) and for vertical flow is Ki dz2 + d2 [_c>z J (50) Commonly h{ is expressed in terms of 0f to reduce the number of variables appearing in equation 49. One means to this end is a redefinition of the conductivity characteristic in terms of a diffusivity coefficient as follows: D{=K{ (51) wherein I)( is referred to as the liquid diffusivity. Numerical values of for a given medium can be obtained directly from the versus 0f and 0f versus h{ curves assuming all interrelations to be single valued as discussed earlier. From equation 51 and the functional relations already defined: *1-=®'!!- <52> Substitution of equation 52 in equation 49 yield Crank and Henry (1949a, b) described a numerical iteration procedure for solving equation 53 in studies related to the diffusion of solutes. Klute (1952) used this procedure for calculating the horizontal movement of moisture as a function of time in an initially dry sand which was wetted from one end. The boundary conditions of the system investigated may be expressed as 0f=0s, 2=0, t >0 0{=0t, 2>0, f=0. (54) The physical interpretation of equation 54 is that the sand was initially at a uniform moisture content 0; in a sand of semiinfinite extent. At some arbitrary time, identified by t—0, the moisture content at the face of the sand was elevated to 0S and held at that level as time increased. Klute’s reported results are shown in figure 22. Computations based on equation 53 indicated that moisture advances along a comparatively deep wetting front. In general form this result represents a significant improvement in the investigation of nonsteady unsaturated flow systems. It has been common practice to assume that a pressure drop equal to the capillary rise exists 0 o w 20 o c2 LU 22 s z CJ> z 25 1 o 30 LU z 40 o => 60 Z3 0 5 10 15 20 DISTANCE FROM WETTED END, IN CENTIMETERS Figure 22.—Moisture content and pressure-head values after horizontal flow into dry media. (After Klute, 1952.)E32 FLUID MOVEMENT IN EARTH MATERIALS across the wetting front. Also it has been presumed that the pore space is completely filled with liquid and that conductivity is constant and equals the permeability behind the advancing front. The latter presumption is derived from the comparison of the pore space with a bundle of cylindrical capillary tubes and was studied in detail by Lambe (1951). An application to infiltration was described by Stallman (1954). The curves of figure 22 indicate that both these assumptions are far from reality. The time period represented is comparatively short, but it can be seen that as time increases, the depth of the “wetting front” increases, and conditions of flow tend to depart increasingly from the assumptions based on the capillary tube hypothesis. Other similar results show clearly that the simple cylindrical capillary tube model is inadequate for calculating, or even visualizing, liquid flow through unsaturated porous media. Although the numerical iteration procedure of Crank and Henry (1949a, b) is a means for solving equation 53, it is laborious. Convergence to a satisfactory endpoint appears slow for the diffusivity characteristics exhibited by most porous media. Philip (1955) devised a more rapidly converging iteration procedure for solving equation 53, subject to the conditions expressed by equation 54, and he tested the result by comparing it with an analytical solution available for a given case. Subsequently, this process was extended (Philip, 1957a) to encompass numerical solution of equation 50 subject to the conditions expressed by equation 54. The latter problem refers physically to downward infiltration of water from the soil surface. In essence, Philip’s procedure for solving equation 50 is comprised of summing successive approximate solutions to a series of differential equations constructed from the basic equations of horizontal and vertical flow: equation 53 and a form of equation 50. His analysis leads to the following expression: -z=At1/2+Xt+it3/2+wt2+ . . . /m(0f)r/2 (55) where — z is the depth beneath the upper surface of a vertical column, comparable to the land surface in a natural environment; A, x, 4/, «, and fm(df) are functions of Of only, and t is the time since the moisture content at the upper end of the column changed. A is the function of 0f found by solving equation 53, which applies to horizontal flow under conditions expressed by equation 54. x is the function of df found by solving an equation developed from the difference between forms of equations 50 and 53. Thus x may be considered as an approximate correction, which accounts only partly for the effect of gravity on vertical flow. The remaining functions of 0f in equation 55 are determined in much the same way. All the values of /m(0f) are found by numerical solution of ordinary differential equations, each of which represents successive improvement in the treatment of gravity effects upon the liquid distribution in the profile. The details of the computing procedures and applications to infiltration problems are described by Philip (1955, and 1957a-f, and 1958a, b). Youngs (1957) compared moisture profiles, produced in the laboratory on two types of porous materials under the conditions of equation 54, with profiles of liquid distribution calculated by Philip’s methods. Good agreement was found between observed and calculated profiles. Gardner and Mayhugh (1958) also applied Philip’s method to solve equation 53 subject to the conditions of equation 54 for the porous media in which D(=D{(exp u(0{— 0fi) (56) where 0fi is the initial moisture content in the medium and Z>f4 is the diffusivity before infiltration corresponding with On, and a is a constant. Their published selected curves for 20. The subscript f» refers to conditions at <>0 and r=0). media may not therefore be assessed accurately by the concepts embodied in equation 46 alone, wherein flow in the liquid phase was presumed to account for all fluid motion. A study of the vapor motion should aid in defining those conditions for which equation 46 alone is a satisfactory approximation of fluid flow through porous media. Much of the literature devoted to description of vapor movement in unsaturated media seems to be concerned with explanations of applying the theory of vapor diffusion. Equations of water-vapor diffusion through gas-filled volumes are adjusted to account for the reduction of space caused by the occurrence of liquid in the pore space. The latter approach has led to much corollary speculation concerning the wide discrepancies between measured and calculated rates of vapor flow. Laboratory studies by Hanks (1958) indicate that the diffusion concept is adequate for estimating flow of vapor through the pore area in a section of dry material. Agreement between his observed and predicted flow is to be expected, for both his test conditions and the conditions postulated in the diffusion equations employed are identical. The dry porous material per- forms only as a composite inert object occupying part of the volume available if diffusion were only through still air. If the gas phase is completely immobile, the rate of vapor flow indicated by the simplest concept of diffusion may be expressed as: «.=-■y(*-*r)A eT(P_Pv) ~k (57) where qy is the rate of flow in g-cnr2-sec_1 y is a tortuosity factor and equals about 0.7 Dv is the coefficient of diffusion of water vapor into still air, in cm2-sec_1 (equals about 0.239 cm2-sec*1 at 8°C), and the remaining terms are as previously defined. Equation 57 was tested by observing vapor flow through a section of dry soil less than 1 cm thick. A wetted section was placed in contact with one end of the dry sample and vapor diffused through to the other end which was held at constant temperature and humidity. The dry section was pretreated with Arquad 2HT to prevent condensation and movement of liquidE34 FLUID MOVEMENT IN EARTH MATERIALS on the solid surface. The rate of vapor transfer was calculated from successive weighings of the sample and sample container assembly. Measurements of vapor transfer were made for both anisothermal and isothermal conditions over a wide range of moisture contents in the wetted section. The value of Dv was adjusted for temperature and pressure (Krischer and Kohnalter, 1940) by the relation A=A^(t02! (58) where the subscript zero refers to values at some standard set of conditions for which D0 is known. Comparison of the vapor-flow rate, predicted by means of equations 57 and 58, with the flow rate observed experimentally indicates that equation 57 is valid for both isothermal and anisothermal diffusion of water vapor through dry soil. In connection with his figure 2, reproduced here as figure 24, Hanks (1958) indicated these relations may not hold exactly for low SOIL-MOISTURE CONTENT AT WETTED END, IN PERCENT Figure 24.—Water vapor diffusion through dry soil from wetted section in contact with sample. (After Hanks, 1958.) moisture contents. Note in figure 24 that the observed vapor-diffusion rate drops off steeply as the moisture tension of the wetted section approaches and falls below the 15-atmosphere moisture content. (Whether the percentage of moisture content shown is a function of dry weight or porosity is not indicated by Hanks.) The curvature in this region may result partly from the measurements employed. Corrections for reduced relative humidity as liquid is removed from the wet-dry interface at the higher moisture tensions would tend to straighten the curve. In part, the curvature might also be due so movement of the “effective” interface position away from the dry section as the liquid in the wetted section is depleted by flow. These effects would be progressively more significant in the observations made as the soil-moisture content of the wetted section was decreased. If both these conditions were appropriately accounted for, the range over which the diffusion rate appears independent of moisture content in the wetted section would probably extend to much lower values of Of in the wetted section. In wet porous media the effect of liquid-vapor interaction may play a predominant role in the rate of vapor transfer. At moisture contents below 60 it has been held that conductivity to liquid flow is zero and there is no interaction between the two fluid phases for the steady state. Yet even at liquid contents just slightly above zero, about 3 percent by dry weight in one group of experiments (Wong and others, undated), vapor movement was significantly greater than can be accounted for by the simple diffusion concept embodied in equation 57. Smith (1943) proposed that migration of liquid could be triggered by condensation, which would locally increase the liquid content above 0o and thereby locally create zones with a conductivity favorable to movement in liquid form. This view is strongly supported (Wong and others, undated) by detailed laboratory observations. Others (deVries, 1958; Philip and deVries, 1957) have also indicated that there must always be some interaction or interdependence between the vapor and liquid phases, although this interaction may be negligible in extremely dry or wet porous media. Experiments reported by Solvason (1955) showed heat flow through porous materials was noticeably affected by the presence of water in liquid and vapor form. Imposition of a temperature gradient across a moist porous medium can readily be seen equivalent to an imposition of differing stress from one section to another. Consider the closed container filled with a moist porous medium, with a constant temperature difference maintained across two ends, and all other surfaces completely insulated as shown schematically in figure 25. If there should be no interaction between the liquid and vapor phase this system should eventually reach a static state, except for a continuous flow of heat from one side to another. According to equation 57, diffusion of the vapor phase through unsaturated media can be zero only if the term dpv/dx is zero over all the pore space. For static conditions in the vapor Vapor flow Warm Figure 25.—Schema of a sealed sample under a thermal gradient.MULTIPHASE FLUIDS IN POROUS MEDIA E35 phase, hi in the liquid phase becomes increasingly negative as temperature increases, according to the differential of equation 6 (p. ElO). According to equation 24 (p. E18), however, static conditions in the liquid phase will result in an ht becoming less negative at a comparatively low rate as temperature increases. Thus the net effect of static conditions in the vapor phase, while maintaining a temperature difference across the sample of figure 25, is a tendency for liquid to flow from the cool toward the warm side. This flow creates a vapor-pressure deficiency on the cool end and a vapor-pressure excess on the warm end, causing a flow of vapor from the warm toward the cool end. Evidently there can be no static occurrence of liquid and vapor in the sample controlled as shown in figure 25. That this is so has been demonstrated experimentally by Rollins and others (1954). Liquid evaporates near the warm end, flows as vapor toward the cool end where it condenses and flows back toward the warm end as liquid to close the cycle. 0 2 4 AVERAGE TEMPERATURE GRADIENT IN DEGREES CENTIGRADE PER CENTIMETER Figure 26.—Vapor flow rates versus temperature gradients, (After Rollins and others, 1954.) Rollins and others (1954) connected the warm and cool ends, of an otherwise sealed system, externally through a capillary tube. The movement of liquid in the connecting tube was considered to be a measure of the vapor motion through the sample. It was postulated that vapor condensed at the cool end of the sample and then entered and flowed through the external connecting tube to the warm end where it was evaporated and again entered the pore system. The flow rates per unit area observed in tests on one sample are shown in figure 26. At high-temperature gradients, calculated flow rates based on the simplified concept of vapor diffusion (as expressed in equation 57) through the restricted pore area were typically only about one-sixth of the observed rates. At low temperature gradients, the observed and calculated rates were in better agreement but still sufficiently different to indicate some deviation between the nature of flow and the physical model used as a basis for calculating rates of vapor movement. One cannot be certain which is responsible for the deviation noted, the model on which the calculations were based, or the method of measuring the rate of vapor flow. A temperature difference placed across the capillary tube in itself might cause the liquid movement observed in the external capillary. Perhaps the external capillary acts to short circuit the flow of the liquid phase. Thus, there is no assurance that the liquid flow rate observed in the external capillary is a true measure of vapor flow through the sample. Assuming that the liquid phase is immobile on the solid surfaces in comparatively dry media, one would not expect gravitational effects to influence the transmission of heat appreciably. In equation 57, the term dpjdx is not dependent on the direction of x with respect to earth gravity. However, as indicated by figure 27, the net rate of heat flow through a clay sample (Woodside and de Bruyn, 1959) may be dependent on whether heat flow was upward or downward. Liquid circulation is directly affected by the gravity-force field, TEMPERATURE GRADIENT (°F-IN •) Figure 27.—Heat conductivity effects caused by gravity flow components. (After Woodside and de Bruyn, 1959.) and the rate at which heat moves through the sample should therefore depend on the amount of heat transported through movement of the liquid phase. In a closed system, circulation of the liquid phase is dependent on the degree of interaction between the liquid and gas phases. Thus the data of figure 27 indicate that such interaction was possibly contributing to the mechanism of vapor transmission. However, the authors note that the observed differences in heat flow may have been due to slight inequalities in densities and moisture contents in the samples tested or to convection within the gas phase. The physical law expressed by equation 57 presumes that changes in the liquid phase do not affect the energy distribution or vapor content in the gas phase. At any rate it has been applied with this assumption as the basis for calculating rates of vapor flow. It seems quite apparent that this assumption is not an adequate approximation for describing the flow of water vaporE36 FLUID MOVEMENT IN EARTH MATERIALS through water-wet media, but this does not preclude use of the diffusion concept for developing a more comprehensive understanding of vapor flow. The law of vapor flow expressed by equation 57 may be expanded to include the interrelations with the liquid phase not accounted for in its development. ANISOTHERMAL FLOW OF LIQUID AND VAPOR GENERAL RELATIONS In comparatively dry or fine-grained material, gradients of temperature and head may be of equal import to the movement and distribution of fluids in the pore space. Such a system cannot be idealized as one of only liquid or vapor motion. A complete description of flow applicable over a broad range of moisture contents must include consideration of both the liquid and gas phases and the interrelations between them. Assuming there is an intimate relation between the liquid and vapor phases necessitates considering the system as being anisothermal. Such flow along a short length of the flow path, Al, is shown schematically in figure 28. The fluid is Change in storage n+APn Liquid phase Liquid = £ ^ Outflow Evaporate 'V Condense Inflow Gas phase Gas= P0v„ Finite length along flow path Figure 28—Schema of combined anisothermal flow of gas and liquid through porous media. composed of two separate phases, liquid and gas. At the inflow end of the section, arbitrarily taken at the right end, the gross velocity of liquid (the volume rate of flow per unit cross-sectional area of the porous medium) is defined as ®r, and similarly the gross velocity of the gas phase is ve. The rate at which mass enters each unit area of the inflow end equals mfin=vtpt-\-vgpg (59) where p{ and pg are the densities of liquid and gas phases, respectively. Within the short interval of time, At, there may be an interchange of mass between phases caused by condensation of vapor from the gas phase, or evaporation of the liquid phase. Thus, at the outflow end of the section the relative velocities or relative distribution of mass between the two phases may be different than at the inflow end. Should flow be in the steady state, no storage changes will take place in the interval Al and Thus, for the steady state Av(p{——Avgpe, where the differences refer to changes between the inflow and outflow ends of the section. For nonsteady flow mfin+Am/ At=mfOV,t where Am/At is the rate of change of fluid stored in the element Al. In the following it will be assumed that all the potential creating flow is derived from differential heads and temperatures applied across the space Al, and that in the gas phase only the vapor is free to move. Clearly, the temperature distribution along Al will be affected by the liquid and vapor flow as well as by any interchange between these fluids, and, conversely, the flow in each phase is dependent on the distribution of thermal energy. The three principal mechanisms involved in heat transport are: 1. Conduction. A temperature gradient across Al causes heat to flow through the fluid-solid system at a rate proportional to the thermal conductivity of the system independent of fluid motion. 2. Convection. This term is here applied to identify heat flow through the medium of moving fluids. The rate of heat flow by convection equals the mass rate of fluid flow times the temperature times the specific heat in each phase. Thus convective heat transport is directly proportional to the fluid velocities. 3. Conversion. Wherever liquid is evaporated or vapor is condensed in the system, heat is liberated or consumed to effect the phase transformation. Thus, heat sources and (or) sinks may be distributed along the length Al, their strength being dependent on the rate at which fluid locally undergoes a phase change. The flow system illustrated in figure 28 is evidently vastly more complicated than isothermal liquid flow, which is completely defined by only one differential equation containing relatively few terms. Anisothermal flow, on the other hand, requires two differential equations, both of a more complicated form; each of which must include a host of interrelations among the laws of fluid and heat flow and among characteristics of the porous medium and fluid. A complete set of equations defining the anisothermal fluid system has not yet been developed. Although derivation of an adequate set of equations for anisothermal fluid movement is beyond the scope of this report, certain phases of the interrelations within this system will be discussed in the following. LIQUID PHASE Assuming (1) Aj- is dependent on only temperature and moisture content, (2) the thermal gradient is parallel with the direction of flow, and (3) the totalMULTIPHASE FLUIDS IN POROUS MEDIA E37 potential gradient is parallel with the direction of flow; the total potential gradient along the direction of flow in the liquid phase may be written d(hx+z)_ d/q dT bhx dof dz , .. dl ~ZT dl+i>6[ dl+dl' U} From equations 21 and 23, ^=-1.55X10-4%/r. (61) At 20°C, r/(/=7.4X10-2. Therefore equation 61 may be written ^=-2.1X10-3Ar. (62) lower liquid heads the thermal effects would constitute more than 10 percent of the liquid flow potential at the postulated temperature gradient of 0.01 °K per cm. For smaller temperature gradients the corresponding estimated hx values would be greater in magnitude. Even though the above estimates are approximate, from them evidently the principal condition determining the degree of liquid phase movement by thermal gradients is that hx be of large magnitude. Thus it appears vertical movement of the liquid phase in the unsaturated zone is not influenced to a significant degree by thermal gradients under natural conditions except in the very fine grained silts and clays, or where the liquid content is comparatively small. VAPOR PHASE Substitution of equations 62 and 60 in equation 35 yields "=—K'[s| f-2-lxI°-*.f+§} m Equation 63 is sensibly the same as the equation for velocity of the liquid phase under anisothermal conditions given by Philip and deVries (1957). It should be noted that the term ddx/dl is dependent on both hx and T. Equation 63 may be applied for estimating the relative effects on flow caused by either head or temperature gradients under various circumstances. Assume, for example, that the direction l is coincident with the zaxis. Then dz/dl=l. Further, assume that dhx/dl—1, a comparatively common situation in deeper parts of thick unsaturated zones containing more moisture than is required for static conditions. The condition dhx/dl=l may be interpreted to mean that a downward velocity of Kx exists continually in the liquid phase. Also it can be seen ddx/dl=0, or the moisture content along the z axis is uniform. Applying these conditions to equation 63, the liquid phase velocity is zero if 2.1X10~3hxdT/dz=l. (64) Therefore if the product hxdT/dz equals about 5X102, the temperature effects equal the gravity effects on vertical liquid flow. If temperature effects are about 10 percent of the total gradient producing flow, which might be considered a lower limit of significance of thermal effects on liquid flow, the product hxdT/dz^50. A value of dT/dz= — 0.01°K=cm-1 would be rather high for the unsaturated zone in most areas. Using this value in the relation hxdT/dz=50 provides an estimate of the lowest value of hx for which vertical temperature gradients significantly affect vertical flow of liquid in a region of uniform moisture content. This evaluation yields Af«—50X102=—5,000 cm. At A law of flow describing movement of the vapor phase has been given in equations 57 and 58. Combining these relations yields 2»= TDqPq T1-3 dpv RT2 3 P—p, dx (65) For the temperatures and pressures associated with the p unsaturated zone, g—" - will very nearly equal one. ■T pv Also, assuming 7=0.66, taking D„=0.239 cm2-sec_1 at T0=281°K, and expressing pv in g-cm-2, rtJ-3{p-p,) 8-2X10 “• (66) Substituting equation 66 in 65 leads to &=—8.2X10-n(—(67) It has been assumed tacitly that qv is a function of temperature and liquid head. For comparing the relative effects of temperature and the pressure head of the liquid phase on vapor movement, it will be convenient to express p„ in terms of T and hx- Equilibrium between the liquid and vapor phases is defined by equation 6, written here again for convenience: hx=RT log (6) If hx in the liquid phase is less than the value given by the right side of equation 6 at any point, vapor will condense there and enter the liquid phase. Conversely, if hx is greater than the value given by the right side of equation 6, liquid will evaporate and enter the vapor phase. The rate of exchange between the two phases is dependent on the amount by which hx deviates from the value given by the right side of equation 6, the rateE38 FLUID MOVEMENT IN EARTH MATERIALS of flow of heat and fluid in both phases, and on the shapes of the liquid-gas interfaces and the solid particles. According to equation 6, as displayed in table 2 (p. E12), /ij for static equilibrium is very highly dependent on pvlp0- Therefore, equation 6 may not be applied as a definition of p, in equation 67 if an exact relation among q„ T, and /if is to be established. However, as an approximation, it will be applied here with the assumption that interchange, or conversion, between the two phases occurs very readily; that is, they coexist under mainly static conditions. The total differential of pjp0 along the assumed direction of flow may be stated as dpjp0_dpjp0 dT . dp,/p0 d/q=J_ dp1_p1 dp„ dx dT dx ' d/if dx p0 dx pi dx From equation 6, pv—p„ehtIBT and d(pjp0)_ P, dT p0 RT2 d(pjp„) p0 dh( p0RT (68) (69) (70) (71) Substitution of equations 69, 70, and 71 in equation 68 yields dpv_(J_dh{__Af_ dT. J_dpo\ h ,RT dx \RT dx RT2 dx dx ) Vo r (72) wherein p„ is dependent only on T. Subject to the validity of equation 6 for nonstatic conditions, equation 72 may be substituted for dpjdx in equation 67. Thus, 2t=—8.16X10-1,(<^)—0f)!r1-3po / 1 dh{ /if dT . 1 dp„\ h ,RT \RTdx RT2 dx'p0 dx) e * or (73) 2t=-8.16X10-^-e^p, (J_ dht_h(_ dT dp0\ , . \RTdx RT2 dx ‘ Po dx) K > At the edge of a comparatively wet section, the head gradient across the wet-dry contact is high. Even though the temperature gradient in such a region may be small, equation 74 predicts that vapor-flow rates toward the dry side may be of comparatively large magnitude through the influence of the liquid phase. Where h{ is lower than about —105, that is in very dry media, and gradients of liquid head are comparatively small along the x direction, equation 74 reduces to dp0 dx’ a good approximation of equation 67 if the ratio pv/p0 is nearly one. For a gradient of 0.01°K-cm'1, the value used for illustrating thermal effects on liquid flow in connection with equation 64, movement of vapor upward from the water table would be about 3 grams per year per sq cm according to equation 75. Experiments (deVries, 1958) have shown that equations 67 and 74 generally underestimate rates of vapor flow by a factor of about 3 to 10. Thus vapor movement upward through comparatively dry soils from the water-table due to temperature gradients may be as much as 1 foot per year in some places. COMPOSITE OF ANISOTHERMAL FLOW EQUATIONS Flow of vapor could be estimated by means of equations 63 and 74 for a great number of individual situations or combinations of temperature, head, and liquid content in porous media. Such estimates, however, are mainly of academic interest because the knowledge required to specify the values of potential over finite lengths must come either from direct observation of a system or from a problem solution already known. The ultimate expression for anisothermal multiphase flow should be constructed in such a fashion that head and temperature relations may be calculated over a finite region of a given porous medium, on whose boundary the head and temperature distribution are known. Philip and deVries (1957) approached this problem by stating the flow conditions as two differential equations. The first was obtained by entering the laws of fluid flow in an equation expressing fluid conservation. Similar treatment of the laws of heat flow produced a second equation. Both equations apply at each point in the fluid-heat flow system. Figure 29 shows the relatively ■ VOLUMETRIC AIR CONTENT (cm3 -cm-3 ) ^-s.iexio-^-^T1-3 (75) Figure 29.—Ratio of calculated anisothermal flow of fluid to calculated vapor diffusion and observed fluid flow as a function of liquid content. (After Philip and deVries, 1957.)MULTIPHASE FLUIDS IN POROUS MEDIA E39 good agreement between observed and calculated rates of fluid flow they obtained. deVries (1958) discussed an extension of the latter work, including a more comprehensive statement of the interchange between the liquid and vapor phases. Results from the latter theoretical investigation are in need of testing by comparison with observation. Unfortunately, laboratory studies reported in the literature have not included a sufficient number or variety of observations and control to permit testing deVries’ approach adequately at this time. Miller and Seban (1955) studied the liquid-vapor complex using enthalpy relations and Darcy’s law. Equations were derived for computing the amount of vapor flowing as a fraction of the total flow, and for computing the position and length of the transition zone between the single and multiphase flow sections. They used liquid propane flowing horizontally through a porous medium, which was thermally and hydraulically insulated except at the ends. Temperature and pressure were held constant at the uninsulated ends, such that the pores at the inlet end were completely filled with liquified gas, and both liquid and gas phases could exist at the outlet. With such control, at some point between the sample ends, the conditions of temperature and pressure permitted the liquid to boil and produce vapor. Downstream from that point both the liquid and gas phases were in motion. Both the experimental and theoretical approach taken seem unique to multiphase flow investigations. Predictions made from their theoretical development compared well with laboratory data. While the heat-flow relation as stated by Philip and deVries (1957) is especially useful for identifying the hydrologic significance of each component part of their equations, the large number of terms contained by their equations greatly complicates the computation of fluid distribution. Use of the enthalpy relations simplifies the expression of heat flow compared with the detailed form required by adding all the components previously discussed. It may be found possible to express aniso-thermal flow as simple functions in terms of bulk energy or matter transport, and thereby reduce the amount and complexity of computation to a minimum. However, it appears premature to consider the form of expression at this time because even the basic understanding of the mechanics of anisothermal flow is far from complete. CRITERIA FOR HYDRAULIC MODELS OF LIQUID FLOW As discussed briefly on page E30, calculation of isothermal movement of liquid over a given region is extremely difficult because the basic differential equations defining flow are nonlinear. The hydraulic model, a scaled replica of the prototype, affords in some problems a convenient substitute for calculation. Through proper model design, systems of comparatively large size can be reproduced in the laboratory on a small scale where liquid movement as a function of time and space may be observed with relative ease. The model should be designed so as to enable one to convert the liquid movement observed in the model to the dimensions of the prototype system. The chief aim of the model design is to reduce the prototype size to some size convenient for laboratory use and to reduce the time scale of the prototype so the laboratory observations may be concluded in a comparatively short span of time. The results obtained in laboratory studies of flow through small samples can be put in terms of flow through prototype systems by applying the rules of model design to the dimensions of the laboratory model. This extends the range over which the laboratory tests can be used as a visual aid to the understanding of unsaturated fluid flow under various prototype conditions. Experimental results on flow in or drainage from vertical columns (King, 1898; Remson and Fox, 1955; and Youngs, 1957,1958a, b), on centrifuge control of samples (Briggs and McLane, 1910; and Marx, 1956), and on flow in the vicinity of structures (Aronovichi, 1955; and Day and Luthin, 1954) cover a wide spectrum of prototype conditions of size, shape, and conductivity. Model criteria may be applied to such results so as to make the observed data more useful as expressions of prototype activity. A few prototype flow systems have been modeled deliberately and the results have been reported in the literature (Marx, 1956, for example). However, in most of the laboratory investigations reported in the literature, the scale relations between model and prototype have not been recognized. In many reports of laboratory studies on flow the conclusions developed do not apply to prototype systems because experiment design did not adequately account for scaling factors. Thus a direct comparison between laboratory observations and the flow regime of the much larger prototype cannot be made; and it is difficult, through brief contact with the literature on unsaturated flow, to fully appreciate the validity of general conclusions drawn from small-scale observations. Leverett and others (1942) described the application of the principles of dimensional analysis to unsaturated liquid flow through porous media. Although attention was focused on modeling the displacement of oil by water, their selection of pertinent variables on which flow depends and their discussion of model design apply equally well to the liquid phase in water-water vapor or water-gas systems where flow is isothermal and predominantly in the liquid phase. A more comprehensive treatment including interchange between liquid and gas phases, or anisothermal effects would require inclusion of flow parameters not considered by Leverett andE40 FLUID MOVEMENT IN EARTH MATERIALS others. Much of the following has been condensed from three papers (Hubbert, 1937; Leverett and others, 1942; and Marx, 1956) with emphasis on models in which either the porous medium and liquid properties are different from the prototype system or in which the external force field applied on the model is different from gravity. DIMENSIONAL SIMILITUDE In ordinary mechanics, motion can be expressed in terms of three fundamental dimensions—length, time, and mass. Although in some cases flow in unsaturated porous media may depend on additional characteristics, it is assumed here that the flow is dependent only on those variables that can he defined by combinations of these three dimensions. Temperature and associated thermodynamic relations and most chemical interactions, for example, are excluded from the liquid flow regime under consideration. Dimensional similitude between a given characteristic F in the prototype and F' in the model, as dictated by the principles of dimensional analysis, can be stated in very general form as follows: F F K§K0(£K£)‘(#)W <*> where A is a characteristic length, t is a unit of time, M is a characteristic unit of mass, and A, B,C are variables defining F, all relating to the prototype. The corresponding characteristics and variables of flow in the model are designated by primes. Equation 76 can be obtained simply by setting a non-dimensional group of variables in the prototype equal to the same group in the model and simplifying the result to the form shown. This is a basic requirement for effecting dimensional similitude between flows in the model and prototype (Bridgman, 1931). The exponents a,b, * * * / are obtained from the dimensional nature of the functions F or F'. For example, if F represents velocity, then a—1, b — —l and c, d, f * * * =0, for by definition velocity is a length divided by time. Each characteristic significant to flow hi the model must satisfy equation 76. Variables like F, A, B, and C can all be expressed in terms of the fundamental dimensions A, t, or M. Thus the model-prototype functions like A might be related as H (£\61 (M\H A' \L'J \t'J \M'J B_=(L\ "2 (t\ s2 (M \ C2 B' \L'J \t'J \M'J C /A\°8 /t\ ”3 /M\ C' \L') Gy \M7) (77) and so on. For any given model-prototype system the ratios 7 and —— individually must have the same value as applied to determine each pertinent characteristic of the flow regime. Equation 76 will be more convenient to apply for some purposes than equations like 77, but both forms will be applied in the following for defining the dimensional interrelations among variables of model and prototype flow. A complete model design is attained only if all the variables on which flow is dependent satisfy the relations expressed in equations 76 and 77. For liquid flow the equation of continuity and a form of Darcy’s law are believed to adequately define the flow regime at any point. The continuity relation for this case was previously given as equation 44: dpw dpj-flj- dp{V{_ bx by bz bt (44) In the form of Darcy’s law given as equation 35 the velocity of flow was defined loosely in terms of liquid pressure head only. This statement is adequate where flow occurs in a uniformly accelerated field. However, where force fields of varying magnitude may be impressed on the flow system, such as may be generated by centrifuging, flow must be defined in its more complete form as a response to energy dissipation as follows: Ptm=~K< sin x} (78) Equations 44 and 78 together describe the flow relations at every point in an isothermal liquid flow system. Flow is dependent on each of the variables included in these expressions in both model and prototype. Dimensional similarity between flow at a particular point in the model and the corresponding point in the prototype is assured by appropriate scaling of each characteristic affecting flow by means of equations 76 or 77. The function F of equation 76 was defined with emphasis on it being of a dimensional nature. This does not preclude the use of equation 76 for showing the model-prototype relation between nondimensional factors such as porosity, ’ are true representations of the corresponding functions in the prototype. If the value of K is large and the inflow and outflow lengths are small in the prototype, the grain size required in the model may be of the same order of magnitude as the length of openings on the boundaries of the model flow system. To this extent, length reduction by scaling is limited. The limits of the characteristics of fluids useful for modeling water movement by stationary models are such that length reductions may not exceed a value of about 30 to 1, and the time scale reduction is therefore limited to about 5.5 to 1. ARTIFICIALLY ACCELERATED MODEL Marx (1956) obtained equations relating the flow from centrifuged samples to flow from prototype systems by analyzing the boundary conditions in each field of flow. The same equations showing the effects of centrifuging on dimensional relations can be obtained directly from equations 85 and 86. If the centrifuged sample is identical with the material of the prototype, and if the liquid is also the same as that flowing in the prototype, K t n Pf K' t' / f Pt (91) Substituting equations 91 in equations 85 and 86, and expressing each result as g/g', from which '/‘-(f)’ (92) (93) Equation 93 can be used in equation 92 for relating theMULTIPHASE FLUIDS IN POROUS MEDIA E43 time and length scales to the model acceleration, which shows that L'=yL and t'=(^jt. (94) The advantages to be derived from artificially accelerating the model are partly apparent from equations 94. The length and time scales can be greatly reduced by centrifuging. In the conventional test for moisture equivalent (Am. Soc. Testing Materials, 1958), g' = 1,000 g, and some soils have been centrifuged at g' —100,000 g in special tests. For the latter, a sample length of 1 cm would be equivalent to a prototype column length of about 3,300 feet, and a centrifuging time of 30 minutes would represent a period of 5 X109 hours, or more than half a million years in the prototype. Like the stationary model, the artificially accelerated model is also subject to limitations. In order to keep the size of centrifuge equipment within reasonable bounds, the sample size must be kept small—of the order of a centimeter in length or diameter. The grain diameter of the medium must therefore be small so the nondimensional characteristics of the centrifuged medium are adequately modeled. On account of the sample size limitations alone, the grain size diameter should be less than 0.01 cm, preferably much less. Rearrangement of unconsolidated material under stress produced by acceleration may void the use of centrifuging for accurate model studies. Compression of both consolidated and unconsolidated models under high acceleration may also cause significant inaccuracies. The influence of rearrangement and compression on flow from centrifuged models cannot now be evaluated because very little appears to be known about these factors. HYDROLOGIC INVESTIGATIONS AND THE UNSATURATED ZONE NATURE OF HYDROLOGIC PROBLEMS The principal goal of most hydrologic studies is to define the distribution of water on and under the earth’s surface as a function of time and space. The occurrence of water at any specified place is dependent on the manner in which water is supplied to that locale and on the ease with which the supply may flow away in the form of discharge. Under natural conditions, supply is largely a function of climate, and the form of discharge is regulated mainly by such factors as topographic characteristics, slope-area-discharge relations of the surface waterways, permeability of the soils and deeper rocks, and other factors. Thus in general terms, the distribution of water might be viewed as dependent on either supply and/or environmental factors. In the writer’s opinion, this division of factors is the funda- mental basis for the evolution of the greatly differing approaches to the studies of ground- and surface-water distribution. Most surface-water investigations are designed to define water distribution as a supply characteristic, relatively independent of environmental conditions. Results from such studies are often expressed in the form of flow-duration curves which define the probability of obtaining a given yield from a particular locale over a given time. The occurrence of ground water, on the other hand, is so greatly dependent on environment that ground-water investigations are almost completely devoted to an interpretation and description of environmental factors alone. It is not the writer’s intent to discredit this difference in approach to hydrologic studies, for like other inventions, it has been mothered by necessity. However, the hydrologist is facing a period in which water distribution will be influenced more than ever by man’s activities. Where man’s use of water rearranges water distribution markedly, the climatic factors become relatively less important in resource evaluation for a given locale, and the environmental factors—both natural and manmade—become relatively more important. Questions arise concerning ground-water yield from extensive dewatering of saturated formations, reduction of rejected recharge by lowered ground-water levels, effects of land-management practices on stream flow, operation of multipurpose water systems, return of irrigation water to ground-water storage, contamination of water supplies by the disposal of waste, that is, sewage from septic systems and from atomic energy plants. These are but a few of a host of problems associated with man’s use of water that require attention. As water use is increased, the significance of factors like these to total water availability is expected to increase. Beyond the common denominator of man’s use in the general class of problems characterized above, each is highly dependent on one environmental factor heretofore given little attention in hydrologic investigations. The solution to all these problems, in quantitative terms, depends upon knowledge of the characteristics of the unsaturated zone. Within the category of environmental factors the unsaturated zone to this day remains as a kind of no-man’s land viewed from afar by both the ground- and surface-water specialists. Yet with the increasing use of water, the unsaturated zone must be studied by hydrologists because it is a dominant link between ground- and surface-water distribution. Furthermore, the hydrologic questions posed require that the unsaturated zone be considered as an integral part of the hydrologic system. Even in ground-water investigations where environmental factors are given greatest weight, the unsaturated zone usually is treated as a passive component.E44 FLUID MOVEMENT IN EARTH MATERIALS As a summary statement to describe bis concept of changes in ground-water storage, Meinzer (1923, p. 28) proposed the term “specific yield” and defined it as follows: The specific yield of a rock or soil, with respect to water, is the ratio of (1) the volume of water which, after being saturated, it will yield by gravity to (2) its own volume. Carrying idealism to an extreme, one might note that the effects of gravity, space, and time are not specified completely, or in detail, in this definition. Thus one might assume that, ideally, it applies to static conditions in an infinitely long column. This speculation leads to the result that specific yield is simply equal to the porosity. But an idealistic interpretation of Meinzer’s definition is not warranted. He wrote (Meinzer, 1923, p. 1-2): A definition is the expression of a concept by means of language, it should include all that is involved in the concept, but nothing more * * * It has the same significance as the definition of the concept; it is neither more nor less precise * * * Through the philosophy evident in Meinzer’s writings on the subject of specific yield, it is apparent that the concept is not defined by a restricted set of idealized environmental conditions. Bather, he included discussion of several factors that affect specific yield as observed in the natural state, that is, recharge or discharge through the land surface, thickness of the unsaturated zone, variations in hydraulic characteristics of the rocks, salinity, and temperature. Thus specific yield relates changes in ground-water level to changes in storage in the field environment. Because the environmental conditions regulating changes in liquid content vary with time, field observations of specific yield will in general differ from one another at the same location. Thus tests on small samples designed to measure specific yield in the laboratory are not adequate unless the field environmental conditions are accounted for in the analysis of the laboratory tests. Specific yield, like the flow-duration curve, originates from measurements of the flow system made under a particular set of circumstances not always known in sufficient detail to permit use of the observed value for other prescribed boundary conditions. It represents a resultant of unknown factors. Yet, the conditions that result in a particular value of specific yield as may be observed in the field, contribute equally to the change in storage caused by any set of conditions man’s use of water may conceivably impose on the unsaturated zone. To afford means for predicting hydrologic changes in the unsaturated zone, or the effect of the unsaturated zone on surface and ground water, the unsaturated zone must be defined in terms of those hydraulic char- acteristics that regulate fluid movement through it. Adhering strictly to the principles of flow known from theoretical considerations of multiphase fluids, one might visualize that a complete definition of the unsaturated zone would require a great amount of detail. However, it is not unrealistic to assume that further study might produce one or a few simple general relationships and yet retain all the fundamental concepts of flow. It seems almost imperative that some form of simplified approach be obtained for analyzing in a practical manner the role of the unsaturated zone on water distribution. PROBLEMS IN DEFINING THE BEHAVIOR OF THE UNSATURATED ZONE Viewing the flow in the unsaturated zone as a predictable response to given stress, it is evident that three basic features of the zone must be defined to make behavioral predictions in advance of stress applications. These features are: 1. The nature of the fluids and the porous media. Fluid and media characteristics must be expressed in terms of energy dissipation or energy states resulting from fluid movement or state of fluid occurrence in porous media. 2. The laws of fluid flow which define the relations be- tween matter and energy distribution or transfer. 3. The stress to be applied along the boundaries of the region of interest. Included in unsaturated zone investigations for example, might be such factors as position of the water table and rates of change, rates of recharge at the land surface, distribution of temperature and salinity in time and space. The extent to which flow in the unsaturated zone, and flow to or from the contiguous saturated zone and land surface, can be predicted is dependent on the degree to which all these features may be defined in any given problem. There exists an interdependence among the features listed. As was noted in earlier discussions in this report the laws of anisothermal liquid, vapor, and gas flow are complicated statements of physical activity that are not yet satisfactorily understood. Where the stress applied on the unsaturated zone is such that simultaneous motion of all three fluids must be analyzed, definition of the system is therefore inadequate. Where the nature of the fluids and porous media needs to be investigated in detail by field measurement, again there is difficulty in defining the problem because field measurement techniques are unequal to the task (Olson and Hoover, 1954). It is evident that the hydrologist’s ability to predict flow in the unsaturated zone is limited partly by fundamental deficiencies in the laws of flow and by inadequate field instrumentation. Nevertheless,MULTIPHASE FLUIDS IN POROUS MEDIA E45 in many hydrologic problems unsaturated flow may be adequately approximated by assuming that the flow is entirely in the liquid phase, and by substituting laboratory tests and analysis for field measurement. Once the hydrologist has gained an adequate and practical definition of the flow characteristics of the unsaturated zone for a given problem through approximation, he faces other barriers that might prevent the successful prediction of flow. Integration of flow relations expressed in the simplest of forms over a finite region is virtually impossible by the simpler techniques of constructing algebraic analytical equations relating flow to time and space. Further, it does not appear that simple analytical techniques are competent for this purpose. At best the laws of flow are of nonlinear form, systems studied in the field are highly nonhomogeneous, and the boundary conditions are highly variable. However, these conditions do not present an impenetrable barrier to the solution of flow problems. It has already been demonstarted that finite-difference techniques, hydraulic models, or electronic analogs may be employed to solve problems of this type satisfactorily. Aside from calculating flow within the unsaturated zone, this methodology may also be applied for computing the boundary conditions attending overland flow and for defining the rate at which water is removed from storage in ground-water reservoirs. RESEARCH REQUIREMENTS It is evident from a review of the literature on the occurrence of multiphase fluids that problems of isothermal flow of the liquid phase in the unsaturated zone can be solved directly by adopting applicable tested theories. However, it appears likely that all solutions to such problems will be attained only through some form of approximation, the degree of acceptable approximation depending on the nature of the individual problem to be solved. The characteristics of the sediments comprising the unsaturated zone are everywhere variable from place to place. Generally it will not be possible to take all the variations into account in a study of flow because the volume and complexity of the computations required becomes too great, or it is impractical or impossible to measure the variations in sufficient detail. A similar difficulty is had with nearly all flow problems involving natural porous media, and it is resolved mainly by limiting the degree of detail observed to that which will yield a realistic estimate of flow relations, and to that which can be processed with funds made available for the work. Thus, permissible detail is limited mostly by the economic import of the flow problem. Though such limitations tend to regulate the extent to which the hydrologist may delve into a given flow problem, a discussion of them is beyond the scope of this report. It will suffice here to recognize that such limitations exist, and that the cost of a problem solution is dependent on the degree to which the complexities of the flow system are described. The fundamental question of greater interest now is how the hydrologist might employ multiphase fluid-flow theory so as to treat the unsaturated zone as a dynamic part of the hydrologic cycle. There is great need for developing an understanding of the general manner in which the unsaturated zone affects flow at the surface and underground. This understanding might be gained most simply by comparing calculated flows in the unsaturated and contiguous zones for a variety of assumed sediment characteristics and boundary conditions. General relationships obtained in this way should be useful as a guide to the selection of those most significant variables affecting flow and requiring measurement in the field. Dewatering by gravity drainage might be cited as an example wherein even a general understanding of the nature of flow in the unsaturated zone would be extremely useful to the hydrologist. Extensive dewatering of aquifers by pumping wells takes many months or years. Predictions of the amount of water recoverable by dewatering are often based on a value of specific yield observed in pumping tests of short duration— from a few hours to a few weeks. Recognizing that dewatering involves downward movement of liquid through the unsaturated zone in response to declining ground-water levels, one realizes that a certain amount of time is needed to reach a stable liquid-content profile above the water table. If the time required is comparatively large, tests of short duration will provide values of specific yield that are perhaps much lower than might be attained over long periods of time. Consequently, predictions of aquifer yield based on shortterm test results may be significantly in error. On the other hand, the results of long term tests may not apply to short term yields. This issue has been clouded materially by conclusions stated from laboratory experimentation. The almost standard illustration begins with completely saturating a long upright column of clean sand, then permitting the saturated column to drain freely from the bottom by gravity flow. The drainage from the column is observed and plotted as cumulative discharge of water versus time of drainage. As time increases, the rate of change of total discharge becomes small, very small when compared with the average rate of outflow, even for periods of drainage of only a few hours. Observations of this type are the basis for concluding that most of the drainage occurs almost instantaneously as the water level isE46 FLUID MOVEMENT IN EARTH MATERIALS lowered and therefore the hydrologist need not concern himself with timewise variations in observed specific yield because they are negligible, as observed. Such a conclusion would be correct for homogeneous and isotropic porous media of high permeability, initially fully saturated to the land surface. However, such a set of circumstances is rarely met in the field. More often the water content profile above the water table is as shown schematically by the solid curve extending above the water-table position I in figure 30. Should the Land surface Figure 30.—Diagram showing steady state profiles of liquid content above water table. water table be lowered to position II, AH below position I, eventually the /if versus 0f curve would assume the location shown by the dashed line. For a homogeneous and isotropic profile, if the only condition changed on the profile is the water-table position, the dashed curve will have the same form as the solid curve and will everywhere be a distance AH lower than the solid curve. The total drainage from the column effected by the water-level change is given by the area between the two curves shown. In the usual laboratory demonstration, however, the total drainage volume observed is represented by all the area above the dashed line bounded at the top by the land surface and at the right side by 0f=l. Thus the laboratory observations on drainage from initially saturated columns places a disproportionate emphasis on the amount of liquid available for drainage under field conditions. Not only are the volumetric relationships distorted by studying fully saturated columns. The degree of initial saturation also affects the rate at which drainage to the water table may occur. The low liquid contents shown in figure 30 are indicative of the conductivity in the profile in general accord with the conductivity relations given by figure 18. It can be seen that the conditions represented by figure 30 are such that a much smaller volume of material is available for drainage and the material is of much lower conductivity than that used in the usual laboratory demonstration referred to above. Thus the usual laboratory observations of the relation of drainage volume versus time may lead one to seriously overestimate the rate of drainage of porous materials under field conditions. If one considers also the effect of stratification in reducing the conductivity (Corey and Rathjens, 1956; and, Day and Luthin, 1953) in a vertical direction in the field, it is clear that laboratory tests have thus far yielded very little useful information on the relation between time and specific yield. The inadequacy of conclusions reported from laboratory investigations stems mainly from an inadequate appreciation of the complexities of earth materials, rock texture, the hydrologic boundary conditions, and the true nature of changes imposed on the system under natural conditions. Laboratory investigations specifically designed through dimensional analysis of characteristic field situations are sorely needed. This need, illustrated here only by detailed reference to gravity drainage, pertains to all areas of analysis involving flow in the unsaturated zone. It should be met by accelerated activity in properly designed laboratory studies utilizing hydraulic or electrical models. The feasibility of such investigations presupposes a knowledge of the variables controlling flow under field conditions, and an adequate definition of the basic laws of flow. The least complicated form of fluid flow'—isothermal liquid motion in granular porous media—has already reached an acceptable stage of definition for engineering use with the exception of accounting for hysteresis effects. Although the latter effects have not been incorporated in methods for calculating flow where the liquid content both increases and decreases, they are important. Christensen (1944) noted an example in which hysteresis caused a shift in the curve of head versus moisture content by a factor of 30. Potentially, this may be a source of considerable error in flow computations. Hysteresis is believed to arise from the nature of the pore shapes, within which the shapes of the liquid-gas interfaces are not single-valued functions of liquid head (Smith, 1933b). Thus the position of the liquid content-liquid head curve is dependent on antecedent flow conditions. Study of this phase of liquid flow should be made to show conclusively whether hysteresis, in fact, does exist for the steady state betweenMULTIPHASE FLUIDS IN POROUS MEDIA E47 the liquid and vapor phases. It is conceivable that hysteresis may be a function of nonsteady conditions between these two phases, and that laboratory measurements of liquid head versus liquid content show hysteresis because they were made before steady state is reached. Carman (1952) indicated that the curve of liquid conductivity versus liquid content shows little if any hysteresis. If this proves to be a valid generalization, the hysteresis function may not be too troublesome in flow calculations. A carefully controlled laboratory observation to refine the understanding of hysteresis phenomenon will be difficult. As DeLollis has noted (1952), accurate steady state measurements of the interrelationships among fluid-solid systems are very difficult or impossible to make because such systems are generally self contaminating. Difficulties notwithstanding, a positive approach to the definition and use of hysteresis characteristics of porous media must be made before the laws of even the simplest form of flow, isothermal liquid motion, are complete. In a preceding section of this report, it was indicated that where the unsaturated zone is comparatively dry, or is comprised of relatively fine-grained materials, flow must be considered as anisothermal. Fukuda (1955) investigated the removal of liquid from the unsaturated zone as it may be effected by air movement through the upper soil layers in response to short and sharp changes in atmospheric pressure. The effect of this type of movement on liquid pressure heads was discussed on pages E21, 22. Fukuda’s study showed that fluid transport in vapor form by convection in the gas phase may amount to as much as a foot of water per year. Fluid transport by convection in the gas phase has received such little attention in hydrologic studies that currently it is impossible to clearly evaluate its importance as a mechanism of recharge or discharge in the unsaturated zone. Even though the gas phase velocity in the unsaturated zone has been assumed negligible in most studies of anisothermal fluid flow, careful and detailed field studies should be made so as to gain a more positive identification of its magnitude and effect on liquid content under field situations. Anisothermal liquid and vapor flow, with the gas phase velocity negligible, has been defined by assembling the laws of liquid, vapor, and heat flow through simple summation (deVries, 1958; Philip and deVries, 1957). Observed flow rates are in good agreement with the rates predicted from theoretical relationships. Provisionally, these comparisons are an indication that the theoretical relations may be adequate, on an empirical basis at least. Some of the component laws contained are yet suspect and work is being done to further test them. Calcium chloride has been used to determine the relation between liquid and vapor movement in response to head or temperature gradients in laboratory tests (Kuzmak and Sereda, 1957b). It was postulated that salt in solution with water would move through a sample only with the liquid phase. Thus by monitoring the salt content in the outflow from a sample initially containing the salty solution, the relative amounts of vapor and liquid reaching the outflow end could be discerned. Conclusions as reported were that, under a temperature gradient movement through the test cell was almost exclusively in the vapor phase, and under a head gradient movement was almost exclusively in the liquid phase. These conclusions should be questioned, however, for there remains the problem of accounting for osmotic pressure effects on the flow system caused by the calcium chloride content (0.0248 g-ml*1) of the liquid, and the relative effects on liquid and vapor movement caused by either the head or temperature gradient as defined by equations 63 and 74 of this report. One of the more perplexing problems associated with anisothermal liquid and vapor flow is centered on the disparity between measured vapor flow rates and rates calculated by the gas-diffusion equations. Decent research has shown that temperature gradients produced locally around grain contacts are much higher than the average gradient imposed across a block of grains. At low moisture contents liquid is concentrated in the vicinity of the grain contacts and the locally high thermal gradients are believed to be more nearly representative of the diffusion potential than are the average gradients. Woodside and Kuzmak (1958) illustrated the local temperature distribution by means of a model of a grain contact area and studied its effect on vapor flow in the vicinity of a pendular ring. Their model gave the temperature distribution in the solid-liquid-vapor system resulting only from heat conduction. Effects on heat distribution from condensation and distillation were not included. This model study showed that temperature gradients in the vapor phase locally in the pore may be as much as 6 times the average gradient across the system. However, the model also predicts a flow rate in excess of that postulated from the diffusion concept even for dry media, and this has been largely discounted by experiment (Hanks, 1958). (See also fig. 24.) Philip and deVries (1957) have developed a theoretical model for estimating anisothermal fluid flow through moist materials and obtained good agreement with observed values. Their theoretical development, however, is still very much in need of substantiation. The calculated values of fluid flow are dependent on the curves of heat conductivity and heat diffusivity versus moisture content, and these data have been takenE48 FLUID MOVEMENT IN EARTH MATERIALS from steady-state heat flow measurements. In past investigations, the heat-flow characteristics of moist porous media have been measured by techniques adapted directly from studies on the physics of solids. The measurement and analytical techniques applied do not account for the recognized mobility of the fluids due to temperature gradients. The excellent work of Jack-son and Kirkham (1958) seems to emphasize that water distribution in the pore space is strongly affected by temperature gradients and that this mobility greatly affects the observed heat characteristics. They applied sinusoidal temperature fluctuations of different amplitudes and periods across soil at various moisture contents. Apparent diffusivities were plotted as a function of applied period, and, this curve extrapolated to zero period, was supposed to yield estimates of the real thermal diffusivity, relatively unaffected by fluid mobility. Effects of the applied periods on apparent diffusivity were quite pronounced. The final curves of thermal diffusivity versus moisture content obtained indicate that the thermal characteristics of moist sand observed by steady state techniques may be in error by more than a factor of 10. In another investigation (Kuzmak and Sereda, 1957a) rates of vapor flow across an open space between saturated materials was observed to be nearly three times the rate calculated from the concept of simple vapor diffusion. The authors maintain that this discrepancy might have resulted from inaccurate temperature measurements. One wonders if more attention might profitably be directed toward estimating the effects on diffusion caused by the boundary conditions in the test apparatus. Uncertainties regarding the mechanism of vapor movement in moist porous media find a parallel in the liquid phase. Generally it has been assumed that movement of the adsorbed film on the solid surface is negligi- ble where liquid occurrence is in the pendular stage. Evidence to the contrary is mounting. Barrer (1954) has made a theoretical study of the mobility of the adsorbed stage and has described liquid motion in this stage as a type of surface diffusion. Carman and Kaal (1951) studied the amount of surface diffusion as a function of the amount of fluid adsorbed, and Carman (1952) found that the relation between flow of the adsorbed film and the flow of fluid held by capillarity qualitatively follows the characteristics of the adsorption isotherm. Philip (1957b, p. 347) discussed mobility of the adsorbed stage briefly. Flood and others (1952) derived an empirical function relating “surface conductivity” to the amount of fluid adsorbed. The latter involved study of the behavior of several types of fluids adsorbed by porous carbon rods. One of their water adsorption-desorption and conductivity curves are shown in figures 31A and 31B. Flow was produced by maintaining a constant vapor-pressure difference across the carbon under isothermal conditions at the sample ends. Presumably, movement of adsorbed liquid is most noticeable at liquid contents below the minimum 6( of the funicular stage because continuity of the thick liquid sections held by capillarity is broken. It is generally accepted that hysteresis effects do not appear at lower df values in the pendular and adsorbed stages. The range of 6j- values over which liquid exists in these stages corresponds with those parts of the adsorption-desorption curves which coincide in figure 31A. According to the simple vapor diffusion theory, if all flow is in the vapor form and the adsorbed film is immobile, the flow rate for this region of 0r as plotted in figure 31B should show a marked decline as d-c increases. This trend is expected because the area available to vapor diffusion decreases and tortuosity increases (Fatt, 1960). However, figure 31B WATER CONTENT PER UNIT DRY WEIGHT FLUID FLOW PER UNIT GRADIENT, IN CUBIC CENTIMETERS PER MINUTE A B Figure 31.—Water content and conductivity of carbon plug showing hysteresis. (After Flood and others, 1952.)MULTIPHASE FLUIDS IN POROUS MEDIA E49 shows the flow rate increasing through all of this region. It therefore seems plausible to attribute much of this movement to mobility in the adsorbed stage. Additional studies are needed relating solid surface area and other characteristics to the slope of the conductivity curve at low vapor pressures so as to provide a more positive basis for showing the significance of the movement of moisture in films. In coarse materials such as the sands, the surface area per unit volume is probably so small that film movement is negligible in comparison with vapor movement. But in fine materials such as silts and clays, which in many places control the vertical movement of fluid in the unsaturated zone, movement of adsorbed films of moisture may account for most of the fluid movement. SUMMARY Evidently a comprehensive statement of the laws of fluid motion in the unsaturated zone has not yet been developed. A few simplified flow models, however, have been devised and tested and are valid for application to restricted ranges of fluid concentration. They may be applied directly for treating the unsaturated zone as a dynamic part of the hydrologic cycle. This is a basic need, especially in studies of water occurrence where man’s use of water markedly rearranges distribution from the natural state. The validity of the simplified models of flow is dependent on whether the assumptions used in developing the model satisfy the environmental conditions. Few suitable comparisons between the simplified flow models and environmental conditions are available in the literature. The need for more comprehensive definitions of fluid motion cannot be evaluated successfully until such comparisons are fully realized. It therefore seems that for the present, the principal effort in unsaturated zone studies should be devoted to such comparisons. The application of the theory of multiphase flow will require the development of computing and modeling techniques for solving the equations of flow, and methods for making field and laboratory measurements of the physical and hydrologic characteristics of earth materials and their environment. Flow problems of the unsaturated zone are not particularly unique, but relatively little has been done to solve equations of the type describing unsaturated flow. Instruments are available for measuring liquid content, density, water salinity, temperature, and other factors in porous media but, in general, only a few have been adapted to meet the demands posed by inaccessability of the unsaturated zone or accuracy requirements of the analytical techniques. One of the most important deterrents to the full use of the published results of laboratory and field studies of multiphase flow is related to incomplete statement of the test conditions or medium characteristics in published reports. This, at any rate, was the writer’s experience in studying the published test data that was considered potentially useful for elucidating the mechanisms of multiphase flow. One investigator may present a very detailed account of flow rates through “a porous medium,” showing clearly the effects of flow caused by a variety of well-controlled and defined energy gradients. But his information cannot be compared, on an adequate base of reference, with another work in which the porous medium was completely described, and only qualitative flow relations were given in summary form even though carefully controlled and complete experiments had been made. It is recognized that publication space is limited, but nonetheless if basic experimental work is to be effective toward solving the problems of multiphase flow, complete published descriptions of experimental test conditions and results are imperative. 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I: World Oil, Production Sec., v. 146, p. 121-128. ------1958b, The generalized Kozeny-Carman equation: Its application to problems of multiphase flow in porous media; Pt. 2: World Oil, Production Sec., v. 146, p. 210-228. Youngs, E. G., 1957, Moisture profiles during vertical infiltration : Soil Sci., v. 84, p. 283-290. ------1958a, Redistribution of moisture in porous materials after infiltration: 1: Soil Sci., v. 86,117-125. ------1958b, Redistribution of moisture in porous materials after infiltration: 2: Soil Sci., v. 86, p. 202-207. o>Extending Darcy’s Concept of Ground-Water Motion GEOLOGICAL SURVEY PROFESSIONAL Prepared in cooperation with the U.S. Atomic Energy Commission, Division of Isotope Development and Division of Reactor Development f ir ‘ ' • TS DEf'ARTMtiiT; JAN 6 1065 LIBRARY mwwPTf of PAPER 4 11 -FExtending Darcy’s Concept of Ground-Water Motion By H. E. SKIBITZKE FLUID MOVEMENT IN EARTH MATERIALS GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-F Prepared in cooperation with the U.S. Atomic Energy Commission, Division of Isotope Development and Division of Reactor Development UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director The U.S. Geological Survey Library has cataloged this publication as follows : Skibitzke, Herbert Ernest, 1921- Extending Darcy’s concept of ground-water motion. Washington, U.S. Govt. Print. Off., 1964. v, 6 p. illus., diagrs. 30 cm. (U.S. Geological Survey. Professional paper 411-F) Fluid movement In earth materials. Prepared in cooperation with the U.S. Atomic Energy Commission, Division of Isotope Development and Division of Reactor Development. Bibliography: p. 6. 1. Water, Underground. 2. Dispersion. I. U.S. Atomic Energy Commission. II. Title. (Series) For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price 20 centsPREFACE The ground-water hydrologist seeks to identify and use the physical parameters controlling ground-water flow. Scientists are seldom required to define a more heterogeneous or complex region than that involved in an underground flow system. In this respect, the hydrologist’s problem is different from that of, say, the electrical engineer or the physicist, who normally analyzes regions constructed with homogeneous characteristics and definable geometries. The heterogeneity of the flow region involved in ground-water studies is so great that it is difficult to find physical characteristics adequate to describe the region. In effect, then, the region must be described by more complex parameters than are generally used in other scientific fields. Two such parameters are particularly significant. The first is the factor of heterogeneity, which somehow must be identified. But the nature of the heterogeneous region can hardly be described through reference to the individual geometric discontinuities. Such a description would require an endless compendium of individual descriptions, a device so obviously impractical that it renders the region not amenable to description by measurement of any of the characteristics visible or accessible from the surface of the region. The second region embraces the flow relations at any given point and these relations often reflect an angular discrepancy between potential gradient and direction of flow. Thus, tensor description is required because any scalar factor is inadequate. The measurement of the two factors—heterogeneity and flow relations—has been one of the principal difficulties in accomplishing an engineering analysis of ground-water flow. Laboratory experiments and study have illuminated some of the basic principles involved in the measurement of these factors. The effects of heterogeneity have been studied by hydraulic experiments on two types of laboratory models, one type constructed of heterogeneous materials and the other of homogeneous materials. A comparison of the experimental results for flow through the two model types shows that for heterogeneous porous material a dissolved constituent is dispersed widely; conversely, for material that is simply homogeneous a dissolved constituent is not dispersed in the moving fluid. This paper discusses results of experiments on the former type material. The paper was prepared and should be read as a companion to USGS Professional Paper 411-G, by Akio Ogata, in which experiments on homogeneous materials are discussed. Because the effects of heterogeneity cannot be expressed quantitatively, experimental procedures must entail the use of a more suitable concept. The observation of the flow is generally accomplished by following a tracer element in its progress downgradient. However, as tracer elements flow downstream they are dispersed to a degree depending on the conditions met in the porous medium—assuming the flow and fluid properties are unchanged. If it were possible to delineate a given series of streamlines, as they would be defined by Darcy’s law, the need for describing the flow might be circumvented provided some measure of heterogeneity could be established. Generally, however, establishment of such measure is impossible because of physical limitations in experimental techniques. Because it is physically impossible to trace out any given streamline, the description of the dispersive mechanism may be of importance. Dispersion in porous media has been studied to some extent, and a general description of the spreading of the flowlines can be predicted for isotropic media. The paper by Ogata describes, for such a medium, an experiment on dispersion transverse to the direction of flow. Good correlation was obtained between analytical and experimental results. The distribution of concentration about the centerline thread of flow was almost symmetrical. The tracer elements illustrate the history of events as the fluid progresses downstream. This seems to imply that if the dispersive effects can be separated the flow system can be com- mIV PREFACE pletely defined within the region. Although the tracer elements show the historical progress of each fluid element and even if it were possible to describe geometrically any flowline, the process of integrating this description throughout a region is extremely formidable—more likely impossible; hence, there is the need to establish a new outlook in describing the flow system, as discussed in some detail in this paper. Further research is required to determine the utility of a macroscopic concept of the flow system to replace the prevailing microscopic concept. The use of the microscopic analysis has led to the conclusion that the Navier-Stokes equation is valid; however, the integration of the point function throughout the region becomes impossible. In other words, each streamline needs to be described as a separate entity. Further, the microscopic concept has led to a statistical analysis of flow employing an ordered porous body and a random flow system, which is in direct contradiction to the conclusion that the Navier-Stokes equation is valid. In essence, then, the microscopic analysis has produced a method of describing a point, whereas the description of a large-scale aquifer is the goal of the hydrologist. The implications of studies already accomplished are that further experiments should be tried in larger regions and that a few small-scale supporting laboratory experiments should be added. Because in many of the tests made to date only dissolved constituents were observed, some different and more suitable type of tracer must be used to reveal the components of the dispersion process. Radiotracers have proven especially well suited for this purpose in most laboratory experiments. The analysis of dispersion processes will necessarily require the use of some tracer techniques, simply because dispersion is the product of the motion of the dissolved component. In a macroscopic sense, of the order involved in ground-water field studies, the dispersion of contaminant does not depend upon any measurable hydraulic factor. Therefore, part of the experimental technique must involve addition of dissolved components to the fluid. Because the dispersion is also dependent upon the location and geometry of the injection site, no naturally occurring tracer element will suffice. Furthermore, once a tracer is decided upon, it will, in general, need to be a radiotracer because the problem of remote sensing is involved in ground-water studies. It seems necessary, then, that techniques be developed for extending the laboratory use of radiotracers into the field.CONTENTS Page Preface__________________________________________________ hi Abstract_______________________________________________ FI The permeability concept__________________________________ 1 The ground-water flow regime_______________________________ 3 Page Dispersion phenomena.................................... F5 Limitations of the calculus_______________________________ 6 References________________________________________________ 6 ILLUSTRATIONS Figures 1-3. Diagrams of flow regime in region— Page 1. Of constant permeability_____________________________________________________________________ F3 2. Where scalar permeability varies continuously________________________ 3. Where nonscalar permeability is continuously distributed_____________ 4. Diagram showing theoretical flowline refraction at a boundary_______________ 5. Laboratory photograph of flowline refraction________________________________ 6. Diagram of the flow regime in region where tensor permeability is discontinuous. 7. Diagram showing tracer dispersion in heterogeneous aquifer__________________ 8. Diagram showing the mean line of dispersion in heterogeneous aquifer-------- v CO CO ^ ^FLUID MOVEMENT IN EARTH MATERIALS EXTENDING DARCY’S CONCEPT OF GROUND-WATER MOTION By H. E. Skibitzke ABSTRACT The tensor nature of permeability and its effect in a given ground-water flow regime have been acknowledged by various investigators. The effect on the spread of a given dissolved substance can be qualitatively discussed on the basis of the tensor characteristic of the permeability coefficient. The correlation between the dispersion coefficient and the tensor characteristic is one possible means of defining the flow regime within a given aquifer. Because of the large-scale changes in magnitude and direction of the tensor components of permeability, the concept of a mean line of dispersion may produce a more significant statement on the characteristics of flow than does the present concept of hydraulic potential and the streamline. THE PERMEABILITY CONCEPT The analysis of a region harboring ground-water flow requires the description of the permeability function in a quantitative sense. In studying most ground-water flow problems, the permeability of the porous earth material through which the flow occurs has been assumed a constant. At best the flow analysis may sometimes acknowledge the permeability as a scalar, variable in space. This practice is illustrated by studies where permeabilities are determined by discharging-well tests of an aquifer and then are plotted on a regional map and used to develop contours of permeability. This simple scalar relationship is one wherein the permeability is constant at a given point—that is, it does not change in the different directions. The permeability can thus be defined by a single number at that point. If another point in the region is considered, a new number may represent the permeability at that point, but again the permeability is the same in any direction. Thus the scalar coefficient—permeability—of the Darcy equation may vary in space, but it will be constant for all directions at any point. Now suppose that the permeability differs in various directions at any point; that is, if the permeability were measured in the east-west direction, it might differ from that measured in the north-south direction, which in turn might differ from the vertical direction. The manner in which the permeability may vary can no longer be described mathematically by a simple coefficient such as the scalar permeability previously used. An adequate description will necessarily lead to a tensor concept for the Dermeability relationship. The permeability concept can be extended further by considering spatial relationships as well as directional characteristics. Assume first that the permeability can be described by a function that would account for variations in direction—that is, by a tensor. It may then be assumed that permeability can vary in any direction specified by the Cartesian coordinate system and, further, that these variations change as the position of measurement is moved laterally or vertically within the aquifer. Measurements made at various points within an aquifer will then define a function varying with position. If these changes are continuous —that is, vary smoothly in a mathematical sense—the change of permeability can be described as a continuous function. This limitation of continuity is unnecessary, however, and its removal leads to the final enlargement of the permeability concept into something that is a discontinuous function. Imagine that the permeability of aquifer material is describable by a tensor relationship and that the tensor then can properly account for permeability variations in direction and magnitude. The tensor may vary not only in space but also in a discontinuous manner. Previously the permeability function was described as varying in space—both in the lateral and the vertical directions—but in a smooth manner, so that a continuous function accomplished its description. For discontinuous variation, permeability measurements at points adjacent to one another would reveal abrupt changes in the permeability tensor. This discontinuity condition leads to the most generalized concept of permeability necessary for describing the porous material constituting an aquifer. Thus an adequate FIF2 FLUID MOVEMENT IN EARTH MATERIALS concept of the permeability function leads to the definition of a variable that changes with position in space as well as with direction at a point, the variations being discontinuous or continuous or both. Often in aquifer materials the variation in permeability is continuous over a small region, at the boundary of which it abruptly changes in a discontinuous manner. An analysis of the permeability factor requires determination of the implications involved in its variations. First, Darcy’s equation must be revised. The Darcy equation has the general form Q=P grad h. (1) To develop a more elaborate description than is generally used, the scalar permeability P must be replaced in equation 1 by the tensor permeability Pt, which represents a coefficient that would vary in space continuously or discontinuously—and generally both. Some comprehension of the tensor relationship may be extracted from a definition given by Margenau and Murphy (1959, p. 161), a part of which is quoted as follows: In many physical problems the notion of a vector is too re- —> stricted. For example, in an isotropic medium, stress S and —> strain X are related by the vector equation S=A: X (2) —> —> X and S having the same direction. If the medium is not —> —> isotropic, S and X are not in general in the same direction; it is then necessary to replace the scalar k by a more general —) mathematical construct capable, when acting on the vector X, of changing its direction as well as its magnitude. Such a construct is a “tensor.” A similar generalization has to be made in the vector equations P=€ E (3) —> —> where P and E represent electric polarization and field strength, and I=n H (4) —> —> where I and H represent intensity of magnetization and field strength; for anisotropic media, the susceptibilities t and /i must be replaced by tensors. Equally complex conditions confront the analysis of ground-water motion. If the permeability varies with direction, there is an angular difference between the hydraulic gradient and flow direction. Because of this difference, the vector descriptions require application of a tensor concept to the permeability coefficient. In the simplest situation the permeability function is a scalar, which in turn is the most degenerate form of tensor. More generally the tensor is a function that may be used to describe mathematically the relations between the hydraulic gradient and the direction and rate of ground-water flow, regardless of the directional characteristics of the rock permeability. This tensor can vary both continuously and discontinuously in space. The tensor relationship changes the continuity conditions for which flow equations can be specified. The Laplace equation, written with a simple scalar permeability varying as a continuous function, describes the continuity condition of the liquid in a moving state. In this form the Laplace equation appears different from the form usually given, and one might argue that it is no longer properly termed Laplace’s equation. As it is commonly written, Laplace’s equation has the form V2A=0 (5) where h is the hydraulic head. When the scalar permeability P varies in space, the equation must necessarily contain P=P(x,y,z) as a variable. Thus equation 5 would be rewritten V-Pvh=0. (6) Upon completing the indicated mathematical operations, equation 6 becomes Pv2h+vh-vP=0. (7) That equation 7 is reducible to equation 5 is readily shown. For the simplest concept of permeability— that is, constant or unvarying with respect to space and —> direction—the permeability gradient VP is obviously zero, and the second term in equation 7 is therefore zero. The remaining term can be divided by the constant permeability P, and the simple Laplacian form of equation 5 is thus left. However, the simple Laplacian form is not applicable to the ground-water conditions commonly found in nature. The continuity equation for ground-water motion becomes even more complex when the Laplace equation is written in terms of a tensor relationship. A formal derivation and a discussion of its significance are not included here. These will be covered in a separate chapter of this Professional Paper. Sufficient for the present is an indication of the complexity in the form of Laplace’s differential equation when written to show, say, the distribution of velocity potentials V in terms of a completely general coordinate system and the fundamental metric tensor gt]. In tensor notation -) the Laplacian relation V2U=0 has the form gliVi.i=o (8) or J_ ^(v/g.9°dU/dz<) =Q ^ (See James and James, 1949, p. 206.)EXTENDING DARCEY'S CONCEPT OF GROUND-WATER MOTION F3 Finding solutions for a relationship similar to equation 9 is much more difficult than for the simple Laplace equation. Actually, it would be virtually impossible to find any significant solutions for the tensor relationship. However, some flow features leading up to and involving the relationship might be briefly examined. THE GROUND-WATER FLOW REGIME It is helpful first to have clearly in mind, as a reference base or point of departure, the ground-water flow regime as it is commonly visualized and depicted— permeability in every place and direction being constant for the porous medium. These simple conditions are presented as case 1 in figure 1. Observe that the flowlines are coincident with the lines denoting hydraulic gradient, and the flow rate is labeled as inversely proportional to the distance between the equipotential contours. Now consider the matter of scalar variations in permeability in space—that is, the simple situation in which the permeability is always a scalar but varies continuously in space. The pertinent conditions are described as case 2, shown in figure 2; the changing permeability produces a flow pattern distorted in that the flow rate is not proportional to the gradient vector grad h. The flow is in general parallel to the gradient vector, but the rate is not proportional to the hydraulic gradients indicated by the spacing of contours denoting equipotential surfaces. This set of conditions, case 2, represents the simplest conceivable departure from the common idealized conditions illustrated in case 1 (fig. 1). A more general departure from the idealizations of case 1 is represented by the conditions shown as case 3 in figure 3. The permeability not only varies continu- Figgre 1.—Case 1: V«/i=0. Diagram of the flow regime in a region in which permeability P is constant throughout. Lines along gradient are streamlines— that is, total flow Q is constant between streamlines. Flow rate is inversely proportional to the distance between contours. Arrows Indicate direction of flow. Figure 2.—Case 2: V.p grad A=0. Diagram of the flow regime in a region in which permeability P is a scalar varying continuously in space. Flow rate is not inversely proportional to the distance between contours. Arrows indicate direction of flow. ously with space but also varies with direction, and it has the characteristics of a tensor. Thus the ground water flows in directions differing from those indicated by the hydraulic gradients. A corollary case deserving brief mention involves the discontinuous condition resulting from the abrupt change between the simple permeabilities of two adjacent parts of an aquifer. The permeability P is a scalar quantity in each part but abruptly changes at the interface between the parts. The flow regime in such an environment has been analyzed and discussed by a number of technical writers but is particularly well covered in the works of Muskat (1946) and Hubbert (1940). The sketches drawn in figure 4 are from Hubbert’s discussion; they show how flowlines are abruptly refracted at the permeability discontinuity. Clearly illustrating the same refraction phenomena is the picture shown in figure 5, taken in the Phoenix research laboratory of the U.S. Geological Survey during Figure 3.—Case 3: V- (Pt grad h)= 0. Diagram of the flow regime in a region in which permeability Pt is a nonscalar continuously distributed in space. Flow direction is not parallel to the gradient. Flow rate is not inversely proportional to the distance between contours. Arrows indicate direction of flow. 741-755 0-64-2F4 FLUID MOVEMENT IN EARTH MATERIALS Figure 4— Diagram showing theoretical flowline refraction across layers of coarse and fine sand. After Hubbert (1940). a hydraulic experiment on an artificial-sandstone model. Uniform flow is from left to right and the refraction of the black-appearing dye stream is clearly evident as it passes through the tapered band of higher permeability sandstone in the center of the model. To be examined as a final departure from the idealizations of case 1 are the completely general conditions presented as case 4 in figure 6. The permeability is now a tensor function and can vary with direction; it also can vary discontinuously in space. The ground-water flow is not only in directions differing from those indicated by the hydraulic gradients but appears to have several directions at each point. The net effect upon the flow system of combining all the individual conditions and factors discussed in the preceding paragraphs is most difficult to illustrate. A possible approach might be to construct in model form a heterogeneous medium having all these factors inherent in the construction. The author has concluded, after many such experiments (see, for example, Skibitzke and Robinson, 1963), that the most general- Figure 5.—Laboratory photograph showing refraction of flowlines in artificial-sandstone model. Figure 6.—Case 4: V-(Pt grad h)= 0. Diagram of the flow regime in a region in which permeability Pt is a tensor function (either scalar or directional) and is discontinuous in space. Flow is dispersed, appearing to have several directions at a point. Flow is not inversely proportional to the distance between contours. Arrows indicate direction of flow. ized state of the heterogeneous medium would lead to flowlines and related features as shown in figures 6 and 7. The details in figure 7 are explained in the following paragraphs. Observe, however, that if the path of a water particle is traced, starting from any point, the flowline begins by briefly indicating laminar motion, according to the classical thinking represented by Darcy’s law. As distance from the starting point increases and the flow pattern becomes more widely dispersed, however, the flowline quickly departs from the classical concepts. The manner in which the flow disperses downstream from a reference point is described by Ogata (1964) in a companion chapter to the present report. Presented and discussed therein are the experimental data collected in a series of experiments designed specifically to show the effects of dispersion. The important conclusion drawn by Ogata is that the spread of a tracer or contaminant migrating from a deposit in a heterogeneous aquifer is completely the result of the heterogeneity, insofar as can be observed.EXTENDING DARCEY’S CONCEPT OF GROUND-WATER MOTION F5 Figure 7.—Diagram showing dispersion of a tracer injected into a heterogeneous aquifer. Permeability P has scalar characteristics. DISPERSION PHENOMENA The preceding conclusion leads to an important concept in describing the nature of flow in a ground-water reservoir. Two significant regions in the flow field can be visualized. In the first region the flow environment is so heterogeneous that the travel paths of a tracer or contaminant follow streamlines winding tortuously through the porous medium. Therefore, within this region, if a tracer-concentration profile were drawn for a section normal to the general direction of flow, it would be highly irregular in form (fig. 7). Further down the path of motion the effects of the intertwining of flowlines tend to blur out the apparent discontinuous features of tracer concentration. The region in which this blurring occurs can be regarded as merely a transition region (fig. 7). Ultimately, the flow reaches the second region to be visualized, which is at some distance downstream from the tracer or contaminant origin. In this second region (fig. 7), the porous medium appears to be more amenable to description by dispersive processes in a mathematical sense. If a coordinate system moving with the liquid is postulated, the mathematical description would be the Fickian diffusion equation, having the form (10) In equation 10, however, the magnitude of D is many powers of ten greater than it is in the ordinary process of molecular diffusion. Until region 2 is reached (fig. 7) and “complete” (in the mathematical sense) dispersion occurs, the discontinuities of flow virtually defy description. Their identification would require a complete knowledge of the permeability of the aquifer material in microscopic detail; in other words, the nature of about each cubic foot of the aquifer would have to be studied. Knowledge of the hydraulic gradient would also be required in the same detail. Obviously these data, required primarily in region 1 (fig. 7), could never be obtained. To summarize, therefore, it must be concluded that around the tracer or contaminant source and proceeding downstream from it there exists first a region of indeterminate flow where the tracer is mixed discon-tinuously. This kind of mixing is illustrated schematically by the upper inset cross section of figure 7. This first region grades gradually through a transition region, where the flow and dispersion discontinuities begin to smooth out, into a second region where the flow can be characterized as simple dispersed flow. The effects are again schematically illustrated in the lower inset cross section of figure 7. However, it may take many times the length of any normal aquifer to arrive at the second region. Conversely, there may well be some situations in which the second region is reached in a matter of a few feet. Even in the simplest and most uniform of aquifers there will still be some intertwining of flowlines within the pore spaces themselves. The flow-regime features sketched in figure 7 typify the nature of flow and dispersion in a heterogeneous aquifer and show that a more generalized description of the flow pattern is needed. If the flow-regime idealizations based upon Darcy’s law are compared with the features shown in figure 7, it may be concluded that the commonly used formula Q—P grad h in reality describes only a generalized average direction of flow. The formula does not describe actual stream paths of tracer or contaminant elements, but describes instead what might be termed a mean path of dispersion. As a beginning step, therefore, in a more realistic analysis of flow through a heterogeneous aquifer, it would appear most desirable to promote this concept, or new definition, that theF6 FLUID MOVEMENT IN EARTH MATERIALS central streamline, as it is identified classically by use of Darcy’s equation, is more nearly the central line about which the dispersed tracer travels. That this is only a beginning step in a more realistic analysis can be inferred from the features shown in figure 6, characterizing the flow regime when the permeability is a tensor function. In this completely general situation the mean line of dispersion will depart from the lines representing the conventionally determined hydraulic gradient. Although the manner of departure is yet to be determined, it is schematically represented in figure 8. Figure 8—Diagram showing the mean line of dispersion in a heterogeneous aquifer when permeability Pt has directional characteristics. LIMITATIONS OF THE CALCULUS One additional point merits brief discussion, and it concerns the nature of infinitesimals when working with geologic bodies the size of regional aquifers. In mathematical analysis the infinitesimal generally must be considered as ever reducible to yet a smaller one without losing any of its inherent physical characteristics. If, for example, observation wells penetrate a regional aquifer on a ^-mile-square grid pattern, it must be assumed in any mathematical analysis of data obtained from such wells that all aquifer elements smaller than the K-mile-square segments will be identical in nature. If these “infinitesimals” remained the same in character as the elemental areas approached zero, a fairly representative mathematical picture of the aquifer would be obtained. If on the other hand the successively smaller infinitesimal elements—considered within the ^-mile-square segments—changed in their physical characteristics as the areas tended toward zero, the mathematical operations involving the calculus would not be valid. Commonly the heterogeneity of porous media is such that the permeability function will contain changes in its nature through all magnitudes of areal size that may be chosen throughout the regional aquifer. It is not unreasonable to expect, therefore, that flow patterns different from those described by simple solutions of Laplace’s equation will be observed. In field practice the smallest infinitesimals that can be studied and described are those whose areas are delimited by the spacing of wells in the observation-well network. Thus any analytical conclusions dependent upon mathematical integration and differentiation processes will be invalid. For these reasons, employing the concept of the mean line of dispersion seems simpler and more promising than attempting to find the direct flow path. In heterogeneous acquifers the latter approach obviously requires a mode of description potentially involving an infinite number of paths. REFERENCES Hubbert, M. K., 1940, The theory of ground-water motion: Jour. Geology, v. 48, no. 8, pt. 1, p. 785-944. James, G., and James, R. C., 1949, Mathematics dictionary: New York, D. Van Nostrand Co., 432 p. Margenau, H., and Murphy, G. M., 1959, The mathematics of physics and chemistry: 2d ed., New York, D. Van Nor-strand Co., 604 p. Muskat, M., 1946, The flow of homogeneous fluids through porous media: Ann Arbor, Mich., J. W. Edwards, 763 p. Ogata, A., 1964, The spread of a dye stream in an isotropic granular medium: U.S. Geol. Survey Prof. Paper 411-G (in press). Skibitzke, H. E., and Robinson, G. M., 1963, Dispersion in ground water flowing through heterogeneous materials: U.S. Geol. Survey Prof. Paper 386-B, 3 p., 1 pi., 2 figs. U. S. GOVERNMENT PRINTING OFFICE : 1964 O - 741-755The Spread of a Dye Stream in an Isotropic Granular Medium GEOLOGICAL SURVEY PROFESSIONAL PAPE^/411-G Prepared in cooperation with the U.S. Atomic Energy Commission, Division oj Isotope Development and Division of Reactor Development *362' JAN 6 1365 LIBRARY of crimiwiA U.S.SJ&The Spread of a Dye Stream in an Isotropic Granular Medium By AKIO OGATA FLUID MOVEMENT IN EARTH MATERIALS GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-G Prepared in cooperation with the U.S. Atomic Energy Commission, Division of Isotope Development and Division of Reactor Development UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director The U.S. Geological Survey Library catalog cards for this publication appear after page Gil. For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 - Price 20 cents (paper cover)PREFACE The ground-water hydrologist seeks to identify and use the physical parameters controlling ground-water flow. Scientists are seldom required to define a more heterogeneous or complex region than that involved in an underground flow system. In this respect, the hydrologist’s problem is different from that of, say, the electrical engineer or the physicist, who normally analyzes regions constructed with homogeneous characteristics and definable geometries. The heterogeneity of the flow region involved in ground-water studies is so great that it is difficult to find physical characteristics adequate to describe the region. In effect, then, the region must be described by more complex parameters than are generally used in other scientific fields. Two such parameters are particularly significant. The first is the factor of heterogeneity, which somehow must be identified. But the nature of the heterogeneous region can hardly be described through reference to the individual geometric discontinuities. Such a description would require an endless compendium of individual descriptions, a device so obviously impractical that it renders the region not amenable to description by measurement of any of the characteristics visible or accessible from the surface of the region. The second factor embraces the flow relationships at any given point and these relations often reflect an angular discrepancy between potential gradient and direction of flow. Thus tensor description is required because any scalar factor is inadequate. The measurement of the two factors—’heterogeneity and flow relations—-has been one of the principal difficulties in accomplishing an engineering analysis of ground-water flow. Laboratory experiments and study have illuminated some of the basic principles involved in the measurement of these factors. The effects of heterogeneity have been studied by hydraulic experiments on two types of laboratory models, one type constructed of heterogeneous materials and the other of homogeneous materials. A comparison of the experimental results for flow through the two model types shows that for heterogeneous porous material a dissolved constituent is dispersed widely; conversely, for material that is simply homogeneous a dissolved constituent is not dispersed in the moving fluid. This paper discusses results of experiments on the latter type material. The paper was prepared and should be read as a companion to USGS Professional Paper 411-F, by H. E. Skibitzke, in which experiments on heterogeneous materials are discussed. Because the effects of heterogeneity cannot be expressed quantitatively, experimental procedures must entail the use of a more suitable concept. The observation of the flow is generally accomplished by following a tracer element in its progress downgradient. However, as tracer elements flow downstream they are dispersed to a degree depending on the conditions met in the porous medium—assuming the flow and fluid properties are unchanged. If it were possible to delineate a given series of streamlines, as they would be defined by Darcy’s law, the need for describing the flow might be circumvented provided some measure of heterogeneity could be established. Generally, however, establishment of such measure is impossible because of physical limitations in experimental techniques. Because it is physically impossible to trace out any given streamline, the description of the dispersive mechanism may be of importance. Dispersion in porous media has been studied to some extent, and a general description of the spreading of the flowlines can be predicted for isotropic media. This paper describes, for such a medium, an experiment on dispersion transverse to the direction of flow. Good correlation was obtained between analytical and experimental results. The distribution of concentration about the centerline thread of flow was almost symmetrical. The tracer elements illustrate the history of events as the fluid progresses downstream. This seems to imply that if the dispersive effects can be separated the flow system can be mIV PREFACE completely defined within the region. Although the tracer elements show the historical progress of each fluid element and even if it were possible to describe geometrically any flowline, the process of integrating this description throughout a region is extremely formidable—-more likely impossible; hence, there is the need to establish a new outlook in describing the flow system, as discussed in some detail in Professional Paper 411-F. Further research is required to determine the utility of a macroscopic concept of the flow system to replace the prevailing microscopic concept. The use of the microscopic analysis has led to the conclusion that the Navier-Stokes equation is valid; however, the integration of the point function throughout the region becomes impossible. In other words, each streamline needs to be described as a separate entity. Further, the microscopic concept has led to a statistical analysis of flow employing an ordered porous body and a random-flow system, which is in direct contradiction to the conclusion that the Navier-Stokes equation is valid. In essence, then, the microscopic analysis has produced a method of describing a point, whereas the description of a large-scale aquifer is the goal of the hydrologist. The implications of studies already accomplished are that further experiments should be tried in larger regions and that a few small-scale supporting laboratory experiments should be added. Because in many of the tests made to date only dissolved constituents were observed, some different and more suitable type of tracer must be used to reveal the components of the dispersion process. Radiotracers have proven especially well suited for this purpose in most laboratory experiments. The analysis of dispersion processes will necessarily require the use of some tracer techniques, simply because dispersion is the product of the motion of the dissolved component. In a macroscopic sense, of the order involved in ground-water field studies, the dispersion of contaminant does not depend upon any measurable hydraulic factor. Therefore, part of the experimental technique must involve addition of dissolved components to the fluid. Because the dispersion is also dependent upon the location and geometry of the injection site, no naturally occurring tracer element will suffice. Furthermore, once a tracer is decided upon, it will, in general, need to be a radiotracer because the problem of remote sensing is involved in ground-water studies. It seems necessary, then, that techniques be developed for extending the laboratory use of radiotracers into the field. It is important at this point to note also that the use of tracers to determine hydraulic parameters is at present a useless endeavour. This has been discussed by Skibitzke (1958) in an earlier paper.CONTENTS Page Preface_________________________________________________ hi Abstract________________________________________________ G1 Introduction_____________________________________________ 1 Adsorption________________________________________________ 3 Mathematical analysis of transverse dispersion with radial symmetry_________________________________________ 4 Page Experimental method_______________________________________ G7 Analysis of experimental data_____________________________ 8 Results____________________________________________________ 9 References________________________________________________ 11 ILLUSTRATIONS Figure 1. Graph of the isotherm characteristic of multilayer adsorption________________________________________________ G4 2. Graph showing comparison between equations 11 and 15________________________________________________________ 8 3. Graph showing theoretical relationship______________________________________________________________________ 9 4-15. Graphs showing radial concentration distribution: 4. Block 11................................................-.........—...........................— 9 5. Block 10_______________________________________________________________________________________________ 9 6. Block 9_______________________________________________________________________________________________ 10 7. Block 8...........................................................--------------------------------- 10 8. Block 7.........................................--------------------------------------------------- 10 9. Block 6_____________________________________________________________________________________________ 10 10. Block 5.....................................................—...................................... 10 11. Block 4__________________________________—--------------------------------------------------------- 10 12. Block 3.....................................----------------------------------_____________________ 11 13. Block 2............................................................................................ 11 14. Block 1............................................................................................ 11 15. Block 14___________________________________________________________________________________________ 11 TABLE Table 1. Computation of ua2/ixDr from actual ratio Cr_a/Cr_0 for various distances s> 11.0 cm............ G9 vFLUID MOVEMENT IN EARTH MATERIALS THE SPREAD OF A DYE STREAM IN AN ISOTROPIC GRANULAR MEDIUM By Akio Ogata ABSTRACT The mechanics of dispersion transverse to the direction of fluid flow was studied Loth analytically and experimentally. A discussion of the overall mechanism is presented and the analytical expressions obtained are compared with experimental data. Correlation between experimental and theoretical results for the spread of a tracer is fair. However, from the data available an initial approximation is made for the dispersion constant in the direction transverse to the direction of flow. INTRODUCTION When a region saturated by a given fluid (say, fresh water) is invaded by another fluid (say, a tracer or salt water), a process of mixing occurs if the two fluids are miscible. Of special import is the mixing of salt and fresh water along coastal regions (Cooper, 1959). Additional occurrences of economic importance are the injection of low-level fission products into ground-water systems and the secondary recovery methods utilized by oil companies. For recovery the fluids can be immiscible and a zone of mixing is not created. In the other two processes, because the carrier is usually water, there is creation and movement of a zone called “the zone of diffusion.” This is a region in which there exists a gradation from the concentration of the original fluid to that of the invading fluid. Recent investigations have been concerned about the zone of diffusion because the final disposition of the contaminant in the transient state is dependent on the creation and movement of this zone. The purposes of the investigations reported herein are to describe the mechanisms involved and to predict quantitatively the final disposition of the contaminant with the passage of time. The dilution of the contaminant may be attributed to three major processes—mechanical or hydrodynamic dispersion, adsorption, and dispersion due to heterogeneous aquifer conditions. Other minor mechanisms, such as ionic diffusion or discontinuous pore channels, have a tendency to cause spreading. However, the magnitude of spread due to these minor mechanisms is small in comparison to that due to the other processes except when the flow rate of the ground water is of the same order as the mean flux due to diffusion. The rate of movement and the path traversed by this zone of dispersion are dependent on the existing flow condition and the method of injection. Thus, assuming that other things are equal—such as density of fluid and external pressures—and that Darcy’s law is valid, the major mechanisms of dilution are classified as follows: 1. Mechanical or hydrodynamic dispersion—due to microscopic velocity variation controlled by viscosity and pore-channel diameters; principle mechanism in homogeneous and isotropic media. 2. Adsorption—due to mass transfer from liquid to solid phase controlled by chemical properties of contaminant and solid phase. 3. Heterogeneity—due to regional or macroscopic changes in the properties of the granular media. Most papers on dispersion in a porous medium published to date have dealt exclusively with “mechanical dispersion” in the direction of flow, which is of primary importance in the analysis of reactor columns. However, in ground-water systems, adsorption and variation in the permeability of the media may be the major causes of the apparent dispersion. Adsorption or ion-exchange processes may cause significant dilution in a short distance owing to the presence of a large surface area available for the transfer of mass. Heterogeneity on the other hand relates to the finite changes in the properties of the aquifer. The occurrences of the discontinuities cause additional spreading that is due to refraction of the streamlines. There is a different type of problem in regions near the seacoast. Although the problems involved are not the same, the mechanism of mixing is similar. Studies have been initiated (Kohout, 1960; Henry, 1959) to analyze the mixing in response to the tidal oscillation in the ocean and the natural velocity with which the ground water flows through the system. These effects, however, have not been separated and the importance of each process cannot yet be evaluated. Investigation is as yet in its initial stage, and a great deal of field data G1G2 FLUID MOVEMENT IN EARTH MATERIALS is being accumulated, especially in Florida. However, the distribution of velocity within the two fluids and the effect of the change in densities along their interface remain to be determined either by in-place measurements or by model studies. The discussion that follows is confined to dispersion of miscible fluids without density variations. It should be useful in determining the final disposition of radioactive waste in the event of spillage or leakage. The accumulated knowledge of the dispersion process in the ground-water system is directly applicable provided the geologic conditions can be described. In essence, the final disposition of any contaminant is dependent on the path traversed. In ordinary circumstances the liquid waste first flows through the unsaturated zone overlying the ground-water system. Most of the contaminants are adsorbed in this region owing to the existing exchange potential of the soil. Because the adsorption depends on the types of materials present, all the contaminants may be fixed to the surface of the soil particles. A few of the fission products, however, may enter the saturated zone in which further dilution may occur. In the saturated zone there are two possible means of reducing the concentration of the contaminated fluid. These are dilution by spreading, caused by mechanical dispersion, and dilution by actual removal of the contaminant from the liquid phase by adsorption. The fate of the contaminant in this region is thus dependent on the rate of flow, the geologic conditions, and the types of materials present. Most of the reported studies of the dispersion phenomena are based on a gross analysis in which is assumed a diffusion process described by a relationship similar to Fick’s law. The magnitude of the constant of proportionality, or the dispersion coefficient, is determined experimentally. These investigations of longitudinal dispersion (for example, de Josselin de Jong, 1958; Orlob and Radhakrishna, 1958) indicate that the magnitude of the dispersion coefficient D is proportional to the magnitude of the velocity u—that is, D=mu. The constant m is dependent on the size and distribution of aquifer material and also on the degree of saturation; thus it varies according to the individual situations considered. This relationship has been obtained from studies of unconsolidated-sand models composed of fairly uniform material. Thus, for application of the results to the existing field condition, or to consolidated materials, correction factors must be introduced. In regions where materials are extremely heterogeneous, it may be possible to increase the constant m to describe the dispersion or spread of contaminant. The mechanism that controls the lateral spread has not yet been extensively correlated experimentally, de Josselin de Jong (1958) and Saffman (1960) concluded from the study of statistical models that the lateral spread is also dependent on the average longitudinal velocity. However, in extremely slow flow, ionic diffusion becomes important (Beran, 1957). Laboratory study has been reported by Simpson (1962), and some laboratory investigations of lateral dispersion have been reported in chemical engineering journals. The investigations reported in these journals deal with flow rates much higher than those found in ground-water reservoirs. In most studies, good correlation is obtained between the lateral-dispersion coefficient and the velocity in the longitudinal direction. For both lateral and longitudinal dispersion in isotropic media, the spread is due primarily to microscopic controls or variations. But, in nonhomogeneous or consolidated materials, macroscopic variations may have a marked effect. When these variations constitute changes in geologic conditions, the term “heterogeneity” applies. Heterogeneous conditions exert a large influence on the breaking up and the spread of a stream of contaminant progressing through the saturated zone. There is little doubt that these macroscopic discontinuities exist in many aquifer systems and depend on environmental conditions at the time the aquifer materials were deposited. Of course, the objective is still to obtain quantitative measures of the spread of a contaminant. Because the prevailing environmental conditions at the time of deposition are not known, it would be virtually impossible to describe quantitatively the individual discontinuities or changes in a large region. But reason suggests that, as for dispersion in homogeneous media, all macroscopic variations may be treated separately by using an averaging process. By this means it may be possible to describe the spread or mixing due to heterogeneity, assuming that the constant m is dependent on the size and distribution of isolated discontinuities. The mechanism of adsorption or ion exchange, which is of primary importance in predicting the fate of the contaminant, is usually not mentioned in studies of dispersion. Adsorption cannot be classified as a dispersion phenomenon; it is primarily a surface chemical process due to the existence of an exchange potential between the dissolved contaminant and the surface of the media through which the fluid flows. This exchange capacity is primarily dependent on the solid material and the type of contaminant present (Kaufman and Orlob, 1956). The dilution of the liquid phase is realized because of the transfer of dissolved solid from the liquid state to the solid state. Although the contaminant may not be permanently fixed, the rate of exchange from liquid to solid isSPREAD OF A DYE STREAM IN AN ISOTROPIC GRANULAR MEDIUM G3 usually large enough that considerable dilution is realized. In this way adsorption may be the most effective method of reducing contamination hazards. Adsorption is discussed further in the next section. Analysis of dispersion with linear adsorption isotherm superimposed on diffusion has been attempted. However, major simplification is usually necessary to carry out analytical details. In laboratory experiments utilizing continuously injected organic dye tracers, a considerable travel delay due to adsorption or ion exchange is usually noted, but ordinarily no quantitative description is attempted. Investigations or Hou-gan and Marshall (1947) are an example of this attempt; however, the fluid was gas and the simplification utilized cannot be applied to liquid flows. The lack of available data on transverse dispersion and effects of heterogeneity prompted the Phoenix research office of the U.S. Geological Survey to construct models for laboratory analysis of dispersion. Owing to time limitations and the urgency of other projects, laboratory-model studies of heterogeneity were intended to furnish qualitative rather than quantitative description of the process (Skibitzke and Robinson, 1963). Both mathematical and laboratory studies were made to analyze transverse diffusion. A value of the transverse-diffusion coefficient has been obtained; however, inasmuch as only one test run was made, this value must be substantiated further. The laboratory experiment and the analysis of the experimental block was carried out by H. T. Chapman and J. M. Cahill of the Phoenix research office. Their diligence and ingenuity in overcoming many difficult technical problems resulted in the successful conclusion of the experimental phase. ADSORPTION Although adsorption is not a dispersion process, a short discussion is presented because the experiment reported requires knowledge of concentration relationships between the liquid and the solid phases. Generally, if the concentration of the tracer used in the experiment is low, a linear relationship can be assumed provided an equilibrium condition is reached. This relationship follows from the assumption that the Langmuir exchange isotherm is valid. All the isotherms reported are semiempirical in nature. Although there are many published reports of transfer from solutions to solid surfaces, the exchange isotherms utilized are valid only for the systems reported, inasmuch as the theoretical expressions are obtained only after a series of simplifications to construct a mathematical model. First, consider the exchange isotherms for equilibrium conditions. The isotherms presented are based on 740-419 O—64--2 the assumption that only a monomolecular film is formed around the solid particles. This fact, in essence, tends to limit the use of the isotherms presented unless extremely low concentrations are used in the experiments. Adsorption of particles from gaseous mixtures onto a solid surface depends primarily on temperature and pressure (Adamson, 1960). In transfer of mass from a solution, concentration of the solution replaces the pressure. The isotherms proposed by Langmuir and Freundlich are those most widely used, because of their simplicity. Freundlich’s isotherm, at first thought to be an empirical relationship, is written F=aC1,n where F= amount of adsorbate per unit amount of solid, a and »=constants, and C= concentration of liquid phase. Langmuir’s isotherm, on the other hand, can be written where F=abCI(\+bC) a=a constant related to the specific surface area of the solid, and 6=a constant related to the heat of adsorption. Roughly speaking, the constants appearing in the Freundlich isotherm may be interpreted to represent the same specific area and heat of adsorption. For both the Fruendlich and Langmuir isotherms, a curve of F versus C shows a convexity towards the C axis. However, there is one major difference: the Langmuir isotherm reaches a saturation value of a as C increases and bC becomes much greater than unity, whereas the Freundlich isotherm does not reach an upper limit. Furthermore, when the concentration is small, or the Langmuir isotherm gives a straight-line relationship between the liquid concentration and the solid concentration, though the Freundlich isotherm does not show this effect and the straight-line approximation is obtained only when n—>1. In this circumstance, the isotherm is a straight line regardless of concentration. Investigations of adsorption on solids from solutions indicate that porous adsorbents usually obey Langmuir’s equation because the pore volume limits the amount adsorbed (Condon and Odishaw, 1958). However, this evidence does not prove that Langmuir’s expression is superior to Freundlich’s expression. OnG4 FLUID MOVEMENT IN EARTH MATERIALS the contrary, many laboratory data show better correlation with the Freundlich isotherm, but when concentrations are low Langmuir’s isotherm seems better suited for a large number of adsorption systems. Later researchers have reported a sigmoid isotherm (fig. 1), which is indicative of a multilayered formation for transfer from the vapor phase to the solid phase (Adamson, 1960). Although very little is known as to whether multilayers actually form, there seems to be a general acceptance of this possibility. The theoretical analysis involves the use of Langmuir’s isotherm for n layers. The expression was developed by Brunauer, Emmett, and Teyerson and is generally known as the BET equation (Condon and Odishaw (1958) or Adamson (I960)). Figure 1. Graph of the isotherm characteristic of multilayer adsorption. In utilizing the equilibrium isotherms discussed previously, it is assumed that there is instantaneous transfer of mass. The assumption of instantaneous adsorption does simplify to a great extent the analysis of the combined effects of adsorption and diffusion. However, for a fluid flowing through a stationary porous bed, this assumption may not be valid unless the time elapsed is large. In nonequilibrium conditions the expression that must be considered is tr*<*>« assuming that rate of transfer is a function of the liquid concentration only. Generally the function jf(C) is not described but an assumed relationship, usually linear, is superposed on the diffusion equation. Two types of nonlinear expressions for j{(J) were discussed by de Vault (1943).’ Generally, j(C) is assumed to be linear; a typical relationship (for example, that given by Merriam and others (1952)) has the form bF frn+u >r+v >r+w 3T bt bx by bz (2) Ji=mass flux due to dispersion only in the direction i, and u, v, w=constant average velocity components in the x, y, and z directions, respectively. In anisotropic media where the dispersion coefficient D depends on the direction, D takes on the propertiesSPREAD OF A DYE STREAM IN AN ISOTROPIC GRANULAR MEDIUM G5 of a tensor. Hence, the components of dispersion at any given point are described by r-n bC+D ^4-D dC -Jx—Du by+Diad2 t n dC+n dC4-n dC 'J*-D21 ^+Dn by+Dn dz r-n *c+n dC+D ■*7‘-£>31 te+D}2 dy+°33 J (3) In other words, the mass flux in any given direction is not only dependent on the concentration gradient in the same direction but also on the gradient in the other two directions. Owing to mathematical difficulties and the lack of experimental data available for anisotropic conditions, it is assumed that the principle direction of dispersion is along the axis of the coordinate system chosen. Equation 2 has been expressed in terms of Cartesian coordinates; however, the equation may be written for any system of coordinates. Using appropriate transformations x=r cos 0 y=r sin 6 2 = 2, equation 2 in cylindrical coordinates is DC= 13 / n dOYJ_ d_ Dt r dr \ r dr J r2 dd (4) The differential equation of diffusion may also be written in spherical coordinates, but it is not as useful as the expression involving Cartesian coordinates for problems of flow through granular media. Generally in flow through porous media the velocity component can be considered to be unidirectional and the principal axes of diffusion are assumed parallel and transverse to the velocity. Thus for unidirectional flow, equation 2 is reduced to DtV2C=^+u i=x, y, and 2. (5) tux v?t n 2D~4IT\ or the moving coordinate z—x—ut, equation 5 can be reduced to the standard diffusion equation DtV2C= dC dt (6) In the cylindirical coordinate system, where radial symmetry exists, the principal axes of dispersion are assumed to be in the directions parallel and transverse to the direction of flow. Thus the dispersion equation can be written dC dC_n &C, Drd_ dt U dx 1 dz2 r dr (7) Equation 7 can be reduced to the diffusion equation by-employing the moving-coordinate system, or i)=x—ut; hence dC=n A d/ &C\ dt z dti2 r dr \ dr) (8) Equation 6 in its one-dimensional form has been solved for various boundary conditions in the investigations of both heat-flow and diffusion processes. Analytical and experimental correlation studies have been carried out by many investigators, as previously stated. These investigations have indicated that the dispersion coefficient is directly proportional to the average velocity of the fluid both transverse and parallel to the direction of flow, but there is no indication that the coefficients are of equal magnitude in both directions. Industrial interest has prompted extensive study of longitudinal dispersion, in both the laminar range (Reynolds No. <1) and the so-called turbulent range (Reynolds No. >1), but few studies have been made either analytically or experimentally of dispersion transverse to the direction of flow. The remainder of this section will therefore be confined to the development of a mathematical description of transverse diffusion. Analytical difficulties are inherent in equation 7, even for Dr=Dx, unless boundary conditions are simple. Thus, to simplify the analysis to a large degree, consider separately the two processes of longitudinal and transverse dispersion. Except in the frontal zone where longitudinal dispersion predominates, the system can be described approximately by n 1 d ( dC\ dC_dC Trdr\dr) Udx dt (9) Von Rosenburg (1956) indicated that the length of the frontal zone, over which the concentration varies about 80 percent, is proportional to the velocity and inversely proportional to the diffusion coefficient. Because the diffusion coefficient is proportional to velocity once the front is established, it remains nearly constant as the stream progresses through the media. Further, if no chemical reaction or adsorption of the contaminant takes place, there is no “holdup” time and the front progresses through the media at a rate equal to the average velocity. The foregoing observations are based on one-dimensional problems; thus, the simplifications may not be wholly justified physically. /G6 FLUID MOVEMENT IN EARTH MATERIALS To simplify the mathematical model it is assumed that the contribution of the front to the overall lateral spread is small, especially in the zone x > ut. In other words, in the mathematical model it is assumed that a piston flow takes place; that is, a sharp front traverses the media with a velocity u, and diffusion occurs normal to the direction of flow. For this simplified model it is apparent that equation 9 holds. By letting z=x—ut and r=t, it is possible to simplify equation 9 further: that is, nia / dc\ dc r dr \ dr) dt Although equation 10 can be computed, the radial-dispersion coefficient cannot be easily determined from the resulting expression. However, the integral can be integrated for the specific situation of either r=0 or r=a. Consider first the distribution of concentration along r=0. Because J0(0) = 1, equation 10 can be written CJ -Da2z/u 77=ffl I e Ji(ad)da. t/o Jo Watson (1948, p. 394) gave the value of the above integral as Physically, this equation means that an observer follows a slug of contaminant at a rate equal to the average velocity u as the contaminant progresses through the system. For the unsteady condition this approximation requires that the boundary conditions be <=0, c=ca, f=0, <7=0, r>a r=0, g?-a Or If the source emits at a constant rate, this problem is the same as diffusion from a semi-infinite cylinder of radius a into the surrounding media. In the steady situation, dC/dt= 0; thus t can be replaced by xju in the boundary condition specified above. The solution of the above system has been obtained and discussed extensively (Goldstein, 1932). This solution written in equivalent form is 7T=a f ’ e~Dc,ltJi(aa)J0(ar)da ^0 Jo S vme-tIm(Z) - m = l S i e-Vm(& 771—0 V J (10) where ri=a/r, ^—-ra/2Dt, and Jx and Ja are Bessel functions of the first kind of order 1 and 0, respectively. The integral solution was written in its alternate form to facilitate computation of the general solution. The author showed this solution in graphical form in chapter B of this series (Ogata, 1961). As previously indicated, the solution of the steady-state system is obtained by simply replacing t by — Because the u experiment that was performed is assumed to have reached steady state, the solution to equation 10 is hereafter written in its steady-state form. g=i-e*P [(-jb0) (ii) If the concentration distribution is experimentally determined along r=0, the radial component can easily be computed using equation 11. However, the radius of a finite source of tracer may be such that within the limits of the experimental model no noticeable change in concentration will occur along r=0. For this situation, it would be preferable to determine distribution along a known value of r, say a. When r—a, equation 10 can be written (j r*—Da2t YT=a I e Ji(aa)J0(ao)da. M) Jo Goldstein (1932) expressed the above integral in terms of the modified Bessel function of the first kind, that is, <12> Inasmuch as the function e~xI0 (x) is well tabulated, the term D can be readily computed from any given experimental data. Here again the value of D can be obtained by a curve-fitting process. Further reflection on the problem of radial dispersion led to the hypothesis that the physical assumptions may not be realistic and that this problem may best be described as diffusion from a moving-disk source in a semi-infinite medium. A second analysis was thus attempted based on the assumption that, owing to radial symmetry, the coefficient of dispersion depended only on the x and r directions. A summary of the results obtained is presented. It should be noted that the complete solution for the unsteady-state system was not computed because of the complexity of the resulting integral. But, for comparison purposes the steady-state system was computed for various values of the ratio DzIDr. In approximating the process of transverse diffusion,SPREAD OF A DYE STREAM IN AN ISOTROPIC GRANULAR MEDIUM G7 it was assumed that the term Dx was negligible in equation 9. It was also assumed that the conditions of the experiment justified the use of the steady-state system. These conditions cannot be checked experimentally although the theoretical solution can be correlated with experimental evidence. Because radial symmetry exists, equation 7 would describe the process. For the instantaneous point source (Carslaw and Jaeger, 1959, p. 260), it is possible to write the solution of the problem immediately. The expression c_u'a' r r [x-«'(*-r)n dT Co 2 Jo 6XP \ mt-r) J ,-D(l-r)X2 t/o (Xp)t/ l(Xffi (13) is obtained by integrating the expression for the continuous point source from zero to a, where a is the radius of the disk source. The terms used in equation 13 are u’=UtJD/Dx p=r^D!Dr X—x^D/Dx a'=a^D/DT Inasmuch as the strength of the source is u'C0, the expression for steady-state conditions is C Co 2 u'-yjiz a' J0(\p)Ji(\a')d\ 4^ \ xw + u2 )f 16Z>2£2J ? u' J."Mp [“(* r “p hs o which is identical to equation 11. Thus it can be stated that for any given ratio DxIDr the approximate solution is valid for large distances from the source. a2 D The value of the parameter -j ~ at which the ap- X Ur proximation is correct to the nearest thousandth is about 10-4. Figure 2 shows a comparison between equations 11 and 15. For r=a the solution based on the disk source gives <7 1/ 1 (ux\C1 /ux I,4a2 Dx 1 or $ \1-'exp w/J* 6xp v1+y d, r\ VI X‘ Dr hY (16) Again, as ^ ^ t becomes small, Hence I, . 4as Dx 1 (2a\2Dx V1+75,'“1+)h) d,t- C IT __ f-a2u\ T ( a2u Y1 Co 2L exp \ 2D,x)Io \2Z?rx/J which is identical to the approximate solution, or equation 12, for r=a. Above analysis of the disk source thus indicates that if the experimental column is sufficiently long the radial-diffusion coefficient can be computed by equation 11 or 12. In addition, by utilizing equation 15 or 16, the longitudinal-dispersion coefficient can be readily computed provided reliable data are obtained near the source. Experimental Method W2 V)2 2)1/2] if was possible toSPREAD OF A DYE STREAM IN AN ISOTROPIC GRANULAR MEDIUM G9 II 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 a2u/4x Dr Figure 3—Graph showing theoretical relationship between Cr-«/CV-o and mV ixDr. compute values of D for each given section of the experimental model. The value of D obtained by this method ranged from 3.9 X10~4 to 6.0 X10-4 cm2 per min, as shown in table 1. The arithmetic average was computed to be 5.0 X10-4 cm2 per min. By utilizing the average value of the dispersion coefficient, an average value of C0 was computed to be approximately 140 counts per minute. Table 1.—Computation of ua*/4xDr from actual ratio Cr_a/Cr-ofor various distances xf>11.0 cm [Average: ?=10.3 cm; Dr=4.97X10-4 cm2 per min] X CV—o Cr—a Cr-JCr-0 Theoretical m2/ixDr q=m2l4Dr DrX10« 11.1 83.0 48.5 0.584 0.81 9.0 5.68 14.8 72.0 43.0 .597 .76 11.2 4.56 18.1 59.2 39.5 .667 .53 9.6 5.32 21.3 54.0 34.5 .639 .61 13.0 3.93 24.3 46.2 34.5 .747 .35 8.5 6.01 27.6 38.5 28.0 .727 .39 10.8 4.73 30.6 39.3 30.3 .771 .31 9.5 5.38 33.3 37.5 30.0 .800 .26 8.7 5.87 36.7 25.0 19.0 .760 .33 12.1 4.22 RESULTS By utilizing the values CQ= 140 cpm and D—5X10-4 cm2 per min, theoretical and experimental values were compared as shown in figures 4 through 15. Note that for most of the sections the correlation is good. However, in some sections all experimental points fall consistantly below the theoretical curve, perhaps primarily because of instrumental errors, though the exact causes are unknown. The approximate analysis of the experimental data does seem to indicate that diffusion coefficients transverse to the direction of flow are of the same order of magnitude as ionic diffusion. A spread to approximately three times the width of the source at a dis- Figure 4— Graph showing concentration distribution along radius r, block 11. x=7.5 cm, a=0.95 cm. O* experimental. tance of 43 cm seems indicated. Because only one test run was made, these figures should be taken as preliminary in nature. Future experiments should take into account the nature of the adsorption isotherms. Generally, it is good practice to ascertain the range of concentration for which the linear part of the Langmuir isotherm can be used. Analysis of the data does show that transverse diffusion is small; thus, if future experiments are desired, the described method can be utilized provided the gross transfer of mass between the liquid and the solid phases is determined for the concentration and for the tracer used. Figure 5.—Graph showing concentration distribution along radius r, block 10. £=11.1 cm, a=0.95 cm. O* experimental.G10 FLUID MOVEMENT IN EARTH MATERIALS Figure 6—Graph showing concentration distribution along radius r, block 9. 2=14.8 cm, a=0.95 cm. Q, experimental. Figure 9—Graph showing concentration distribution along radius r, block 6. 2=24.3 cm, a=0.95 cm. Q» experimental. 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 RADIAL DISTANCE (r/a) Figure 7.—Graph showing concentration distribution along radius r, block 8. 2=18.1 cm, a =0.95 cm. O, experimental. Figure 10____Graph showing concentration distribution along radius r, block 5. 2=27.6 cm, a=0.95 cm. O, experimental. RADIAL DISTANCE (r/a) Figure 8.—Graph showing concentration distribution along radius r, block 7 2=21.3 cm, a=0.95 cm. Q, experimental. „ concentration distribution along radius r, block 4. Figure 11.—Graph showing & 30 6 cm, a=0.95 cm. Q, experimental.SPREAD OF A DYE STREAM IN AN ISOTROPIC GRANULAR MEDIUM Gil Figure 12.—Graph showing concentration distribution along radius r, block 3. 2=33.3 cm, a=0.95 cm. O, experimental. Figure 13.—Graph showing concentration distribution along radius r, block 2. 2=36.7 cm, a=0.95 cm. O. experimental. Figure 14.—Graph showing concentration distribution along radius r, block 1. 2=39.8 cm, a=0.95 cm. Q, experimental. RADIAL DISTANCE (r/a) Figure 15.—Graph showing concentration distribution along radius r, block 14. 2=43.3 cm, a=0.95 cm. Q» experimental. REFERENCES Adamson, A. W., 1960, Physical chemistry of surfaces: New York, Intersci. Publishers, 629 p. Beran, M. J., 1957, Dispersion of soluble matter in flow through granular media: Jour. Chem. Physics, v. 27, no. 1, p. 270-274. Carslaw, H. S., and Jaeger, J. C., 1959, Conduction of heat in solids: Oxford Univ. Press, 510 p. Condon, E. U., and Odishaw, Hugh, 1958, Handbook of physics: New York, McGraw Hill Book Co., 172 p. Cooper, H. H., Jr., 1959, A hypothesis concerning the dynamic balance of fresh water and salt water in a coastal aquifer: Jour. Geophys. Research, v. 64, no. 4, p. 461-467. de Josselin de Jong, G., 1958, Longitudinal and transverse diffusion in granular deposits: Am. Geophys. Union Trans., v. 39, no. 1, p. 67-74. De Vault, Don, 1943, Theory of chromatography: Am. Chem. Soc. Jour., v. 65, p. 532-540. Goldstein, S., 1932, Some two dimensional diffusion problems with circular symmetry: London Math. Soc. Proc., ser. 2, v. 34, p. 51-88. Henry, H. R., 1959, Salt intrusion in fresh-water aquifers: Jour. Geophys. Research, v. 64, no. 11, p. 1911-1920. Hougen, O. A., and Marshall, W. R., 1947, Adsorption from a fluid stream flowing through a stationary granular bed: Chem. Eng. Process, v. 43, no. 4, p. 197. Kaufman, W. J., and Orlob, G. T., 1956, An evaluation of ground-water traces: Am. Geophys. Union Trans., v. 37, no. 3, p. 297-306. Kohout, F. A., 1960, Cyclic flow of salt water in the Biscayne aquifer of southeastern Florida: Jour. Geophys. Research, v. 65, no. 7, p. 2133-2142. Merriam, C. N., Jr., Southworth, R. W., and Thomas, H. C., 1952, Ion exchange mechanism and isotherms for deep bed performance: Jour. Chem. Physics, v. 20,no. 12, p. 1842-1846. Ogata, A., 1961, Transverse diffusion in saturated isotropic media: U.S. Geol. Survey Prof. Paper 411-B, 8 p. Orlob, G. T., and Radhakrishna, J. N., 1958, The effects of entrapped gases on hydraulic characteristics of porous media: Am. Geophys. Union Trans., v. 39, p. 648-659. Saffman, P. G., 1960, Dispersion due to molecular diffusion and macroscopic mixing in flow through a network of capillaries: Jour. Fluid Mech., v. 7, no. 2, p. 194-208. Simpson, E. S., 1962, Transverse dispersion in liquid flow through porous media: U.S. Geol. Survey Prof. Paper 411-C, 30 p. Skibitzke, H. E., 1958, The use of radioactive tracers in hydro-logic field studies of ground-water motion: Internat. Assoc. Sci. Hydrology, General Assembly, Toronto, Canada, 1957, 1957, v. 2, p. 243-252. ------ 1964, Extending Darcy’s concept of ground-water motion: U.S. Geol. Survey Prof. Paper 411-F (in press). Skibitzke, H. E., Chapman, H. T., Robinson, G. M., and McCullough, R. A., 1961, Radio tracer techniques for the study of flow in saturated porous material: Internat. Jour. Appl. Radiation and Isotopes, v. 10, no. 1, p. 38-46. Skibitzke, H. E., and Robinson, G. M., 1963, Dispersion in ground water flowing through heterogeneous materials: U.S. Geol. Survey Prof. Paper 386-B, 3 p. Von Rosenburg, D. U., 1956, Mechanics of steady-state single phase displacements from porous media: Am. Inst. Chem. Eng. Jour., v. 2, p. 55. Watson, G. N., 1948, Treatise on the theory of Bessel functions: Cambridge Univ. Press, 804 p. oThe U.S. Geological Survey Library has cataloged this publication as follows: Ogata, Akio, 1927- The spread of a dye stream in an isotropic granular medium. Washington, U.S. Govt. Print. Off., 1964. v, 11 p. diagrs., table. 30 cm. (U.S. Geological Survey. Professional paper 411-G) Fluid movement in earth materials. Prepared in cooperation with the U.S. Atomic Energy Commission, Division of Isotope Development and Division of Reactor Development Bibliography: p. 11. (Continued on next card) Ogata, Akio, 1927- The spread of a dye stream in an isotropic granular medium. 1964. (Card 2) 1. Dispersion. 2. Water, Underground. 3. Fluids. I. U.S. Atomic Energy Commission. II. Title. (Series) Mathematics of Dispersion With Linear Adsorption Isotherm DOCUMENTS rcp8PTVf^n OCT 1 W64 I Lit, , 'H- UNIVERSITY Of CAUTCt^lA .S.S.D.Mathematics of Dispersion With Linear Adsorption Isotherm By AKIO OGATA FLUID MOVEMENT IN EARTH MATERIALS GEOLOGICAL SURVEY PROFESSIONAL PAPER 411-H A theoretical consideration of the effect of a reacting media on the dispersion of fluids in porous media UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1964UNITED STATES DEPARTMENT OF THE INTERIOR STEWART L. UDALL, Secretary GEOLOGICAL SURVEY Thomas B. Nolan, Director The U.S. Geological Survey Library has cataloged this publication as follows: Ogata, Akio 1927- Mathematics of dispersion with linear adsorption isotherm. Washington, U.S. Govt. Print. Off., 1964. iii, 9 p. diagrs, 30 cm. (U.S. Geological Survey. Professional paper 411-H) Fluid movement in earth materials. Bibliography: p. 9. 1. Dispersion. 2. Adsorption. I. Title. (Series) For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402CONTENTS Page Abstract__________________________________________________ HI Introduction_______________________________________________ 1 Adsorption isotherms_______________________________________ 1 Field equation--------------------------------------------- 2 Review of some previous analyses___________________________ 3 Mathematical analysis______________________________________ 4 Special situations_________________________________________ 7 Discussion_________________________________________________ 8 References_________________________________________________ 9 ILLUSTRATIONS Figure 1. 2. Concentration distribution for system with irreversible linear adsorption isotherm. Plot of /-function---------------------------------------------------------- Page H4 7 SYMBOLS Dt Dispersion coefficient in the direction i; i—x, y, z. C Concentration of the liquid phase. C0 Concentration of the liquid at a given point, say, 2=0. N Concentration of the solid phase. N0 Concentration of the solid phase at saturation. k, m Constants associated with adsorption isotherm. u Average velocity of the liquid. t Time variable. x Space variable. A Integration constant. I0(x) Modified Bessels function of the first kind of order zero. Jo(2) Bessels function of the first kind of order zero, erfc (2) Complementary error function. J(x,y) Goldsteins J-function. L(t) Laplace transform of function L~lcf>(p) Inverse Laplace transform. U(t—h) Unit step function. a a2/32Fm/4|2 or a2/32^2m/4. b a/3/2. a UxfiD. 0 2 t/D/u. t km(t—y2/i?'). 7 x2/2^lJ. £ Dummy variable. inf FLUID MOVEMENT IN EARTH MATERIALS MATHEMATICS OF DISPERSION WITH LINEAR ADSORPTION ISOTHERM By Akio Ogata ABSTRACT The simultaneous occurrence of dispersion and adsorption is noted frequently in laboratory models. In most investigation, especially that of the exchange process, the adsorption phase dominates; thus, prediction is based on a reduced system. The discussion presented in this paper, however, is based on the concept that the orders of magnitude of the adsorptive and the dispersive processes are the same. In addition, because of the meager knowledge of the macroscopic adsorption, the isotherm is assumed to be linear. This leads to a system which may be described by the equations &C_ dC dC.M dx1 U dz dt ™=k(C-mN). The paper deals entirely with the solution of the above equations for flow in a porous column. INTRODUCTION In the transport of two miscible fluids through isotropic porous media, the mixing and the dilution of any contained contaminant is dependent on two mechanisms, namely, mechanical dispersion and adsorption. The mechanical dispersion is defined to include all mechanisms that tend to cause spreading of the fluid stream which generally arises because of variations in fluid velocity throughout microscopic segments of the porous medium. Although variations in the magnitude and direction of the fluid velocity is noted throughout macroscopic segments of heterogeneous materials, the theory of dispersion cannot be extended to describe this system. The adsorption mechanism is defined as the removal or addition of the contained contaminant by the solid matrix through which the fluid flows. That is, the solid material may be thought to act as a mathematical source or sink depending on the concentration of the liquid flowing through the porous medium. The physicochemical aspects of the problem of adsorption are generally approached through thermodynamic principles based on analysis of the total free energy available at any given state. However, because of the complexities of these phenomena and the porous medium, such analysis has as yet produced no general conclusions on how to make use of the adsorption isotherms that characterize naturally occurring porous media. In other words, because of the extreme range of conditions met in aquifers, adsorption study of small segments of the flow system—microscopic analysis— has not produced results directly applicable to the entire flow regime. This indicates that for each soil complex or aquifer material, the adsorption characteristics of various candidate tracer materials must be predetermined so that the results can be given macroscopic application. To circumvent this difficulty, most quantitative analyses of laboratory experiments are generally based on an assumed adsorption isotherm. In most situations the tracer or contaminant concentration is sufficiently small that a linear isotherm can be assumed. This is the condition on which the following analysis is based. Adamson (1960), however, discussed various findings of recent years in which the linear assumption does not hold, especially where high concentrations are reached. Because in both laboratory tracer studies and actual disposal systems the concentrations are kept at a minimum, it is believed that the adoption of a linear isotherm is a reasonable approximation. Linear isotherms have been utilized by chemical engineers in the study of exchange processes. There is, of course, no proof that the same type of isotherm would be applicable for adsorption in a nonactivated bed. However, for want of better or more detailed information, the assumption of a nonlinear isotherm would only increase the mathematical complexity without furthering the systematic progress of the investigations. ADSORPTION ISOTHERMS A linear isotherm under isotropic exchange conditions may be generally classified as representing either a HIH2 FLUID MOVEMENT IN EARTH MATERIALS reversible or irreversible reaction. The simplest situation is the linear isotherm which represents an irreversible system, as described by the expression where N is the solid-phase concentration defined as the mass per unit volume of solid material, C is the liquid-phase concentration defined as the mass per unit volume of liquid, t is the time variable, and k is the proportionality constant. Equation 1 is essentially Langmuir’s isotherm for low concentrations and for equilibrium conditions. For complete discussion of Langmuir’s isotherm, the reader is referred to Adamson (1960). It is noted that equation 1 describes a system in which the solid acts as a mathematical sink. That is, the amount transferred—-or the rate of change of solid concentration with time—depends only on the liquid concentration. Physically this system cannot exist unless some method is available to remove all material transferred from the liquid to the solid phase. In terms of the overall mathematical treatment, however, inclusion of the terms represented in equation 1 does not add to the complexity of the problem. More generally, isotherms of this nature may be expressed as ^=F(C) (2) where F(C) is a function of liquid concentration only and must be determined for each system considered. In all situations, the medium tends to show a fixed capacity to adsorb a given substance which in turn is set by the level of the liquid concentration. That is, bN/bt becomes zero, indicating establishment of equilibrium between the liquid- and the solid-phase concentrations. An example of this type of isotherm is that used by Amundson (1948), mathematically represented by the equation \ ™=kC(N0-N). (3) In equation 3, a is a constant and N0 is the saturation capacity of a unit volume of granular material. It is noted that equation 3, like equation 1, shows a straight-line relation within the range of the porous medium’s capacity to adsorb. The saturation capacity N0 is dependent on the concentration of the liquid phase and thus is assumed to be a known constant in any given analytical study. Merriam and others (1952) and Hougen and Marshall (1947) indicated that the adsorption isotherm can be depicted by the expression ^ = k(C-mN) (4) where m is a constant. Note that unlike equation 3, equation 4 indicates a possible negative rate of change of solid concentration. This indicates that the adsorption process is reversible in the sense that when mN )> C the rate of change is negative, hence providing for a reverse effect or diminution of the solid-phase concentration. In other words, in a continuous-injection system, a continuous adsorption and desorption process is possible. The problems in which adsorption has been investigated extensively are those involving gas or liquid flow through activated porous media. There is general agreement that the dispersion under these conditions is so small that it may be neglected in describing the outflow-concentration profile. This assumption eliminates the second-order term in equation 5 and the mathematical treatment is thereby simplified to some degree. In a liquid-solid system, for the conditions imposed in waste-disposal problems, the relative magnitudes of the two processes are not known. It appears that, for the initial passage of the contaminant, the condition of bN/bt>^>D(b2C/bx2) would be valid. However, under conditions of repeated injection, the available free energy may be reduced to a great extent by the preceding injection; hence, the relative magnitudes may be of the same order. The immediate significance of this relates to the predicting of the usefulness of any given analysis, in that the historical events may figure importantly in evaluating the fate of the contaminant. In the following analysis the isotherm described by equation 4 is utilized. This assumption admittedly restricts the applicability of the resulting analytical solution, but our knowledge is too meager to specify these limitations. It is believed that, as more data become available, the applicability of the solution will become more clearly defined. FIELD EQUATION The actual description of mass transport through porous media is a matter of conjecture since there is no method available to measure all energy states of the system. The differential equation presented is thus based on a macroscopic concept so that the primary result gives a description of the liquid-concentration distribution at a given point in space for a given time. The mathematical model considered is that of flow through a column of semi-infinite extent. It is, thus, assumed that (а) porous medium is homogeneous and isotropic, (б) flow field is parallel to the x axis (say), andMATHEMATICS OF DISPERSION WITH LINEAR ADSORPTION ISOTHERM H3 (c) mass-transfer rate from liquid to solid is linear and can be expressed by equation 4—that is, bN/bt=k(C—mN). Assuming that the three components of mass transport are dispersion, convection, and adsorption, the law of conservation of mass may be stated as b2C bC=bC bN bx2 u dx bt bt (5) where the relationship between N and G is given by equation 4 and the transport due to dispersion is described by an equation similar to Fick’s law in a moving medium—that is, the total transport is The symbols appearing or inherent in equation 5 are defined as follows: D: dispersion coefficient, u: average velocity or interstitial velocity, k, m: constants relating the adsorption isotherm, G: concentration of the liquid phase, N: concentration of the solid phase, x: space variable, and t: time. The boundary conditions chosen are parallel to those of other investigators. Say, at x=0, the liquid concentration is maintained at some constant value, C=C0. This condition depicts dispersion phenomena from a plane source discharging into a semi-infinite medium. Mathematically, these boundary conditions can be stated: and C(x,0) = 0 for z>0 C(0, t) = Co for <>0 lim C(x, <) = 0for<>0 *->oo JV(x, 0) =0 for x>0. (6) Before developing an analysis of equation 5, subject to conditions 6, it will be of interest to review other solutions obtained in the analysis of exchange phenomena. REVIEW OF SOME PREVIOUS ANALYSES Amundson (1948), in describing the mass transport, assumed that dispersion is small; hence, equation 5 becomes bC bC l bN u bx*~ bt "a bt =0 where a is the volume of voids in the medium. The assumed adsorption isotherm is given by equation 3. The boundary conditions are as specified in 6, except for the condition N(x, 0)=Nt(x). For this system, the solution obtained is Co(t—x/u) exp | H r; -x/u) l exp|^J rvo-MU)] a 'o }+exp| H ru-x/u) n Co(rj)drjj H which is valid for t> x/u. Because dispersion is neglected, it is noted that N(x,t)=Nt{x) for t < x/u, which means that no transfer from liquid to solid occurs until mass is transported by convection to the point of interest. Because the solution is given in terms of functions of the coordinates, there is no immediate applicability unless these functions are defined. A special circumstance considered by Amundson is for the conditions bC/dt Iffi] “*• [(■+? ") jp] where The adsorption term intro- duces a “holdup time” and sets an upper limit to C/C0 depending on the value of k. For example, C/C0=% does not correspond to the point <=— as indicated by solution of the dispersion equation when jfc=0. In addition, the slope of the effluent curve with respect to time is smaller and an upper limit other than C/CQ=1 is reached because of the continual removal of substance from the system. The effect of the adsorption • • • "UX term L shown in figure 1 for the special case of ij= ^-=1. Crank (1956), on the other hand, gave a solution of equation 5 for u—0, using the isotherm depicted by equation 4. In his discussion he cited situations of diffusion in a plane sheet and in a cylinder. The solutions are, however, too complex to be discussed in this paper. Interested persons are referred to Crank’s book, in which a complete discussion of numerical computation and solution in graphical form is presented. MATHEMATICAL ANALYSIS Let us consider now the dispersion and adsorption phenomena utilizing equations 4 and 5, subject to conditions 6. There are various analytical methods that can be employed to obtain the solution of the differential equation. However, for this type of problem perhaps the simplest and most straightforward method is that employing the operational device called Laplace transforms. Because of its usefulness in analysis, various texts have been written on the theory and application of the Laplace transform. For discussion and application of complex variables and transform calculus, the reader is referred to McLachlan (1955), Churchill (1958), or Doetsch (1961). The Laplace transform C(x, p) is defined by the expression 7J(x, p)=J^ e~vtC{x,t)dt. Hence, substitution of the integral transform into equa-MATHEMATICS OF DISPERSION WITH LINEAR ADSORPTION ISOTHERM H5 tions 4 and 5 gives the expressions and n d27J d7j 79 t TT7 DW'-uto=pV+pW pN=k(JO—mN). (7) (8) It is noted that the transformation reduces the second-order partial differential equation into a second-order ordinary differential equation and reduce^, the first-order equation to an algebraic equation. In establishing equations 7 and 8, two of the boundary conditions, C(x, 0)=0 and N(x, 0)=0, have been utilized. The remaining two boundary conditions written in terms of G are t7=C„J e~v,dt=Co/p and lim (7=0. £->co The simultaneous equations 7 and 8 may be readily solved^ by solving for N in equation 8 and substituting for N in equation 7. This process gives a single equation, D d2(7 dV dx2 u dx~P O+iTBs)5- The resulting equation is a linear second-order ordinary differential equation with constant coefficient for which the solution can be readily obtained. The complete solution is 77=A exp [^±^V“2+4DP (1+^k)} Application of the boundary conditions specifies that since C\ °°, p) =0, only the negative part of the argument is retained, and since C(0, p) = C0/p, A=C0/p. Thus, the required solution of the transformed system is V=J exp \h [«-V“2+4flp (1+pT^)]}‘ O) The initial step in determining the inverse transform is to refer to any table of inverse Laplace transforms— for example, Erdelyi and others (1954). If the required V does not appear in the tables, the inversion theorem is utilized which states that C(x, Jr e*‘C(x, z)dz (10) where r specifies a given path in the complex plane z. The choice of T is dependent on the singular points of the function given in equation 9 (McLachlan, 1955). The function involved does not, however, lend itself for ready determination of the type of singularity involved in the computation. Hence, the following mathematical manipulation is a process of simplification so that 732-758—64----2 various theorems in conjunction with a table of inverse transforms may be utilized to obtain the desired results. From any table of integration, it is noted that 4= r"e-£s-*s/SJd£=e-2». Vtt Jo Hence, equation 9 can be written in terms of an infinite integral to eliminate the radical term—that is, f f [I+* (1+pT^)H * where a=ux/2D and j82=_4D/u2. Accordingly assuming C is continuous, substituting C into equation 10 leads to the expression C = 2e“ f Co AH Jo exp 2 iri L { i2 J Jrexp [zt-fPz f di where p is replaced by z. Let us consider only the complex part and denote this by the symbol F. Letting (zJrkm)=\, the complex part can be written F==2^r!1,Jrexp [x‘-“Wx-*w><1+*/x>4*’] x^k Rearranging the argument of the exponential term, the expression can be written F=exp kmt— (k—^ f __ l\ (. c?p\ , a^k2m~\ d\ JrexpLXV 4£V + 4£2X J X-im' Thus, the complex part of the function F is 1 f J\ A o?0\ , a*pkm-\ d\ 2ri JreXPLX(t“4F2) + “ii^rJ (11) The nature of equation 11 makes further simplification possible by the use of the shift theorem (McLachlan, 1955). The shift theorem specifies that if there exists a function defined as X(i—A)=^.J e‘«-»4>{z)dz, where X(() is continuous or piecewise continuous for t^>h, its transform is obtainable by the standard method. The value of the integral, however, is zero when t(t)= J e~vt{t)dt = (p). Replacing (t) in the above expression by (t—h) gives 5(p)=J e-r(‘-h)(t—h)dt = e~rt{t — h)dt, or = J“ er*4>(.t-h)U(t-h)dt where U(t—h) is a unit step function with properties 17=0 for t(t—h) U(t—h). The function (t) thus can be written $(<) = 7o(2aVO -\-kmekm‘J e~kmTIo(^a -ft) dr. Further, letting a=a/£, where a=a2/32&2wi/4, the shift theorem requires that the inverse of equation 11 be written 62/£2 where 62=a2/32/4. Since the solution gives zero values for t<^a2P2/4£2, or J£2, the particular solution of equations 4 and 5, subject to conditions 6, becomes Thus, if $(p) exists, the inverse transform of (t—h) can readily be obtained since L~1(p) = (t). In this specific problem, the function for which the inverse must be determined is p — km exP (— «2/82p/4f2) exp (a2/S2&2m/4|2). As previously stated, L~1(p)=(t); hence, the required computation can be written in operational form, *(0=Irli=jS»exp (oVp) where u2=a2/32Pm/4£2. Because the exact expression of

can then be directly obtained from the table of inverse Laplace transforms— that is, Lr1 exp (a2/p)/p=/0(2aVO and lrx 1/(p + 6) = exp (bt). Thus, the first term of 0(t) is I0(2a^/t). The second term, on the other hand, can be evaluated from the convolution or superposition theorem which states that if Lf(t) =f(p) and Lg(t) = /o(2V7p) and utilizing the definition of the 7-function, equation 13 can be written C 2e^ f ” T)d{. (14) ^0 -\TT j x 2 VIS Hence, the second term can be shown to be L~l J a2Ip ji eM-r)/0(2O Jr)dT where y—ky2/^ and r=km(t—y2/^2). To determine whether the boundary conditions are satisfied, it is noted that p—km p 7(0, y) = \\ J(r, 0) = e~TMATHEMATICS OF DISPERSION WITH LINEAR ADSORPTION ISOTHERM H7 and lim J(t, y) = 0; lim J{t, y) = 1. t—>oo y-$ co Substituting these known conditions, it can be seen that the boundary conditions for r=0 and f=0 are satisfied. Also when x—><», C/C0—>0 and, for large times—that is, t——J(t, y) = 1; hence, substituting leads to C/G0=l. Thus, all conditions are satisfied and by direct substitution, equations 4 and 5 can be shown to be satisfied. Equation 14, although highly complex, cannot be reduced any further; however, numerical integration can be carried out if k and m can be approximated. No attempt is made to calculate this function in this paper, although it would be of interest to note two special situations for k=0, m^O and k^O, m=0. First consider k=0, m^0. Equation 14 simply becomes Co 2e V* £! C exp 2 y/Dt 16D2, which is the solution for the circumstance of no adsorption—that is, for For m=0 and k^0, equation 14 reduces to uh? 16Z>* C 2ea C0~ V? 2 y/Dt U2X2 16D2 df, which is the solution for bN/bt=kC. Furthermore, since J* e-£2-»2/£22C dC dN dx2 u dx dt‘ (16) The mass transfer to the solid phase is again assumed to be expressed by equation 4 and the boundary conditions are those given in 6. The conditions under which these assumptions are valid are not known; however, these solutions may be useful in analyzing some laboratory models. For both circumstances expressed by equations 15 and 16, the method utilized is the Laplace transformation. Hence, applying the Laplace transform to equation 15, the subsidiary equations obtained are »%+PC+pN=°'I and _ f" pN=k(C—mN) J Equations 17 are now first-order differential equations with constant coefficient; the solution can be shown to be r=A pxn r * p(p + km+k)l *L u p+krn J From the initial condition C(x, 0) = C0 for x^>0, the constant A is again determined to be A—Co/p. Further, letting s=p-\-km and utilizing the inversion theorem, the solution can be expressed Q -kmt-tk-km) cTe Or, (J — kmt — (1—m) [‘K)+*?/■] km dz z — km Accordingly, utilizing the shift theorem, the function for which the inverse is to be determined is k2mx / j e “ / (s — km). The transformation is carried out in the same manner as that used in obtaining equation 14. The required solution is, thus, -\-km exp j~km ^ kmr^j dr j for t^>xlu. The above relation written in terms of the previously described (/-function is simply 7T=J(y, r) (18) Co ktJC / where V=~ and T=km It——J. Equation 18, in terms of a definite integral, can be written 5-,-""^) J. [V4” ('-;>] *• <19> It is noted that the function obtained above is the same as that derived by Hougen and Marshall with the replacement of f°r t. This function has been computed and is presented in graphical form as figure 2. Further, note that for t~~, equation 19 reduces to which indicates that measurable traces of solute will appear at t=x/u provided k is small. In most experimental setups, the holdup time is much greater; thus the expression is generally not useful in the analysis of experimental data. Let us consider now the second special circumstance in which dN/dt^S>dC/dt. Here the transformed or the subsidiary equations become and Solving for O' gives n«PC dC pN =k(C — mN). C=y exp (l-Vl + 4.Dpl;/w2(p+A;TO))J- The inverse obtained in the same manner described previously is + e~kmXI„ ^|V/3hxj dxjd£ where /3=Dka2/u2. In terms of the J-function, the preceding relation becomes Cte r »exp l_e_aVe]J(Vi r)d( (20) Co \TT J0 where y=|3/£2 and r=kmt. However, once again the function is not reducible and in its integral form can be written a O pa-kmt r a rail* ____ - 7-~ I exp (p-a*/*?)] e~Ha{2 Vfcm«X)dXd£. (21) Cq •y 7T J */0 DISCUSSION To effectively predict the fate of a tracer or contaminant as it progresses through porous material, two phenomena must be predicted beforehand. These consist of dispersion due to departures from the flowMATHEMATICS OF DISPERSION WITH LINEAR ADSORPTION ISOTHERM H9 predicted by use of Darcy’s law and the loss of material through adsorption by the porous medium. The first effect can generally be predicted for laboratory conditions, because flow is controlled to fit the theoretical conditions postulated. The adsorption effect, on the other hand, cannot be controlled even under conditions met in the laboratory. It can only be said to be dependent on the tracer and the soil constituents. Because of this unknown factor that inevitably appears in all types of studies involving tracers, the need for finding a near-perfect tracer is apparent. But discovery of this near-perfect tracer is yet to be made, although there appear to be a few elements that display only minor attraction to the silica sands commonly used in the laboratory. However, when natural soils are used as the porous medium, adsorption may play an important role in determining the distribution of tracer material at some point downstream. This then requires that the laboratory data be analyzed to estimate the rate at which the tracer or contaminant is transported from the liquid to the solid. Studies of exchange processes show that, in the low-liquid-concentration range, a good approximation is obtained by assuming a linear adsorption isotherm. On the basis of these findings the preceding mathematical analyses were developed utilizing the linear isotherm described by equation 4. The solutions obtained are extensions of previous expressions which were developed and used to represent ion-exchange phenomena in fixed columns. However, the final solutions, equations 14, 18, and 20, are such that no real purpose would be accomplished by numerical computation unless realistic values of k and m in equation 4 were available. It is expected that, as some data indicating the range of k and m become available, computation of equation 14, 18, or 20 will be attempted. From the experimental and analytical standpoint, equation 18 would be the most appropriate for obtaining an approximation of the constants k and m, provided extremely slow flow can be obtained in the experimental model. REFERENCES Adamson, A. W., 1960, Physical chemistry of surfaces: New York, Interscience Publishers, 629 p. Amundson, Neal R., 1948, A note on the mathematics of adsorption in beds: Jour. Physical and Colloidal Chem., v. 52, no. 10, p. 1153. Churchill, Ruel V., 1958, Operational mathematics: New York, McGraw-Hill Book Co., 331 p. Crank, J., 1956, The mathematics of diffusion: London, Oxford Univ. Press, 347 p. Doetsch, Gustav, 1961, Guide to the application of Laplace transforms: New York, D. Van Nostrand Co., 255 p. Erd61yi, Arthur, Magnus, Wilhelm, Oberhettinger, Fritz, and Tricomi, F. G., 1954, Table of integral transforms, vol. I: New York, McGraw-Hill Book Co., 391 p. Goldstein, S., 1953, On the mathematics of exchange processes in fixed columns: Royal Soc. (London) Proc., v. 219, p. 151. Hougen, O. A. and Marshall, W. R., 1947, Adsorption from a fluid stream flowing through a stationary granular bed: Chem. Eng. Process, v. 43, no. 4, p. 197. Luke, Yudell L., 1962, Integrals of Bessels functions: New York, McGraw-Hill Book Co., 419 p. Marshall, W. R., Jr., and Pigford, R. L., 1947, Application of differential equations to chemical engineering: Newark, Del., Univ. Delaware Press, 135 p. McLachlan, N. W., 1955, Complex variable theory and transform calculus: London, Cambridge Univ. Press, 388 p. Merriam, C. Neale, Jr., Southworth, Raymond W., and Thomas, Henry C., 1952, Ion exchange mechanism and isotherms from deep bed performance: Jour. Chem. Physics, v. 20, no. 12, p. 1842. O '