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Decision analysis applied to ground water exploration 1979

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DECISION ANALYSIS APPLIED TO GROUND WATER EXPLORATION by BENEDICT ANEKWE AGINAH B.Eng., Ahmadu Bello University, Zaria, 1972. A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE In THE FACULTY OF GRADUATE STUDIES (The Department of C i v i l Engineering) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA January 1979 (c) Benedict Anekwe Aginah, 1979 In p r e s e n t i n g t h i s t h e s i s in p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree at the U n i v e r s i t y o f B r i t i s h Co lumb ia , I agree that the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and study. I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s thes is f o r s c h o l a r l y purposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . It i s u n d e r s t o o d tha t copying o r pub l i ca t ion o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l not be a l l o w e d w i thout my w r i t t e n p e r m i s s i o n . Department o f The U n i v e r s i t y o f B r i t i s h Co lumbia 2075 Wesbrook Place Vancouver, Canada V6T 1W5 Date ' j ^ ^ T i " j ABSTRACT An outline of the essential steps needed i n ground water exploration i s given. Since d r i l l i n g for ground water involves a lot of uncertainty, the main concepts of Bayesian decision theory are br i e f l y reviewed. Three models for analyzing ground water decision problems are developed with an emphasis on the well-owner's u t i l i t y or desirability to actually venture to invest on a water-drilling project. Finally, use of .the decision models is Illustrated by applications to a) Ryder Lake District (in B r i t i s h Columbia) - an area where water supply i s a problem, with the only source being from underground; and to b) Inches Creek study area where approximately 4500 gallons per minute of ground water i s needed for salmon enhancement f a c i l i t i e s . i i TABLE'OP CONTENTS Page ABSTRACT i i LIST OP TABLES v LIST OP FIGUFES v i ACKNOWLEDGEMENT v i i CHAPTER 1. INTRODUCTION 1 2. SUMMARY OP GROUND WATER EXPLORATION 4 2.1 Geologic: ̂  Considerations 4 2.2 Past Records 5 2.3 Hydrologic Considerations 5 2.4 Test D r i l l i n g and Sample Analysis 6 2.5 Surface and Subsurface Geophysical Methods 7 2.6 Logging Techniques Used In Ground Water Exploration .... 7 2 . 6 . 1 Spontaneous Potential 8 2 .6 .2 Resistivity 8 2 .6 .3 Other Logging Methods 9 2.7 Pump Tests 11 2.8 Observation Wells 11 2.9 Water Quality 12 2.10 Ground Water Recharge 12 3. DECISION ANALYSIS UNDER UNCERTAINTY IN GROUND WATER TERMS ... 13 3.1 U t i l i t y Theory 14 4. MODELS FOR ANALYZING GROUND WATER DECISION PROBLEMS 17 4.1 Cast I 17 4 .1 .1 U t i l i t y of Not D r i l l i n g 20 i i i TABLE OF CONTENTS (continued) Page CHAPTER 4. 4.2 Cast II 20 4.2.1 Case 11(a): No Relationship Between Yield and Depth 20 4.2.2 Case 11(b): Probability Relationship Between Yield and Depth 20 4.2.3 Expected U t i l i t y of Not D r i l l i n g 25 4.3 Case III - Decision to Purchase Imperfect Informtion .. 27 5. APPLICATIONS 30 5.1 Ryder Lake Dist r i c t 30 5.1.1 Introduction 30 5.1.2 Location 30 5.1.3 Climate 30 5.1.4 S u r f i c i a l Geology 31 5-1.5 Water Supply 31 5.1.6 Quality of Water 34 5.1.7 Decision Model Applications to Ryder Lake Area . 34 5.1.7-1 Case I of Model 34 5.1.7.2 Case 11(a) of Model 38 5.1.7.3 Case 11(b) of Model 40 5.2 Inches Creek 40 5.2.1 Location 40 5.2.2 Objective of Study 44 5.2.3 Aquifer Recharge 44 5.2.4 Application of Decision Model Case III 44 6. DISCUSSION AND CONCLUSIONS 51 BIBLIOGRAPHY 53 i v LIST OF TABLES Table Page 5 . 1 Drilled Well Records - Ryder Lake Area 35 v LIST OF FIGURES Figure "" ;Page 4.1 Decision Tree Schematic (Case I) 18 4.2 U t i l i t y versus Yield 19 4.3(a) Decision Tree Schematic (Case II) 21 4.3(b) Decision Tree Showing Expected Values 21 4.4 U t i l i t y "Cost" versus Depth 23 4.5 Yield versus Depth Probability Band 24 4.6 Schematic Decision Tree 26 4.7 Decision Tree - Purchase Of Additional Information • 28 5.1 Map Showing Drilled Well Locations - Ryder Lake Area 33 5.2 U t i l i t y versus Yield (Ryder Lake Area) 36 5-3 Yield versus Cumulative Probability - Ryder Lake Area .... 38 5.4 U t i l i t y "Cost" versus Depth (Ryder Lake Area) 40 5.5 Depth versus Cumulative Probability - Ryder Lake Area 4l 5.6 Yield versus Depth Probability Band - Ryder Lake Area 42 5-7 Inches Creek Location Map 44 5.8 Yield versus Cumulative Probability (Prior) - (Inches Creek). 45 5.9 Production Yield versus Test Yield Probability Band (Inches Creek) - 46 5.10 Production Well Cost versus Yield - (Inches Creek) 48 5.11 Decision Tree showing Purchase of Imperfect Information (Inches Creek) 49 v i ACKNOWLEDGEMENT The author wishes to express his sincere gratitude to his supervisor Professor S.O. Russell for his invaluable guidance, assistance and constant encouragement throughout the preparation of this thesis. Special thanks are also extended to Mr. E.C. Halstead and Mr. H.M. Liebscher (both of the Hydrology Research Division, Vancouver, B.C.) for their advice and assistance i n data collection; and to Mr. Ron Grigg for assistance i n computer programming. And, f i n a l l y , the author would also like to thank the C i v i l Engineering Department for financial assistance. v i i CHAPTER 1 INTRODUCTION Human consumption of ground water has been increasing steadily over the years, especially i n the past seventy-eight years. This has been as a result of increased use of irri g a t i o n , industry, and the r i s i n g standards of l i v i n g . Today, ground water resources, which constitute more than ninety-five percent of the world's to t a l fresh water supply, are generally uncontaminated in contrast to the increasingly polluted nature of many of i t s surface water sources. Though ground water generally averages out to be a l i t t l e harder and more mineralized than surface water i n the same lo c a l i t y , yet i t s quality i s more uniform during the year. The temperature of ground water, like i t s chemical quality, i s also relatively uniform throughout the year. This makes i t preferable for many uses especially for the fishing industry, and also for cooling purposes i n the summer, when surface water i s warmer. The importance of ground water does not mean that wells should be d r i l l e d just anywhere. There are many uncertainties involved; for one cannot say exactly what the outcome of a d r i l l i n g project would be even after the hydrologist (the expert) has predicted a good aquifer. The outcome could be a dry hole or an undesirable yield. A systematic and formal analysis to take care of the risk and uncertainty i s therefore very worthwhile. Decision analysis, also known as s t a t i s t i c a l decision theory, management science, operations research, and Bayesian decision theory, i s a discipline consisting of various methods, techniques, and attitudes to help the decision maker to choose wisely under these conditions of uncertainty. This analysis has already been applied to o i l and gas exploration (Grayson, i960 and Newendorp, 1975), forest management and geological investigations (Halter 1 2. and Dean, 1 9 7 D , water quality management (Hershmann, 1 9 7 4 ) and also i n the search for-minerals. It requires that the explorer (expert.) associate specific probabilities with the possible outcomes (dry hole, or various yields); and this i s where the element of risk comes i n . Where there are past records, s t a t i s t i c a l methods are used to c a l c u l a t e , p r o b a b i l i t i e s , Next, the owner of the project assesses his u t i l i t y values or desirability of the various outcomes. Finally, expected u t i l i t y values, which form the basis for decision, are computed. The objective of this thesis, therefore, i s to apply decision theory i n the search for ground water. Different decision models have been developed and applied to Ryder Lake D i s t r i c t , some f i f t y - f i v e miles east of Vancouver i n Bri t i s h Columbia; and also to Inches Creek where approximately 4500 gallons per minute of ground water i s needed for salmon enhancement f a c i l i t i e s . Ryder Lake District depends solely on ground water for i t s water supply. And so, people interested i n ..settling there have always wanted to know how good the chances of obtaining water are before involving themselves i n expensive d r i l l i n g programs. A summary of a l l the steps needed i n ground water exploration i s given i n Chapter 2. Chapter 3 describes,in a nutshell, the procedures of carrying out a decision analysis, more especially as i t applies to the search for ground water. This chapter also explains the use of u t i l i t y theory which i s one of the backbones of decision analysis. Three different models that can be used i n analyzing ground water decision problems are developed and explained i n Chapter 4. Chapter 5 illustrates how the above models can be applied to real-world situations such as i n the Ryder Lake District and i n Inches Creek study area. A description of the computer program used i n the analysis i s also given i n this chapter, while the discussion of results and the conclusions are given i n Chapter 6. CHAPTER 2 SUMMARY OF GROUND WATER EXPLORATION In the past, the only method of prospecting for ground water was "water witching" or "dowsing". But this method has proved most unreliable and a more s c i e n t i f i c approach had to be found. In 1 9 6 3 , the U.S. Geological Survey published a report"*' summarizing a general approach to ground water exploration. The following paragraphs are taken from that report. 2.1 Geologic. Oonsiderations. "Certain clues are helpful i n locating ground water supplies. For instance, ground water i s l i k e l y to occur i n larger quantities under valleys than under h i l l s . In arid regions, certain types of water-loving plants give the clue that there has to be ground water at shallow depths underneath to. feed them. Any area where water shows up attthe surface - In springs, seeps, swamps, or lakes - has to have some ground water, though not . .:,co necessarily i n large quantity or of usable quality. "But the most valuable clues are the rocks. Hydrologlsts and geologists use the word rock to mean both hard, consolidated formations, such as sandstone, limestone, granite, or lava rocks, and loose unconsolidated sediments such as gravel, sand, and clay. They use the word aquifer for a layer of rock that carries a usable supply of water. Gravel, sandstone, and limestone are the best water carriers but they form only a fraction of the rocks i n the earth's outer crust. Not a l l of them yie l d useful supplies of water. The bulk of the rocks consist of clay, shale, and crystalline rocks - a term used for the great variety of hard rocks that form most of the earth's crust. Clay, shale, and crystalline rocks are a l l poor producers, but they may yield enough water for domestic stock uses in areas where no better aquifers are present. 4 . 5. "The h y d r o l o g i s t o r g e o l o g i s t f i r s t o f a l l p r e p a r e s a g e o l o g i c a l map and c r o s s - s e c t i o n s showing where t h e d i f f e r e n t r o c k s come t o t h e l a n d s u r f a c e and how t h e y a r e a r r a n g e d b e n e a t h : t h e s u r f a c e . He w i l l o b s e r v e how t h e r o c k s have b e e n a f f e c t e d by e a r t h p r e s s u r e s i n t h e p a s t . The g e o l o g i c map and s e c t i o n s and t h e accompanying e x p l o r a t i o n s show j u s t w h i c h r o c k s a r e l i k e l y t o c a r r y w a t e r and where t h e y a r e b e n e a t h t h e s u r f a c e . " 2.2. F a s t R e c o r d s "Next, he w i l l g a t h e r a l l t h e i n f o r m a t i o n he c a n on e x i s t i n g w e l l s - t h e i r l o c a t i o n , d e p t h , d e p t h t o w a t e r , and amount o f w a t e r pumped, and what k i n d o f r o c k s t h e s e w e l l s p e n e t r a t e . Much o f what he i s i n t e r e s t e d i n i s below t h e d e p t h o f o r d i n a r y e x c a v a t i o n s , and he cannot a f f o r d t o d r i l l a w e l l o r t e s t h o l e i n e v e r y p l a c e where he needs i n f o r m a t i o n . "Records o f w e l l s where t h e d r i l l e r has c a r e f u l l y l o g g e d t h e d e p t h and t y p e s o f d i f f e r e n t r o c k s t r a t a a r e h e l p f u l . A r e a l l y u s e f u l w e l l r e c o r d w i l l i n c l u d e t h e f o l l o w i n g : samples o f t h e r o c k ; i n f o r m a t i o n on w h i c h s t r a t a y i e l d w a t e r and how f r e e l y ; t h e s t a t i c w a t e r l e v e l i n e a c h s u c c e s s i v e l y d e e p e r s t r a t u m ; and d a t a f r o m a pumping o r b a i l i n g t e s t o f e a c h w a t e r - b e a r i n g s t r a t u m showing how much w a t e r was y i e l d e d , and how much t h e w a t e r l e v e l l o w e r s a t t h e g i v e n r a t e o f pumping o r b a i l i n g . " 2.3. H y d r o l o g i c C o n s i d e r a t i o n s "The h y d r o l o g i s t w i l l t h e n make a c o n t o u r map o f t h e w a t e r t a b l e he measures t h e d e p t h f r o m t h e l a n d s u r f a c e t o t h e w a t e r t a b l e a t w e l l s . Next, he d e t e r r n i n e s e i t h e r f r o m a t o p o g r a p h i c map o r by s u r v e y i n g , how much t h e l a n d i s above s e a l e v e l . F i n a l l y , he draws l i n e s t o c o n n e c t a l l t h e p o i n t s o f e q u a l e l e v a t i o n s o f t h e w a t e r t a b l e , so t h a t t h e map shows t h e shape o f t h e w a t e r t a b l e i n t h e same way t h a t a t o p o g r a p h i c map shows t h e shape o f t h e l a n d s u r f a c e . 6. "The water-table map i s especially important because i t gives a clue not only to the depth below which ground water i s stored, but also to the direction i n which the water i s moving. I f there i s any slope to the water table, the water moves i n the direction of the slope." 2.4. Test D r i l l i n g and Sample Analysis "Where there are no wells or not enough information on existing ones, the hydrologlst may have to put down some test holes . . . The samples of the earth material brought up by d r i l l i n g are examined and analyzed to determine which strata are water-bearing and how large an area they underlie. "Thus, there i s no magic about the hydrologist's work. It i s based on common sense and sc i e n t i f i c observation. He uses a l l the clues he can get - what he can see of the rocks as they are exposed at the land surface or i n road cuts, quarries, tunnels or mines and what he can learn from wells. "These ground water studies vary i n completeness with the need for information. I f the need i s mostly for domestic supplies, an area the size of a county can be studied i n a summer. The report and maps can be prepared the following winter. "The hydrologist's report and maps w i l l show where water can be obtained, what kind of water i t i s chemically, and i n a very general way how much i s available. I f a large supply Is needed or i f there are problems with the present supply, more detailed studies must be made, either i n the area where a large need exists or, i n some cases, where a future need i s anticipated. Whatever the scope of the study, the report i s designed to provide a sound basis for whatever may follow:\it, whether i t may be d r i l l i n g home and farm wells, or large-scale water projects for a city, for industry, or for an irrigation project." 7. 2.5 Surface and Subsurface Geophysical Methods If the exploration project i s economically important enough and i f the geologic framework of the area i s favourable, surface geophysical methods such as earth r e s i s t i v i t y and seismic surveys could be used to locate aquifers. The earth r e s i s t i v i t y method i s useful for the detection and delineation of near-surface aquifers often outlining the courses of buried valleys, while seismic prospecting provides f a i r estimates of layer depth. On the other hand, subsurface geophysical methods would also give more information about an aquifer. But before these methods are applied, an exploratory hole has to be d r i l l e d through the formations, obtaining samples while d r i l l i n g , and recording a log of the borehole. Well logging consists of recording characteristic properties of the various strata i n terms of depth. The next common well log i s the d r i l l e r ' s description of the geologic character of each stratum, the depth at which changes i n character were observed, the thickness of the strata, and the depth to water. 2.6 Logging Techniques Used In Ground Water Exploration Electric logging i s the most common borehole geophysical operation. It verifies and supplements the descriptive logging of the hole which the d r i l l e r records as d r i l l i n g proceeds. An electric log consists of a record of the apparent r e s i s t i v i t i e s of the subsurface formations and the spontaneous potentials generated i n the borehole, both plotted in terms of depth below the ground surface. These two properties are related indirectly to the character of the subsurface formations and to the quality of water contained i n them. They can be :... measured only i n mud-filled, uncased boreholes. 8. 2 .6 .1 . Spontaneous Potential The spontaneous potential or self-potential (SP) curve i s a record of natural voltages developed i n most d r i l l e d wells between dissimilar fluids contained i n the rocks penetrated and the borehole. The equipment used consists of two lead electrodes, one moving i n the d r i l l hole and the other stationary at the surface. The recorder plots m i l l i v o l t changes i n electric potential between these two electrodes as a function of depth. The source of spontaneous potential i n a d r i l l hole i s generally accepted to be the sum of electro-chemical and electro-kinetic potentials. The spontaneous potential curve may be used to calculate formation water r e s i s t i v i t y , locate bed boundaries, distinguish between shales and sandstone or limestone i n combination with other logs, and for stratigraphic correlation. The SP log i s affected by hole diameter, bed thickness, water or mud r e s i s t i v i t y , density, and chemical composition, mud cake thickness, mud f i l t r a t e invasion and well temperature. Although correction factors and curves are available to reduce or eliminate these effects, considerable information obviously must be available to make the necessary corrections. The SP log i s rarely used quantitatively i n groundwater hydrology, but i t i s widely run for qualitative lithological information. SP deflections are read from a shale baseline on the right to maximum negative deflections. The shale baseline i s drawn through as many deflection.minima as possible. A sand line may then be drawn through negative deflection maxima and i f f l u i d s a l i n i t y i s constant, these lines w i l l be parallel to each other and the zero baseline. 2.6...2. Resistivity Theoretically the r e s i s t i v i t y values recorded on a log are a measurement of the resistance of a cube of material measuring 1 meter along each edge, hence the units are ohms meter 2 per meter or simply ohm-meters. Since most 9. rocks consist of nonconductive particles, the nature of the pore spaces and i n t e r s t i t i a l fluids determines the character of the r e s i s t i v i t y curve. The numerous types of r e s i s t i v i t y curves made by commercial logging companies are differentiated by the configuration of the electrodes and the resulting differences i n the thickness of rock units measured and the depth of investigation. The single-point r e s i s t i v i t y log, along with the SP, i s the most widely used logging technique i n water wells. It detects very- thin beds and fracture zones.(Davis, S.N. and Dewiest, R.J.M., 1966). One principal use of the r e s i s t i v i t y curve i s that by merely glancing at i t , the water-well d r i l l e r can deterniine the depth and thickness of . almost every bed penetrated except the thinnest ones. When i t i s known that the quality of the water remains nearly the same for a l l the aquifers penetrated, changes i n r e s i s t i v i t y can generally be interpreted as being caused by changes i n porosity, or by a clayey condition. But simultaneous use of the SP or gamma ray curve w i l l assist i n determining which of the two situations actually exists. 2.6.3. Other Logging Methods Apart from electric logs, there are also radiation logs, acoustic logs, caliper logs, temperature logs, f l u i d conductivity logs, and f l u i d movement logs. Like many geophysical logs, any radiation log may be used to determine the depth and thickness of beds, and for subsurface mapping. Other applica- tions are: logging of cased holes (with the gamma ray and/or neutron, curve); identification of clay and shale beds (with the gamma ray curve); i d e n t i f i - cation of aquifers (with a combination of gamma ray and a neutron curve); and estimation of the porosity of aquifers (with any neutron curve or a gamma-gamma log). Radiation logs cannot be used to estimate the total dissolved solids (TDS) i n aquifer waters unless the solids are primarily 10. chlorides and exceed 40,000 parts per million (ppm). The applications of acoustic logs i n groundwater hydrology are: determination of porosity (from velocity measurements); location of fractured zones i n dense rocks (from amplitude measurements); and determination i n cased holes where cement makes good bond against casing and formation (from amplitude measurements). Neither the identification of rocks nor the estimation of TDS are possible from acoustic measurements:. Caliper logs have the following main applications to hydrology, namely: location of fractures, with a caliper having a single sharp feeler arm; "Ir -:.:Jy~.o location of washouts (hole enlargements) and other openings; guide to establish correction factors for measurements affected by hole size (in particular, r e s i s t i v i t y and neutron); and guide to well construction. Temperature logs are used i n the following: determination of the temperature of aquifer waters i n wells i n thermal equilibrium; location of sources of waters and thieving beds; study of seasonal recharge to a groundwater system; and study of the distribution of waste during disposal proj ects. The f l u i d conductivity log i s a record as a function of depth of the conductivity - or i t s r e c i p r i c a l , the r e s i s t i v i t y of the borehole f l u i d . Its main applications i n hydrology are: location of the point(s) of entry of formation water(s) into a well; location of the point.(s) of entry of injected water into permeable beds; and estimating the TDS of water in wells as a function of depth. Fluid movement logging methods determine the direction and velocity of natural or artificially-induced flow within a well (Guyod, H., 1972). 1 1 . 2 . 7 . Pump Tests If j after a l l the above-mentioned exploratory methods have been applied., and groundwater i s encountered during the test d r i l l i n g , the d r i l l e r can give a rough estimate of the yield of the well by balling. But, i f large quantities of water are needed and the funds are available, a pump test would be worthwhile i n order to obtain an exact yi e l d and the drawdown characteristics of the well. Before the pump test, however, the well i s developed by screening. The test data can also be used to determine the coefficient of storage of the aquifer. 2 . 8 Observation Wells Observation wells are used to monitor drawdown and pumpage character- i s t i c s of production wells. In order to obtain uniform distribution of drawdown, observation wells should not be locateda too close to the pumped well. They should be located about 100 feet to 300 feet from the pumped well (for unconfined aquifers) and about 300 feet to 700 feet (for confined squifers). .'A longer pumping duration is also required (Johnson, U.O.P., 1 9 7 2 ) . The number of observation wells to be employed depends upon the amount of information that i s desired and upon the funds available for the test program. The data obtained by measuring the drawdown at a single location outside the pumped well permit calculation of the average permeability and trarsmissibility of the aquifer and i t s coefficient of storage (Domenico, P.A. 1 9 7 2 ) . I f two or more observation wells are placed at different distances, the test data can be analyzed i n two ways by studying both the time-drawdown and the distance-drawdown relationships. Usually both these methods of analysis give a check on the results and enhance, the dependability of the conclusions. It Is always best to have as many observation wells as conditions allow. 12. 2 .9 Water Q u a l i t y Samples o f the water encountered i n the w e l l should be analyzed i n order t o ensure t h a t i t meets the r e q u i r e d standards f o r whatever purpose i t i s needed - whether f o r d r i n k i n g , i n d u s t r i a l use o r f o r i r r i g a t i o n (Todd, 1959). 2.10 Ground Water Recharge In order t o av o i d complete d e p l e t i o n o f the a q u i f e r , the va r i o u s modes of recharge are o f utmost importance. I n many p l a c e s , t h e major sources o f recharge t o a q u i f e r s are d i r e c t p r e c i p i t a t i o n on i n t a k e areas and/or downward p e r c o l a t i o n o f stream r u n o f f . There a r e , however, a r t i f i c i a l recharge t e c h - niques which i n some circumstances can be employed i f needed (Walton, 1970) . CHAPTER 3 DECISION ANALYSIS UNDER UNCERTAINTY IN GROUND WATER TERMS Decision making under uncertainty implies that there are at least two possible outcomes that could occur i f a particular course of action i s chosen. Or, i n other words, decision making under uncertainty occurs where the prob- a b i l i t i e s of the outcomes of any choice are not completely known. For example, when the decision to d r i l l a water well i s made, i t i s not known with certainty what the outcome would be. Even If water was encountered, i t i s not entirely certain what the yield of the well would be. A summary of the steps used i n solving decision analyses problems are as follows: 1 . To define the possible outcomes that could occur for each of the available decision choices, or alternatives. 2. To evaluate profit or loss (or any other measure of value or worth) for each outcome. 3 . To determine or estimate the probability of occurrence of each possible outome. 4. To calculate a weighted average profit (or measure of value) for each decision choice, where the weighting factors are the respective probabilities of occurrence of each outcome. This weighted average profit i s called the expected value of the decision alternative, and i s the comparative criterion used to accept or reject the alternative (Schlaifer, R., 1 9 6 9 ) . Usually, the most d i f f i c u l t problem i s obtaining the probabilities of occurrence of the various outcomes. Where no past s t a t i s t i c a l data are available, the geologist or hydrogeologlst after studying the area concerned, gives his subjective probability estimates which w i l l certainly be based on 1 3 his personal biases, emotions, and past experience. Herein l i e the elements of risk and uncertainty. For example, he could say that the probability of d r i l l i n g and hitting water Is 75% or even 20%. I f the owner of the d r i l l i n g project i s not satisfied with the geologist's probability estimate, he could purchase additional information in the way of d r i l l i n g a test hole, collecting samples and analyzing them to obtain permeabilities of the materials or even running r e s i s t i v i t y and spontaneous potential tests. Depending on the outcome of the additional information, the uncertainty involved would be reduced and new probability estimates could be obtained. These new estimates are obtained by updating the prior estimates using Bayesian Analysis (Benjamin and Cornell, 1970). 3.1. U t i l i t y Theory The concept of mathematical expectation, or expected monetary value (EMV), i s the traditional approach to decision making under conditions of uncertainty. Use of this criterion consists of multiplication of a probability of occurrence with the financial payoff for each possible outcome. For example, i f p i s the probability that a particular outcome w i l l occur and v i s the payoff (profit or loss) to be realized by the decision maker i f the outcome occurs, then p x v i s the "expected value" of the outcome. If there are two or more possible outcomes the expected values for each outcome are summed algebraically, with the decision being to accept the act i f the sum i s positive. I f several decision alternatives are being considered, the criterion i s to select the alternative which w i l l maximize expected monetary value. The expected monetary value concept implies that the decision maker i s totally impartial to money. But this i s not true because people are 15. not impartial to money. Rather, they have specific attitudes and feelings about money which depend on the amounts of money, their personal risk preferences, and any immediate and/or longer term objectives they may have. A decision maker's attitudes and feelings about money may change from day to day, and may even be influenced by such factors as his business surroundings, and the overall business climate at a given time. The noted Swiss mathe- matician, Daniel Bernoulli (1700-1782) was one of the f i r s t to suggest that monetary values alone do not adequately represent a person's value system. He suggested that the u t i l i t y (desirability, usefulness) of money i s inversely proportional to the amount he already has (Newendorp, P. 1975). The derivation of u t i l i t y theory i s based on eight axioms (von Neuman and Morgenstern). A person's u t i l i t y curve i s unique to him and increases with an increase i n pref erability. U t i l i t y values, or index numbers are dimensionless and the magnitude of the u t i l i t y scale i s arbitrary. U t i l i t y values are therefore used to replace monetary values and.hence expected u t i l i t i e s are calculated as before. The problem i n implementing u t i l i t y theory i s that at present there are no effective methods to construct or determine the u t i l i t y curve. Previous research on this problem has centred on the development and use of testing procedures to obtain the data needed to construct a u t i l i t y c:.. curve. These procedures generally have been based on offering the decision : .. maker a choice between a gamble having a desirable outcome (X) and a less desirable outcome (Z), or a no-risk alternative (Y) of intermediate desirability. The testing would seek to determine the decision maker's point of indifference between accepting the gamble (X ̂ occurring with probability p 'and Z occurring with probability 1 - p) or the no-risk alternative. The indifference point represents an'equality of the decision 16. maker's u t i l i t y for the gamble and the no-risk alternative; that i s p x-U(X) + ( 1 - p) x U(Z) = U(Y) ( 3 - D where U(X) = u t i l i t y value of outcome X. By ar b i t r a r i l y assigning numerical values to two of the above u t i l i t i e s , the third could be computed. With careful design of the testing sequence, these three numerical u t i l i t i e s would be used to compute successive u t i l i t i e s . After determining a sufficient number of u t i l i t i e s , a u t i l i t y curve would be drawn through the data points (Grayson, I960). In ground water terms, the u t i l i t y curve would be that of the well- owner and not of the hydrogeologist or d r i l l e r . This u t i l i t y curve could show either the well-owner's preferability for obtaining various water yields or the desirability of having to d r i l l to any depths. The next chapter w i l l show how this u t i l i t y theory can be applied to the development of three models for analyzing ground water decision problems. CHAPTER 4 MODELS FOR ANALYZING GROUND WATER DECISION PROBLEMS 4.1 Case I: Well Cost Known: Yield Not Known: This case would involve a trade-off between the cost of d r i l l i n g and the possible returns as regards the yield obtained from the well. And therefore, the u t i l i t y of d r i l l i n g and obtaining various yields and the u t i l i t y of not d r i l l i n g at a l l would be needed i n order to be able to make a decision. The u t i l i t y curve i s usually that of the owner of the well project and not that of the d r i l l e r nor that of the hydrogeologist. Figure 4.2 i s an example of one such u t i l i t y curve showing that beyond a particular yield, y(gpm), the well-owner's relative desirability ( u t i l i t y ) to d r i l l the well would be zero. But thereafter, his preference or u t i l i t y for d r i l l i n g would increase with an increase i n the yield. U t i l i t y curves such as in f i g . 4.2 are obtained by asking the well-owner questions such as: "Which alternative would you prefer - alternative (1) i n which you would obtain say yigpm for certain, or alternative (2) - a gamble i n which you have say a 75-25 chance of obtaining y2gpm or nothing (a dry hole)'?" y 2 Is very much greater than yi . If he replies that he feels the two alternatives are about equal, that i s , he i s "indifferent" between the two, then these alternatives would have the same u t i l i t y to him. If the u t i l i t y of y2gpm i s set equal to say 100 u t i l e s , and the u t i l i t y of a dry hole i s set equal to 0 utiles (or any arbitrary units could be chosen), then the u t i l i t y of yigpm would be calculated using the following equation: U(yi) = 0.75[U(y 2)] + 0.25[U(O)] (4.1) 17. FIG.4.1 i DECISION TREE SCHEMAT IC 19. 1 0 0 Yield (gpm) FIG. 4 . 2 5 UTILITY VERSUS YIELD. 20. By asking a series of such questions with different values of yields and probabilities, enough points could be obtained to plot his u t i l i t y curve, which is entirely unique to him. If the probabilities of obtaining the various yields are say pj for yield y l s p 2 for yield y 2 etc, and the u t i l i t i e s of the same yields are Ui, U 2 etc. as i n f i g . 4.1, then the expected u t i l i t y value of d r i l l i n g would be given by: n EUVn = , E p. U. (4.2) 1=1. 1 1 These probabilities of the various yields can be obtained i n either of two ways: 1. By asking a hydrogeologist who knows about the area i n question, and 2. By use of - cumulative probability curves where data are available. 4.1.1 U t i l i t y of Not D r i l l i n g The expected u t i l i t y value of not d r i l l i n g , EUV^, i s obtained by z. asking the well-owner a question such as: "If the only possible outcomes were the best (optimum yield) or the worst (dry hole), what would the chance of success have to be before you would accept to d r i l l ? " I f he says 80%, for example, then EUV^ would be equal to 80. . whichever value i s greater, EUVD or EUV^, indicates the best decision, that i s , either to d r i l l or not to d r i l l . 4.2 Case II: Well Cost Known; Well Depths Not Known; Yields Not Known; Stop Once an Aquifer i s Encountered 4.2.1 Case 11(a) - No Relationship Between Yield and Depth Here, there are several depths the well could be d r i l l e d to. But, for each particular depth, (say di with a probability p(^1 of getting water), there would be yields y1 y n with probabilities p y i .... p ^ 21. Utility Values FIG. 4.3(a)* DECISION TREE SCHEMAT IC . Utility Values FIG.4.31b) 8 DECISION TREE SHOWING EXPECTED V A L U E S . 22. and u t i l i t y values LT1.... U n associated with each yield value. The u t i l i t y versus yield curve would be obtained as i n Case I (f i g . 4 . 2 ) . The expected u t i l i t y values (EUV ••... EUV^) at the chance nodes A 1 ... A n are again calculated as in Case I using equation ( 4 . 2 ) . These expected u t i l i t y values would be the same for the various depths i f there were no relationahip between depth and yield. Since the cost of d r i l l i n g a hole i s charged per foot d r i l l e d plus mobilization and demobilization, there would be a u t i l i t y "cost" (UC^) associated with d r i l l i n g to any depth. This u t i l i t y "cost" or relative desirability of d r i l l i n g to any depth decreases with increase i n depth. Figure 4.4, therefore, shows the well-owner's u t i l i t y "cost" curve obtained by again asking him questions similar to those used i n obtaining f i g . 4.2 (Case I ) . The expected u t i l i t y value of d r i l l i n g to a l l the different depths XEUV^) (with probabilities of obtaining water p^ 1 ... p ^ and corresponding u t i l i t y "costs" UC d l ... UC^) i s again calculated using equation ( 4 . 2 ) . The difference between the expected u t i l i t y value of yie l d (EUV ) and the expected u t i l i t y value of depth, gives the expected net u t i l i t y value of d r i l l i n g decision (ENUV^). Or, E i W D =- EUVy - EUVd 4.2.2. Case 11(b): Probability Relationship Between Yield and Depth Where there i s a relationship between yie l d and depth i n the form of a probability band with a mean, lower and upper limits such as i n f i g . 4.5, then the expected u t i l i t y values (EUVyi E U V y n ^ a t t h e c n a n c e n o d e s A A n would be different because of the uncertainty involved. The probability of "yield" for a given value of "depth" i s assumed to have a Depth (feet) n FIG.4.4 : UTILITY "COST" VERSUS DEPTH . Depth (feet) FIG.4 . 5 ! YIELD VERSUS DEPTH PROBABILITY BAND. 25. skewed normal distribution between the upper and lower bounds. Using the u t i l i t y versus yield curve ( f i g . 4.2) and fig.-!4.5, different values of expected u t i l i t y of yields for a l l the various depths would be obtained. From each expected u t i l i t y value for a particular depth i s subtracted the u t i l i t y "cost" for that particular depth, to obtain an expected net u t i l i t y value (ENUV^) for that depth. This i s done for . a l l the different depths. The expected net u t i l i t y value of d r i l l i n g decision ((ENUVp) i s obtained by multiplying each expected net u t i l i t y value for a particular depth (ENUV^-) by the corresponding probability (P d l) of obtaining water at that depth, and summing over the entire range of depths. Or, F.NUV- = t -{p,.(0W,.)] ( i j* 3 ) D L * a i • di J 4.2.3. Expected U t i l i t y of Not D r i l l i n g To obtain the expected u t i l i t y value of not d r i l l i n g , the well-owner is asked a question such as: "You are offered two alternatives as follows: Alternative A: You do not d r i l l at a l l , but you obtain an outcome very close to the "best" - no risks involved. Alternative B: A gamble i n which you have a probability p of obtaining the "best" outcome and a probability (1 - p) of obtaining the "worst". At what probability values would you be indifferent between accepting alternative A or B?" In this particular case, the "best" outcome -would be to d r i l l to zero depth and s t i l l obtain the maximum yield. The u t i l i t y associated with this would be 200 (100 + 100) - obtained by combining the u t i l i t y curves of figs. 4.2 and 4.4. On the other hand, the "worst" outcome would be to 26. FIG.4.6= SCHEMATIC DECISION T R E E 27. d r i l l to the maximum depth only to obtain a zero yield. And again, combining figs. 4.2 and 4 . 4 , the u t i l i t y associated with this "worst" outcome would be zero (0+0). I f the well-owner's point of indifference were actually at p and (1 - p), then the expected u t i l i t y value of not d r i l l i n g (EUV^) would be given by the following equation: EUV^' = p x 200 + (1-p) x 0 (See the decision tree of f i g . 4 . 6 ) , Here again the decision to d r i l l or not to d r i l l would be made depending on which act has the greater expected net u t i l i t y value (that i s either ENUVD or EUV^). .4.3 Case III: Decision-to Purchase - Imperfect Information The importance of purchasing additional information i s to better define (or reduce) the uncertainty associated with the decisions to be made. For example, the decision to d r i l l a 700 foot water well could be deferred u n t i l say, a seismic and/or r e s i s t i v i t y survey i s run to better define the structure and i t s physical dimensions. Other examples of information purchased to better reduce, uncertainty are logging surveys, analysis of samples, and pump tests i n order to decide how many more wells have to be d r i l l e d to meet a specific water demand. If the additional information i s perfect (that i s , there i s no error i n the interpretation and i t w i l l t e l l precisely the true state of nature), a relatively straightforward analysis w i l l suggest whether i t i s feasible to purchase the information. But, i f the information i s imperfect, the analysis of whether to purchase the information becomes more complex. Figure 4.7 i s a schematic decision tree for the analysis of decisions to purchase imperfect information. Alternative, time-zero investment strategies i n l i e u of purchasing the adultional information purchase addi- t i o n a l i n f o r - mation . (before deci- ding which decision choice to accept). Various possible i n t e r - pretations, or evidence^ that could become available from the information that i s purchased (2 or more branches). Pr o b a b i l i t y of evidence occuring i s the denomi- nator term of Bayes' Theorem-for each possible evidence or interpretation of the purchased information. States of nature (out comes) that can occur for the choices (2 or more branches). Probability terms derived by solving Bayes' Theorem. F I G . 4.7-D E C I S I O N T R E E U S E D TO D E T E R M I N E T H E F E A S I B I L I T Y OF P U R C H A S I N G A D D I T I O N A L I N F O R M A T I O N . 2 9 . I f there were more than two possible interpretations of the information (E), the number of branches i n Section (A) would be increased accordingly. Similarly, for Sections (C) and (D) i f there were more choices and more possible states of nature (U ...U ). The probabilities on the chance node branches in Section (D) are obtained by solving Bayes''...Theorem. The probability terms represent the revised perceptions of the likelihoods of the various states of nature, given the new evidence or interpretation. The probability terms in Section (A) represent the denominator terms of Bayes' Theorem. Case III could be combined . with either Case I or Case II, and the analysis carried out as before. CHAPTER 5 APPLICATIONS 5.1 Ryder Lake District 5.1.1 Introduction Since the only source of water In the Ryder Lake District i s from undergroundj prospective settlers i n the area have always wanted to know what are the chances of obtaining the quantity of water they need before investing i n d r i l l i n g water wells. Obviously, this i s a big problem having to do with uncertainty. Therefore, a formal analysis using decision theory, w i l l throw more light on the decision to be made instead of the dependence on sheer intuition as i n the past. 5.1.2 Location Ryder Lake District i s a r o l l i n g h i l l y area with elevations that rise to more than 2,700 feet above sea-level. It i s located within Chilliwack District Municipality, and l i e s between longitudes 121° 51' and 121° 56'30" and latitudes 49° 05'30" and 49° 07'30". It i s bounded on the south by the Chilliwack River and on the east by the Skagit Range of the Cascade Mountains, and i s about 55 miles east of Vancouver. It has an area of approximately 26 square kilometers and a population of roughly 1,000. It i s partly a residential and partly farming community. 5.1.3 Climate The Ryder Lake area i s characterized by a heavy winter r a i n f a l l and a dry summer. About two-thirds of the annual average to t a l precipitation of about 56 inches occurs from October to March inclusive. Rainfall during the growing season - April to September - i s inadequate in most years for the maximum development and yield of crops. The heavy sustained rains from October to March replenish the groundwater reservoirs. During this period, 30. 31. l i t t l e water, apart from runoff, i s lost by evaporation and transpiration. The s o i l and the unconsolidated surface deposits above the water-tables are kept wet and maximum i n f i l t r a t i o n results. 5.1.4 S u r f i c i a l Geology The oldest known unconsolidated deposits i n the Ryder Lake area are the Huntingdon gravels. They appear to be stream deposits l a i d down during the retreat of the Cordilleran Ice (Vashon) Sheet and prior to the advance of the Sumas Ice. These gravels are overlain by sediments transported by the Sumas Ice Sheet which originated i n the Cascades some 11,000 years ago. Sumas t i l l , composed mainly of sand t i l l , boulders, gravel and clay is formed i n layers up to 50 or 60 feet thick, and i n places s t r a t i f i e d , overlying bedrock. A mechanical analysis of a fine fraction of this Sumas t i l l gave an average.result of 63 percent sand, 33 percent s i l t and 4 percent clay (Halstead, E.C., 196l) . The bedrock consists of shales and a r g l l l i t e s that may yield some ground water from joints and fracture zones. 5.1.5 Water Supply Groundwater i s the only source of water i n this area. And i n order to tap this water, wells had to be dug or d r i l l e d . The type of well depends partly on the depth to water but more on the financial resources of the well owner. About 60 percent of the inhabitants have dug wells to a maximum depth of about 20 feet i n unconfined or perched aquifers i n Sumas t i l l . These dug wells are commonly lined with concrete t i l e s or wood curbing, but those dug i n t i l l may not require l i n i n g as the compact t i l l w i l l stand without caving or slumping. Most of these wells do not yield sufficient supplies and often go dry i n summer. 32. Those of the inhabitants who could afford the b i l l , have d r i l l e d wells,;(See Table,,5.1 and f i g . 5-1). Drilled wells are the most effective type for the recovery of groundwater and are required especially where large yields are needed, such as for municipal or irrigation use. Drilled wells are lined with a casing commonly more than six inches i n diameter, and may be completed as open-end, screened, or gravel-packed wells. Cable-tool and rotary d r i l l i n g rigs are used, commonly the former because of the following advantages: 1) Economics: a) Lower i n i t i a l equipment cost, and hence lower depreciation. b) Lower daily operating cost, including maintenance, personnel, and water requirements. c) Lower transportation costs. d) Lower rig-up time and expense. e) D r i l l i n g rates comparable to rotary i n hard rocks at shallow depths. 2) Better cutting samples. 3) Easy identification of water-bearing strata. 4) No circulating system. 5) Minimum contamination of producing zones. (Campbell, M.D., and ! . :.- Lehr, J.H., 1973). There are, however, two groups of people i n the area that have constituted themselves into Water Users Communities. They are the Uplands Water Users Community and the Southside Water Users Community. The former obtains i t s water directly by channelling a l l the flows from a group of springs known as Eden Banks Springs. These springs produce nothing less than about 10,000 gallons of water per day which i s more than sufficient for the eighteen homes (lots) and one slaughter house they are supposed FIG. 5.1 1 MAP SHOWING DRILLED WELL L O C A T I O N S - R Y D E R L A K E A R E A - UJ i. 34. to serve. The Southside Water Users Community,:.made up of 20 homes (lots), also obtain their supply from a spring which flows into a dug well about 15 feet deep. Surprisingly, none of the supplies has gone dry so far. 5-1.6 Quality of Water The hardness of the groundwater i n this area ranges between 43 and 135 parts per million (ppm) (Halstead, 196l) . The water i s generally medium to soft, but there are some exceptions. Where hard water Is found, i t s total hardness i s not excessive and does not limit the use of the water. The water also f a l l s within safe limits for Irrigation use. Some might be rejected because of i t s high iron content and the probable damage i t could cause to the distribution system. 5.1.7 Decision Model Applications To Ryder Lake Area The only available well data for the study area as shown i n Table 5.1 was used i n a l l the calcuations and graphs. A Probability Matrix Program (set up i n the C i v i l Engineering Department for manipulating probability matrices and vectors with options for multi- plication, addition, subtraction, updating and rescaling) was used for the expected value computations. 5.1.7.1 Case I of Model F i r s t , a cumulative probability versus yield curve was produced using data from Table 5-1. Secondly, a u t i l i t y (of dri l l i n g ) versus yield curve ( f i g . 5-2) was obtained as i n f i g . 4.2 and using equation (4 .1) . The optimum domestic water requirement was taken as 1 gpm; and 5 gpm (a value below which no d r i l l i n g licence would be Issued) was assigned a u t i l i t y value of 100, that i s U(5 gpm) = 100, and U(0 gpm) = 0. Shown below i s a sample calculation of points plotted to obtain the u t i l i t y (of dr i l l i n g ) curve. TABLE 5.1 DRILLED WELL RECORDS - RYDER LAKE DISTRICT 35. Address Depth (ft) Dia. (In) Yield gpm Comments 1. Ryder Lake Rd. & No 2 Rd 132 6 Ik Quartz Lenses, (February 1977) Fractures % 132' 2. 48455 Elk View Road 744 6 ik 9 - 284 Bedrock 3. 48470 Elk View Road 104 6 2 0 - 1 0 t i l l ; (October 1975) 10 - 104, bedrock 4. 49185 Elk View Road 47 6 Dry 0 - 8 gravel; 8 - 1 7 t i l l ; (November 1972) 17 - 47, gravel 5. 5014 Farnham Road 249 Dry 0 - 4 2 sand, gravel (May. 1976) 240 - 249, clay hardpan 6. 49612 Atkins Road 110 Dry 0 - 4 1 gravel; 50 - 110 (December 1974) packed sand and gravel 7. Extrom Road (Location?) 365 Dry 345 - 365 gravel, sand (December 1974) some shale, 0 - 1 7 find sand 8. 47320 Extrom Road 500 3k 0 - 8 loam (July 1971) 340 - 500 shale 9. 46925 Extrom Road 269 6 1 0 - 3 0 hard packed sand and (1970) clay; 263 - 269 gray clay 10. 47200 Extrom Road 740 6 Dry 0 - 37 t i l l ; (May 1975) 281 - 740 bedrock 11. 46650 Thornton Road 443 6 Dry Sand and gravel (0 - 443) (July 1975) 12. 46880 Jinkesson Road 63 Dry 0 - 17 s i l t e d gravel and t i l l ; - (July 1958) 53 - 63 dry sand 13- 5296 Tesky Road 61 Dry Sand (September 1971) static 106 14. 5392 Tesky Road 343 6 2 , 0 - 2 6 t i l l ; (September 1975) 94 - 343 bedrock, shale 15. 47005 Russell Road Dry 0 - 60 t i l l and boulders (June 1977) static 19g,245 - 265 gravel, sand 16. 46655 Russell Road 343 6 lk 0 - 6 loam (December 1974) static 101' 17. 6l80 Promontory Rd . 127 6 3 0-20 dug well (1959) 112 - 118 w.b. sand 18. 588 Bailey Road 123 Dry 0 - 2 2 peaty loam 121 - 123 coarser sand 19. End of Parsons Road 380 6 Dry 0 - 155 sand, gravel (September 1974) 300 - 380 s i l t y sand & clay continued overleaf TABLE 5.1 (Continued) 36. 20. 6235 Parsons Road 320 (September 1974) 21. Lindel Road 455 (May 1976) 22. Lindel Road 85 (August 1976) 6 1 0 - 6 Dirt 6 - 320 shale 6 5g 0 - 2 1 broken shale 21 - 445 shale 6 3 0 - 9 overburden 9 - 8 5 shale 37. 2 3 Y ie ld ( g p m ) FIG. 5.2-UTIL ITY VERSUS YIELD (RYDER LAKE AREA) .38. 1st Question; posed to one of the well-owners, gave the following result: Alternative 1: Obtain 1 gpm for certain, or Alternative 2: A gamble in which there i s an 80-20 chance of obtaining 5 gpm or a dry hole (0 gpm). Using equation (4 . 1 ) , U(l gpm) .= 0.8[U(5 gpm] + 0.2[U(0 gpm)] . -. = 0..8.x 100 + 0.2 x 0 = 80 2nd Question; gave the following result: Alternative 1: Obtain 3 gpm for certain, or Alternative 2: A gamble i n which there i s a 50-50 chance of obtaining 5 gpm or 1 gpm. Again using equation (4 .1) , U(3 gpm) = 0.5[U(5 gpm)] + 0.5[U(1 gpm)] = 0.5 x 100 + 0 . 5 x 80 = 90 By inputting the curves i n figs. 5.2 and 5.3 into the computer program, and multiplying, the expected u t i l i t y value of d r i l l i n g , EUV^ was found to be 33-66. On the other_.hand, the expected u t i l i t y value of not d r i l l i n g , EUVj^, was obtained as 85 using the method of asking questions outlined i n the preceding chapter. Comparing both expected u t i l i t y values, the ultimate decision for this case would be not to d r i l l . 5.1.7.2 Case 11(a) of Model: No Relationship Between Yield and Depth F i r s t , the u t i l i t y versus yield curve ( f i g . 5.2) and the yield versus cumulative probability curve ( f i g . 5.3) were fed into the computer program and multipled to obtain an expected u t i l i t y value (EUV as regards to yield) 39. FIG.5.3= Y I E L D V E R S U S C U M U L A T I V E P R O B A B I L I T Y . - R Y D E R L A K E A R E A - of 33.66, as i n Case I. Secondly, the u t i l i t y "cost" versus depth curve ( f i g . 5.4) and the depth versus cumulative probability curve ( f i g . 5.5) were fed i n and again multiplied to obtain an expected u t i l i t y value (EUV^) as regards to depth) of 32.39. The difference of 1.27 between both values was found to be the expected net u t i l i t y value of the act "to d r i l l " . 5.1.7.3 Case 11(b) of Model: Probability Relationship Between Yield and Depth For this case, the u t i l i t y versus yield curve ( f i g . 5.2) and the yield versus depth curve ( f i g . 5.6) i n the form of a probability band were fed into the program and multiplied to obtain an expected u t i l i t y value i n the form of a matrix. From this matrix was subtracted the u t i l i t y "cost" versus depth curve ( f i g . 5.4) (in matrix form) to obtain expected net u t i l i t y values for a l l the various depths (also i n matrix form). The depth versus cumulative probability curve ( f i g . 5.5) was f i n a l l y fed i n . The matrix of the'expected net u t i l i t y values for the different depths was multiplied by the cumulative probability matrix to obtain the overall expected net u t i l i t y of d r i l l i n g decision of 43.17. Using equation (4.4) with p = 0.7, the expected u t i l i t y value of not d r i l l i n g (EUV^) was calculated to be 140. Comparing the expected net u t i l i t y values of the act "to d r i l l " , namely, 1.27 for Case 11(a), 43.17"' for Case 11(b) and the expected u t i l i t y value of the act "not to d r i l l " (140), the decision to be made would then be "not to d r i l l " . . 5.2 Inches Creek 5.2.1 Location Inches Creek study area i s part of the a l l u v i a l fan and flood plain deposits at the mouth of Norrish Creek on the north of the Fraser River about 80 kilometres east of "Vancouver (British Columbia) ( f i g . 5-7). Because  0 0.25 0.50 0.75 1.00 Cumulative Probability FIG.5.5= DEPTH VERSUS CUMULATIVE PROBABILITY. - R Y D E R L A K E A R E A -  44. of the hydrogeologlcal setting of Inches Creek, i t has become an important natural spawning ground for coho and chum salmon. 5.2.2 Objective of Study The objective of the study i s to provide approximately 4500 gpm of groundwater needed for salmon enhancement f a c i l i t i e s (spawning, hatchery, and incubation) for the Fisheries Department. 5.2.3 Aquifer Recharge The recharge to the aquifer i n Inches Creek area i s partly from precipitation and partly from Inflow from Norrish Creek. 5.2.4 Application Of Decision Model Case III The i n i t i a l problem in Inches Creek study area was the lack of past d r i l l e d well data from which probabilities of occurrence of the various well yields could be obtained. The hydrogeologists, therefore, had to carry out preliminary' geologic investigations and were able to give the following aquifer yield estimates: Minimum yield = 1000 gpm Most probable yield = 2500 gpm Maximum yield = 5000 gpm Applying a triangular distribution to the above, a yield versus cumulative probability (prior) curve ( f i g . 5-8) was obtained. In order to obtain some more information about the yield of the aquifer and hence the number of production wells that would be needed, a test hole was d r i l l e d and pump-tested at a .total cost of approximately $2,500. Based on the new test yields, the hydrogeologist, from past experience, was able to predict corresponding production well yields and hence the probability band shown in f i g . 5 .9.  0 Cumulative Probability FIG. 5 . 8 ' Y IELD VERSUS CUMULATIVE PROBABILITY ( PRIOR) - INCHES C R E E K - 47. FIG.5.9' PRODUCTION YIELD VS .TEST YIELD PROBABILITY BAND. - INCHES CREEK - 48. To obtain a monetary value versus yield curve such as i n f i g . 5.10, the owner of the project was asked how much he would be prepared to pay for various yields, for certain, i f he were buying ready-made wells. Figure 5.11 shows the decision tree layout. At the terminal H, the outcome of the various yields would be the dollar values C i , C 2, ....C5, obtained from f i g . 5-10. But at G, the corresponding outcomes would be (Ci + GYp), .... (C5 + C T), where C T i s the cost of the test well. Figure 5 . 9 3 when applied to the computer program (used i n Ryder Lake Analysis) produces probabilities [(p ̂ ) of production yields given the various test yields] i n the form of a matrix. Multiplying the row matrix produced from f i g . 5.10 (after adding C^) by the above matrix gives another row matrix of test expected monetary values TEMV"! TEMV5. The probabilities (P-ti"'"'" p t 5 ^ of obtaining the various test well yields are found., by using Bayes' Theorem. These probabilities multiplied by their corresponding (TEMV) values and summed gave the net test expected monetary value (NTEMV) of $i:,.l62. By using figs. 5.8 and 5-10, an expected monetary value (EMV) of $1,662 was obtained for the decision node C. And hence the decision to d r i l l a test hole has been proved to be ju s t i f i a b l e . 49. FIG.5.10= P R O D U C T I O N W E L L C O S T V E R S U S Y I E L D . - I N C H E S C R E E K - G .5.11: DECISION TREE SHOWING PURCHASE OF IMPERFECT INFORMATION - INCHES C R E E K - o CHAPTER 6 DISCUSSION AND CONCLUSIONS The decision models developed i n this thesis are to enable prospective water well owners to make the right decisions under condition of uncertainty, that i s , whether to invest i n d r i l l i n g or not; or whether to f i r s t of a l l spend extra money i n d r i l l i n g test holes i n order to gain more information about an aquifer before actually embarking on d r i l l i n g the required production well(s). The decision criterion, however, i s based on expected u t i l i t y which is a summation of the products of probabilities of obtaining the various yields and u t i l i t y values. The u t i l i t y values are those of the decision maker and entirely represent his preferences. One major problem, therefore, l i e s In obtaining a f a i r l y accurate u t i l i t y curve. And so far, there has not been a set-down procedure for achieving this. Where there are no past well records as i n the Inches Creek area, the probabilities of obtaining various yields w i l l certainly be those of the expert hydrogeologist. And, of course, these probabilities w i l l vary from one hydrogeologist to another. There i s no doubt, then, that the accuracy of the results w i l l be very much affected by these two parameters - u t i l i t y and probability (the source of uncertainty). The results for the Ryder Lake District indicate the decision of not to d r i l l any water wells at a l l while i n the Inches Creek area, the decision to d r i l l two production wells i n order to meet the 4500 gallons per minute requirement was made only after d r i l l i n g a test hole. The above decisions could have been made without going through a formal, systematic analysis as outlined i n the thesis. But what i f the decision maker has made a wrong decision. How would he exonerate himself? What would be his criterion for . 51 52. the decision he has made? Hence, i n order to protect himself, he definitely w i l l need to follow a rational process, considering a l l risks involved before arriving at a f i n a l decision. Finally, use of the techniques i n this thesis w i l l enable the j u s t i - fication of major decisions especially those dealing with public resources such as the Fisheries. BIBLIOGRAPHY 53- "A Primer on Ground Water", U.S. Geological Survey, Washington, D.C, 1963. Benj.amin, J.R., and Cornell, C.A., Probability St a t i s t i c s , and Decision for C i v i l Engineers. McGraw-Hill Book Co., New York, 1970. Campbell, M.D., and Lehr, J.H., Water Well Technology. McGraw-Hill Book Co., New York, 1973. Davis, S.N., and DeWiest, R.J.M., Hydrogeology. John Wiley and Sons Inc., New York, 1966. Domenlco, P.A., Concepts and Models i n Groundwater Hydrology. McGraw-Hill Book Co., New York, 1972. Grayson, C.J., Jr. Decisions Under Uncertainty, D r i l l i n g Decisions by O i l and Gas Operators. Harvard Business School, Division of Research, Boston, I 9 6 0 . Guyod, H., "Application of Borehole Geophysics to the Investigation and Development of Ground Water Resources". Water Resources Bulletin, Vol. 8_, No. 1., February, 1972. Halstead, E.C., "Groundwater Resources of Sumas, Chilliwack, and Kent Municipalities, B r i t i s h Columbia", Geological Survey of Canada, Paper 60-29, 1961. Halter, A.N. and Dean, G.W., Decisions Under Uncertainty with Research Applications. South-Western Publishing Co., Cincinnati, 1971. Hershman, S.H., "An Application of Decision Theory to Water Quality Management". M.A.Sc. Thesis, Dept.v;;:of-.Civil. Engrg..;. University of B r i t i s h Columbia, Vancouver, 1974. Johnson, E.E. Ground Water and Wells. Johnson Division, Universal O i l Products Co., Saint Paul, Minnesota, 1972. Newendorp, P.D. . Decision Analysis for Petroleum Exploration. Petroleum .". Publishing Co., Tulsa, 1975. Schlaifer, R. Analysis of Decisions Under Uncertainty. McGraw-Hill Book Co., New York, 1969. Todd, D.K. Ground Water Hydrology, John Wiley and Sons, Inc., 1959- Von Neuraan, J. and Morgenstern, 0. Theory of Games and Economic Behaviour Princeton, New Jersey, 1953- Walton, W.C. Groundwater Resource Evaluation. McGraw-Hill Book Co., New York, 1970.

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