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Decision analysis applied to ground water exploration Aginah, Benedict Anekwe 1979

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DECISION ANALYSIS APPLIED TO GROUND WATER EXPLORATION  by  BENEDICT ANEKWE AGINAH B.Eng., Ahmadu B e l l o 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 t h i s 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 advanced  an the I  Library  further  for  this  thesis  degree shall  agree  scholarly  in p a r t i a l  fulfilment  o f the  at the U n i v e r s i t y  of B r i t i s h  Columbia,  make that  permission  purposes  by  his representatives.  of  this  written  thesis  available  may be g r a n t e d It  for financial  of  The U n i v e r s i t y  of British  2075 Wesbrook Place Vancouver, Canada V6T 1W5  ' j ^ ^ T i " j  Columbia  copying  of this  that  study. thesis  by t h e Head o f my Department o r  is understood gain  I agree  f o r r e f e r e n c e and  for extensive  permission.  Department  Date  it freely  requirements f o r  shall  that  copying  n o t be a l l o w e d  or publication w i t h o u t my  ABSTRACT  An outline o f the e s s e n t i a l steps needed i n ground water exploration i s given. the  Since d r i l l i n g f o r ground water involves a l o t of uncertainty,  main concepts of Bayesian decision theory are b r i e f l y reviewed. Three  models f o r analyzing ground water decision problems are developed with an emphasis on the well-owner's u t i l i t y or d e s i r a b i l i t y t o a c t u a l l y venture to invest on a w a t e r - d r i l l i n g project. i s I l l u s t r a t e d by applications to  F i n a l l y , use of .the decision models  a) Ryder Lake D i s t r i c t ( i n 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 f o r salmon enhancement facilities.  ii  TABLE'OP CONTENTS  ABSTRACT  Page  i i  LIST OP TABLES  v  LIST OP FIGUFES  vi  ACKNOWLEDGEMENT  vii  CHAPTER 1.  INTRODUCTION  1  2.  SUMMARY OP GROUND WATER EXPLORATION  4  3.  2.1  Geologic: ^ Considerations  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  DECISION ANALYSIS UNDER UNCERTAINTY IN GROUND WATER TERMS 3.1  4.  4  U t i l i t y Theory  ... 13 14  MODELS FOR ANALYZING GROUND WATER DECISION PROBLEMS  17  4.1  17  Cast I 4.1.1  U t i l i t y o f Not D r i l l i n g iii  20  TABLE OF CONTENTS (continued)  Page  CHAPTER 4.  4.2  Cast I I 4.2.1  4.2.2  20  Case 11(a):  Case 11(b):  No Relationship Between Y i e l d and Depth  20  P r o b a b i l i t y Relationship Between Y i e l d and Depth  4.2.3 4.3 5.  Expected U t i l i t y o f Not D r i l l i n g  Case I I I - Decision to Purchase Imperfect Informtion  25 ..  27  APPLICATIONS  30  5.1  Ryder Lake D i s t 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 5.1.6  Water Supply Quality o f Water  31 34  5.1.7  Decision Model Applications t o Ryder Lake Area .  34  5.1.7-1  Case I o f Model  34  5.1.7.2  Case 11(a) of Model  38  5.1.7.3  Case 11(b) o f Model  40  5.2  6.  20  Inches Creek  40  5.2.1  Location  40  5.2.2  Objective o f Study  44  5.2.3  Aquifer Recharge  44  5.2.4  Application o f Decision Model Case I I I  44  DISCUSSION AND CONCLUSIONS  51  BIBLIOGRAPHY  53 iv  LIST  OF  TABLES  Table  5.1  Page  Drilled Well Records - Ryder Lake Area  v  35  LIST  OF  FIGURES  Figure  "" ;Page 18  4.1  Decision Tree Schematic  (Case I)  4.2  U t i l i t y versus Y i e l d  4.3(a)  Decision Tree Schematic  4.3(b)  Decision Tree Showing Expected Values  21  4.4  U t i l i t y "Cost" versus Depth  23  4.5  Y i e l d versus Depth Probability Band  24  4.6  Schematic Decision Tree  26  4.7  Decision Tree - Purchase Of Additional Information  5.1  Map Showing D r i l l e d Well Locations - Ryder Lake Area  33  5.2  U t i l i t y versus Y i e l d (Ryder Lake Area)  36  5-3  Y i e l d versus Cumulative Probability - Ryder Lake Area  5.4  U t i l i t y "Cost" versus Depth (Ryder Lake Area)  40  5.5  Depth versus Cumulative P r o b a b i l i t y - Ryder Lake Area  4l  5.6  Y i e l d versus Depth Probability Band - Ryder Lake Area  42  5-7  Inches Creek Location Map  44  5.8  Y i e l d versus Cumulative Probability (Prior) - (Inches Creek).  45  5.9  Production Y i e l d versus Test Y i e l d Probability Band  19 (Case II)  21  (Inches Creek)  •  ....  28  38  -  46  5.10  Production Well Cost versus Y i e l d - (Inches Creek)  48  5.11  Decision Tree showing Purchase of Imperfect Information (Inches Creek)  49  vi  ACKNOWLEDGEMENT  The author wishes t o express h i s sincere gratitude t o h i s supervisor Professor S.O. Russell f o r h i s invaluable guidance, assistance and constant encouragement throughout the preparation o f t h i s t h e s i s .  Special thanks are  also extended t o Mr. E.C. Halstead and Mr. H.M. Liebscher (both o f the Hydrology Research D i v i s i o n , Vancouver, B.C.) f o r t h e i r advice and assistance i n data c o l l e c t i o n ; and t o Mr. Ron Grigg f o r assistance i n computer programming. And, f i n a l l y , the author would also l i k e t o thank the C i v i l Engineering Department f o r f i n a n c i a l assistance.  vii  CHAPTER  1  INTRODUCTION  Human consumption of ground water has been increasing s t e a d i l y over the years, e s p e c i a l l y i n the past seventy-eight years.  This has been as a r e s u l t  of increased use of i r r i 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 n i n e t y - f i v e percent of the world's t o t a l fresh water supply, are generally uncontaminated i n 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 l o c a l i t y , yet i t s quality i s more uniform during the year.  The temperature of ground water,  l i k e i t s chemical q u a l i t y , i s also r e l a t i v e l y uniform throughout the year. This makes i t preferable f o r many uses e s p e c i a l l y f o r the f i s h i n g industry, and also f o r 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; f o r one  cannot say exactly what the outcome of a d r i l l i n g project would be even a f t e r the hydrologist (the expert) has predicted a good aquifer. be a dry hole or an undesirable y i e l d .  The outcome could  A systematic and formal analysis to  take care of the r i s k 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 d i s c i p l i n e consisting of various methods, techniques, and attitudes to help the decision maker to choose wisely under these conditions of uncertainty. has already been applied to o i l and gas exploration (Grayson,  This analysis  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.  I t requires that the explorer (expert.) associate  s p e c i f i c p r o b a b i l i t i e s with the possible outcomes (dry hole, or various y i e l d s ) ; and t h i s i s where the element o f r i s k 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 h i s u t i l i t y values or d e s i r a b i l i t y of the various outcomes.  F i n a l l y , expected u t i l i t y values, which form the  basis f o r decision, are computed. The objective of t h i s t h e s i s , therefore, i s to apply decision theory i n the search f o r 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 B r i t i s h Columbia; and also to Inches Creek where approximately 4500  gallons per minute of ground water i s needed f o r salmon enhancement  facilities. water supply.  Ryder Lake D i s t r i c t depends s o l e l y on ground water f o r i t s 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 n u t s h e l l , the procedures of  carrying out a decision analysis, more e s p e c i a l l y as i t applies to the search f o r 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 d i f f e r e n t  models that can be used i n analyzing ground water decision problems are developed and explained i n Chapter 4.  Chapter 5 i l l u s t r a t e s how the above  models can be applied to real-world situations such as i n the Ryder Lake D i s t r i c t and i n Inches Creek study area.  A description of the computer  program used i n the analysis i s also given i n t h i s chapter, while the discussion of r e s u l t s and the conclusions are given i n Chapter 6.  CHAPTER  2  SUMMARY OF GROUND WATER EXPLORATION In the past, the only method of prospecting f o r ground water was "water witching" o r "dowsing".  But t h i s method has proved most u n r e l i a b l e  and a more s c i e n t i f i c approach had t o be found. In 1 9 6 3 , the U.S.  Geological Survey published a report"*' summarizing  a general approach t o ground water exploration.  The following paragraphs  are taken from that report. 2.1  Geologic. Oonsiderations. "Certain clues are h e l p f u l i n l o c a t i n g ground water supplies. For  instance, ground water i s l i k e l y t o occur i n l a r g e r quantities under v a l l e y s than under h i l l s .  In a r i d regions, c e r t a i n types of water-loving plants give  the clue that there has t o 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 t o have some ground water, though not . .:,co necessarily i n large quantity o r of usable q u a l i t y . "But the most valuable clues are the rocks.  Hydrologlsts and geologists  use the word rock t o mean both hard, consolidated formations, such as sandstone, limestone, granite, or l a v a rocks, and loose unconsolidated sediments such as gravel, sand, and clay.  They use the word aquifer f o r a layer of rock that  carries a usable supply of water.  Gravel, sandstone, and limestone are the  best water c a r r i e r s but they form only a f r a c t i o n o f the rocks i n the earth's outer crust.  Not a l l of them y i e l d useful supplies o f water.  The bulk of  the rocks consist o f clay, shale, and c r y s t a l l i n e rocks - a term used f o r the great variety of hard rocks that form most of the earth's crust.  Clay,  shale, and c r y s t a l l i n e rocks are a l l poor producers, but they may y i e l d enough water f o r domestic stock uses i n 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 s h o w i n g 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 a c c o m p a n y i n g 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.  Fast Records "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 can 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 kind of rocks these wells penetrate.  Much o f what he i s i n t e r e s t e d i n i s  b e l o w 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 n e e d s 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  types o f d i f f e r e n t rock s t r a t a are h e l p f u l . A r e a l l y u s e f u l w e l l record w i l l include the following:  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  water-bearing  s t r a t u m s h o w i n g 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.  Hydrologic  Considerations  "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 deterrnines 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 land surface.  6. "The water-table map i s e s p e c i a l l y important because i t gives a clue not only to the depth below which ground water i s stored, but also t o the d i r e c t i o n i n which the water i s moving.  I f there i s any slope t o the water  t a b l e , the water moves i n the d i r e c t i o n o f 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 e x i s t i n g ones, the hydrologlst may have t o 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 s t r a t a are water-bearing and how large an area they underlie. "Thus, there i s no magic about the hydrologist's work. common sense and s c i e n t i f i c observation.  I t i s based on  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 f o r information.  I f the need i s mostly f o r 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 o f 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 d e t a i l e d studies must be made, e i t h e r i n the area where a large need exists or, i n some cases, where a future need i s anticipated. Whatever the scope o f the study, the report i s designed to provide a sound basis f o r 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 f o r a c i t y , f o r industry, or f o r an i r r i g a t i o n project."  7. 2.5  Surface and Subsurface Geophysical Methods I f the exploration project i s economically important  enough and i f  the geologic framework o f 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 f o r the detection and delineation of near-surface aquifers often o u t l i n i n g the courses of buried v a l l e y s , while seismic prospecting provides f a i r estimates o f 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 t o 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 l o g o f the borehole.  Well logging consists  of recording c h a r a c t e r i s t i c properties o f the various s t r a t a i n terms o f depth.  The next common w e l l l o g i s the d r i l l e r ' s d e s c r i p t i o n o f the geologic  character o f each stratum, the depth at which changes i n character were observed, the thickness o f the s t r a t a , and the depth to water. 2.6  Logging Techniques Used In Ground Water Exploration Electric  It v e r i f i e s  logging i s the most common borehole geophysical operation.  and supplements the descriptive logging o f the hole which the  d r i l l e r records as d r i l l i n g proceeds. An e l e c t r i c  log consists o f 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 p l o t t e d i n terms o f depth below the ground surface. These two properties are r e l a t e d i n d i r e c t l y to the character o f the formations and to the q u a l i t y o f water contained i n them. measured only i n mud-filled, uncased boreholes.  subsurface  They can be :...  8. 2 . 6 . 1 . Spontaneous Potential The spontaneous p o t e n t i a l or s e l f - p o t e n t i a l (SP) curve i s a record o f natural voltages developed i n most d r i l l e d wells between d i s s i m i l a r f l u i d s contained i n the rocks penetrated and the borehole.  The equipment used  consists o f 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 e l e c t r i c  p o t e n t i a l between these two electrodes as a function of depth.  The source of  spontaneous p o t e n t i a l i n a d r i l l hole i s generally accepted to be the sum o f electro-chemical and e l e c t r o - k i n e t i c p o t e n t i a l s . The spontaneous p o t e n t i a l curve may be used to calculate formation water r e s i s t i v i t y , locate bed boundaries, d i s t i n g u i s h between shales and sandstone or limestone i n combination with other logs, and f o r stratigraphic correlation.  The SP l o g 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 w e l l temperature.  Although correction factors  and curves are available t o reduce or eliminate these e f f e c t s , considerable information obviously must be available to make the necessary corrections. The SP l o g i s r a r e l y used quantitatively i n groundwater hydrology, but i t i s widely run f o r q u a l i t a t i v e l i t h o l o g i c a l 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 l i n e may then be drawn through negative d e f l e c t i o n  maxima and i f  f l u i d s a l i n i t y i s constant, these l i n e s w i l l be p a r a l l e l to each other and the zero baseline. 2.6...2. R e s i s t i v i t y 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 o f material measuring 1 meter along each edge, 2  hence the units are ohms meter per meter or simply ohm-meters.  Since most  9. rocks consist o f nonconductive p a r t i c l e s , the nature o f the pore spaces and i n t e r s t i t i a l f l u i d s determines the character of the r e s i s t i v i t y curve. The numerous types o f r e s i s t i v i t y curves made by commercial logging companies are d i f f e r e n t i a t e d by the configuration o f the electrodes and the r e s u l t i n g differences i n the thickness o f rock units measured and the depth o f 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.  I t detects very-  t h i n beds and fracture zones.(Davis, S.N. and Dewiest, R.J.M., 1966). One p r i n c i p a l use o f 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 o f . almost every bed penetrated except the thinnest ones. When i t i s known that the q u a l i t y 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 a s s i s t  i n determining which o f the two situations a c t u a l l y e x i s t s . 2.6.3. Other Logging Methods Apart from e l e c t r i c logs, there are also r a d i a t i o n logs, acoustic logs, c a l i p e r logs, temperature logs, f l u i d conductivity logs, and f l u i d movement logs. Like many geophysical logs, any r a d i a t i o n l o g may be used t o determine the depth and thickness o f beds, and f o r subsurface mapping. tions are:  Other a p p l i c a -  logging of cased holes (with the gamma ray and/or neutron, curve);  i d e n t i f i c a t i o n o f clay and shale beds (with the gamma ray curve); i d e n t i f i cation o f aquifers (with a combination o f gamma ray and a neutron curve); and estimation o f the porosity of aquifers (with any neutron curve or a gamma-gamma l o g ) .  Radiation logs cannot be used t o estimate the t o t a l  dissolved s o l i d s (TDS) i n aquifer waters unless the solids are primarily  10.  chlorides and exceed 40,000 parts per m i l l i o n (ppm). The applications of acoustic logs i n groundwater hydrology are: determination of porosity (from v e l o c i t y measurements); l o c a t i o n 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). the estimation of TDS  Neither the i d e n t i f i c a t i o n of rocks nor  are possible from acoustic measurements:.  Caliper logs have the following main applications to hydrology, namely: l o c a t i o n of fractures, with a c a l i p e r having a single sharp f e e l e r arm; "Ir -:.: y~.o J  l o c a t i o n of washouts (hole enlargements) and other openings; guide to e s t a b l i s h correction factors f o r measurements affected by hole s i z e ( i n p a r t i c u l a r , r e s i s t i v i t y and neutron); and guide to w e l l construction. Temperature logs are used i n the following: determination of the temperature of aquifer waters i n wells i n thermal equilibrium; l o c a t i o n of sources of waters and t h i e v i n g beds; study of seasonal recharge to a groundwater system; and study of the d i s t r i b u t i o n of waste during disposal proj ects. The f l u i d conductivity l o g 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: l o c a t i o n of the point(s) of entry of formation water(s) i n t o a w e l l ; l o c a t i o n of the point.(s) of entry of i n j e c t e d water i n t o permeable beds; and estimating the TDS  of water i n  wells as a function of depth. F l u i d movement logging methods determine the d i r e c t i o n and v e l o c i t y of natural or a r t i f i c i a l l y - i n d u c e d flow within a w e l l (Guyod, H.,  1972).  11.  2.7.  Pump Tests I f j a f t e r 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 o f the y i e l d o f the well by b a l l i n g .  But, i f large  quantities of water are needed and the funds are a v a i l a b l e , a pump test would be worthwhile i n order t o obtain an exact y i e l d and the drawdown c h a r a c t e r i s t i c s of the well. developed by screening.  Before the pump t e s t , however, the w e l l i s  The test data can also be used to determine the  c o e f f i c i e n t o f storage of the aquifer. 2.8  Observation  Wells  Observation wells are used to monitor drawdown and pumpage characteri s t i c s of production wells.  In order to obtain uniform d i s t r i b u t i o n o f  drawdown, observation wells should not be locateda too close t o the pumped well.  They should be located about 100 feet to 300 feet from the pumped  w e l l (for unconfined aquifers) and about 300 feet to 700 feet (for confined s q u i f e r s ) . .'A longer pumping duration i s 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 f o r the test program.  The data obtained by measuring the drawdown at a single l o c a t i o n  outside the pumped w e l l permit c a l c u l a t i o n o f the average permeability and trarsmissibility of the aquifer and i t s c o e f f i c i e n t o f storage (Domenico, P.A. 1972).  I f two or more observation wells are placed at d i f f e r e n t distances,  the test data can be analyzed i n two ways by studying both the time-drawdown and the distance-drawdown r e l a t i o n s h i p s .  Usually both these methods o f  analysis give a check on the r e s u l t s and enhance, the dependability of the conclusions.  I t 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 t h e w a t e r e n c o u n t e r e d i n t h e w e l l s h o u l d be a n a l y z e d i n  o r d e r t o ensure t h a t i t meets t h e r e q u i r e d s t a n d a r d s 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 I n o r d e r t o a v o i d complete d e p l e t i o n o f t h e a q u i f e r , t h e v a r i o u s modes  o f r e c h a r g e a r e o f utmost i m p o r t a n c e .  I n many p l a c e s , t h e major s o u r c e s o f  r e c h a r g e t o a q u i f e r s a r e 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 a r e a s 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 r e c h a r g e t e c h -  n i q u e s w h i c h i n some c i r c u m s t a n c e s c a n be employed i f needed ( W a l t o n , 1 9 7 0 ) .  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 p a r t i c u l a r course of action i s chosen. Or, i n other words, decision making under uncertainty occurs where the proba 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 w e l l i s made, i t i s not known with certainty what the outcome would be.  Even I f water was  encountered,  i t i s not e n t i r e l y c e r t a i n what the y i e l d of the w e l l 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 f o r each of the available decision choices, or a l t e r n a t i v e s .  2.  To evaluate p r o f i t or loss (or any other measure of value or worth) f o r each outcome.  3.  To determine or estimate the p r o b a b i l i t y of occurrence of each possible outome.  4.  To calculate a weighted average p r o f i t (or measure of value) f o r each decision choice, where the weighting factors are the respective p r o b a b i l i t i e s of occurrence of each outcome.  This weighted  average  p r o f i t i s c a l l e d the expected value of the decision a l t e r n a t i v e , and i s the comparative c r i t e r i o n used to accept or r e j e c t the alternative ( S c h l a i f e r , R.,  1969).  Usually, the most d i f f i c u l t problem i s obtaining the p r o b a b i l i t i e s of occurrence of the various outcomes.  Where no past s t a t i s t i c a l data are  a v a i l a b l e , the geologist or hydrogeologlst a f t e r studying the area concerned, gives h i s subjective p r o b a b i l i t y estimates which w i l l c e r t a i n l y be based on 13  h i s personal biases, emotions, and past experience. of r i s k and uncertainty.  Herein l i e the elements  For example, he could say that the p r o b a b i l i t y of  d r i l l i n g and h i t t i n g water Is 75% or even  20%.  I f the owner of the d r i l l i n g project i s not s a t i s f i e d with the geologist's p r o b a b i l i t y estimate, he could purchase a d d i t i o n a l information i n the way  of d r i l l i n g a test hole, c o l l e c t i n g samples and analyzing them  to obtain permeabilities of the materials or even running r e s i s t i v i t y spontaneous p o t e n t i a l t e s t s .  Depending on the outcome of the a d d i t i o n a l  information, the uncertainty involved would be reduced and new estimates could be obtained.  These new  probability  estimates are obtained by  the p r i o r estimates using Bayesian Analysis (Benjamin 3.1.  and  and  updating  C o r n e l l , 1970).  U t i l i t y Theory The concept of mathematical expectation, or expected monetary value  (EMV), i s the t r a d i t i o n a l approach to decision making under conditions of uncertainty.  Use of t h i s c r i t e r i o n consists of m u l t i p l i c a t i o n of a  p r o b a b i l i t y of occurrence with the f i n a n c i a l payoff f o r each possible outcome. For example, i f p and  v  i s the p r o b a b i l i t y that a p a r t i c u l a r outcome w i l l occur  i s the payoff ( p r o f i t or loss) to be r e a l i z e d by the decision maker  i f the outcome occurs, then  p x v  i s the "expected value" of the outcome.  I f there are two or more possible outcomes the expected values f o r each outcome are summed a l g e b r a i c a l l y , with the decision being to accept act i f the sum i s p o s i t i v e .  the  I f several decision alternatives are being  considered, the c r i t e r i o n i s to select the a l t e r n a t i v e which w i l l maximize expected monetary value. The expected monetary value concept implies that the decision maker i s t o t a l l y impartial to money.  But t h i s i s not true because people are  15. not impartial to money.  Rather, they have s p e c i f i c attitudes and feelings  about money which depend on the amounts of money, t h e i r personal r i s k 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 h i s business and the o v e r a l l business climate at a given time.  surroundings,  The noted Swiss mathe-  matician, Daniel B e r n o u l l i (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 ( d e s i r a b i l i t y , usefulness) o f money i s inversely proportional t o 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 prefe r a b i l i t y .  U t i l i t y values, or index numbers are  dimensionless and the magnitude o f the u t i l i t y scale i s a r b i t r a r y .  Utility  values are therefore used t o 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 e f f e c t i v e methods to construct or determine the u t i l i t y  curve.  Previous research on t h i s problem has centred on the development and use of t e s t i n g procedures t o obtain the data needed to construct a u t i l i t y c:.. curve.  These procedures generally have been based on o f f e r i n g the d e c i s i o n : ..  maker a choice between a gamble having a desirable outcome (X) and a less desirable  outcome  desirability.  (Z), or a no-risk alternative (Y) o f intermediate  The t e s t i n g would seek t o determine the d e c i s i o n maker's  point of indifference between accepting the gamble (X ^occurring with probability alternative.  p 'and Z  occurring with p r o b a b i l i t y  1 - p) or the no-risk  The indifference point represents an'equality o f the decision  16. maker's u t i l i t y f o r the gamble and the no-risk a l t e r n a t i v e ; that i s  p x-U(X) + ( 1 - p) x U(Z)  where  U(X) = u t i l i t y value of outcome  =  U(Y)  (3-D  X.  By a r 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 t h i r d could be computed.  With careful design of the t e s t i n g 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 s u f f i c i e n t 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 w e l l owner and not of the hydrogeologist or d r i l l e r . show e i t h e r the well-owner's  This u t i l i t y curve could  p r e f e r a b i l i t y f o r obtaining various water y i e l d s  or the d e s i r a b i l i t y of having to d r i l l to any depths. The next chapter w i l l show how t h i s u t i l i t y theory can be applied to the development of three models f o r analyzing ground water decision problems.  CHAPTER 4 MODELS FOR ANALYZING GROUND WATER DECISION PROBLEMS 4.1  Case I: Well Cost Known:  Y i e l d Not Known:  This case would involve a trade-off between the cost o f d r i l l i n g and the possible returns as regards the y i e l d obtained from the w e l l . And therefore, the u t i l i t y o f d r i l l i n g and obtaining various y i e l d s 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 o f the owner of the well project and not that o f the d r i l l e r nor that o f the hydrogeologist. Figure 4.2 i s an example o f one such u t i l i t y curve showing that beyond a p a r t i c u l a r y i e l d ,  y(gpm),  the well-owner's r e l a t i v e d e s i r a b i l i t y  ( u t i l i t y ) to d r i l l the well would be zero.  But thereafter, h i s preference  or u t i l i t y f o r d r i l l i n g would increase with an increase i n the y i e l d . U t i l i t y curves such as i n f i g . 4.2 are obtained by asking the well-owner questions such as:  "Which a l t e r n a t i v e would you prefer - a l t e r n a t i v e (1)  i n which you would obtain say yigpm  f o r c e r t a i n , or a l t e r n a t i v e (2) -  a gamble i n which you have say a 75-25 chance of obtaining (a dry hole)'?"  y  2  Is very much greater than  y2gpm or nothing  y i . I f he r e p l i e s that he  feels the two alternatives are about equal, that i s , he i s " i n d i f f e r e n t " between the two, then these alternatives would have the same u t i l i t y to him. I f the u t i l i t y of y2gpm  i s set equal t o say 100 u t i l e s , and the  u t i l i t y o f a dry hole i s set equal to 0 u t i l e s (or any a r b i t r a r y 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 )] + 0.25[U(O)] 2  17.  (4.1)  FIG.4.1 i DECISION T R E E S C H E M A T I C  19.  1 0 0  Yield (gpm)  FIG. 4 . 2  5  UTILITY VERSUS YIELD.  20. By asking a series of such questions with d i f f e r e n t values o f y i e l d s and p r o b a b i l i t i e s , enough points could be obtained t o plot h i s u t i l i t y curve, which i s e n t i r e l y unique t o him. I f the p r o b a b i l i t i e s o f obtaining the various y i e l d s are say pj for y i e l d are  Ui, U  y 2  l s  p  2  for yield  y  e t c , and the u t i l i t i e s of the same y i e l d s  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 EUV  =  n  ,E  p.  1=1.  U.  1  (4.2)  1  These p r o b a b i l i t i e s of the various y i e l d s can be obtained i n either of two ways: 1.  By asking a hydrogeologist who knows about the area i n question, and  2. 4.1.1  By use of - cumulative p r o b a b i l i t y curves where data are available. 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 o f not d r i l l i n g ,  EUV^,  i s obtained by z.  asking the well-owner a question such as:  " I f the only possible outcomes  were the best (optimum y i e l d ) or the worst  (dry h o l e ) , what would the chance  of success have t o be before you would accept t o d r i l l ? " for example, then  EUV^  I f he says 80%,  would be equal t o 80. .  whichever value i s greater,  EUV  D  or EUV^, indicates the best decision,  that i s , e i t h e r to d r i l l or not to d r i l l .  4.2  Case I I : 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 Y i e l d and Depth Here, there are several depths the w e l l could be d r i l l e d  each p a r t i c u l a r depth, (say di with a p r o b a b i l i t y there would be y i e l d s  y  1  y  n  to.  But, f o r  p ^ o f getting water),  with p r o b a b i l i t i e s  (  1  p  y i  .... p ^  21. Utility Values  FIG. 4.3(a)* DECISION T R E E  SCHEMATIC.  Utility Values  FIG.4.31b)  8  DECISION T R E E SHOWING E X P E C T E D  VALUES.  22.  and u t i l i t y values  T  L .... U 1  associated with each y i e l d value.  n  The u t i l i t y versus y i e l d curve would be obtained as i n Case I The expected u t i l i t y values (EUV A  1  ... A  n  ••... EUV^)  (fig.4.2).  at the chance nodes  are again calculated as i n Case I using equation ( 4 . 2 ) .  These  expected u t i l i t y values would be the same f o r the various depths i f there were no relationahip between depth and y i e l d . Since the cost o f 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 t o any depth.  This u t i l i t y "cost" or r e l a t i v e  d e s i r a b i l i t y 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 s i m i l a r to those used i n obtaining f i g . 4.2 (Case I ) . The expected u t i l i t y value o f d r i l l i n g t o a l l the d i f f e r e n t depths XEUV^) (with p r o b a b i l i t i e s o f obtaining water u t i l i t y "costs"  UC  dl  ... UC^)  p^  1  ... p ^  and corresponding  i s again calculated using equation ( 4 . 2 ) .  The difference between the expected u t i l i t y value o f y i e 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^).  EiW  4.2.2.  D  Or,  =- EUV  y  - EUV  d  Case 1 1 ( b ) : P r o b a b i l i t y Relationship Between Y i e l d and Depth Where there i s a r e l a t i o n s h i p between y i e l d and depth i n the form  of a p r o b a b i l i t y band with a mean, lower and upper l i m i t s such as i n f i g . 4.5, then the expected u t i l i t y values A  A  n  (EUVyi  E U V  yn^  a  t  t  h  e  c  n  a  n  c  e  n  o  d  e  s  would be d i f f e r e n t because o f the uncertainty involved. The  p r o b a b i l i t y of " y i e l d " f o r a given value o f "depth" i s assumed to have a  Depth ( f e e t )  n  FIG.4.4 U T I L I T Y " C O S T " V E R S U S D E P T H . :  Depth ( f e e t )  F I G . 4 . 5 YIELD V E R S U S D E P T H P R O B A B I L I T Y BAND. !  25.  skewed normal d i s t r i b u t i o n between the upper and lower bounds. Using the u t i l i t y versus y i e l d curve ( f i g . 4.2) and fig.-!4.5, d i f f e r e n t values of expected u t i l i t y of y i e l d s f o r a l l the various depths would be obtained.  From each expected u t i l i t y value f o r a p a r t i c u l a r depth  i s subtracted the u t i l i t y "cost" f o r that p a r t i c u l a r depth, to obtain an expected net u t i l i t y value  (ENUV^)  f o r that depth.  This i s done f o r  . a l l the d i f f e r e n t 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 f o r a p a r t i c u l a r depth  (ENUV^-)  by the corresponding p r o b a b i l i t y  (P  d l  ) of obtaining  water at that depth, and summing over the entire range of depths.  F.NUV- = D  Or,  t -{p,.(0W,.)] *ai • di L  ( i j  J  *  3 )  4.2.3. Expected U t i l i t y o f Not D r i l l i n g To obtain the expected u t i l i t y value o f not d r i l l i n g , the well-owner i s 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 t o the "best" - no r i s k s involved.  Alternative B:  A gamble i n which you have a p r o b a b i l i t y  p of  obtaining the "best" outcome and a p r o b a b i l i t y (1 - p) o f obtaining the  "worst".  At what p r o b a b i l i t y values would you be i n d i f f e r e n t between accepting alternative  A  or B?"  In t h i s p a r t i c u l a r case, the "best" outcome -would be to d r i l l to zero depth and s t i l l obtain the maximum y i e l d . t h i s would be  The u t i l i t y associated with  200 (100 + 100) - obtained by combining the u t i l i t y curves  of f i g s . 4.2 and 4.4.  On the other hand, the "worst" outcome would be t o  26.  FIG.4.6= SCHEMATIC DECISION  TREE  27. d r i l l to the maximum depth only to obtain a zero y i e l d .  And again, combining  f i g s . 4.2 and 4 . 4 , the u t i l i t y associated with t h i s "worst" outcome would be zero (0+0). I f the well-owner's point o f i n d i f f e r e n c e were a c t u a l l y at p and (1 - p), then the expected u t i l i t y value o f 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 t o 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 e i t h e r ENUV  D  .4.3  or  EUV^).  Case I I I :  Decision-to Purchase - Imperfect  Information  The importance o f purchasing a d d i t i o n a l information i s t o better define (or reduce) the uncertainty associated with the decisions t o 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 p h y s i c a l dimensions.  Other examples of information purchased t o  better reduce, uncertainty are logging surveys, analysis o f samples, and pump tests i n order to decide how many more wells have t o be d r i l l e d to meet a s p e c i f i c water demand. I f the a d d i t i o n a l information i s perfect (that i s , there i s no e r r o r i n the i n t e r p r e t a t i o n and i t w i l l t e l l p r e c i s e l y the true state of nature), a r e l a t i v e l y straightforward analysis w i l l suggest whether i t i s f e a s i b l e t o purchase the information.  But, i f the information i s imperfect, the analysis  of whether t o purchase the information becomes more complex. Figure 4.7 i s a schematic decision tree f o r the analysis of decisions to purchase imperfect information.  A l t e r n a t i v e , time-zero investment s t r a t e g i e s i n l i e u o f purchasing the a d u l t i o n a l information  S t a t e s o f nature (out comes) t h a t can occur f o r the c h o i c e s (2 o r more branches).  Various p o s s i b l e i n t e r p r e t a t i o n s , o r evidence^ t h a t c o u l d become a v a i l a b l e from the information that i s purchased (2 o r more branches).  purchase a d d i tional information .  (before d e c i ding which d e c i s i o n choice to a c c e p t ) .  P r o b a b i l i t y o f evidence o c c u r i n g i s the denominator term o f Bayes' Theorem-for each p o s s i b l e evidence o r i n t e r p r e t a t i o n o f the purchased i n f o r m a t i o n .  FIG.4.7-DECISION T R E E U S E D TO D E T E R M I N E  P r o b a b i l i t y terms d e r i v e d by s o l v i n g Bayes' Theorem.  THE FEASIBILITY  OF P U R C H A S I N G A D D I T I O N A L  INFORMATION.  29.  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. S i m i l a r l y , f o r Sections (C) and (D) i f there were more choices and more possible states of nature  (U  ...U ).  The p r o b a b i l i t i e s on the chance node branches i n Section (D) are obtained by s o l v i n g Bayes''...Theorem.  The p r o b a b i l i t y terms represent the  revised perceptions o f the likelihoods of the various states of nature, given the new evidence or i n t e r p r e t a t i o n .  The p r o b a b i l i t y terms i n Section  (A) represent the denominator terms of Bayes' Theorem. Case I I I could be combined . with e i t h e r Case I or Case I I , and the analysis carried out as before.  CHAPTER  5  APPLICATIONS 5.1  Ryder Lake D i s t r i c t  5.1.1  Introduction Since the only source of water In the Ryder Lake D i s t r i c t i s from  undergroundj prospective s e t t l e r s 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 w e l l s . to do with uncertainty.  Obviously, t h i s i s a b i g problem having  Therefore, a formal analysis using decision theory,  w i l l throw more l i g h t on the decision to be made instead of the dependence on sheer i n t u i t i o n as i n the past. 5.1.2  Location Ryder Lake D i s t r i c t i s a r o l l i n g h i l l y area with elevations that r i s e  to more than 2,700 feet above sea-level.  It i s located within Chilliwack  D i s t r i c t Municipality, and l i e s between longitudes 121° and latitudes 49°  05'30"  and 49° 07'30".  51'  and 121°  56'30"  I t 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. I t i s p a r t l y a r e s i d e n t i a l and p a r t l y 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 t o t a l p r e c i p i t a t i o n of  about 56 inches occurs from October to March i n c l u s i v e .  R a i n f a l l during the  growing season - A p r i l to September - i s inadequate i n most years f o r the maximum development and y i e l d of crops.  The heavy sustained rains from  October to March replenish the groundwater r e s e r v o i r s .  30.  During t h i s period,  31. l i t t l e water, apart from runoff, i s l o s t by evaporation and t r a n s p i r a t i o n . 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 r e s u l t s . 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 o f the C o r d i l l e r a n Ice (Vashon) Sheet and p r i o r to the advance of the Sumas Ice.  These gravels are o v e r l a i n 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  i s formed i n layers up to 50 or 60 feet t h i c k , and i n places s t r a t i f i e d , overlying bedrock.  A mechanical analysis o f a f i n e f r a c t i o n o f t h i s Sumas  t i l l gave an average.result o f 63 percent sand, 33 percent s i l t and 4 percent clay (Halstead, E.C., 1 9 6 l ) . The bedrock consists o f shales and a r g l l l i t e s that may y i e l d some ground water from j o i n t s and fracture zones. 5.1.5 Water Supply Groundwater i s the only source of water i n t h i s area. to tap t h i s water, wells had t o be dug or d r i l l e d .  And i n order  The type of w e l l depends  p a r t l y on the depth to water but more on the f i n a n c i a l resources o f the w e l l 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 l i n e d 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 caving or slumping.  without  Most of these wells do not y i e l d s u f f i c i e n t supplies  and often go dry i n summer.  32. Those o f the inhabitants who could a f f o r d the b i l l , have d r i l l e d wells,;(See and f i g . 5-1).  Table,,5.1  D r i l l e d wells are the most e f f e c t i v e type f o r the  recovery of groundwater and are required e s p e c i a l l y where large y i e l d s are needed, such as f o r municipal or i r r i g a t i o n use.  D r i l l e d wells are l i n e d  with a casing commonly more than s i x 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 r i g s 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 d a i l y 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 i d e n t i f i c a t i o n of water-bearing s t r a t a .  4)  No c i r c u l a t i n g 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 d i r e c t l y 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 o f water per day which i s more than s u f f i c i e n t f o r the eighteen homes (lots) and one slaughter house they are supposed  UJ i.  FIG. 5.1  1  MAP  SHOWING  DRILLED  -RYDER  WELL  LAKE  LOCATIONS  AREA-  34. to serve.  The Southside Water Users Community,:.made up o f 20 homes ( l o t s ) ,  also obtain t h e i r supply from a spring which flows i n t o a dug w e l l about 15 feet deep.  Surprisingly, none o f the supplies has gone dry so f a r .  5-1.6 Quality of Water The hardness o f the groundwater i n t h i s area ranges between 43 and 135 parts per m i l l i o n (ppm) (Halstead, 1 9 6 l ) . medium to s o f t , but there are some exceptions.  The water i s generally Where hard water Is found,  i t s t o t a l hardness i s not excessive and does not l i m i t the use o f the water. The water also f a l l s within safe l i m i t s f o r I r r i g a t i o n use.  Some might  be rejected because of i t s high i r o n content and the probable damage i t could cause t o the d i s t r i b u t i o n system. 5.1.7  Decision Model Applications To Ryder Lake Area The only available w e l l data f o r 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 p r o b a b i l i t y matrices and vectors with options f o r m u l t i p l i c a t i o n , addition, subtraction, updating and rescaling) was used f o r the expected value 5.1.7.1  computations.  Case I of Model  F i r s t , a cumulative p r o b a b i l i t y versus y i e l d curve was produced using data from Table 5-1.  Secondly, a u t i l i t y (of d r i l l i n g ) versus y i e l d 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 c a l c u l a t i o n of points p l o t t e d to obtain the u t i l i t y ( o f d r i l l i n g ) curve.  35. TABLE 5.1  DRILLED WELL RECORDS - RYDER LAKE DISTRICT  Address  Depth (ft)  Dia. Y i e l d (In) gpm  Comments  132  6  Ik  Quartz Lenses, Fractures % 132'  2. 48455 E l k View Road  744  6  9 - 284 Bedrock  3. 48470 E l k View Road  104  6  ik 2  47  6  Dry  0 - 8 gravel; 8 - 1 7 17 - 47, gravel  1. Ryder Lake Rd. & No 2 Rd (February 1977)  0-10 till; 10 - 104, bedrock  (October 1975)  4. 49185 E l k View Road (November 1972)  till;  5. 5014 Farnham Road (May. 1976)  249  Dry  0 - 4 2 sand, gravel 240 - 249, clay hardpan  6. 49612 Atkins Road  110  Dry  0 - 4 1 gravel; 50 - 110 packed sand and gravel  7. Extrom Road (Location?)  365  Dry  345 - 365 gravel, sand some shale, 0 - 1 7 f i n d sand  8. 47320 Extrom Road  500  3k  0 - 8 loam 340 - 500 shale  (December 1974) (December 1974) (July 1971)  9. 46925 Extrom Road (1970)  269  6  1  0 - 3 0 hard packed sand and clay; 263 - 269 gray clay  10. 47200 Extrom Road (May 1975)  740  6  Dry  0 - 37 t i l l ; 281 - 740 bedrock  11. 46650 Thornton Road  443  6  Dry  Sand and gravel (0 - 443)  Dry  0 - 17 s i l t e d gravel and t i l l ; 53 - 63 dry sand  (July 1975)  12. 46880 Jinkesson Road - (July 1958)  63  13- 5296 Tesky Road  61  14. 5392 Tesky Road  343  Dry Sand s t a t i c 106  (September 1971)  6  2,  0-26 till; 94 - 343 bedrock, shale  (September 1975)  15. 47005 Russell Road  Dry  (June 1977)  16. 46655 Russell Road  static  343  6  (December 1974)  17. 6l80 Promontory Rd . (1959)  127  18. 588 Bailey Road  123  19. End of Parsons Road  380  (September 1974)  6  6  19  0 - 60 t i l l and boulders g,245 - 265 gravel, sand 0-6  lk  loam  s t a t i c 101'  3  0-20 dug w e l l 112 - 118 w.b. sand  Dry  0 - 2 2 peaty loam 121 - 123 coarser sand  Dry  0 - 155 sand, gravel 300 - 380 s i l t y sand & clay  continued overleaf  36. TABLE 5.1 (Continued)  20. 6235 Parsons Road (September 1974)  320  6  1  21. Lindel Road (May 1976)  455  6  5g  85  6  3  22. Lindel Road (August 1976)  0 - 6 Dirt 6 - 320 shale 0 - 2 1 broken shale 21 - 445 shale 0 - 9 overburden 9 - 8 5 shale  37.  2  3 Yield (gpm )  FIG. 5 . 2 - U T I L I T Y V E R S U S Y I E L D ( R Y D E R L A K E A R E A )  .38. 1st  Question;  posed to one of the well-owners, gave the following r e s u l t :  Alternative 1:  Obtain 1 gpm f o r c e r t a i n , or  Alternative 2:  A gamble i n 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 r e s u l t : Alternative 1:  Obtain 3 gpm f o r c e r t a i n , 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 x 100 + 0 . 5  By inputting the curves i n f i g s . 5.2  + 0.5[U(1 x 80  and 5.3  =  gpm)]  90  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  drilling,  was obtained as 85 using the method of asking questions outlined i n  EUVj^,  the preceding chapter. Comparing both expected u t i l i t y values, the ultimate decision f o r t h i s case would be not to d r i l l . 5.1.7.2  Case 11(a)  of Model:  No Relationship Between Y i e l d and Depth  F i r s t , the u t i l i t y versus y i e l d curve ( f i g . 5.2)  and the y i e l d versus  cumulative p r o b a b i l i t y 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 y i e l d )  39.  FIG.5.3= Y I E L D  VERSUS  - R Y D E R  CUMULATIVE  LAKE  AREA  -  PROBABILITY.  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 p r o b a b i l i t y curve ( f i g . 5.5) were fed i n and again m u l t i p l i e d t o 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 For  Case 11(b)  of Model:  Probability Relationship Between Y i e l d and Depth  t h i s case, the u t i l i t y versus y i e l d curve ( f i g . 5.2)  and the y i e l d  versus depth curve ( f i g . 5.6) i n the form of a p r o b a b i l i t y band were fed into the program and m u l t i p l i e d to obtain an expected u t i l i t y value i n the form of a matrix.  From t h i s matrix was subtracted the u t i l i t y "cost" versus  depth curve ( f i g . 5.4)  ( i n 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  p r o b a b i l i t y 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 f o r the d i f f e r e n t depths was m u l t i p l i e d by the cumulative  p r o b a b i l i t y matrix to obtain the o v e r a l l 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 not d r i l l i n g  (EUV^)  p = 0.7,  the expected u t i l i t y value of  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 f o r Case 11(a), 43.17"' f o r Case 11(b)  and the expected u t i l i t y  value of the act "not t o 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 p l a i n deposits at the mouth of Norrish Creek on the north of the Fraser River about 80 kilometres east of "Vancouver ( B r i t i s h Columbia) ( f i g . 5-7).  Because  0  0.25  0.50  Cumulative FIG.5.5= DEPTH  0.75  Probability  V E R S U S CUMULATIVE  -RYDER  1.00  LAKE  AREA  PROBABILITY. -  44. of the hydrogeologlcal s e t t i n g of Inches Creek, i t has become an important natural spawning ground f o r 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 f o r salmon enhancement f a c i l i t i e s (spawning, hatchery, and incubation) f o r the Fisheries Department. 5.2.3  Aquifer Recharge The recharge to the aquifer i n Inches Creek area i s p a r t l y from  p r e c i p i t a t i o n and p a r t l y from Inflow from Norrish Creek. 5.2.4  Application Of Decision Model Case I I I The i n i t i a l problem i n Inches Creek study area was the lack of past  d r i l l e d w e l l data from which p r o b a b i l i t i e s of occurrence of the various w e l l y i e l d s could be obtained.  The hydrogeologists, therefore, had to carry out  preliminary' geologic investigations and were able to give the following aquifer y i e l d estimates: Minimum y i e l d  =  1000  Most probable y i e l d  =  2500 gpm  Maximum y i e l d  =  5000  gpm  gpm  Applying a t r i a n g u l a r d i s t r i b u t i o n to the above, a y i e l d versus p r o b a b i l i t y (prior) curve ( f i g . 5-8)  was  cumulative  obtained.  In order to obtain some more information about the y i e l d of the aquifer and hence the number of production wells that would be needed, a t e s t 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 y i e l d s , the hydrogeologist, from past  experience, was able to predict corresponding production w e l l y i e l d s and hence the p r o b a b i l i t y band shown i n f i g .  5.9.  0 Cumulative  Probability  FIG. 5 . 8 ' 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 ( PRIOR) - INCHES  CREEK-  47.  F I G . 5 . 9 ' PRODUCTION Y I E L D -  V S . T E S T YIELD PROBABILITY BAND.  INCHES  CREEK -  48. To obtain a monetary value versus y i e l d curve such as i n f i g . 5.10, the owner o f the project was asked how much he would be prepared to pay f o r various y i e l d s , f o r certain, i f he were buying ready-made wells. Figure 5.11 shows the decision tree layout.  At the terminal  outcome of the various y i e l d s would be the d o l l a r values obtained from f i g . 5-10. (Ci 5.9  But at G,  + GYp), .... ( C 5 + C ) , T  3  where  C  H, the  C i , C , ....C , 2  5  the corresponding outcomes would be T  i s the cost o f the test w e l l .  Figure  when applied to the computer program (used i n Ryder Lake Analysis)  produces p r o b a b i l i t i e s [(p ^ ) o f production y i e l d s given the various test y i e l d s ] i n the form of a matrix. f i g . 5.10 (after adding  p  TEMV"!  TEMV . 5  The p r o b a b i l i t i e s  of obtaining the various test w e l l y i e l d s are found., by  using Bayes' Theorem. (TEMV)  from  C^) by the above matrix gives another row matrix  of test expected monetary values (P-ti"'"'" t 5 ^  Multiplying the row matrix produced  These p r o b a b i l i t i e s m u l t i p l i e d by t h e i r corresponding  values and summed gave the net t e s t expected monetary value (NTEMV)  of $i:,.l62. By using f i g s . 5.8 and 5-10, an expected monetary value (EMV) o f $1,662 was obtained f o r the decision node  C.  And hence the decision t o  d r i l l a test hole has been proved to be j u s t i f i a b l e .  49.  FIG.5.10= P R O D U C T I O N  WELL COST  -INCHES  C R E E K -  VERSUS  YIELD.  G  o  .5.11: D E C I S I O N T R E E SHOWING P U R C H A S E OF IMPERFECT -  INCHES  CREEK-  INFORMATION  CHAPTER DISCUSSION  AND  6 CONCLUSIONS  The decision models developed i n t h i s thesis are to enable prospective water w e l l owners to make the r i g h t 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 a c t u a l l y embarking on d r i l l i n g the required production well(s).  The decision c r i t e r i o n , however, i s based on expected u t i l i t y which  i s a summation of the  products  of p r o b a b i l i t i e s of obtaining the various  y i e l d s 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 e n t i r e l y 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 f a r , there has not been a set-down procedure  for achieving t h i s . Where there are no past w e l l records as i n the Inches Creek area, the p r o b a b i l i t i e s of obtaining various y i e l d s w i l l c e r t a i n l y be those of the expert hydrogeologist.  And, of course, these p r o b a b i l i t i e s w i l l vary from  one hydrogeologist to another.  There i s no doubt, then, that the accuracy  of the r e s u l t s w i l l be very much affected by these two parameters - u t i l i t y and p r o b a b i l i t y (the source of uncertainty). The r e s u l t s f o r the Ryder Lake D i s t r i c t 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 a f t e r d r i l l i n g a t e s t hole.  The above decisions  could have been made without going through a formal, systematic analysis as outlined i n the thesis. decision.  But what i f the decision maker has made a wrong  How would he exonerate himself?  .  51  What would be h i s c r i t e r i o n f o r  52. the decision he has made?  Hence, i n order to protect himself, he d e f i n i t e l y  w i l l need to follow a r a t i o n a l process, considering a l l r i s k s involved before a r r i v i n g at a f i n a l decision. F i n a l l y , use of the techniques i n t h i s thesis w i l l enable the j u s t i f i c a t i o n of major decisions e s p e c i a l l y those dealing with public such as the F i s h e r i e s .  resources  53-  BIBLIOGRAPHY "A Primer on Ground Water",  U.S. Geological Survey, Washington, D.C,  1963.  Benj.amin, J.R., and C o r n e l l , C.A., P r o b a b i l i t y S t a t i s t i c s , and Decision f o r C i v i l Engineers. McGraw-Hill Book Co., New York, Campbell, M.D.,  and Lehr, J.H.,  Water Well Technology.  McGraw-Hill Book  1973.  Co., New York, Davis, S.N.,  1970.  and DeWiest, R.J.M.,  Hydrogeology.  John Wiley and Sons Inc.,  1966.  New York,  Domenlco, P.A.,  Concepts and Models i n Groundwater Hydrology.  Book Co., New York,  McGraw-Hill  1972.  Grayson, C.J., J r . Decisions Under Uncertainty, D r i l l i n g Decisions by O i l and Gas Operators. Harvard Business School, D i v i s i o n of Research, Boston, Guyod, H.,  I960.  "Application o f Borehole Geophysics to the Investigation and  Development of Ground Water Resources". No. 1.,  Water Resources B u l l e t i n , V o l . 8_,  February, 1972.  Halstead, E.C., "Groundwater Resources of Sumas, Chilliwack, and Kent M u n i c i p a l i t i e s , B r i t i s h Columbia", Paper 60-29,  1961.  Halter, A.N. and Dean, G.W., Applications. Hershman, S.H.,  Geological Survey of Canada,  Decisions Under Uncertainty with Research  South-Western Publishing Co., C i n c i n n a t i ,  1971.  "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, Johnson, E.E.  1974. Ground Water and Wells.  Products Co., Saint Paul, Minnesota,  Johnson D i v i s i o n , Universal O i l 1972.  Newendorp, P.D. . Decision Analysis f o r Petroleum Exploration. Publishing Co., Tulsa,  1975.  Petroleum .".  S c h l a i f e r , R.  Analysis of Decisions Under Uncertainty.  Co., New York, Todd, D.K.  1969.  Ground Water Hydrology,  Von Neuraan, J . and Morgenstern, 0. Princeton, New Jersey, Walton, W.C. New York,  John Wiley and Sons, Inc., 1959Theory of Games and Economic Behaviour  1953-  Groundwater Resource Evaluation. 1970.  McGraw-Hill Book  McGraw-Hill Book Co.,  

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