_ uU.s. DEPpostToRy OCT 09 1980 Petroleum-Resource Appraisal and Discovery Rate Forecasting in Partially Explored Regions An Application to the Denver basin By L. J. DREW, J. H. SCHUENEMEYER, and D. H. ROOT Mathematical Foundations By D. H. ROOT and]. H. SCHUENEMEYER An Application to Supply Modeling By E. D. ATTANASI, L. J. DREW, and J. H. SCHUENEMEYER GEOEOGICAL SURVEY PROFESSIONAL PAPER B. C UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON, D.C. : 1980 UNITED STATES DEPARTMENT OF THE INTERIOR CECIL D. ANDRUS, Secretary GEOLOGICAL SURVEY H. William Menard, Director Library of Congress Cataloging in Publication Data Petroleum resource appraisal and discovery rate forecasting in partially explored regions Geological Survey Professional Paper 1138 CONTENTS: A. Drew, L. J., Schuenemeyer, J. H., and Root, D. H., An application to the Denver basin B. Root, D. H., and, Schuenemeyer, J. H., Mathematical foundations C. Attanasi, B. E. D., Drew, L. J., and Schuenemeyer, J. H., An application to supply modeling Supt. of Docs. No.: I 19.16:1138 1. Prospecting-Denver Basin, Colo. and Wyo.- Mathematical models. 2. Petroleum-Denver Basin, Colo, and Wyo.-Mathematical models. I. Drew, Lawrence J. II. Series: United States, Geological Survey. Professional Paper 1138 TN271.P4P47 622'.18282°09788 80-607018 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 Petroleum-Resource Appraisal and Discovery Rate Forecasting in Partially Explored Regions- An Application to the Denver basin By L.. J. DREW, J. H. SCHUENEMEYER, and D. H. ROOT C EOLEOGCIE AL_ SURVEY -PROEESS LO NAE-PARPER I| +S8-A A model of the discovery process that can be used to predict the size distribution of future petroleum discoveries in partially explored basins UNITED STATES GOVERNMENT PRINTING QOFFICE, WASHINGTON, D.C. : 1980 CONTENTS Page .. ! *-. sco eee ele cei neon cena debes s neo cee aces Al Introduction .=. -_ 2. . - nL n o Ls n na ise j _________________________________________ 1 Area of influence of a drill hole: . sc-... ___ RA 1 Estimation of effective basin area____________ J _________________________________________ 2 Concept of discovery efficiency nn. lillie cer een il anis. n Fag. 5 Estimation of discovery efficiencies in the Denver basin ________________________________ 5 Predicting future discoveries within a size clase 7 Predicting future discoveries across size classes - mri ssri 8 Discussion cde, cause dae caer cen 10 References CLIBU - >. . e l Pe ece ceo neden oie anna een 11 ILLUSTRATIONS FiGures 1-11. Graphs showing: | 1. Average area exhausted per well (mcludlng both development and exploratory) for different target sizes and number of wells _. .-. .._ -e 12s Une eeonineh che ous Bp de bo eet cenno A2 2. Total area exhausted versus target size for selected numbers of wells 3 3. Area exhausted and effective basin size versus cumulative wells for targets ____________________--- 4 4. Cumulative fraction of targets discovered versus fraction of basin exhausted for two different hypothesized efficiencies" }.... cP. 2% 2 nou ens ae dul s Latae rans an oo bee s ane 5 5. Possible anomalous discovery rate that initially appears to have a discovery efficiency less than 1-_________ 6 6. Computed discovery efficiencies for classes 9 through 15 for years 1963-69 6 7. Cumulative discoveries versus area exhausted for class size 9 (256,000 to 512,000 bbl) ____________________ 6 8. Cumulative discoveries versus area exhausted for class size 10 (512,000 to 1,024,000 bbl) __________________ 7 9. Cumulative discoveries versus area exhausted for class size 12 (2,048,000 to 4,096,000 bbl) ______________-_ 8 10. Cumulative discoveries versus area exhausted for class size 13 (4,096,000 to 8,192,000 bbl) --------------- 9 11. Seventeen successive predictions of the total volume of petroleum to have been discovered by the end of 1969 in classes O9.:through 15 -... o rel. c apes an 9 12. Histogram comparing actual versus predicted petrolleum discovered within size classes 9 through 15 for discoveries made between January 1, 1955, and December 31, 1969, based on data to December 31, 1954 _________________- 10 13. Graph showing 22 successive predictions of the total volume of petroleum to have been discovered by the end of 1974 in classes 9 LhroUuglt 48 - .. : 3 aL - ce CHG - cel ena Bier enn - ena 10 14. Histogram comparing actual versus predicted petroleum discovered within size classes 9 through 15 for discoveries made between January 1, 1956, and December 31, 1974, based on data to December 31, 1955 -_________________ 10 TABLES Page TABLE 1. Conversion factors used in computation for development well conversions A3 2. Expected number of discoveries per 1,000 square miles searched for efficiencies of 1 and 2.5 ______________________ 5 3. Sample calculations for future discoveries in-size class 12 _._. LIl lay 8 Page PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING IN PARTIALLY EXPLORED REGIONS- AN APPLICATION TO THE DENVER BASIN By L. J. Drew, J. H. ScnurnEmEvER, and D. H. Root ABSTRACT A model of the discovery process can be used to predict the size distribution of future petroleum discoveries in partially explored basins. The parameters of the model are estimated directly from the historical drilling record, rather than being determined by assump- tions or analogies. The model is based on the concept of the area of influence of a drill hole, which states that the area of a basin exhausted by a drill hole varies with the size and shape of targets in the basin and with the density of previously drilled wells. It also uses the concept of discovery efficiency, which measures the rate of dis- covery within several classes of deposit size. The model was tested using 25 years of historical exploration data (1949-74) from the Denver basin. From the trend in the discovery rate (the number of discoveries per unit area exhausted), the discovery efficiencies in each class of deposit size were estimated. Using pre-1956 discovery and drilling data, the model accurately predicted the size distribu- tion of discoveries for the 1956-74 period. INTRODUCTION Before the cost of petroleum obtained from future discoveries can be determined, a model of the discovery process must be used to predict not only the total vol- ume of petroleum to be discovered with any given level of exploratory drilling but also the size distribution of these discoveries. This detailed information is required because, as the discovery process proceeds, the incre- mental size distribution of discoveries progressively contains a higher percentage of smaller deposits. Cost rises not only because progressively smaller volumes of petroleum are found per unit of exploratory effort but also because smaller deposits usually have higher per unit development costs. The purpose of this investigation was to develop a model of the discovery process that will predict the in- cremental size distribution of future discoveries in any partially explored region as a function of exploratory drilling effort. In the construction of this model, a structure was specified whose parameters can be esti- mated directly from historical drilling and discovery data rather than by subjective judgment or the use of an analogy. The number of discoveries predicted in each size class is determined by the value of two pa- rameters. The first parameter, which is commonly es- timated for all size classes, is the effective basin area, which is that region within which drillers are willing to site wells. The second parameter, which is uniquely estimated for each size class, is the discovery efficiency parameter. This parameter measures the effectiveness of exploratory drilling relative to the outcome of a ran- dom search process. The higher the value of the discov- ery efficiency parameter in any given size class, the earlier most of the deposits in that class will be discov- ered. The model of the discovery process described in this report is related to the model proposed by Arps and Roberts (1958) but differs in several significant as- pects. First, the discovery efficiencies are estimated within each deposit size class rather than assumed to be a single efficiency for all size classes as determined by subjective judgment. Second, the effective basin area is estimated from discovery time series data, in contrast to the approach taken by Arps and Roberts in which the boundaries of the region to be explored were selected by expert judgment. In the model used here, it was possible to estimate these parameters directly from discovery time series data by using the concept of the area of influence of a drill hole (Singer and Drew, 1976). Using this ap- proach, it was possible to predict accurately the number of deposits to be discovered within a group of size classes in the Denver basin from only a small amount of historical data. The Denver basin was cho- sen to test this model because it contains only one major producing interval (the D and J members of the Cretaceous Dakota Sandstone), and so a reliable data base could be constructed within a reasonable period of time (Schuenemeyer and Drew, 19776). AREA OF INFLUENCE OF A DRILL HOLE The area exhausted by a drill hole varies with the size of the targets that can occur in the basin. A basin may be fully explored with respect to large targets but Al A2 only 10 percent explored with respect to small targets. The shape of the targets also has an effect; for example, a pattern of exploratory and development holes may have fully searched a basin for circular targets having areas of 2 mi", although irregularly shaped targets also having an area of 2 mi are still undiscovered. The amount of area exhausted by any given hole can range from a maximum of the area of the target being considered when the areas of influence of any pre- viously drilled holes do not overlap to a minimum of zero when all points within the area of influence of the hole have been exhausted by previously drilled holes. Singer and Drew (1976) and Root and Schuenemeyer (1980) discuss this concept in detail, and Singer (1976) and Schuenemeyer and Drew (1977a) describe the computations. The degree to which the Cretaceous Dakota Sandstone interval in the Denver basin was physically exhausted by the end of each year within the period 1949-74 was calculated for targets up to 4.3 mi? in area (fig. 1). The levels of exhaustion shown are for elliptical targets with an axial ratio of 0.38, the mean axial ratio of deposits discovered during 1949- 74. For example, the 1,096 wells drilled through the end of 1952 exhausted on the average 0.545 mi' per well with respect to targets 1.09 mi" in area. By the end of 1954, this average declined 12.4 percent to 0.478 mi* per well. This decline in the effective exhaustiveness of wells is a consequence of the progressive crowding of wells. For larger targets this crowding is even more severe. For example, for targets 4.3 mi" in area, the corresponding area exhausted per well declines by 26 percent from 1.71 to 1.26 mi" per well during 1952-54. In 1969 the Union Pacific Railroad released for ex- ploration approximately 4,400 mi" of virtually unex- plored land in the basin (Oil and Gas Journal, 1969). The release and exploration of this large block of land during subsequent years are reflected in figure 1 by a slight increase in the level of the 1974 curve over the 1968 curve. This increase is a result of the wider spac- ing of holes, which normally occurs during the early phase of exploration of any region. ESTIMATION OF EFFECTIVE BASIN AREA The effective basin area is defined as that part of a total basin within which drillers are willing to site wells. Given this definition, there is no reason to as- sume that the area searched will have any particular shape. One basin may be more or less evenly explored from rim to rim. In others, many wells may be clus- tered in small parts of the total basin area. In such basins, the effective basin area may not even be a con- PETROLEUM-RESOURCE APPRAISAL AND ASCOVERY RATE FORECASTING 2.0 T T T T 1.9 - A 1.8 |- = 1.7 |- = 1096 wells = HF (1952) 1.5 |- 1932 wells 5 (1953) 3567 wells (1954) 5779 wells (1955) 7591 wells ¥ (1956) 9004 wells (1957) 11731 wells | (1959) .6 |- 17891 wells - (1968) .s |- 22577 wells o (1974) AREA EXHAUSTED PER WELL, IN SQUARE MILES PER WELL 1 L 1 1 0 1 2 3 4 5 TARGET SIZE, IN SQUARE MILES FIGURE 1.-Average area exhausted per well (including both devel- opment and exploratory) for different target sizes and number of wells. nected region." Using the concept of area of influence of a drill hole to measure the crowding of exploratory and development drilling, a method has been developed to estimate the effective basin area. As a simplified example of this method, suppose that an irregular area is marked off on the floor, and then disks are placed at random inside the area so that they have as much chance of landing one place as another and they may overlap. After several disks have been placed, the boundaries of the area are erased, but the disks are left on the floor. The problem then is to calcu- late the area, B, inside the boundaries that have now been erased. Suppose the disks each have area "@a." In the current analysis, the centers of the disks are the locations of the wells. The disk area is taken to be the 'A region is connected if and only if any two points in the region can be joined by a curve line that does not go outside of the region (region is in one piece). AN APPLICATION TO THE DENVER BASIN area of the largest target under consideration, and B is the effective basin area. The average target area cor- responding to class 15, the largest deposit size under consideration, is 4.3 mi". Let the function A (n) meas- ure the area covered by the first n disks. The value of this function will then be A (n)=na if none of the disks overlaps and A(n) < na when they do. Assume, fur- thermore, that the disks are small compared to B, the effective basin area. It can then be shown that even though A (n) is a random variable, it grows in a regular fashion as n increases (Root and Schuenemeyer, 1980). On the average A(n) grows according to a negative exponential: A(n) = (1) If an area A (n) has already been exhausted, then the additional area that will be exhausted by m additional wells sited at random within B is given by A(n+m)-A(n)=(B -A (n))(1-e-4/8), (2) A proof of these identities is provided by Root and Schuenemeyer (1980). In general, any arbitrary shape having the same area can be used without changing the rate of growth of A (n). In the following calculations, the targets are assumed to be elliptical. Given the area exhausted by the exploratory and development wells drilled through time (fig. 2), an estimate of the effective basin area for the largest class can be obtained in the case for which all wells are sited at random by solving equation 1 for B. In practice, however, development wells are not drilled at random but instead in a closely spaced regu- lar pattern. In the Denver basin, most development wells have been drilled one-quarter mile apart. As a consequence, a development well cannot have the same average capacity to exhaust a region as an exploratory well. We therefore treat a development well as a frac- tion of an exploratory well. At any time, then, the equivalent number of exploratory wells that have been drilled is specified by # = W +Dd(a) (3) where A = equivalent number of exploratory wells, W = number of true exploratory wells, D = number of development wells, and d (a) = conversion factor for development wells. a = target size The conversion factor, d(a), is a function of the target size, the target shape, and the spacing of development wells. The value of this factor for very small targets is A3 TOTAL AREA EXHAUSTED, IN THOUSANDS OF SQUARE MILES 0 1 1 1 1 | 1 0% ;5 1 2 3 4 5 6 TARGET SIZE, IN SquaARE MiLES Fiaure 2.-Total area exhausted versus target size for selected numbers of wells. equal to one, and, as the target size increases, d(a) decreases toward zero. If the spacing between holes is one-quarter mile and the axial ratio is 0.38, then targets with an area not greater than 0.0187 mi? will have d(a)=1. This factor is chosen so that two target- like ellipses having the same orientation and centers one-quarter mile apart will have an area of overlap, averaged over all orientations of (1-d(a)a. Conver- sion factors for various sizes of ellipses with axial ratio 0.38 are presented in table 1. TABLE 1.-Conversion factors used in computation for development well conversions Average Conversion producing area Class Size (10° barrels) factor d(a) (mi®) 9 lcs. adil oles a- 256- 512 0.54 0.339 10°. csc seco. 512- 1,024 A5 513 fr ___ "_- 1,024- 2,048 .37 153 124... ._ __ 2,048- 4,096 .31 1.091 19 A. erie. Ld. _ 4,096- 8,192 25 1.670 14 2- eee: rre 8,192-16,384 .23 2.070 1D ee n RPT Hee a+ the aie 16,384-32,768 16 4.300 A4 Given the number and location of exploratory and development holes drilled, we can now solve for the effective basin area using equation 4, A(n)=B(=e~*¥*>: (4) For estimating the effective basin area, A is computed using d(a=4.3)=0.16, the conversion factor for the largest target size. If an area A(A) has already been exhausted within an effective basin area B, then the additional area to be exhausted with respect to a target of area a by an ad- ditional number of equivalent exploratory wells m is A (A +m)-A (A)=(B -A (5) The estimate m is obtained in a manner similar to that of n, except that the future ratio of the number of de- velopment holes to exploration holes must be assumed. We assumed that the future ratio would be the same as the past ratio. With the results of the calculations from equation 4, PETROLEUM-RESOURCE APPRAISAL AND PISCOVERY RATE FORECASTING both the actual area exhausted and the estimated ef- fective basin area can be graphed as a function of cumulative wells drilled (fig. 3). The estimated effec- tive basin area increased rapidly through the drilling of the first 9,000 wells (1949-1957). This rapid rate of growth implies that the drillers rapidly expanded their views of where deposits could be discovered in the basin. During the following 10 years, when an ad- ditional 8,300 wells were drilled, the effective area of the basin grew very little, which implies that explora- tion during this period was being confined, for the most part, to the region established by 1957. From 1969 through 1973, the effective basin area again began to grow more rapidly. This growth was caused by the re- lease in 1969 of approximately 4,400 mi" in the Denver basin that had been held by the Union Pacific Railroad. The crowding that resulted in the change in slope in 1957 recurred in 1973-implying that the area to be searched was beginning to be reestablished. It is reasonable to assume that, had such a large region not been withheld from exploration, the effective basin area curves would have shown only one change in slope, from growth to stabilization. 18 T T T T T T T T T T Sy s r-1y -d I frt ye-- A 15 |- w w J. 3 w 12 |- CC T4 a Estimated effective HHH enate _L sei Lu: \ basin size / Aren 6 foe exhausted w 9_\ E o 2 / .. *= - -or- (- 0 T 1 1 ‘ 1 1 L ll L 1 | III 1 -_ Loot 1 Ill £. JCJ 5 10 15 20 WELLS, IN THOUSANDS FicurE 3.-Area exhausted and effective basin size versus cumulative wells for 4.3-mi* targets. AN APPLICATION TO THE DENVER BASIN A5 CONCEPT OF DISCOVERY EFFICIENCY . Historical exploration records show that as. explora- tion proceeds, the rate of discovery declines; that is, exploratory drilling begins in the more productive parts of a basin and later moves to the less productive parts of that basin. The concept of discovery efficiency is used to measure this phenomenon. A discovery effi- ciency of 1 is used as a reference point that defines the outcomes of a random search process. In such a search, the number of deposits discovered in each size class per unit area exhausted remains constant throughout the total exploration history of one region. For example, if the ultimate effective basin area were 10,000 mi and 100 deposits existed in a certain size class at the start of the search process, we would expect 10 deposits to be discovered with the exhaustion of each 1,000 mi? of the region. If the exploration process were better than random, we would expect a larger number of deposits to be found during the earlier stages of exploration. For example, if the discovery efficiency were 2.5 in the pre- ceding example, the exhaustion of the first 1,000 mi would produce 23.2 discoveries versus 10 discoveries for the random search. The number of discoveries ex- pected to be made in this example for each 1,000 mi exhausted is shown in table 2. These values were com- puted from the following discovery process model: f = 1-(1-A/BY (6) where f =the fraction of the deposits that have been found, A = area that has been exhausted, B = effective basin area, and c = the discovery efficiency. A graph of the cumulative number of discoveries as a function of area exhausted for two different efficiencies is schematically diagrammed in figure 4. TABLE 2.-Expected number of discoveries per 1,000 square miles searched for efficiencies of 1 and 2.5 Expected number of discoveries per 1,000 mi" searched Area searched (mi?) Efficiency = 1 _ Efficiency = 2.5 23.2 19.6 16.2 1 1 " Emre to to on- ho os ho i- 1.0 T T T T T T T T 0.9 |- a 0.8 |- a 0.7 |- c=2.5 { 0.6 |- © PS1 c=1.0 | 0.4 |- ~ 0.3 |- f - 0.2 |- A FRACTION OF TARGETS DISCOVERED 0.1 |- * 0.0 1 1 1 1 1 1 1 1 1 0.0 0.1%:0.2-0.3 0.4 0.5 -06 0.7. 08 0.9 1.0 FRACTION OF BASIN EXHAUSTED Fraure 4.-Theoretical graph of cumulative fraction of targets dis- covered versus fraction of basin exhausted for two different efficiencies. While it is theoretically possible for a discovery effi- ciency to be less than 1, this value is of no relevance because it implies a steadily increasing discovery rate through the end of the exploration process; in the long run, drilling would terminate when the discovery rate reaches its peak! In the short run, however, a sudden geologic insight or chance fluctuation could cause the discovery rate to increase for a period of time (fig. 5). If the method for computing efficiencies described below results in an efficiency of less than 1, the efficiency is set equal to 1. ESTIMATION OF DISCOVERY EFFICIENCIES IN THE DENVER BASIN From the trend in the discovery rate (the number of discoveries per unit area exhausted), the efficiencies in each class of deposit size were estimated. In the estima- tion of the efficiencies it is necessary to choose a basin size. The assumption that the entire basin is available for drilling ignores two effects; namely, that explora- tion tends to follow past discoveries, and drillers tend to explore the shallower parts of a formation before the deeper. These effects combine to produce a greater crowding of wells than would occur with random drill- ing. This greater crowding results in an underestima- tion of the area that will ultimately be explored. We have found a way to adjust the model that com- pensates for the fact that the area being considered for exploration grows through time. For estimating the A6 CUMULATIVE DISCOVERIES AREA EXHAUSTED FiGurE 5.-A possible anomalous discovery rate that initially ap- pears to have a discovery efficiency less than 1. efficiency of exploration for fields of each size class, the basin size used was the basin size estimated for that size class. For estimating future discoveries, the basin size used was that estimated for the largest size class. The theoretical description of the search of an expand- ing area is still under investigation. Efficiencies were estimated annually" for the years 1952-69. Seven efficiency estimates (1963-69) for seven deposit size classes are given in figure 6. No trend with time was found within each series of esti- mates. No estimate was made for class 16 (82.7-65.5 million barrels) because only one deposit was discov- ered in this, the largest class, and an estimate would be statistically meaningless. Efficiency increases as de- posit size increases (fig. 6). Efficiencies less than 1 oc- curred in classes 9, 10, and 11, but for the purpose of estimating future discoveries, it was assumed that in the long run no efficiency can be less than 1. If the estimated efficiency was less than 1, then for purposes of prediction, it was set equal to 1. The relation between the discovery efficiency and the shape of the cumulative discovery curve can be seen by comparing the efficiencies estimated in figure 6 with the discovery curves shown in figures 7-10. For exam- ple, the discovery efficiency within class 12 was esti- mated to be approximately 2.6, and the discovery curve has a markedly decreasing slope (fig. 9). An efficiency of this magnitude implies that when half the effective *The details of the calculation procedure are presented in Root and Schuenemeyer (1980). PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING DISCOVERY EFFICIENCY 0 1 1 1 1 1 1 1 9 10 11 12 13 14 15 DEPOSIT SIZE CLASS FiGurE 6.-Computed discovery efficiencies for classes 9 through 15 for years 1963-69. 110 T T T T T T 100 |- gs * - 90 |- o m 80 |- o 2 70 |- $ 8 - 60 |- A = 50 |- F a 40 |- 5 30 |- =] NUMBER OF DISCOVERIES 20 |- 5 10 |- x lp 1 1 1 1 1 1 0 640 1280 1920 _ 2560 3200 - 3840 AREA EXHAUSTED, IN SQUARE MILES 4480 FiGUrE 7.-Cumulative discoveries versus area exhausted for class size 9 (256,000 to 512,000 bbl). basin area has been exhausted, 84 percent of the depos- its containing between 2.048 and 4.096 million barrels of recoverable petroleum will have been found. The efficiency of discovery for the deposits in class 9 (0.256 to 0.512 million barrels) was approximately 1.4 (fig. 7). This value means that deposits of this size were found at nearly a uniform rate through 1969, as the nearly straight cumulative discovery curve for this class of deposits shows (fig. 7). The discovery efficiency for the deposits in class 10 (0.512 to 1.024 'million barrels) was estimated to be less than 1.0 (fig. 8) be- AN APPLICATION TO THE DENVER BASIN AT 100 T T T T 90 |- 80 |- 70 |- 60 |- 50 |- # NUMBER OF DISCOVERIES 10 |- & o 0 1 1 L. 1 1800 2400 1 3000 3600 4200 4800 5400 AREA, IN SQUARE MILES FiGUrE 8. -Cumulative discoveries versus area exhausted for class size 10 (512,000 to 1,024,000 bbl). cause the discovery rate increased rapidly during the exhaustion segment from 1,500 to 3,000 mi". As we have previously discussed, this phenomenon can be only temporary, and, as shown in figure 8, the discov- ery rate declined during the second half of the exhaus- tion sequence. No obvious geologic cause could be iso- lated to explain this temporary increase in the rate of discoveries. PREDICTING FUTURE DISCOVERIES WITHIN A SIZE CLASS The model of the discovery process shown in equa- tion 6 was used to predict the ultimate number of de- posits to be discovered and the number of deposits to be discovered for a given area exhausted. Each of these predictions is made within a deposit size class. The predictions made for some later segment of exploration are designed to answer the question: How many depos- its may one expect to find by drilling a certain number of additional wells, given the previous exploration his- tory? In order to validate the model given in equation 6, resource appraisal estimates have been made within each deposit size class from the discovery time series for various years beginning in 1952. The last year of historical data used in the prediction is called the pre- diction year. The number of discoveries estimated from the end of the prediction year until 1969 and until 1974 is compared with the actual number. Before a forecast can be made, the following esti- mates are calculated: (1) the ultimate number of depos- its within each size class; (2) the area that will be phys- ically exhausted by the number of wells actually drilled between the time the forecast is to be made and the end of 1969 and 1974; and (3) the discovery effi- ciency. The number of deposits estimated to have existed originally in each size class is obtained from equation 6. Given the fraction of the effective basin area still remaining to be explored and the corresponding dis- covery efficiency for each class, the fraction of deposits remaining to be discovered is (1-A/B Y . The number of deposits then estimated to have occurred originally in each size class is obtained by dividing the number al- ready discovered in a class by 1-(1-A/BY. From this calculation we then can construct a resource appraisal for the basin for any segment of the exploration data time series. f & The estimate of the basin area for class 15 is ob- tained from equation 4. Once the basin area has been determined, we then assume that the ratio of explo- ratory to development wells remains constant. Then we can calculate from equation 5 the additional area to A8 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING 36 |- 32 |- 0 28 |- o 20 |- 0 NUMBER OF DISCOVERIES 0 12 |- co T ° 0 0 p $r3t 0 1 1 1 1 1 1 1 1 o 1000 2000 3000 4000 5000 6000 7000 8000 9000 AREA EXHAUSTED, IN SQUARE MILES FIGURE 9.-Cumulative discoveries versus area exhausted for class size 12 (2,048,000 to 4,096,000 bbl). be exhausted for a given number of additional wells. The basin size estimated for deposits of size class 15 is used in this calculation. Using these estimates, the number of discoveries expected within each size class from the end of the prediction year through 1969 and through 1974 are computed. Sample calculations for future discoveries through 1969 in deposit size class 12 are presented in table 3 for the prediction year 1957. In the computation of ad- justed future wells, the number 2 appears as a divisor because the ratio of exploratory to development drill- ing has been found through time to be approximately equal to 1. PREDICTION OF FUTURE DISCOVERIES ACROSS SIZE CLASSES An estimate of the total volume of petroleum to be discovered was obtained by summing the future oil dis- covered over size classes 9 through 15 (deposits be- tween 0.256 and 32.768 million barrels). This estimate was then compared with the volume actually discov- ered to test the validity of the prediction. In addition to estimating the total volume of petroleum to be discov- ered, it is necessary for purposes of economic analysis TaBLE 3.-Sample calculations for future discoveries in size class 12 Basic Data Description i. Symbér & Value - _ Area exhausted to December 31, 1957 __ A (1957) 4,165 mi" Target O8. { el cio lel a 1.09 mi' Estimated effective basin size _________- B 10,624 mi* Correction factor for development wells __ d 0.31 Discovery efficiency c 2.577 Number of discoveries to December 31, -__- N 34 Average volume of recoverable oil _____. 2.863 x 10®bbls Future wells from January 1, 1958, to December 81, 1969. ___. 9,765 Calculations - ~ 9765 + 9765 (0.31 Ajusted future Welle =+; & s- _,7,,,L, U- 6.396 A(1969) = (B -A (1957))(1-e-@/") + A(1957) = 7,273 mi" Unexplored fraction of basin (1957) = 1 - Aqlggfl): 0.608 Uhesplored factionor basin = T- 4 (B’s—9) = 0.315 Undiscovered fraction of targets (1957) = (0.608) = 0.277 Undiscovered fraction of targets (1969) = (0.315) = 0.051 Fraction of targets discovered (1958-69) = 0.226 Estimate of ultimate number of targets = 47.0 iaNaeig 1-0.277 Estimated number of discoveries (1958-69) = (47.0)x (0.226) = 10.6 Future oil discovered (1958-69)=10.6x2.863x 10° = 30.3 x 10° bbls R AN APPLICATION TO THE DENVER BASIN A9 30 T T T I 25 |- 20 |- 0 10 |- NUMBER OF DISCOVERIES T I I I T 0 I 1 1 1 E 3000 4000 5000 1 6000 7000 8000 9000 10,000 AREA EXHAUSTED, IN SQUARE MILES Figure 10.-Cumulative discoveries versus area exhausted for class size 13 (4,096,000 to 8,192,000 bbl). to predict accurately the number of deposits to be dis- covered in each size class. The prediction of the total oil to be discovered is the sum of seven quantities, each subject to independent random fluctuations that on the average tend to cancel each other; thus the total is a more stable quantity. On the average, estimated efficiencies tend to increase with deposit size class (fig. 6). Deviations from this trend are attributed to random fluctuations. The prediction of the total amount of oil to be discov- ered by drilling 18,762 wells changes as the initial drilling and discovery sequence on which it is based changes (fig. 11). Thus, using only 5 years of data (1949-54), or 19 percent of the wells drilled through 1969, the basin size and efficiencies are sufficiently well determined to permit accurate estimation of the oil to be discovered by the next 15,195 wells. In order to use this estimate for economic calculations, it is neces- sary to be able to predict the size distribution of depos- its. Predicted and actual future discoveries agree closely across size classes using 1954 as the prediction year (fig. 12). This agreement is particularly good in the larger, more important size classes. The calculations made through 1969 were repeated for the 22,577 wells drilled through 1974 (figs. 13 and 14). During the 5-year period beginning in 1969, the land released by the Union Pacific Railroad was ex- plored. The late release of this much land might have caused the model slightly to underestimate the amount of oil to have been discovered through the end of 1974. T I | I l I || T ll I ll 1 af 1 F-13997 I 6 Actual total discoveries _____.----_-._---~ a through 1969 Lok © sel aan amt os - Z / C= s |- /~Predicted total s] - / discoveries fe / G 4 w? 4} Cumulative discoveries 3 O in g c CC w ef 2} x 2 u. Z 30 T 2 |- A 2 / o "l' 8s 3 © co o noog 0 a: | t w w to © 0 e 6 8 8 $ £ o _I $ 1 § 2 rs o 1 T l 1 I | r]- 1 II I r]- 1 1 4 ll 1 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 CUMULATIVE WELLS, IN THOUSANDS FiGurE 11.-Seventeen successive predictions of the total volume of petroleum to have been discovered by the end of 1969 in classes 9 through 15. A10 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING 150 s s f= 3 I I | I 48 1 1 | | | | [] Y 125 - ___‘ | Actual future 7 160 |-- T T w +L E Predicted future L3” | ~Actusl future $ 100 |- | mn -I Ig 125 |- Predicted future *y as s nat Saat. i Af CH 151 1a N Am l a \. Actual past 2 is |- 14 i ac [| - |G 100 |- [| \\/°”’°° sl o S| | _" 1g" £4 m [N n 1 | l ta 2 NIN 2 silk =O [+] N te- e} NN 2 is i p DM Aa 71 19 NN NN s A fer gol | -< |N |N [N i NN NN - & 2 Sg 4 ~} N DN " ND z NN NN > 5 as- _ |: [] N N |N =| ~ NN NNY = [| 1 | N ~ |N g NN NN NN q NH NN BDN NN NN N pN DNN -§O 0 | l 9 10 11 12 13 14 15 | t t t t: : N DEPOSIT CLASS SIZE 1 N \|\ | | Ficurg 12.-Histogram comparing actual versus predicted petro- o 9 10 11 12 13 14 15 leum discovered within size classes 9 through 15 for discoveries made between January 1, 1955, and December 31, 1969, based on data to December 31, 1954. w g 7-5 [III || |l Ii] II||IIIII|1||II|I||1 G3 Actual total discoveries oro Is 1 6.9|- through 1974-- =- E a /~Predicted total i o g 4% |-- 74 discoveries €. 2 c Cumulative discoveries mg / [re [(- - ° o 3.0 go - w g- $ Hale a ' _s ift : 4 } Z o l l a a a a a a ao ® a =- bad «- = al ®je - «- a me Lac acl a cet ( alii cl 0 0.0 2.5 5.0 7.5 10.0 12.5 15.0 17.5 20.0 22.5 CUMULATIVE WELLS, IN THOUSANDS Ficur® 13.-Twenty-two successive predictions of the total volume of petroleum to have been discovered by the end of 1974 in classes 9 through 15. DISCUSSION In assessing the petroleum potential of an area as described above, no use was made of the geologic and geophysical knowledge acquired during the explora- tion of the area. Several reasons exist for not making direct use of such information. Geologic and geophysi- cal data are difficult to assemble because they are scat- tered among many exploration companies and they are often confidential. Even if the data were gathered, a resource estimate for a partially explored area based upon them would have to be taken on faith because the data set would be too large and complex to communi- cate. Moreover, the reasoning by which one proceeds from geologic and geophysical knowledge of a basin to an estimate of the number and sizes of undiscovered L DEPOSIT SIZE CLASSES 14.-Histogram comparing actual versus predicted petro- leum discovered within size classes 9 through 15 for discoveries made between January 1, 1956, and December 31, 1974, based on data to December 31, 1955. fields and the number of wells required to find them is largely subjective and intuitive and hence difficult to expose. $ Although direct use of geologic and geophysical in- formation is not feasible, this information is contained implicitly in the record of drilling and discovery insofar as it affected the siting of wells. For example, in the Denver basin, the effective basin size is determined by geologists and is revealed through the siting of wells. The effective basin area was estimated here to be about 10,000 mi", or about one-fifth of the whole Denver basin. This area was deduced without directly examin- ing the reports on prospects in the other fourth-fifths of the basin. The examination of those reports was left to those who were siting the exploratory holes. From the discovery rate, one can estimate how well exploration prospects are evaluated. The decline in the discovery rate (quantity of oil found per unit area searched) gives a quantitative measure of the superior- ity of actual exploration over random drilling. This superiority, measured by the parameter "efficiency of exploration," shows to what extent the better prospects are drilled early. The efficiency of exploration in the Denver basin was found to be about two to four times better than that of random drilling in the search for targets in larger classes (2% 10®bbl to 32x 10%bbl) and to be about the same as that of random drilling in the search for small targets. Thus, both the area to be ex- plored and the efficiency with which it is being ex- plored can be deduced from the drilling and discovery AN APPLICATION TO THE DENVER BASIN record. When the area to be searched and the efficiency of search are known, one can extrapolate the past dis- covery record to obtain a reasonably accurate forecast of the number and sizes of oil fields that will be discov- ered by a given amount of future exploratory drilling. For the Denver basin, the discovery record of the first 3,638 exploratory wells was extrapolated to provide forecasts of discoveries to be made by the next 7,929 exploratory wells. The future discoveries were divided into six different size classes. In most of these size classes, close agreement was observed between the predicted and actual discoveries. Estimates of future oil discoveries are difficult to make in the larger size classes because of the paucity of data. However, at the time in the discovery sequence when sufficient data are available to estimate future oil discoveries, most of the large deposits have been discovered, and the interest is in predicting the number of moderate size deposits re- maining to be discovered. All REFERENCES CITED Arps, J. J., and Roberts, T. G., 1958, Economics of drilling for Cre- taceous oil and east flank of Denver-Julesburg basin: American Association of Petroleum Geologists Bulletin, v. 42, no. 11, p. 2549-2566. Oil and Gas Journal, 1969, Union Pacific, Pan Am enter exploration pact in Rockies: Oil and Gas Journal v. 67, no. 35, p. 86. Root, D. H., and Schuenemeyer, J., 1980, Petroleum-resource ap- praisal and discovery rate forecasting in partially explored regions-Mathematical foundations: U.S. Geological Survey Professional Paper 1138-B, p. B1-B9. Schuenemeyer, J. H., and Drew, L. J., 1977a, An exploratory drilling exhaustion plot program: Computers and Geosciences, v. 3, no. 4, p-617-631. 1977b, Data reduction and analysis for a petroleum explora- tion model (abs.): American Association of Petroleum Geologists Conference, Exploration Data Synthesis, Tucson, March 1977, p. +: Singer, D. A., 1976, Resin-A Fortran IV program for determining the area of influence of samples or drill holes in resource target search: Computers and Geosciences, v. 2, no. 2, p. 249-260. Singer, D. A., and Drew, L. J., 1976, The area of influence of an exploratory hole: Economic Geology, v. 71, no. 3, p. 642-647. Petroleum-Resource Appraisal and Discovery Rate Forecasting in Partially Explored Regions- Mathematical Foundations By. .D. H. ROOT and ]. H.:SCHUENEMEYER @EOLOUG SSURVE LPROFESSIO N AE-PAPE R 1-1-0 85 The basic mathematical properties of a model of the discovery process used to predict the size distribution of future petroleum deposits in a partially explored region UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON, D.C.: 1980 dregs CONTENTS Page AbsLFact . . : .s SES nue ce nan. cease eee cle . Leute. . e od. nena, B1 Introduction.... 1. - .. ini ne coe bae o. eon ene ene aoa Ma caccl age... £ Estimation of basin site ASL incoln. csf. oll ll AL_ 2 Practical difficulties and modifications. cc DML 0 .L cc:: 4 Efficiency ofblexploration .-..... leo a DLO. at- 5 Results of simulation | --..«_.r..-l_. 3.2. c sutllt L1. Don {on cz offe O :o v References.cited cl. _...... OOo r coal cui At 9 ILLUSTRATION Page Frourp 1.. Estimates of effective basin size from simulated c ~.. t.. } [ cl. (RNO et. B8 TABLE Page Tapir 1» - Estimated medianefficiencies [co ___ Unlock ae norton ea ina s, B7 III PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING IN PARTIALLY EXPLORED REGIONS- MATHEMATICAL FOUNDATIONS By D. H. Root and J. H. ScnurnEmEvyER ABSTRACT A stochastic model of the discovery process has been developed to predict, using past drilling and discovery data, the distribution of future petroleum deposits in partially explored basins, and the basic mathematical properties of the model have been established. The model has two exogenous parameters, the efficiency of exploration and the effective basin size. The first parameter is the ratio of the probability that an actual exploratory well will make a discovery to the probability that a randomly sited well will make a discovery. The second parameter, the effective basin size, is the area of that part of the basin in which drillers are willing to site wells. Methods for estimating these parameters from locations of past wells and from the sizes and locations of past discoveries were derived, and the prop- erties of estimators of the parameters were studied by simulation. INTRODUCTION This paper presents the mathematical derivations for the estimators of the parameters used in a model of the discovery process proposed by Drew, Schuene- meyer, and Root (1980). The model was designed to summarize the past exploration and discovery experi- ence in a region in order to predict the quantity of recoverable oil that future exploration could be ex- pected to find. The model was developed because a significant amount of the petroleum remaining to be discovered in the United States and elsewhere is in regions where there has already been some exploratory drilling. During exploration, a large amount of data is gen- erated, including seismic surveys, magnetic surveys, geologic maps, well logs, discovery sizes and locations, and drilling dates and locations of discovery wells and dry holes. Because it is not feasible to use all the avail- able data, a subset must be selected. The data incorpo- rated in the model were the dates and locations of past exploratory wells together with the locations of discov- ery wells and the sizes of any deposits that were found. The geologic and geophysical data are thus implicitly included in that they guide the drilling and discovery process. If exploration were completely random, then the probability of a given oil field being found by a single exploration well is the ratio of the area of the field to the area being searched. A model developed by Arps and Roberts (1958) assumes that the probability of an exploratory well discovering a given oil field is the product of a constant, the efficiency of exploration, and the ratio of the area of the field to the search area. In that model the measure of exploration was the number of exploratory wells. The model proposed by Drew, Schuenemeyer, and Root (1980) is similar to the model of Arps and Roberts but uses a different measure of past exploratory effort. Rather than the number of ex- ploratory wells, it uses the cumulative area exhausted by these wells (Singer and Drew, 1976). The area exhausted by a well is related to the area and shape of the deposits under consideration. For example, for cir- cular deposits having a radius of 1 mi, a dry explo- ratory well tells the explorer that the center of such a target cannot lie within 1 mi of the well and 3.14 mi* have been exhausted so far as such deposits are con- cerned. A pattern of dry wells will exhaust, with re- spect to such targets, an area equal to the area covered by all circles of radius 1 mi centered at each well. Be- cause these circles may overlap, the exhausted area may be less than the number of wells times the area of a circle. If the deposits are elliptical having a major axis of, say, 4 mi and a minor axis of 2 mi, then a dry well exhausts all of the area within 1 mi of the well, none of the area more than 2 mi away, and partially exhausts the area between 1 and 2 mi from the well. A deposit centered within 1 mi from the well would have been hit regardless of its orientation, assuming cer- tainty of recognition. If the center of the deposit is be- tween 1 and 2 mi from the exploratory well, the deposit would have been hit for some orientations but not for others. The degree to which a point is exhausted is the probability that a randomly oriented deposit centered at the given point was hit. This concept of area ex- hausted can be used as a measure of the extent of ex- ploration for targets of any given size and shape. B1 B2 When an exploratory well is successful, then any point within 2 mi of the discovered deposit would have been partially exhausted for elliptical targets 4 mi by 2 mi. The degree of exhaustion at a point is the prob- ability that a randomly oriented 4-mi by 2-mi ellipse centered at the point would have intersected the dis- covered deposit. This example illustrates that the area exhausted is a function of the size and shape of the deposits being considered and that more area is exhausted for larger deposits than for smaller ones. In the analysis of the discovery data, deposits are divided into size classes. All deposits within a given size class have approximately the same area, and each of these classes is analyzed independently. The following dis- cussion focuses on elliptical deposits of the same size and shape. If exploratory wells are randomly sited, then the pe- troleum richness of the area exhausted remains ap- proximately equal to the richness of the unexplored area so that the fraction of all the deposits that are still undiscovered remains approximately equal to the frac- tion of the search area that has not been exhausted. If the exploration is better than random, then the petro- leum richness of the area exhausted is greater than that of the area remaining to be searched. In order to predict future discoveries, it is necessary to know the ultimate area of exhaustion that may be expected. This search area, defined as that part of the basin where operators are actually willing to site wells, is called the effective basin, and it may be only a small fraction of the entire basin. Because the size of the effective basin is an unknown quantity that typically grows through time, it was estimated from growth of the physical exhaustion time series. ESTIMATION OF BASIN SIZE We assume that within a large geologic basin there is a smaller effective basin to which the search for oil is restricted. Depth is not considered; the targets and the basin are assumed to be two-dimensional. The oil fields are assumed to be nonoverlapping ellipses all of the same size and shape inside the effective basin. The orientation of the targets, the angle between the major axis of the ellipse and the east-west line measured counter clockwise from the east-west line, may be anywhere between 0° and 180°. The search consists of the searcher selecting a point (drilling a well) in the effective basin. If the point is in a target, then the searcher is told the location of the boundaries of the target. This information approximates what would be learned in practice by subsequent development drilling. PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING There is an (x, y) coordinate system on the basin having x increase to the east and y increase to the north. The position of a target is completely specified by three coordinates (x, y, 0), where x and y specify the location of the center and 0 gives the angle between the major axis of the ellipse and the x-axis of the coordinate system measured counterclockwise from the x-axis. We define a search solid, S, to be all points (x, y, 0), where (x, y) is a point in the geologic basin and 0 is between 0° and 180°. Then any target in the geologic basin corre- sponds to a unique point in S and vice versa. We can visualize S as a cylinder 180 units high and having a base of the same size and shape as the geologic basin. Thus the A-axis is vertical, and the x-y plane is horizon- tal. When a search point (x', y') in the effective basin is selected and is not in a target, then every point (x, y, 0) in S that corresponds to an ellipse in the geologic basin that contains the search point, (x', y'), is eliminated from further consideration. The set of points in S con- demned by an unsuccessful search point at (x', y') can be visualized as the volume swept out by a horizontal target-size ellipse centered at (x', y', 0°) with major axis lying along the x-axis as it rises to (x', y', 180) while rotating 1°, counterclockwise from above, for each de- gree it rises. Each point in this volume corresponds to an ellipse in the geologic basin that would contain (x', y'). When a search point is successful, then the points in S that are eliminated are all those points corre- sponding to ellipses that would intersect the target which was found. The introduction of the space S has changed the problem from search points looking for area targets to search volumes in S looking for point targets in S. The cumulative volume in S that has been eliminated by successful and unsuccessful search points in the effective basin, EB, is measured in acre- degrees. This unit, the acre-degree, is the unit in which we will measure the extent of exploration. Inside the search volume, S, we define the effective search volume, ES, as all points (x, y, 0) in S where (x, y) is a point in the effective basin. The size of the effec- tive basin is not known to the data analyst. We will give a method for estimating the effective basin size from well locations under simplifying assumptions and then describe how the method is modified to fit more realistic assumptions. At first, it is assumed that wells are sited randomly in EB according to a uniform prob- ability distribution. From this assumption we derive an estimator of effective basin size which is then modified to account for the fact that development wells are always close to existing wells. Let EB be the effective basin, let B be the Lebesgue measurable sets on EB, and assume EB is endowed with the uniform probability measure. Let Q = MATHEMATICAL FOUNDATIONS B3 EB xEB x... , a countable product of copies of the effec- tive basin. Let B,, be the smallest & field containing all sets of the form A; x. ..xA,, XEB. .. where each A,; is a Lebesgue measurable set in EB. Let B, be the trivial & field on . A function defined on O can be B,, measura- ble only if it is dependent on at most the first m coordi- nates of a point of . Let d;, j = 1, 2,. . . be a sequence of positive numbers. Let D,(P) denote a disk of area d; centered at the point P. If (P;, P;,. . .) is a point in 0, define alj,n; P,, P;,. . .) to be the total area covered by D;(P,),. . ., D;(P,,). Because a(j,n) is a continuous func- tion of P;, ., P,, it is measurable with respect to B,. Using the notation just defined we now establish the basic theorem for the estimation of effective basin size. Theorem: If d;->0 as j-» and n and j are made to approach infinity in such a way that nd,->A where A is an arbitrarily selected positive number and if EB is a bounded set whose boundary has zero area, then W -> 1 in probability. The symbol EB is used for both the effective basin and its area. Proof: The proof proceeds by first showing that E(a(j,n)->EB (1-e-4*") and then showing that 0*(al(j,n))->0. Let a sequence r, j=1,2. . . . of positive numbers be defined by the equation #'=d;,. Define EB(r;+) to be all points (x,y) for which there is a point (x',y') in EB such that (x'-x)' + (y'-y)" d+ (1-d/EB)atj,.n-1). (2) Next we condition both sides with respect to B, _;; recall that EG(alj.n) | Bi-)lBi-) = Ela(j.n)|B.-)>d; + (1 - d;/EB) Ela(lj,n - 1)| Bu-»). (3) Applying our inequality (1) to E(a(j,n—1)|Bn_2) gives E(a(j.n-1)|B.-2) > d; + (1-d;/EB)a(j,n-2). (4) Combining (3) and (4) gives E(alj,n)| B, -2)>d, +(1-d;/EB)(d; +(1-d;/EB)a(j,n-2)) =d; +(1-d,/EB)d,+(1-d,;/EB)a(j,n-2). Continuing this process through »-2 more condition- ings gives n E(a(j,n)) = E(a(lj,n)| B,) > d, % (1-d,/EB)® _ (5) k =o and n d; kgo (1-d;/EB)"® = EB-EB(1-d;/EB)"+*. (6) Because we assume that nd,-A we have that the RH.S. of (6) approaches EB-EB e-4/"*. Thus we may conclude lim n,.J E(a(j,n)) > (7) when nd, -A as n, j ->» Referring back to equation 1, we now seek a lower bound for {9 f(P)dP . Recall that f is d,/EB on EB(r;-) and it takes smaller values on OEB(z) so we assume that as much of S(j,n-1) is in 0 EB (r;) as possible to find a lower bound. Thus [ PP + | f(PdP > | f(PP SNEB(G-) SNOEB(G) SNEB(r;-=-) The area of S(j,n-1)NEB(G@-) is at least a(j,.n-1) -oEB (r ;), where OEB (r ;) is used to denote the area of JEB (z). Thus {g f(P)dP > (a(j,n-1) - JEB [EPP = S which yields the inequality, B4 E(alj,n) |Bi-1) < d, + a(j,n-1) - (a(j,n-1) - dEB(n))d,/EB =d, + a( j,n-1)(1-d;/EB) + (OEB(r;))d;/EB. As before we condition both sides on ,-; and obtain Elalj,.n)|Bi-2) < d, + (1-d,/EB)E(alj,n-1)| Bu-») + (OEB(r;)d,/EB = d, + (1-d;,/EB)(d, + (1-d,/EB) x aljn-2) + (OEB(r;)d,;/EB) + (OEB(n))d,/EB = d, + (1-d,/EB)d;, + (OEB (r;))d,;/EB + (1-d;/EB)" a( j,n-2) + (1-d,;/EB) (OEB (n))d,/EB. We repeat this process for B.-s, Bi-, . . . ,Bo to obtain n-1 E(a(j,n)) < d, % (1-d,/EB)* k=0 n-1 + (OEB(n))d,/EB x % k=o (1=-d;/EBY. (8) Since the boundary of EB has 0 area and since EB is a bounded set it follows that JEB (15) -O as ; -O. Hence as n, j ->» the second term in (8) goes to 0. The limit of the first term was calculated before (see equation (6)) to be EB (1-e-4/*8). Hence we have shown that lim E(a(j,n)) < EB(1-e*/*") nj (9) when nd, -> A. Combining (9) and (7) we have that E(a(j,n)) -EB(1-e-*/*") when n,j ->» and nd ;, -A. To complete the proof of the theorem, it is sufficient to show that the variance of a(j,n), o®(a(lj,n)) ->0 as n,j -» and nd -> A. This will be done by establishing (10) j,n))=, P1, < and (Pi,. Piy..=*=> . Pay. <2) = P' be two points in Q which agree in each of the first n coordinates ex- cept for the &". Then |a(j,n)(P) =- a(lj,n)(P")| = d; be- cause they differ only in the position of a single disk of area d;. Hence [Eat n) | BNL :- .. Biel dB) Ci, ..: & d;. Hence it follows that lE(a(j,n)|Bk—1) (Pi,: . ; Px=1) * Eiolnlfd Pl. a(n)> d, we will get unreasonably large estimates for the effective basin size. The problem of disk size is further complicated by MATHEMATICAL FOUNDATIONS B5 the effect of target size on the effective basin size. For example, a very large area may be sparsely drilled for very large targets, whereas the search for small targets might be restricted to a smaller area. The assumption of random location of wells is not applicable to devel- opment wells, which are always sited near existing wells, so the disks around development wells cover very little additional area compared to what the aver- age randomly located disk would cover. In applying the estimator, , defined by equation (11) in the analysis of the discovery of targets of a par- ticular size class, d was taken to be the area of the targets in question, a(n) was taken to be the acre- degrees in the search solid which had been exhausted divided by 180, and n in the exponent was taken to be the number of wildcat wells plus a fraction, «, of the number of development wells. The fraction, a, is chosen to satisfy the condition that if D, and D, are target-size and target-shape ellipses having the same but ran- domly selected orientation and centers 0.25 mi apart (40-acre spacing), then (1 + a)d is the expected total area covered by D, and D,. Thus a is between 0 and 1 and decreases as target size increases. EFFICIENCY OF EXPLORATION* We assume that there are N target ellipses all of the same shape and area in EB and that they correspond to N points in ES, the effective search solid. Let T; be the fraction of ES that had been exhausted at the time of the i*" discovery. Thus 0 1. Suppose a;-< a < 1; then *A part of the results in this and the next section appeared in Schuenemeyer and Root (1977) and are reproduced here with permission from American Statistical Association. fic(a|Tk—l = @r-1) € Aim=~~ 4 | = a, 1] Aa-0 Aa = lim 1 _P[T; @ and Aa-0 - Aa TIC—1 se (1k._1] X P[Tk > a I TIC—l = ak_1] N-Ak+1 j lim: a | *> (_A_a> Aa-0 Aa | j=1 J 1-a N-k+1-3 N-k+1 1-a-A a x 1-a 1-a N-k+1 = N A11 (l—a 1a 1->a;-; We isolate the factor lim 1 C P[Tk< a+Aa| Tk>a and TIC—1 = ak_1] Aa-o - Aa and call it the instantaneous rate of discovery for the R*" discovery, given that the next discovery is the &" discovery. In purely random exploration, this instan- taneous rate is N-k+1 1-a We say that exploration is c times as efficient as random exploration if the instantaneous discovery rate for the k,, discovery is gof NEAL Y . 1-a The instantaneous discovery rate for the k,, discovery determines the conditional density of T, given T;. For if g, (a ITk_1) is the conditional density for T, given Ti-, and A;(a |Tk_1) is the instantaneous discovery rate, then A,, and g, must satisfy the relation hk(a'Tk_1) == 51:01 Tic—1) 1-]"¢ a (s | Th- whenever a > T,,. Thus A,, and g, each determine the other. Using the above notation we now establish the B6 main theorem from which the estimator of the effi- ciency is derived. Theorem: If the efficiency is c > 0, then T;, . ..., Ty have the same joint density as the order statistics of N independent random variables X,, .., X, taking values on the interval [0, 1] and having the density f(x) = c(l-x¢f -'. Moreover if T,. < a < T,,, for some fixed "a," then k -% i1in(1-T;) has the same density as the sum of k independent ran- dom variables Z,, . . , Z,, each having the density y> 0 < z < - In(1-a). he) ={ 1-(1-a)° 0 otherwise Proof: Let X;, . . , X, be independent random variables with common density [(G) =; 0 otherwise 0 a and Y,-;a and Yk_1a and ¥;.,a and Yk_.(F(a)) Jj=1 (j > (F(a +4a)-F (a)) (1 -F (a 44a))" ***? + (F(a))"“ (1—F(a))A\'—k+| PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING T (N-R+1) (F(a+Aa)—F(a))(1_F(a+Aa)),v_k (1_F(a)).\'—k+l ~ lim Aa-o Aa _ (N-k+1) f(a) (I =Fiay' * (1_F(a))x~k+1 _ f(a) _ e(1 -a) ~* _ c(N-k+1) 1-F (a) (1-a)"" 1-a Hence Y,,... Yy and T,,. ., Ty have the same joint density. The last part of the theorem follows from the fact that Z,; has the density of -In(1-X,) given that X,% a and -3In(1-T;) has the same probability all T; 1x2(1-Y¥,;) which has the same all Y; ln(1-X;). all X,; f f(a,, Sfx ,a‘\')dak+1 P daN a a P[Tk< a and Tk+1>a] kl e® L R &. (1_ai)c_l - a minis. w. -_ (4 Two important features of this joint density are that it is independent of N and that a sufficient statistic for "c" is given by k TT i=1 (4-a;. k s : hed. Hence -y In (1-a;) is also a sufficient statistic i=1 for c given that there are k discoveries. From the previ- ous theorem we can calculate that MATHEMATICAL FOUNDATIONS * B7 E(Z,) 1 II MM a- E II M an Inrn(1-t;) =:1- k - i=1 & 4 1 ;. (L-af In(l-a) c 1-(1-ay We can thus obtain a point estimate, ¢, for c based on @i, . . ,@; by choosing é so that k C <4 \y And=ap= 1 , Leal Ind-0 qs k=1 € 1=(1-=a) Such a ¢ will exist only if —_%_ I; In(1-a;) < ___—1n(21—a) y k=1 and this inequality need not always hold. In the practical problem of oil exploration (see Drew and others, 1980) it is reasonable to assume that c>1. Therefore, in the analysis of oil exploration data we take c to be the solution of (12) if (12) has a solution and if that solution is greater than or equal to 1; otherwise ¢ is taken to be 1. The unconditional joint density of the areas exhausted by the time that "a" of the effective search solid has been exhausted is I k!c" TT (21-4100—l hai -. ,a;) = i=1 (I= N Y1-(1=a¥y:((1-ayw-*, x(k)( (1-a))" ((1-a) and its integral over all (a; , . . , a,,) is the probability of exactly k discoveries. If one attempts to find maximum likelihood estimates simultaneously for "N" and "c," the maximum can occur at N = + » and c = 0. To avoid this, it is necessary to include in the model the condi- tion that the point targets in the effective search solid cannot be too close together because they must corres- pond to nonoverlapping ellipses in the effective basin. Because this condition is difficult to work with, it is decided to estimate the efficiency first by the method described above and then to choose N to maximize N k (1-(1-a)°} ((1-ay)*, which means N is the smallest integer such that RESULTS OF SIMULATION A simulation study was conducted to investigate the behavior of the estimated effective basin size. The well locations were chosen randomly within the basin, and the basin size was estimated using equation (11). Cir- cular disks were assumed; the ratio of the area of a single target to the basin size was 1 to 2,500. The re- sults for three replications are presented in figure 1; basin size estimates are plotted for every tenth well. The estimated basin size is infinite until disks overlap. In order to study the distribution of ¢ as estimated by equation (12), a simulation experiment was conducted for various values of c and N. Estimates for c were calculated at 0.1, 0.2, ..., 0.9 fraction of the area searched, given that at least one target had been dis- covered. The median efficiencies from these simula- tions are presented in table 1. Each simulation was replicated 100 times; however, the median estimates corresponding to c=2 or c=4 sometimes are based on fewer replications when the area searched is less than 0.4 since at least one discovery is required for an esti- mate. The estimated efficiency was set to zero when discoveries were present, but no solution to (12) TABLE 1.-Estimated median efficiencies Efficiency = 1.0 Fraction of Medians Medians Medians area searched N = 50 N = 70 N = 90 0.1 22.2: L.. 0.33 0.00 0.82 Ta Serie .o nes be-: on cess cou. .88 A5 1.07 O ref r ches tees 1:17 1.12 .92 ALi nene eon r e eee enne .83 .94 1.08 egs aie o enne bans .97 .96 1.04 $6... Sone. .L - st Ctg .99 1.13 1.04 aer ieee conn anke .94 1.06 1.00 Tm oe ci o. candle sed cedh . .92 1.03 1.10 iP *. re cued dne ab cane a ame 1.01 1.01 1.03 Efficiency = 2.0 Fraction of Medians Medians Medians area searched N = 20 N = 30 N = 40 0M AE ecs - nenas ean re bae ale mes 1.27 2.30 0.00 Ta An- 260, 204 1.33 1.83 1.95 To cen cednes s 2.10 2.36 1.81 ih . winn presen nen den os 2.08 2.17 2.16 N er einen hees abe duns 1.95 2.11 2.20 MD Pulao ri rss 2.02 1.91 2.02 ace we sen cera ens 2.17 2.01 1.98 TD Ruch cD ate Bs 2.16 1.98 2.02 19 urease 2.05 1.97 2.03 Efficiency = 4.0 Fraction of Medians Medians Medians area searched N = N = 5 N = 10 OL Z. err cedar - e cc cone ae nk 5.30 6.18 3.22 g 3.31 3.64 4.173 4.43 4.21 4.29 3.80 4.39 4.47 4.02 3.84 4.32 4.25 4.21 4.53 4.19 4.41 4.52 4.31 4.35 4.55 4.33 4.37 4.54 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING B8 '0'T sem ozrts xstp oy; pue qo¢'z sem ozts urseq ouf, 'paseq sem ajeuutso ay} yoy Uo s[JaM JO 1aquinu a,} qsutege poyjord azts urseq pageuwmsa Sumoys suor}etnuts azts Uutseq -T #unol4 $7173M 0007 0081 0091 OOL O0TL 0001 008 009 dd 007 0 T T T I T I T I I T I I T T T I T T T I T I T | T I T | T I T | I I I _ T T T coo- | A i] A4 - I F - OSZL _ I / _ - _ - O0St a L i a - .A - OSLL i # L payouees I 'f as russied GZ § E- e : i - 0007 E om eme ............... ...H £ ar "' $,. [>* 7 A ir usosued & "uing « Mrk %s. 1+ : r os 24 /1\\l cip IY ls P x2 _ \\ \/ "A £} # sl Gare v * Anel o ns .se 'v A "pcv' :> i% m als Pl "mo J t ; 7 97 "'A i 3 Neaquet Uit a, \|.\|.|\ payouees \' "1 f £ usosuad p f pape nee rani e cos? - OSLT | russied Qg peyouees i ma jussiad GZ |- | - 0008 -< - OSZE E- - OosE f - O§LE fs - O00P E. - OScH 1 1 1 1 i 1 1 | | | 1 | | 1 1 | 1 1 L 1 1 1 1 | 1 1 1 | 1 1 1 | 1 1 1 | 1 1 1 dost 3ZIS Q3LVWILS3 MATHEMATICAL FOUNDATIONS A B9 existed. The fractions of the area searched were gen- erated from 1-(1-u;)'* where u; is uniform (0,1), i=1,2, . . . ,N. Smaller values of N were associated with larger c's because in petroleum exploration the larger, less numerous deposits tend to be found with higher efficiency. REFERENCES CITED Arps, J. J., and Roberts, T. G., 1958, Economics of drilling for Cre- taceous oil and east flank of Denver-Julesburg basin: Ameri- can Association of Petroleum Geologists Bulletin, v. 42, no. 11, p. 2549-2566. Drew, L. J., Schuenemeyer, J. H., and Root, D. H., 1980, Petroleum- resource appraisal and discovery rate forecasting in partially explored regions-An application to the Denver basin: U.S. Geological Survey Professional Paper 1138-A, p. A1-A11. Schuenemeyer, J. H., and Root, D. H., 1977, Computational aspects of a probabilistic oil discovery model: Proceedings of the Statisti- cal Computing Section, American Statistical Association, p. 347-851. Singer, D. A., and Drew, L. J., 1976, The area of influence of an exploratory hole: Economic Geology, v. 71, no. 3, p. 642-647. Petroleum-Resource Appraisal and Discovery Rate Forecasting in Partially Explored Regions - An Application to Supply Modeling By E. D. ATTANASI, L. J. DREW, and J. H. SCHUENEMEYER GFEFOLOGICAL SURVEY. PROEESSTONAL PAPER | Investigation of field exploration behavior with distributed lag models and a discovery process model, and application of models to predict forthcoming reserves UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1980 FiGurE TABLE p $o po' CONTENTS Page ADStraCt C1 InfrOduCtI0N 22% ...00. -__ corer noel -o pim be acu Be re nica nos ade ne bane nos coals 1 Previous SHIUIGS -. ob oon ceca ae.. 2 Operator behavior at the field 1.0 c. DL LMR L_LLL_ 3 Firm behavior. n n o.oo, - secre- outh 4 Formation of-expectatiohns -->... . _. -. __ =-. .._ l Pllc. n null 4 models... := -->... _ _- caree cels olan n oal enn noc rene - see nage 5 Operator drilling behavior.... -. . 7 EmMpiTIcal resUIts w. - . +82 : . - 2 o eee eee doce e+ o 2 a nals clean anelli ane o an nan nene aad a. 8 Application of the discovery process model predictions ______________________________ 11 Supply Of FeSeTryeS.- _-. Co ens. annal coe eon 15 Contlusi0ns 2. - cer o renee ern aon so 18 References Cited ==.22 2... sy oun oin onin co 19 ILLUSTRATIONS Page Shapes of normalized time profiles for weights associated with values of the lag variable __________________________ C7 Temporal effects of a one-time unit change in the value of total discoveries on wildeat drilling ______________________ 11 Temporal effects of a one-time unit change in the value of total discoveries on drilling expenditures ________________ 11 Residuals for well-drilling models. :.. 3002-000... _...... .so doesnot ot inc 14 Residuals for well-drilling expenditures models - --=. .-.. .. ___.. lull. lll nA oue rel Lene nl dill ove. 15 TABLES Page Estimated coefficients of- infinite lag motels | l Coll cen serie C9 Coefficients of polynomial distributed lag models semiannual observations ________________________________________ 10 Coefficients of polynomial distributed lag models quarterly data _____________________________________LL_L_LLLLL___ 10 Forecast performance ofantegratedisystam oo... Sec lleno Olly ut Biblen ices, 16 I PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING IN PARTIALLY EXPLORED REGIONS- AN APPLICATION TO SUPPLY MODELING By E. D. ATrANAst1, L. J. DrEw, and J. H. ScnurneEmEyEr ABSTRACT This study examines the temporal properties and determinants of petroleum exploration for firms operating in the Denver basin. Ex- pectations associated with the favorability of a specific area are modeled by using distributed lag proxy variables (of previous discov- eries) and predictions from a discovery process model. In the second part of the study, a discovery process model is linked with a behav- ioral well-drilling model in order to predict the supply of new re- serves. Results of the study indicate that the positive effects of new discov- eries on drilling increase for several periods and then diminish to zero within 2% years after the deposit discovery date. Tests of alter- native specifications of the argument of the distributed lag function using alternative minimum size classes of deposits produced little change in the model's explanatory power. This result suggests that, once an exploration play is underway, favorable operator expecta- tions are sustained by the quantity of oil found per time period rather than by the discovery of specific size deposits. When predictions of the value of undiscovered deposits (generated from a discovery proc- ess model) were substituted for the expectations variable in models used to explain exploration effort, operator behavior was found to be consistent with these predictions. This result suggests that operators, on the average, were efficiently using information con- tained in the discovery history of the basin in carrying out their exploration plans. Comparison of the two approaches to modeling unobservable operator expectations indicates that the two models produced very similar results. The integration of the behavioral well-drilling model and discovery process model to predict the ad- ditions to reserves per unit time was successful only when the quar- terly predictions were aggregated to annual values. The accuracy of the aggregated predictions was also found to be reasonably robust to errors in predictions from the behavioral well-drilling equation. INTRODUCTION Interest in the state of the U.S. domestic petroleum industry has resulted in a critical examination of the usefulness of current economic models. There is some question as to whether these models can provide useful information aboout petroleum supply availability, supply price sensitivity, and the effects of alternative policy options on future supply possibilities. The per- formance of current models in terms of predicting fu- ture oil and gas discoveries and supply has thus far been disappointing (MacAvoy and Pindyck, 1975). The poor performance of these models may be caused in part by their almost universal reliance on aggregated data, which obscures the effects of physical exhaustion on the supply response. Although a major reason for constructing empirical economic models is to predict price responsiveness of future supply, the historical data upon which current models are based generally do not contain sufficient price variation to accurately predict future price- supply responses within a region. The price variations present in available data relate to the quality of crude 'oil rather than to the incremental costs of exploration and development of individual deposits. Moreover, the effect of price changes on expected supplies is moderated by the level of resource depletion for a par- ticular basin. Because of the potential interaction be- tween price and resource depletion, the use of highly aggregated data will likely lead to spurious correlation between changes in supply and incremental price changes. Very few attempts have been made to model economic behavior at the level of the exploration play. In this report, an exploration play is defined as the increase in wildcat drilling attributed to and following the discovery of a significant (large) deposit in a forma- tion that was not known to yield significant amounts of oil. Although the major obstacle to modeling has been the lack of basin-specific exploration data, many work- ers believe that field behavior is too erratic or unsys- tematic to model successfully. This study has two parts. The first discusses the temporal properties of operators' exploration behavior for the Denver basin. In particular, the field behavior of operators in terms of responses (wildcat drilling and drilling expenditures) to new discoveries is examined and then compared with results from another basin. Properties of operators' behavior include the duration of the response and the nature of new deposits that must be discovered in order to sustain favorable operator expectations associated with a given area. The stochastic process characterizing the temporal dis- tribution of previous discoveries is used as a basis for modeling operator expectations associated with the C1 C2 value of deposits remaining to be found. The stochastic nature of this scheme is compared with the process characterizing the predictions of the value of remain- ing deposits calculated with discovery process models similar to the ones proposed by Arps and Roberts (1958) and Drew, Schuenemeyer, and Root (1980). In the second part of the study, the discovery process model is linked with the drilling model in a recursive system of equations in order to predict the supply of new reserves. Whereas the purpose of the behavioral drilling model is to explain operator action within a given time period, the discovery process model predicts the number and size distribution of new discoveries as a function of the number of wells drilled. By linking the two models, we can predict the quantity of new reserves discovered per unit time. These predictions are compared to the actual sequence of discoveries. PREVIOUS STUDIES Industry-level petroleum supply models have histor- ically treated exploration as a process using wildcat wells that generates proved reserves. As such, little effort was devoted to formulating testable behavioral hypotheses or extending the theory of the firm to in- clude the exploration function. In early models, no con- sideration was given to integrating the role of the ini- tial resource base into the analysis. Because of geologic differences from area to area and very limited histori- cal data series, the results of those industry-level studies have been somewhat disappointing. Data have been aggregated to the extent that significant behav- ioral and physical relations are concealed by the model specifications. Moreover, although almost all of these models have attempted to determine the price sensitiv- ity of exploration and petroleum supply, price has varied little within specific geologic regions during the time periods chosen for analysis. Alternatively, engineering-type process models (National Petroleum Council, 1972; Federal Energy Administration, 1974; 1976) designed to forecast oil and gas supply have frequently reduced exploration behavior to mechanical rules, assuming, for example, that firms' exploration expenditures are a certain percent of the previous period's net profits. Rather than investigating each of the models in detail, the following discussion focuses on motivation and differences in approach. Perhaps the first widely publicized econometric study of oil and gas exploration was by Fisher (1964). Fisher, who noted that the supply of exploratory drill- ing differs greatly from the supply of new discoveries, used a three-equation model to attempt to explain (1) the annual number of wildeat wells drilled, (2) success ratio (proportion of wildcat wells drilled which resulted PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING in a discovery), and (3) the average size of the discov- ery. Fisher distinguished between exploration at the intensive and extensive margins. Exploration at the extensive margin yields discoveries that are charac- terized by a relatively low frequency and a large size, whereas exploration at the intensive margin yields relatively small discoveries that occur with greater frequency. Consequently, Fisher specified both the suc- cess ratio and the discovery size as a function of eco- nomic variables and also asserted that short-term reac- tion to increases in price results in a shift of explora- tion to the intensive margin. In the three-equation model, the number of explo- ratory wells drilled is specified as a function of the price of oil, geophysical crew time, and lagged values of the average size of oil and gas discoveries, depth, the success ratio, and regional dummy variables. The equation for the success ratio includes the price of oil, geophysical crew time, and lagged values of the success ratio, depth, and average size of oil and gas discoveries. For the final equation, the average-size oil discovery is specified as a function of the price of oil and the previ- ous period's values of the average size of oil and gas discoveries and the success ratio. In order to increase the number of degrees of freedom, short-time series data for large, geologically heterogeneous areas were pooled. Intercept dummy variables were included to account for differences in individual areas. Although the model specification was not derived from a firm's decision process, the explanatory power of the esti- mated equations was adequate. For predictive pur- poses, however, the price responsiveness of the new discoveries is unlikely to be the same for all the areas from which the data were taken. Furthermore, varia- tions in the historical price data used by Fisher were the result of differences in the quality of the oil rather than incremental production costs. Erickson and Spann (1971), working to model the supply of natural gas, elaborated on Fisher's original formulation by in- cluding an equation for the average annual size of gas discoveries. The functional form of the gas supply model is similar to the functional forms used by Fisher: in modeling oil supply. MacAvoy and Pindyck (1973, 1975) modeled and in- tegrated the components of natural gas demand and supply. Like Fisher, they modeled the supply of new discoveries by predicting the number of wells drilled, success ratios for oil and gas exploration, and the ex- pected size of oil and gas discoveries for individual Pe- troleum Administration Districts. The supply of re- serves may be calculated from the success ratio, wells drilled, and expected size. Both the success ratio and the expected discovery size represent (decline) extrapo- lations from a calculated reference size and success an AN APPLICATION TO SUPPLY MODELING ratio. These variables are also sensitive to field prices and signify whether firms are operating at the inten- 'sive or extensive margins. The well-drilling equation is based upon the expected returns and the variance of returns which are, in turn, calculated from the ex- pected size of discovery and the success ratio. In par- ticular, the annual number of wells drilled is a function of the petroleum district dummy variables, expected returns, the variance of returns; and a drilling cost index. Although recent revisions (Pindyck, 1978) of the model are logically similar to previous versions, the success ratio and expected size of oil discoveries were made independent of price and dependent upon re- gional physical characteristics. These revisions led to reductions in previous estimates of the price respon- siveness of new oil discoveries. Two other approaches to modeling new discoveries were made by Khazzoom (1971) and Epple (1975). Khazzoom (1971) estimated the volume of gas discov- ered from the two-period ceiling averages for the price of natural gas, price of crude oil, price of natural gas liquids, and previous period's volume of gas discovered. The model does not explicitly include a variable associ- ated with the exhaustion of undiscovered deposits. Using a unique approach, Epple (1975) considered ex- ploration as a production process that used wells and oil-bearing land as inputs. Exhaustion of the oil- bearing resource is explicitly considered to hp repre- sented by the productivity and input cost of oil-bearing land. The model attempts to predict the value of new discoveries of crude oil and natural gas by using a joint production function. Specifically, Epple assumes a form for the production function," then derives the firm's op- timizing conditions under the assumption that it maximizes the net present value of exploration effort. Parameters of the oil-bearing land supply function are estimated from aggregated U.S. oil and gas exploration statistics. Although the analytical approach that Epple takes is novel and some aspects of resource exhaustion are considered, the use of aggregated data and the ab- sence of detail relating to the spatial distribution of deposits restrict the applicabilty of the model. Along similar lines, Uhler (1976) developed a stochastic production function for the discovery of new petroleum reserves. The marginal exploration cost function, derived from the production function, in- cludes the following variables: an index of field knowl- edge, an index of physical exhaustion, and wells drilled. One distinguishing feature of the analysis is that it considers a disaggregated area. Because the study concentrates on the marginal cost of new discov- 'The supply of oil-bearing land is specified as L = Ro» where R is unit rent, w is past exploratory effort; with r > O and B > O and defined as estimated parameters and e is the Naperian constant. C3 eries, it does not attempt to explain the behavioral de- terminants of wildcat drilling. Cox and Wright (1976) abstracted data from the ex- ploration process by considering all drilling ex- penditures as investment in reserves, which are treated as inputs in the production of crude oil. The objective of their study was to link investment in re- serves to government policies such as import quotas, prorationing, and special income tax provisions. Al- though the empirical results are impressive, the au- thors do not consider exploration investment apart from reservoir development investment. Furthermore, they do not consider the influence of uncertainty or resource exhaustion in their investment decision models. In summary, economists appear to have taken two approaches to modeling firm exploration behavior. For the first approach, the number of prospects, labor, and capital services serve as inputs to the petroleum re- serve production process called exploration. For the second, reserves are considered an input to the crude oil production process. Exploration is then regarded as similar to firm investment in physical capital equip- ment. To some extent, both approaches are correct, depending upon the nature of the firm under considera- tion. Firms engaged primarily in exploration regard discovered reserves as the end product, whereas verti- cally integrated firms might view reserves as inputs. Exploration can be regarded as an investment or input in the production of an inventory of reserves that re- sults in crude oil production. As an intermediate prod- uct, reserves will undergo additional modification in. the production process in order to generate final prod- ucts. OPERATOR BEHAVIOR AT THE FIELD LEVEL With a few exceptions, the studies just reviewed have been concerned with modeling exploration behav- ior when data are highly aggregated. Frequently, data limitations necessitate such spatial and temporal aggregation. However, the testable behavioral hypoth- eses generated from these models are, in general, quite limited. In order to examine operator behavior, in more detail, data specific to the Denver basin are used in this study. In this area deposits were usually found in a single formation; the only exploration play that oc- curred during te period examined was stratigraphic in nature. Because the initial discoveries were made ear- lier than the beginning of the historical time series of data used in this study, the behavior examined relates to exploration at the intensive margin. With the excep- tion of the Union Pacific Railroad acreage (Drew and others, 1980), mineral rights were regularly bought and sold. The following discussion concerns the explo- C4 ration decisionmaking of firms, specifically, how they formulate and respond to expectations of undiscovered deposits remaining in the basin. FIRM BEHAVIOR Investment in production of petroleum reserves may take several forms. Additional reserves can be devel- oped by extension drilling (leading to revised estimates of reserves), investment in improved recovery methods (enhanced recovery) from existing reservoirs, and wildcat drilling for the discovery of new deposits. Within the individual firm there are tradeoffs in costs that must be considered when using any one of these three sources of additional petroleum (White and others, 1975). For example, development drilling can be done with borrowed funds, whereas only equity cap- ital can be used to pay for exploratory drilling. The theory of firm behavior under production uncer- tainty has only recently been treated extensively in the economic literature (see Fama, 1972; Leland, 1974). If it is assumed that firm managers attempt to maximize the net present value of the firm (perhaps in terms of outstanding capital stock), a capital market valuation model might be posited. For purposes of explaining overall firm exploration decisions, such a model may be particularly applicable because generally only equity capital is used for exploration expenditures. Among the alternatives available to the firm for obtaining ad- ditional reserves-that is, wildcat drilling, extension drilling, and enhanced recovery-wildeat drilling clearly is the most uncertain. Firms commonly restrict their exploration activities to a small set of areas where they have had previous experience or where the firm plans a prolonged exploration program. For a given area, early exploration investment can be viewed as investment in the current set of prospects that pro- vides additional information about other prospects in the area. The decision regarding when and in what areas to initiate an exploration program and at what times to allocate expenditures between exploration, ex- tensions, and enhanced recovery are made at the firm level. However, the decision on when and where to drill strategic prospects in an area is frequently the respon- sibility of the field managers (Kaufman, 1963). This description of the optimizing process of a firm is non- technical. For a technical discussion of the optimal ex- ploration, production, and capital investment policy when the firm operates in geologically diverse regions, see Attanasi (1978). At the field level, the operator is faced with the prob- lem of allocating a fixed amount of funds to a number of prospects over a given planning period. For larger firms operating in many areas, a marginal amount of exploration funds might be reallocated within the typi- PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING cal planning period. The optimal search policy for a given set of prospects may be expressed as a function of the expected profit, marginal opportunity costs, and expectations associated with the size (and spatial) dis- tribution of remaining deposits (Attanasi, 1978). Field expectations are formed when operators make explora- tion decisions based upon unknown or uncertain pa- rameters. For the shortrun model considered here, un- observable operator field expectations appear to be the most difficult components of the determinants of explo- ration to measure. FORMATION OF EXPECTATIONS Expectations are formed when economic agents (firm managers or consumers) are required to make deci- sions based upon the unknown value of a particular variable. The expectations or predictions that are formed are then used in the decision process. In tests of the theory of the formation of expectations in economic models, the modeler may attempt to gather data di- rectly from economic decisionmakers or construct a prediction scheme based upon observable characterts- tics of the specific decision problem such as market prices or output. Traditionally, these prediction schemes, although typically rather simple, rely almost entirely upon past values of the variables to be forecasted. A serious limitation of nearly all the exist- ing expectations models is that their estimates are formed without consideration of the decisionmaker's criterion function. Embodied in the decisionmaker's criterion function are attitudes toward risk and conse- quences of under- and over-estimating the value of the uncertain variable. Originally, expectations models were developed in connection with agricultural price supply responses (Ezekiel, 1938; Nerlove, 1958). Farm- ers must commit land and labor to the production of specific crops before the market price of the crop is known. The farmer may base his predictions of the current period's price on the previous period's realized price. In particular, suppose p? is the expectations var- iable and p, is the observed value at time ¢. The static expectations formulation is defined as (1) If the current period's expectation is a weighted combi- nation of the realization and expectation of the previ- ous period, that is, 0 < a < 1, (2) pt = pts + a (fri - pFf-1) then expectations are said to be formed adaptively. In the equation above, « is the coefficient of adaptation and controls the degree to which expectations will con- AN APPLICATION TO SUPPLY MODELING form to the previous period's realization. Expectations are said to be regressive if PF = + aabe-a + @sBi_s + . ... , (3) with a,> a;> a; the weights are positive and approach zeros with the passage of time. Alternatively, expecta- tions are extrapolative, if pF =a, (131-1 “fit—2) + a; (13,72 ag laps) + .. .,(4a) and | PZ = aip-i + (@2-@i)D;-; + (as“a1)$r—-3 * ;... . (4b) where the weights a; are also positive and decline. In terms of the final coefficients of the laggfd terms of p,, the coefficients (equation 4b) are negative and ap- proach zero. Turnovsky (1969) has shown how a Bayesian updating scheme for uncertair4 values of the distribution parameters of the stochastic decision vari- ables results in a general expectations formation scheme yielding as special cases staticjextrapolative, and adaptive expectations. | As suggested by these formulations and others fre- quently found in the literature, the estimated form of the unobservable expectations variable is based on the autoregressive process® of the historical realizations of the decision variable. In fact, a relatively unrestrictive definition of rational expectations is that the generated predictions and realizations follow the same autore- _ gressive scheme. A reason for this definition is that autoregressive forecasting schemes provide optimal predictors for a wide range of stochastic processes. However, these predictors are still constructed without recourse to the decisionmaker's criterion function. Moreover, the criterion functions for the prediction and decision problems will, in general, diffel; because the consequences of errors in the estimates differ. Distrib- uted lag models are frequently applied when the unob- servable expectations variable is approximated by the autoregressive formulation of the historical realiza- tions of the variable of interest. If it is assumed that the decisionmakers have knowledge of the underlying stochastic structure that generates the time series of the variable to be forecasted, then the distributed lag proxies are equivalent to calculating the conditional expected value of the uncertain variable (Pesando, 1976). DISTRIBUTED LAG MODELS - Unobservable expectations variables are frequently modeled with a distributed lag function of the histori- finite linear aggregate of previous values of the process. The nature of the stochastic process might correspond to an autoregressive model, moving average model, or combination of the two models. For more detailed definitions, see Box and Jenkins (ITO, p. 7-11). * The term autoregressive process used here refers to a model tha}can be expressed as a C5 cal realizations of the variable of interest. Simply, dis- tributed lag models provide the basis for analyzing the temporal response of the dependent variable (y,) to a change in a specific independent variable (x,). In par- ticular for any given T, it is assumed that c + ..., (5) where __ &, =u, i=0,1,... ,T OX) and e, is a stochastic error term. Two fundamental problems arise if one were simply to apply ordinary least squares to equation (5) to esti- mate all w's. First, if T is large, there may be too many parameters, leaving few degrees of freedom to make statistical inferences about parameter values with any degree of confidence. Second, there is likely to be a high degree of collinearity among the lagged values of the independent variables. Consequently, in order to con- serve available degrees of freedom and improve effi- ciency of the statistical estimates, a lag generating function is chosen to correspond to a given time profile that characterizes the response of the dependent vari- able to changes in a particular independent variable. By assuming a specific analytical form for the generat- ing function, the number of parameters required to be estimated can be made quite small. For example, sup- pose the decisonmaker's expectation x} is adjusted adaptively as an observation of a realization &,, is made, that is, x =x, %],) (6) where 0 < a < 1. If \ is defined by A = 1- a, then the expectations variable may be calculated by the follow- ing equations: #3 MSX 1 xii: = (1_A)[£’71 + X£1_2 + AzxA1_3 +. shale ]. (7) Hence, given the derived structure, a single parameter A can be used to generate values of the unobservable expectations variable from the historical data associ- ated with x, .;. Suppose it is assumed that Yr = xf. (8) Then recursively substituting (7) into (8) and the lag- ged form of (8) into the resulting equation, the estima- tion form of equation (8) is yt:(1—)\)-72t—1 + AYr-1 (9) C6 or using the lag operator L the equation is sme (1—Afit—l % t= x1" Consequently, by deriving the specific form of the lag generating function, the estimation of the distributed lag structure of the model can be reduced to a small number of parameters. Jorgenson (1966) has shown that any arbitrary lag function can be approximated by the rational form* - AU) Yt ————B(L) X; (10) n ail B(L) = % b.L. (0 j =0 m where - A(L) = % L == A rational lag function means that the function can be expressed as the ratio of two polynomial (lag) func- tions. That is, any finite or infinite response function can be approximated by the rational form (10). For example, a form of the geometric function results when A(L) = a I and B(L) = 1 - AL, that is, (aI) X;. ALL s (11) (A-AL) Vit iT Alternatively, by specifying the denominator in (10) to be equal to a constant (unity), a finite distributed lag function results. In general B(L) of higher degree than 2 or 3 might be difficult to identify uniquely. The statistical methods used to estimate such models are complex because the estimation routines are highly nonlinear. In addition, the distribution of the descrip- tive statistics are known only for large samples (Dhrymes, 1971). In terms of temporal behavior, the implications of particular distributed lag schemes are not always ob- vious from the estimated structural equation form. For model comparison and selection, certain standard properties of the implied function, which are more in- tuitively interpretable, are frequently considered. Two such properties are the normalized time profile of the distributed lag response function and the average lag length. The time profile represents the temporal distri- bution of the effects on the dependent variable of a unit change in the independent variable. The general shape of the time profile indicates how rapidly and in what direction a change in the independent variable affects the dependent variable. Changes in the units of meas- urement of either the dependent or independent vari- "The general form of the rational distributed lag function may be derived from an exten- sion of the adaptive expectations hypothesis (see Dhrymes, 1971). PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING ables affect the shape of the time profile. Consequently, when the time profiles for alternative distributed lag models are compared, they must be taken to use the same units of measurement. The time profile may ex- hibit a variety of forms (fig. 1). Some common ones include an exponentially decaying function, an in- verted "V" shape or a function with weights oscillating between positive and negative values. In some cases, the general shape of the time profile can be inferred from the functional form of the distributed lag model. However, the time profile weights can always be calcu- lated using the estimated function and tracing the ef- fects on the dependent variable of a unit change in the specific independent variable. Another property of distributed lag functions that may serve as a basis for comparison is the average lag for an n period lag function, which is calculated as n > 65 1-0 . n- > wi p¥ t=0 where ¢ is the period subscript and w is the associated weight. The average lag O turns out to be a simple weighted average of the time periods where the weighting is proportional to the time profile weights. The average lag reflects how the various values of the time profile weights are temporally distributed. Values of the average lag must be interpreted in relative terms. That is, larger values of 0 indicate that much of the distributed lag effect is felt at larger values of t, and smaller values of 0 indicate that the weights asso- ciated with earlier lag periods are relatively larger than for the later ones. The time profile and average lag represent prop- erties of the distributed lag model that are easily in- terpreted. Frequently, an individual researcher may have subjective prior knowledge of the form for the time profile and attempt to use this prior knowledge to evaluate the empirical model. For example, suppose a distributed lag model is estimated representing the ef- fects of increases in advertising on product sales. One might reasonably expect the time profile to indicate the effect of a one-time increase in advertising to be expo- nentially decaying, but it would be unreasonable to expect the function to indicate explosive or monotonically increasing effects. The time profile pro- vides a means whereby the estimated distributed lag model can be evaluated for consistency with expected economic behavior. OPERATOR DRILLING BEHAVIOR Assuming the type of decentralized decisionmaking that was previously described, the field manager's re- (12) AN APPL110ATION TO SUPPLY MODELING 1.4 T T T T T T T T T T T T LAG WEIGHTS =-0.2F- ~ -0.3 i 1 1 1 1 1 i 1 1 i 1 1 1 2 3 6 8 9° #0 {11 12 LAG PERIOD 1.-Shapes of normalized time profiles for weights associated with values of the lag variable. Analytical specifications of the distributed lag functions are adjacent to each of the time profiles with L defined as the lag operator. source allocation problem might then be specified as a sequential decision or adaptive control problem. With- out going into an involved mathematical deFvation, the optimal shortrun search strategy might be specified as a function of profit expectations, perceived field risks, and expected opportunity costs of foregone alternatives. Field managers are assumed to use avail- able information to predict the spatial and size distri- bution of undiscovered deposits. The optimal sjaarch ef- fort might be specified as a function of expected profit # and expectations associated with deposits left to be found x*. Depending on the nature of the area, a vari- able k might also be included to index the current state of field knowledge relating to the original distribution of deposits and extent of physical exhaustion. The op- timal search effort may be specified by the following function: yi=f(m, X, ko. (13) To estimate a specific functiohal form of equation (13), the variables #, and &, must be defined. It seems reasonable to assume that there is a lag between the C7 firm's exploration allocation decisions across regions and the reporting of profits from which #, is assumed to be estimated. In the empirical models to be presented, the expected profits variable was computed as the lag- ged value of returns per wildcat well. Returns per ex- ploratory well were calculated by multiplying the quantity of the dollar value of oil found per successful wildcat less the cost of the well by the success ratio. This variable was assumed to reflect the general level of a basin's exploration profitability. The variable &, is assumed to reflect the relative degree of knowledge operators possess about a particular set of targets and the extent of physical exhaustion. This relative index of experience and exhaustion k was calculated on the basis of the weighted average of the cumulative number of wells drilled in a specific target area over the previous three periods.* The index as defined would have greater significance and intuitive appeal if the area under consideration exhibited several plays or target formations. Because a large block of acreage in the Denver basin was restricted from exploration, the index was calculated on the basis of areal extent with- held rather than on the basis of target formation (see footnote 4). The index specified in this fashion captures the effects of sampling deposits without replacement and also will increase as economically exploitable de- posits are exhausted. The wildcat drilling data and re- serves data were obtained from the Well History Con- trol File of Petroleum Information, Inc. Drilling costs were calculated from various annual issues of the Joint Association Survey of the U.S. Oil and Gas Producing Industry (JAS) (American Petroleum Institute, 1953, 1955-56, 1959-73). The study area was partitioned into 88 smaller units (of 625 mi? or 1,619 km?) for pur- poses of assigning target depths for wildcat wells drilled within each cell. Well costs were calculated from the inferred target depth, and costs per foot were taken from the JAS annual summaries for Colorado. Although some natural gas deposits were found, operators were searching predominantly for new oil deposits. The oil found was of a relatively uniform quality. During the period from 1949 to 1972, prices were relatively stable and ranged from $2.63 per barrel to $3.46 per barrel. However, during 1973 prices for new oil increased to over $8.90 per barrel. Initially, the general estimated form of equation (13) was +B: m-+f (D,) + yhki+€, (14) 'If P, is the proportion of wells drilled in the i*" formation or sub basin at time ¢ and C; is the cumulative number of wells drilled in the i*" formation or subarea, then the experience- m Bl exhaustion index is defined by k= % Ci + (I3)( % p/). _ The Denver ba- i=1 }=1 sin had two subareas and a single producing formation. The areas were partitioned into acreage excluded from exploration and held by the Union Pacific Railroad: other acreage corresponded to the rest of the basin which was open for exploration. C8 where f (D,) represents the general lag structure and x* | = f(D,). The variable y, represents search effort ex- pended as measured by either wildcat wells drilled or estimated drilling expenditures, #,-; is the expected value of returns per wildcat well lagged two periods, D, is the value of new discoveries equal to or greater than a given minimum size class of deposit for a specific time period, and &, is the index of field experi- ence described earlier. The lag function of the dollar value of the oil found in certain size classes of deposits per time period is assumed to reflect changes in the operator's expectations regarding the distribution of deposits remaining to be found in the basin. The vari- able D, was defined with respect to a minimum size class of deposits for two reasons: Larger deposits gen- erally have lower production costs; moreover, the oc- currence of larger deposits is frequently taken to be indicative of the presence of deposits of a similar size. Estimated empirical models are presented in the fol- lowing two sections. In the first section, alternative forms of the lag function were tested; the final model took the form of a finite lag function with lag weights constrained to lie in a second-order polynomial. Infer- ences about operator behavior are made from the esti- mated models. On the basis of these results, inferences about operator behavior observed in the Denver basin are compared with inferences about behavior observed in the Powder River basin (Attanasi and Drew, 1977). In the next section, the estimated quarterly model is compared to the model estimated from semiannual ob- servations for consistency in interpretation. The dis- covery process model similar to the model presented in Arps and Roberts (1958) is then used to calculate the value of deposits remaining to be found. The perfor- mance of this type of model and its apparent efficiency in the use of information contained in the drilling data suggest that the model predictions are equivalent to the expected value of the uncertain variable. A more restrictive definition of a rational expectation is that it efficiently uses all available information about the un- certain variable. It would therefore be useful to com- pare the statistical properties of the ad hoc distributed lag model with the performance of an empirical drill- ing model that usés the (rational) predictions of the value of deposits remaining to be found calculated with the discovery process model. EMPIRICAL RESULTS Several forms of the distributed lag function f (D,) were fit to the historical drilling data in order to arrive at a reasonable description of the discovery process. In evaluating alternative empirical models, several criteria are frequently applied. The model structure and estimated coefficients should first be consistent PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING with economic theory. Other criteria relate to the stan- dard statistics that describe the quality of the fit of the model, that is, coefficient "?" statistics, coefficient of determination and the standard error of the regression. Finally, the predictive performance of the model can be examined by using part of the historical sample to es- timate the model and the remainder of the sample to compare model forecasts with actual sample values. The specific lag forms that were tested included the geometric, second-order rational, and finite polynomial distributed lag functions. The geometric lag structure is derivable if it is assumed that expectations are formed adaptively." If, in addition to an adaptive ex- pectations hypothesis, it is assumed that reactions to changes in expectations exhibit some inertia-that is, only a partial immediate adjustment (see Johnston, 1972, p. 302-303)-a second-order rational lag model results. Motivation for the finite polynomial lag func- tion was not based on any specific set of assumptions that explain the process by which expectations are formed. However, the empirical results for the finite lag distribution will provide a useful comparison to the results obtained for alternative infinite lag forms, that is, the geometric and second-order rational functions. An infinite lag distribution means that the effect on the dependent variable of a change in the lag variable continues over an infinite number of time periods. The parameters for the geometric and second-order lag functions were obtained using a nonlinear iterative regression technique that yields maximum likelihood estimates (Pierce, 1971). For the initial test runs, D, was defined as the aggregate value of discoveries per time period of deposits of at least 500,000 barrels of reserves. Prior to carrying out the estimation proce- dures, the data were tested for seasonal variations, which were removed from the quarterly data using the Census X-11Q procedures (Shiskin and others, 1967). Table 1 presents the parameters estimates for models based on semiannual and quarterly observations. Simi- lar results were obtained when #,, was used in place of ,~, and when alternative assumptions were imposed on the error term. Comparisons of the coefficients of the lag terms associated with the models estimated from "For this derivation see the preceding discussion and Attanasi and Drew (1977, p. 57). The estimated form of the geometric lag function was B 1 D t + 327?le + vite, (1-AL) 4=Bo + where y;, D;, 9,4, R, were defined earlier and B,, A;, A, B», and y are estimated coefficients. L is the lag operator. The stochastic component of the model is embodied in e; and was permitted to have autoregressive and moving average components. *The second-order rational lag function that was estimated took the following form: = B U = Y; # Be + + Bam + vi, + 6 t o d- xLixt t 2TTi~2 t t where the variables y,, D,, #,-;, k, were defined earlier and the estimated coefficients are Bo, Bi, B», A1, As, and y. L is the lag operator. The error term e, was assumed to have autoregres- sive and moving average components. AN APPLICATION TO SUPPLY MODELING TABLE 1.-Estimated coefficients of infinite lag models [Numbers in parentheses are "?" statistics, and R* is the coefficient of determination} Geometric: Drilling: Semiannual ____-488.0 _ 0.00102 0.852 -0.0173 0.066 _ 0.735 (4.1) (8.2) (52.5) (8) (5.8) Quarterly ________ =114 0003: 899 -.0052 s 806 (1.4) (11.7) (100.2) (1.0) (6.7) Drilling expenditures: Semiannual .._. -2768.0 00450 881 -.0690 320 766 (4.6) (7.9) (68.8) (7) (5.7) Quarterly _._______ -40.4 00161 9 -.0241 0258 830 (.7) (12.2) (130.6) (1.0) (3.6) Second order: Rational Lag Drilling: Semiannual ....-3944 0.00064 1.284 -0.389 -0.0266 0.0589 0.766 (3.5)(3.9) (7.9) (2.6) (1.3) (5.4) Quarterly ._______ -15.3 00019 1.491 -.543 -.00324 _ 840 (1.3)(4.0) (10.2) (3.8) (T) (7.6) Drilling expenditures: miannual _.-2370.0 0028 1.270 -.354 -.100 290 .166 (3.9)(3.1) (5.8) (1.8) (1) (5.0) Quarterly ________ 28.4 00096 1.363 -All -.0192 0262 _ .843 (.5)(3.9) (7.0) (2.2) (8) (3.6) the semiannual and quarterly data show surprising similarity. Parameter coefficient estimates of the ex- pected profits and the field exhaustion variables which are negative and positive, respectively, are inconsis- tent with prior theoretical expectations. The negative expected profit coefficient is, in part, due to multicol- linearity between #, and D,. Because the empirical es- timates were inconsistent with the theoretical restric- tions, both infinite lag forms were rejected. There seems to be no a priori reason for restricting the distributed lag function to an infinite lag distribu- tion. A finite lag model was estimated in which the lag weights or coefficients were constrained to lie within a polynomial function (Almon, 1965). As discussed ear- lier, behavioral interpretations associated with the derivation of such lag models are limited. The form of the model used to study wildcat drilling expenditures is 3 m Yi C Bi t Bi ms + i210 @-iDie- + yr + €. (15) Coefficients ,.; are the weights associated with (¢+i)th period and reflect the temporal effects of a change in D;. on operators' exploration effort. Assuming that a finite lag model is appropriate, it is important to de- termine the appropriate lag length for D,; and the de- gree of the polynomial containing the lag weights. Owing to the presence of serial correlation in the data, the application of ordinary least-squares regres- sion procedures in the linear equations would have re- sulted in inefficient parameter estimates. The Cochrane-Orcutt procedure, which is described in de- tail in Kmenta (1971, p. 287-289), was used to gen- erate efficient estimates. This procedure amounts to the iterative application of ordinary least squares on data that have been transformed using an estimate of the serial correlation coefficient p. Ordinary least C9 G btain an 11! ata are squares is initially applied to the raw data to estimate of p from the residuals. The e proce- ange in p. The consequence of understating or overstating the true lag structure is the introduction of a specification error (Schmidt and Waud, 1973). If the finite la weights are interpreted as representing the response of operators' search effort to new di then certain restrictions should be placed or weights to be consistent with economic the coveries, the lag this fashion if they oscillated between positi negative signs, or if they monotonically inc propriate lag length be determined after es the model for a number of different lengths and that the final choice be made on the basis of maximizi adjusted coefficient of determination or mini 'zing the standard error of the regression. In this study, the lag structure was determined by searching ove 1; various combinations of lag lengths, and the choice of the final form was based on minimizing the standard error of the regression equation. In this search, D,; represented the value of new discoveries found in deposits of at least 500,000 barrels. Estimates of coefficients for the finite distributed lag models are presented i ‘ tables 2 and 3. For the model based on semiannual data, the appropriate choice of lag length was five periods', with the coefficient for the fifth period restricted to zero. The appropriate lag length for the model based on quar- terly observations was found to be 11 periods with the coefficient for the final period restricted to 0. When the initial period is also taken into consideration, the em- pirically determined lag lengths for the two models (semiannual and quarterly) are consistent. The time profile indicated by the lag weights have the shape of an inverted "V". Using the semiannual data, the finite lag models were reestimated with D, redefined to represent the total value of new discoveries of deposits greEer than imating 2.5 million barrels and 5 million barrels. A co parison of the estimated models and associated statistics across * One problem that arises in determining the appropriate lag length and that has not been addressed in the literature is whether the models of varying lag length should be estimated using the maximum sample size or using identical sample sizes. For the model specifications that were tested, the five-period model estimated from the maximum sample size had the minimum standard error of the regression, and the seven-period model had the minimum error when alternative models were estimated using identical samples. C10 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING TABLE 2.-Coefficients of polynomial distributed lag models, semiannual observations [Specifications A and B are based on 46 observations from exploration activity in the Denver basin. Specification C is based on 29 observations from exploration activity in the Powder River basin. R*, adjusted coefficient of determination; D.W., Durbin-Watson statistics; S.E., standard error of regression. Numbers in parentheses are absolute values of "¢" statistics] Wells 213.9 0.0340 -0.271 0.279 0.579 0.634 0.441 ~0005 ,: : Als et cl lode i Aideen «Hews 0.756 2.3 60.5 (2.2) (.6) (1.1) (1.6) (3.5) (4.1) (4.2) (4) Fall's 184.6 0330 -227 309 1600 645 445 .005 0.005 -0.001 -0012 _ .749 _ 23 61.4 (1.5) (.6) (.8) (1.6) (3.5) (4.1) (4.2) (2) (2) (2) (2) C 47.5 0054 -.061 19 302 269 094 -.053 -.051 -.006 117 .940 2.1 13.6 (5.4) (2.4) (1.5) (6.8) (11.3) (11.2) (2.7) (5.0) (4.9) (1.4) (5.1) Expenditures A msict=s 2098.0 0.5180 _ -1.420 -0.072 0.744 1.028 0.780 C0165 _ . ae 0.756 - 26 325.8 (3.2) (1.8) (1.0) (1) (T) (1.1) (1.2) (2.1) Bjs..z. 1577.0 5090 -.976 272 994 1.189 .858 092 0.080 0.036 -0.257 156 24 325.9 (2.1) (1.7) (D (3) (1.0) (1.3) (1.4) (.6) (5) (1.5) (9) C 9324 0410 __ -1.740 25 1271 1313 381 -.545 -.571 077 ose - 164 22 160.1 (7.8) (1.6) (3.8) (T) (3.8) (4.3) (4.2) (4.4) (1.5) (3.4) TABLE 3.-Coefficients of polynomial distributed lag models, quar- terly data [R', coefficient of determination; D.W., Durbin-Watson statistic; S.E., standard error of regression] Drilling Drilling expenditures Coefficients "C" statistics Coefficients "¢" statistics 2.2 801.0 2.9 1.7 .0275 .9 A -.0546 1.7 .3 126 4 2.1 401 1.3 3.2 .620 2.1 4.0 181 2.4 4.3 .884 2.5 4.4 .930 2.6 4.4 .919 2.6 4.4 .850 2.6 4.4 124 2.5 4.4 .540 2.5 4.4 .299 2.5 .912 1.98 94.89 these deposit size categories indicated little difference in terms of the overall model form or the explanatory power of the model specifications.® It might reasonably be concluded that operator expectations regarding the value of remaining deposits were sustained by the quantity of oil found per time period rather than by the discoveries of specific size deposits. This finding is con- sistent with results of a similar study carried out for the Powder River basin (Attanasi and Drew, 1977). Both the profit and field experience exhaustion vari- ables are statistically more significant in the ex- penditures equation than in the drilling equation. In order to determine if the experience-exhaustion index exhibited systematic intertemporal effects, a lag structure was imposed on the variable k,, and the equa- tion was reestimated. Results of these specifications are presented in table 2. None of the lag coefficients was significant in both the drilling and drilling ex- penditures equations, nor was there any improvement in the explanatory power of the equations. ® The adjusted coefficients of determination (R®) for a five-period lag when the minimum size classes of deposits are defined as 2.5 million and 5 million barrels are 0.7434 and 0.7455 for well drilling and 0.7131 and 0.7550 for well drilling expenditures, respectively. Neither drilling or drilling expenditures are perfect measures of exploration effort. Drilling activity, al- though easily interpreted in terms of sampling effort, cannot differentiate between the wells drilled under varying depths or qualitative degrees of difficulty. Al- though drilling expenditures may more completely re- flect the economic dimension of search effort, they can- not fully represent the temporal behavior because the costs of land acquisition and preliminary geophysical work are not included. The negative coefficients asso- ciated with the first periods of the distributed lag models based on semiannual observations appear to be the result of two factors. First, drilling and drilling expenditures do not reflect the proper magnitude of search effort in the early stages of exploring a prospect. Second, adjustments in onshore exploration activity would be more readily captured if observations were associated with shorter time periods, as shown by the model based on quarterly data. The estimated responses can also be compared to re- sults obtained for exploration activity in the Powder River area (specifications C, table 2). In the study of that area, the geometric, second-order rational lag and finite lag models were also estimated. The model that most appropriately described the data was again found to be a finite lag distribution. The empirically deter- mined lag length for the Denver basin was five semiannual periods with the fifth-period coefficient constrained to zero. The appropriate lag structure for the Powder River basin was determined to be four periods with the coefficient for the fourth period uncon- strained. Major differences between the estimated re- sponse functions are related to the differences in the magnitudes of the lag function weights and the significance of the field experience-exhaustion vari- able. The lag coefficient on the expenditures equations are comparable. However, the values of the lag coeffi- cients for the drilling equation for the Powder River basin area are about half of those for the Denver basin. AN APPLICATION TO SUPPLY MODELING This result, in part, reflects the fact that wells were two or three times as expensive to drill in the Powder River basin. Because there were three exploration plays and four different target formations in the Powder River basin, the field experience-exhaustion index varied widely. The estimated coefficients are consistent with this observation and indicate that exploration effort in the Powder River basin was significantly influenced by the current state of experience or exhaustion. Differ- ences in the estimated relations appear to be reason- ably consistent with differences in economic and geologic factors that are reflected in the specific basin's exploration history. In order to compare the implied temporal properties of the alternative lag schemes, the time profiles of the geometric, second-order rational lag and polynomial (finite) distributed lag schemes are presented in figures 2 and 3. Infinite lag functions consistently overesti- 40 T r- f--- -+- -3 T T T 35 |- a 0.00019 1-1.49L+0.54L2 30 |- - Finite lag model 25 ia 20 15 |- 10 [- LAG WEIGHTS, IN HUNDRED-THOUSANDTHS 1 1 1 1 1 1 1 1 A2 3.45 6. "o A6 1 LAG PERIOD .. 12 13 14 FiGur® 2.-Temporal effects of a one-time unit change in the value of total discoveries (in thousands of dollars) of at least 500,000 barrels on wildcat drilling. Specifications of the lag function are adjacent to each of the time profiles with L defined as the lag operator. 17 I I T I I I I I I I C11 0.00161 1-0.92L 6.00096 1-1.36L+0.41L2 LAG WEIGHTS, IN HUNDREDTHS #. te lag odel 1% 2% za 4 "b. % 7 s 9. 10 LAG PERIOD Figure 3.-Temporal effects of a one-time unit change i n the value of total discoveries (in thousands of dollars) of at least 500,000 barrels on drilling expenditures. Specifications of the 1 are adjacent to each of the time profiles with L defin operator. g function as the lag mate operators' responses® because, in the infinite lag models, the negative coefficients on the profit (table 1) inflate the role of the lag function in exploration effort. APPLICATION OF THE DISCOVERY PROCESS PREDICTIONS variables inducing S MODEL A fundamental premise of the approach to field ex- ploration modeling taken here is that oper induced to explore as long as there are high tions associated with the value of remaining fitors are expecta- deposits. In order to test this premise, a discovery process model was used to provide predictions of the undiscovered re- coverable resources as a function of the c ulative number of wells drilled. Two discovery process models that are very similar in their initial assumptions and that have been estimated for the Denver basin are the models described in Arps and Roberts (1958) Schuenemeyer, and Root (1980). d Drew, * These comments are relevant if the distributed lag model is interpreted to represent operators' temporal response functions to new discoveries. C12 Both models proceed from the assumption that the probability of finding a deposit in a particular size class is proportional to the number of undiscovered deposits of that size and proportional to the ratio of the area of the typical deposit of the deposit class to the relevant area of the basin under exploration. The model of Arps and Roberts (1958) specifies that the rate of discovery (within a size class of deposits) declines exponentially as a function of the number of exploratory holes drilled. Alternatively, Drew, Schuenemeyer, and Root (1980) use a somewhat different functional form to express the decline rate. They are able to estimate objectively the discovery efficiencies and relevant basin size, whereas Arps and Roberts used prior subjective infor- mation to make these estimates. The discovery process model described below is similar to the Arps-Roberts model in functional form, although the data used to estimate the model are more recent and accurate. Be- cause of its computational simplicity, the model of Arps and Roberts was used instead of the model by Drew, Schuenemeyer, and Root (1980). Both models appear to predict the historical discovery sequence quite well (Drew, Schuenemeyer, and Root, 1980)." One function of the discovery process model is to pro- vide a framework for predicting future recoverable re- sources as a function of the cumulative number of wells drilled. An important component of the model is the estimate of the ultimate or total number of deposits within the particular size class of deposits, given as F;(u) where i denotes the deposit size class and u is an arbitrary large number of wells (perhaps infinite) that exhausts or finds all the deposits in the ith size class. This estimate positions the extrapolation of the discov- ery decline curve. Given the parameters of the discov- ery process model, historical data may be used to ob- tain an estimate of F;(u). During the discovery history of the basin, estimates of F,(w) may be made and operator exploration behavior tested in order to deter- mine (1) if exploration is induced by the expectations associated with the remaining deposits and (2) if explo- ration is sustained by expectations attached to specific size classes of deposits or the cumulative value of re- maining deposits. o The Arps-Roberts discovery process model has the following form: F,(w)=F;(u)(1 -e **) (16) where F; (w) is the predicted number of deposits within size class i, F;(u) is the ultimate number of deposits in the class, w is the cumulative number of wildcat wells, '* Root and Schuenemeyer (1980) show the Arps and Roberts (1958) model and the limit discovery process model of Drew, Schuenemeyer, and Root (1980) to converge in the limit. ~PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING and 0; is a constant that includes the basin size, the area of the typical deposit from a given size class of deposits, and the exploration efficiency (Arps and Roberts, 1958, p. 2563). The basin size B was taken to be 5,700,000 acres (approximately 23,068 km") (Arps and Roberts, 1958). The exploration efficiency was as- sumed to be 2 for all size classes of deposits, and the area A,; associated with a deposit size class : was taken from Drew, Schuenemeyer, and Root (1980).''" More specifically for the i'" deposit size class, 0, = 2A; (17) B Although exploration efficiency does vary across size classes of deposits, tests of the model using the con- stant 2 for all deposit size classes did not produce any appreciable loss in predictive accuracy. Arps and Roberts (1958) indicate that predictions of the model are equivalent to finding the (mathematical) expected value of the number of discoveries to be made with a given search effort. Consequently, when this discovery process model is used to predict the value of undiscov- ered deposits, the specific forecasts will be referred to as "rational" predictions or "rational" expectations. The "rational" predictions of the number of remain- ing deposits for each size class of deposits was gen- erated in the following manner. An estimate of F;(u) was made by calculating F;(w=F,;(w)/(1-e **), (18) where F;(w) is the number of deposits already found with w wells within the ith size class. The number of deposits remaining to be found is the difference be- tween F;(u) and F;(w). By assuming a current price of oil and an average size to deposits within each size class, the value of oil remaining to be found in the basin is predicted after summing the values across size classes. The estimation value of remaining deposits is 11 : Area in acres Deposit size per square kilometer in (values in parentheses millions of barrels are in square kilometers) (.010) (.016) (.025) (.047) (.079) 032- (138) ABB Cig. .d. - 54.100 (221) 128- 956 97.300 (394) 250- 216.960 (.878) 500- 1.00 328.320 (1.329) 4 (1.950) (2.823) (4.325) (5.361) (11.137) (25.900) AN APPLICATION TO SUPPLY MODELING updated each period using both the most recent discov- ery data and the data from previous periods. This scheme for generating "rational" expectations can nat- urally be interpreted to suggest that each period operators using available information sequentially update their estimates of the value of undiscovered de- posits remaining in the basin. Empirical tests that are carried out are related to how closely levels of explora- tion effort correspond to the predicted value of remain- ing deposits and whether exploration is correlated more with predictions for specific size classes of depos- its or with the cumulative value of deposits remaining. The models explaining exploration behavior were specified to include the lagged profit term and the pre- dicted value of undiscovered deposits. Because the dis- covery process model implicitly takes into considera- tion the effects of physical exhaustion, the variable k, was omitted from the model specifications. Estimates of F;(u) can be quite erratic when there are relatively few wildcat wells drilled in the basin. Consequently, the data set that was used included the period from the third quarter in 1951 through 1973. Again, the Cochrane-Orcutt procedure was applied to handle problems of autocorrelation (Kmenta, 1971). The esti- mated equations and the statistics describing the qual- ity of the estimates are presented in the following equations: for drilling, = 14.160 + 0.009347,-, + 0.10172 yt (0.4) (1.4) (3.3) (19) R,=0.885 S$.E.=20.72 Durbin-Watson statistic = 1.45; for drilling expenditures, y.= -49.010 + g 0.02559,4 , +0. (58202 xt (20) (0.4) (1. (3.6) R,=0.898 S.E.=104.36 Durbin-Watson statistic = 1.72. In these equations R* is the adjusted coefficient of de- termination, S. E. is the standard error of the regres- sion, and the numbers in parentheses below the coeffi- cients are the associated "¢" statistics. The variable x# represents the total value (across all classes) of depos- its remaining to be found in the basin. Regressions using particular size classes of deposits for the variable associated with operators' expectations were also car- ried out. Although they produced slight differences in explanatory power, the improvements were not significant. From these results it can be concluded that, with reference to exploration, operators appear to react C13 to the total perceived economic value of undiscovered deposits rather than to the presence of a specific size of deposits. Predictions generated by the discovery process model could also be used as arguments in a distributed lag model. This use might be rationalized by assuming that the current estimates of the value of undiscovered deposits to which operators react are weighted av- erages of previous estimates. Several model specifications of this type have been estimated. How- ever, the results indicated no substantive improvement in the statistics describing the quality of the models. In some cases, the signs of the lag coefficients were diffi- cult to rationalize. Consequently, the model estimates are not presented or discussed in the subsequent analysis. The qualitative statistics that describe the fits of the ad hoc distributed lag model and the model based on the predictions of the discovery process model are com- parable. It would be of particular interest to determine if the predictive performances of the models are simi- lar. For a situation of equal reliability, the use of the simpler model is to be preferred since it has a some- what less ad hoc foundation and appears to use availa- ble observations for parameterization efficiently. The motivation for comparing the performance of the models beyond their respective qualitative statistics is to identify systematic relative weaknesses. Procedures for the comparative evaluation of econometric models have only recently been discussed in detail in the liter- ature (Dhrymes and others, 1972). During the process of constructing the present models, several evaluation criteria have already been applied. These criteria con- cern how well the model structure conforms to accepted economic theory and if the estimates are theoretically consistent. Other techniques compare the pattern of the model predictions with the historical data. One method is to examine whether the model predicts the turning points that actually occur in the dependent variable. In this regard, the performance of both models was relatively poor. Generally, the predicted turning points were lagged one period from the histori- cal turning points. For the well-drilling equations, both models predicted only 2 of the 27 turning points correctly; for the expenditures equations, the "ra- tional" model predicted 5 of 26, and the distributed lag model predicted 3 of 26 correctly. One explanation for the relatively large number of turning points in the data is that “j? data appear to contain a residual sea- sonal variation that was not removed. Another technique frequently used in model evalua- tion is to examine the pattern in the residuals. Figures 4 and 5 plot the residuals of the two models over a common sample period. The plotted residuals do not C14 PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING 20 |- x o x 0 10 -_* > xO o o xO 0 o xO x o x WELLS le] =10 -o * o x o .: 6 *~* x -30 |- 4 x Ox EXPLANATION 7 x Residual of "rational" model o Residual of distributed lag model 1 1 1 1 1 1 1 1 1 1 64 65 66 67 68 69 70 71 72 73 YEAR FIGURE 4.-Residuals for well-drilling models. indicate systematic variation in either figure or for either model specification. The presence of a systematic component in the residuals would indicate that the model was misspecified. That is, if a significant vari- able was omitted from the specification, then a pattern in the residuals would correspond to systematic varia- tions of the omitted variable. The most extreme re- siduals are accounted for by the "rational" model. Al- though the residuals vary widely for drilling from the third quarter of 1968 through 1970, the residuals for the drilling expenditure models indicate less erratic behavior. Economic theory rarely specifies the functional forms that are used in applied econometric modeling. Consequently, in the process of model selection, several alternative functional forms are generally examined, and the final form is chosen on the basis of goodness of fit. This procedure is not entirely satisfactory, because specification errors resulting from incorrect or omitted variables are still possible. Specification errors may lead to biased estimates and predictions. A technique used to detect specification errors is to test for stability or structural changes in the model parameters during the original sample period (Jorgenson and others, 1970; Dhrymes and others, 1972). In particular, the full data sample is split into two subsamples or periods, and the models are reestimated. The resulting parame- ters are tested to determine if statistically significant changes have taken place in their values. The procedure used in detecting structural changes is the following: Suppose Q; is the sum of squared re- siduals associated with each subsample. Further, let Q ; be the sum of squared residuals of the regression based on the pooled data. Under the hypothesis that the sets of regressors are equal for the two subperiods and the pooled sample, the test statistic is distributed accord- ing to the F distribution and is calculated (Chow, 1960) as F = (Q1‘szk » (21) Q@;/(m +n -2k) with degrees of freedom (k,m+n-2k). The variables m and » are the number of observations in each subsam- ple, and k is the number of parameters that must be estimated in each model. For the distributed lag models, the two subperiods were from the fourth quarter of 1952 to the first quar- ter of 1962 and from the second quarter of 1962 to the fourth quarter of 1973. The sample periods used for tests of the "rational" model were from the fourth quar- ter of 1953 to the third quarter of 1962 and from the AN APPLICATION TO SUPPLY MODELING C15 350 |- 250 |- 100 |- * xO xO x DOLLARS, IN TEN THOUSANDS -300 1 1 1 1 1 1 1 1 1 xO o xO bt s- beck EXPLANATION a x Residual of "rational" model 0 Residual of distributed lag model L 1 1 1 1 1 1 1 1 1 63. | 64 _ 65. ~ 66. =67 -be: op: 1970: _ 71492" | 73 YEAR F1GURE 5.-Residuals for well-drilling expenditures models. fourth quarter of 1962 to the fourth quarter of 1973. The level of type I error was set at 0.05; that is, one could expect to reject the null hypothesis about 1 time in 20 even if it were true. Rejection of the null hypothesis would indicate temporal parameter changes that might be the result of model specification error. For the distributed lag model the "F" statistics were 1.34 and 0.58 for the drilling and drilling ex- penditures equations, respectively. The model based on the predictions of the discovery process model had F statistics of 1.57 and 0.74 for the drilling and drilling expenditures equations, respectively. Thus there were no instances when the test statistic exceeded the cor- responding critical level and the null hypothesis was not rejected, and the results of the tests for structural change provided no evidence that specification errors might be present in either set of models. In conclusion, the test results comparing the distrib- uted lag model and the model based upon predictions from the discovery process models indicated little dif- ference in the potential predictive performance of the models. This finding is significant for three reasons. First, it appears that the predictions of the discovery process model contain most, if not all, of the informa- tion included in the distributed lag functions of D, and the experience-exhaustion variable k,. Because of its relative simplicity, it may be more appropriate to use this model instead of the distributed lag model in applied situations. Second, if the discovery process model efficiently uses available information to gen- erate predictions, then the specific distributed lag proxy used here does seem to capture operator expecta- tions. Third, the reasonably high explanatory power of the "rational" model appears to indicate that the operators are using information contained in the discovery history of the basin efficiently and operating in a fashion consistent with these predictions. SUPPLY OF RESERVES The behavioral drilling models presented in the pre- ceding sections appear adequate to explain the histori- cal pattern of operator behavior in the basin. As such, they served as a basis for making inferences concern- ing the determinants of firm behavior. The estimated C16 behavioral models may also be applied in conjunction with the discovery process models to forecast the tem- poral pattern of new reserves, given certain assump- tions about crude oil prices and drilling costs. The usefulness of forecasts of future reserves from undis- covered deposits is obvious. Furthermore, the genera- tion of the forecasts by the analytical model allows the predictions to be conditioned on prices, well costs, and the current state of physical exhaustion. However, the accuracy of the forecasts is generally constrained by the nature of the models and the relevant behavioral and technical relations. The estimated discovery process model mechanically provides predictions of the number of deposits within each size class as a function of the cumulative number of wildcat wells. To obtain a prediction of the reserves found for a given increment in wells, the field size (in barrels of oil) is multiplied by the incremental value in the predicted number of deposits. By linking the dis- covery process model to the behavioral drilling model, the predicted number of wildeat wells (per unit time) can be used to generate a forecast of the amount and value of reserves forthcoming. Because of the non- linear nature of the discovery process model, the link- ing of it and the behavioral drilling model must be relatively simple. Consequently, the integrated model is recursive in nature. Although the discovery process model is independent of time, the behavioral drilling model is not. The drill- ing model was estimated with quarterly data, and con- sequently forecasts of the number of wildcat wells are also on a quarterly basis Moreover, the model is specified so as to describe shortrun operator decisions. In general, the ability of econometric models to provide accurate forecasts deteriorates rapidly as the number of periods that the forecasts are made into the future increases. The extent of the deterioration can, to some degree, be determined by carrying out simulation ex- periments with the integrated equation system. In par- ticular, these tests attempt to determine how rapidly the accuracy of the forecasts deteriorate, the robust- ness of the predictions of one endogenous variable when errors are in the other endogenous variable, and whether forecast errors are compensating or noncom- pensating as the results from individual time periods were aggregated. A measure used to determine the quality of the inte- grated model's forecasting performance was the root- mean-square (RMS) prediction error (Thiel, 1966). It is defined as RMS= 1 n A II M 3 (Pi —Ai)2, 1 (22) A PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING where A; is the actual value of the predicted variable and P; is the prediction. Also used as measures of per- formance were the mean error (over several forecast periods) and the ratio of the RMS prediction error to the actual mean value of the variable predicted. The integrated model is specified as a recursive equ- ation system. Because of its simplicity, the "rational" well drilling specification was used rather than the dis- tributed lag model. Variables that are endogenous or determined within the system are y,= number of wells drilled in period t and R,= reserves discovered in period ¢. Variables that are taken as given or determined exogenously are cw,= cost per well in period i, p= price of crude oil, and U,= the initial value of undiscovered deposits. Among the predetermined variables that are taken as given are the initial values of the lagged endogenous variable y,, and #;. The discovery success ratio is not predicted, and consequently the profit variable was re- defined to have the following form: 7Tt—2———L—~Rt—2 i-%._-CWH-s, Yt-2 (23) where R,, p,, cw, , and y, were defined as above. In the original estimation of the "rational" model, F;(w) was reestimated each period as new information became available or as discoveries were made. Reestimation of F;(u) each period for the integrated model would result only in the initialized value of the variable. Therefore, the final empirical estimates of the ultimate numbers of deposits for each size class were used. Because of the redefinition of the profit variable and the use of the final empirical estimates for F;(w), the "rational" well drilling equation was reestimated. The new equation is of the following form: y, = 8.386 +0.063397%,-,+ 0.10858 x (0.2) (1.7) (8.0) (24) R*=0.881 S.E.=21.1 Durbin-Watson statistic = 1.53, where the numbers in parentheses below the model coefficients are the "¢" statistics, R is the adjusted co- efficient of determination, and S. E. is the standard error of the regression equation. Using the Cochrane- Orcutt procedure to handle serial correlation, the first-order correlation coefficient was estimated to be 0.814. The predictive form of equation (24) (Kmenta, 1971) is y,=0.814y-i + (1 -0.814)8.386+0.06339(77,_,-0.8147,-3) +0.10358(#-0.814y¢ ,). (25) AN APPLICATION TO SUPPLY MODELING The predicted reserves found per time period, R,, is calculated by taking the cumulative number of wells to period ¢-1, calculating the predicted (total) reserves, adding the number of wells drilled in period ¢ to the cumulative well count, recalculating the predicted (to- tal) reserves, and taking the difference between two total reserve predictions. The value of undiscovered deposits at the beginning of time ¢+1 was forecast by taking the difference between the predicted amount of discovered reserves at ¢ and the sum of F;(w) across classes and multiplying the result by the crude oil price at the time. Predictions of y,, R,-,, and x*,, are em- ployed in (23) and (25) to generate the predicted number of wells for period y,,,, which in turn is used to predict R, ,;,. The historical period from the first quarter of 1960 to the fourth quarter of 1968 served as the base period for examining the integrated model's forecasting ability. This period was chosen because the estimated ultimate number of deposits in each size class had stabilized (Drew and others, 1980). The test period was cut off at the end of 1968 in order not to include the perod when the Union Pacific Railroad acreage began to be drilled. We believed that the initiation of drilling on the new acreage represented an entirely exogenous influence, and the model could not be expected to reproduce it. In order to determine how the accuracy of the fore- casts was affected by the length of the forecast period, that is, how far ahead forecasts should be made, three sets of simulations were generated for the 9-year period. These include forecasts of 4, 8, and 12 periods (quarters) ahead. For example, during the 36 periods there were nine sets of 4-period forecasts. Using 8-period forecasts, four sets were generated, and for the 12-period forecasts, three sets were generated. Quar- terly observations, particularly for new discoveries, have substantial stochastic components. Therefore, it is also of interest to determine if prediction errors dur- ing the forecast period offset each other when results for the time periods are aggregated. Results of the experiments are presented in table 4. The table indicates the RMS prediction error, the mean (actual-predicted) prediction errors over the forecast time period, and the mean value of the actual variable that is being forecasted. The pattern of the prediction errors for wells indicates the degree by which lengthening the forecast period increases the RMS prediction errors and mean prediction error. The over- all mean values of the RMS to the actual variable value are 0.141, 0.240, and 0.257, and the ratios of the value of mean period prediction error to the mean ac- tual values are 0.033, 0.162, and 0.189 for the 4-, 8-, and 12-period forecasts, respectively. The obvious ef- fect of increasing the forecast period is the accumula- "C17 TABLE 4.-Forecast performance of integrated system Dependent Variable Wells Reserves Forecast RMS Mean Actual RMS Mean Actual length' Period _ errors' errors mean errors errors mean A NL cout 7.551 3.589 109.250 2.103 0.786 3.163 2. <- ice. 12.795 -9.213 92.250 3.183 -.025 3.161 I 13.183 .868 110.000 3.036 -.573 4.111 4: siv. 8.509 5.915 90.500 2.969 277 2.555 § i... 5.227 3.679 86.250 1.928 .330 2.096 6 coc... 20.195 14.769 89.250 1.927 1.346 1.142 a eae. 12.703 9.698 68.750 3.381 -.519 2.374 B ceil: 9.783 6.882 54.250 1.298 -.526 2.069 9. ccs cet 19.146 _ -10.847 70.750 .585 -.458 1.837 Mean Values 12.121 2.816 85.694 2.268 .071 2.567 5. I:... 22.953 14.986 100.750 2.729 0.776 3.462 2 19.434 12.399 100.250 2.987 .059 3.333 § 13.424 5.803 87.750 1.915 .805 1.619 4 .L... 28.126 23.671 61.500 2.549 -.275 2.221 Mean Values 20.984 14.215 87.563 2.545 .341 2.659 CI {sick. 20.438 13.099 103.833 2.860 0.341 3.678 2 cits. 15.643 11.746 88.667 2.345 153 1.931 $ 29.878 23.761 64.583 2.102 -.141 2.093 Mean Values 21.986 16.202 85.694 2.436 .318 2.567 Forecast lengths A, B, and C are 4, 8, find»? quarters, respectively. *Root-mean-square value is given by ( N _21(P, - A;)*)°5, where P; is the pre- i= dicted value and A, is the actual value. o *Mean error is based on actual minus predicted value. tion of prediction errors. It does not appear that the percentage errors increase linearly with the forecast period. To some degree, it appears that errors will be offsetting if the forecast period is taken to be suffi- ciently short, as evidenced by the relatively low mean error to actual value of 0.033 for the 4-period forecast simulation. As the forecast period is increased, the number of wells predicted appears to be systematically overestimated. Results for the predicted amount of reserves found in new deposits are also presented in table 4. The discov- ery process model, used to predict the number of depo- sits and reserves forthcoming, produces relatively smooth estimates of forthcoming reserves. However, the actual number and sizes of new deposits, particu- larly for as short a period as a quarter, are highly erra- tic in nature. As a result, individual prediction errors as measured by the RMS prediction errors can be ex- pected to be relatively large. However, if the model is operating reasonably well, in aggregating the predic- tions over time, errors will be compensating so that the mean prediction error should be relatively small. Re- sults of the simulations appear to be consistent with this conjecture. The RMS prediction errors are very . large, even for the set for which only 4-period forecasts were made. However, the mean prediction errors taken over the entire period are relatively small. The ratios of the mean prediction error values to the actual mean values are 0.028, 0.128, and 0.124 for the simulations based on the 4-, 8-, and 12-period forecasts, re- spectively. Individual errors in predicting forthcoming C18 reserves tend to offset each other as the errors are aggregated over time. The supply of reserves for in- crements in the drilling rate are too random to be pre- dicted on a quarterly basis. Aggregation of quarterly forecasts to obtain annual values results in reasonably accurate predictions. The positive values of the mean forecast error for the 8- and 12-period simulation ex- periments does not necessarily mean that forecasts of the discovery process models are generally biased. The positive mean prediction error is probably the result of the overestimates for the number of predicted wells drilled as discussed earlier. In summary, the prediction errors of the behavioral well-drilling model seem to be biased positively. How- ever, the magnitude of the RMS and mean errors are quite comparable to the performance of well-drilling equations reported in other studies (MacAvoy and Pin- dyck, 1975) for similar forecast periods, that is, 4-8 periods ahead. The relatively high RMS errors for the additions to reserves appear to be more the result of the erratic or stochastic nature of the historical arrival of discoveries than systematic bias in the discovery pro- cess model. The relatively small mean prediction errors, particularly for the 4-period forecasts, indicate that if the quarterly forecasts of additions to reserves were aggregated, perhaps on an annual basis, and compared to the historical realizations, the RMS pre- diction errors would also be much smaller. The rela- tively small mean prediction error associated with the additions to reserves seems to suggest that the fore- casts are reasonably robust even though there were errors in predictions of the number of wells to be drilled. Although the linking of the discovery process model with the behavioral well-drilling model produced an analytical means of translating the forecasts of re- serves per unit exploration effort, that is, wells drilled to reserves per unit time, several limitations of the analysis should be kept in mind. First, the economic model was specified to describe operator field decisions that are short term in nature. That is, the nature of decisions that are modeled are marginal adjustments in the rate of exploration rather than decisions to enter or exit a geologic basin. Second, because of the short- run nature of the models, it would be misleading to attempt to draw general conclusions about the effects of a general price change on drilling activity within the basin. That is, a general price change would induce some longrun adjustments to take place in the firm's internal allocation of resources across several regions. The type of price change that the behavioral well- drilling model might more appropriately capture cor- responds to a change in the relative price of oil, for instance, the price of ail found in the Denver basin as PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING opposed to another basin. In order to predict the effects of a general change in the price of oil on drilling behav- ior for a particular basin, the behavioral well-drilling equation should be respecified to reflect the firm's long- run decisions and include a variable that would denote the firm's alternative exploration opportunities in other geologic basins or its alternative opportunities for obtaining additional reserves. CONCLUSIONS The purpose of this study was to examine operator exploration behavior at the field level. In the first part of the study an empirical model was specified and esti- mated. Distributed lag proxy variables were used to model operator expectations associated with the distri- bution of deposits remaining to be found in the basin. The estimated drilling models permitted inferences to be made about operator field behavior. First, if the form of the distributed lag function is interpreted to represent the operators' responses to new discoveries, then the nature of function is important. The appropri- ate form of the function was a finite polynomial lag function with an inverted "V" shape; that is, the effects of new discoveries increased for several periods and then diminished rapidly to 0 within 2% years after dis- covery of the deposit. The independent variable used in the distributed lag model was the total value of depos- its found in a given period with a minimum size of 500,000 barrels. Second, tests using alternative minimum size classes of deposits, that is, 2.5 million and 5 million barrels, in the distributed lag function produced no improvements in the fits. This result im- plies that, once an exploration play is underway, operator field expectations are sustained by the quan- tity of oil found per time period rather than by the discovery of specific size deposits. Comparison of these results and the regression coefficients showed them to be quite consistent with the results of a similar previ- ous study based on the discovery history of the Powder River basin (Attanasi and Drew, 1977). In the second part of the study, the discovery process model was applied to generate sequential predictions of the value of undiscovered deposits, which were used in the behavioral well-drilling model in place of the dis- tributed lag operator-expectations proxy variable. The discovery process model efficiently uses information contained in the historical time series to estimate the future discoveries. Consequently, it was assumed that such predictions would closely correspond to "rational" expectations where "rational" is used in the sense that all available information in the historical data is used efficiently in the estimation process. Using the predic- tions from the discovery process model as the unob- servable expectations variable associated with the AN APPLICATION TO SUPPLY MODELING value of undiscovered deposits, the behavioral well- drilling equation was reestimated. The estimated "ra- tional" model explained historical operator behavior well. This result seemed to indicate that operators, on the average, were efficiently using information con- tained in the discovery history of the basin and were behaving in a fashion consistent with these predic- tions. Comparison of the performance of the distributed lag model with that of the "rational" model indicated little differences in the predictive performance of the two models. The predictions of the discovery process model, used as "rational" expectations, appear to con- tain most if not all of the information included in the distributed lag functions and experience-exhaustion variable. In the final section of the study, the behavioral well-drilling model and the discovery process model were integrated into a recursive equation system. The purpose of linking the models was to gain the ability to forecast additions to reserves in the time domain or on a unit time basis. The forecast performance of the inte- grated system was tested by examining the accuracy for various lengths of forecast periods. As might be ‘expected, as the number of forecast periods increased, the accuracy of the forecasts deteriorated. However, the level of accuracy was still comparable to levels at- tained in other studies. Because of the stochastic na- ture of the discovery process, aggregation of quarterly ‘predictions of additions to reserves to annual values tended to increase accuracy as measured by the mean error of prediction. The accuracy of the aggregated ipredictions appeared to be reasonably robust to the presence of prediction errors in the behavioral well- drilling equations. Finally, the integrated model pro- vides for the first time a means of explicity incorporat- ing physical and economic exhaustion in the generated profit predictions and predictions of the value of re- maining deposits into a behavioral model that de- scribes the exploration effort (wells and expenditures) of operators. | REFERENCES CITED Almon, Shirley, 1965, The distributed lag between capital appropria- tions and expenditures: Econometrica, v. 33, no. 1, p. 176-196. American Petroleum Institute, 1953, 1955-56, 1959-73, Joint Asso- ciation Survey of the U.S. Oil and Gas Producing Industry: i Washington, D.C., American Petroleum Institute. Arps, J. J., and Roberts, T. G., 1958, Economics of drilling for Cretaceous oil on the east flank of the Denver-Julesburg basin: American Association of Petroleum Geologists Bulletin, v. 42, | no. 11, p. 2549-2566. Attanasi, E. D., 1978, Firm size and petroleum exploration, Council of Economics of American Institute of Mining, Metallurgical, | and Petroleum Engineers, Annual Meetings, Denver, Proceed- ings: p. 57-62. Attanasi, E. D., and Drew, L. J., 1977, Field expectations and the C19 determinants of wildeat drilling: Southern Economic Journal, v. 44, no. 1, p. 53-67. ; Box, G. E. P., and Jenkins, G. M., 1970, Time series analysis: San Francisco, Holden-Day, Inc., 553 p. Chow, G. C., 1960, Tests of equality between sets of coefficients in two regressions: Econometrica, v. 28, no. 3, p. 591-605. Cox, J. C., and Wright, A. W., 1976, The determinants of investment in petroleum reserves: American Economic Review, v. 66, no. 1, p. 153-167. Dhrymes, P. J., 1971, Distributed lags: Problems of estimation and formulation: San Francisco, Holden-Day, Inc., 414 p. Dhrymes, P. J., Howrey, E. P., Hymans, S. H., Kmenta, Jan, Leamer, E. E., Quandt, R. E., Ramsey, J. B., Shapiro, H. T., and Zar- nowitz, Victor, 1972, Criteria for evaluation of econometrica models: Annals of Economic and Social Measurement, v. 1, no. 3, p. 272-328. Drew, L. J., Schuenemeyer, J. H., and Root, D. H., 1980, Petroleum- resource appraisal and discovery rate forecasting in partially explored regions; Part A, An application to the Denver basin: U.S. Geological Survey Professional Paper 1138-A, 11 p. Epple, D. N., 1975, Petroleum discoveries and government policy: An econometric study of supply: Cambridge, Mass., Ballinger, 139 p. Erickson, E. W., and Spann, R. M., 1971, Supply response in a regu- lated industry: The case of natural gas: Bell Journal of Econom- ics and Management Science, v. 2, no. 1, p. 94-121. Ezekiel, M. J., 1938, The cobweb theorem: Quarterly Journal of Eco- nomics, v. 53, no. 1, p. 255-280. Fama, E. F., 1972, Perfect competition and optimal production deci- sions under uncertainty: Bell Journal of Economics and Man- agement Science, v. 3, no. 1, p. 151-174. Federal Energy Administration, 1974, Project independence blue- print: Final task force report: Oil: Possible levels of future pro- duction: Washington, D.C., U.S. Government Printing Office, 162 p. 1976, National energy outlook: Washington, D.C., U.S. Gov- ernment Printing Office, 190 p. Fisher, F. M., 1964, Supply and costs in the U.S. petroleum industry: Baltimore, Johns Hopkins University Press, 177 p. Johnston, J. J., 1972, Econometric methods [2d ed.]: New York, McGraw-Hill, 437 p. Jorgenson, D. W., 1966, Rational distributed lag functions: Econometrica, v. 34, no. 1, p. 135-149. Jorgenson, D. W., Hunter, J. S., and Nadiri, M. L., 1970, The predic- tive performance of econometric models of quarterly investment behavior: Econometrica, v. 38, no. 2, p. 213-224. Kaufman, G. M., 1963, Statistical decision and related techniques in oil and gas exploration: Englewood Cliffs, Prentice-Hall, 307 p. Khazzoom, J. D., 1971, The FPC Staff's econometric model of natural gas supply in the United States: Bell Journal of Economics and Management Science, v. 2, no. 1, p. 51-93. Kmenta, Jan, 1971, Elements of econometrics; New York, MacMill- an, 655 p. Leland, H. E., 1974, Production theory and the stock market: Bell Journal of Economics and Management Science, v. 5, no. 1, p. 125-144. MacAvoy, P. W., and Pindyck, R. S., 1973, Alternative regulatory policies for dealing with the natural gas shortage: Bell Journal of Economics and Management Science, v. 4, no. 1, p. 454-498. 1975, The economics of the national gas shortage (1960-1980): Amsterdam, North-Holland, 259 p. National Petroleum Council, 1972, U.S. energy outlook: Washing- ton, D.C., U.S. Government Printing Office, 260 p. Nerlove, M. L., 1958, Distributed lags and estimation of long-run supply and demand elasticitiese-theoretical considerations: Journal of Farm Economics, v. 40, no. 2, p. 301-311. C20 Pesando, J. E., 1976, Rational expectations and distributed lag ex- pectations proxies: Journal of the American Statistical Associa- tion, v. 71, no. 1, p. 36-42. Pierce, D. A., 1971, Fitting dynamic time series models-some con- siderations and examples: Annual Symposium on the Interface Between Computer Science and Statistics, 5th Stillwater, Okla., Proceedings, p. 1-2. Pindyck, R. S., 1978, Higher energy prices and the supply of natural gas: Energy Systems and Policy, v. 2, no. 2, p. 177-209. Root, D. H., and Schuenemeyer, J. H., 1980, Petroleum-resource ap- praisal and discovery rate forecasting in partially explored re- gions; Part B, Mathematical foundations: U.S. Geological Sur- vey Professional Paper 1138-B, 9 p. Schmidt, P. J., and Waud, R. N., 1973, The Almon lag technique and monetary versus fiscal policy debate: Journal of the American Statistical Association, v. 68, no. 1, p. 11-19. PETROLEUM-RESOURCE APPRAISAL AND DISCOVERY RATE FORECASTING Shiskin, Julius, Young, A. H., and Musgrove, J. C., 1967, The X-11 variant of the Census Method II seasonal adjustment program: Washington, D.C., U.S. Department of Commerce, 66 p. Thiel, Henrie, 1966, Applied economic forecasting: Amsterdam, North-Holland, 474 p. Turnovsky, S. J., 1969, A Bayesian approach to the theory of expec- tations: Journal of Economic Theory, v. 1, no. 3, p. 220-227. Uhler, R. S., 1976, Costs and supply in petroleum exploration-the case of alberta: Canadian Journal of Economics, v. 9, no. 1, p. 72-90. White, D. A., Gerrett, R. W., Marsh, G. R., Baker, R. A., and Gehman, H. M., 1975, Assessing regional oil and gas potential, in Haun, J. D., ed., Methods of estimating the volume of undis- covered oil and gas resources: Tulsa, American Association of Petroleum Geologists, p. 143-159. GPO 689 -143/ M7 6 N a. fedlment-Trappmg Characterlstlcs of the Helley—Smlth ; dload Sampler \_ GEOLOGICAL SURVEY PROFESSIONAL PAPER 1139 U.s, Dfipflsl‘TORY DOCUMENTS DEPARTMENT l ) 1980 wie oD FUG 01 1980 - JULLA B0 L1BRAR - UNIVERSITY OF CA‘ IFORK ”A I WWW“. i Vh *" wa “a“ AU _ __. Hast Fork River, Wyoming - upstream of bedload transport research facxlzt} A Field Calibration of the Sediment-Trapping Characteristics of the Helley-Smith Bedload Sampler By WILLIAM W. EMMETT GEOLOGIC AL SURVEY PROEESSIONAL PAPER I| 1359 Studies of bedload transport in river channels UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON -: 1980 UNITED STATES DEPARTMENT OF THE INTERIOR CECIL D. ANDRUS, Secretary GEOLOGICAL SURVEY H. William Menard, Director Library of Congress Cataloging in Publication Data Emmett, William W. A field calibration of the sediment-trapping characteristics of the Helley-Smith bedload sampler. (Geological Survey professional paper ; 1139) Bibliography: P. 29-30. Supt. of Docs, no.: 1 19.16:1139 1. Bedload-Measurement. 2. Hydrology-Instruments. I. Title. II. Title: Helley-Smith bedload sampler. III. Series: United States. Geological Survey. Professional paper ; 1139. GB850.E45 551.3°03°028 80-607050 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 FicurE 1. bo 19-27. CONTENTS Page ADSLTACE . core lee ee naren ia arn n oh ane Ane an na nang mine noo ae e chun s capa a aie iba ae ae c aand 1 Introduction ~-: leon onlines. oon s geste tono Ls o ute Rte UL 1 Pedidad .=: c Sele IL. ned naval neem n angina alena nal ia aand oe women 1 Helley-mith bedload sampler _-_ 2 Description of field-test facility .. .. __. _. __. __ onc Alcon cone 3 East Fork River .o. l_ c ace l L BJ Conveyor-belt bedload Lrap -.. lll. Duy nlc 4 Sampling techniques with Helley-Smith bedload sampler ___________________________- 8 Results of field-calibration tests =.. -.:: L. Lull LUI ei 9 Bedload-transport rates by individual particle-size categories, basic data ____________ 10 Comparison of Helley-Smith results to conveyor-belt results, basic data ______________ 14 Modifications to the DaSIG MALA -- 2 . _ _ _ 2 222.2 G ARIEL GLI La n ween nv 19 Summary and recommendatioMs _ 2.2 IL.... illi ge 27 Acknowledgments _ ___ -. : cool Coro o Les n ATLAN IL EL EL anne ame a 27 Réferefices tited... .. . .z. >. _ ll sono o AEE PLL D. ane eras 27 ILLUSTRATIONS Sketch of the Helley-Smith bedload SaMpIGT | -L enon am can Plan and side-elevation drawings of 7.62-em Helley-Smith bedload Graph showing vertical-velocity profiles upstream of sampler, and in sampler orifice of 7.62-cm Helley-Smith bedload SAMpIET 22200002 o- on Ge ane eo rh ab aar Eb EE can anak co - an bean an aman an on anale bie ene nes ann an oui Map of East Fork River drainage area upstream of the project site Planimetric map of East Fork River in vicinity of bedload trap, showing median diameter of bed material at sections AIOng TVET Lo c. lc Ll enate eaik su rani ung aa aar ans a al alisa aa onne ores ne aa sen deiet c Cross section of the East Fork River at the bedload Photographs of the East Fork River and conveyor-belt bedload Graph of hydraulic-geometry relations for the East Fork River at the bedload trap ___________________________-_----- Photogaphs showing use of the Helley-Smith sampler at the bedload trap Graphs showing relation of bedload-transport rate in individual-size category as function of total bedload-transport rate for Helley-Smith sampler and conveyor-belt sampler: 10. 9:06-0.12 mm size class.. . 220 c LLE HAIER EE ere GU - ay iro - +h amide a ion ana fig aie - auch alee nea al aln a ne anl mim size Class- 0 l.. 10. cogent a u Po CAn LIL Ue heartened cued ane ese ene. 120.25-0.50 min size Clase >. .._ cc cull coo ool nl LLE UERL IIN tne Prune neon tne ue ae aes 1:00 mim size Clase >.. . lull lenee oen 22 CLL L Loot enlace bungee ane ben aree conan somes an eee ba ae 14; :1:00-2:00 mim size clase 2.22200. 102 0.0L 00 0.22 Eade PULL CGL Serre nD ie oun ce an ans 15: 2.00-4:00 mim size clase _L 0. 1 _ LEL OLG UEO e nw aes ane 16: mim size Class...... .:. . nel Uda AI LIU v an daa eeu es eben ane aaa ads 17. 5.00-16.-0 mm size class ._ 201 clis lL O EIL CELL - Ue eins rae nan ce ab ee - unl a nae ens 18s: 16.0-32.:0 imm size Icell Llc LOR Lebel -be Graphs showing comparison of bedload-transport rate by individual particle-size category for Helley-Smith sampler and conveyor-belt sampler: £9;0,06-0. 12: mim: size Clase : cel _. ool ern LLL IL- Lo oe nene nin en nia ar an ul nal ae ne ne cennet e ne nile tensa a's ao ame 20-0.12-0 25 mm size Class . __ ___.. cl.. LIl LLL CEIE L IEA r Ear oun caca nha amer eel 21.10, 2520.50 mim #ize Cle . -L __ _L 2. cn LL Loc Ullal aoa qee Gnar e an naren ng onne ain a ih a's care a e's aie a alee a aln t an o a o ein a be nel 22 10-50-1.00 min size class >. -.:... 22000000, LL. As rede cen PEP Po ou «Geh b ap en aaa baw aioe at eep ee 2371.00-2:00 mim size Clage 2 -. 2. .c cD ILL DL Lc ele ol neo redone ne d ahr ce nnege bere cre to bere cas 24, 200-400 mim #1ze CASE -_... . Lcol cul L eL ool enn n io rene noun ne one bara ohipenen ant nee on ao ane an anale ae enn and eens 257 mim #ize CIASS -_ --..! ne nil. nou - ne ans ob be bo ae ane ne ae rene ameiecnte use ber 26..8:00- 16.0 mm site .. .L 00, cL l aie 1 20 e 1 o Blum be o alain e an a n e a an me fil fia mid we als ae boul a a a ali a ae ae 21, 16.0-82.0 mm size class. :=: LLL LIEU L Ira ISU IEC Guna rs onus con n an ane seee ne ans IV CONTENTS FicureEs 28-39. Graphs showing comparison of total bedload-transport rate: Page 28. Helley-Smith sampler and conveyor-belt sampler; conveyor-belt sampler designated as independent ariable -se none ie eel n cein es cea s neben nde cecersess coweerdie 20 29. Helley-Smith sampler and conveyor-belt sampler; Helley-Smith sampler designated as independent yariable " 22.0025 3-2 nln ure l cool ee cati are ae ne- n ania nea cena aa anna coa bans ce enna. an 20 30. Helley-Smith sampler and conveyor-belt sampler corrected to condition of stream-wide slot; conveyor- belt sampler designated as independent variable =_ {O00 L. 21 31. Helley-Smith sampler and conveyor-belt sampler corrected to condition of stream-wide slot; Helley- Smith sampler designated as independent vartable _ 010 21 32. Helley-Smith sampler corrected for stage difference and conveyor-belt sampler corrected for condition of stream-wide alot? >=. s 2 xl dn . oe Pie et no bare bere r b an enue. an naan r as enue 21 33. Helley-Smith sampler corrected for stage difference and conveyor-belt sampler corrected for condition of stream-wide slot; relation corrected for variance of the independent variable in the least-squares regression ""}. c. ece 22 oo PALL Luo ol nar Leal cae re enne an aka ane unk ss 22 34. Sediment particles larger than 0.25 mm; Helley-Smith sampler and conveyor-belt sampler corrected for condition of stream-wide sIGL: .-. . . .. 222... 0 ld cep re ade 22 35. Sediment particles larger than 0.50 mm; Helley-Smith sampler and conveyor-belt sampler corrected for condition of streath-wide SIGL : _.. ..... ... . 8 ~.. . 2 ool u lens erice pean, o ob e anal fone t ae ain ana daa ree ae aca eae 23 36. Sediment particles larger than 0.50 mm; Helley-Smith sampler and conveyor-belt sampler corrected for condition of stream-wide slot; relation corrected for variance of the independent variable in the least-squares repression ;.. .. .. ... 2 ote noelle noe ene ben ane nabe n. Adie n aus, bee e ua neni ren bean's 283 37. Sediment particles larger than 0.25 mm; Helley-Smith sampler corrected for stage difference and conveyor-belt sampler corrected for condition of stream-wide SIOt 28 38. Sediment particles larger than 0.50 mm; Helley-Smith sampler corrected for stage difference and conveyor-belt sampler corrected for condition of stream-wide SIOt 24 39. Sediment particles larger than 0.50 mm; Helley-Smith sampler corrected for stage difference and conveyor-belt sampler corrected for condition of stream-wide slot; relation corrected for variance of the independent variable in the least-squares regression L 24 40-48. Graphs showing comparison of bedload-transport rate by individual particle-size category, Helley-Smith sampler and conveyor-belt sampler corrected for condition of stream-wide slot: £0:-0.00640-19 mim size clase -.. : 2 -). .l 0.12 22 sync Loool n 2 en Pen an dan ol o onn on anna agra d ou ances aes L pae nan oe 25 d1- 0.12-025 imm size class -. _- __. ¢.... loo. .l. nil ll ELU v inin enne l anno ine ooo tonga gene 25 42> Clase. 2. ..... n- 1 2. .cc coule co cole LNL Ed ne clan en nine nee leann s oa s oe ain ad oe ne alea e aay 25 43; 050-1200 mm sizer@lase -- . . -. :.. s 2 lll rels enon EALTH a bee ae an nnn penne eons oon dgn an one bene ia 25 44. mmusize I L IIN IL Lennie oen r eave beans apenas na Bane 26 45:-2:00-4.00 imm size Clase =.. L IATL CALLIE Noone aeon end nge sae pune tate 26 £6.74 00-8:00mmsize clase .> ..... .., _... 2. oon o e Ud Ae Ped ie naa s ane dined ale aa ne eee ee vien 26 147. 5.00216.0 mim size Clase ". 2200. 2.20 o.. Linnell iad re dei ive nra pne andr bee ner ar aaa n nene eo nne e 26 48/-16:0-32.0 mm size Class" 210... _L d. __ ODL oa aoa [te gl OL oie o e ix ret ado tile aus 27 TABLES Page TABLE 1. Size distribution of composited bed material, East Fork River, Wyoming, at bedload-transport research project ______ 7 2. Size distribution of transport-weighted composite bedload (1976 conveyor belt), East Fork River, Wyoming, at bedload- $ransport research project -.. .. m ul ner n ned nene aan el adage nas ao oe adi ean 7 3. Comparison of bed material and bedload particle sizes .._. 7 4. Summary data of river hydraulics and bedload transport, conveyor-belt sampler _________________________________-- 30 5. Particle-size distribution of bedload sediment, conveyor-belt sampler 82 6. Summary data of river hydraulics and bedload transport, Helley-Smith sampler ________________________________--- 36 7. Particle-size distribution of bedload sediment, Helley-Smith sampler ______________________________________________ 38 8. Summary of statistical data: log-transformed linear regression of transport rate by particle-size class versus total transport rate. conveyor-belt sampler -. _ ;... coo f 0 O0 002 LU Lo oe opine gi oe oe deen ne nde nek e 41 TABLE 10. 11. 12. 13. CONTENTS Page Summary of statistical data: log-transformed linear regression of transport rate by particle-size class versus total transport rate, sampler 12 GLO -o anes cans cn aik oman 41 Listing of comparable data sets used in direct comparison of results from conveyor-belt sampler with results from Helley-Smith -..... conn ire ane reneareleuresn cues as nen in abe aus bes sie oo nemen 41 Summary of statistical data: log-transformed linear regression of transport rate, Helley-Smith sampler versus conveyor-belt sampler tbasic data) .. . . . .-.. :- rou: cee ans o _ oo ae ale an main i me ole o he in be and bbe mle tn ih h a in n a ie i tig al mae tain ara e ae 43 Summary of statistical data: log-transformed linear regression of transport rate, Helley-Smith sampler versus conveyor-belt sampler (comparison of various modifications to the basic data) ______________________________---- 43 Summary of statistical data: log-transformed linear regression of transport rate, Helley-Smith sampler versus conveyor-belt sampler (conveyor-belt data corrected for conditions of stream-wide slot) _____________________--- 44 SYMBOLS Coefficient in regression equation. Exponent in regression equation. Reference to conveyor-belt sampler. Sediment-particle size, in millimeters. Mean depth of effective discharge, in meters. Reference to Helley-Smith sampler. Unit transport rate of sediment in dry weight per second, in kilograms per second per meter. Probability distribution. Complete river discharge, in cubic meters per second. Discharge over 14.6-m width of bedload trap; includes all flow over active width of the streambed, in cubic meters per second. SD SE Correlation coefficient. Standard deviation. Standard error. Mean velocity of effective discharge, in meters per second. Effective width of channel, = 14.6 m. Independent variable. Dependent variable. Variance of the error in the independent variable. Variance of independent variable. Summation. (superscript) reference to mean value of parameter. A FIELD CALIBRATION OF THE SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER By Wiruram W. Emmett ABSTRACT For sediment particle sizes between 0.50 and 16 millimeters, the Helley-Smith bedload sampler has a near-perfect sediment-trapping efficiency. For particle sizes smaller than 0.50 millimeters, the Helley-Smith sampler has a high bedload sediment-trapping effi- ciency because part of the sediment retained by the sampler has been transported in suspension and cannot be quantified separately from the bedload. For particle sizes larger than about 16 millimeters, the Helley-Smith sampler has a low sediment-trapping efficiency, but this may be related to the paucity of coarse particles in transport in the calibration tests, rather than a reflection of an actual low trap efficiency for large-size particles. INTRODUCTION Schoklitsch (1950), in reference to bedload transport, stated "there is not too much known about it." His statement was not without reason; the problems asso- ciated with measurement of bedload transport in allu- vial channels are significant. Hubbell (1964) has described many of the problems encountered with measurement of bedload and also provided a current (at that time) state-of-the-art report on apparatus and techniques for measuring bedload. The reader is re- ferred to the discussion by Hubbell for an overview of various bedload-sampling devices and the merits and shortcomings of each device. Bedload samplers of the direct-measuring type are simplest and most widely used. A direct-measuring bedload sampler intercepts sediment that is in transport over a small incremental width of streambed and accumulates the sediment in a chamber within the sampler. The sampling efficiency of a bedload sampler is defined (Hubbell, 1964) as the ratio of the weight of bedload collected during a sampling time to the weight of bedload that would have passed through the sampler width in the same time had the sampler not been there. Ideally, the ratio is 1.0, and the weight of every particle-size fraction in the collected sample is in the same proportion as in the true bedload discharge. This report presents information on a field calibra- tion of the sediment-trapping efficiency of the Helley- Smith bedload sampler, developed since the Hubbell report. Because the Helley-Smith bedload sampler presently is in widespread use (probably in excess of over 200 samplers worldwide), the data of this report are of particular significance. However, test conditions for field calibration during this study were limited, and thus results are certain not to be applicable to all situa- tions in which the Helley-Smith bedload sampler is being used or being proposed for use. BEDLOAD Bedload is that sediment carried down a river by rolling and saltation on or near the streambed. Though bedload may best be defined as that part of the sedi- ment load supported by frequent solid contact with the unmoving bed, in practice it is the sediment moving on or near the streambed rather than in the bulk of the flowing water. In the sediment-transport process, individual bed- material particles are lifted from the streambed and set into motion. If the motion includes frequent contact of a particle with the streambed, the particle consti- tutes part of the bedload. If the motion includes no contact with the streambed, the particle is literally a part of the suspended load, regardless of how close to the streambed the motion occurs and whether or not the particle is capable of being sampled by existing suspended-sediment sampling equipment. Depending on the hydraulics of flow in various reaches of a chan- nel, particles may alternate between being a part of the bedload or a part of the suspended load. Likewise at a given cross section of channel, particles that are a part of the bedload at one stage may be a part of the sus- pended load at another stage. Any particle in motion may come to rest, and for bedload, the downstream progress is likely to be a succession of movements and rest periods. Particles at rest are part of the bed mate- rial. Obviously, there is an intimate relation between the bed material, bedload, and suspended load. Owing to the somewhat nebulous definition of bed- load, it becomes an exceedingly difficult task to build measuring equipment which samples only bedload. Any device which rests on the streambed is perilously close to sampling bed material, and any device which 1 2 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER protrudes upwards from the streambed, or by necessity is raised or lowered through the flow, may sample some part of the suspended load. This paper utilizes the practical definition of bed- load; that is, bedload is the sediment moving on or near the streambed. A sampler used to measure the transport rate of bedload, by the practical definition, is designated a bedload sampler. This designation does not preempt the fact that some amount of suspended load also may be measured. This possibility and its implications are dis- cussed in a later section of this report. HELLEY-SMITH BEDLOAD SAMPLER Helley and Smith (1971) introduced a pressure- difference bedload sampler that is a structurally modified version of the Arnhem sampler (Hubbell, 1964). The Helley-Smith bedload sampler has an ex- panding nozzle, sample bag, and frame (fig. 1). The sampler was designed to be used in flows with mean velocities to 3 m/s and sediment sizes from 2 to 10 mm. The sampler has a square 7.62-ecm entrance nozzle and a 46 ecm-long sample bag constructed of 0.2-mm mesh polyester, though more recently it has become stan- dard practice to use a sample bag of 0.25-mm mesh polyester. The standard sample bag has a surface area of approximately 1,900 em". Details of the sampler noz- zle and frame assembly are shown in figure 2. The original design included a brass nozzle, aluminum-tubing frame weighted with poured molten lead to a total weight of 30 kg, aluminum tail fins, and bolted construction. More recent versions of the sam- pler have stainless-steel nozzles for greater durability, steel-plate tail fins, solid-steel round-stock bar frame selected to maintain a 30-kg total weight, and all- welded construction. The sample bag attaches to the Sample bag CS FIGURE 1.-Sketch of the Helley-Smith bedload sampler. All dimensions in centimeters 8.893-I F7.62—>’ [+-8.80-*] |-7.62| 4 t (e -=- F— 230.82 ---+ 3.18 Aluminum tubing ea 40.64 |-€ _} 11.43 8.89 P o <- 25.40 -| 93.98 Ficur®E 2.-Plan and side elevation drawings of 7.62-cm Helley- Smith bedload sampler nozzle (above) and sampler (below). rear of the nozzle with a rubber "O" ring. A sliding bracket on the top frame member allows for cable- suspended lowering and raising of the sampler. Posi- tion of the bracket along the frame controls the sam- pler attitude; normal attitude is a slightly tail-heavy position (about 15-degree angle). Since this original design, several structurally dif- ferent versions of the sampler have been built to adapt the sampler to various field uses. One version, devel- oped by the author, has been scaled up from the 7.62- cm sampler. The orifice is twice scale (15.24 ecm), and the frame is one and one-half scale. The larger frame assembly allows for greater weighting; total weight of the larger sampler has generally been either 45 kg or 75 kg, but one sampler constructed for use on the Ama- zon River weighs 250 kg. The large-nozzle sampler is generally used to sample larger sediment sizes, and the heavier samplers become necessary as deeper and swifter rivers are sampled. Perhaps the most extensively used version of the sampler is the 7.62-ecm nozzle adapted to a wading rod, DESCRIPTION OF FIELD-TEST FACILITY 3 rather than having a frame and tail-fin assembly. To minimize weight and to facilitate use of this model, the nozzle is generally of cast aluminum and equipped with a sectionalized tubular aluminum wading rod. A laboratory hydraulic calibration of the Helley- Smith bedload sampler has been conducted (Druffel and others, 1976). Hydraulic efficiency of a bedload sampler has been defined (Hubbell, 1964) as the ratio of the mean velocity of water discharge through the sampler to the mean velocity of the water discharge which would have occurred through the area occupied by the opening in the sampler nozzle had the sampler not been there. In the laboratory study, velocity pro- files were measured in the sampler nozzle and at var- ious locations upstream from the sampler. Typical ve- locity profiles are illustrated in figure 3. The results of this study showed the hydraulic efficiency of the 7.62- cm and the 15.24-cm Helley-Smith bedload sampler is approximately 1.54. This value of hydraulic efficiency was found to be constant for the range of flow condi- tions in the experiments, a range applicable to many natural streamflow conditions. The study, along with other observations by the au- thor, indicates the sample bag can be filled to 40 per- cent capacity with sediment larger than the mesh size (0.2-0.25 mm) of the bag without reduction in hydrau- lic efficiency. However, sediment with diameters close to the mesh size of the sample bag both plugs the sam- ple bag and escapes through the mesh, causing an un- predictable decrease in hydraulic efficiency and loss of the sample. Data on the hydraulic characteristics of the sampler provide qualitative information about probable per- formance, but such data cannot be used directly to w CC u I: 8 T T T T -T T T T y y - EXPLANATION U * - & R A _ 5 7 FO Undisturbed velocity Toprot s G ho's orifice p O 6 F A Velocity profile < 2 7.6 cm upstream y~BF of orifice a > < Velocity profile ~ 4p _ in orifice L uu O 22:8 _ < '.— 2.9 | o —, > < 1 O C 1 L 1 1 1 & 20 40 60 80 100 120 ~ VELOCITY (V )}, IN CENTIMETERS PER SECOND FIGURE 3.-Vertical-velocity profiles upstream of sampler and in sampler orifice of 7.62-cm Helley-Smith bedload sampler. evaluate the sediment-trap efficiency of the unit. Con- trolled experiments are still needed to define sediment-trap efficiency; this report describes the re- sults of one such field calibration of the sampler. DESCRIPTION OF FIELD-TEST FACILITY An open slot in the streambed of the East Fork River, Wyoming, continually excavated of trapped debris by a conveyor belt, provided a bedload trap and direct quan- titative measurement of bedload-transport rates for comparison with bedload-transport rates measured with the Helley-Smith bedload sampler. The following sections describe the test stream, conveyor-belt bed- load trap, and procedures followed in using the Helley-Smith bedload sampler. EAST FORK RIVER The East Fork River heads in the Wind River Range of Wyoming west of the Continental Divide and east and south of Mt. Bonneville (fig. 4). From a series of small alpine lakes and an altitude of approximately Mount d WYOMING Bonneville @- Li: Area of \ ; ® report \ (n)? 2 {he S V’x sae? [> $: Km” apa a ..J .o sabor" yx 109°34' W / Project .|\.. f/_)' $s 3 7 "442°4o' N ) Af . ** tis, ~. .' 'Drainage boundary m_z. i. £ 5 10 _ 15 KILOMETERS FicurE 4.-Map of East Fork River drainage area upstream of the project site. 4 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER 3,400 m, the East Fork River descends about 1,250 m in 50 river km to the project site described in this re- port. Downstream from the study reach, it continues another 50 km to its confluence with the New Fork River, tributary to the Green River. The project site is at lat 42°40'23" N., long 109°34'16"W. The drainage area of the East Fork River at the project site is 466 km". About half of this basin area lies within the Wind River Mountains and is underlain by granitic and metamorphic rocks, mostly of Precambrian age; the other half of the basin area is provided by a major tributary, Muddy Creek, that en- ters the East Fork River about 5 km upstream of the project and drains an upland of rolling hills underlain by lower Tertiary sandstone and shale of the Wasatch Formation. Much of the sand portion of the sediment load for the East Fork River comes from the Muddy Creek basin, but most of the water during high flow comes from melting snow of the mountain area. The high-flow season is generally late May to mid-June, and little bedload movement occurs at other times in the year. In the vicinity of the project, the East Fork River meanders in a flood plain averaging 120 m in width, which, in turn, is confined within a glacial outwash terrace of sand and gravel, the tread or surface of which is some 5 m above the flood plain. This terrace and outcrops of the Wasatch are sources of fresh sand and gravel debris wherever the river impinges later- ally against them. The level of the flood plain corresponds with the bankfull stage of the river, at which the water has an average depth of about 1.2 m. The bankfull discharge is about 20 m*/s, which, in the annual flood series, has a recurrence interval of about 1.5 years. The water- surface slope in the vicinity of the project area is 0.0007, averaged over 1.5 km of river length. Composition of the streambed of the East Fork River at the project site is predominantly sand, but in the 5-km reach of river from Muddy Creek to the project, gravel bars are spaced at regular intervals of about five to seven channel widths. Eight bed-material samples were collected at each of 29 sections along approxi- mately a 200-m reach upstream and downstream of the bedload trap. Data of the composite size distribution of the 232 samples (about 200 kg) are included in table 1 and indicate a median bed-material particle size of 1.25 mm. The median bed-material particle sizes at each of the 29 sections are shown on the planimetric map of figure 5. The occurrence and location of gravel bars is apparent, as median particle sizes vary from 0.6 to 25 mm and indicate a large range of particle sizes avail- able for transport. However, the majority of median particle-size data indicate an overwhelming abun- dance of medium to coarse sand available for transport. Only limited information is available describing bed- forms and their characteristics. At low flows when bed- load is negligible, ripples exist over the sandy portions of the streambed. Isolated sediment particles may be in motion, but generally the ripples are stationary. From intermediate flows to the highest discharges observed, the bed is either flat or has long, low dunes and is fairly resistant to local scour around a foreign object placed on the bed. The better defined bedforms, as recorded on sonar tracings, indicate an amplitude of about 10 em, a wave length of about 10 m, and a period of about 30 minutes. These characteristics of the dunes are sub- stantiated by cyclic trends in measured bedload- transport rates. However, the measured bedload- transport rates cannot be used to quantitatively describe bedforms, because dune fronts traveled dia- gonally to the flow, whereas bedload measurements were taken orthogonally to the flow and integrated the passage of bedforms over time. CONVEYOR-BELT BEDLOAD TRAP Across the East Fork River, a concrete trough was constructed in the bed, orthogonal to the flow direction, that would constitute an open slot into which would fall any sediment moving near or on the streambed. The trough is 0.4 m wide and 0.6 m deep; the level of the lip or top surface corresponds to the natural bed, lower in elevation at the thalweg than near the banks. Figure 6 is a cross section at the bedload trap; although at the trap the entire wetted perimeter is bounded by concrete construction, only at the definite angles at changes in boundary projections is the cross section different than the preconstruction cross section. Along the bottom of the concrete trough passes an endless belt of rubber, 0.3 m wide; it is threaded around some drive and guidance cylinders, then returns over- head, where it is supported by a suspension bridge across the river. Thus, sediment falling into the open slot drops on the moving belt, then is carried laterally to a sump constructed in the riverbank, where it is scraped off the belt. From the sump, sediment is exca- vated by a series of perforated buckets on an endless belt. The buckets lift the sediment to an elevation 3 m above the riverbank and dump the load into a weighing hopper. When the hopper is periodically evacuated by opening a bottom door, accumulated sediment falls on a horizontal endless belt that carries it in a downstream direction 12 m and dumps the load on a transverse endless belt, which, in turn, carries the de- bris toward the river and dumps it into the flowing water, to be carried downstream in a normal manner. In this way, trapped sediment is collected, weighed continuously, and returned to the river. Figure 7 pro- vides some general views of the river and the bedload trap. DESCRIPTION OF FIELD-TEST FACILITY 5 Channel stationing, in meters 120 x \ \ Medi i ee ian bed material ”OX //6'2 particle size, in millimeters wer //’( \ _-6.0 100% -~" 34 -__- /\\ _- 1.8 N PAT 0.8 go* __X~ $ 1.2 & .A & , 90 8..." ~ 08 £ y x P 3 \ eek X ~ <>. \ we 308 o 10 20 30 METERS a +4 C_ sms 1 n 2 1T __ _d 4 80_ ~.. fe. x 0.7 > f. Y- Flood plain -T | ““ 1.6‘\\;;7£ ‘.( 1.0\\ / 1.1 > 2544 o 4 10.5 3" "s. /eo\ J. 0s 07 12 1418 (\\ PX aJ | l- E \<50 5 .% 1 77 I, 1 #. S-st g-] | x +-+ . > ~+--|-+ 40 *~ Low terrace l I I 10 | x Bedload trap Figure 5.-Map of East Fork River in vicinity of bedload trap; data show median diameter of bed material at sections along river. 2.5 2.0 1.5 s 0.5 GAGE HEIGHT DATUM, IN METERS Gate numbers _10 1. 1 M N 1 I 1 L N 1 1. . 2° 4 C6. g f0 12 A14 16°18 20 DISTANCE FROM LEFT BANK ABUTMENT, IN METERS Figure 6.-Cross section of the East Fork River at the bedload trap. The concrete slot across the riverbed may be closed by a series of eight gates, each 1.83 m in length. The gated length of the slot is thus 14.6 m, constituting the full width of the bed active in bedload transport. The gates are actuated hydraulically and may be opened or closed individually. When the gates are open, the open slot or trap is 0.25 m wide. At low and moderate dis- charges, all gates are open so that the load accumu- lated in the weighing hopper represents the total for the river. At high discharges, gates are opened indi- vidually, and the transport rate for the whole river is computed by adding the rates recorded in the eight gates individually opened. The hopper collecting the debris stands on a large scale that may be read visually. The belt-and-bucket-transport system can ac- commodate a load received at a rate as great as 100 kg/min. The weight of the trapped load is recorded each minute as it accumulates in the hopper, so the weights represent a wet sample. Numerous comparisons of the weight of samples when wet and after drying give a consistent ratio of dry/wet weight of 0.85. Mean transport rates are determined by averaging the 1-minute recordings over a sampling duration of 30 minutes to several hours. Samples of the trapped sediment for size analysis are scooped from the endless belt as the weighing hopper is periodically emptied. Samples were collected every time the hopper was emptied; each sample retained weighed about 2 kg. These samples were taken to the laboratory where they were dried, sieved, and weighed 6 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER FicurE 7.-Conveyor-belt bedload sampler, East Fork River, Wyo- ming. A, View across river showing suspension bridge and drive mechanism of conveyor-belt bedload sampler; flow is relatively low. B, View downstream at suspension bridge; flow is relatively by size fractions. For small samples (single emptying of the hopper), the entire sample was used in the sieve analysis. For large samples (multiple emptying of the hopper), the entire sample was sieved for gravel-size sediment (>2.0 mm) and the remainder split to about 1 kg for sieving of the material smaller than 2.0 mm. In all instances, the sample retained was large enough to be representative of all sizes of material collected, and the sieving procedure maintained this accuracy throughout the analysis. For comparison with the bed-material size data in table 1, table 2 lists a transport-weighted particle-size distribution for the whole of bedload sampled in 1976. The median particle size of bedload is 1.13 mm, compared to 1.25 mm for bed material. Although the median particle size of bedload and bed material is nearly the same, the bed material consists high. C, Bedload trap on streambed is visible below suspension bridge; gates are in closed position. D, Vertical-lift assembly, weighing hopper, and conveyor belt for return of sampled sediment to stream. of some larger particles that are rarely moved. For bed- load and bed material, table 3 lists particle size at given particle-size categories (given percentage, by weight, finer than values). Table 3 clearly indicates that some bed-material particle sizes are seldom in- volved in the sediment-transport process. Discharge measurements by current meter are made nearly every day during the sampling season from the suspension bridge at the project site. At low flow, all discharge, Q, is within the 14.6-m width of the gated slot; at bankfull (@~20 m*/s) discharge, the water spreads over the full 19-m width of channel, but only 5 percent of this discharge is in the near-bank zones be- yond the 14.6-m wide bedload trap; at maximum dis- charge (45 m*/s), about 8 percent of the discharge is beyond the ends of the bedload trap. Though overbank flow onto the flood plain occurs in other reaches of the DESCRIPTION OF FIELD-TEST FACILITY 7 TABLE 1.-Size distribution of composited bed material, East Fork River, Wyoming, at bedload-transport research project Percentage, by weight, Sieve Percentage, by weight, 1 finer than sieve diameter retained on sieve (mm) "d m 3 1 1 1 i 1 1 1 1 1 1 1 1 1 1 1 e to O o oo ho -i- to m on on to li- ia i- tn O O im O im i- o~Non coo; ~1t0hoghNOkRi-,. . . ( o ho 0 ip oo tn & im bo in do ~1 as ho go do ho do do h to O & w 1 90 op 1 -1 ~1 Or G2 OT Or & G0 bo i a TABLE 2.-Size distribution of transport-weighted composite bedload (1976 conveyor belt), East Fork River, Wyoming, at bedload- transport research project Percentage, by weight, Sieve Percentage, by weight, finer than sieve diameter retained on sieve (mm) "d u 3 1 1 1 i i 1 1 1 1 1 1 i 1 I 1 O to O C w w 1 to 1p tp co 1g Onh to - © w w o NC N 1 N SDF, 0; _ 0 oo in to to i- -i ho on to oo to i- o ho ib 0 m in to 0 a TaBLE 3-Comparison of bed material and bedload particle sizes Particle-size category Particle size (mm) (4 incteentage tinct want Bed material Bedload dy ALLA... c ald 0.27 0.32 did PSLE PACT < + har 42 47 pa AI.. mls. 53 58 e PL I_ _ Nl. 69 73 lore in y 1.25 1.13 f eno en - 3.20 1.73 §§ reer e Mere n s 209 8.00 2.37 As f rerun c 17.6 3.42 (Beco ere . nie ananas 37.6 7.01 river, at the project site a high natural bank on the right side and a short embankment on the left prevent any overbank flow. Essentially, all bedload is ac- counted for, and all the flow passes through the 19-m width of channel at the measuring section. The hydraulic-geometry relations for the East Fork River at the bedload trap are shown in figure 8. In reality, the concrete trough and abutments of the bed- load trap force small "kinks" in the hydraulic- geometry relations; the relations shown in figure 8 have been smoothed and reflect the hydraulic charac- teristics of the river if the bedload trap were not in- stalled. For interpretative studies of bedload transport, the hydraulic conditions above the 14.6-m width of bed- load trap are more significant than the whole-channel hydraulic conditions. These hydraulic conditions will be termed "effective hydraulics," and it is the effective hydraulic parameters that are listed in subsequent tabulations of data in this report. The reader may ob- tain corresponding stream-wide conditions by refer- ence to figure 8. The bedload trap was installed in fall and spring, 1972-73. Robert M. Myrick was project engineer for construction of the trap and is due much of the credit for subsequent successful operation of the installation. Data collection began in the spring of 1973 and has continued during spring months since then. The data- collection program for the conveyor-belt bedload sam- pling was initially under the direction of Luna B. Leopold but gradually has drifted toward co-direction by Leopold and the writer. Basic data for the bedload 5 T T T Err u 50 2 F Surface width / 4 20 10 COP OULU 1 LL_L__LL_L_L_LLL 4 1 2 5 10 20 50 100 DISCHARGE, IN CUBIC METERS PER SECOND DEPTH IN METERS; VELOCITY IN METERS PER SECOND WIDTH IN METERS; AREA IN SQUARE METERS Ficur® 8. -Hydraulic-geometry relations for the East Fork River at the bedload trap. 8 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER trap have been previously published (Leopold and Emmett, 1976, 1977), and the same authors are pres- ently preparing an interpretative report utilizing these data. Some of the data have been incorporated into a report (Mahoney and others, 1976) which compiled in- formation necessary for calibrating unsteady-flow sediment-transport models. This latter report princi- pally contains data of cross-sectional changes (scour and fill) at a number of sections in the 5-km reach of channel upstream of the bedload trap. Two doctoral dissertations (Lisle, 1976; Andrews, 1977) provide ad- ditional information about the fluvial characteristics of the East Fork River in the vicinity of the bedload trap. Use of the conveyor-belt facility for field calibration of the Helley-Smith bedload sampler covered the period 1973-76, but principal data collection for the field- calibration purpose was May-June, 1975. The writer had sole responsibility for the Helley-Smith bedload- sampling program. SAMPLING TECHNIQUES WITH HELLEY-SMITH BEDLOAD SAMPLER Although the Helley-Smith bedload sampler is in widespread use by the U.S. Geological Survey and other Federal and State agencies, and by university and private organizations, it has not been officially sanctioned by the Federal Inter-Agency Sedimentation Committee (Water Resources Council) nor certified for its technical performance by the U.S. Geological Sur- vey. This certification is awaiting completion of rigor- ous laboratory testing of the sediment-trapping characteristics of the sampler under direction of the U.S. Geological Survey and the Federal Inter-Agency Sedimentation Committee. Although laboratory test- ing is now underway, it appears that the early 1980's is a reasonable target date for completion of testing and certification (or possible rejection) of the sampler. Widespread use of the sampler, but lack of certifica- tion, combine to create some confusion with sampling procedures. No formal technique manual for use of the sampler exists, and instructions for its use are gen- erally passed on by word of mouth from user to new- comer. Even on an interim basis, this procedure is ac- ceptable only if the user passes along instructions based on reliable past use of the sampler. The writer has collected more than 10,000 individual bedload samples with the Helley-Smith sampler. This experience, combined with gained insight of temporal and spatial variabilities in bedload-transport rates, has enabled him to establish a sampling procedure for the Helley-Smith sampler which gives consistent re- sults. The spatial or cross-channel variations in bedload- transport rates are significant. Frequently, all or most of bedload transport occurs in a narrow part of the total width of channel. Though this narrow width of significant transport is generally stationary, it can shift laterally with changes in hydraulic conditions or sediment characteristics. Therefore, knowledge of where maximum or all bed load transport had occurred previously is not a criterion for eliminating a portion of channel width from the sampling program. At least 20 equally spaced, cross-channel sampling stations are necessary to insure that zones of both maximum and minimum transport are adequately sampled. (For large rivers and small rivers, the technique may be modified so that sections are not spaced greater than 15 m apart, nor is there apparent need for spacing sections closer than 0.5 m.) Temporal variations in bedload-transport rates may also be large. This variation with time is obvious for the stream channel with movement of dunes, but even in gravel-bed rivers with no apparent dunes or migrat- ing bedform, bedload transport may occur in slugs of sediment and show distinct cyclic trends with time. The frequency of the cyclic trend is dependent on the velocity and wavelength of the bedform or slug of sed- iment. Obviously, a precise procedure would be to sam- ple at each cross-channel station until a reliable mean transport rate was established at each cross-channel location, but time requirements prohibit this detail. The adopted procedure, a compromise betwen effort expended and idealized precision (in reality, little pre- cision is lost), is to conduct two traverses of the stream and to sample at least 20 sections on each traverse. The spatial factor is covered by the 20 sections; the tem- poral factor is covered both because of the time ex- pended during a single traverse of the stream and the time lag at each section as the second traverse is con- ducted. A comparison of values of mean transport rate, determined by multiple traverses of the stream, shows little change in the mean value by the addition of more than two traverses. Further, because of changes in the river hydraulics with time, and with each traverse of the river being time consuming, it is often impossible to conduct more than two traverses of the river and have the data considered as instantaneous or existing simultaneously. Each sample collected with the Helley-Smith bedload sampler requires about 2 to 3 minutes for lowering, sampling, raising, emptying, and moving to a new cross-channel location. A typical traverse thus requires about 1 hour; two traverses re- quire about 2 hours. The time required to complete the double traverse generally allows a minimum of several cycles to be sampled in the cyclic trend of transport; this appears adequate to average temporal variations in transport. For the East Fork River sampling program, all bed- RESULTS OF FIELD-CALIBRATION TESTS 9 load occurs over the 14.6-m length of the gated slot in the streambed. Eight gates constitute this width; bed- load sampling with the Helley-Smith sampler was made at the 1/6, 1/2, and 5/6 points of each gate (cen- troid of each third of gate length). Thus, 24 cross- channel sections constituted the cross-channel fre- quency of sampling for the East Fork River. Two traverses of the stream total to 48 individual Helley- Smith type samples, which are averaged to give a mean bedload-transport rate and used in the compari- son with a mean bedload-transport rate for the conveyor-belt sampler. The suspension bridge across the East Fork River at the bedload trap provided access across the river. The Helley-Smith sampler was lowered by cable to the streambed, timed for a duration of 30 seconds, and re- trieved. By lowering the sampler from the upstream side of the bridge and placing the sampler on the streambed just upstream from the conveyor belt, si- multaneous collections of data could be made with both sampling devices. Though efforts were made to have simultaneity in sampling, in reality, varying lengths of time were required to complete data collection by the two sampling methods. A slightly different mean stage or discharge may be recorded for the time period of Helley-Smith type sampling versus that for the conveyor-belt sampling. The differences were not con- sistent in biasing one method of sampling and were always minor. A later section of this report shows the results of corrections made to the measured bedload data to compensate for slight mean-discharge differ- ences for data sets that were designed to be contempo- rary. Generally, each Helley-Smith type bedload sample was individually bagged and later air dried, sieved, and weighed. Data thus collected could be later analyzed for cross-channel variability in transport rate and particle size or composited by gate length or whole-stream width for a comparison with the conveyor-belt data. Although many data are available for a gate-by-gate comparison of the conveyor-belt and Helley-Smith sampling methods, all data of this report are for stream-wide mean values. Thus, each point of comparison involves 48 Helley-Smith bedload samples and, generally, several hours of conveyor-belt opera- tion. Totally, 100 runs were made with the conveyor belt, and 83 runs were made with the Helley-Smith sampler. All data are useful in separate analyses of percentage of total load in each particle-size class, and these analyses may be compared by method of collec- tion. In addition, concurrent runs by the two methods can be compared directly. Comparison of total bedload-transport rates as measured by the two meth- ods does not require knowledge of particle-size distri- bution. In this instance, there are 74 matched sets of data available for direct comparison. For a comparison on a given particle size basis, various runs combine to give 61 simultaneous or matched data sets. Both the separate analyses and the direct comparisons are the results described in the next section. Figure 9 illustrates some of the techniques and pro- cedures involved with use of the Helley-Smith bedload sampler. RESULTS OF FIELD-CALIBRATION TESTS All basic data of the study are summarized in tables 4-7. Measured and computed river hydraulic data and measured bedload-transport rates are listed in tables 4 and 6 for the conveyor-belt sampler and Helley-Smith sampler, respectively. Particle-size distributions of bedload for the transport rates listed in tables 4 and 6 are given in tables 5 and 7 for the conveyor-belt sam- FigurE 9.-Bedload sampling with the Helley-Smith bedload sam- pler. A, Preparing to lower sampler at relatively high flow rate. B, Preparing to empty sampler. 10 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER pler and the Helley-Smith sampler, respectively. For all analyses of this report, transport rates for given particle-size classes were obtained by multiplying total bedload-transport rate (tables 4 and 6) times the per- cent retained, by weight, in each particle-size class (from tables 5 and 7). This voluminous set of data was stored in computer memory during analysis of data, but since it can be easily duplicated, tabulations of it are not reproduced in this report. Although sieve analyses were conducted by half-phi increments (factor of V2), transport rates by particle-size class were analyzed by whole-phi increments (factor of two). This gave nine particle-size classes ranging in particle size from 0.06 mm to 32 mm; for the conveyor-belt sampler, several particles larger than 32 mm were measured, but these did not constitute a large enough sample to establish a reliable data set. BEDLOAD-TRANSPORT RATES BY INDIVIDUAL PARTICLE-SIZE CATEGORIES, BASIC DATA Relations of the bedload-transport rate in each particle-size class as functions of total bedload- transport rate were determined for both methods of sampling. The statistical procedure utilized was a least-squares linear regression of log-transformed data, giving a power equation of the form: Y =' AX". or more specifically: J» (size class) = A;, (total)", where j, is the dry weight unit bedload-transport rate in kilograms per meter-second. Data of the statistical analyses are presented in ta- bles 8 and 9 for the conveyor-belt sampler and the Helley-Smith sampler, respectively. Graphs of basic data and statistical analyses are illustrated in figures 10-18 for various particle-size classes; for each method of sampling, graphs show the least-squares fit to the data and, superimposed, the least-squares fit to the - data of the alternate method of sampling. Of special interest is the percentage of total bedload occurring in each particle-size class. Utilizing mean values of X and Y data and summarizing from tables 8 and 9: Particle-size class Mean percgniage of total bedload (mm) in particle-size class (¥/X) Helley-Smith Conveyor-belt .... /.: _E 0.35 0.32 Age cg cl elce Bel 3.24 1.74 ade ADO scl itl rong 22.89 18.49 02.200 20 26.84 27.89 1:00-:2:00 20.07 21.89 2.00- 4.00 __ A 13.87 4.00- 8.00 __ 5.56 8.00-16.00 __ 1.49 16.00:32 .65 74 Mean percentages in the above table do not add to 100, because the mean value of total bedload is variable. That is, larger particles move only during higher transport rates, and the mean value of total bedload transport is, obviously, greater during those instances. The effect is to decrease the apparent mean percentage of total bedload in the larger particle-size classes; the adequacy in sampling of large particles will be dis- cussed subsequently. Before continued discussion, it is also of interest to note the rate of change in the above percentages as the actual bedload-transport rate increases or decreases. This rate of change is described by the exponent of the regression equations, B. Summarizing from tables 8 and 9: Particle-size class Rate of change in percentage of total bedload (mm) i in particle-size class (B) Helley-Smith Conveyor belt 90.00-: 0:12: : cose ll cont. de ad lod 0.727 0.663 AR - B0. ein ERICA Higa n Bice 599 553 vae C100 2.0000 die iano s 698 742 d0- 100- 1 tint 1.050 1.000 1.00 2.00 ..... .. .and t cel mst 1.213 1.173 2,00--4.00 ...;... sls inl 1.344 1.278 4008.00 GGA 1.193 1211 £:00-16.00 -.. s..... . sll .867 .995 16:00-92:00 o 387 .926 Because the mesh size of the sample-collection bag used on the Helley-Smith sampler was 0.20 mm, data of the first two particle-size categories tabulated above should be disregarded. Probably quite by coincidence, the amount of 0.06 to 0.12 mm size sediment trapped by the conveyor-belt sampler (insignificantly at 0.3 percent) is nearly identical to the amount of same-size material that was trapped in, rather than washed through, the Helley-Smith sample-collection bag. The Helley-Smith sampler collects nearly twice as much sediment in the 0.12 to 0.25 mm size class as the conveyor-belt sampler. However, not only is the catch in the Helley-Smith sampler not valid because of the mesh size of the collection bag, but also analysis of suspended-sediment size data indicates this particle- size class represents the dominant particle sizes of sus- pended sand. Thus, the Helley-Smith sampler, which protrudes into the flow, is receiving an abundance of this size suspended sediment, some of which is trapped but the majority of which is washed through the sam- ple bag. For sediment in the 0.25 to 0.50 mm particle-size class, both samplers must retain all sediment which is supplied to them. The Helley-Smith sampler shows a greater mean percentage of total bedload in this size class than does the conveyor-belt sampler, but again, analyses of suspended-sediment data show appreciable quantities of this size sediment in suspension. Cer- tainly the collection of some suspended sediment by the BEDLOAD-TRANSPORT RATES BY INDIVIDUAL PARTICLE-SIZE CATEGORIES, BASIC DATA 11 1.0 t T T T (A) HELLEY - SMITH SAMPLER (0.06 - 0.12 MM SIZE CLASS ) 0.1} { EXPLANATION ] 1973 o - 1974 PR @ 1975 7 O 1976 2 0.001} _ 5 w < P" 0 PH m > o 0.0001} On - u 3 £3 i Least squares From (B) below te J: o € {i 0.00001 + 4 + 4 GC = s= & . m < iP 1.0 T T T T €.2 3 g (B) CONVEYOR - BELT SAMPLER g 3 (0.06 - 0.12 MM SIZE CLASS ) < = 0.1 4 oC ax E EXPLANATION 3 u - 1973 a o - 1974 a a e 1975 | O 1976 0.001} - 0.0001} 7 Least squares 0.00001 i i ; M 0.00001 0.0001 0.001 0.01 0.1 1.0 TOTAL BEDLOAD TRANSPORT RATE, IN KILOGRAMS PER METER - SECOND FIGURE 10.-Relation of bedload-transport rate in individual-size category as function of total bedload-transport rate; 0.06-0.12 mm size class. Helley-Smith sampler is an explanation for its greater mean percentage in this size category, but a quantita- tive description of how much of it is attributable to this effect is not possible. It is most important to recognize that the Helley-Smith sampler does receive suspended sediment, and the absolute quantites of it are depen- dent on the sizes of sediment in transport and hydrau- lic characteristics of the flow-factors which are differ- ent for every stream and thus cannot be calibrated. Complete analysis of suspended-sediment size data for the East Fork River shows no significant quantity of suspended sediment larger than 0.50 mm, For mate- rial capable of being moved in suspension (<0.50 mm), its significance as bedload decreases as bedload- 1.0 T T T T (A) HELLEY - SMITH SAMPLER (0.12 - 0.25 MM SIZE CLASS ) 0.1} E EXPLANATION |_| 1973 o 1974 e 1975 | O 1976 C 0.001 |- i 5 w < 3 Least squares f Ao | // 3 9 0.0001 Lee { 5 § »" From (B) below _L_) w & | € {§ 0.00001 1 1 1 i 2 § L 2 E 1.0 t T u T ou» --. é E (B) CONVEYOR - BELT SAMPLER 3 S (0.12 - 0.25 MM SIZE CLASS ) < 9 0.1} A cc x g EXPLANATION S ® - 1973 a am: o 1974 | to 7 ® 1975 O 1976 0.001} - 0.0001} a = Least squares DO 00 0.00001 1 i i i 0.00001 0.0001 0.001 0.01 0.1 1.0 TOTAL BEDLOAD TRANSPORT RATE, IN KILOGRAMS PER METER- SECOND FIGURE 11.-Relation of bedload-transport rate in individual-size category as function of total bedload-transport rate; 0.12-0.25 mm size class. transport rate increases. This is reflected in the rate of change values (exponent B) tabulated above. The val- ues for suspended-sediment size particles are less than unity, indicating that as total bedload-transport rate increases, the percentages of sediment in those size classes decrease. For sediment in the four particle-size classes ranging in size from 0.50 to 8.0 mm, significant bedload transport occurs, and the significance increases as the total bedload-transport rate increases. The dominant particle-size class of bedload is 0.50 to 1.0 mm and ac- counts for a little over one-fourth of the total bedload (recall also the size distribution of composited bedload, table 2). The greatest rate of change in percentage of 12 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER 1.0 T T T T (A) HELLEY - SMITH SAMPLER ( 0.25 - 0.50 MM SIZE CLASS ) 0.1} _ EXPLANATION = - 1973 O 1974 op @ 1975 § O 1976 2 sont Least squares _s | 4 Fa ys c From (B) below S O 0.0001} 7 w 2 & 8 o & - | E E 0.00001 1 1 1 1 2 § us 2 g 1.0 T t T T * us E § (B) CONVEYOR - BELT SAMPLER g 5 (0.25 - 0.50 MM SIZE CLASS ) o 3 3 0.1} - c % Z EXPLANATION é = - 1973 a orl o 1974 | C2 ' @ 1975 0 1976 a From (A) abov 0.001} # _ Least squares 0.0001} o s it 0.00001 <4. i i i 0.00001 0.0001 0.001 0.01 0.1 1.0 TOTAL BEDLOAD TRANSPORT RATE,IN KILOGRAMS PER METER- SECOND FiGurE 12.-Relation of bedload-transport rate in individual-size category as function of total bedload-transport rate; 0.25-0.50 mm size class. total bedload in a given particle-size class occurs for particles in the size class of 2.0 to 4.0 mm, followed by size classes 1.0 to 2.0 mm and 4.0 to 8.0 mm. These rates of change values combine with the mean percent- age values such that at high bedload-transport rates the percentage of total bedload is actually greatest in particle-size categories of 1.0 to 2.0 mm and 2.0 to 4.0 mm, This leads to a median particle size of composited bedload being 1.13 mm (table 3). For sediment sizes greater than 8.0 mm, only about V&-2 percent of the total bedload occurs in the particle- size categories of 8 to 16 mm and 16 to 32 mm. The mean transport rate for these size particles is about 1.0 T T T T (A) HELLEY- SMITH SAMPLER (0.50 - 1.00 mm SIZE CLASS ) 0.1} _ EXPLANATION a 1973 [e] 1974 ae @ 1975 7] O 1976 0.001} { // Least squares From (B) below/% 74 0.0001} / 3 0.00001 4 4 4 4 1.0 T T T T (B) CONVEYOR - BELT SAMPLER ( 0.50 - 1.00 MM SIZE CLASS ) KILOGRAMS PER METER - SECOND EXPLANATION 1973 1974 1975 1976 BEDLOAD TRANSPORT RATE IN PARTICLE-SIZE CLASS, IN 0.01 C e o a 0.001} From (A) above { 0.0001} Least squares 0.00001 1 1 i i 0.00001 0.0001 0.001 0.01 0.1 1.0 TOTAL BEDLOAD TRANSPORT RATE, IN KILOGRAMS PER METER- SECOND FIGURE 13.-Relation of bedload-transport rate in individual-size category as function of total bedload-transport rate; 0.50-1.00 size class. 0.0004 kg/m-s (tables 8 and 9). The average 32-mm particle weighs about 55 grams, and the average 16-mm particle weighs about 6.8 grams. These num- bers can be manipulated to show that, streamwide, only about three 32-mm particles and twenty-five 16-mm particles pass down river every 30 seconds, the duration of sampling with the Helley-Smith sampler. Since the Helley-Smith sampler covers only about 0.5 percent of the stream width (76.2 mm nozzle/14.62 m wide), the Helley-Smith sampler has somewhat less than a 2 percent chance of collecting a 32-mm particle and about a 15 percent chance to collect a 16-mm parti- cle. This is additionally reflected in the number of ob- BEDLOAD-TRANSPORT RATES BY INDIVIDUAL PARTICLE-SIZE CATEGORIES, BASIC DATA 13 1.0 t - -r 4 (A) HELLEY - SMITH SAMPLER (1.00 - 2.00 mm SIZE CLASS ) 0.1- _ EXPLANATION # 1973 o 1974 tor ® 1975 1 0 1976 Least squares 2 y 0.001 - [a] R w < -) O H s From (B) below / n a 0.0001) / { 4 9 4 o } te | a &© 0.00001 i i 1 i 2 G J = - oc C £ 1.0 o © : : t 2 (B) CONVEYOR- BELT SAMPLER g (1.00 - 2.00 mm SIZE CLAssS ) 2 S < _ E > 0.1} _ G EXPLANATION < o = 1973 & o 1974 e 0.01 1 a f e 1975 o 1976 0.001 f- - From (A) above Least squares 0.0001] 7 o/o 0.00001 1 i a i 0.00001 0.0001 0.001 0.01 0.1 1.0 TOTAL BEDLOAD TRANSPORT RATE, IN KILOGRAMS PER METER- SECOND FiGurE 14.-Relation of bedload-transport rate in individual-size category as function of total bedload-transport rate; 1.00-2.00 mm size class. servations recorded in tables 8 and 9. While nearly 50 percent of the runs with the conveyor-belt sampler in- cluded trapping a particle of size 16-32 mm, fewer than 25 percent of the runs with the Helley-Smith sampler included trapping a particle of that size. Thus, the transport rate for large particles in the East Fork River was too minimal to allow reliable calibration for particles larger than about 8 mm, perhaps to 16 mm. It should also be pointed out that the rate-of-change data for the two coarsest size categories are mislead- ing. Since the largest particles move only at high transport rates, many low transport runs are not in- cluded in the analysis for these size particles. By this 1.0 T T T T (A) HELLEY - SMITH SAMPLER (2.00 - 4.00 MM SIZE CLASS) 0.1} a EXPLANATION = - 1973 o 1974 0,01 [- ® 1975 a O 1976 2 0.001} _ A < a= 0 8 o 3 g o.0001} h 5 8 / / Least squares o ¥ From (B) beIOW/ o o & | / ® < { 0.00001 : 4 4 + = 5 5 5 £ et a & 1. t t t - $8 ** g: 5 (B) CONVEYOR - BELT SAMPLER $ g (2.00 - 4.00 mm SIZE CLASS ) <2): s 0.1} - gC . be g EXPLANATION PS = - 1973 é apt.. O 1974 3 lind ° © 1975 O 1976 0.001} - / From (A) above Least squares B o 0.00001 i O _ i i 0.00001 0.0001 0.001 0.01 0.1 1.0 TOTAL BEDLOAD TRANSPORT RATE, IN KILOGRAMS PER METER- SECOND FIGURE 15.-Relation of bedload-transport rate in individual-size category as function of total bedload-transport rate; 2.00-4.00 mm size class. fact alone, large particles begin their significance at high transport rates and increase from there. Because zero values cannot be used in log-transformed re- gressions, values of rate of change (slope of the regres- sion equation) comparable to the smaller particle-size categories cannot be quantitatively determined. This section of the results has concentrated on analysis of bedload-transport rates by individual particle-size categories as functions of total bedload- transport rate. Its primary purpose is to place reliabil- ity limits on the comparability of data collected and was used to show that for particle sizes less than 0.50 mm, the influence of suspended sediment casts doubts 14 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER 1.0 T T T T (A) HELLEY - SMITH SAMPLER (4.00 - 8.00 mm SIZE CLAss ) 0.1f 1 EXPLANATION # - 1973 gall o 1974 | we e 1975 o 1976 2 0.001 - _ 74 < =d O 8 o fl; S 0.0001 - - te 4 o g g 0.00001 { 4 f 4 < € Lu z g 1.0 t T t T t€ w & § (B) CONVEYOR - BELT SAMPLER 5 g (4.00 - 8.00 Mm SIZE CLASS ) 3 2 ¢ & onl a a s EXPLANATION & = - 1973 a t o 1974 | H 0.01 e 1975 0 1976 0.001 - - Least squares 0.0001} { 0 0 0.00001 1 G- 4 i 0.00001 0.0001 0.001 0.01 0.1 1.0 TOTAL BEDLOAD TRANSPORT RATE, IN KILOGRAMS PER METER- SECOND FIGURE 16. -Relation of bedload-transport rate in individual-size category as function of total bedload-transport rate; 4.00-8.00 mm size class. on comparability (not reliability) of data collected with the Helley-Smith sampler. For particle sizes less than 0.20 mm (mesh size of the bag), data collected with the Helley-Smith sampler should be discarded. For particle sizes larger than 8.0 to 16 mm, paucity of individual particles moving probably prohibits the Helley-Smith sampler from collecting a representative sample, and data should be treated with caution. The above analyses and discussion are not applicable to a direct comparison between the Helley-Smith sam- pler and the conveyor-belt sampler, because the analysis utilized all available data rather than matched data sets. For example, many data collected at 1.0 T T T T (A) HELLEY - SMITH SAMPLER (8.00 - 16.00 MM SIZE CLASS ) 0.1} -_ EXPLANATION # - 1973 o 1974 9901 @ 1975 , 1 o 1976 '% ¥ 2 0.001} Least - a squares 2 e < P a) o uJ 5 g 0.0001} a L 6 /7 a 2 From (B) below/ D‘ - w 5 | 1 /l 1 1 < gg 0.00001 2 G a = 2 E 1.0 T T T T cc A_ t 2 (B) CONVEYOR - BELT SAMPLER 2 & (8.00 - 16.00 mm SIZE CLASS ) 2 6 < 3 0.1} - F 5 g EXPLANATION 3 u - 1973 a o 1974 | _ 0.01} e 1975 O 1976 0.001 - 7 Least squares 0.0001} ~ o © g O 0.00001 i i i T 0.0001 Y"" aig 0.1 1.0 TOTAL BEDLOAD TRANSPORT RATE, IN KILOGRAMS PER METER- SECOND 0.00001 0.001 FigurE 17.-Relation of bedload-transport rate in individual-size category as function of total bedload-transport rate; 8.00-16.0 mm size class. low transport rates in 1976 with the conveyor-belt sampler were not obtained with the Helley-Smith sampler; this created a data base with a mean transport rate that is different between the two methods of sampling. The next section discusses a di- rect comparison of the two methods of sampling. COMPARISON OF HELLEY-SMITH RESULTS WITH CONVEYOR-BELT RESULTS, BASIC DATA Data collected concurrently with both the Helley- Smith sampler and the conveyor-belt sampler may be compared directly one against the other; a total of 74 COMPARISON OF HELLEY-SMITH RESULTS TO CONVEYOR-BELT RESULTS, BASIC DATA 15 1.0 t t y ; (A) HELLEY- SMITH SAMPLER (16.00 - 32.00 MM SIZE CLASS ) 0.1 - ' EXPLANATION # - 1973 o" t974 0.01} @ 1975 - o 1976 0 E 0.001 - 0 _ 3 + o KQ é Least squares o R P & q; g 0.0001 |- { a Cg / From (B) below E | < & 0.00001 4 + o 1 2 5 i ac 5 5g, 1.0 f i : f 5 & (B) CONVEYOR - BELT SAMPLER 3 $ (16.00 - 32.00 MM SIZE CLASS ) <2! =! 0.1} - CC x 5 EXPLANATION 3 m 1973 a pal o 1974 6 f ® 1975 o | O 1976 Ip o 0.001 - 9 o ~ _ From (A) above ~O 0.0001 | - o Least squares o 0.00001 1 0 i i 0.00001 0.0001 0.001 0.01 0.1 1.0 0 TOTAL BEDLOAD TRANSPORT RATE, IN KILOGRAMS PER METER- SECOND o FigurE 18. -Relation of bedload-transport rate in individual-size category as function of total bedload-transport rate; 16.0-32.0 mm size class. such matched sets of data exists. Some of these data sets are composed of a single run with the conveyor- belt sampler and two runs during the same time with the Helley-Smith sampler, thus giving two points of comparison. Of the 74 data sets, some runs with the conveyor-belt sampler are lacking particle-size analyses. For comparisons made at given particle-size classes, 61 matched sets of data are available. Table 10 lists matching of data sets as used in the present and next sections of this report. Comparisons of the bedload-transport rate in each particle-size class were made with the Helley-Smith sampler results expressed as functions of the conveyor-belt sampler results. As in the previous sec- tion of this report, the statistical procedure utilized was a least-squares linear regression of log- transformed data, giving a power equation of the form: Y = Ax", or, for this analysis: j,(Helley-Smith) = A;, (conveyor belt)", where j, is the dry-weight unit bedload-transport rate in kilograms per meter-second. Results of the statistical analysis are presented in table 11. The sample means of the log-transformed transport rates (log Y and log X) and the standard de- viations of the transformed values (SD(log Y) and SD(log X)) are given in table 11 (top) for each particle-size class; also given are the computed inter- cept (log A) and slope (B) for the transformed data, the estimated variances for these parameters (see Draper and Smith, 1966, section 1.4), and the values of the multiple correlation coefficient, r*, for the regressions. Table 11 (bottom) lists the means for the transport rates before the log transformation (¥ and X), the ratio Y/X, and the values of A and SE(A) (computed by tak- ing the antilogs of log A and SE (log A), respectively). The quantities SE(log A) and SE(A) should be in- terpreted as follows: a confidence interval for the inter- cept log A can be constructed by taking the lower limit of the interval to be log A - constant (SE (log A)) and the upper limit to be log A + constant (SE (log A)). The limits of the corresponding interval for A are obtained by taking A + (SE(A)) constant and A x (SE(A)) con- stant. For a 95-percent confidence interval, the value of the constant is 1.96. Graphs of basic data and statistical analyses are il- lustrated in figures 19-27 for various particle-size categories. Graphs show the least-squares fit to the data and, superimposed, the line of perfect agreement. Summarized below are salient data of table 11, utiliz- ing the mean values of the regression statistics. Particle-size Mean ratio in transport rate; Rate of change in ratio class Helley-Smith:conveyor belt of transport (mm) (¥Y/X, in percent) (B) 0.06- 0.12 _______________- 123.08 0.928 AS-: BD 211.66 151 aB BO 149.98 .802 B0- 1.00 ...:. 98.70 .934 1.00- 2.00 ________________ 89.36 .868 200-400: L.____.__________ 86.43 .803 4.00-.8.00 :.._.______:-_.. 93.81 139 8.00-16.00 ________________ 93.58 747 16.00-82.00 ________________ 55.67 501 If values of the exponent B were equal to 1.0, regres- sion relations would be linear. To test the hypothesis - & 2 1.0 T T T T cc / ] & EXP A 2 LANATION f 2 , - ual _ m- 1978 - I C 8 o 1974 / 5 5 Q @ 1975 a 1 o 1976 / = # 0.01f 4 LJ T G Aerfect agreement & & # € "4 # & so01f ( - - < east S 5 squares O 3 9 9 7 [3 0.0001} { s ® 2 o a { sond /o 0.06 - 0.12 MM ' 0.00001 - 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND Figure 19.-Comparison of bedload-transport rate by individual particle-size category, Helley-Smith sampler versus conveyor- belt sampler; 0.06-0.12 mm size class. E 1.0 T I T T e 2 EXPLANATION / s 3 jl. * 197 pae mcg g o 1974 s a S e 1975 / 1 8 0 1976 a | / a fre 0.01} ~ €'5 al" 7a C oc p E '= < "4 3 0.0001} 54 & { a s #* § 4.00 - 8.00 MM ao 0.00001 F i i i 0.00001 _ 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND FicurE 25.-Comparison of bedload-transport rate by individual particle-size category, Helley-Smith sampler versus conveyor- belt sampler; 4.00-8.00 mm size class. 1.0 T T T T E 1.0 T T T T # / u EXPLANATION Perfect 4 m 1973 agreement/ 2s _.: f o 1974 { = § e 1975 5 o o 1976 be S & 2 a _ oc < < 2 f 3 E C0 oc u E 3 g - s.001f - - 5 Least square 3 EZ w 3 € /j g c 0.0001} _ o / < 0 a 2 - an f 2.00 - 4.00 MM : 0.00001 _ 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND FicurE 24.-Comparison of bedload-transport rate by individual particle-size category, Helley-Smith sampler versus conveyor- belt sampler; 2.00-4.00 mm size class. For particle sizes less than 0.50 mm, the effect of the Helley-Smith sampler catching suspended sediment is apparent. Sampling efficiency as determined for parti- cle sizes less than 0.25 mm should be discounted be- cause of mesh-size limitations of the sample-collection bag. Data show that for sediment in the particle-size class of 0.25 to 0.50 mm, the Helley-Smith sampler is € -> o EXPLANATION Parfact / ec E onl = - 1973 agreement/ Y 1 o N o 1974 i- 2 s S e 1975 yf 1 8 O 1976 & | / -A cc 0.01} o - I a = T o w 8 B Least squares ag - { - 1.0 T T T T o F4 ao EXPLANATION 7 5 = 1973 7 6 0.1}- o 1974 i E 3 e 1975 5 9 o 1976 = "I" Least uy lik oor squares o | © E ‘ I w 3. 8 CO or / $& / & $ - s.o01}f / - - I 5 & » Perfect agreement & 9 ,/ a = / f 0.0001] / 4 2 / a -" a 74 TOTAL BEDLOAD co 0.00001 i i i R 0.00001 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND FigurE 30.-Comparison of total bedload-transport rate; Helley- Smith sampler (dependent variable) versus conveyor-belt sam- pler corrected to condition of stream-wide slot (independent vari- able). 1.0 T +- ~- T 3 / 2 EXPLANATION A w = 1973 z 0.1}- Oo 1974 7 £8 3 < © 1975 > Q O 1976 g wn o 1 L R S E 0.01 wou Least squares < F 5s 7** 3 w 0.001] A h C. 3 / 2 & " Perfect agreement a O / c O / - 3 ye s g 0.0001} / - / a / to / 7 TOTAL BEDLOAD 0.00001 i i i i 0.00001 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY HELLEY-SMITH SAMPLER, IN KILOGRAMS PER METER - SECOND Figure 31.-Comparison of total bedload-transport rate; conveyor-belt sampler corrected to condition of stream-wide slot (dependent variable) versus Helley-Smith sampler (independent variable). It was noted in an earlier section of this report that the mean river stage, or discharge, during the period of operating the Helley-Smith sampler was often, but only slightly, different than the mean discharge during the operation of the conveyor-belt sampler. Regression of the conveyor-belt-determined transport rates against effective discharge (data of table 4) shows that bedload-transport rate is approximately proportional to the square of effective discharge. An appropriate correction factor to transpose the Helley-Smith trans- port data to the same hydraulic base as the conveyor- belt data would be Q /('If § Q's ) - Values of this correction factor along with Helley- Smith-determined transport data, both unaltered and modified by the correction factor, are listed in table 10. Modification (5) involves regressing corrected Helley-Smith data as just described against conveyor- belt data as modified in (3) above. The regression equa- tion determined is HS = 0.78 CB" (5) and indicates a mean sediment-trapping efficiency of 123 percent. The graph of these results is shown in figure 32. Reversal of the dependent and independent vari- ables, as in modifications (2) and (4), shows some inadequacies of a least-squares statistical procedure in the analysis of some data sets. Use of least-squares techniques usually implies that data used as the inde- pendent variable are free of error. This is not the case with the present data, for, indeed, the data collected 2 1.0 -T T T T 5. a EXPLANATION a e m 1973 z 9 f o 1974 3 = § e 1975 w a $ o 1976 Least u a F bor- squares o | T © % s E a § t a me & - 0.001} /* { & & / Pert 9 8 Perfect agreement is " f 0.0001) R 3 7 E # & TOTAL BEDLOAD co 0.00001 i 1 L L 0.00001 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND Figure 32.-Comparison of total bedload-transport rate; Helley- Smith sampler corrected for stage difference versus conveyor- belt sampler corrected for condition of stream-wide slot. 22 with the conveyor belt are subject to some of the same variability as are data collected with the Helley-Smith sampler. When the independent data have an error as- sociated with them, the least-squares technique will invariably yield a relation which has underestimated the exponent or slope value of the relation. There does exist, however, a statistical procedure for adjusting slope value by a correction factor: 1+ _G ~A, 012—072 where a;" is the variance of the error in the indepen- dent variable and a," is the variance of the indepen- dent variable (see Snedecor and Cochran, 1967, p. 165). Variance of the error in the conveyor-belt data can be obtained for those runs in which all gates were opened simultaneously and a minute-by-minute record was kept of the bedload-transport rate being measured. Necessary basic data for this determination can be found for conveyor-belt data collected in 1976 by refer- ence to Leopold and Emmett (1977). Variance of the conveyor-belt data is obtained as part of the least- squares regression of the present analysis. A mean value of the least-squares correction factor, as ob- tained, is 1.069. This correction factor is based on measured total bedload; its uniform applicability to all particle-size classes is not known. Modification (6) applies this correction factor to the regression of modification (5) and results in HS = 0.96 CB", (6) and the mean sediment-trapping efficiency remains at 123 percent. This is graphed in figure 33. It was earlier shown that collection of suspended sed- iment influences trap efficiency of small-size particles for the Helley-Smith sampler. Modification (7) uses the corrected conveyor-belt data of (2) but excludes sedi- ment particles smaller than 0.25 mm. The resulting regression is 0.89 HS = 0.85 C. (7) and indicates a mean sediment-trapping efficiency of 123 percent. The graphic relation is shown in figure 34. Modification (8) is identical to modification (7) except that it excludes all sediment particles smaller than 0.50 mm, the upper limit of particle sizes associated with suspended sediment. The new regression equation is HS = 0:82 CB"", (8) and a resulting mean sediment-trapping efficiency is 109 percent. The relation is graphed in figure 35. 1.0 T T T T P4 EXPLANATION $3 = 1973 | o 1974 © 1975 O 1976 Least squares SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER /, - erfect agreement KILOGRAMS PER METER - SECOND BEDLOAD TRANSPORT RATE BY HELLEY-SMITH SAMPLER, IN 0.0001} P A A A. TOTAL BEDLOAD 0.00001 i 1 * a 0.00001 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND FicurE 33.-Comparison of total bedload-transport rate; Helley- Smith sampler corrected for stage difference versus conveyor- belt sampler corrected for condition of stream-wide slot; relation corrected for variance of the independent variable in the least- squares regression. E 1.0 T T T T g. &: EXPLANATION 3 s gal @ 1973 vl = 2 o 1974 s 8 ® 1975 T & o 1976 is 0.01 - [e- & C € 5 ags - o § t a c & 0.001} f. 7 m I 6 6 /Perfect agreement 5 9 2 s * £ 0.0001|- a o / < 9 a TOTAL BEDLOAD a 0.00001 1 i i 1 0.00001 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND FigurE 34.-Comparison of total bedload-transport rate for sedi- ment particles larger than 0.25 mm; Helley-Smith sampler ver- sus conveyor-belt sampler corrected for condition of stream-wide slot. Modification (9) is similar to modification (8) except that it also includes correction for variance of the inde- pendent variable. The new result is HS: = 1.92 CB:", (9) MODIFICATIONS TO THE BASIC DATA 23 1.0 T T T T / EXPLANATION 0.0001 / 7 TOTAL BEDLOAD 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND 2 s a - = 1973 n 0.1}- = o o 1974 t C 8 @ 1975 51, 9 o 1976 w = i | { m in a 2 Least squares I w -< CO oc “24 a i -' 9.001; 7 c s m T / co CC g 8 Perfect agreement _] < € -- = o < 0 xl a Ld co 0.00001 0.00001 FicurE 35.-Comparison of total bedload-transport rate for sedi- ment particles larger than 0.50 mm; Helley-Smith sampler ver- sus conveyor-belt sampler corrected for condition of stream-wide slot. and the mean trap efficiency remains at 109 percent. Figure 36 illustrates this regression. Modification (10) involves regressing corrected Helley-Smith data against corrected conveyor-belt data (5) but also excludes sediment particles smaller than 0.25 mm. The relation determined is HS = 0.87 CB" (10) and includes a mean sediment-trapping efficiency of 122 percent. The relation is illustrated in figure 37. Modification (11) is similar to (10) except that it ex- cludes sediment particles smaller than 0.50 mm. The regression is HS = 0.84 CB®®"® (11) and yields a mean sediment-trapping efficiency of 109 percent. This relation is illustrated in figure 38. Modification (12) utilizes the regression of (11) and further includes correction for the variance of the inde- pendent variable. This regression yields a comparison of the Helley-Smith sampler to the conveyor-belt sam- pler of HS = 1.05 CBR" (12) and gives a mean sediment-trapping efficiency of the Helley-Smith sampler of 109 percent. The relation is illustrated in figure 39. The 12 modifications to the basic data give some re- gression equations that are quite different from some 2 1.0 T t T T oc g EXPLANATION 5 @ - 1973 9.3 0.1}- o 1974 3 I E & e 1975 3 9 o 1976 I w 81 :. 3 g 0.01 4 f © Least squares C0 or w E m & - o.001}f z { E & / 6 5 & 9 7° Perfect agreement a = / at 0.0001}- // P (4 / rol / = / a / TOTAL BEDLOAD c 0.00001 i i i i 0.00001 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND FigurE 36. --Comparison of total bedload-transport rate for sedi- ment particles larger than 0.50 mm; Helley-Smith sampler ver- sus conveyor-belt sampler corrected for condition of stream-wide slot; relation corrected for variance of the independent variable in the least-squares regression. bo 1.0 ; t p- ; 5 t # EXPLANATION < m 1973 w 0.1} a T o O 1974 Least E 3 @ 1975 squares C o 1976 3 a 0.01} ~ 2 | Fals - *a c $ - o.o01f / - E & /Perfect agreement 2 S w a S / f- 0.0001 a 3 # E /* for TOTAL BEDLOAD co 0.00001 i 1 1 i 0.00001 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND Figure 37.-Comparison of total bedload-transport rate for sedi- ment particles larger than 0.25 mm; Helley-Smith sampler cor- rected for stage difference versus conveyor-belt sampler cor- rected for condition of stream-wide slot. of the others, though reference to figures 28-39 indi- cates that throughout the measured range of transport rates, the variation is not as large as might be imag- ined. The real importance lies in extrapolation of rela- tions some distance from the range of measured data, bo J 1.0 T T T T EXPLANATION = 1973 | SC o 1974 e 1975 0 1976 0.01}- = Least squares 0.001] 74 -_ KILOGRAMS PER METER - SECOND / Perfect agreement 0.0001] / - TOTAL BEDLOAD 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND BEDLOAD TRANSPORT RATE BY HELLEY-SMITH SAMPLER, IN 0.00001 0.00001 FicurE 38.-Comparison of total bedload-transport rate for sedi- ment particles larger than 0.50 mm; Helley-Smith sampler cor- rected for stage difference versus conveyor-belt sampler cor- rected for condition of stream-wide slot. < 1.0 T T T T r a EXPLANATION i- # 1973 ~ O "'-.. o- 1974 § = & < 8 @ 1975 ‘i’ { O 1976 I G m - g.01} &e - t H Least squares I w £ Z a wo H xg BA /. { m 4 § g /Perfect agreement w £ 0.0001 4 a # E 7 a TOTAL BEDLOAD ea _ 0.00001 r i A i 0.00001 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND FicurE 39.-Comparison of total bedload-transport rate for sedi- ment particles greater than 0.50 mm; Helley-Smith sampler cor- rected for stage difference versus conveyor-belt sampler cor- rected for condition of stream-wide slot; relation corrected for variance of the independent variable in the least-squares regres- sion. because the difference in relations (1) to (12) generally increases in significance as the transport rate increases or decreases from the mean transport rate as deter- mined in the study. It becomes a decision as to which SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER modifications to the basic data should be employed, and the reasoning for their use. The writer believes all of the modifications are objec- tive and relevant, but, lest a reader disagrees, this is the reason the basic data are presented first. Though all of the modifications may be relevant, it is apparent in the comparison of equations (5) to (3), (10) to (7), and (11) to (8) that the effect of correcting the Helley-Smith results to accommodate the slight differences in river stage is very modest and may be neglected. The correc- tion for excluding suspended-sediment size particles (<0.50 mm) has no bearing on a comparison of the two methods of sampling on a particle-size class basis, when particle sizes are greater than 0.50 mm. Thus these two corrections need not be further considered. The correction for error in the independent variable affects slope value of the relation only; it does not in- fluence the value of mean sediment-trapping efficiency. It is not known with certainty if the slope correction factor can be applied equally to all particle-size classes; it thus becomes subjective in its application. This di- lemma may be solved by not using the correction fac- tor, utilizing mean sediment-trapping efficiency as de- termined, and recognizing that the rate of change in trap efficiency (exponent value of the regression equa- tion) should be increased by an amount less than 10 percent of its value. This leaves only the correction applied to some of the conveyor-belt data (table 10) to normalize these data to the condition of a continuous stream-wide open slot. Utilizing only this modification to the data, table 13 lists a summary of the statistical data generated when regressing the Helley-Smith sampler data against the conveyor-belt sampler data on a particle-size class basis. Graphings of data and least-squares relations are shown in figures 40-48. A summary of the pertinent statistics from table 13 are listed below. Particle-size Mean ratio in transport rate Rate of change in ratio class Helley-Smith:conveyor-belt of transport (mm) (¥/X, in percent) (B) $06- 0.12 143.03 1.030 P= 428 ln al 246.00 .868 B0Gr 8 BO oul. olsen on's 174.34 914 50-100; 114.74 1.016 100-200) :. 103.88 .923 200- 4.00 100.47 848 400- 8.00 . ...... .C 109.06 T75 8$.00-16.00 -__.:..._.l_l.c.l.l 109.04 788 16.00-82:00 72.38 501 Information on particle-size classes 0.06 to 0.12 mm and 0.12 to 0.25 mm is discarded. These are particle sizes nominally in suspension; further, the mesh collec- tion bag for the Helley-Smith sampler cannot accom- modate these sizes of sediment. MODIFICATIONS TO THE BASIC DATA 25 i 1.0 T T T T ce / 5 EXPLANATION B 2 m 1973 Fie un 0.1}- D. 1974 / 7 © 9 / a g © 1975 / a 9 o 1976 / § 1 # g oc 0.01} A 7 LJ E / T w 4" Perfect agreement - ® to fer / E a "4 e.. // 4 C & Least 0 5 squares 5 0 > w < "4 (< 0.0001] [ a @ < ye a 2 0.06 - 0.12 mm co 0.00001 i 1 i i 0.00001 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND FIGURE 40.-Comparison of bedload-transport rate by individual particle-size category, Helley-Smith sampler versus conveyor- belt sampler corrected for condition of stream-wide slot; 0.06- 0.12 mm size class. / 0.0001] / - / 2 1.0 -T T t a 5 y". g EXPLANATION / T6 . "; m 1973 { Ce o 1974 / 5 o e 1975 §." o 1976 /' S &© CL 4 ir /Perfect agreement , = Least squares C oc w £ S g - s.001]f - - 2. 5 EXPLANATION 3 / n 0.1} = - 1973 h o E 3 o 1874 squares _ OZ" 5 0 @ 1975 w E | sah O 1976 a/ G 2'F 34 = 5 s & C oc & 8 m & - o.001}f { m 1.0 ; y j f & / a / C / G EXPLANATION of. n 0.1} | z 2 = 1973 - & = 9 o 1974 3 0 1 8 e 1975 3 2 El o.01}- 0 1976 C 4 5 s , & c a #E oe 3 & - ooo1} - Least squares?" _, | m / & / € FA A EXPLANATION ip" ig. S ® 1973 y". 7 ~ 2 = 8 o 1974 e e 1975 & | o 1976 8 - oc 0.01} - u LL = & . E C0 oc oR Least squares & $ sof 7 E & o 9 3 3 /(3‘ a 2 ~ a g $0001 //Perfect agreement e < / 9 / -) a f\ 2.00 - 4.00 MM ca 0.00001 i l ; i 0.00001 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND FIGURE 45.-Comparison of bedload-transport rate by individual particle-size category, Helley-Smith sampler versus conveyor- belt sampler corrected for condition of stream-wide slot; 2.00- 4.00 mm size class. near ideal for conditions involving no suspended- sediment transport. For particle-size classes between 0.50 mm and about 16 mm, there is good agreement between the transport rate measured with the Helley-Smith sampler and that measured with the conveyor-belt sampler. Exponent SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER > 1.0 T T T T *> r4 5 / s EXPLANATION Perfect "4 3 3 oil m 1973 agreement// E z 8 o 1974 T = 9 e 1975 a Q / J o O 1976 / a g o.01}- A a & o F T § rak - t or Least squares u W p & & & - s.o01f = pale 3 4 w 3 = 4 - 0.0001} hud _ o < F / 2 8.00 - 16.0 MM co 0.00001 R 1 i R 0.00001 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND FiGUrE 46. --Comparison of bedload-transport rate by individual particle-size category, Helley-Smith sampler versus conveyor- belt sampler corrected for condition of stream-wide slot; 4.00- 8.00 mm size class. 2 1.0 T T T T #* € EXPLANATION F6 < o # 1973 $6 o s & E 2 o 1974 / ma / s 9 ® 1975 Z o P?” o 1976 o~ uy - cc 0.01 = ~ C0 oc uo E Least s 3&2 o.0o1} _ Squares # - I & ce CC 2 8 # & 9 @* Perfect agreement 2 = < ~ / & 0.0001 |- o* % 7 o / 4 / a / fer 7 4.00 - 8.00 MM co 0.00001 i i i i 0.00001 0.0001 0.001 0.01 0.1 1.0 BEDLOAD TRANSPORT RATE BY CONVEYOR BELT, IN KILOGRAMS PER METER - SECOND FicuUurE 47.-Comparison of bedload-transport rate by individual particle-size category, Helley-Smith sampler versus conveyor- belt sampler corrected for condition of stream-wide slot; 8.00- 16.0 mm size class. values of the regression equations indicate that trap efficiency of the Helley-Smith sampler decreases somewhat as transport rate increases, but correction for the error in the independent variable makes the actual decrease less than the apparent decrease in trap efficiency. Considering all factors involved in the ex- REFERENCES CITED 27 3 1.0 T T T T e / & / E EXPLANATION // a 0.1} 6 1973 Perfect / [ z 9 o 1974 agreement / = 2 @ 1975 // 3 & o 1976 # BZ 0.01 ea & C 7 § - & C0 oc p E & & soo - - '...... 6-02-78 ____________ 18.8 17.8 1.11 1.09 .0235 74 6-03-73 17.5 16.6 1.06 1.07 ©0240; s. 11.8 11.3 .81 .96 ©1837: 500.3 6-07-78 ____________ 16.7 15.9 1.03 1.06 ©0260. -_. Ll.. 6-08-78 ____________ 20.3 19.2 1.17 1.19 0218 .98 b-25b-174) 5.44 5.34 A8 16 0056 54 5-26-74 ____________ 10.3 9.92 T4 92 0822 .09 D-27-14 22.9 21.5 1.27 1.15 1758 1.03 5-28-74 ____________ 32.0 29.8 1.60 1.28 2255 1.40 5-20-74 ...._.s._uuk 45.0 41.5 2.01 1.41 29012 1.52 8-80-14 34.6 32.2 1.68 1.31 .0786 1.51 b-81-14 24.4 22.9 1.33 1.18 0647 1.40 6-01-74 ____________ 25.9 24.3 1.38 1.20 0206 94 6-02-74 27.2 25.5 1.43 1.22 0130 .99 6-03-74 31.9 29.7 1.59 1.27 0172 .88 6-04-74 .. .2: 20000 29.9 27.9 1.52 1.25 .0285 .92 6-05-74 ____________ 28.3 26.5 1.47 1.23 0305 .81 8-27 15: L. i a pre 2.44 2.44 .28 .61 0021 *>. } (S22. 5-80-15 2.04 2.04 .24 .87 0016. _- 6-02-75. .c 5.98 5.82 51 178 0484 T4 6-03-75 9.52 9.13 70 .90 ©7091 .c. sculls 6-04-75 _ 10.5 10.0 74 .92 .0812 1.16 6-05-75 11.2 10.7 18 94 0972 1.26 6-06-75 ____________ 21.3 20.0 1.21 1.13 8114 1.36 6-07- 7B... c. L.- uu. 26.6 24.8 1.40 1.21 2069 1.28 6-08-75 ... 27.5 25.6 1.44 1.22 1783 1.41 6-09-75 ....:._._.__.l... 26.2 24.3 1.38 1.20 .0833 1.35 15.3 14.4 .96 1.03 0348 .~ ° 6-11-75 10.6 10.1 15 .92 0110 :: - ...=. 613-15 16.7 15.8 1.02 1.06 0277 .50 6-14-75 ..:. .L 27.6 25.7 1.44 1.22 0926 1.27 615-75 :g. 31.4 29.0 1.57 127 1190 1.05 6-10-75 ...! ......._L 32.8 30.3 1.62 1.236 1190 1.19 6-17-15 23.8 22.2 1.30 1.17 0796 1.36 6-18-75 .Q 13.5 12.8 .88 .99 0106... -. .._... 6-19-75 _L 10.5 10.1 15 92 0097 13 6-21-75 7.48 7.23 .59 .84 0032 70 6-22-75 - 7.25 7.01 .58 .83 0047 64 6-28-75 8.55 8.24 .65 .87 0062 T7 6-24-75 11.3 10.8 18 94 0194 .98 6-25-75 .L. 23.2 21.7 1.28 1.16 .0838 1.10 6-20-75 13.8 13.1 .90 1.00 0396 .99 7-01-18 :s a cul gis 24.8 283.1 1.34 1.18 2159 1.63 T=08-175 n..... 23.0 21.5 1.27 1.16 0317 .91 b-18-76 ____________ 10.4 9.87 178 .87 .0838 .98 5-19-76) .... .....lc.4 15.7 14.8 1.01 1.00 1359 1.04 5-20-76 s sul olde. 20.3 18.9 1.19 1.09 1163 .96 5-20-76 ____________ 21.0 19.6 1.22 1.10 1295 1.04 ye2l-176 :..:¢.c2.... 24.0 22.4 1.33 1.15 1769 1.52 p92-176 .i.._..l.l.0.l 18.6 17.5 1.13 1.06 0754 1.56 5-26-76 .. 10.3 9.77 T7 .87 0130 71 TABLES 4-13 TABLE 4.-Summary data of river hydraulics and bedload transport, conveyor-belt - sampler-Continued River discharge Unit Sen HALL LLL LINE enc Mean Mean bedload- Bedload Date! Total," Effective," depth," velocity," transport size," Q ' D v rate," (m'/s) (m/s) (m) (m/s) Jo (mm) (kg/m-s) b-27-76 ____________ 15.2 14.3 ;:09 .99 .0232 .59 5-27-76 __________-- 14.5 13.7 .96 .97 0301 .61 5-27-76 13.8 13.0 .93 .96 .0233 T7 5-28-76 ___________- 20.1 18.8 1.18 1.08 0437 .95 5-28-76 ____________ 21.2 19.8 1.23 1.10 0454 1.11 b-20-76 ____________ 22.0 20.5 1.25 1.12 0712 1.30 5-20-76 ____________ 22.4 20.9 1.27 1.12 0618 1.67 b-80-76 _._..________ 22.4 20.9 1.27 1.12 0774 1.29 5-31-76 ____________ 17.7 16.6 1.09 1.04 0621 1.09 5-31-76 ____________ 16.8 15.8 1.06 1.02 0405 .98 6-01-76 ____________ 15.2 14.3 99 .99 .0361 .81 6-01-76 ____________ 14.7 13.9 .97 .98 0325 .80 6-02-76 ____________ 19.1 17.9 1.15 1.07 0576 94 6-02-76 ____________ 19.0 17.8 1.14 1.06 .0463 1.04 6-08-76 ____________ 23.2 21.6 1.30 1.14 .0834 1.18 6-04-76 ____________ 23.4 21.8 1.30 1.14 0871 1.40 6-05-76 ____________ 23.0 21.4 1.29 1.13 .0918 1.76 6-05-76 ____________ 24.0 22.4 1.33 1.15 0784 1.51 6-06-76 ____________ 24.2 22.6 1.33 1.16 .0908 1.30 6-07-76 ____________ 26.5 24.6 1.41 1.19 .0869 1.35 6-08-76 22.7 21.1 1.28 1.13 0570 1.24 6-09-76 ____________ 20.1 18.8 1.18 1.08 0513 1.03 6-09-76 ____________ 20.3 18.9 1.19 1.09 0346 1.08 6-10-76 ____________ 19.5 18.2 1.16 1.07 0290 1.06 6-11-76 ____________ 14.6 13.8 :97 .98 0253 .84 6-11-76 ____________ 15.4 14.5 1.00 .99 .0289 1.05 6-11-76 ____________ 16.6 15.7 1.05 1.02 0280 1.02 16.1 15.2 1.03 1.01 0629 1.07 6-11-76 ____________ 15.3 14.4 :99 .99 0236 19 6-12-76 ____________ 13.9 13.1 .93 .96 .0181 .81 6-12-76 L___________ 13.2 12.5 .90 94 0169 T7 6-12-76 ____________ 11.8 11.2 .84 .91 .0162 .81 6-12-76 ____________ 11.0 10.5 .81 .89 0145 .82 6-12-76 ____________ 10.1 9.64 16 .86 0106 .82 6-12-76 ____________ 8.89 8.50 70 .83 .0084 T7 6-13-76 ____________ 6.80 6.55 .59 16 0028 A9 6-14-76 ____________ 5.13 4.97 .50 .69 0022 Al 6-14-76 ____________ 4.19 4.65 AT .67 0020 .58 6-15-76 ___ 3.96 3.87 A2 64 00083 .66 6-15-76 ____________ 3.51 3.44 39 61 0004 .88 6-16-76 ____________ 5.13 4.97 50 .69 0009 .50 6-18-76 ____________ 3.99 3.90 A2 .63 0006 A2 6-19-76 ____________ 4.30 4.20 A4 .65 0009 A4 6-20-76 ___________. 4.70 4.57 AT .67 .0023 A83 6-21-76 ____________ 10.0 9.53 16 .86 .0181 .68 'Dates correspond to dates listed in table 5. 'Complete river discharge including overbank flow. "Discharge over 14.6-m width of bedload trag includes all flow over the active width of the streambed. 'Mean depth over effective width W; D = _> __. vw *Mean velocity of effective discharge; V = @ = ,__Q_ WD 14.6D "Unit transport rate of solids in dry weight per second, over 14.6-m width of bedload trap. 'd;, is median diameter of grains; complete grain-size data are given in table 5. 31 32 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER TaBus 5.-Particle-size distribution of bedload sediment, conveyor-belt sampler Date! 5-26-73 5-28-73 5-29-73 6-01-73 6-02-73 6-03-73 6-06-73 - 6-07-73 6-08-73 5-25-74 5-26-74 5-27-74 5-28-74 5-29-74 @" 16.1 6.50 7.30 16.1 17.8 16.6 11.3 15.9 19.2 5.34 9.92 21.5 29.8 41.5 A" - 0.0122 0.0009 0.0011 0.0190 0.0235 0.0240 0.0133 0.0260 0.0218 0.0056 0.0822 0.1758 0.2255 0.2912 Percent by weight finer than sieve size (mm) indicated 0.06 _. 0.1 (4) (4) (4) 0.3 (") (") (4) 0.1 0.3 neve a n Pean. f .3 (* () () A (* (* (* .3 .5 .5 §1~-t ...ll. 0.1 ; 4A (") () () 4 (") (6) (4) A 1.1 1.0 .3 0.1 4 : R (") (W (") 1.3 (W () W 1.0 2.8 2.1 .6 I .3 f 1.3 (") (4) (4) 2.6 () () (") 1.9 8.9 4.3 1.4 .5 .5 > 3.7 (") (") (") 74 ()) (") (") 4.8 27.0 13.0 4.6 1.9 1.4 f 9.9 (5) (") (") 22.1 (") (") W 13.8 46.2 38.3 15.6 6.9 5.6 A 22.1 (5) (*) (") 48.2 W (") () 34.0 63.8 62.6 32.4 18.8 17.3 1. 36.8 (5) (") (") 62.6 (") (") (") 51.2 75.9 75.3 48.8 34.1 33.5 1.4 52.0 W (") (") 73.0 (5) (") W 65.6 85.1 83.5 62.3 50.4 47.4 2.00 . - 67.6 W (") (") 83.3 (") (*) W 71.9 92.3 90.8 75.7 66.8 59.4 2.83 _ - 80.0 (") (") (") 90.9 (") (") (") 87.9 96.1 95.8 85.5 78.8 69.0 4.00 . - 88.8 (") (") (") 96.2 (") (") (4) 94.9 98.2 98.7 $22 88.2 76.5 5.00... :... 95.7 (") (") (") 98.4 (0) () (5) 98.1 99.2 99.5 95.1 93.2 81.0 8.00 _____. 98.2 (") (") (") 99.7 (") () ( 99.5 99.7 99.8 96.5 96.1 83.6 113 . _ 99.4 (4) (") W 100.0 (") (") (") 100.0 100.0 100.0 97.2 91.1 85.7 16.0 _ 100.0 (") (") (My t gece. . (") (4) (M) ce io ian cee. ao 97.7 98.5 86.5 22.6 (*) (") (") (") (") (") 98.2 99.4 87.5 32.0 (") (") (") (") (") (") 100.0 100.0 87.9 45.0 (*) (4) W (") ( (M) neve t ci Ol 89.2 64.0 () (W) (* (") W (Bek soos ge ede oan . aeons ns bae oe doe NME] eaten iol Phe 100.0 d; .. 0.39 (") (5) (") 0.31 (") 0.36 0.21 0.26 0.36 0.45 0.48 du =- .61 (") (") an A4 (") 52 .29 37 .50 .66 .69 eeser .96 (") (*) (") .60 (") 72 Al 48 15 1.02 1.04 1.35 (®) (") (") 74 () 98 .54 59 1.03 1.40 1.52 ex.. 1.88 () (©) (6) 1.08 () 1.39 13 75 1.51 1.92 2.44 w- 3.27 (") (") (") 2.06 () 2.44 1.36 1.45 2.67 3.39 8.71 thy E12 oue 5.38 (") (") () 3.61 (4 4.03 2.51 2.63 5.57 6.89 48.9 Date! ___. 5-30-74 5-31-74 6-01-74 6-02-74 6-03-74 6-04-74 6-05-74 - 5-27-75 5-30-75 6-02-75 6-03-75 6-04-75 6-05-75 6-06-75 @* A 32.2 22.9 24.3 25.5 29.7 27.9 26.5 2.44 2.04 5.82 9.13 10.0 10.7 20.0 g ewe 0.0786 0.0647 0.0206 0.0130 0.0172 0.0285 0.0305 0.0021 0.0016 0.0484 0.0791 0.0812 0.0972 0.3114 Percent by weight finer than sieve size (mm) indicated 0.1 0.1 0.2 0.1 0.1 0.2 (") (") 0.5 W 0.2 0.2 0.2 4 2 A 3 .2 A (5) (") .I W .3 .3 :2 .3 0.1 A .9 .6 A .6 W (*) 1.1 (") :B A .3 .6 .3 .8 1.5 1.2 .8 1.0 W (") 1.9 (") .8 .6 4 1.2 1.1 2.5 3.5 3.2 1.9 2.2 (") (") 4.3 (*) 1.6 1.0 .8 2.6 4.0 9.8 11.5 14.0 8.5 10.0 () (") 14.2 (") 4.8 3.2 2.3 7.6 10.0 23.3 26.1 31.0 23.3 30.1 (4 (") 30.8 (") 14.4 10.4 8.2 18.2 22.4 37.2 38.5 43.3 38.8 45.0 (5 (W 48.1 W 29.1 24.2 21.9 32.3 36.4 52.6 50.4 54.2 53.4 58.3 (5 (?) 60.8 (") 43.1 38.6 36.1 47.1 49.6 65.3 60.7 63.7 65.8 69.3 W (") 73.7 ( 59.0 55.5 51.7 63.0 64.2 77.3 71.1 73.3 77.4 80.0 (") (W 85.7 () 76.7 73.5 69.3 76.3 77.3 86.1 80.4 81.9 85.7 87.9 (") (") 92.2 () 89.4 86.9 82.6 86.4 ? 92.5 88.0 90.1 91.9 93.7 () (W 96.5 (0 96.5 95.4 91.6 92.6 96.5 .8 95.0 95.7 96.9 W ( 99.0 (") 98.6 98.6 95.8 96.5 98.3 . 98.5 97.9 99.0 (") (") 99.8 (5 99.7 99.5 97.9 98.3 99.4 100.0 99.6 100.0 W (") 100.0 (") 99.8 99.8 98.8 99.7 100.0 ays 1000 :~ () PRE L (") 100.0 100.0 99.6 100.0 See ; Poke (0 () (Ae meats. 100.0 mk ri is ge o onn ll () (0 (yt pe Pep Pett oon : (0 (* (y 0st oot aie Ltt ta it to o - veer sex W Pso (420 009 Nol ne den p po Pope e Particle size (mm) at given percent finer dy 0.43 0.38 0.29 0.27 0.27 0.31 0.30 (? () 10.26 (? 0.36 0.40 0.43 dis rind? 60 42 .40 37 43 40 () (h AT (* .82 .59 .62 i - as R07 97 67 .64 56 65 56 (? () .55 (? .82 .92 .98 Hamp Bd 1.40 .94 .99 .88 .92 .81 () (0 .74 (0 1.16 1.26 1.36 this .- > 240 2.04 1.40 1.62 1.48 1.38 1.23 (5) (? 1.11 (5) 1.58 1.68 1.83 Cs --- 3.66 3.47 2.59 3.31 3.07 2.63 2.36 (? () 1.90 (5) 2.40 2.60 2.97 is - 6.83 6.40 8.84 7.52 5.66 5.23 4.50 (0 (0 3.45 (6 3.61 3.90 5.21 TABLES 4-13 TaBig 5.-Particle-size distribution of bedload sediment, conveyor-belt sampler-Continued 33 bos s | Gog Ts | 61075 - 611-75 - 618-75 6-14.75 615075 \ 6-17-75 6-18-75 6-19-75 6-21-75 6-22-75 25.6 24.3 14.4 10.1 15.8 25.7 29.0 30.3 22.2 12.8 10.1 7.23 7.01 0.1733 0.0833 0.0348 0.0110 0.0277 0.0926 _ 0.1190 0.1190 0.0796 0.0106 0.0097 0.0032 0.0047 Percent by weight finer than sieve size (mm) indicated 0.06 __ 0.2 0.2 0.2 (6 () 0.4 0.3 0.2 0.2 0.2 () 0.2 0.6 0.5 08 ::|. (A .8 .3 (") (") .5 A 2 .3 .2 (*) A $ .6 A4 °. .5 A () (0 .8 .6 A 4 .2 (*) .6 1.6 13 A8 2 (ub .9 .6 () () 1.7 .9 .6 .T .3 () 1.2 2.9 24 25 ..) :9 1.8 1.5 ( ( 5.9 2.0 1.3 1.4 .9 (5 3.5 7.0 7.0 A5 5s 28 4.0 4.7 () () 28.7 5.7 5.5 5.1 4.5 () 16.6 20.6 22.9 .50 __ 8.1 8.4 9.8 () () 50.2 14.4 13.8 14.0 11.8 (") 30.0 35.0 39.0 41" .-228 19.5 21.5 () (0 68.2 27.8 30.4 28.2 24.0 () 48.9 50.2 54.9 1.00 -- 38.8 32.9 36.0 (* (6) 77.7 40.8 47.8 43.9 36.9 (©) 63.1 63.8 68.6 1.41 __ 54.8 50.0 52.1 (6) (6) 84.6 54.3 62.0 56.2 51.8 (6) 76.4 74.1 78.7 2.00 __ 70.8 68.0 68.2 (") () 90.8 68.0 74.7 69.3 68.4 (6) 87.4 83.7 86.7 2.83 __ 82.3 81.6 80.8 () (") 95.5 79.6 84.4 80.5 81.2 (* 93.8 90.5 92.2 4.00 __ 90.4 91.0 89.6 (0 () 98.6 88.3 91.8 89.7 90.7 () 97.3 95.4 96.2 5.66 __ 94.5 95.4 94.2 (* (0 99.6 93.6 94.9 94.2 95.8 () 98.9 97.3 98.1 8.00 -- 97.1 98.0 97.3 (6) () 100.0 96.8 97.3 97.1 98.2 () 99.6 99.0 99.0 11.3 -__- 98.6 99.2 98.9 () () 99.1 98.6 98.5 99.4 (6) 100.0 100.0 99.5 16.0 ___ 99.6 99.7 99.8 () () 99.7 99.5 99.3 100.0 () v (BASC 100.0 22.6 ____100.0 100.0 100.0 () (0 f 100.0 100.0 (4 nly us § AE . ® &. (6) (0 (") east men od y - (*) I (M) isa tite. ,,,,,, (0 () (* Fakes 0.39 0.36 (6) 0 0.24 0.34 _ 0.34 0.35 0.37 0 0.27 0.22 0.22 .64 .61 () () 30 52 _ .53 53 57 (") .35 .32 .31 1.05 .18 () (0 .39 .86 0.78 .83 .95 (4 .55 .50 46 1.41 1.35 (0 (* .50 127 - 1.05 1.19 1.36 (6) 73 170 .64 1.88 1.86 (* (©) 66 1.85 _ 1.53 1.78 1.86 (6 1.05 1.04 .91 3.06 3.17 (* () 1.38 3.29 - 2.80 3.19 3.10 () 1.78 2.04 1.77 5.43 6.07 (* (0 2.70 6.44 _ 5.72 6.12 5.27 (* 3.11 3.85 3.53 Date! ___ 6-23-75 0 6-24-75 0 6-25-75 6-26-75 7-01-75 7-08-75 5-18-76 5-19-76 5-20-76 5-20-76 5-21-76 5-22-76 5-26-76 5-27-76 @t... 8.24 10.8 21.7 13.1 23.1 21.5 9.87 14.8 18.9 19.6 22.4 17.5 9.77 14.3 Mage.. 0.0062 0.0194 0.0838 0.0396 0.2159 0.0317 0.0838 0.1359 0.1163 0.1295 0.1769 0.0754 0.0130 0.0232 Percent by weight finer than sieve size (mm) indicated 0.06 ___ 0.4 0.6 0.4 0.4 0.2 0.2 0.5 0.3 0.3 0.3 0.3 1.4 0.3 wo... >7 T A .5 .3 .3 T .5 4 4 .3 1.5 A Ag: } 10 1.0 4 A .5 .5 .9 4 .5 .6 4 1.6 .6 385. ; 14 1.5 T 1.1 .8 8 1.3 1.0 .9 .9 .T 1.9 .9 Ab-. 41 3.2 1.5 1.8 1.7 2.2 2.2 1.8 1.7 1.6 1.1 2.6 2.6 85 ..- 155 12.7 5.9 5.5 6.2 10.3 7.8 5.7 5.8 4.9 3.8 5.8 12.0 .50 ___ 29.8 24.2 17.8 15.6 11.7 24.1 22.0 18.3 18.5 15.3 11.1 12.5 31.6 :A .s 45.7 36.4 33.5 32.1 20.3 40.2 38.0 34.9 37.1 33.4 22.9 22.9 50.1 1.00 ___ 61.9 50.7 46.5 49.4 30.7 53.5 50.7 48.3 51.9 48.3 33.9 33.2 61.0 1.41 .__ 75.0 64.1 58.7 66.9 43.6 66.6 64.4 62.3 66.9 63.0 46.8 45.8 70.9 2.00 ___ 85.1 77.6 70.6 82.5 59.1 76.1 78.3 75.9 80.2 76.6 61.5 60.4 80.9 281... 92.1 88.1 80.7 92.4 73.2 84.6 88.3 86.1 88.8 86.3 73.8 73.9 88.7 4.00 ___ 96.6 94.9 89.0 97.6 85.4 91.0 94.2 92.4 93.7 91.8 82.0 84.6 94.1 5.66 ___ 98.5 98.2 93.5 99.1 92.3 94.2 97.3 96.4 96.5 95.2 87.6 92.6 97.3 99.8 96.9 99.7 96.5 96.5 98.7 98.5 98.0 97.1 90.3 97.0 97.3 100.0 98.7 100.0 98.3 97.9 98.7 99.0 98.6 97.9 94.3 98.8 98.6 @@ on Lisicl 99.0 98.6 99.5 99.0 99.2 98.1 96.8 99.4 100.0 ,,,,,, 99.2 99.2 100.0 99.4 99.9 98.4 97.5 100.0 ______ 100.0 100.0 100.0 100.0 99.9 100.0 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, 100.0 Particle size (mm) at given percent finer 0.28 0.34 0.34 0.33 0.30 0.31 0.34 0.34 0.36 0.38 0.33 0.29 0.25 .39 A8 .50 .60 42 44 48 A7 .51 .59 57 .39 .33 .68 .74 15 1.13 .64 .67 T1 .68 73 1.03 1.05 .53 45 .98 1.10 .99 1.63 .91 .98 1.04 .96 1.04 1.52 1.56 41 .59 1.44 1.69 1.36 2.30 1.35 1.43 1.51 1.35 1.48 2.20 2.24 1.14 .91 2.44 3.22 2.09 3.83 2.16 2.40 2.61 2.31 2.59 4.51 3.93 2.28 212 4.03 6.43 3.24 6.89 6.28 4.29 4.89 4.60 5.53 12.23 6.63 4.31 4.27 34 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER Tame 5.-Particle-size distribution of bedload sediment, conveyor-belt sampler-Continued Date! __ 5-27-76 5-27-76 5-28-76 5-28-76 5-29-76 5-29-76 5-30-76 _ 5-31-76 5-31-76 6-01-76 6-01-76 6-02-76 6-02-76 6-03-76 @" sll 13.7 13.0 18.8 19.8 20.5 20.9 20.9 16.6 15.8 14.3 13.9 1789 17.8 21.6 A* 0.0301 0.0233 0.0437 0.0454 0.0712 0.0618 0.0774 0.0621 0.0405 0.0361 0.0325 0.0576 0.0463 0.0834 Percent by weight finer than sieve size (mm) indicated I » std 4 0.3 0.3 0.3 0.6 0.3 0.2 0.2 0.2 0.2 0.2 0.2 1.0 a .5 .5 4 A v4 A .3 .3 .3 .3 .3 .B 11 - 4 .5 .6 .6 .9 .5 A A A .6 .5 4 12 h 1.4 1.0 1.2 1.0 2.2 .9 .6 9 .6 14 .8 .6 1.5 > 4.4 3.3 3.5 2.3 2.8 1.8 1.3 7.6 6.0 11.7 10.0 6.4 8.2 A 19.5 13.2 13.6 8.5 8.1 6.1 5.6 21.1 19.0 27.3 25.5 18.2 19.4 f 40.8 30.7 29.0 20.3 19.1 14.6 16.5 36.7 36.5 45.3 45.1 36.1 33.6 % 56.3 47.5 42.5 34.7 32.3 24.8 30.1 47.5 49.3 57.0 58.2 48.8 44.6 ; 65.8 58.4 51.4 46.3 42.2 38.7 41.3 58.0 61.6 67.5 69.3 59.8 55.9 1. - 74.5 68.7 60.2 58.5 52.6 43.9 53.1 68.8 73.5 77.5 79.2 69.6 67.5 2.00 . 83.0 78.8 69.8 71.2 64.3 56.6 65.9 78.2 83.1 85.6 87.3 78.5 76.9 2.8 89.6 87.2 78.2 81.6 15.5 70.1 77.5 86.1 90.4 91.8 93.0 86.3 83.9 4.0 94.3 92.8 85.8 89.6 85.1 81.7 86.8 92.4 95.6 95.5 v 91.5 89.1 5.66 . 97.2 96.8 91.6 94.8 92.2 90.3 93.5 96.4 98.4 97.9 95.2 92.9 8.00 . : 98.6 95.8 97.9 96.2 94.3 97.5 98.2 f 98.9 96.5 95.1 11.9. : 100.0 97.3 98.8 98.0 95.8 98.8 99.1 99.6 98.3 95.9 160 00.0 «- 97.8 100.0 99.1 96.9 99.7 99.5 100.0 99.8 95.9 22.6 _ 100.0 Tame 99.7 98.2 100.0 99.8 100.0 97.2 32.0 . £ ed + 3 100.0 100.0 /= @...... 100.0 AL 100.0 45.0 . 32. Phase tims Lu IDL gl Lol idl Ae CL URC a u Lorne tee as nee rep T toga ce ut aie, ien aawell cul. | on h n ee dene 6E0 2002s ce Skew faces. toc. se: iz re be min AIL | din NC "ol aes 3 Particle size (mm) at given percent finer 0.27 0.27 0.30 0.30 0.33 0.34 0.32 0.34 0.28 0.30 0.30 0.33 0.30 .39 .38 45 46 .53 49 A5 A7 40 Al 42 48 46 .55 .59 11 178 1.05 .83 .68 .69 .58 .60 .64 .69 14 il .95 111 1.30 1.67 1.29 1.09 .98 .81 .80 .94 1.04 1.18 1.24 1.67 1.68 2.04 2.47 1.95 1.76 1.55 1.30 1.23 1.50 1.69 1.85 2.46 3.68 3.11 3.84 4.35 3.58 3.64 2.95 2.63 2.44 2.90 3.59 4.06 4.71 7.35 5.74 7.07 9.38 6.24 6.91 5.37 5.37 4.173 5.11 7.81 11.03 Date! __ 6-04-76 6-05-76 6-05-76 6-06-76 6-07-76 6-08-76 6-09-76 _ 6-09-76 6-10-76 6-11-76 6-11-76 6-11-76 6-11-76 6-11-76 Q* 21.8 21.4 22.4 22.6 24.6 21.1 18.8 18.9 18.2 13.8 14.5 15.7 15.2 14.4 A* 0.0871 0.0918 0.0784 0.0908 0.0869 0.0570 0.0513 0.0346 0.0290 0.0253 0.0289 0.0280 0.0629 0.0236 Percent by weight finer than sieve size (mm) indicated | . 0.2 0.2 0.2 0.2 0.2 0.2 0.3 0.2 0.2 0.3 0.3 0.3 0.2 0.3 .09 . .3 2 .2 .2 .3 .3 A .3 .8 A A A .3 .5 ; .6 A .3 .3 A A .5 .5 A .6 .5 .5 A 4 18 . .8 .6 .5 .6 T .6 .8 T 4 .8 ye .8 4 11 (25 ...: AA 1.2 1.1 1.2 1.3 .2 1.8 1.6 1.6 11 1.7 1.7 2.0 2.5 .35 _. 5.4 4.6 4.4 5.3 5.9 9 7.5 6.7 7.5 7.1 6.8 6.7 9.3 10.6 50 .- 14.1 12.3 13.7 16.9 18.6 .$ 19.9 17.5 19.0 21.3 17.6 17.6 22.1 26.5 71+ 27.5 23.3 25.8 30.1 33.3 I 36.3 33.5 35.3 42.5 34.2 34.9 37.0 45.5 100..... 889 33.1 36.4 40.9 42.6 pl 48.8 47.0 47.8 57.7 48.1 49.1 47.8 59.2 50.3 43.2 47.8 52.8 51.2 .0 61.0 60.3 59.9 70.5 61.6 62.3 58.5 71.0 2.00 . 61.3 53.9 59.5 65.1 60.3 .0 73.2 72.5 72.0 81.2 75.0 75.0 TLT 81.5 2.83 .____ 70.3 64.2 69.4 75.3 69.3 , 83.6 82.2 82.3 88.8 86.6 85.4 81.6 89.3 4:00..:-. 77.6 73.8 110 83.0 77.8 90.8 89.2 90.0 93.8 84.4 92.1 89.8 94.6 5.00%... 87.1 83.2 86.0 89.6 A 96.1 94.2 95.0 97.3 98.5 96.8 95.4 98.2 8:00 ..:. 93.4 92.0 92.4 94.6 98.8 97.1 98.1 98.9 99.8 99.3 98.2 99.4 $1.05... 96.5 : 96.1 97.5 99.7 98.2 99.0 100.0 100.0 100.0 99.3 100.0 16.0 ...:. 08.2 99.0 98.9 100.0 100.0 99.6 cel encaps nees 1000 .< sate t- 220: :.. 99.0 100.0 99.2 Fuses aunt eel 100.0 all s) t ee nn, Stl ade 30. BOB . oi l | ( O0. Bbq inte eas . t weirs. o Pi Spates o weh s engine ap de non andy oul Lo ui RER pL arc e (ote Pad 45.0 _____100.0 ze. 1000 ain Age dene . (inin &T O PWL 10. Doub do ihan nuts area Mie te $410 ne Ail . o CD C eke re en .o n t C eee ce iy A asec. 3 im u Poema (oN Particle size (mm) at given percent finer 0.36 0.37 0.35 0.34 0.36 0.32 0.33 0.32 0.32 0.33 0.33 0.30 0.29 .57 54 49 A7 .51 A6 A8 A6 45 A8 A8 43 Al 1.07 .96 .83 15 .82 .69 74 70 .63 12 T1 .68 .59 1.76 1.51 1.30 1.35 1.24 1.03 1.08 1.06 .84 1.05 1.02 1.07 19 2.91 241 1.99 2.39 1.84 1.58 1.61 1.63 1.21 1.54 1.51 1.67 1.18 5.83 5.18 4.22 5.33 3.50 2.89 3.08 3.03 2.25 2.59 2.69 3.10 2.22 9.35 9.97 8.31 9.83 7.42 5.16 6.08 5.65 4.40 4.14 4.81 5.49 4.10 TABLES 4-13 35 TaBug 5.-Particle-size distribution of bedload sediment, conveyor-belt sampler-Continued hw _ 6.1276 61276 61276 61276 612276 613076 614-76 6-14-76 b-lB76 6-15276 6-16-76 6-18-76 6-19.76 6-20-76 6-21-76 q: A81 12.5 11.2 10.5 9.64 8.50 6.53 4.97 4.65 3.87 3.44 4.97 3.90 4.20 4.57 9.53 o.di81 - o.0i69 o.0i62 0.0145 0.0106 0.0084 0.0028 0.0022 0.0020 0.0003 0.0004 0.0009 0.0006 0.0009 0.0023 _ 0.0181 Percent by weight finer than sieve size (mm) indicated $06....;.: 0.4 0.2 0.3 0.2 0.8 2.3 1.7 0.4 0.2 1.0 0.6 0.3 f 15 .3 A 3 1.2 3.7 2.7 .6 (4 1.5 .8 15 . .T .5 .6 .6 1.8 5.9 4.6 1.2 .8 2.4 18 3 f 1.1 9 o 4 38 _ 10.3 7.8 2.1 1.8 5.4 4.6 1.9 . 2.6 24 2.3 I 104 _ 20.7 14.2 7.2 5.1 13.5 10.9 3.6 .8 11.1 10.0 8.9 .8 282 0 41.2 28.6 20.1 14.7 37.8 35.9 17.4 s 27.5 24.8 22.9 .0 512 0 22 46.9 37.5 29.6 61.9 62.1 37.7 ¢ 46.9 43.7 43.0 A 120 - 77.8 63.7 53.5 44.1 77.7 77.0 51.4 1. 60.2 58.9 58.8 .8 884 - 85.3 172.5 63.0 53.5 85.0 84.1 60.0 1.41 71.6 72.1 72.2 s 90.8 _ 89.9 78.5 70.6 61.1 89.7 89.2 68.6 2.00 81.7 83.3 82.7 954 - 93.0 83.3 76.9 68.6 92.4 92.8 77.6 2.83 89.8 91.2 89.5 919 _ 95.1 87.4 f 74.8 93.6 94.8 85.2 4.00 94.9 96.7 - 99.3 _ 96.8 91.3 80.3 94.3 95.3 91.1 5.66 97.6 98.7 100.0 _ gg 3 95.0 85.5 94.8 95.6 95.7 8.00 99.0 1000 ~~ 900 . - UHS: L 99.6 97.9 90.4 95.6 96.9 98.5 11.3 99.3 2 100.0 99.2 94.3 96.5 98.0 99.3 16.0 100.0 pene: holes 100.0 96.2 98.2 98.7 100.0 rb ea cnl. cd As 100.0 f 100.0, - 2:0 aO eas e Dien nl tin. U . | lelnes nies - F ; tess we 45.0 e C n ennis o 0 i Nin - 3 sp ades arre n AOT - dO eet lest Coll.. cid - Iris oen lls Ae meals | . die. seid Particle size (mm) at given percent finer 0.29 0.30 0.30 0.30 0.29 019 _ 0.11 0.13 0.22 0.25 0.20 0.17 0.19 0.18 0.27 .40 42 42 [43 41 28 ~ 22 .26 32 37 .30 .26 128 27 35 158 .61 161 .62 159 30° - 32 .40 48 157 to 134 .36 35 (48 277 .81 .82 .82 T7 49 "41 153 .66 .88 .50 42 (44 (43 .68 1.15 1.15 1.17 117 1.08 68. -_- 58 [74 1.09 1.69 153 155 153 1.92 2.19 2.04 2.06 213 1.90 1.03 _ 94 2.14 3.58 5.13 1.19 .96 1.26 1.00 2.68 4.02 3.71 3.48 4.17 3.30 192 277 5.650 0 11.78 12.65 5.83 6.18 11.67 3.14 5.31 'Dates correspond to dates listed in table 4. "Discharge over 14.6-m width of bedload trap; Indicates sieve analysis not available. includes all flow over the active width of the streambed. "Unit transport rate of solids in dry weight per second, over 14.6-m width of bedload trap. SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER TaBLs 6.-Summary data of river hydraulics and bedload transport, - Helley-Smith sampler River discharge Unit oen ae laned n. Mean Mean bedload- Bedload Date! Total," Effective," depth," velocity," transport size," Q' D ¥. rate," Uso (m'/s) (m'/s) (m) (m/s) FA (mm) (kg/m-s) 6-01-78 ____________ 16.6 15.8 1.02 1.06 0.0065 0.45 6-02-78 18.5 17.5 1.10 1.09 0277 58 6-03-7§ : 18.0 17.0 1.08 1.08 0425 71 6-06-78 ____________ 11.9 11.4 .81 .96 0074 .56 6-07-78 .__..L_IL__. 16.4 15.6 1.02 1.06 .0383 .60 6-08-78 ____________ 19.7 18.6 1.15 111 0059 A2 6-09-78 ____________ 28.3 26.4 1.47 1.23 1106 1.53 6-09-73 ____________ 27.6 25.8 1.45 1.23 1624 1.13 5-24-74 ____________ 2.95 2.95 .31 64 0015 122 §&-25-T4 L____L______ 5.44 5.34 A8 +17 0046 Al 8-26-74 L. 10.8 10.4 16 94 0451 50 5-26-74 ____________ 10.1 9.73 13 .92 0583 .65 B-27-14 23.3 21.9 1.29 1.17 1702 .98 5-28-74 ____________ 29.7 27.7 1.52 1.25 1532 1.07 5-28-74 ____________ 32.6 30.3 1.62 1.29 2533 1.15 5-20-74 45.3 41.8 2.03 1.42 3025 1.14 8-80-14 36.4 33.8 1.74 1.33 1227 1.76 cll, 25.1 23.5 1.35 1.19 .0768 1.14 6-01-74: . 25.8 24.2 1.38 1.20 0200 .68 6-02-74 27.5 25.7 1.44 1.23 0106 A0 6-03-74 ._..;__._._? 31.9 1.60 1.28 0233 A9 6-04-74 ____________ 30.0 28.0 1.53 1.26 0307 .59 6-05-74 ____________ 28.3 26.4 1.47 1.23 0447 10 5-25-75 ____________ 2.35 2.35 27 .60 0061 .82 B-25-45 ° 2.35 2.35 27 .60 0024 A8 b-20-75 2.41 2.41 .27 .60 0050 74 5-20-75 sA... 2:38 2.38 ~27 .60 0075 .57 5-27-15. 2.44 2.44 .28 .61 0034 10 5-27-15) 2.44 2.44 .28 .61 0062 A9 6-01-75 .c... coin, 3.26 3.25 .84 .66 0080 A9 6-01-75 ____________ 3.26 3.25 34 .66 0086 .61 6-02-75 ____________ 5.38 5.28 A7 16 0649 .64 6-02-75 5.35 5.25 AT 16 0749 .87 6-03-75 ____________ 9.06 8.75 .68 .89 0093 18 6-04-75 ____________ 10.1 9.73 13 92 .0983 99 6-05 10.6 10.2 15 .93 0512 1.06 6-OB-T5..._L_COLL.} 10.5 10.1 15 .93 .0829 1.05 6-06-75 ..--.......; 20.9 19.7 1.20 1.13 1076 .90 6-06-75 ____________ 21.3 20.1 1.21 1.14 0860 .97 6-07-45... 26.9 25.2 1.42 1.22 1262 .95 6-07-75 ____________ 27.0 25.3 1.43 1.22 0474 .68 6-07-75 _. 26.6 24.9 1.41 1.21 2168 .93 26.4 24.7 1.40 1.21 1896 127 6-08-75 ____________ 27.4 25.6 1.44 1.22 2027 1.24 6-08-75 ____________ 27.5 25.7 1.44 1.23 1025 1.28 6-09-75 ____________ 21.1 25.4 1.43 1.22 1185 .82 6-09-75 ____________ 27.1 25.4 1.43 1.22 1054 1.50 6-10-75 ....__._L.__. 17.0 16.1 1.04 1.07 0674 1.11 6-10-75: 16.7 15.9 1.03 1.06 0298 1.41 6-11-75 . cccslceisal 10.9 10.5 T7 .94 0210 1.15 6-11-75 ____________ 10.8 10.4 16 94 0225 1.02 6-12-15 ...e lke, 10.7 10.3 16 .93 0112 A9 6-12-15 .:.... 10.6 10.2 15 .98 0125 A8 6-13-75 16.7 15.9 1.03 1.06 0337 +10 6-13-75 16.7 15.9 1.03 1.06 0252 A4 6-17-75 ____________ 24.5 23.0 1.33 1.19 0879 1.58 6-17-15 24.5 23.0 1.33 1:19 1052 1.38 6-18-45 13.7 13.1 90 1.00 0265 57 6-18-75 ____________ 13.7 13.1 90 1.00 0195 .59 6-19-75 10.9 10.5 T7 94 0163 70 TABLES 4-13 TABLE 6.-Summary data of river hydraulics and bedload transport, Helley-Smith sampler-Continued River discharge Unit roa a aera ae mass Mean Mean _ Bedload- Bedload Date! Total," Effective," degbth, + velo‘cllty," transport sidze,‘ (M's) (m'/s) (m) (m/s) trie (mm) (kg/{11451 6-19-75 .___________ 10.9 10.5 47 94 0114 .65 6-20-75 ___________. 7.14 6.95 .57 .83 0040 55 6-20-75 ____________ 7.14 6.95 57 .83 0045 50 6-21-75 _._._..__._.:_.. 7.70 7.48 .61 .85 0039 52 6-21-7§8 7.170 7.48 61 .85 0051 .65 6-22-75 ____________ 7.48 7.27 .59 .84 0075 .81 7.48 7.27 .59 .84 0099 .56 6-28-48) 8.44 8.17 .64 .87 0096 .92 ..._.._._..._ 8.44 8.17 .64 .87 0154 1.08 6-24-75 11.3 10.9 19 95 0254 .82 6-24-75 .___________ 11.8 10.9 19 .95 .0183 178 §-2B-76 ...... _.... 21.5 20.3 1.22 1.14 0926 .88 6-25-75 ____________ 21:7 20.4 1.23 1.14 0814 99 6-20-75 ...__.____sll 23.3 21.9 1.29 1.17 0912 1.19 6-25-76 23.3 21.9 1.29 1:17 0725 1.56 6-26-75 ____________ 15.9 10.1 .99 1.05 0684 .82 6-20-75 :0...1000001 15.7 14.9 .98 1.04 .0976 16 6-26-75 ____________ 11.8 11.3 .81 .96 0513 1.04 6-26-75 ..._._._.___._. 11.6 11.1 .80 .95 .0683 1.03 b-28-76 20000000100. 21.4 20.1 1.23 1.11 0390 .65 b-28-76 .._ 21.1 19.9 1.23 1.11 .0822 64 5-20-76 ___._________ 22.4 21.0 1.27 1.13 1205 1.14 5-20-76 22.4 21.0 1.27 1.13 0977 .56 'Dates correspond to dates listed in table 7. 'Complete river discharge including overbank flow. *Discharge over 14.6-m width ofbedgoad trap; includes all flow over the active width of the streambed. , 'Mean depth over effective width W; D = L "Mean velocity of effective discharge; V = _Q_ = WD - 14.6D "Unit transport rate of solids in dry weight per second, over 14.6-m width of bedload trap. "d;, is median diameter of grains; complete grain-size data are given in table 7. 38 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER TaBur 7.-Particle size distribution of bedload sediment, Helley-Smith sampler Date! __ 6-01-73 6-02-73 6-03-73 6-06-73 6-07-73 6-08-73 6-09-73 - 6-09-73 5-24-74 5-25-74 5-26-74 5-26-74 5-27-74 5-28-74 Nige. .8 17.5 17.0 11.4 15, 18.6 26.4 25.8 2.95 5.34 10.4 9.73 21.9 27.7 0.0277 0.0425 0.0074 0.0383 0.0059 0.1106 0.1624 0.0015 0.0046 0.0451 0.0583 0.1702 0.1532 Percent by weight finer than sieve size (mm) indicated T 0.1 0.3 0.2 0.1 0.3 0.1 0.1 0.1 0.5 0.0 0.1 0.1 z A .5 .3 .3 A 4 .2 .2 1.1 0.0 A .2 f 1.1 .8 .5 .6 [f 2 .8 y 2.2 .5 A .8 : 3.1 1.8 1.1 1.7 .5 A .6 1.6 4.1 1.0 r .6 ; 9.6 5.2 3.2 7.5 *I 1.1 1.7 6.9 15.7 4.0 1.6 1.9 .35 - 30.7 17.5 10.3 26.3 A 3.1 4.6 16.5 42.1 12.9 4.5 5.3 .50 . 57.7 41.1 26.7 44.4 [4 7.6 12.3 27.1 60.7 33.8 14.4 14.7 Aly 71.6 62.7 50.1 62.0 .3 17.4 31.7 37.8 72.5 53.9 31.4 30.2 1.00 . 77.9 73.1 65.4 70.6 A 29.5 45.7 46.2 80.9 66.7 51.2 46.9 1.41 83.0 81.3 77.3 77.7 .6 45.2 57.9 52.9 87.2 77.6 66.7 61.8 2.00 87.9 88.5 86.9 84.1 f 66.1 TL.1 61.1 92.2 88.6 80.3 74.9 2.83 92.2 93.8 93.2 89.0 83.2 82.4 714 95.7 94.7 89.7 85.6 4.00. ..-: 95.9 97.5 97.4 92.7 93.1 91.6 78.4 97.5 98.0 95.2 95.1 $.060-..«. 98.2 98.8 99.1 95.2 96.8 96.2 83.9 98.5 99.3 97.9 98.0 8.00 f 99.6 99.8 . 98.9 98.6 87.0 99.4 100.0 99.2 99.1 11.3 99.8 100.0 99.7 99.5 88.3 100.0 100.0 99.5 16.0 100.0"; :- 100.0 100.0 100.0 99.8 22.6 _ Hirers 3 100.0 32.0 decent Be l uy. id 1. In mesede oo, its despees bi nce on ull Inepan's alu nese ir mies nd Parm 45.0 . 3 $0 Crewel nl . S lint: Mis ioc 111. o fo ol p o eee i a ne | awakens oon o tin cite i tii freee back, (named Particle size (mm) at given percent finer 0.25 0.28 0.23 0.25 0.21 0.42 0.36 0.23 0.18 0.25 0.27 0.36 0.35 34 Al .30 .35 .28 .68 54 .35 .25 .33 .87 .52 52 A6 .57 A2 AT .35 1.14 T7 .65 .33 43 .50 16 19 .58 T1 .56 .60 A2 1.53 1.13 1.22 Al .50 .65 .98 1.07 16 .99 .80 .91 .50 1.96 1.69 2.27 .56 .65 .94 1.36 1.53 1.60 1.79 2.00 1.82 T7 2.90 2.98 5.80 1.18 1.22 1.65 2.26 2.68 3.09 3.18 5.47 3.00 2.32 4.66 5.05 12.3 2.60 2.26 3.30 3.93 3.98 Date! __ 5-28-74 5-29-74 5-30-74 5-31-74 6-01-74 6-02-74 6-03-74 - 6-04-74 6-05-74 5-25-75 5-25-74 5-26-75 5-26-75 5-27-75 @"..." $08 41.8 33.8 23.5 24.2 25.7 29.7 28.0 26.4 2.35 2.35 2.41 2.38 2.44 T 0.2533 0.3025 0.1227 0.0768 0.0200 0.0106 0.0233 0.0307 0.0447 0.0061 0.0024 0.0050 0.0075 0.0034 Percent by weight finer than sieve (mm) indicated h 0.1 0.2 0.1 0.1 0.2 0.3 0. 0.2 0.2 0.3 0.8 0.7 0.5 f .2 3 E1 2 .5 T 3 A A .6 .8 4 1.1 s .3 A .2 .3 1.0 1.3 4 .T y4 .6 .8 14 1.1 s .5 .6 A .6 1.7 2.2 1. .6 1.1 .9 1.5 1.5 1.6 95 ....0 918 1.3 1.2 .8 8.6 10.7 6. .8 2.9 3.0 4.6 3.6 3.8 40 4.1 2.9 .6 30.0 41.1 29. .5 11.3 12.0 19.8 12.0 11.4 (60... .s 11.7 11.8 6.6 T 42.6 66.7 49. .6 32.0 28.7 54.2 28.8 27.7 A1..2:t 264 26.1 14.3 .9 49.4 72.3 62. .0 51.0 45.5 81.7 47.8 50.5 1.00 ___. 43.1 44.5 26.0 .3 56.5 75.8 72. 9 64.4 56.0 92.4 64.6 72.8 1415... 59.7 59.0 40.6 .0 63.9 79.3 78. .3 75.2 70.4 96.9 81.8 91.8 200... 74.2 71.7 55.4 .3 72.1 83.2 83. .8 85.0 87.1 : : 98.9 2.83 .____ 84.8 81.5 69.2 34 79.6 87.1 88. .8 91.6 98.5 99.5 4.00 .____ 91.8 88.5 81.5 4 86.3 91.0 92. .6 : 99.7 100.0 5.66 95.6 93.6 89.6 v 90.9 94.1 94. .0 100.0 $:00..-.. 97.6 96.8 95.2 ve 94.5 96.9 97. .6 11.3 - 98.6 98.4 98.0 .0 96.7 98.6 98. .T 16.0 . : 99.3 99.6 .8 98.0 99.3 99. 4 226 . 99.8 100.0 .0 99.4 100.0 99: T 32.0 100.0 we 100.0 100. .0 45.0 . a melts oen MA LCL Ao Ives sade se NELL lly unl deen ees Buti (uy 0.38 0.44 0.30 0.22 0.21 0.23 0.23 0.28 0.28 0.25 0.27 0.26 0.27 56 75 51 29 27 30 .32 39 39 33 6 40 84 1.25 84 41 33 39 A5 53 57 42 56 47 56 1.14 1.76 1.14 68 40 49 .59 70 82 48 74 57 70 1.66 2.53 1.69 1.48 49 77 .84 1.02 1.24 56 1.01 69 88 3.18 4.42 3.04 3.54 2.16 2.05 1.92 1.93 1.85 15 1.48 97 1.19 TABLES 4713 39 TaBug 7.-Particle size distribution of bedload sediment, - Helley-Smith sampler-Continued Date! __ 5-27-75 6-01-75 6-01-75 6-02-75 6-02-75 6-03-75 6-04-75 - 6-05-75 6-05-75 6-06-75 6-06-75 6-07-75 6-07-75 6-07-75 @* :s.- 2.44 3.25 3.25 5.28 5.25 8.176 9.73 10.2 10.1 19.7 20.1 25.2 25.3 24.9 T ge. 0.0062 0.0080 0.0086 0.0649 0.0749 0.0093 0.0983 _ 0.0512 0.0829 0.1076 0.0860 0.1262 0.0474 0.2168 Percent by weight finer than sieve size (mm) indicated 0.4 0.4 0.4 0.9 0.3 0.5 0.3 0.4 0.5 0.5 0.6 0.3 .6 15 4 1.3 8 7 A .6 4 .5 .% 3 .8 T .6 i% 15 9 .6 .8 9 f 11 15 1.7 1.1 1.0 2.7 T 1.4 .9 1.3 1.4 ¥4 1.7 2T 8.1 3.9 2.8 6.4 .6 3.0 2.0 3.2 3.8 3.5 5.3 1.9 20.2 18.0 121 16.5 t 7.9 5.5 9.0 9.5 8.8 13.4 4.7 39.1 36.7 26.8 32.6 .8 17.5 14.6 21.8 21.5 21.1 29.6 14.4 58.4 55.6 42.4 46.6 .0 33.1 31.7 39.9 374 38.7 52.3 34.8 72.8 68.7 55.2 58.8 A 47.6 47.6 54.6 51.1 52.2 70.4 54.4 80.9 80.8 69.1 70.6 .0 63.0 63.3 67.5 63.1 64.5 84.0 69.8 85.1 90.8 83.6 82.4 ; 79.2 78.9 79.0 75.0 76.0 92.9 81.8 87.0 96.3 92.9 91.7 90.6 89.9 87.8 85.2 85.3 96.9 89.4 88.1 98.6 97.9 97.3 96.9 96.8 94.2 92.8 92.7 98.7 94.1 89.0 99.5 99.1 98.5 99.0 99.0 97.1 96.3 96.0 99.5 96.5 89.8 99.8 99.7 99.5 99.8 99.8 98.6 98.4 98.0 100.0 98.2 90.9 100.0 100.0 99.5 100.0 100.0 99.4 99.3 99.1 99.2 93.8 1000 ° | ...l tale o 99.6 100.0 99.5 100.0 100.0 Si - l | c Ate mal t AOOO L t eid 2. eac 2 1000 _ ..o... 100.0 0.22 0.26 0.28 0.23 0.36 0.30 0.34 0.29 0.28 0.28 0.25 0.36 32 34 .39 85 .53 48 52 44 43 44 .38 52 AT 49 .60 .53 15 74 16 .65 .67 .66 .55 11 .61 .64 .87 18 .99 1.06 1.05 .90 .97 .95 .68 .93 .82 .90 1.27 1.19 1.41 1.47 1.46 1.32 1.49 1.43 .90 1.26 1.84 1.56 2.03 211 2.35 2.27 2.31 241 2.71 2.69 1.42 2.20 16.4 2.54 3.15 3.34 3.66 3.48 3.53 4.32 4.87 5.00 2.93 4.48 Paw _ bort 60575 60575 60075 600.75 6.10075 610-75 6-11-75 6-11-75 6-12-75 6-12275 6-13-75 6-13-75 6-17-75 @* 24.7 25.6 25.7 25.4 25.4 16.1 15.9 10.5 10.4 10.3 10.2 15.9 15.9 23.0 0.1896 0.2027 0.1025 0.1185 0.1054 0.0674 0.0298 _ 0.0210 0.0225 0.0112 0.0125 0.0337 0.0252 0.0879 Percent by weight finer than sieve size (mm) indicated 0.3 0.2 0.5 0.4 0.3 0.4 0.4 0.7 0.5 1.0 0.9 0.8 0.3 .3 A .6 .5 3 .5 5 .9 T 1.3 1.2 1.1 4 15 15 .8 .8 .5 # 4 11 1.0 1.6 1.5 1.5 .5 'T 4 1.1 1.1 .6 1.0 1.0 1.6 1.4 2.3 22 28 4 1.6 2.0 3.3 3.5 2.8 25 3.9 5.3 5.1 10.5 9.5 9.4 1.8 £3 4.9 7.9 13.9 9.9 7.1 10.4 16.1 15.9 35.5 34.9 37.9 6.2 8.8 9.9 14.1 25.6 16.4 15.0 15.7 24.0 25.0 49.5 52.0 58.3 10.7 22.7 22.7 25.3 43.5 24.1 27.2 23.6 34.2 35.6 61.1 69.1 70.0 18.1 39.1 39.1 37.9 58.7 34.0 44.1 34.3 45.1 49.0 68.3 83.4 15.9 27.9 55.2 56.8 54.9 718 47.2 63.1 50.2 57.6 64.0 74.7 90.9 80.6 43.3 71.0 73.8 72.4 81.4 62.9 80.1 69.8 71.4 79.2 80.7 94.6 84.8 63.8 83.4 85.8 82.7 88.9 76.9 91.5 85.1 82.8 90.2 87.4 96.5 88.6 80.0 91.8 93.3 91.0 94.4 89.0 97.2 94.3 91.8 96.8 93.8 98.1 s . 96.0 97.1 94.8 97.2 94.2 99.1 98.3 96.7 ; 97.9 99.1 98.2 98.9 98.0 99.2 97.6 99.8 99.8 99.0 99.5 100.0 99.4 99.7 99.6 100.0 99.2 100.0 100.0 99.6 100.0 100.0 100.0 pos." c.... 100.0. 100.0 100.0 .. [S.. . clime cual PTR o aise. dy 0.41 0.36 0.29 0.27 0.29 0.31 0.27 0.25 0.25 0.21 0.21 0.23 0.21 0.23 .60 .53 .38 49 .52 .50 .35 35 .28 .28 B1 .28 .64 .92 .93 .60 1.03 .84 1.02 173 .69 35 .35 A7 .34 1.18 1.24 1.28 .82 1.50 111 141 1.15 1.02 49 48 .70 44 1.58 1.66 1.71 1.18 2.10 1.46 1.83 1.69 1.44 .85 .65 1.19 .61 2.05 2.67 2.97 2.24 3.41 2.21 2.15 2.95 2.29 2.36 1.03 2.24 1.89 3.17 4.55 5.15 4.25 6.02 887 4.17 4.87 3.52 4.31 2.13 3.81 4.94 5.20 40 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER Tamu 7.-Particle size distribution of bedload sediment, Helley-Smith sampler-Continued Date' .. 6-17-75 6-18-75 - 6-18-75 6-10-75 | 6-19-75 6-20-75 6-20-75 6-21-75 G@LT5 6-22-75 6-327) 6275 621.75 6.2475 C 23.0 13.1 13.1 10.5 10.5 6.95 6.95 7.48 7.48 7.27 7.27 8.17 8.17 10.9 if" i:f.c2 +0.1058 0.0265 0.0195 0.0163 0.0114 0.0040 0.0045 0.0039 0.0051 0.0075 0.0099 0.0096 0.0154 0.0254 Percent by weight finer than sieve size (mm) indicated 0.5 0.7 0.4 0.3 0.5 0.4 0.9 0.7 0.5 0.4 0.6 0.4 0.4 7 .8 4 .6 .5 "4 9 T .5 4 .8 .5 .6 .8 1.0 .8 .8 .9 .8 1.4 1.1 .T :f 1.0 5 .8 1.2 1.5 1.2 13 1.4 1.2 2.3 1.8 1.2 .9 1.5 1.1 1.2 4.2 4.9 4.8 5.8 9.0 7.8 11.1 115 5.8 6.5 6.1 4.5 4.0 22.2 24.4 19.7 23.8 30.8 29.4 33.3 30.9 20.8 27.1 21.0 16.5 17.0 42.4 42.3 33.1 39.1 47.1 49.8 48.6 43.5 34.6 45.9 32.1 26.9 33.5 61.5 58.8 50.7 53.7 58.4 67.3 60.6 52.2 46.0 59.2 42.2 35.8 46.2 78.5 74.3 67.4 65.8 68.3 77% 68.5 59.4 56.7 68.5 52.7 47.0 55.1 88.9 86.1 81.2 78.3 74.1 84.5 74.5 66.5 70.7 76.0 64.7 61.0 65.1 94.2 93.5 90.2 89.1 80.1 90.2 80.1 74.1 86.4 83.9 77.1 76.2 77.7 96.7 97.1 94.4 95.0 84.6 93.9 85.2 82.4 95.4 92.3 87.0 88.2 88.8 98.4 99.0 96.9 97.8 89.6 96.3 89.4 89.2 98.8 96.1 92.4 95.5 95.4 99.1 99.5 98.3 99.0 92.8 97.6 92.6 93.2 99.5 98.2 94.8 99.1 98.8 99.3 100.0 99.6 100.0 97.3 100.0 97.7 97.5 100.0 99.4 97.5 100.0 99.8 99.4 / 1000 - .-.... $66 100.0 1000 _| ._.. 100.0 98.7 e 100.0 100.0 est a i a arai i n o a anal ce 1000. c.: nnd ccs, dy isl. 0.31 0.26 0.25 0.25 0.24 0.22 0.23 0.21 0.21 0.24 0.24 0.24 0.26 0.26 dis 48 .33 .32 .33 .31 .29 .30 .28 .28 .32 .30 .32 .35 .35 das .87 45 44 .52 46 .39 .39 .37 40 151 Al .55 .69 .52 ds eaid: 1.38 .57 159 .70 .65 155 .50 .52 .65 .81 .56 .92 1.08 .82 Ue 1.98 15 .81 .95 .98 .89 .67 .85 1.31 1.22 .87 1.43 1.54 1.41 3.33 1.18 1.32 1.56 1.67 2.72 1.39 2.62 3.06 1.88 2.02 2.52 2.47 2.40 e 5.61 2.20 2.25 3.03 2.83 6.49 3.26 6.41 6.34 2.176 3.55 5.77 3.86 3.88 Date' .. 6-94-75 6-25-75 - 6-25-75 6-35-75 6-30-75 6-26-75 | 6-26-75 6-26-75 - 6-26-75 | 5-28-16 5-26076 5-20-76 - 5-49-76 @*. 10.9 20.3 20.4 21.9 21.9 15.1 14.9 11.3 11.1 20.1 19.9 21.0 21.0 maxed 0.0183 0.0926 0.0814 0.0912 0.0725 0.0684 0.0976 0.0513 0.0683 0.0390 0.0822 0.1205 0.0977 Percent by weight finer than sieve size (mm) indicated 9.06..... 0.5 0.4 0.3 0.4 0.4 0.4 0.4 0.3 0.1 0.2 0.3 0.4 = 1 4 A .5 15 .5 .5 A 2 % 4 .5 g .6 .6 S .6 5 .6 4 t .5 .5 .8 Bse... / 44 1.0 1.0 1.1 .9 1.0 .9 .T .5 .8 .8 1.2 5 4.0 3.1 3.0 3.2 2.7 2.8 2.4 1.8 1.0 8.7 3.0 4.9 os... _- 12.3 11.4 9.3 7.8 9.6 9.0 5.6 2.9 14.4 10.2 19.7 34.7 28.8 26.2 19.1 15.3 22.8 22.8 15.3 9.4 35.1 22.0 42.9 Holle. malls 43.6 40.2 31.1 23.8 41.3 45.1 31.9 24.5 55.5 35.2 63.5 87% 53.8 50.4 43.0 34.9 61.8 68.2 48.3 48.4 66.3 45.6 73.6 far..c..*670 64.5 61.1 56.9 46.6 77.1 82.4 64.1 69.4 74.9 57.3 80.9 2.00 __ 8.3 77.0 71.9 71.6 58.8 88.1 90.8 78.5 84.3 83.1 71.2 86.9 2.83 __ 89.6 87.0 80.3 82.0 70.1 94.9 95.5 89.0 93.0 89.5 83.6 91.5 4.00 __ 96.0 93.7 88.5 89.3 79.9 98.5 98.2 95.2 97.8 93.6 91.5 94.9 5.66 __ 98.4 97.3 91.6 93.7 89.9 99.5 99.4 98.3 99.1 96.3 96.5 97.5 8.00 99.5 99.1 97.3 96.5 95.6 99.8 99.8 99.7 99.8 97.9 98.8 99.2 11.3 - 100.0 99.7 99.4 97.8 98.3 100.0 99.8 100.0 100.0 98.5 100.0 99.7 16.0 - 100.0 100.0 98.8 99.3 99.8 98.9 99.8 226. 98.8 100.0 100.0 99.4 100.0 32.0 _ pes nial tC alld tane. 3 & 45.0 _ 100.0 a neon & 640 seas n ately . A t ae meee 1 o H ak (ont o etn e toot s c , size (mm) at given percent finer 0.28 0.28 0.29 0.30 0.29 0.30 0.34 0.41 0.22 0.28 0.29 0.25 .39 40 45 .51 43 43 .51 .60 .30 .87 43 .33 .58 .63 .80 1.00 .63 .61 176 .83 43 .50 .70 45 .88 .99 1.19 1.56 .82 .176 1.04 1.03 .65 .64 1.14 .56 1.43 1.60 1.70 2.41 1.07 .95 1.44 1.31 1.59 .96 1.70 .74 2.53 3.28 3.09 4.55 1.73 1.50 2.36 1.99 10.0 2.10 2.89 1.69 4.42 6.69 6.51 7.61 2.85 2.170 3.94 3.15 23.6 4.67 4.95 4.03 'Dates correspond to dates listed in table 6. *Discharge over 14.6-m width of bedload trap; includes all flow over the active width of the streambed. "Unit transport rate of solids in dry weight per second, over 14.6-m width of bedload trap. TABLES 4-13 TaBLE 8.-Summary of statistical data: log-transformed linear regression of transport rate by particle-size class versus total transport rate, conveyor-belt sampler' Particle j, (size class) = A , (total)" [¥ = AX") size Number z Mi Correlation a ms class of data Y X A B coefficient, Y/X (mm) points (kg/m-s) (kg/m-s) r (percent) 0.06- 0.12 __86 0.000092 0.029164 0.000961 0.663 0.912 0.32 A2- 25 _.88 000520 029855 003627 553 890 1.74 .25b- .50 __88 005519 029855 074800 742 954 18.49 .50- 1.00 _-88 008326 029855 278676 1.000 .994 27.89 1.00- 2.00 __88 006535 029855 401883 1.173 992 21.89 2.00- 4.00 __88 004139 029855 368044 1.278 .985 13.87 4.00- 8.00 __88 001659 029855 116466 1.211 .957 5.56 8.00-16.0 _ __86 000460 030946 014618 995 .808 1.49 16.0 -32.0 _ __43 000379 049484 006132 926 .875 #7 32.0 -64.0 ____6 000800 107516 653752 3.007 760 74 'Basic data as collected. 9.-Summary of statistical data: log-transformed linear regression of transport rate by particle-size class versus total transport rate, Helley-Smith sampler' j, (size class) = A j, (total)" [¥ = AX") Particle- size Number - - Correlation lies class of data ¥ X A B coefficient, Y/X (mm) points (kg/m-s) (kg/m-s) r (percent) 0.06- 0.12 __82 0.000116 0.032706 0.001390 0.727 0.913 0.35 12- .25 __83 001027 031693 008111 599 .858 3.24 .25- .50 __83 007254 031693 080705 .698 912 22.89 .50- 1.00 __83 008510 031693 319448 1.050 978 26.84 1.00- 2.00 __83 006360 031693 A19373 1.213 973 20.07 2.00- 4.00 __83 003362 031693 348418 1.344 939 10.61 4.00- 8.00 __80 001185 034352 066206 1.193 870 3.45 8.00-16.0 _ __68 000356 039981 005802 .867 114 .89 16.0 -32.0 _ __18 000421 065133 001210 .387 385 .65 'Basic data as collected. TaBug 10.-Listing of comparable data sets used in direct comparison of results from conveyor-belt sampler with results from Helley-Smith sampler Effective discharge," Discharge Unit bedload transport rate," Q' (m*/s) ratio 4, (kg/m-s) Date equaredy Conveyor Helley- 'er \ Conveyor belt Helley-Smith belt Smith (T) ® ® ® ® 0 0d ue a un 16.1 15.8 1.04 0.0190 0.0190 0.0065 0.0067 6-02-18; .. o He .. ue delim ne in mo aniem e 17.8 17.5 1.03 0235 0235 0277 0287 6:05- 18. := -. oon cu us 16.6 17.0 .95 0240 0240 0425 0405 00 ci 11.3 11.4 .98 0133 0133 0074 0078 6-07-18 52... SBL LZ a roe na neonl bone 15.9 15.6 1.04 0260 0260 .0383 .0398 6-08-18... .o Leu. dpa on 19.2 18.6 1.07 0218 0218 0059 0063 0-20-14 -.: linen anos 5.34 5.34 1.00 0056 0056 0046 0046 $2014 .:. naar 9.92 10.4 .91 .0822 .0632 0451 0410 9-20-14. coll 9.92 9.73 1.04 0822 .0632 0583 0606 fea- T4... sc oue cakes aree ase 21.5 21.9 .96 1758 A853 1702 1640 pep- T4. . RLS to in oa ala mad 29.8 27.7 1.16 2255 1735 1532 1773 en TK s eT Ee o) o a ata a h tua tie Wii 29.8 30.3 D7 2255 1735 2533 2450 0fr20-T4 sL TEE IEG ie cule 41.5 41.8 .99 2912 2240 3025 .2982 5230-14 ...s yc. edn Pe bes mea 32.2 33.8 91 0786 0605 1227 1114 0.0 up 22.9 23.5 .95 0647 0498 .0768 .0729 6-01-14 culte cy ut 24.3 24.2 1.01 0206 0206 0200 0202 - .. . 2 2 LIU ae c Ha nine. 25.5 25.7 .98 0130 0130 0106 0104 6-035 IEG diy 290.7 29.7 1.00 0172 0172 .0233 .0233 6=04-74:.. .so. L t page wens, 27.9 28.0 .99 0285 0285 0307 0305 6-05-14: .. UL .a, aan 26.5 26.4 1.01 0305 0305 0447 0450 dled eee cuca a ew 2.44 2.44 1.00 0021 0021 0034 0034 $227: 15 __ nll ege odie ae arise aoe 2.44 2.44 1.00 0021 0021 0062 0062 6DP~TBD :.. ool i ane wou 5.82 5.28 1.22 0484 0372 0649 0789 6-02-75 ...l. .A ua 5.82 5.25 1.23 0484 0372 .0749 0920 6_08- TD.; . ... s.. 2 e + Has vea e d e wg a id the me w oa ap 9.13 8.175 1.09 .0791 0608 0093 .O101 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER 10.-Listing of comparable data sets used in direct comparison of results from conveyor-belt sampler with results from Helley-Smith sampler-Continued Effective discharge," Discharge Unit bedload transport rate, Q' (m*/s) ratio J, (kg/im-s) Date ono mmm. equared. Conveyor Helley- @'ca \' Conveyor belt Helley-Smith belt Smith Q3”) ® ® ® ® sere ede cle aoe e regne's 10.0 9.73 1.06 0812 .0625 .0983 1038 6200-70... >a 42.2 ae an arg bhale me whe ms 10.7 10.2 1:10 0972 0748 0512 0563 6-054 75 . s 2T n. crn Ga an n ih n income 10.7 10.1 1:12 0972 0748 .0829 0930 6-06-15. .s: NLN Ll 20.0 19.7 1.03 3114 .2395 1076 1109 6-06-75: ..:..s 5. Lous cabe bn na 20.0 20.1 99 3114 .2395 0860 0851 6-0 7210... .are nute t edhe ba epee ure a 24.8 25.2 .97 .2069 1592 1262 1922 .. es t re Tea an ane 24.8 25.3 .96 2069 1592 0474 0455 6-07-15. . c en eich bec ca nave. 24.8 24.9 .99 2069 1592 .2168 2151 6-07-15 ere. cru db de aha aos 24.8 24.7 1.01 2069 1592 .1896 1911 nll 25.6 25.6 1.00 1733 1388 2027 2027 6-08-75... sert isto ela cote ces 25.6 25.7 .99 1733 13838 1025 1017 6-09-15. . :.. ELEV ELLY a+ Cous isu s uue 24.3 25.4 .92 .0838 .0722 1185 1085 600-7D. . ece ced ogle esen aas 24.3 25.4 92 .0838 0722 1054 .0965 610=-175.:0000910.001 MCR ece: 14.4 16.1 .80 0348 .0268 0674 .0539 6-10-75... ce sone 14.4 15.9 .82 0348 .0268 .0298 0244 6-11-75 :.. ua . dl oe aaa 10.1 10.5 .93 0110 0110 0210 0194 611275... 0... ue Aden L ues aire 10.1 10.4 .94 0110 0110 0225 0212 6-13-15... . yl luo uel stero i2ll 15.8 15.9 .99 0277 0213 0337 0333 « AS -Go be nook 15.8 15.9 .99 0277 0213 0252 0249 6:17 78... 34 cal.. Be ain the a fh a wee a a ene 22:2 23.0 .98 .0796 .0612 0879 .0819 611475... .so ap e ho o oan d 4 8 g a uel 22.2 23.0 .98 0796 0612 1052 0980 12.8 13.1 .95 0106 0106 0265 0253 6-19-70... cours aees s 12.8 13.1 .95 0106 0106 0195 .0186 6-19-15. ...l aun ells 10.1 10.5 .93 0097 0097 .0163 0151 610-75... n filed d 10.1 10.5 .98 0097 0097 0114 0106 6-21-75 . rv duas 7.23 7.48 .93 0032 0032 0039 0036 6-21-75 _ [.e nle nea s ga wen ahs 7.23 7.A8 .98 0032 0032 0051 0048 TD.. 2 ELL olin an Sire nba alie 7.01 1:27 .93 0047 0047 0075 0070 622° TB.. Art eae vea ul 7.01 727 .93 0047 0047 0099 0092 _ on- ss 8.24 8.17 1.02 0062 0062 0096 0098 6-20-10. s NAL Leaded kv dena ns 8.24 8.17 1.02 0062 0062 0154 0157 624 TD.. IIE n raga oa aw die ack 10.8 10.9 .98 0194 0194 0254 0249 ceo 10.8 10.9 .98 0194 0194 .0183 0180 E0 ioe 21.7 20.3 1.14 .0838 0645 0926 1058 6-25-15... o .ll OCIT ec ce aan 21.7 20.4 1.13 .0838 0645 0814 0921 620-15 nec eee ope ene s 21.7 21.9 .98 .0838 0645 0912 .0895 6-20-70. ELSE ens 21:7 21.9 .98 .0838 0645 0725 0712 6-20-15. Aiea u eet ane 13.1 15.1 15 .0396 .0396 0684 0515 6-20-10; .l cL ee dle menses aan 13.1 14.9 44 0396 .0396 0976 0754 bead TD : L LL oue e oe ane anale aem 13.1 11.3 1.34 0396 .0396 0513 0689 icu oes 13.1 111 1.39 0396 0396 .0683 0951 Sear I0. cols s Puls de agen s 18.8 20.1 .87 0437 .0336 0390 0341 .. l.. . 18.8 19.9 .89 04837 .0336 .0822 0734 LE. a tic 19.8 20.1 .97 0454 0454 0390 0378 icon teed lle l. 19.8 19.9 99 0454 0454 .0822 0814 D420 76 . . st a L eee anc 20.5 21.0 .95 0712 0548 1205 1148 5-20-10... .s ss en ene agen 20.5 21.0 .95 0712 0548 0977 .0931 $=20-70.:. pa cen 20.9 21.0 .99 0618 .0618 1205 1194 sect n ost sono, 20.9 21.0 .99 0618 .0618 0977 .0968 'Discharge over 14.6-m width of bedload trap; includes all flow over the active width of the streambed. *Regression of transport data from table 4 indicates the transport rate is proportional to the square of the effective discharge. To correct for stage (discharge) differences between otherwise comparable conveyor-belt and Helley-Smith data sets, the appropriate correction applied to Helley-Smith transport data is the factor (Q cn/lQ'ns). "Unit transport rate of solids in dry weight per second, over 14.6-m width of bedload trap. 'Basic data as collected. *Conveyor-belt data corrected to conditions of stream-wide slot. ®Helley-Smith data corrected for stage difference with conveyor-belt data; see note 2. TABLES 4-13 43 TABLE 11.-Summary of statistical data: log-transformed linear regression of transport rate, Helley-Smith sampler versus conveyor-belt sampler (basic data) Particle- Number j, (Helley-Smith) = A 7, (conveyor-belt?" [¥ = AX") size class of data zz (mm) points Log ¥ Log X SD (log Y) SD (log X) Log A B 1%, Var (log (A)) - SE (log (A)) Var (B) -3.713838 -3.804032 0.430278 0.389728 -0.1841 0.928 0.7394 0.078713 0.280558 0.005384 -2.821671 -3.147199 280158 299980 -.4586 .T51 .6463 .052254 228591 .005229 -1.961669 -2.137697 342941 361216 -.2473 .802 1135 020564 143400 .004377 -1.882042 -1.876378 540334 509253 -.1289 .934 1755 .016175 127180 004284 -1.975948 -1.927100 584104 .584409 -.3024 .868 1550 .016799 129610 .014148 -2.189868 -2.126512 .584247 623490 -.4831 .803 1336 .019445 139445 .003965 -2.668660 -2.640915 .589985 .682776 -.T161 139 7331 025106 158450 003378 -3.248296 -3.219476 632180 761091 -.8438 147 .8085 .028324 168298 002591 -3.330789 -3.076402 481029 458405 -1.7910 .501 2275 .684715 827475 .070886 j» (Helley-Smith) = A 7, (conveyor-belt)" [¥ = AX") Particle- Number oe - sto ze size class of data Yy X Y/X A SE (A) B SE (B) (mm) points (kg/m-s) (kg/m-s) (percent) 0061012 61 0.000193 0.000157 123.08 0.654 1.91 0.928 0.073 ; : 001508 000713 211.66 .348 1.70 751 .072 010923 .007283 149.98 .566 1.39 802 .066 013121 .013293 98.70 743 1.34 934 .065 010569 .011828 89.36 498 1.35 868 .064 .006459 007473 89.43 .329 1.38 803 .070 002145 .002286 93.81 192 1.44 739 .058 000565 .000603 93.58 143 1.47 747 .051 .000467 .000839 55.67 .016 8.54 501 .266 TABLE 12.-Summary of statistical data: log-transformed linear regression of transport rate, Helley-Smith sampler versus conveyor-belt sampler (comparison of various modifications to the basic data)" Conditions applied j, (Helley-Smith) = A 7, (conveyor-belt)" [¥ = AX") regess‘ieon Number - - Correlation - _- statistics of data Y X coefficient, Y/X (see notes) points (kg/m-s) (kg/m-s) A B r (percent) * 74 0.041516 0.038041 0.550 0.790 0.881 109.14 ((:4) 74 041516 038041 1.150 1.013 .881 109.14 (* 74 041516 033349 162 .856 .881 124.49 (***) 74 041516 033349 1.594 1.102 .881 124.49 ((*) 74 041023 033349 784 .867 .887 123.01 (*:*. 5) 74 041023 033349 960 $27" _.. 123.01 ('* 61 048285 039375 .846 .885 906 122.63 * 61 034967 032015 .819 .916 .891 109.22 (***) 61 034967 032015 1.016 910 . sully 109.22 (**} 61 048045 039375 .873 .896 908 122.02 (** 3 61 034763 032015 .843 D27 .893 108.58 (t: ** ") 61 034763 032015 1.053 91 _ 108.58 'Total bedload. *Basic data as collected. "Regression treats Helley-Smith data as independent variable. 'Conveyor-belt data corrected to conditions of stream-wide slot. *Helley-Smith data corrected for stage difference with conveyor-belt data. "Includes correction for variance of independent variable in least-squares regression. "Excludes sediment particles smaller than 0.25 mm. "Excludes sediment particles smaller than 0.50 mm. 44 SEDIMENT-TRAPPING CHARACTERISTICS OF THE HELLEY-SMITH BEDLOAD SAMPLER 13.-Summary of statistical data: log-transformed linear regression of transport rate, Helley-Smith sampler versus conveyor-belt sampler (conveyer-belt data corrected for conditions of stream-wide slot) Particle Number j, (Helley-Smith) = A 7, (conveyor-belt)" [¥ = AX") size class of data (mm) points L ___. Var SE Log Y Log X SD (log Y) SD (log X) Log A B 18 (log(A)) (log(A)) Var (B) 0.06- 0.12 -3.1713838 -3.869253 0.430278 0.356737 0.2701 1.030 0.7287 0.100988 0.317786 0.006690 12- .25 -2.821671 -3.212598 280158 258133 -.0341 .868 .6392 .074816 273525 .007203 .25- .50 -1.961669 -2.263078 342941 .319925 .0527 .914 1275 025758 160494 005307 .50- 1.00 -1.882042 -1.941755 .540334 468626 .0899 1.016 1758 020139 141912 005052 1.00- 2.00 -1.975948 -1.992475 .584104 .545829 -.1372 .923 1437 .021207 145626 .004975 2.00- 4.00 -2.189868 -2.191892 584247 584976 -.3323 .848 7200 .025468 189586 .004734 4.00- 8.00 -2.669660 -2.106331 .589595 .645916 -.5720 175 7204 030541 174759 003949 8.00-16.0 -3.248206 -3.285883 .632180 722824 -.6607 .188 8107 032110 179194 .002839 16.0 -32.0 -3.330789 -3.190414 481029 458413 -1.7338 .501 2215 135338 .857518 .070884 Particle: Number j, (Helley-Smith) = A 7, (conveyor-belt)" [¥ = AX") size class of data =s Te so me (mm) Points Y X Y/X A SE (A) B SE (B) (kg/m-s) (kg/m-s) (percent) 0.000193 0.000135 143.03 1.863 2.08 1.030 0.082 .001508 .000613 246.00 .924 1.88 .868 .085 .010923 .006265 174.34 1.129 1.45 .914 .073 .013121 .011435 114.74 1.230 1.39 1.016 .071 .010569 .010175 103.88 .129 1.40 .923 .071 .006459 .006428 100.47 465 1.44 .848 .069 002145 .001966 109.06 .268 1.50 irk .063 .000565 .000518 109.04 .218 1.51 .188 .053 .000467 .000645 72.38 .018 7.20 .501 .266 mus o € ir 1 Ad UNIVER&ITY OF - #--- An Economic Analysis of Selected Strategies . for Dissolved-Oxygen Management: + Chattahoochee River, Georgia By JOHN E. SCHEFTER and ROBERT M. HIRSCH P GEO LOGICAL -_SURYVEY PROEESSION AL «PAPER 1 1 4 0 A method for evaluating the cost effectiveness of alternative strategies for dissolved- oxygen management is demonstrated UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON : 1980 UNITED STATES DEPARTMENT OF THE INTERIOR CECIL D. ANDRUS, Secretary GEOLOGICAL SURVEY H. William Menard, Director Library of Congress Catalog-card No. 80-600113 For sale by the Superintendent of Documents, U.S. Government Printing Office Washington, D.C. 20402 w FWi nkt Aa CONTENTS Page Abstract 1 | Estimation of costs: benefits foregone and waste treatment Introduction E 1 costs Nature of the problem 1 Estimates of benefits, given 1954-56 drought conditions __ Buford Dam 1 Recreation Relationship of flow and dissolved oxygen ___________ 3 Hydroelectric power and peak generating capacity ___ Managing the dissolved-oxygen concentration __________ 3 Estimation of average annual benefits _________________ The range of alternatives F Estimation of added waste-treatment costs _____________ The alternatives considered 4 | The reregulation project Study overview 5 | Least-cost method of producing a given minimum dissolved- The dissolved-oxygen model 5 oxygen concentration Results from the dissolved-oxygen model ______________ 6 Sensitivity of the least-cost solution to the cost of depend- The hydrologic simulation model _._. _---_______-s1_L°°0___ 7 able peak-generating capacity ______________________ \ Operation of the Buford Dam hydroelectric generating Institutional constraints and associated costs ______________ facility: assumptions and definitions _________________ g | Concluding remarks Description of the hydrologic simulation model __________ 1p | References cited Results from the hydrologic simulation model ___________ 11 | Appendix A- Analysis of the reregulation project ___________ ILLUSTRATIONS FicurE 1. Graph showing dissolved-oxygen concentrations at the Fairburn, Ga., station monitor and mean daily discharge at the Atlanta, Ga., station during July 1977 2. Schematic map of the study reach of the Chattahoochee River, Ga 3, 4. Graphs showing: 3. Minimum dissolved-oxygen concentration as a function of minimum flow at the Atlanta, Ga., gage, given that 48 percent of the waste flow receives nitrification, for 1990 conditions 4. Relationships of parameters of aQ, +b=D to percentage of total waste flow receiving nitrification _________________ 5-7. Iso-dissolved-oxygen curves showing combinations of percentage of total waste flow receiving nitrification and minimum dis- charge at Atlanta, Ga., that are predicted to result in minimum dissolved-oxygen concentrations of 3, 4, and 5 mg/L: 5. For 1980 conditions 6. For 1990 conditions 7. For 2000 conditions 8. Diagram showing simulated releases at Buford Dam, Ga., during weeks 33 and 40 of 1954, given a minimum flow at Atlanta, Ga., of 1,290 and 1,600 ft8/S, For 1990 conditions 9-12. Graphs showing: 9. Pool elevations of Lake Sidney Lanier, Ga., over the period of simulation, given that the minimum flow at Atlanta, Ga., is set at 1,290 and 1,600 ft®/s, for 1990 conditions 10. Benefits from recreation on Lake Sidney Lanier, Ga., given various pool elevations 11. The relationship between average annual benefits foregone and minimum flow at Atlanta, Ga., for 1980, 1990, and 2000 12. The estimated annual cost of adding a nitrification process to secondary waste-treatment plants, as a function of the percentage of the total waste flow receiving nitrification IMI Page 13 14 14 14 16 17 18 21 21 28 24 25 25 Page 13 14 15 18 19 IV f CONTENTS FicurkEs 13, 14. Diagrams illustrating: 13. The method of determining the least-cost combination of the percentage of waste flow receiving nitrification and the minimum flow at Atlanta, Ga., required to produce a minimum dissolved-oxygen concentration of 4 mg/L, in 1990 14. The solution for least-cost combinations of percentage of wastes receiving nitrification and minimum flow at Atlanta, Ga., necessary to produce a minimum dissolved-oxygen concentration of 5 mg/L, for 1980, 1990, and 2000 : 15. Graph showing the costs (benefits foregone plus added waste-treatment cost) of attaining various minimum dissolved- oxygen concentrations under different policies for 1990 16. Diagram illustrating the relationship between estimated annual dependable-peaking-capacity benefits and minimum flow at Atlanta, Ga., with and without reregulation, for 1980, 1990, and 2000 TABLES TasLx 1. The expected average daily flow from waste-treatment plants discharging to the Chattahoochee River between Atlanta, Ga., and Whitesburg, Ga., in 1980, 1990, and 2000 2-4. Withdrawals from: 2. Lake Sidney Lanier, Ga 3. The Chattahoochee River, Ga., from Buford Dam to Morgan Falls Dam 4. The Chattahoochee River, Ga., from Morgan Falls Dam to the Atlanta gage 5-7. Hydrologic-simulation-model results: 5. Of water discharge and electric-power production of Buford Dam, Ga., for 1990 conditions _______________________ 6. For years 1980, 1990, and 2000 7. And estimated benefits for 1990, given certain minimum flows at Atlanta, Ga., under 1954-56 drought conditions ___. 8. The average daily flow from waste-treatment plants discharging to the Chattahoochee River between Atlanta, Ga., and Whitesburg, Ga., and the annualized cost of adding a nitrification process to the plants 9. Combinations of percentage of wastes receiving nitrification and minimum flow at Atlanta, Ga., that will provide minimum dissolved-oxygen concentrations of 3, 4, and 5 mg/L at least cost, for 1980, 1990, and 2000 _________________________ 10. Costs of dependable peak generating capacity above which the indicated minimum dissolved-oxygen concentration can be achieved at least cost by operating Buford Dam, Ga., so as not to forego benefits from dependable peak generating capacity _._ 11, 12. Economic-efficiency loss if least-cost combination of the percentage of wastes receiving nitrification and the minimum flow at Atlanta, Ga., is selected under the assumption that the cost of peak generating capacity: 11. Is $23.34/kW/yr but actual cost is $49.35/kW/yr 12. Is $49.35/kW/yr but actual cost is $23.34/kW/yr 13. Percentage of wastes that must receive nitrification to provide minimum dissolved-oxygen concentration of 3, 4, and 5 mg/L at least cost, given that the minimum flow at Atlanta, Ga., is constrained to 860 for 1980, 1990, and 2000 _____ Page 19 21 24 26 Page 11 11 12 12 12 16 18 20 22 AN ECONOMIC ANALYSIS OF SELECTED STRATEGIES FOR DISSOLVED- OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GEORGIA By JOHN E. SCHEFTER and ROBERT M. HIRSCH ABSTRACT A method for evaluating the cost-effectiveness of alternative strategies for dissolved-oxygen (DO) management is demonstrated, us- ing the Chattahoochee River, Ga., as an example. The conceptual framework for the analysis is suggested by the economic theory of pro- duction. The minimum flow of the river and the percentage of the total waste inflow receiving nitrification are considered to be two variable inputs to be used in the production of given minimum concentration of DO in the river. Each of the inputs has a cost: the loss of dependable peak hydroelectric generating capacity at Buford Dam associated with flow augmentation and the cost associated with nitrification of wastes. The least-cost combination of minimum flow and waste treatment necessary to achieve a prescribed minimum DO concentration is iden- tified. Results of the study indicate that, in some instances, the waste- assimilation capacity of the Chattahoochee River can be substituted for increased waste treatment; the associated savings in waste- treatment costs more than offset the benefits foregone because of the loss of peak generating capacity at Buford Dam. The sensitivity of the results to the estimates of the cost of replacing peak generating capaci- ty is examined. It is also demonstrated that a flexible approach to the management of DO in the Chattahoochee River may be much more cost effective than a more rigid, institutional approach wherein con- straints are placed on the flow of the river and (or) on waste-treatment practices. INTRODUCTION This study has two primary purposes: (1) to demonstrate a method of evaluating the cost effec- tiveness of alternative strategies for the management of the concentration of dissolved oxygen (DO) in a river; (2) to demonstrate how the results of a U.S. Geological Survey (U.S.G.S.) River Quality Assessment can be ap- plied within the context of economic analysis to a DO management problem. Results of the U.S.G.S. Chat- tahoochee River Quality Assessment are utilized to estimate the costs associated with selected strategies for maintaining three different minimum DO concentra- tions in the Chattahoochee River between Atlanta, Ga., and West Point Lake, Ga. NATURE OF THE PROBLEM During 1977 the dissolved-oxygen (DO) concentration in the Chattahoochee River at Fairburn, Ga., 25 miles downstream of Atlanta, was less than 5.0 mg/L (milligrams per liter), 10 percent of the time (Stamer and other, 1978). The periods of low DO concentrations occurred primarily in the summer and autumn. During October the DO concentration was less than 5.0 mg/L 31 percent of the time-more often than in any other month. The occurrences of low DO concentrations correspond closely with the occurrences of low discharge of the river. This relationship can be seen in figure 1, which shows (top) the average daily DO concentration at Fair- burn and (bottom) the average daily discharge at Atlan- ta, which is about 1.5 days traveltime upstream of Fair- burn. BUFORD DAM Both graphs in figure 1 display a 7-day periodicity. The periodicity of the Atlanta hydrograph is a conse- quence of the pattern of releases at Buford Dam. Figure 2 is a schematic map of the Chattahoochee River. In this figure, the various impoundments, gages, water-supply withdrawal points and waste-water discharge points of interest to this study are identified and located by river mile. The multipurpose Buford Dam impounds Lake Sidney Lanier, which has a storage capacity of 1.9 million acre- feet at normal pool elevation. In a study of the benefits of the Buford Dam-Lake Sidney Lanier project, the U.S. Army Corps of Engineers (1977) estimated that 74 per- cent of the average annual benefits come from recrea- tion, 17 percent from hydroelectric power, and the re- mainder from flood control, navigation (in the Apalachicola waterway), water supply (for the Atlanta metropolitan area), and low-flow and water-quality maintenance (for the Chattahoochee River from Atlanta to West Point Lake). Buford Dam has an installed hydroelectric generating capacity of 105 MW (megawatts), which is used primari- ly during periods of peak demand. Electricity is generated primarily about 6 hours per day on weekdays. During these peak hours water is released from Lake Sidney Lanier at a rate as high as 10,000 ft*/s, and at 1 ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GA. sof - t AT- T tat -t- pests T- FAIRBURN STATION| MONITOR co I | AVERAGE DAILY DISSOLVED OXYGEN, IN MILLIGRAMS PER LITER 4 l =< BSM cic _ _| fs ti tal C- e _[- St" ( 4000 T o= GT Totar 1 -T ~ 3 T Tr - - -1--T H ATLANTA STATION 3500 3000 |- PEAK HYDROPOWER| RELEASE 2500 2000 1500 I 1000 |- MINIMUM |HYDROPOWER RELEASE DISCHARGE, IN CUBIC FEET PER SECOND 500 |- % Wiss ed- I0 duc d'id 22%>23: 24. .25 26.27 26 29 30 31 1 Peachtree Creek Waste-water 3 discharges Wrede p 369 Fairburn gage --->, Shoals Reach of 06 © Whitesburg gage West Point Lake INDEX MAP INTRODUCTION 3 River miles - 360 [- 340 |- 320 |- 300 |- 280 |- 260 |- 240 |- 220 - 200 Ficurs 2.-Schematic map of the study reach of the Chat- tahoochee River, Ga. (DO, dissolved oxygen.) | other times (morning, lates night, and weekends) the | rate of release is approximately 600 ft?/s. The extreme fluctuation in the flow of the river due to these releases is somewhat dampened by Morgan Falls Dam, located 10 miles above Atlanta, and by the natural attenuation of the flood wave over the 46 miles between Buford Dam and Atlanta. There is some tributary inflow between Buford Dam and Atlanta, but there are also water-supply withdrawals in this reach. The effects of | the low release rates at Buford Dam that occur from late Friday night through midday Monday are somewhat mitigated but are very evident in the Sunday and Mon- day flows at Atlanta. RELATIONSHIP OF FLOW AND DISSOLVED OXYGEN There are three mechanisms whereby increased river discharge may affect the minimum DO in the river. The | first is dilution: higher discharge causes a lower waste concentration, which results in a higher DO concentra- tion throughout the DO sag. The second is a change in re-aeration: higher discharges generally cause less ex- change of oxygen from the air to the water per unit volume of water and result in a lower DO concentration throughout the sag. The third is the decrease in travel time to the shoals, which are located between 30 and 50 miles below Atlanta. Shoals have a pronounced re- aerating ability; the sooner the shoals are reached the less the wastes are able to exert their oxygen demand and, thus, the higher is the minimum on the DO sag. The net effect of these three mechanisms appears to be, both empirically and in model results (Stamer and others, 1978), that higher river discharges lead to higher minimum DO concentrations in the sag below Atlanta. MANAGING THE DISSOLVED-OXYGEN CONCENTRATION Stamer and others (1978) reported that on June 1-2, 1977, when the river flow at the Atlanta gage was 1,150 the minimum DO in the river was 4.0 mg/L and the DO was less than 5.0 mg/L along approximately a 20 mile reach. At that time, the flow of waste water into the river was 185.3 The average concentration of the ultimate biochemical oxygen demand (BOD) of the waste water was 44 mg/L, and the average ammonia- nitrogen concentration was 11 mg/L. A model developed by Stamer and others (1978) predicts that under condi- tions anticipated for the year 2000 and with secondary waste treatment (370 ft?/s of waste water, BOD. concen- trations of 45 mg/L, and an ammonia-nitrogen concen- tration of 15 mg/L) the minimum DO concentration given the same river flow would be 1.1 mg/L and the DO concentration would be less than 5.0 mg/L along a 50 mile reach. This model also predicts the change in the minimum DO given a change in the flow at Atlanta. For 4 ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GA. example, if the flow were 1,800 ft?/s instead of 1,150 ft3/s in 2,000, the minimum DO concentration would be 2.6 mg/L, and a reach of 43 miles would have a DO con- centration less than 5.0 mg/L. The model developed by Stamer and others (1978) also predicted minimum DO concentrations given other degrees of waste treatment. For example, if the BOD: concentration of the waste effluent were 15 mg/L rather | than 45 mg/L and the ammonia-nitrogen concentration were 5 mg/L rather than 15 mg/L, the minimum DO con- centration would be 5.1 mg/L rather than 2.6 mg/L, given a flow at Atlanta of 1,150 ft?/s. These model results clearly indicate that both modification of the hydrograph at Atlanta and modifica- tion of waste inputs from treatment plants located just below Atlanta are possible approaches to manipulating the present and future DO concentrations in the Chat- tahoochee River. THE RANGE OF ALTERNATIVES A number of techniques can be conceived that might be used alone or in combination to manage the DO con- centration in the Chattahoochee River. The techniques include: 1. Improved sewage treatment, so that less water is re- quired in the Chattahoochee River for water-quality maintenance purposes. 2. Construction of a sewage storage facility to hold the sewage for release during peak flows of the river. 3. Construction of a water-supply storage facility so as to permit increased withdrawals from the river dur- ing peak flow periods for use during low flow periods; this would leave more water available for water-quality maintenance during low flow periods. 4. Developing sources of water supply outside of the Chattahoochee River basin, so that more water could be available for water-quality maintenance. 5. Reducing the rates of water use (and, thus, sewage discharge), especially during low flow periods; this reduction could be accomplished by a number of ra- tioning and (or) water-pricing schemes. 6. Dredging Morgan Falls Reservoir so as to increase its capacity and thus permit a more steady flow of the Chattachoochee River at Atlanta without affecting the dependable peaking capacity of Buford Dam. 7. Construction of a reregulation structure (dam and reservoir) between Buford Dam and Morgan Falls Dam so as to permit a more steady flow at Atlanta. 8. Changing the operating procedure of Buford Dam so as to release less water (and generate less electrici- ty) during periods of peak demand for electricity and release more water at other times. The full range of these techniques, both separately and in various combinations, may warrant consideration in the selection of an efficient method of improving the water quality of the Chattahoochee River below Atlanta. THE ALTERNATIVES CONSIDERED To reduce this study to a manageable size, given the resources available, only the following techniques are considered (separately and in combination): 1. Add nitrification to the treatment process at some or all of the treatment plants discharging into the Chattahoochee River or its tributaries between Atlanta and Whitesburg. The effluent concentra- tions given that secondary treatment is used are assumed to be 45 mg/L BOD. and 15 mg/L NH,-N. Adding nitrification is assumed to result in concen- trations of 27 mg/L BOD. and 3 mg/L NH,-N. 2. Dredge Morgan Falls Reservoir and construct a re- regulation structure between Buford Dam and Morgan Falls Dam. 3. Change the operating procedure of Buford Dam so as to give explicit consideration to the release of water from Lake Sidney Lanier for water-quality maintenance purposes. Monetary costs are of course associated with the first and second techniques. Also, a change in the operation of Buford Dam may entail changes in the benefits presently derived from that project. There may be changes in the pool elevation of Lake Sidney Lanier that would affect recreation benefits and the amount of elec- trical energy produced per unit volume of water releas- ed. The relative proportion of high-valued peak power and lower-valued nonpeak (or base) power may change. Most importantly, as more water is reserved for low- flow maintenance less water is dependably available for peak power generation and the dependable peak generating (peaking) capacity of the generators at Buford Dam may change. The loss of this dependable peaking capacity will, it is assumed, entail the construc- tion of peaking facilities elsewhere. Any change in the sum of these benefits as a result of change in the opera- tion of Buford Dam for purposes of maintaining water quality is considered to constitute a cost incurred for that purpose. In this study, an attempt is made to identify the least- cost combination of the three techniques (nitrification, change in the operation of Buford Dam for water-quality maintenance, and improved reregulation) that will achieve a given level of water quality as measured by the DO concentration in the Chattahoochee River. The least-cost combination of the three techniques are iden- tified for three DO-concentration standards, 3, 4, and 5 mg/L, to obtain estimates of the cost (in terms of in- creased treatment costs and benefits foregone) of THE DISSOLVED-OXYGEN MODEL 5 achieving different DO concentrations in the river below Atlanta. Also, the quantity of waste discharged to the river will increase along with the population of the Atlanta region over time. Thus, for any given level of waste treatment and DO standard, the water required for water-quality maintenance will increase with time. For this reason, separate estimates of the costs of the least-cost com- bination of the three techniques are presented for the years 1980, 1990, and 2000. Estimates of the costs do not include any change in the flood-control, navigation, and downstream hydroelectric-power-generation benefits as a result of a change in the operation of Buford Dam. Because the changes in operation considered are relatively minor, in- volving no change in the volume of the flood pool, no change in flood-control benefits would be expected. Navigation and downstream hydroelectric-power benefits would change only as a result of a major change in the seasonal pattern of releases from Buford Dam. The changes in operation of Buford Dam contemplated herein are substantial at the time scale of hours and days but not at the time scale of seasons. The only costs con- sidered are the change in the benefits associated with recreation of Lake Sidney Lanier and generation of elec- tric power at Buford Dam, the cost of adding nitrifica- tion to secondary waste-treatment facilities, and the cost of constructing and dredging reregulating facilities. Just as costs are incurred in achieving or maintaining a given level of water quality in the Chattahoochee River, benefits may also be gained from so doing. Economic-efficiency criteria state that the net benefits to be obtained from an increase in the DO concentration of a river will be a maximum at that level of concentra- tion where the cost of providing the last increment of DO concentration (for example, to 4.6 mg/L from 4.5 mg/L) is just equal to the benefits to be obtained by im- proving the DO concentration by that amount. Estima- tion of the benefits to be obtained by improving the DO concentration of the river is beyond the scope of this study, and no attempt is made to identify that level of DO concentration that will maximize net benefits. STUDY OVERVIEW The model used to relate the minimum flow of the Chattahoochee River at Atlanta, the proportion of the wastes discharged that receive nitrification, and the DO concentration in the Chattahoochee River below Atlanta is described in the next section. This model provides estimates of the combinations of minimum flow at Atlanta and nitrification that will provide a given minimum DO concentration in the river. A hydrologic simulation model that relates the flow of the Chattahoochee River at Atlanta and the pool eleva- tion of Lake Sidney Lanier to the operation and depend- able hydroelectric peaking capacity of Buford Dam is described next. This model also provides estimates of the maximum sustainable minimum flow at Atlanta and thus delimits the combinations of minimum flow and nitrification that are potentially capable of producing a given minimum DO concentration in the river. The methods used to obtain estimates of the change in hydroelectric power and recreation benefits and of the waste-treatment costs are next described. Following this, the method of identifying the least-cost combina- tion of additional waste treatment (nitrification) and flow augmentation is described. Finally, the sensitivity of the least-cost combination to the estimate of the cost of replacing peak generating capacity is explored, and an analysis of the consequences of certain institutional constraints on the cost of attaining a given DO concen- tration is provided. This study does not represent an attempt to prescribe either specific operating rules for Buford Dam or a specific waste-treatment plan for the Atlanta region. This study only provides an examination of the relation- ship (or trade off) between the use of the Lake Sidney Lanier and Chattahoochee River waters for enhance- ment of its DO concentration on the one hand and hydroelectric-power generation on the other. That is, we asked to what extent can the waste-assimilation capaci- ty of the river be substituted for an increased waste treatment with what concomitant decrease in treatment costs and at what cost, if any, in terms of hydroelectric- power and recreation benefits foregone? This question is explicitly posed, and one scheme for exploring it is presented herein. THE DISSOLVED-OXYGEN MODEL Stamer and others (1978) describe a dissolved-oxygen model (DOM) of the Chattahoochee River from the Atlanta gage at river mile (rm) 302.97 to the Franklin gage at rm 235.46. This model is used herein to estimate the minimum DO concentration in this reach as a func- tion of (1) the minimum flow at the Atlanta gage (Qa) and (2) the percentage of total wastes receiving nitrifica- tion (P) in addition to secondary treatment at the sewage-treatment plants along the reach. Model runs were conducted using three different rates of waste-water discharge corresponding to the rates ex- pected for the years 1980, 1990, and 2000. In table 1 is given the name, location (by river mile), and expected flow rate for each of the sewage-treatment plants along the reach. The estimates of the waste water flow rates were based on information published by the Atlanta Regional Commission (Atlanta Regional Commission, 6 ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GA. TABLE 1.-The expected average daily flow in cubic feet per second, from waste-treatment plants discharging to the Chattahoochee River between Atlanta, Ga., and Whitesburg, Ga., 1980, 1990, and 2000 Ri Expected average daily flow iver Plant name mile 1980 1990 2000 Cobb-Chattahoochee _______ 300.56 24 29 31 300.24 131 150 161 South Cobb 294.78 38 51 48 Ufoy Creek ../... ... 291.60 42 46 44 Sweetwater Creek _________ 288.57 ---- 3 3 Camp Creek 283.78 15 22 27 Annewakee Creek _________ 281.46 --- 6 6 Regional Interceptor _______ 281.45 _ ---- --- 42 Bear Creek _- 274.48 --- 7 8 Total: '_Ps _ s l 250 314 370 1977). All waste waters are assumed to have a DO con- centration of 6 mg/L when discharged from the treat- ment plants. In the model, the Chattahoochee River at the Atlanta gage is assumed to have a BOD, concentration of 4.0 mg/L, an ammonia-nitrogen concentration of 0.02 mg/L, and DO at its saturation concentration of 9.3 mg/L. The tributary BOD. concentrations range from 3.0 mg/L to 7.0 mg/L; ammonia-nitrogen concentra- tions, from 0.01 mg/L to 0.12 mg/L; and DO concentra- tions are assumed to be at or near saturation. River water temperatures range from 20.8° C to 27.1° C. All these temperature, BOD«, ammonia, and DO values are based on those observed in June 1977. The model assumes steady flow conditions. Stamer and others (1978) verified that,even though the flows in this reach are often quite unsteady (see fig. 1), their model provides a satisfactory representation of the DO system in a given "parcel" of water as it moves downstream. were conducted so as to provide a basis for the develop- ment of a general expression of the relationship between hoochee River. A typical run may be described as follows: The flow at the Atlanta gage (rm 302.97) is 1,800 ft?/s. The Atlanta | Water Works (rm 8300.62) withdraws 109 ft?/s, leaving a | flow of 1,691 ft?s to the confluence of the Chatta- , hoochee River and Peachtree Creek (rm 300.52). Over the next 26.14 miles from this point, eight waste water treatment plants discharge effluent at the rates specified in table 1. The total flow from these plants is 314 ft?/s. In addition, a total of 93 ft?/s of tributary flow (the 7-day 10-year low flow of each tributary) enters the mainstem over the 65 miles between Peachtree Creek and the Franklin gage. Thus, the flow along the entire reach varies from 1,800 ft?/s down to 1,691 ft?/s and back up to 2,098 at the downstream end. In the particular model run being considered here, seven of the eight treatment plants are assumed to employ only secondary treatment, whereas the R. M. Clayton plant (rm 300.24) employs nitrification in addi- tion to secondary treatment. The flow from the R. M. Clayton plant is predicted to be 150 ft*s in 1990, whereas the total flow from all eight plants is predicted to be 314 ft?/s. Thus, 48 percent of the wastes receive nitrification (P=48). Given that Qa is set at 1,800 ft?/s and that P = 48, the model results show a minimum DO concentration of 5.0 mg/L in the study reach. Another run of the DOM was conducted, identical with the run just described, except that the flow at the Atlan- ta gage was 850 ft*/s (resulting in a flow at Peachtree Creek of 741 ft?/s and a flow at the Franklin gage of 1,148 ft?/s). Given a Qa of 850 ft?/s and a P set at 48, the model estimated a minimum DO concentration of 2.8 mg/L. According to Stamer and others (1978), the relation- ship between minimum DO and flow at Atlanta is very nearly linear (see fig. 3). Thus the results of the two model runs just described may be summarized by an equation of the form aQa + b = D, where Qa is the minimum flow at the Atlanta gage, in. cubic feet per second, and D is the minimum DO over the reach, in milligrams per liter. The equation may be con- sidered valid only for Qa values in or near the range of 850 to 1,800 ft?/s. Inserting the appropriate values for the slope (a) and the intercept (b) results, for the ex- ample described, in 0.0023Qa + 0.89 = D. RESULTS FROM THE DISSOLVED-OXYGEN MODEL Pairs of runs (one for Qa = 1,800 ft?/s; the other for Qa f | = 850 ft?/s) similar to the two just described were con- For 1990, as an example, 14 different runs of the DOM | ducted for a total of seven different cases. In each of these cases, the combination of treatment plants pro- lat10 | viding only secondary treatment and those providing Qa, P, and minimum DO concentration in the Chatta- | | J T T I I I m &n o I D, IN MILLIGRAMS PER LITER, IN THE ATLANTA-TO-FRANKLIN, REACH o- } | | 1 | * 1000 1200 1400 Op, IN CUBIC FEET PER SECOND | 1600 1800 FicurE 3.-Minimum dissolved-oxygen concentration (D) as a function of minimum flow at the Atlanta, Ga., gage (QA), given that 48 percent of the waste flow receives nitrification, for 1990 conditions. THE HYDROLOGIC SIMULATION MODEL T secondary treatment plus nitrification (that is, the value of P) was varied. The results of these 14 runs are presented in the two graphs in figure 4. In figure 4a, the slope parameter (a) is plotted against the percentage of the total waste flow receiving nitrification (P). In figure 4b, the intercept parameter (b) is plotted against P. These figures suggest that both a and b are strongly related to P. The relationship between a and P was ex- pressed by a linear regression (R2 = 0.99) and that be- tween b and P, by a piecewise linear regression. (Each of the two regressions of b on P had an R = 0.99.) The im- plication of these good fits (high R?) is that P is a very good predictor of the relationship between Qa and D as provided by the DOM and that the locations of those sewage-treatment plants chosen to provide nitrification is only minor importance. Thus, in the context of this study, the location of the plants providing nitrification may be ignored, and the treatment levels can be characterized by P -the percentage of the wastes receiv- ing nitrification. The regression lines in figure 4 thus describe the relationship between Qa, D, and P. 0.004 | | | Relationship between a and P o |< Elo 0.003 -| u ac | O ® =|. 0:002 <] uu oc | WJ (Du. a} (€] =| c =|3 0.001 ® o | | | | 5 T I T T Relationship between 5 and P b b, MILLIGRAMS PER LITER g 1 | | 1 40 60 so PERCENTAGE OF TOTAL WASTE FLOW RECEIVING NITRIFICATION (P) FIGURE 4. - Relationship of parameters of aQa +b=D to percentage of total waste flow receiving nitrification (P). 100 Figure 6 provides a useful graphical description of this relationship. It shows the combination of treatment (P) and minimum flow (Qa) necessary to achieve a minimum DO concentration (D) of either 3, 4, or 5 mg/L. These curves are denoted "iso-DO" (iso-dissolved-oxygen) curves. The same procedure as that just described was used to approximate the relationship between D, P, and Qa for the years 1980 and 2000. The results are depicted in figures 5 and 7. Note that P refers to the percentage of the total waste flow receiving nitrification and that this total increases with time. The consequences of the expected increase in waste water flow can be seen by comparing the required amount of nitrification in 1980, 1990, and 2000 given, for example, Qa = 1,500 ft*/s and D = 4 mg/L. It is estimated that, given these conditions, the percentage of the total waste effluent that must receive nitrification would increase from 22 in 1980, to 36 in 1990, and to 48 in 2000. Since the total waste flow is increasing (see table 1), this means that the flow receiving nitrification would have to be 55 ft?/s (0.22 x 250) in 1980, 118 ft?/s (0.36 x 314) in 1990, and 178 ft?/s (0.48 x 370) in 2000. Given any minimum DO standard (D*), the combina- tion of P and Qa selected must lie on or above the iso-DO curve representing D* milligrams per liter. But, not all combinations of P and Qa along these iso-DO curves are technically feasible. For example, from figure 6, setting Qa equal to 1,800 ft?/s and P equal to 50 will provide a minimum DO concentration of 5 mg/L in 1990. As will be seen, it is not possible to sustain this minimum flow at Atlanta under all hydrologic conditions. In addition, it is necessary to associate a cost with each combination of P and Qa so as to permit identification of the least-cost combination. This cost is related in part to the minimum flow at Atlanta, which in turn is related to the operation of Buford Dam. The hydrologic simulation model (HSM) used both to identify the feasible values of Qa and to pro- vide a basis for estimating the costs (benefits foregone) associated with these values is presented in the next see- + tion. THE HYDROLOGIC SIMULATION MODEL The hydrologic simulation model (HSM) was developed to determine, under a set of assumptions that shall be specified, the pattern of releases from Lake Sidney Lanier that are necessary to achieve a given dependable minimum flow at the Atlanta gage (Q»). The pattern of release has effects on the benefits associated with each of the project purposes, and the HSM is designed to provide a basis for estimating the change in the project benefits as a result of a change in the pattern of release. 8 ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GA. 1800 1700 1600 1500 --- s] 1400 1300 1200 1100 Qa, IN CUBIC FEET PER SECOND 1000 900 |-- I I I I I I | Cee | | | | 1 | 0 10 20 30 40 50 60 70 80 90 100 PERCENTAGE OF TOTAL WASTE FLOW RECEIVING NITRIFICATION (P) FIGURE 5. - Iso-dissolved-oxygen curves showing combinations of the percentage of total waste flow receiving nitrification (P) and minimum dis- charge at Atlanta, Ga., (Qa) that are predicted to result in minimum dissolved-oxygen concentrations (D) of 3, 4, and 5 mg/L, for 1980 conditions. 1800 1700 1600 1500 1400 1300 1200 Qa, IN CUBIC FEET PER SECOND 1100 1000 900 | 1 | 0 10 20 30 40 50 60 70 80 90 100 PERCENTAGE OF TOTAL WASTE FLOW RECEIVING NITRIFICATION (P) FIGURE 6. - Iso-dissolved-oxygen curves showing combinations of the percentage of total waste flow receiving nitrification (P) and minimum dis- charge at Atlanta, Ga., (Qa) that are predicted to result in minimum dissolved-oxygen concentrations (D) of 3, 4, and 5 mg/L, for 1990 conditions. THE HYDROLOGIC SIMULATION MODEL 0 The key relationship that the HSM describes is that between the dependable minimum flow at Atlanta and the dependable hydroelectric peak generating (peaking) capacity of Buford Dam. The "amount" of each of these "products" that can be dependably provided by the Buford Dam project is a function of the inflows to Lake Sidney Lanier and tributary flows to the Chattahoochee River above Atlanta over an extended (at least two- year) drought. The meaning of the word "dependable" is of para- mount importance to an understanding of the HSM. Dependable minimum flow is defined as that rate of flow that can be provided at all times throughout a period in which the flows (for example, both into Lake Sidney Lanier and tributary flow between Buford Dam and Atlanta) are those that occurred in the most severe extended drought in the historic record. Similarly, dependable peaking capacity is defined as that peaking capacity that can be provided at all times throughout a period in which the flows are those that occurred in the most severe extended drought in the historic record. The most severe extended drought occuring in the study area during the 49-year historic record was a 132 week period comprising June 1954 through December 1956. As there is no reason to believe that a more severe drought will not occur in the future, that which is defined as "dependable" herein may not be "dependable" in the future. Rather than attempt to estimate the prob- ability of more severe droughts or to justify this defini- tion of "dependable" on some economic grounds, it is ac- cepted simply on the basis that previous studies of the Buford Dam project and of the Chattahoochee River (U.S. Army Corps of Engineers, 1977; Atlanta Regional Commission, 1977) have relied on the same convention. OPERATION OF THE BUFORD DAM HYDROELECTRIC GENERATING FACILITY: ASSUMPTIONS AND DEFINITIONS Though Buford Dam has an installed hydroelectric generating capacity of 105 MW, the rate of production of electrical energy varies with the pool elevation of the reservoir and with the rate of flow of the water past the turbines; that is, it varies with the pattern of releases from Lake Sidney Lanier. The calculation of hydroelectric-power production is based on the follow- ing formula (Joe DeWitt, U.S. Army Corps of Engineers, Savannah District, oral commun. 1978): Pe=82.645(0.12390 + 0.000925 (E - 1,055)) Q, where Pe =power, in kilowatts, E =pool elevation, in feet above sea level, and Q - =flow through the powerplant, in cubic feet per second. It is assumed that all water released from Lake Sidney Lanier is used for the production of electric energy. The HSM is designed to first pattern the release of water from Lake Sidney Lanier so as to maximize the dependable summer-peak generating capacity of Buford 1800 1700 1600 |- 1500 1300 1200 1100 Qa, IN CUBIC FEET PER SECOND 1000 900 1 I I | 0 10 20 30 40 PERCENTAGE OF TOTAL WASTE FLOW RECEIVING NITRIFICATION (P) 50 60 70 80 90 100 FicurE® 7. - Iso-dissolved-oxygen curves showing combinations of the percentage of total waste flow receiving nitrification (P) and minimum dis- charge at Atlanta, Ga., (Qa) that are predicted to result in minimum dissolved-oxygen concentrations (D) of 3, 4, and 5 mg/L, for 2000 conditions. 10 Dam. Given that this has been accomplished, the model allocates the release of water within any given week so as to maximize the peak energy production. Both of these maximizations are conducted subject to the con- straints that the given downstream water-supply needs and minimum flow at Atlanta (Qa) are satisfied. | | | Definitions used for the HSM are as follows: Peak energy. All electric energy generated between 2 p.m. and 8 p.m. | on weekdays. | Nonpeak base energy. All electric energy other than peak energy. | Dependable peak generating (peaking) capacitfi. The minimum rate | of electric-energy production during the peak hours of the summer | periods of the 132 week simulation period. | Summer. Early June (week 22) through late September (week 33). To understand the design and assumptions of the | HSM, it is helpful first to understand the intertemporal | distribution of the demand for electric energy. The | quantity of electric energy demanded generally reaches a peak during the afternoon and early evening on | weekdays and falls to a low during the early morning | hours and on weekends. Though the "height" of these | peaks varies throughout the year, the peak demand for | electric energy is typically the greatest during the sum- | mer months. The electric-utility companies attempt to | maintain sufficient generating capacity to meet the | maximum peak demand, which will occur typically dur- | ing the afternoon or evening of a summer weekday. _ | Hydroelectric turbines are especially useful for peak- 1 ing purposes as they require very little startup time and can be brought online quickly. Because of this capability, | the limited water available is not generally used to pro- | duce base power, except when water must be released to | meet downstream needs or to vacate the flood pool. _ | The release of water from Lake Sidney Lanier is | therefore assumed to be patterned so as to maximize the | dependable summer peaking capacity of Buford Dam, | for it is during the summer that the electric-utility com- | pany (the Georgia Power Co.) that purchases power | from the dam is most likely to require maximum | generating capacity. If no releases are necessary (for ex- ample, when tributary flows are high and the pool eleva- | tion of Lake Sidney Lanier is below 1,070 feet above sea | level), it is assumed that no base electric energy is pro- duced. Consequently, it is assumed that Buford Dam provides no dependable base generating capacity. DESCRIPTION OF THE HYDROLOGIC SIMULATION MODEL The HSM is designed to answer the following ques- | tions: 1. What is the range of minimum flows at the Atlanta gage (Qa) that could have been achieved under the 1954-56 drought hydrology? | 2. Given a minimum flow at Atlanta (Qa), what plan of | operation of Buford Dam will maximize the depend- able peaking capacity of the dam? ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GA. 3. What is the dependable peaking capacity of Buford Dam, given this plan and Qa? 4. What is the peak and nonpeak electric-energy pro- duction, given this plan and Qa? 5. What is the history of pool elevations of Lake Sidney Lanier, given this plan and Qa? The HSM is designed to find a plan of releases from Buford Dam and Morgan Falls for the 132-week period that maximizes the dependable peaking capacity of Buford Dam subject to the following flow and storage constraints: dS;,/dt=T;i- Wi-Qi, (1) Q1 < 10,000, (2) Si(to) = 8.835 x 109, (8) x 109, (4) 81 <8.35x 10", (5) Qi +T;> Ws, (6) ds;/dt=Q1+T;- Wa- Q2, (7) Sz(to) = 0, (8) S3(t) > 0, - (9) S, < 1.09 x 108, and (10) Q: +Ts;- Wsz Qa. (11) The decision variables are all time varying, defined for values > 0 and are as follows: S1 - =storage in Lake Sidney Lanier, in cubic feet; S;, - =storage in Morgan Falls Reservoir, in cubic feet; Qi - =release from Buford Dam (Lake Sidney Lanier), in cubic feet per second; and Q2 - =release and spill from Morgan Falls Dam, in cubic feet per second. The initial storage conditions, beginning of week 22, 1954, are Si(t,) =initial storage in Lake Sidney Lanier, in cubic feet, and S2(t,) = initial storage in Morgan Falls Reservoir, in cubic feet. | The final storage conditions, end of week 49, 1956, are S(t) =final storage in Lake Sidney Lanier, in cubic feet, and S;(t) =final storage in Morgan Falls Reservoir, in cubic feet. The time-varying model parameters are Ti - =inflows to Lake Sidney Lanier, in cubic feet per second (constant over a week; values are those used in U.S. Army Corps of Engineers, 1977); T; - =tributary inflows between Buford Dam and Morgan Falls Dam, in cubic feet per sec- ond, (constant over the week; values equal one half of the tributary flow values reported in U.S. Army Corps of Engineers, 1977); THE HYDROLOGIC SIMULATION MODEL T3 tic Gage, in cubic feet per second (constant | 2. = tributary inflows Morgan Falls.to the Atlan- | 1. 11 end-of-week pool elevation for each week; release rate and power production for the 30 peak over the week, values equal one half of the tributary flow values reported in U.S. Ar- my Corps of Engineers, 1977); Wi cubic feet per second (constant over week; varies with time of year and year of analysis; see table 2); = withdrawals from the Chattahoochee River, Buford Dam to Morgan Falls Dam, in cubic feet per second (constant over week; varies with time of year and year of analysis; see table 3), and = withdrawals from the Chattahoochee River, Morgan Falls Dam to the Atlanta gage, in cubic feet per second (constant over week, varies with time of and year of analysis; see table 4). The time constant model parameter is Qa - =Minimum flow at the Atlanta gage, in cubic feet per second. The model constraints are as follows: W» Ws 1. the continuity equation for Lake Sidney Lanier; 2. limitation on release from Buford Dam, 10,000 ft3/s, the channel capacity below the dam; initial storage conditions for Lake Sidney Lanier, equal to initial storage for the same period in the U.S. Army Corps of Engineers base plan of operations; final storage conditions for Lake Sidney Lanier, equal to final storage for the same period in the U.S. Army Corps of Engineers base plan of operation. capacity constraint for Lake Sidney Lanier, 8.35 x 10° ft3, corresponds to pool elevation of 1,070 ft. above sea level (normal pool elevation); flows in the Buford Dam to Morgan Falls Dam reach, must be greater than or equal to the withdrawals in the reach at all times; the continuity equation for Morgan Falls Dam; . initial and final storage in Morgan Falls Res- ervoir (arbitrary); 8. storage, and flows at Atlanta gage, must be greater than or equal to the specified minimum flow Qa, at all times. RESULTS FROM THE HYDROLOGIC SIMULATION MODEL The results of any run of the HSM, where a run is specified by a choice of years (1980, 1990, or 2000) and a choice of Qa values, are the values of the following variables: 11. =withdrawals from Lake Sidney Lanier, in | capacity constraint on Morgan Falls Reservoir | hours in each week; release rate and power production for the 72 nonpeak weekday hours in each week; and release rate and power production for the 66 weekend (nonpeak) hours in each week. These results are summarized as total nonpeak energy, total peak energy, and dependable peaking capacity. To illustrate the results of the HSM, two examples are described. Both are based on water-supply withdrawals estimated for the year 1990 (tables 2, 3, and 4). In the first case, the required minimum flow at Atlanta (Qa) is 3. 4. | set at 1,290 ft?/s. Values for two different weeks of operation are considered in detail in this comparison: those of week 33 (mid-August), 1954, and week 40 (early October), 1954. In both weeks, the tributary flows (T» and T;) were equal to zero. The releases and hydroelectric-power production under each run are given in table 5. The release patterns for these weeks are shown in figure 8. Comparison of the two cases brings out two important points about the consequences of increasing the required minimum flow at Atlanta. The first is that the releases from Buford Dam are redistributed with respect to the TABLE 2. - Withdrawals, in cubic feet per second, from Lake Sidney Lamier, Ga. Year Weeks 1980 1990 2000 :.. s.. _ ices 12.6 23.6 77.3 TOT Taran lone en chee 13.3 24.9 81.9 Tesao Lene cence cali 14.0 26.2 86.0 SAU ean ene . s 14.7 27.5 90.2 P TBEDD Ine nana wes 15.5 29.0 95.3 S600 __ 14.0 26.2 86.0 AOZAA LLL n 13.3 24.9 81.9 45=B2 Le. 12.6 23.6 77.3 AVETAGC .-.. ack 13.6 25.5 83.7 TABLE 3. - Withdrawals, in cubic feet per second, from Chattahoochee River, Ga., Buford Dam to Morgan Falls Dam Year 1990 2000 1s18 _ 114.9 160.4 375.7 A41 T 2 ole bae ena r 121.6 169.8 397.7 Lon nee eer 127.7 178.3 417.6 S20 en onan PLE LL tach 134.0 187.1 439.3 T OD oo one reece eo io eles nnd 141.5 197.6 462.7 S0 _.. l ec ae c anes mice 127.7 178.3 417.6 40=44 {-c 2 aL ke 121.6 169.8 397.7 4D=-D2 ) .. ete a ens r mee 114.9 160.4 375.7 Average .wzl.Li_al..... 124.3 178.6 406.6 12 TABLE 4. - Withdrawals, in cubic feet per second, from Chattahoochee River, Ga., Morgan Falls Dam to the Atlanta gage Year 1990 Weeks 1980 2000 1+18 "L_. 33.3 44.4 107.8 14-17 n. __ 85.2 46.9 114.1 ELO 37.0 49.3 119.9 r S cM an ee paring in bra ane bn ire wae 38.8 51:7 125.8 L T TOD ae ai n iol cn a ie ar mae erie 41.0 54.6 132.8 50-50 Leceellccl 37.0 49.3 119.9 40244: ___ 85.2 46.9 114.1 4B-BAL__LazLLJOL_L__L_ZC 88.3 44.4 107.8 AVETAUE zl 36.0 48.0 116.7 TaBur 5.-Hydrologic-simulation-model results of water discharge and electric-power production at Buford Dam, Ga., for 1990} conditions | ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIV.ER, GA. drawdown during the summer-recreation season than will a high Qa. From the standpoint of recreation, a plan of operation with Qa = 1,600 ft?/s has a more desirable result than does a plan with Qa = 1,290 ft?/s. After running the HSM for a range of different Qa values for any given year, two values of Qa emerge as having special significance. The first of these is the max- imum sustainable Qa value (1,600 ft*/s for 1990). It is, of course, the maximum sustainable Qa only under the specific assumption of the HSM. In particular, it is re- quired that all water-supply requirements be met and that, under the 1954-56 drought hydrology, the minimum storage in Lake Sidney Lanier is not allowed to fall below 1.07 million acre-feet (pool elevation 1,0483.9 ft), which is 56 percent of the storage at normal | pool elevation (1,070 ft). Minimum flow at the Atlanta gage (Qa) _____________________ 1,290 ft?'s 1,600 ft?s Week 33, 1954 Discharge, in cubic feet per second: Average 2,290 1,780 During: Peak hours Lt 10,000 6,480 Nonpeak 198 198 Weekends i 1,070 1,380 Electric-energy production, in megawatt-hours: Total 4,270 3,500 Peak 3,330 2,320 Nonpeak, weekdays _____________________ 160 160 Nonpeak, weekends 780 1,020 Week 40, 1954 Discharge, in cubic feet per second: Average 1,510 1,820 During: Peak hours 5,760 6,810 Nonpeak hours 170 170 Weekends ...: _. 1,030 1,340 Electric-energy production, in megawatt-hours: Total 2,720 3,360 | Peak 1,860 2,250 | Nonpeak, weekdays __________-__________ 130 130 | Nonpeak,- weekends 730 980 time of the week: weekend flows increase and peak, flows either decrease (if summer) or increase (if nonsum- mer). The second point is that releases are redistributed with respect to time of year: weekly average flows dur- ing the summer season decrease, and flows during the' remainder of the year increase. In figure 9 is depicted the 132-week record of simulated pool elevations for these two cases. Given that Qa is set equal to 1,600 ft?/s, the pool elevation varies less throughout each year and tends to be higher during the summer months. When Qa is low, less water need be saved for flow maintenance in the autumn, and thus more may be used for summer-peak power production. Consequently, a low Qa will result in more reservoir The other value of Qa that is of interest is that value below which no additional dependable peaking capacity can be gained by further decreasing Qa. For example, this value is 1,290 ft?/s for 1990. Given this minimum flow requirement, it is possible to fully utilize the generating turbines with a release of 10,000 during all summer peaking hours. The dependable peaking capacity in this case is equal to the generating capacity for a flow of 10,000 ft?/s and a pool elevation of 1,043.9 ft (the minimum pool elevation for the three summers of the simulation period). These two values of Qa and the associated values for dependable peaking capacity are given for each of the three years in table 6. The HSM, then, was used to delimit the feasible range of Qa as it identified the maximum sustainable Qa. It also provided estimates of dependable peak generating capacity, weekly peak and nonpeak power production, and the weekly pool elevation of Lake Sidney Lanier | upon which to base estimates of the change in benefits given a change in Qu. Tasux 6. -Hydrologic-simulation-model results, for 1980, 1990, and 2000 (E LL nle cnr -l neben 1980 1990 2000 Maximum sustainable value of Qq" ____ 1670 1600 1230 Dependable peaking capacity at maxi- mum sustainable Q@a* ._______-____ 66 66 58 Value of Q) associated with maximum dependakfie peaking capacity" ______ 1380 1290 870 Maximum value of dependable peaking caparity!: _-_ "% __ L2 98 98 97 ! In cubic feet per second. * In megawatts. ESTIMATION OF COST: BENEFITS FOREGONE AND WASTE-TREATMENT COSTS ESTIMATION OF COSTS: BENEFITS FOREGONE AND WASTE-TREATMENT COSTS In this section, the method used to obtain estimates of | the recreation and hydroelectric benefits associated 1 with a given Qa under (here, 1954-56) drought condi- | tions is described first. Then, it is argued that these | drought condition benefits are not representative of the ‘ benefits associated with any given Qa under more nearly | average hydrologic conditions, and the method used to | approximate average annual benefits is described. | These estimates of the benefits associated with a given | Qa permit estimation of the costs, in terms of benefits foregone, associated with a change in Qa and, thus, with ' a change in the operation of Buford Dam. | Also described is the method used to obtain estimates 1 13 1 of the cost of adding a nitrification process to secondary waste-treatment facilities and, thus, of increasing the {greentage of the total waste flow receiving nitrification (P). It was necessary to select an interest, or discount, rate with which to amortize both the benefits of the hydroelectric peak generating capacity of Buford Dam and the capital cost of adding a nitrification process to the waste-treatment facilities. If the peak generating capacity of Buford Dam is diminished by operating rules requiring releases from Lake Sidney Lanier for water- quality maintenance purposes, this lost capacity will x have to be replaced (it is assumed) by an electric utility company in the private sector of the economy. The Georgia Power Co. is currently constructing a hydroelectric pump-storage peaking facility (its "Rocky 10,000 -y m § § ¢. | N \a RN bY § ¢ I Y NN 8 mw ¥ Y YS 2 § § § g RN N N RN DY Py DY § f § § s. $ m § § § N § I y" § a § §! § $CC § > g § G § & § - § res Nt Dy N A D N D N 9 § C R gq § p *I T [ | nle < § _C Vs § J Y. § t 2000 -R Q S S -N Q § Q 3 Y .Y I § Y °- § - G § - LO LLL... g k SX Q §X\\J \W:\ \ 3 2 3 1 1: 1.2 :g 29094 3 :.} fs 5 1 tor 1 t jina r 22 ? aaedd < P aA 'P A P AP. aA PAP A/P A. P $ a pA par aA P Alp aA p asp 7 > Mon. - Tues. - Wed. Thurs. _ Fri. Sat. - Sun. - Mon. Mon. - Tues. Wed. Thurs. Fri. Sat. _ Sun. _ Mon. Day of week I WEEK 33, Q,=1290 CUBIC FEET PER SECOND WEEK 33, Qa=1600 CUBIC FEET PER SECOND $ & 8000 - ~ co E C o © 6000 - -I N tn § < PN a - 4000 - "4 N o \ S 2000 - * s PN W RN 0 DY N A \\\ N §K\ S 2 a j 2.00% 2:52 2 2 2 b G acr 1 9 akg m 9 2 p jar 1... pA PA P A P oA IP A P.A PsA P PAPAPAPAPAPAPAPT'meOfdaV Mon. Tues. - Wed. Thurs. Fri. Sat. - Sun. Mon. Mon. - Tues. Wed. Thurs. _ Fri. Sat. - Sun. _ Mon. Day of week WEEK 40, Q, = 1290 CUBIC FEET PER SECOND WEEK 40, Qa= 1600 CUBIC FEET PER SECOND FIGURE 8.-Simulated releases at Buford Dam, Ga., during weeks 33 and 40 of 1954, given a minimum flow at Atlanta (Qa) of 1,290 ft*/s and 1,600 ft3/s, for 1990 conditions. 14 ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GA. Mountain Project") and is amortizing the capital cost of this facility using a discount rate of 11.24 percent (C.R. Thrasher, Georgia Power Co., written commun., June 5, 1978). Though the choice of a discount rate is somewhat subjective and requires a value judgment, a rate of 10 percent was chosen as being indicative of the opportuni- ty cost of capital in the private sector of the economy. All estimates of benefits and costs are presented in terms of first-quarter 1976 dollars. ESTIMATES OF BENEFITS GIVEN 1954-56 DROUGHT CONDITIONS RECREATION Estimates of the benefits from recreation at Lake Sidney Lanier are based on data obtained from a U.S. Army Corps of Engineer publication (1977). According to the Corps of Engineers, the recreation benefits ob- tained from Lake Sidney Lanier vary with both the pool elevation of the reservoir and the season of the year. They have published (1977) estimates of both the peak- and the offpeak-season recreation associated with pool elevations ranging from 1,055 to 1,080 feet above sea level. For example, the Corps of Engineers estimated that a pool elevation of 1,070 feet has associated peak- season benefits of $17,820,900 and offpeak season benefits of $13,011,100. For purposes of this study, it was assumed that the peak-season benefits were distributed uniformly over the 22 weeks from May 1 through September 30 and that the offpeak-season benefits were uniformly distributed over the 30 weeks from October 1 through April 30. Thus, a pool elevation. of 1,070 feet would have associated with it recreation benefits of $810,041 per week during the peak season and $433,703 per week during the offpeak season. The weekly recreation benefits associated with each pool elevation are graphed in figure 10. The HSM provided the weekly pool elevation of Lake Sidney Lanier given that Buford Dam was to be operated so as to achieve a specified minimum flow at Atlanta. The weekly recreation benefits associated with each of the weekly pool elevations were summed over the 132 weeks of the simulation period and averaged to obtain an estimate of the average annual recreation benefits from Lake Sidney Lanier (under 1954-56 drought conditions) given a specified minimum flow at Atlanta. HYDROELECTRIC POWER AND PEAK GENERATING CAPACITY To place a dollar value on the generating capacity of, and electric energy produced at, Buford Dam, it is necessary to ask: what is the least-cost method of pro- ducing an equivalent amount of electric energy by an alternative technique and what is the cost? A detailed in- vestigation of alternative techniques and their associated costs is beyond the scope of this study, but it is necessary to briefly discuss some of the details involved in such an investigation. 1075 | 1070 1065 1060 1055 1050 1045 POOL ELEVATION, IN FEET ABOVE SEA LEVEL 1040 1954 1955 1956 YEAR FicurE 9.-Pool elevation of Lake Sidney Lanier over the period of simulation, given that the minimum flow at Atlanta, Ga., (QA) is set at 1,290 ft?/s and 1,600 for 1990 conditions. ESTIMATION OF COSTS: BENEFITS FOREGONE AND WASTE-TREATMENT COSTS 15 For purposes of analysis, it is useful to separate the cost of producing electric energy into two components: the capacity cost and the energy cost of production. The energy cost consists of the fuel (for example, coal) cost of producing a unit (for example, a kilowatt-hour) of electric energy. The capacity cost stems primarily from the capital investment in the generating facility. If an electric-utility company is to invest in a generating facility, it must receive a rate of return on its investment at least equivalent to that which could have been earned if the money had been invested elsewhere; this is the so- called opportunity cost of capital and is determined by the interest or discount rate. The initial capital cost and useful life of a generating facility, along with the dis- count rate, are the main determinates of the capacity cost of producing electric energy at that facility. As it is currently operated, Buford Dam is used primarily for the generation of electric energy during periods of peak demand. Though it has been assumed herein that the dam provides no dependable base generating capacity, it does produce some energy dur- ing nonpeak hours because water is sometimes released 900 , I 800 |- 700 Peak-season S. (Weeks 18-39) -_ 500 [- 400 |- I 300 Off-peak season (Weeks 1-17 and 40-52) Jc 200 |- WEEKLY RECREATION BENEFITS, IN THOUSANDS OF DOLLARS 100 [- 6 | | | | | I 0 1055 1060 1065 1070 1075 1080 POOL ELEVATION OF LAKE SIDNEY LANIER, IN FEET FIGURE 10. - Benefits from recreation on Lake Sidney Lanier, given various pool elevations. during these hours to satisfy downstream flow re- quirements. Any nonpeak energy produced at Buford Dam has an energy value equivalent to the cost of pro- ducing it by some least-cost alternative method. Similar- ly, the electric energy produced during peak periods has an energy value equivalent to the energy cost of produc- ing it by some least-cost alternative. To assign a capacity value to the generating capacity of Buford Dam and an energy value to the electric energy produced there, it is necessary to make an assumption as to the least-cost alternative source of capacity and energy. It was assumed that any peaking capacity lost at Buford Dam because of a change in its operating rules could be replaced by a facility similar in cost to the Georgia Power Co.'s 675-MW "Rocky Moun- tain" facility, which is scheduled to come online in 1983. Using data obtained from the Georgia Power Co. (C.R. Trasher, Georgia Power Co., written commun., June 5, 1978) and assuming a 10-percent discount rate, it is estimated that the capacity cost of electric energy pro- duced by this pump-storage facility will be $23.34/kW/yr (in first-quarter 1976 dollars). The dependable peaking capacity of Buford Dam was assigned this value. Electric energy produced at Buford Dam was assigned different values depending upon whether it was pro- duced in a period of peak demand or in a period of base demand. According to estimates provided by the Atlan- ta Regional Office of the Federal Power Commisison to the U.S. Army Corps of Engineers (1977), the energy cost of electricity produced by coal-fired thermal electric powerplants in the Atlanta area was 7.75 mills/kWh dur- ing the first quarter of 1976. Because any electricity pro- duced at Buford Dam during periods of base demand could be substituted for electricity produced by coal- fired thermal electric plants, the base electricity pro- duced at the dam was assigned an energy value of 7.75 mills/kWh. However, if peak electricity produced at Buford Dam is to be replaced by electricity generated at a facility similar in cost to the Georgia Power Co.'s "Rocky Mountain" facility, such electricity must be assigned a higher energy value. The Georgia Power Co. estimates that 1.4 kWh of electricity must be expended in pumping for storage (in offpeak periods) to generate 1.0 kWh of electricity in peak periods (Georgia Power Co., 1972). Given that base-period electricity has an energy cost of 7.75 mills/kWh, then peak-period elec- tricity furnished by the "Rocky Mountain" pump-storage facility will have an energy cost of 10.85 mill/kWh ( =7.75 mills/kWh x 1.4). Accordingly, peak-period electricity produced at Buford Dam was assigned an energy value of 10.85 mills/kWh. It should be noted that the U.S. Army Corps of Engineers has assumed that the alternative to produc- ing peak energy at Buford Dam is to produce it by a coal- 16 TaBue 7. -Hydrologic simulation-model results and estimated benefits for 1990, given certain minimum flows at Atlanta, Ga., under 1 954- 56 drought conditions. Minimum flows, in cubic feet per second onun 120 0 beet PR WHIREC -.. cee nee s ut ket sen esau enlil. Confluence of Chattahoochee River and Peachtree 1,290 1,600 1,180 1,490 Average energy output, in megawatt-hours per year PIAA! aree eer enne nee been aan oe a 132,500 134,400 Annual nonpeakc 34,500 45,800 Annual peak c_ T_ c 98,000 88,600 Dependable peaking capacity, in megawatts (See minimum flows given for Atlanta) _ 98.1 65.9 Benefits, in millions of dollars per year Nunpeak energy 21 .35 Peak energy c 1.06 .96 Dependable peaking capacity ________ 2.29 1.54 Recreation 10510001 ___ ;_ 22 i_ 24.38 25.00 28.00 27.85 Total }} fired thermal electric powerplant. Using estimates pro- vided by the Atlanta Regional Office of the Federal Power Commission, the Corps of Engineers valued the dependable generating capacity of Buford Dam at $49.35/kW/yr. They assigned an energy value of 7.75 mills/kWh to electric energy produced at the Dam (U.S. Army Corps of Engineers, 1977). The sensitivity of the results of this study to the value assigned to dependable peak generating capacity is examined in a following see- tion. Given results of any run of the HSM, it is possible to compute the estimated annual energy benefits and dependable-peaking-capacity benefits (under the assumed drought conditions) associated with a par- ticular Qa. Energy benefits were calculated as the sum of average annual peak energy production multiplied by its value (10.85 mills/kWh) plus average annual nonpeak energy production multiplied by its value (7.75 mills/kWh). Dependable peaking-capacity benefits are equal to the dependable peaking capacity times its value ($23.34/kW/yr). In table 7 is summarized the results of the HSM runs and the benefit calculations for the two cases (Qa = 1,290 ft?/s and Qa = 1,600 ft?/s) described in the previous see- tion. Given 1990 water-supply requirements and the drought conditions, the effects on annual benefits as a result of changing Qa to 1,600 ft?/s from 1,290 ft?/s are nonpeak-energy benefits increase by 30 percent, peak- energy benefits decrease by 9 percent, dependable- peaking-capacity benefits decrease by 33 percent, and recreation benefits increase by 3 percent. Total benefits ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GA. are decreased by one-half of one percent. In terms of benefits foregone, the cost of increasing Qa to 1,600 from 1,290 ft?/s in 1990 is estimated to be $150,000/per year. ESTIMATION OF AVERAGE ANNUAL BENEFITS The method of estimating the benefits derived from Buford Dam under different operating rules given 1954-56 drought conditions is described in the preceding section. It is necessary to specify drought con- ditions to obtain an estimate of the maximum sus- tainable Qa and of the dependable peaking capacity associated with each Qa. It is not appropriate, however, to base an estimate of average annual benefits on worst- case (drought) conditions. The estimates of the average annual benefits to be ob- tained under different minimum flows at Atlanta would be more appropriately based on a simulation of dam operations over the entire available hydrologic record, including the worst-case drought. Such a simulation, however, would be an extended task. Also, only the _ change in average annual benefits (that is, benefits foregone) as a result of a change in Qa is of interest here. Thus, the estimates of the benefits foregone associated with a change in Qa are based on the simplifying assumption that the change in average annual benefits due to a change in Qa is solely the result of the associated change in the dependable peaking capacity of Buford Dam. From table 7, note that the sum of peak-energy, nonpeak-energy, and recreation benefits increases with Qs. Conversely, dependable-peaking-capacity benefits decrease with an increase in Qa. This offsetting relation- ship does not hold for years of more nearly average or above average flows. In any year, base- and peak-energy benefits and recreation benefits are a function both of the flows in that year and of Q. But, dependable-peaking-capacity benefits are a function only of Qa since they are deter- mined only on the basis of the limiting (1954-56) hydrologic conditions. When water is more plentiful, setting Qa at a high value (1,600 ft?/s) rather than a low value (1,290 ft/s) does not have much effect on reservoir operations or on benefits. With plentiful water, it becomes possible to simultaneously satisfy the objec- tives of maximizing peak-energy production, holding lake levels stable (near 1,070 ft) for recreation, and pro- viding high minimum flows at Atlanta. As an example, consider the period from June 1959 through May 1960. During this period the average flow to Lake Sidney Lanier was 2,229 ft*/s, whereas during June 1954 through May 1955, the average flow was 1,311 ft?/s. After adjusting for storage, the reported (35-year) average flow at the U.S. Geological Survey ESTIMATION OF COSTS: BENEFITS FOREGONE AND WASTE-TREATMENT COSTS gage below Buford Dam is 2,168 ft?/s. Clearly, the period from June 1959 through May 1960 had more nearly average flows than did the years 1954-56. The HSM was run using this 1959-60 record and the following constraints: 1. All water-supply requirements (1990 levels) are satis- fied. 2. The release through the turbines during all peak- power periods (52 weeks, 30 hours per week) is 10,000 ft?/s; and 3. Reservoir storage is not to exceed 1.917 million acre- ft (at 1,070 ft pool-elevation.) The simulation was conducted for Qa values of 1,290 ft3/s and 1,600 ft/s. The annual recreation benefits associated with the two minimum flows differ by less than $1,000. The results of the simulation associate a minimum pool elevation of 1,065.6 ft with a Qa of 1,290 ft?/s and a minimum elevation of 1,064.6 with a Qa of 1,600 ft?/s. As can be seen by referring to figure 10, recreation benefits are nearly the same for all elevations between 1,064 and 1,071 ft. Peak-energy production is nearly the same given a Qa of either 1,290 or 1,600 ft/s. In both cases, there is. a 10,000 ft3/s flow through the power plant for 30 hrs per week during the full year at heads that differ by no more than 1 foot. As a result, the peak-energy benefits associated with the two different values of Qa differ by ; less than $2,000. | Base-energy production is also virtually the same for | both values of Qa. Whether Qa is set at 1,290 ft?/s or at | 1,600 ft?/s, the same total amount of water must be | released during base-power periods over the course of | the year to keep the reservoir level from rising above | 1,070 ft. The heads being nearly the same, the difference l in base-energy benefits is very small. | Given the 1959-1960 flows, the only benefits significantly affected by the choice of Qa are the depend- able-peaking-capacity benefits. Given the 1959-60 hydrologic conditions, an increase in Qa to 1,600 ft*/s from 1,290 ft?/s decreases the dependable-peaking- capacity benefits by $0.75 million per yr (a 32,8300 kW loss in capacity multiplied by the estimated capacity value of $23.34/kW/yr), as happens under 1954-56 drought conditions. © The sum of the changes in all three other types of benefits is a function of both Qa and the hydrology of that particular year. As a result of an increase in Qa to 1,600 ft%/s from 1,290 ft?/s the increase in the peak- and nonpeak-energy benefits and the recreation benefits ranges from a total of $0.56 million per year under the most adverse hydrologic conditions to zero for average or above-average years. Thus, the assumption that all benefits other than the 17 dependable-peaking-capacity benefits are invariant with Qa results in a slightly high estimate of the benefits foregone given an increase in Qa; but, for simplicity, this assumption was adopted, and the relationship between Qa and average annual benefits foregone, as graphed in figure 11, was computed on this basis. ESTIMATION OF ADDED WASTE-TREATMENT COSTS The location and flows of the waste-treatment plants discharging wastes into the Chattahoochee River be- tween Atlanta and Whitesburg were specified in table 1. These configurations, for each of the three years, are based on data obtained from the Atlanta Regional Com- mission (1977). In this study, the location and flows of the treatment plants are not considered to be decision variables; they are taken as given. Rather, the percentage of the total waste flow receiving nitrification (P) is considered to be the decision variable. Data on waste-treatment costs (Giffels/Black and Veatch, 1977) were used to develop estimates of the capital, operation, and maintenance costs of adding a nitrification process to secondary waste-treatment plants. The capital costs were annualized using a 10-percent discount rate and then added to the annual operation and maintenance costs to obtain the estimated annual cost of adding the nitrification process to each treatment plant. These costs are presented in table 8. The costs of nitrification were estimated under the assumption that the required equipment would be operated year-round, though nitrification may not be required to maintain a given DO standard under some water-temperature conditions. Thus, the cost estimates presented in table 8 may be biased upwards. The data presented in table 8 were then used to develop estimates of the minimum annual cost of sub- mitting any given percentage of the total waste flow to nitrification. This was accomplished by identifying the plant or combination of plants that could provide nitrificaton for a given percentage of the total waste flow at a minimum cost. The total annual nitrification cost of this plant or combination of plants was then plot- ted against the percentage of the wastes receiving nitrification in 1980, 1990, and 2000 to obtain the cost curves depicted in figure 12. These cost curves are, of course, predicated on the particular treatment plants listed in tables 1 and 8. At this point, it seems desirable to summarize what has been so far accomplished herein. A dissolved-oxygen model was used to derive iso-DO curves, which delineate the combinations of P and Qa potentially capable of pro- > ducing a given level of DO. A hydrologic simulation model was used to delimit the feasible values of Qa and to provide a basis for estimating the costs (benefits 18 ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GA. foregone) associated with any given Qa. Estimates of the benefits foregone as a result of an increase in @a and of the costs of increasing P have been developed. Given this information, it is now possible to identify the least- cost combination of P and Qa capable of producing a given level of DO. THE REREGULATION PROJECT The U.S. Army Corps of Engineers has considered a project involving the construction of a reregulation structure on the Chattahoochee River just below Buford Dam and the dredging of the reservoir behind Morgan Falls Dam. This project would permit a more steady (and higher minimum) flow at Atlanta for any given level of peak generating capacity at Buford Dam. Con- versely, an increase in Qa would result in less dependable-peaking-capacity benefits foregone if the reregulation structure were built. A version of the HSM in which it is assumed that this project is completed is described in appendix A. The estimated costs of the project were obtained from a U.S. Army Corps of Engineers publication (1977). As is il- lustrated in the appendix (fig. 16), these costs exceed the project benefits, whether peak generating capacity is assigned a value of $23.34/kW/yr or of $49.35/kW/yr. Thus, the reregulation project would not be included in a least-cost scheme for providing a given level of DO, and it received no further consideration here. LEAST-COST METHOD OF PRODUCING A GIVEN MINIMUM DISSOLVED-OXYGEN CONCENTRATION The problem at hand can be usefully considered as one of finding the least-cost method of producing some given minimum DO concentration using two variable inputs: (1) some minimum flow rate at Atlanta (Qa) and (2) some percentage of the total waste load receiving nitrification (P) in addition to secondary treatment. The curves labeled D=3, D=4, and D=5 in figure 6, for example, give the various combinations of P and Qa that are potentially capable of producing the indicated minimum DO concentration in 1990. If it is desired to "produce" a TaBLE 8. -The average daily flow from waste-treatment plants discharging to the Chattahoochee River between Atlanta, Ga., and Whitesburg, Ga., and the annualized cost of adding a nitrification process to the plants (in First-quarter 1976 dollars) 1980 1990 2000 Plant name Riv? Average Annual Average Annual Average Annual mie low cost ow cost ow cost (ft/s) ($1,000) (ft/s) ($1,000) (ft/s) ($1,000) Cobb-Chattahoochee .________.___._'__.-- _ 2 300.56 24 458.40 29 518.27 31 538.85 R. M. Clayton 300.24 131 1704.19 150 1932.45 161 2069.03 South Cobb 294.78 38 694.43 51 851.60 48 819.79 Utoy Creek 291.60 42 722.43 46 771.07 44 746.75 Sweetwater Creek :_-______Lt_icstt ":- CH- BB8.57 = = = o 8 147.69 a 143.95 Camp Creek 283.78 15 359.82 22 444.02 27 507.63 Annewakee Creek CC I° I2 281.46 ~«_---~ |. ls 6 193.84 6 193.84 Regional Interceptor ABI4Db l " o_ OSLO 42 726.14 Bear Creek 2T4 A48 = ------ 0 7T 207.34 8 216.69 2.0 | M - ® c 2 & 5 O u E6, El » 1.5 |- . °G b a. 1T 2 3 " #10 |- 2000 3 i 1990 < 2 1980 C0 é u 0 O I &. 06 - s < o LL & E iL O -> % o | | | 800 1000 1200 1400 1600 1800 Qa, IN CUBIC FEET PER SECOND FicurE 11.-The relationship between average annual benefits foregone and minimum flow at Atlanta (QA) for 1980, 1990, and 2000. Q,, IN CUBIC FEET PER SECOND LEAST-COST ANNUAL COST, IN MILLIONS OF DOLLARS PER YEAR to I o | | | | 0 20 40 60 80 PERCENTAGE OF TOTAL WASTE FLOW RECEIVING NITRIFICATION (2) 100 FIGURE 12. -The estimated annual cost of adding a nitrification process to secondary waste-treatment plants, as a function of the percentage of the total waste flow receiving nitrification (P), for 1980, 1990, and 2000. METHOD OF PRODUCING A GIVEN MINIMUM DISSOLVED-OXYGEN CONCENTRATION 19 minimum DO concentration of, say 4 mg/L in 1990, it only remains to find that feasible point (combination of P and Qa) on the iso-DO curve labeled D = 4 in figure 6 that has associated with it a lower total cost in terms of benefits foregone and treatment costs than does any other feasible point of the curve. Given the assumptions embedded in the HSM, the up- per limit on the minimum flow that it is feasible to sus- tain at Atlanta is 1,670 ft?/s, 1,600 ft?/s, and 1,230 ft?/s in 1980, 1990, and 2000, respectively. Note that, from figure 6, it is feasible to attain a minimum DO concentra- tion of 3 mg/L in 1990 without nitrification (P =0), given a limit of 1,600 ft?/s on Qa, because the maximum necessary Qa is only 1,430 ft?/s. However, a minimum DO concentration of 4 mg/L requires, if P=0, a minimum flow of about 1,750 ft?/s, whereas the max- imum sustainable Qa is only 1,600 ft?/s in 1990. If the minimum flow is set at the maximum sustainable in 1990, the upper end of the feasible range of the iso-DO curve for 4 mg/L requires that 24 percent of the total waste load receive nitrification (P=24). The upper limit of the feasible range of an iso-DO curve is set by the lesser of either (1) the maximum necessary Qa or (2) the maximum sustainable Qa. Every point on an iso-DO curve represents some com- bination of P and Qa; thus each such point has an associated total cost. That cost can be determined using the: output of the HSM and the estimated cost of 1600 |- B " 1500 |- ms 1400 |- $5. 1300 |- 4 x 4 -l SH sin § a ¢ *% £5 5 s 5 S 8 is < o c S er cP §F §5 .f § ' \ g 8 5 ; £ £ 7 1100 |- 3 5 = i § 2 s S. ¥ -. CP # = 2 5.s = o- = 8 Ell £5) 18 . \" § E i aL vape E a a »| t 8 |g £ x IZ 8 is E a s 8 - 3 "52g! |S & ¥ Cs « 5 me" 1000 |- ——°— -—% N2 52.25 x § =° g— S, it S| |~% ® "C A 900 a I I 1 I I i wD I I Ed 0 10 20 30 40 50 60 70 80 90 100 PERCENTAGE OF TOTAL WASTE FLOW RECEIVING NITRIFICATION (P) FIGURE 13. -An illustration of the method of determining the least-cost combination of the percentage of waste flow receiving nitrification (P) and the minimum flow at Atlanta, Ga., (QA), required to produce a minimum dissolved-oxygen concentration of 4 mg/L, for 1990. Costs are in million dollars per year. 20 ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GA. nitrification and of dependable peak generating capaci- ty. Consider, for example, point A in figure 6; here P=58 and Qa = 1,290 ft/s. From figure 11, it can be seen that, given this Qa, no benefits are foregone in 1990. From figure 12, it can be seen that the additional waste- treatment costs associated with this P are equal to about 2.13 million dollars per year. Thus, point A (P=53, Qa = 1,290 ft*/s) has associated with it a total cost of 2.13 million dollars per year. Next consider point B in figure 6: here, P= 24 and Qa = 1,600 ft?/s. From figure 11, it can be seen that at a Qa of 1,600 ft?/s the benefits foregone equal 0.75 million dollars in 1990. The additional waste- treatment costs incurred, given that 24 percent of the total wastes are to receive nitrifiction, equal 1.08 million dollars per year. Thus, the total cost associated with point B is 1.83 million dollars per year. By calculating the total cost associated with each point on the iso-DO curves depicted in figure 6, the com- bination of P and Qa that will "produce" a given minimum DO concentration at least cost can be found. For 1990, the least-cost method of attaining a minimum DO concentration of 4 mg/L was determined to be associated with point B in figure 6. It can be more readily seen that point B does repre- sent a (1990) least-cost combination of P and Qa by in- specting figure 13. The curve labeled D=4 in figure 13 corresponds to the similarly labeled iso-DO curve in figure 6. The "kinked" curves in figure 13 connect combinations of P and Qa that are associated with equal total costs; these curves are known as iso-cost curves. It has already been determined that point B in figure 13 (and the same point in fig. 6) has an associated total cost of 1.83 million dollars per year. Every combination of P and Qa along the iso-cost curve that passes through point B has an associated total cost of 1.83 million dollars per year. For example, at point C (P=40, Qa = 1,360 ft?/s) on this iso- cost curve, the peak-generating-capacity benefits foregone, given a Qa of 1,360 ft?/s, are (from fig. 11) 0.16 million dollars per year; the additional treatment cost, given that 40 percent of the wastes are to receive nitrification, is (from fig. 12) equal to 1.67 million dollars per year. The total cost of the combination of P and Qa at point C is, then, 1.83 million dollars per year. Iso-cost curves can be derived for any given level of cost, and eight such curves are depicted in figure 13. Note that for any given level of Qa, total cost will in- crease as P is increased, because of increased treatment costs. Note also that for any given level of P, total costs will increase with Qa owing to increased benefits foregone, but such increases will occur only for those Qa greater than 1,290 (in 1990). For those Qa less than 1,290 ft?/s, there are no foregone benefits -that is, there is no decrease in the dependable peak generating capaci- ty of Buford Dam associated with an increase in Qa (see table 6). Thus, for a given level of P, the iso-cost curve is vertical below a Qa of 1,290 (in 1990) and represents only the nitrification costs associated with that level of P. Finally, note that as both P and Qa are increased, total cost increases, and thus the iso-cost curves passing through those points associated with more of both P and Qa represent higher levels of cost. That is, the iso-cost curves lying farther to the northeast of the origin in figure 13 represent higher levels of cost. The least-cost combination of P and Qa capable of pro- ducing a given minimum DO concentration is represented by that point where the lowest possible iso- cost curve just touches the iso-DO curve for that minimum DO concentration; in figure 13, this occurs at point B (P =24, Qa = 1,600 ft?/s). All other combinations of P and Qa capable of producing a minimum DO concen- tration of 4 mg/L in 1990 are associated with higher total costs. The same procedure as that depicted in figure 13 was used to determine the least-cost method of producing a minimum DO concentration of both 3 mg/L and 5 mg/L in 1990. The results are presented in table 9 along with the least-cost combinations for producing the three minimum DO concentrations in 1980 and 2000. In table 9 are also presented the separate components of total cost which are benefits foregone and the cost of adding the nitrification process to the waste-treatment plants. Note that in a comparison of the least-cost combina- tions of a given DO standard across years, the DO stand- ard of 5 mg/L provides the only case examined where TABLE 9.-Combinations of percentage of wastes receiving nitrification and minimum flow at Atlanta, Ga., that will provide minimum dissolved-oxzygen concentrations of 3, 4, and 5 mg/L at least-cost, for 1980, 1990, and 2000 Minimum Percentage of Minimum Costs, in million dollars per year dissolved- wastes re- flow at oxygen . ceiving nitri- Atlanta _ Nitri- Benefits Total concentration fication (P) (QA) fication foregone (mg/L) (ft/s) 1980 o 0 1380 0.00 0.00 0.00 A 0 1670 0.00 0.71 0.71 © 62 1380 2.03 0.00 2.03 1990 o im- Suse, 0 1430 0.00 0.34 0.34 Mielec lcs? 24 1600 1.08 0.75 1.83 earn f 63 1600 2.18 0.75 3.47 2000 O 52 870 2.58 0.00 2.58 3 70 870 3.16 0.00 3.16 ks 90 870 5.10 0.00 5.10 LEAST-COST METHOD OF PRODUCING A GIVEN MINIMUM DISSOLVED-OXYGEN CONCENTRATION 21 1700 | | | T I _ 1600 - 2 s & 1500 |- = LLJ W oc 1400 |- = L a tj 1300 |- = L LL _s & 1200 |- § € 3 2] co 5: o \ € S & & x& 5 O 1100 |- 2 7 'C 2 = z s 5 o, s [2 ® T & & 1000 |- 5 8 2 3 =I O =- = © 3 o ® =. o " = 900 S = 3 ad | | | 1 I 0 10 20 30 40 50 60 70 100 PERCENTAGE OF TOTAL WASTE FLOW RECEIVING NITRIFICATION (P) FicurE 14. -Illustration of the solution for least-cost combinations of percentages of waste Atlanta, Ga., (QA) necessary to produce a minimum dissolved-oxygen concentration of 5 mg/L, for 1980, 199 dollars. the combination switches from no dependable-peak- generating-capacity benefits foregone in 1980 to max- imum sustainable flow in 1990 and then back to no benefits foregone in 2000. Comparing the least-cost combinations for all other DO standards across time reveals that they require the minimum flow at Atlanta be set at either the maximum necessary or the maximum sustainable in 1980 and 1990 and then be reduced to 870 ft3/s in 2000. 'The solutions for the least-cost combinations required to achieve a minimum DO concentration of 5 mg/L in 1980, 1990, and 2000 are depicted in figure 14. Note that the least-cost solution for 1990 would occur at that com- bination of P and Qa represented by the point at the "kink" in the iso-cost curve if the slope of the upper por- tion of the iso-cost curve were only slightly "flatter." This is, the least-cost combination of P and Qa, given a DO standard of 5 mg/L, nearly requires that Buford Dam be operated so as to forego no benefits from dependable peak generating capacity in 1990, just as it does require that it be operated in 1980 and 2000. This suggests two related questions: First, how sen- sitive is the least-cost solution to the value of the parameters that determine the slope of the iso-cost curves? Second, how much difference would it make, in terms of added cost, if the least-cost solution were not chosen? It is to these questions that we now turn. s receiving nitrification (P) and minimum flow at 0, and 2000. Costs, in millions of SENSITIVITY OF THE LEAST-COST SOLUTION TO THE COST OF DEPENDABLE PEAK GENERATING CAPACITY Given the shapes of the iso-cost and iso-DO curves derived in this study, the least-cost combination of P and Qa is found either at the upper end of the feasible range of the iso-DO curve or where the "kink" in an iso-cost curve just touches the iso-DO Curve. That is, the least- cost combination will require either that Buford Dam be operated so as to maintain the minimum flow at Atlanta at the maximum (necessary or sustainable) or that it be operated so as to forego no benefits from dependable peak generating capacity. An increase in the cost of dependable peak generating capacity relative to that of nitrification would be suffi- cient to decrease the slope of the iso-cost curves. Any given level of total cost will be attained at a lower Qa after an increase in the cost of peak generating capacity because the benefits foregone as a result of the loss of such capacity will be greater at each Qa that would cause such a loss. However, given some positive cost for dependable peak generating capacity, that Qa below which no capacity benefits are foregone will remain the same. Thus, the iso-cost curves associated with higher costs of peak generating capacity will lie beneath and have a lesser slope than will such curves associated with lower capacity costs. 22 ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GA. Given a sufficient increase in the cost of peak generating capacity relative to that of nitrification, the least-cost combinations of attaining any given minimum | peak generating capacity of Buford Dam is DO concentration will switch from those requiring a maximum (necessary or sustainable) minimum flow at Atlanta to those that require that no dependable-peak- generating-capacity benefits be foregone at Buford Dam. The dependable-peak-generating-capacity costs that cause such a switch in the least-cost combination of P and Qa are presented in table 10. TABLE 10.-Costs, in first-quarter 1976 dollars per kilowatt-hour per year, of dependable peak generating capacity above which the indi- cated minimum dissolved-oxygen (DO) concentration can be achieved at least cost by operating Buford Dam, Ga., so as not to forego benefits from dependable peak generating capacity Year Minimum dissolved-oxygen concentration (mg/L) 1990 1980 2000 3 () 69 21 4 34 33 22 5 17 25 16 ! It is not necessary to forego peak generating capacity even if Q4 is set at the maximum necessary (1,380 ft?/s) to achieve a minimum DO concentration of 3 mg/L. We have assumed that the replacement cost of dependable peak generating capacity at Buford Dam is equal to the $23.34/kW/yr estimated cost of the "Rocky Mountain" hydroelectric-power pump-storage facility. Consequently, the least-cost combination requires that no dependable peak generating capacity be foregone in order to provide a minimum DO concentration of either 3, 4, or 5 mg/L in 2000 and to provide a minimum DO concentration of 5 mg/L in 1980. But, note that our estimate of $23.34/kW/yr is close to those costs that would require no peak generating capacity be foregone to provide a minimum DO concentration of either 3 or 4 mg/L in 2000 and to provide a minimum DO concentra- tion of 5 mg/L in 1990. For these DO standards in these years, the least-cost combination of P and Qa is quite sensitive to the estimate of the cost of dependable peak generating capacity. As was previously noted, the U.S. Army Corps of Engineers (1977) has assumed that any loss of depend- able peak generating capacity at Buford Dam would be replaced using thermal electric generating facilities at a cost of $49.35/kWi/yr. Using such a replacement cost, the least-cost combination of P and Qa requires that Buford Dam be operated so as to forego no benefits from dependable peak generating capacity in providing a minimum DO concentration of either 4 or 5 mg/L. The least-coast combination would require that the dam be operated so as to maintain the maximum necessary Qa in 1980 and the maximum sustainable Qa in 1990 if the DO standard were set at 3 mg/L. But, no peak generating capacity would be foregone in 1980 given that the maximum Qa necessary to maintain a minimum DO concentration of 3 mg/L is only 1,380 ft*/s. Suppose that the replacement cost of the dependable $49.35/kW/yr but that the choice of the least-cost com- bination of P and Qa is based on an estimated cost of $23.34/kW/yr. Conversely, suppose that the replace- ment cost is really $23.34/kW/yr but that the least-cost combination is chosen under the assumption that the replacement cost is $49.35/kW/yr. In each case, the actual total cost will be greater than the calculated total cost of that which is (mistakenly) thought to be the least- cost combination of P and Qa. The difference between the actual and calculated total costs is a measure of the loss in economic efficiency that would result from the use of an erroneous estimate of the cost of peak generating capacity. The economic-efficiency losses that would result if the cost of peak generating capacity were actually $49.35/kW/yr but the least-cost combination were calculated and selected using an estimated cost of $23.34/kW/yr are listed in table 11; also presented is the correct least-cost combination of P and Qa if the cost is actually $49.35/kW/yr. If the minimum DO concentra- tion were set at 5 mg/L in 1990, for example, the calculated least-cost combination would require that 63 percent of the total wastes receive nitrification and that the minimum flow at Atlanta be set at 1,600 But, the correct least-cost combination would require that 78 percent of the waste receive nitrification and the minimum flow at Atlanta be set at 1,290 ft/s. If we have underestimated the cost of peak generating capacity by $26.01/kW/yr (= $49.35 - $28.34), our (erroneous) least- cost combination of P and Qa results in a $800,000 per year efficiency loss given a 5 mg/L-DO standard in 1990. This efficiency loss would result from too little nitrifica- tion and too much peak generating capacity lost relative to the "correct" least-cost combination. The economic-efficiency losses that would result if the cost of peak generating capacity were actually $23.34/kW/yr but the least-cost combination were calculated and selected using an estimated cost of $49.35/kW/yr are listed in table 12. In this case, if the minimum DO concentration were set at 5 mg/L for 1990, the calculated least-cost combination would require that 78 percent of the wastes receive nitrification and that the minimum flow at Atlanta be set at 1,290 But, the correct least-cost combination would require that only 63 percent of the waste receive nitrification and that the minimum flow at Atlanta be set at 1,600 ft?/s. If the cost of peak generating capacity is overestimated by $26.01, the (erroneous) least-cost combination of P and Qa results in a $40,000/yr efficiency loss in 1990, given a 5 mg/L DO standard. INSTITUTIONAL CONSTRAINTS AND ASSOCIATED COSTS 23 TABLE 11. -Eeonomic-efficiency loss, in million dollars per year, if the least-cost combination of the percentage of wastes receiving nitrifica- tion (P) and the minimum flow at Atlanta, Ga., (Q4), in cubic feet per second, is selected under the assumption that the cost of peak generating capacity is $23.34/kWi/iyr but the actual cost is $19.35/kW/yr, for 1980, 1990, and 2000 Minimum Least-cost combination dissolved- Economic oxygen Calculated Correct efficiency concentration loss (mg/L) P QA P QA 1980 PELCO 0 1,380 0 1,380 0.00 4 .de 0 1,670 31 1,380 .87 D 62 1,380 62 1,380 .00 1990 B ssn ECs 0 1,430 0 1,430 0.00 B Ee sels 24 1,600 52 1,290 54 8 63 1,600 78 1,290 .80 2000 iet 52 870 52 870 0.00 4 ee eases 70 870 70 870 00 5 90 870 90 870 .00 TABLE 12. -Economic-efficiency loss, in million dollars per year, if the least-cost combination of the percentage of wastes receiving nitrifica- tion (P) and the minimum flow at Atlanta, Ga., (Q4), in cubic feet per second, is selected under the assumption that the cost of peak generating capacity is $49.35/kW/iyr but the actual cost is $23.34, /kW!yr, for 1980, 1990, and 2000 Minimum Least-cost combination dissolved- Economic oxygen Calculated Correct efficiency concentration loss (mg/L) £ QA £ QA 1980 0 1,380 0 1,380 0.00 L intents 31 1,380 0 1,670 A2 bae cul. 62 1,380 62 1,380 00 1990 Bs.eccl .gs 0 1,430 0 1,430 0.00 52 1,290 24 1,600 80 $ 78 1,290 63 1,600 04 2000 § e 52 - 870 52 870 0.00 4 _ c_. 70 870 70 870 .00 -as 90 870 870 00 90 capacity cost, only three cases have an economic- efficiency loss associated with the choice of one estimate of the cost over the other-for a DO standard of 4 mg/L in 1980 and 1990 and for a standard of 5 mg/L in 1990. Note also that the "switching costs" presented in table 10 fall between $28.34/kW/yr and $49.35/kW/yr in only these three cases. For all other cases, the least-cost com- bination of P and Qa is the same, given a peak- generating-capacity cost of either $23.34/kW/yr or $49.35/kW/yr. If a decision maker is uncertain as to the cost of peak generating capacity and is risk adverse he might prefer to minimize the maximum possible economic-efficiency loss by choosing to base the selection of the least-cost combination on an estimated capacity cost of $49.35/kW/yr. However, we believe that it is inap- propriate to assume that any peak generating capacity lost at Buford Dam would be replaced by thermal electric facilities. We prefer to base our calculations on the assumption that the peak generating capacity would be replaced by a facility similar in cost to the "Rocky Moun- tain" hydroelectric-power pump-storage facility. The Georgia Power Co. apparently found hydroelectric- power pump-storage to be the least-cost method of ob- taining additional peak generating capacity. INSTITUTIONAL CONSTRAINTS AND ASSOCIATED COSTS The least-cost combinations of P and Qa that are presented in table 9 are based on the assumption that there is complete flexibility in the choice of P and Qa. In reality, constraints may exist in the form of laws or regulations that restrict the range of choice of P and (or) Q. The questions then become what is the least-cost plan, given these constraints, and what is the cost of that plan? Currently, the Georgia Department of Natural Resources requires that a minimum flow of 750 be maintained in the Chattahoochee River immediately upstream of the confluence of Peachtree Creek (U.S. Ar- my Corps of Engineers, 1977). This translates to a minimum-flow requirement of 860 at the Atlanta gage. If this requirement sets the Qa at 860 and no higher, then the problem of finding the "least-cost" method of producing a given minimum DO concentra- tion is reduced to simply finding the minimum level of nitrification (that is, the minimum P) that will provide that DO concentration given this constraint on Qa. For example, given a DO standard of 4 mg/L in 1990 and a Qa of 860 ft3/s, the least-cost combination is indicated by point D in figure 13. Given the constraint on Qa, 72 per- | cent of the total waste must receive nitrification if a Note that, given the two estimates of peak generating minimum DO of 4 mg/L is to be attained. The cost associated with this (constrained) least-cost combination is given by the iso-cost curve that passes through point D in figure 13-3.2 million dollars per year. The same procedure was used to find the least-cost method of pro- viding a minimum DO concentration of both 3 mg/L and 5 mg/L in 1990, given a Qa of 860 ft?/s. The results are presented in table 13 and graphically in figure 15. (Points D and B in figure 15 correspond to the similarly 24 ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GA. TABLE 18. of wastes that must receive nitrification to provide minimum dissolved-oxzygen concentration of 3, 4, and 5 mg/L at least-cost, given that the minimum flow at Atlanta, Ga., is con- strained to 860 ft?/s for 1980, 1990, and 2000 Minimum __ Percentage of Minimum Costs, in million dollars per year np yastes'tre‘: 33:53; Nitri- Benefits conzggfantiun cei‘icggow (E) f(Q fication foregone Total (mg/L) ( ”fig 1980 0 «danntes 39 860 1.35 0.00 1.35 M 59 860 1.91 0.00 1.91 D ese _ sus 84 860 3.12 0.00 3.12 1990 B 52 860 2.10 0.00 2.10 . 72 860 3.20 0.00 3.20 $ 92 860 4.30 0.00 4.30 2000 B Mellen 52 860 2.58 0.00 2.58 E 70 860 3.16 0.00 3.16 D 90 860 5.10 0.00 5.10 labeled points in figure 13.) The least-cost methods of producing minimum DO concentrations of 3 and 5 mg/L in 1980 and 2000, given that Qa is constrained to 860 ft?/s, are also described in table 13. As another example of a constraint and its associated cost, suppose that a requirement existed that Q» be set at 860 ft"/s and that all wastes receive nitrification (P=100). Because no dependable-peak-generating- capacity benefits are foregone under this plan, the total costs are those of adding a nitrification process to all secondary treatment plants; these annualized costs total. 3.95 million dollars in 1980, 5.05 million dollars in 1990, and 5.95 million dollars in 2000. The total costs under this plan in 1990 are also depicted in figure 15. Note that, from figures 5 through 7, the constraint that P=100 and Qa = 860 ft?/s will result in a minimum DO concentration that is greater than 5 mg/L in each of the three years considered. In each of these two examples, the difference between the lower costs of the "unconstrained" least-cost plan and the higher costs of the corresponding "constrained" least-cost plan is due solely to the imposition of the con- straint. This additional cost provides an estimate of the cost of obtaining any benefits (monetary or non- monetary, tangible or intangible) that might result from the constraint. CONCLUDING REMARKS This study has placed a DO management problem in a conceptual framework suggested by the economic theory of production. The minimum flow of the Chatta- hoochee River and the percentage of the waste inflow 6 A/ | | +-- ; %. g |___ p__ JM percent nitrification __ _, 3] COST, IN MILLIONS OF DOLLARS PER YEAR to I JL | | & 3 4 5 & MINIMUM DO, IN MILLIGRAMS PER LITER FicurE 15.-Costs (benefits foregone plus added waste- treatment cost) of attaining various minimum dissolved- oxygen (DO) concentrations under different policies, in 1990. Note that cost is independent of the dissolved-oxygen standard, given a constraint of 100 percent nitrification. receiving nitrification are considered to be two variable inputs that can be used to produce a given concentration 'of dissolved oxygen in the river. Results of the U.S.G.S. Chattahoochee River Quality Assessment project were used to establish the production relationship between minimum flow, waste treatment, and DO concentration. Each of the inputs has a cost: the loss of dependable- peaking-capacity benefits associated with flow augmen- tation and the cost associated with nitrification of wastes. An attempt was made to find the least-cost com- bination of minimum flow and waste treatment necessary to achieve a prescribed minimum DO concen- tration. No attempt was made to identify the benefits associated with various concentrations of DO in the APPENDIX A- ANALYSIS OF THE REREGULATION PROJECT river. Thus, no attempt was made to provide an estimate of the minimum DO concentration that would maximize the net benefits from producing dissolved oxygen in the river. It was not an objective of this study to prescribe a specific set of operating rules for Buford Dam and a waste-treatment plan for the Atlanta region. An objec- tive was to demonstrate a method for evaluating the cost effectiveness of alternative strategies for DO management; the method is the primary message. The Chattahoochee River was used as an example because of the availability of U.S.G.S. data and models that could be used to derive the DO production relationship. Another objective was to demonstrate how the results of a U.S.G.S. Intensive River Quality Assessment could be applied to a water-quality management problem. The DO curves presented in figures 5-7 were derived using the DO model of the Chattahoochee River developed by Stamer and others (1978). These curves describe the physical relationships between flow augmentation, nitrification, and DO and are useful in themselves. When cast with an economic framework, they provide a basis for decisionmaking. In regard to the Chattahoochee River, the results in- dicate that for certain DO standards and between now and 2000 the waste-assimilation capacity of increased flows in the Chattahoochee River can be substituted for increased waste treatment. It is estimated that the sav- ings in waste-treatment costs experienced by so doing will more than offset the benefits foregone because of the loss of peak generating capacity at Buford Dam. However, these results were demonstrated to be, in some cases, sensitive to the value assigned to peak generating capacity and may also be sensitive to (among other things) estimates of the discount rate and the costs of nitrification. There is a strong indication that a flexible approach to the management of DO in the Chattahoochee River may be much more cost effective than a more rigid, institu- tional approach. Examples of such rigid approaches are prohibitions of flow augmentation for water-quality management or blanket requirements for high levels of waste treatment without regard to concomitant costs and resulting water-quality levels. An institutional con- straint on flow augmentation or waste-treatment prac- tices will not in general be consistent with the attain- ment of a prescribed DO standard at least cost; that is to say, such constraints will usually have an associated cost (or economic-efficiency loss). Finally, note that our criterion for evaluating dif- ferent DO-management strategies has been solely one of economic efficiency: What is the minimum-cost method of meeting a given DO standard? Equity, or distribu- tional, considerations have been completely ignored. 25 For example, to attain a minimum DO concentration of 5 mg/L in the Chattahoochee River in 1980 and 1990, the least-cost strategy requires that a little over 60 per- cent of the total waste flow receive nitrification and that, consequently, about 40 percent of the flow receive only secondary treatment. If the additional cost of nitrification is borne only by the taxpayers in the service area of those plants required to add the nitrification pro- cess, the taxpayer serviced by those plants at which nitrification is not required do not bear any of the addi- tional waste-treatment cost incurred in meeting the DO standard. As another example, consider that in choosing between combinations of P and Qa that will produce a given level of DO, some combinations require that more dependable peaking capacity be foregone and less addi- tional waste-treatment costs be incurred than do others. Those individuals that bear the costs of replacing the peaking capacity and those that experience the savings in treatment costs because the peaking capacity has been foregone are not necessarily the same individuals. The choice of a least-cost method for attaining a given minimum DO concentration has distributional or equity implications that have not been considered in this study. REFERENCES CITED Atlanta Regional Commission, 1977, Atlanta region areawide waste- water management plan [revised]: Atlanta Regional Commission in cooperation with U.S. Army Corps of Engineers, Savannah District; Georgia Department of Natural Resources, Environmen- tal Protection Division; and the U.S. Environmental Protection Agency, 28 p. Georgia Power Co., 1972, Application to the Federal Power Com- mission for a license for the Rocky Mountain project: October 1972, Exhibit W, pages 44 and 45 of 47 pages. Giffels/Black and Veatch, 1977, Overview plan with environmental assessment, Detroit Water and Sewerage Department [revised], vol. 1 of Comparative waste-water collection and treatment costs: Detroit, Giffels/Black and Veatch, 118 p., 2 sections, rept. code PF 101. Stamer, J. K., and others, 1978, Magnitudes, nature, and effects of point-nonpoint discharges in the Chattahoochee River basin, Atlanta to West Point Dam, Georgia: U.S. Geological Survey Open-File Report 78-577, 130 p. U.S. Army Corps of Engineers, 1977, Lake Lanier restudy- Preliminary base plan of operations: Savannah, Ga., U.S. Corps of Engineers, Savannah District, 11p. 2 figs., exhibits A-E. APPENDIX A-ANALYSIS OF THE REREGULATION PROJECT If the proposed reregulation structure were built just below Buford Dam (capacity 8,400 acre-feet) and if the Morgan Falls reservoir were dredged to a capacity of 3,500 acre-feet, then more peak hydroelectric power could be produced given any required minimum flow at the Atlanta gage. By storing more of the water released from Buford Dam during peaking hours, most or all the 26 water-supply and minimum-flow needs throughout the entire week can be met. The HSM was modified to simulate this situation, and two variables were added to account for the addition of the reregulating structure: Sr (storage in ft?) and Qr (discharge from the reregulating structure in ft?/s). The additional flow and storage constraints of the HSM are dSr/dt=Q1 - Qr (12) Sr(to)= 0, (13) Sr(ty> 0, and (14) Sr<3.66 x 108. (15) Some other constraints in the original HSM are changed, as follows: Qr +T;> Ws, (6a) dS;/!dt=Q@r+T;,- W- Qz, and (Ta) S,<1.52 x 108. (10a) Added and changed constraints are 12. continuity equation for the reregulating reservoir; 13. and 14. initial and final storage in reregulating reservoir (arbitrary); 15. capacity constraint for reregulating reservoir, 3.66 x 108 ft*=8,400 acre feet; 6a. the withdrawals below Buford Dam but above Morgan Falls Dam, must be satisfied by the release from the reregulating reservoir plus tributary flows; Ta. continuity equation for Morgan Falls reservoir, the inflow being the release from the reregulating reservoir plus tributary flow minus withdrawals; and the capacity of Morgan Falls reservoir, increased by dredging to 1.52 x 108 ft*= 3,500 acre feet. This modified HSM was run to determine the relation- ship between dependable peaking capacity and Qa for each of the three years. 10a. Figure 16 shows the dependable-peaking-capacity benefits as a function of Qa with and without the reregulating structure and Morgan Falls reservoir dredging. Also these benefits minus the cost of these im- provements are shown in this figure. According to the U.S. Army Corps of Engineers' Lake Sidney Lanier restudy (1977), the capital cost of the reregulating structure is $11.50 million and the opera- tion and maintenance costs are $65,800 per year. Based on a discount rate of 10 percent and a life of 100 years, the annualized cost of the facility is $1.22 million per year. The Corps reports the initial cost of the Morgan Falls reservoir dredging is $1.65 million and an- nual maintenance-dredging costs are $15,000/per year. Based on a 10 percent discount rate and a 100 year life, the annualized cost of the Morgan Falls reservoir dredg- ing is $0.18 million per year. Thus, the annual cost of both projects is $1.40 million per year. ECONOMICS OF DISSOLVED-OXYGEN MANAGEMENT: CHATTAHOOCHEE RIVER, GA. The figures show that the costs of these improvements | exceeds the gain in dependable-peaking-capacity benefits. These calculations were made under the assumption that the cost of dependable peaking capacity is $23.34/kW/yr. Even if this cost were assumed to be $49.35/kW/yr, the costs of the improvements would ex- ceed the gain in dependable-peaking-capacity benefits. There may, however, be other benefits from the project, such as enhancement of the river for recreational use or the mitigation of channel erosion. 3 T T T T ........................................................ DEPENDABLE PEAKING CAPACITY BENEFITS, IN MILLIONS OF DOLLARS PER YEAR 0 | | | | 800 1000 1200 1400 1600 Q,, IN CUBIC FEET PER SECOND 1800 FIGURE 16. -The relationship between estimated annual depend- able-peaking-capacity benefits and minimum flow at Atlanta, Ga., (QA), with and without reregulation, for 1980, 1990, and 2000. Solid line is without reregulation; dashed line is with reregulation; dotted line is benefits with reregulation minus the annualized cost of reregulation. # U.S. GOVERNMENT PRINTING OFFICE: 1980 O- 3l1-344/126