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A digital model of fluctuations in wells produced by fluctuations in nearby surface waters McTaggart-Cowan, Gillian Hope 1967

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A DIGITAL MODEL OF FLUCTUATIONS IN WELLS PRODUCED BY FLUCTUATIONS IN NEARBY SURFACE WATERS by GILLIAN HOPE McTAGGART-COWAN B . S c , U n i v e r s i t y o f T o r o n t o , 1965 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n t h e Department o f GEOPHYSICS We a c c e p t t h i s t h e s i s as c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d THE UNIVERISTY OF BRITISH COLUMBIA A p r i l , 1967 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that die Library shall, make i t f r e e l y available for reference and study„ I further agree that permission for extensive copying of t h i s thesis for scholarly purposes may be granted by the Head of my Department or by his representatives. I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission,, Department of Geophysics i i ABSTRACT F l u c t u a t i o n of water l e v e l s i n w e l l s due to f l u c t u a t i o n s of the nearby surface water (by t i d e s , s h i p s , dams) i n t e r -f e r e w i t h w e l l a n a l y s i s by standard f i e l d techniques. There-fore an attempt has been made to approximate these v a r i a t i o n s a n a l y t i c a l l y . These c a l c u l a t e d values could then be used to approach a s t a t i c water l e v e l . The ba s i c ground water theory was considered f o r con-f i n e d and unconfined a q u i f e r s . The same l i n e a r approximation f o r the equation of the v a r i a t i o n s of the water l e v e l was obtained. This was a d i f f u s i o n equation. The s i n u s o i d a l s o l u t i o n was considered i n conjunction w i t h the t i d e s , considered as a sum of sine waves. However t h i s approach was l i m i t e d to the t i d e s and was q u i t e lengthy. A b e t t e r method was developed which approximates the earth as a d i g i t a l f i l t e r , f u l l y described by the d i f f u s i o n equation and i t s g e o l o g i c a l and geometrical parameters. An a p p l i c a t i o n of t h i s technique showed good time c o r r e l a t i o n w i t h the a c t u a l data and good s e n s i t i v i t y to the r a t i o of the appropriate parameters. A c c o r d i n g l y , i t i s proposed that a method i n c o r p o r a t i n g t h i s f i l t e r could be used as a t o o l i n the assessment of p r e l i m i n a r y w e l l s . i i i ACKNOWLEDGEMENTS The author would l i k e to express her g r a t i t u d e to Dr. R.D. R u s s e l l who suggested the problem and gave valuable advice and a s s i s t a n c e throughout the course of t h i s study, and to W.L. Brown, the h y d r o l o g i c a l engineer whose problem t h i s i s , who helped g r e a t l y i n the r e a l i z a t i o n of the p r a c t i c a l l i m i t a t i o n s of w e l l a n a l y s i s and o f f e r e d many valuable suggestions and the i n v a l u a b l e data. The help r e c e i v e d from W.O. Wigens of the Canadian Hydrographic Se r v i c e was g r e a t l y appreciated as were the d i s c u s s i o n s with Dr. R.M. E l l i s , Dr. P.H. LeBlond, and Dr. K.O. Westphal. Without the U n i v e r s i t y of B r i t i s h Columbia Computing Centre, and the U n i v e r s i t y of B r i t i s h Columbia and N a t i o n a l Research C o u n c i l s c h o l a r s h i p s t h i s work would never have been accomplished by the author. F i n a l l y she wishes to acknowledge the FMC Chemicals Li m i t e d at Squamish who contracted with Mr. Brown to obtain a water supply f o r the p l a n t . i v TABLE OF CONTENTS Page ABSTRACT . . . . . . . a . . . . . . . . . . . . . . . . « . . . . . . . « > • • • • « • « • • • • i i ACKNOWLEDGEMENTS i i i LIST OF FIGURES v INTRODUCTION 1 THEORY • O O « « f t « O » 0 O 0 O O » * * O « « ' « * * * « » « * » O f t * 0 « « O « * * O « » 0 t O O O 3 Ground Water Flow 3 Confined A q u i f e r 9 Unconfined A q u i f e r 12 THE WELLS .»»....•.....•.<.••••.......<<>...»..»*«.«••.> 17 TIDAL COMPONENT APPROACH 21 FILTER APPROACH 25 The F i l t e r 30 Computer Program 33 Output • •••••• 3^ CONCLUSIONS ^0 BIBLIOGRAPHY 42 SYMBOL TABLE 44 V F i g . 1: F i g . 2: F i g . 3: F i g . 4: Fi g . 5: F i g . 6: A B F i g . 7: A B C D E F i g . 8: F i g . 9: F i g . 10: F i g . 11: F i g . 12: A B C D E LIST OF FIGURES Following Page Location of FMC Chemicals Ltd. and the Wells 2 Position of Wells with respect to Closest Shore 2 Confined and Unconfined Aquifers 6 Confined and Unconfined Aquifers with respect to Tide Water 6 What Might Exist at Squamish 6 T i d a l Input Data for July 9 , 1 0 , 1965 18 T i d a l Input Data for July 24 ,25,26,1965 . . 18 Data for Well 1 for July 10, 1965 18 Data for Well 3 for July 10, 1965 18 Data for Well 1 for July 2 5 , 2 6 , 1965 18 Data for Weil 2 for July 2 5 , 2 6 , 1965 18 Data for Well 3 for July 2 5 , 2 6 , 1965 18 Impulse Response 30 Non-Dimensional Impulse Response 30 Sketch of Position of Fi g . 9 with respect to a Plot of log F(x,t) V C r S U S 10 ^  t • • o o o o * o 0 9 f o o » f t o o o o e o o o o o o o o e 3 0 Flow Diagram of Computer Program 33 Calculated Values for Well 1 for July 10 . 37 Calculated Values for Well 3 for July 10 . 37 Calculated Values for Well 1 for July 2 5 $ 2 6 • « 9 « « « o * o « « « « a o » a « 9 « « o « e o o e « « a s s e « e « 37 Calculated Values for Well 2 for July Calculated Values for Well 3 for July i v F o l lowing Page F i g . 13: F i n i t e Length C o r r e c t i o n to Impulse Response 39 F i g . 14: Impulse Responses f o r the Three Wells .... 4 l - 1 -INTRODUCTION The water l e v e l i n a w e l l near surface water i s observed to s u f f e r forced o s c i l l a t i o n s due to f l u c t u a t i o n s i n the l e v e l of the surface water. Thus w e l l s near t i d a l waters have t i d a l o s c i l l a t i o n s due to the changes i n water height at the shores c l o s e s t to the w e l l s . F l u c t u a t i o n s can a l s o be produced by marine t r a f f i c , p i l i n g up of water by wind, the e f f e c t of dams on r i v e r l e v e l s , and any other la r g e wave generating mech-anism. The changing water l e v e l i n w e l l s i s a problem of the h y d r o l o g i c a l engineer. In h i s a n a l y s i s of the c a p a b i l i t i e s of a w e l l he must ca r r y out pumping t e s t s which must be r e -l a t e d to a s t a t i c base l i n e , u s u a l l y the undisturbed water t a b l e . Obviously i n such cases as are mentioned above there are c o m p l i c a t i o n s . Accordingly the object of t h i s research has been to attempt to develop some method to synthesize these w e l l o s c i l l a t i o n s so that they can be removed from the w e l l t e s t i n g data to e f f e c t a constant base l i n e . The a c t u a l problem under c o n s i d e r a t i o n i s that of a s s i m i l a t i n g the w e l l l e v e l s given the f o r c i n g f u n c t i o n i n -put, i n t h i s case the t i d e , and s e v e r a l other g e o l o g i c a l and geometrical parameters, which i n many cases can be considered to be constants. The parameters r e q u i r e d are -2-standard ones used i n w e l l a n a l y s i s . I t would a l s o be convenient to be able to f i n d the r a t i o of the g e o l o g i c a l parameters given the f o r c i n g f u n c t i o n i n p u t , the w e l l o s c i l l a t i o n s observed, and the geometrical para-meters. This i s e s s e n t i a l l y the inverse of the fundamental problem. In t h i s case w i t h one small p r e l i m i n a r y w e l l the productiveness of f u r t h e r development of the w e l l could be estimated. The problem of f l u c t u a t i o n s i n w e l l s has received l i m i t e d a t t e n t i o n i n the l i t e r a t u r e . Most a r t i c l e s that deal with i t solve an appropriate d i f f u s i o n equation f o r a sine-wave f o r c i n g f u n c t i o n at a v e r t i c a l face. In an attempt to f i n d a more general and more p r a c t i c a l s o l u t i o n the system was considered to be a d i g i t a l f i l t e r w ith the f o r c i n g f u n c t i o n as input and the w e l l f l u c t u a t i o n as output. Due to l i m i t a t i o n s of both data and time, only one area was stu d i e d . This included the three w e l l s of the PMC Chemicals L i m i t e d , Squamish. This area i s shown i n Figures 1 and 2. The n o n i d e a l i t y of the w e l l s serves to make t h i s a more s i g n i f i c a n t t e s t of the proposed method and i n d i c a t e s some of the major problems i n v o l v e d . Included i s the general F o r t r a n computer program used i n the f i n a l a n a l y s i s of the data and an o u t l i n e of i t s s p e c i f i c case use. 9 Well * 1 9 Well tS. 9 Well *3 Sco-W: KacU-100 FIGURE 2: POSITION OF WELLS WITH RESPECT TO CLOSEST SHORE FROM DATA TAKEN ON A TRIP TO THE WELLS ON OCTOBER 22, 1966 THEORY Ground Water Flow The study of ground water concerns the subsurface water i n the zone of saturation. The motion of this f l u i d i s governed by the laws of hydrodynamics. However i t i s very complex due to the variations i n the pores themselves, adsorption, and other microscopic features. Therefore the macroscopic averaged view i s considered. Use i s made of a " f i c t i t i o u s flow v e l o c i t y " or s p e c i f i c discharge, ^, through a given cross-section of the porous media. Russian authors tend to use the term seepage v e l o c i t y for t h i s quantity. $ i s thus a flow rate through a unit surface. One very important experimental consideration i s Darcy's Law. This states that, macroscopically, the Darcy v e l o c i t y or s p e c i f i c discharge, V, i s proportional to the negative gradient of the head of the flow. The generalized Bernoulli equation i s z i 1 + P I / Y + v i 2 / 2 g = z 2 ' + P 2 A + V2 2/2g + f r i c t i o n loss term where z' i s the elevation of the head of water p i s the pressure Y i s the weight of a unit volume of water g i s the acceleration of gravity v i i s approximately equal to v 2 since the v e l o c i t y head terms are i n general n e g l i g i b l e i n ground water flow; i n other words, the average k i n e t i c energy d i f f e r e n c e s are u s u a l l y n e g l i g i b l e . Thus h = z' + p/y + a r b i t r a r y constant where h i s the piezometric head and the a r b i t r a r i n e s s of the constant depends on the datum chosen. Then h2 - hi = Ah = - f r i c t i o n l o s s term Experiment confirms that f r i c t i o n l o s s i s p r o p o r t i o n a l to the distance between observation p o i n t s , L, and to the speed of the flow. Since head l o s s i s p o s i t i v e , A h i s negative and the h y d r a u l i c gradient i s -Ah/L. Thus Darcy's Law of experimental observation f o r two p o i n t s separated by d i s t a n c e L i s V = -KAh/L where K i s the " h y d r a u l i c c o n d u c t i v i t y " , " e f f e c t i v e p e r m e a b i l i t y " , " c o e f f i c i e n t of p e r m e a b i l i t y " , "seepage c o e f f i c i e n t " , or " t r a n s m i s s i o n c o e f f i c i e n t " . -5-In the general case, for any component of the v e l o c i t y at any point, $ = -KVh where h = h(x,y). K may be expressed as ky/u where y i s the dynamic v i s c o s i t y and k i s the " i n t r i n s i c permeability", c h a r a c t e r i s t i c of the medium only. K i s not a constant but depends on the compressibility of the f l u i d which i n turn depends on the temperature, pressure, concentrations of solutes, and other factors. Usually i n the case of ground water temperature effects can be neglected but the concentra-tions can be important. However i n t h i s discussion these effects are considered to be n e g l i g i b l e and, as i s often done, K i s assumed constant. Darcy's Law has since been derived from the general Navier-Stokes equation for the case of viscous flow with s t a t i s t i c a l considerations and s i m p l i f i c a t i o n s of the micro-flow problems. This law i s v a l i d for the laminar regime only. In the turbulent case non-linear equations are needed to describe the flow. . Actually digression starts before the flow i s usually c a l l e d turbulent. However i n the case under con-sideration v e l o c i t i e s and pore diameters are such that the laminar assumption i s v a l i d . Darcy's Law also assumes that the medium i s completely - 6 -s a t u r a t e d w i t h t h e f l u i d and t h a t no gas i s b e i n g e v o l v e d f r o m s o l u t i o n . $ and h a r e c o n s i d e r e d t o be c o n t i n u o u s f u n c t i o n s o f t h e d i s t a n c e a l o n g t h e d i r e c t i o n o f f l o w . U n d e r n a t u r a l c o n d i t i o n s K i s a n i s o t r o p i c and f o r a s i n g l e homogeneous f l u i d : v Y = -K 3h/3x v = -K 3h/3y v = -K 3h/3z z z K , K , K a r e r e f e r r e d t o as t h e t r a n s m i s s i o n c o e f f i c i e n t s x ' y ' z i n t h e x , y , and z d i r e c t i o n s , r e s p e c t i v e l y . I f t h e m e d i a c a n be assumed i s o t r o p i c w i t h r e s p e c t t o p e r m e a b i l i t y v x = - K 3 h / 3 x , e t c . , and V c a n be e x p r e s s e d as a p o t e n t i a l V = - g r a d <{> where <(> = Kh ( c a s e o f i r r o t a t i o n a l m o t i o n ) . I n t h i s work K w i l l be u s e d as a c o n s t a n t and t h e m e d i a w i l l be assumed i s o t r o p i c w i t h r e s p e c t t o p e r m e a b i l i t y . T h i s a s s u m p t i o n , w h i l e n o t s t r i c t l y t r u e , i s t h e u s u a l a p p r o a c h t o s u c h p r o b l e m s . S i n c e o f t e n t h e f l o w i s c o n -s i d e r e d t o be one d i m e n s i o n a l , t h e K u s e d i s o f t e n e q u i v a l e n t t o , s a y , K x . Two s e p a r a t e c a s e s must be c o n s i d e r e d i n i n v e s t i g a t i n g t h e m o t i o n o f g r o u n d w a t e r : t h e c o n f i n e d a q u i f e r and t h e u n c o n f i n e d a q u i f e r ( F i g u r e 3). The c a s e o f t h e c o n f i n e d a q u i f e r i s t h a t o f a w a t e r b e a r i n g bed b e i n g b o u n d e d above and b e l o w by i m p e r m e a b l e s t r a t a . I n t h i s c a s e t h e FIGURE 3 A : CONFINED AQUIFER FIGURE 3 B : UNCONFINED AQUIFER FIGURE 4A: CONFINED AQUIFER FIGURE 4B : UNCONFINED AQUIFER WITH RESPECT TO WITH RESPECT TO TIDE WATER TIDE WATER •ZZZZZZZZ2." ; MSL ^ Silt Ltujii-Con-f ined . A<^u!$e.c — FIGURE 5: WHAT MIGHT EXIST AT SQUAMISH - 7 -piezometric head refers to the height water w i l l r i s e i n a well that penetrates the aquifer. The water leve l s i n such wells actually define t h i s piezometric surface. In the un-confined case the lower boundary i s impermeable but the upper boundary i s known as the water table and i s the bounding surface of ground wa^er *flow. It may be defined as the imaginary surface along which the water pressure equals the atmospheric pressure. This i s the d e f i n i t i o n of the general piezometric surface. Again, i n p r a c t i c a l terms, t h i s i s equivalent to the elevation of the water surface i n wells that penetrate the zone of saturation. There i s a c a p i l l a r y r i s e zone above these piezometric surfaces which i s .usually n e g l i g i b l e with respect to the t o t a l height of the water column. The c o e f f i c i e n t of storage, S, i s the volume of water released from or taken into storage, by an aquifer, per unit surface area of the aquifer per unit change i n the component of head normal to that surface. For the confined case the water i s released due to the s l i g h t expansion of the water i t s e l f and the s l i g h t decrease i n porosity due to the d i s -t o r t i o n of grains of material composing the aquifer skeleton. In t h i s case the value of S i s usually taken to be d i r e c t l y proportional to the thickness of the aquifer. For the uncon-fined case or "water table aquifer" the water released from or taken into storage i s due to gravity drainage or r e f i l l i n g and compressibility of water and aquifer material. In t h i s case -8-the l a t t e r factor Is usually n e g l i g i b l e . For long periods of pumping S i s approximately equal to the s p e c i f i c y i e l d of the material through which the water table f a l l s and i s inde-pendent of the thickness of the unconfined aquifer. It. i s int e r e s t i n g to note that f i e l d hydrologists tend to use an average value about .001 for the confined case and .2 for the unconfined case. The value of S tends to be more c r i t i c a l for analysis of the unconfined case. These two values could be an order of magnitude out i n either d i r e c t i o n . The c o e f f i c i e n t of t r a n s m i s s i b i l i t y T i s the other geological parameter. It i s defined as the rate of flow of water, at the p r e v a i l i n g water temperature, i n gallons per day, through a v e r t i c a l s t r i p of the aquifer one foot wide extending the f u l l saturated height of the aquifer under a hydraulic gradient of one/one. The value for T i s usually found from pump tests under assumptions of aquifer homogeneity, isotropy, and i n f i n i t e areal extent. The discharging well i s assumed to penetrate and receive water from the entire thickness of the aquifer. The c o e f f i c i e n t of t r a n s m i s s i b i l i t y i s assumed constant at a l l times and at a l l places. There are several other assumptions depending upon the p a r t i c u l a r test used. Despite the r e s t r i c t i v e assumptions such values of T have wide application to ground water problems. S can also be found from the graph used to determine T. - 9 -In order to develop the appropriate d i f f u s i o n equations the confined and unconfined aquifer must now be treated separately. Confined Aquifer For the confined aquifer, also referred to as the "artesian" or pressure system, the compressibility of the aquifer i s Important. The compressibility, 6 , i s expressed as _ -AV/V where B i s aquifer compressibility ¥ i s volume p i s pressure. It i s assumed that the compressive stress acts i n a v e r t i c a l d i r e c t i o n which, i n general, i s normal to the plane of the aquifer. Changes in horizontal directions are also considered n e g l i g i b l e . When the piezometric surface i s lowered a unit distance the amount of water released from the column by Ap i s S; hence S = d¥. The volume per unit area of the aquifer i s equal to the aquifer thickness, b. The pressure change for a unit depth change i s equal to the weight density, y. Therefore dp = -y -10-and thus 3 = S/yb. For an e l a s t i c m a t e r i a l dV/V = dp/p where p i s d e n s i t y . Therefore dp = pfldp or dp = (pS/by) dp. Using the c o n t i n u i t y equation dp/dt = - V - ( p ^ ) pS 5p _ / 9 ( PVy ) . 3( pVy) , 8(pVy.) i by 3t " " 1 9x 9y 9z J * - ( 9 l | p l + i i g Z x i ) s l n C e v z i s sma l l . Using v1 = -Kvh and assuming K and p are constant p = yh and n 2 v, S 9h V h = Kb 9 t ' Kb i s Q'ften r e d e f i n e d as T the " t r a n s m i s s i b i l i t y " , or "co-e f f i c i e n t of t r a n s m i s s i b i l i t y " , or " t r a n s m i s s i v i t y " , Thus V * h = §• T 3 t ' For a more rigorous development see DeWiest's paper (1966) and Copper's (1966). -11-A one dimensional s o l u t i o n of t h i s f o r a sine wave of frequency w i s : h(x,t) = h 0 e - x A ) S / 2 T s i n (u>t - x/u)S/2T ) where the boundary c o n d i t i o n i s h(o,t) = h 0 s i n (u>t). When the confined a q u i f e r extends under an o s c i l l a t i n g 'head' there i s another amplitude f a c t o r i n v o l v e d . In the confined system o_, the compressive s t r e s s , and p are i n e q u i l i b r i u m with the downward forces on the plane of contact with the c o n f i n i n g bed. Thus p + az = constant f o r constant head. Here dp + do z = ydH where H i s the stage of the t i d e . The r e l a t i v e amplitude of t i d a l f l u c t u a t i o n s i n the w e l l or " t i d a l e f f i c i e n c y " i s : dp/y _ dp _ dp/do z dH ~ dp + do z " 1 + dp/da z * The volume of water i n the media i s V = 6¥ where V i s the w volume of the medium and 6 i s p o r o s i t y . Considering the s o l i d p a r t i c l e s as e s s e n t i a l l y incompressible dV = dV w and then dV _ dV *w " e v ° With B w, the c o m p r e s s i b i l i t y of the water, and a, the v e r t i c a l c o m p r e s s i b i l i t y of the granular skeleton of the medium -12-t r e a t e d as a continuum, and a z , the i n t e r g r a n u l a r s t r e s s , d¥/¥ a = — 3 d o z 0 _ -ada 7 Thus f o r t h e - s i t u a t i o n of a c o n f i n e d bed under t i d e water ( F i g u r e 4a) and not d i r e c t l y connected to the t i d e water the e f f e c t of the same s i n e wave t i d e i s : h , v H _ ( a ^h„ e " x / a ) S / 2 T s i n (cut - x/u,S/2T) Water l n a w e l l p e n e t r a t i n g a c o n f i n e d a q u i f e r w i l l a l s o change e l e v a t i o n i n v e r s e l y as the a i r pressure changes. The barometric e f f i c i e n c y of a t r u e c o n f i n e d a q u i f e r ( F i g u r e 3a) can be shown to be j u s t 1 - t i d a l e f f i c i e n c y . In the p a r t i c u l a r example to be c o n s i d e r e d the t i d a l e f f i c i e n c y , u s i n g the average value f o r S, i s approximately u n i t y . Unconfined A q u i f e r In the unconfined case the approximation of s p e c i f i c y i e l d f o r the storage c o e f f i c i e n t i s u s u a l l y used where S - S y = w y / ¥ and Sy i s s p e c i f i c y i e l d W y i s volume of water that can be d r a i n e d by g r a v i t y ¥ i s volume of rock or s o i l . T h i s i s e q u i v a l e n t to the e f f e c t i v e p o r o s i t y . Sy Sj! = 9 -13-where S r i s s p e c i f i c r e t e n t i o n and 0 i s t o t a l p o r o s i t y . The u l t i m a t e value of the storage c o e f f i c i e n t is the s p e c i f i c y i e l d ; but, while the water t a b l e i s being lowered and f o r some time a f t e r i t comes to an approximate s t a t e of r e s t , S i s v a r i a b l e . I n t e r s t r a t i f l e d beds of s i l t or c l a y accentuate the l a g or delay i n the r e l e a s e of water from storage and thus increase the v a r i a b i l i t y of S. The storage c o e f f i c i e n t and i t s u l t i m a t e value, S , can be worked out from pumping t e s t data i f there i s a s t a t i c base l i n e . S depends more on time and l i t t l e on the distance from the w e l l except i n the case of very l e n t i c u l a r formations which probably do occur i n the area under c o n s i d e r a t i o n . S i s much l e s s l i k e l y to be constant i n the case of the unconfined flow. For the flow i n an unconfined a q u i f e r the water t a b l e i s the upper boundary of the surface of flow and i s a "stream surface". I f the r a t e of i n f i l t r a t i o n i s not n e g l i g i b l e the stream l i n e s i n s t e a d of l y i n g i n the water t a b l e cross i t at various angles, s i m i l a r to r e f r a c t i o n , with the angles being the g reatest where the water t a b l e i s steepest. In general i t i s extremely d i f f i c u l t to solve these boundary c o n d i t i o n s a n a l y t i c a l l y since the shape of the water t a b l e determines the d i s t r i b u t i o n of the flow underneath and the shape i s propor-t i o n a l to t h i s d i s t r i b u t i o n . To study the flow, the Dupuit assumptions are used: the -14-water t a b l e Is of low slope and the flow of ground water Is l n the v e r t i c a l plane above a h o r i z o n t a l impermeable bottom. I t i s thus assumed that the v e l o c i t y along the water t a b l e depends on the slope or angle of i n c l i n a t i o n of the water t a b l e ( i n s t e a d of the s i n e , which i s the a c t u a l case). This holds when the tangent i s approximately equal to the s i n e ; that i s , when the angle i s s m a l l . This can a l s o be expressed as: the v e l o c i t y of flow depends on the tangent of the h y d r a u l i c g r a d i e n t . I t i s a l s o assumed that flow i s h o r i z o n t a l at the water t a b l e and everywhere below and the v e l o c i t y i s uniform from top to bottom; that i s , everywhere i n the v e r t i c a l s e c t i o n . These are s i m i l a r to the c o n d i t i o n s and assumptions f o r flow i n the confined a q u i f e r . D i s c u s s i n g the flow of f l u i d : V = Su where V i s seepage v e l o c i t y u i s p a r t i c l e v e l o c i t y S i s equivalent to e f f e c t i v e p o r o s i t y . Using B e r n o u l l i ' s equation i n the form p/pg + z + v 2/2g = constant. h = p/pg + z where h i s "head" thus h + v 2/2g = constant. From Darcy's Law ^ = - KVh. For the one dimensional case, assuming S constant, the r a t e of change of volume of water i n the a q u i f e r i s S3h/3t dx w -15-d i r e c t i o n °^  flou> u where w Is width of a q u i f e r . The net volume of water l e a v i n g the element per u n i t time i s w 3(hv)/3x dx. Since the f l u i d may be assumed incompressible - S 3 h / 3 t = 3(hV)/3x. With $ = -K3h/3x, i n the one-dimensional case being considered, and assuming K constant „ . h / a f _ -K 3 2 h 2 -S3h/3t - 2 3 7 ^ ' I t i s noted that DeWiest (1965) a l s o has t h i s V 2 h 2 term i n h i s more general approach. The one dimensional case i s the approximation that w i l l be used on the data. This equation may be l i n e a r i z e d by;assuming h = h + h' o where h Q i s constant (Mean Sea Level i s the datum used) h' i s small with respect to depth of a q u i f e r . Then S3h'/3t = Kh Q3 2h'/3x 2 and 3 2h'/3x 2 = | 3h'/3t where T = Kh . T O I t i s noted that the greatest change i n head i n the flow - 1 6 -system must be very s m a l l , l e s s than two percent, to make the l i n e a r i z a t i o n a v a l i d approximation. This equation has the same s o l u t i o n f o r the sine wave as d i d the confined case. Thus both s i t u a t i o n s a r r i v e at the same approximate flow equation: 3 2h / 3 x 2 = | 3h/3t where h i s height of d e v i a t i o n w i t h respect to some mean water l e v e l . But the values f o r S are very d i f f e r e n t . -17-THE WELLS The data used l n t h i s t h e s i s came from three w e l l s s i t u a t e d (Figures 1 and 2) at the head of Howe Sound i n the d e l t a of the Squamish R i v e r . In t h i s l o c a t i o n the w e l l to t i d e water distance can vary g r e a t l y due to the extensive mud f l a t s (low t i d e ) . Thus these sample w e l l s are f a r from the i d e a l i z e d case of a v e r t i c a l ground to surface water contact. The area has been leased and f i l l e d by the FMC Chemicals Lim i t e d which had the three w e l l s constructed. The w e l l s produced s a l t water that was o r i g i n a l l y high i n hydrogen sulphide. However w i t h use the amount of hydrogen sulphide has dropped by about t h i r t y per cent i n a year and the s a l i n i t y has l i k e w i s e decreased. This l a s t f a c t s t r o n g l y suggests that there i s no d i r e c t contact with t i d e water. As shown i n the diagram the area has had dredging and f i l l i n g operations. This does not a f f e c t the c l o s e s t distance to surface water ( i n t h i s case the Squamish River) but might suggest the unconfined s i t u a t i o n . However i t i s the p r a c t i c a l experience of those working i n the f i e l d that such dredgings appear to s i l t up very r a p i d l y forming a somewhat impermeable s h i e l d to the d i r e c t i n f l u x of water. A s i l t y l a y e r 40 to 5 0 feet below the surface a l s o suggests the confined a q u i f e r s i t u a t i o n with t h i s l a y e r marking the upper boundary of the - 1 8 -a q u i f e r . The c h i e f f o r c i n g f u n c t i o n f o r the observed o s c i l l a t i o n s i n these three w e l l s i s the t i d e , which can be s i g n i f i c a n t l y modified (by the p r e v a i l i n g weather c o n d i t i o n s . The data a v a i l a b l e c o n s i s t of simultaneous readings of t i d e and w e l l l e v e l s f o r one ten hour period and one twenty-four hour p e r i o d . The readings were taken every quarter hour (Figures 6 and ? ) . In order to c a r r y out computation t h i s data was supple-mented by data from the Canadian Hydrographic S e r v i c e ' s tide-gauge records. The gauge was i n s t a l l e d at the docks i n Squamish during t h i s same p e r i o d . The two sets of records d i d not agree on the overlapping p o r t i o n s , t h e r e f o r e the t i d e -gauge records were co r r e c t e d to the company records using a mean d e v i a t i o n . These correc t e d v a l u e s , sampled at f i f t e e n minute i n t e r v a l s , are included i n Figure 6. I t i s noted that the company records were not taken to the p r e c i s i o n of the tide-gauges. The observations were, made manually using measurements on a s t i c k f o r t i d e records and distance from the surface datum to water l e v e l f o r w e l l records. The l a t t e r was accomplished by means of a measured length of cord with a f l a t object on the end. These tech-niques probably account f o r the i n c o n s i s t e n c i e s i n the data. These i n c o n s i s t e n c i e s appear to be s u f f i c i e n t to a l t e r the time of attainment of mean sea l e v e l e s p e c i a l l y at the end 5 i n v o >-3 O 0 0 < Q Z) Q. 8 < Q o o < UJ 0 - CO 1- 0-0. O o 4-0 v9 —T— 0) > ^ <i> •« _ "3 "I "t FIGURE 6B: TIDAL INPUT DATA FOR JULY 2 4 , 2 5 , 2 6 , 1965 —ho * -hoa ' ti 0 0 » I S O O 2,300 0 3 0 0 07 0 0 11 0 0 1 S O O o'oo 33 OO 0 3 0 O 1 07 OO 1 . Wat er Level "i9^ «U~v: \ toot F IGURE 7A : DATA FOR WELL 1 FOR JULY 10, 1965 104+ \0£ + 98+ —1 = 1 1 = 1 0 9 0 0 IIOO 13 OO ISOO - t , WaAc<- L d v e L "PGE , ^ 1 ^ „ . T i m e — 1 crvv* \ Wo FIGURE 7B: DATA FOR WELL 3 FOR JULY 10, 1965 10V+ ioa+ 96.+, 1 1 1 . 1 09 OO WOO 13OO 15 OO W a i f Level Ti me. V S T 0 0 o 0 0 o 0 0 O N 0 O V O C M in OJ 0 0 3 CM O ll_ 0 o ULI OL O < Q LU O £ 3 J 1/3 y 0- ' O o 8 a) > a) +5 ui 0 cn i in vo i—I VO OJ in OJ 15 Di. O LL OO LU O < LU LU ID o o J > o 1-2 J! CD. CD -19-of the records f o r w e l l s 1 and 2 on the twenty-four hour watch. The datum on a l l water l e v e l charts Is the P a c i f i c Great Eastern Railway datum. One hundred feet corresponds to zero feet Geodetic datum, or mean sea l e v e l . The g e o l o g i c a l parameter T has been measured from pump-ing t e s t s . Table 1 Well No. T at w e l l T out from w e l l T f u r t h e r out 1 2 3 14,000 70,000 59,000 a l l i n U.S. gals/day/foot 23,000 60,000 An average value was quoted of 35,000 U.S. gals/day/foot or about 3.249 f t 2 / m i n . Prom these values i t has been suggested that w e l l s 2 and 3 may l i e i n an o l d f i l l e d channel (higher p e r m e a b i l i t y ) and that 1 may be j u s t outside t h i s channel and p o s s i b l y i n a mop'e ; l e n t i c u l a r r e g i o n . Due to the f a c t that t h i s i s an o l d d e l t a region and that the w e l l logs show d e f i n i t e changes i n composition and tex t u r e with depth and interbedding with organic matter, the p r o b a b i l i t y of t h i s being a l e n t i c u l a r formation seems - 2 0 -high. The amount of hydrogen sulphide a l s o a t t e s t s to the presence of decaying organic matter but obviously there has been some i n f l u x of water since pumping began due to the fa c t that the hydrogen sulphide l e v e l has dropped considerably. The accompanying decrease i n s a l i n i t y suggests that t h i s i n f l u x i s not by a d i r e c t connection with the s a l t water of the Sound. This would i n d i c a t e a confined s i t u a t i o n . Never-t h e l e s s the extreme proximity of the s a l t water with the long t i d a l f l a t s makes the unconfined case seem to be a p o s s i b i l i t y . The sediment extends down to bedrock at about 6 6 5 f e e t . The beds are of hard compacted s i l t y sand with interbeds of s i l t which would not make S f o r the unconfined case a good approximation to Sy. The three w e l l s studied only penetrate to about 1 0 0 feet but are w e l l w i t h i n the a q u i f e r . -21-TIDAL COMPONENT APPROACH In order to d i s t i n g u i s h between the confined and un-confined cases and to suggest an e f f e c t i v e value f o r surface water to w e l l d i s t a n c e , x, the a n a l y t i c a l cosine s o l u t i o n was i n v e s t i g a t e d . Using t i d a l component data, which was a v a i l a b l e f o r Squamish, the d i f f u s i o n equation approximation, and various values f o r x and the.two f o r S an attempt was made to simulate w e l l data. A s o l u t i o n using a cosine f o r c i n g f u n c t i o n , as i s ge n e r a l l y considered i n t i d a l a n a l y s i s , i s : h(x,t) = c e " / a ) S / 2 T x cos (wt + /u>S/2T x + A more general s o l u t i o n of the l i n e a r equation i s : h(x,t) = I c n e ~ / u ) n s / 2 T x c o s + /o , n s/2T x + * n) n At x = 0 h ( 0 , t ) = I c n cos (to t + * n ) n This can represent the t i d e at the head of Howe Sound. The corresponding equation i n terms of t i d a l n o t a t i o n i s : h = I f n H n cos (aJj + E n - g n) n where f n v a r i e s with a period of about 18.61 years and reduces H to the d e s i r e d year -22-H n i s the mean amplitude of co n s t i t u e n t n of t i d e a' n i s the speed of co n s t i t u e n t n i n degrees per hour (Table 2) E n i s a numerical f a c t o r from the equations of e l l i p t i c motion and i s the phase of e q u i l i b r i u m c o n s t i t u e n t n at Greenwich. (The e q u i l i b r i u m theory i s based on the t i d e produced i f no land masses e x i s t e d and water had n e g l i g i b l e i n e r t i a such that i t responded i n s t a n t -aneously to the forces) g n i s the phase l a g and i s the d i f f e r e n c e between the phase of the a c t u a l c o n s t i t u e n t n and that of the e q u i l i b r i u m t i d e at Greenwich f o r component n. A l l values are given with respect to 24 s o l a r hours and a l l use a s t a r t i n g point of high water (hence the cosine n o t a t i o n ) . Table 2 Speed Numbers of a few of the main c o n s t i t u e n t s (degrees/hour) M2 28.9841 average lunar semi d i u r n a l force S2 30.0000 average s o l a r semi d i u r n a l force N2 28.4397 average v a r i a t i o n i n lunar semi d i u r n a l force with change i n Moon's distance 15.0411 average d i u r n a l force due to both lunar and s o l a r d e c l i n a t i o n 0]_ 13.9430 average v a r i a t i o n i n d i u r n a l f o r c e w i t h changes i n Moon's d e c l i n a t i o n - 2 3 -P-i 14.9589 average v a r i a t i o n i n d i u r n a l force w i t h changes i n Sun's d e c l i n a t i o n By comparing the two n o t a t i o n s : ton i s the frequency of t i d a l component n <j/n i s the E n - g n value f o r t i d a l component n c n i s the equivalent to f n H n f o r t i d a l component n t i s i n l o c a l standard time (Greenwich Mean Time + 8 hours f o r P a c i f i c Standard Time). c n and ip n were obtained from the Canadian Hydrographic S e r v i c e and the Admiralty Tide Tables and were a p p l i e d to the general s o l u t i o n f o r any x. This method i s a very long one since a great number of t i d a l c o n s t i t u e n t s are r e q u i r e d to obta i n the r e q u i r e d accuracy. A more d i r e c t method of es t i m a t i n g x and S was c a r r i e d out. By l o o k i n g at the a n a l y t i c a l s o l u t i o n as a wave equation i t i s noted that the phase v e l o c i t y , v , i s : "n v n ~ /(o nS/2T The phase v e l o c i t y i s the speed at which the c r e s t s of the waves t r a v e l . This can be estimated using a rough value f o r x and the average l a g time observed between the high water at the coast and i n the w e l l . The values of v n from the a n a l y t i c a l equation f o r the p r i n c i p a l d i u r n a l and semi-diurnal t i d a l c o n s t i t u e n t s were c a l c u l a t e d using the average value of T given. The average l a g time f o r the w e l l s , about 85 minutes, r e s u l t e d i n a value f o r x of about 40 feet c o r r e s -ponding to an S of .2 and of s e v e r a l hundred feet f o r S equal to .001. C l e a r l y (Figure 2) the value of .001 f o r S i s the r e a l i s t i c one. Thus the system appears to be the confined case. Another e s t i m a t i o n of x can be made without completing the summation. The a t t e n u a t i o n f a c t o r e'^^n2^^ x s h o u l d correspond roughly to the " e f f i c i e n c y f a c t o r " given i n the report on the w e l l s . This e f f i c i e n c y i s the r a t i o of the d i f f e r e n c e between the highest and lowest recorded t i d e values to the same d i f f e r e n c e i n the w e l l . A distance of s e v e r a l hundred feet f o r x gave an a t t e n u a t i o n f a c t o r comparable to the e f f i c i e n c y value. Since f u r t h e r d i r e c t a n a l y t i c a l s t u d i e s were not very p r a c t i c a l , p a r t i c u l a r l y i n the more general w e l l f l u c t u a t i o n problem, the s i t u a t i o n was approached from the more f l e x i b l e p oint of view of a f i l t e r . -25-FILTER APPROACH This approach to the problem t r e a t s the medium as a d i g i t a l f i l t e r , defined by the d i f f e r e n t i a l equation. Thus an input of f l u c t u a t i o n s i n the surface water r e s u l t s i n an output of f l u c t u a t i o n s at the w e l l a f t e r c e r t a i n m o d i f i c a -t i o n s i n amplitude and time l a g have been imposed. In general terms f i l t e r i n g may be represented as y = d*z where d i s the impulse response of the f i l t e r y i s the output s i g n a l z i s the input (or f o r c i n g ) s i g n a l d*z means a con v o l u t i o n (see below) y and z are both f u n c t i o n s of time. An impulse response i s the response of the f i l t e r to an impulse i n p u t , 6 ( t ) : 6(t) = 0 f o r t * 0 l i m \z 6(t) dt = 1 e+0 u Using the impulse response the input s i g n a l can be considered to be a s e r i e s of impulses of magnitude z ( t ^ ) and the area of each impulse to be -26 -z ( t 1 ) A t ± where At^ i s the i time i n t e r v a l . The system response at time t to a unit impulse at time t ^ i s d(t - t^) and thus the response to the above impulse at t ^ i s y(t) = z(t ) d(t - t ± ) A t ± The t o t a l response at t due to a l l the impulses before t i s m y(t) = I z(t.) d(t - t.) At. 1=0 1 1 1 where t = t corresponds to the upper l i m i t of the summation. Usually At^ Is a constant value denoted as At or T. This summation i s represented by y = z*d It i s necessary to fi n d d. - 2 7 -In general a d i f f e r e n t i a l equation of order n has the form: ( a n ( ^ ) n + a ^ ( g f ) " " 1 + ... + a Q ) y ( t ) = z ( t ) By t a k i n g F o u r i e r Transforms of both sides ( a n (ju>) n + a n _ 1 ( j o ) ) 1 1 - 1 + ... ) Y(u>) = Z(u>) where j 2 = -1 Y(u) = F{y} Z(to) = F{z] and F { v ( t ) } = jm v ( t ) e" J t o t dt = V(u>) f " M V ( u ) } = i- J " V U ) e J u t da) v ( t ) Thus y = z*F~l^n(3«)H + a n ^ ( J M ) " - l + ...} Therefore D = F " 1 U n ( j c o ) n + an^(ja»)n-l + . . . 1 i s the re q u i r e d impulse response of the f i l t e r defined by the general d i f f e r e n t i a l equation. This theory was a p p l i e d to the o r i g i n a l d i f f u s i o n equation using Laplace Transforms. This m o d i f i c a t i o n was used since i n t e r e s t i s r e s t r i c t e d to p o s i t i v e time because -28-t h i s i s a causal system. S 3h T 3t The Laplace Transforms are: L { v ( t ) } = v ( t ) e - s t dt = V(s) v ( t ) = L _ 1 { V ( s ) } where v ( t ) = 0, t < 0 s = a + joo To f i n d the impulse response of the f i l t e r defined by the equation the f o l l o w i n g boundary c o n d i t i o n s were a p p l i e d : 1) h(x,0) = 0 2) l i m h(0,t) = 6(t) t+0 3) l i m h(x,t) = 0 Using the d i f f e r e n t i a l equation i n the form 3h 9t = 0 where a = T S and t a k i n g the transform with respect to t , where L t { h ( x , t ) } = h(x,s) The transform of the d i f f e r e n t i a l equation i s a d 2 ^ x > s ) - sH(x,s) + h(x,0) = 0 -29-Applying the f i r s t boundary condition a d i h U i s l _ s R ( X j S ) a Q Taking the transform with respect to x L x{h(x,s)} = h(p,s) ap 2h(p,s) - apR(0,s) - aE x(0,s) - sh(p,s) = 0 h(p s) = aph(0 > s) + aR^O.s) Using the second boundary condition, R(0,s) = L t(h(0,t)} = 1. Thus £ ( p . . , . £ ^ % i ^ f i Then h(x,s) = L'MMp.s)} = 1/2(1 + R (0,s))e /a r ,n /s7a x p - T / s « x' /a e / n „ \ i ^ s 7 a X + 1/2(1 - / I h x ( 0 , s ) ) e But lim h(x,t) = 0, therefore x-»-«> lim h(x,s) = 0 x-*-» and thus 1 f / § h x(0,s) = 0 -30-Therefore "Tat x 2 h(x,t) = x e 2/a /Tt7 Therefore the impulse response F ( x , t ) f o r t h i s f i l t e r i s : Sx 2 F ( x , t ) = X / W 7 — where S i s dimensionless and about .001; T i s of the order of 3 (average value i s 3.2^9) f t 2 / minute; x i s i n f e e t ; and t i s i n minutes. The convolution of t h i s impulse response i n d i g i t a l form w i t h the s i g n a l A(t) i n d i g i t a l form i s : m H(x,t) = I F ( x , n i ) A ( t - m ) T n=l where m i s number of f i l t e r p o i n t s or number of data i n t e r -v a l s i n f i l t e r f u n c t i o n T i s data i n t e r v a l H(x,t) i s the p r e d i c t e d output s i g n a l . The F i l t e r The impulse response, F ( x , t ) , i s the b a s i s of the computer a n a l y s i s of the data. The a c t u a l impulse response shown i n Figure 8, has the problem of an i n f i n i t e t a i l . FIGURE 9 : NON-DIMENSIONAL IMPULSE RESPONSE LOG F C x , "t) LOG -fc FIGURE 1 0 : SKETCH OF POSITION OF FIGURE 9 WITH RESPECT TO A PLOT OF LOG F ( x , t ) VERSUS LOG t - 3 1 -Por the purpose of the d i g i t a l approach the a c t u a l impulse response was used out to an a r b i t r a r y time with respect to the area under the curve and the remainder was approximated by a r e c t a n g u l a r t a i l having an area equal to that of the remaining i n f i n i t e t a i l . This area approximation i s import-ant as i t d i r e c t l y c o n t r o l s the time l a g of the f i l t e r . Before c o n s i d e r i n g the a c t u a l r e c t a n g u l a r approximation i t i s worth c o n s i d e r i n g a non-dimensional approach to the impulse response. With a data i n t e r v a l of f i f t e e n minutes many s i g n a l s , such as the wake of a s h i p , may be considered to be approximations to impulse f o r c i n g f u n c t i o n s . Accord-i n g l y a non-dimensional p l o t of the impulse response could be of i n t e r e s t . Figure 9 i s such a p l o t . This l o g - l o g p l o t i s based on the f o l l o w i n g d e f i n i t i o n of dimensionless q u a n t i t i e s . The equation f o r the response to an impulse of he i g h t - d u r a t i o n , h^, above the mean water l e v e l i s Sx 2 ; "TppF h(x,t) = h+x / S e -/WiT t 3 / 2 where h + , i n the u n i t s used i n t h i s t h e s i s , i s i n f o o t -seconds. D e f i n i n g the non-dimensional parameter 2 * Sx 2  5 ~ Wt the equation becomes -32-and Taking logarithms log h(x,5) = log ( 5 ) + X + log h + l o g t = log U~ 2) - X where X = log 4T Sx 7 By p l o t t i n g log (-—^ ) against log ( C ~ 2 ) Figure 9 was obtained. This plot can be applied to a p a r t i c u l a r case by s h i f t i n g the axes by (X + log hf) and (-X) respectively (see Figure 10). Referring again to Figure 9 the graph may be extended l i n e a r l y since for £~ 2 values greater than about 30 the curve i s e s s e n t i a l l y a straight l i n e of slope -3 /2 . This graph may be used for a series of disconnected "impulses". The t o t a l response i s then the sum of the in d i v i d u a l responses with the appropriate time lags. How-ever t h i s approach becomes unwieldy when the signal becomes complex and one must turn to the convolution approach for any continuously varying input signal. Consequently the adaption of the f i l t e r function to the computer must now be further investigated. At t h i s point i t - 3 3 -i s probably e a s i e r to consider the computer program as a whole. Computer Program A program has been w r i t t e n to construct the f i l t e r approximation, perform the c o n v o l u t i o n , and output the pr e d i c t e d water l e v e l s at the w e l l s . A flow diagram of the program i s shown i n Figure 11 where SH i s mean water l e v e l with respect to the same datum as the data NDP i s the number of data p o i n t s X i s the distance from w e l l to mean water l e v e l , i n feet S i s the storage c o e f f i c i e n t T i s t r a n s m i s s i b i l i t y i n f e e t 2 per minute TAU i s data (time) i n t e r v a l i n minutes V E R F i s the constant used i n the approximation of the impulse response. The re c t a n g u l a r t a i l pf the f i l t e r i s constructed such that i t has a height equal to that of the l a s t f i l t e r point c a l c u l a t e d using the a n a l y t i c a l form of the impulse response and a length such that i t s area i s that remaining i n the t a i l . The a c t u a l use of the program r e q u i r e s data i n p u t , i n the form shown i n the f o l l o w i n g t a b l e . FIGURE 11: FLOW DIAGRAM OF COMPUTER PROGRAM INPUT SK, NDP, *|X, S, T, TAU, VERF INPUT RAW SOURCE DATA COMPUTE VALUES DF SOURCE DATA > WITH RESPECT TO *IEAN SEA LEVEL COMPUTE FIRST 15 POINTS OF FILTER (. ACCORDING TO EQUATION COMPUTE RECTANGULAR TA T 1 FILTER USING :ONVOLUT-ION OUTPUT CALCULATED; RESPONSE AT WELL IS AREA IN REMAINDER OF FILTER/ < VERF 9 COMPUTE NEXT FILTER POINT USING E Q U A T I O N - 3 4 -Table 3 INPUT CARD DATA 1 SH 2 NDP 3 X 4 S 5 T 6 TAU 7 VERF * PUT IN THE DECIMAL POINT CARDS 8 and f o l l o w i n g , f o r as many cards as necessary: the source input data i n the form of 11 numbers of 7 columns length per card: the 7 columns in c l u d e the decimal point and 6 d i g i t s , i n c l u d i n g zeros and blanks. The value f o r T i n f e e t 2 per minute can be found by m u l t i p l y i n g the value i n U.S. g a l l o n s / d a y / f t . by 9 . 2 8 x 1 0 " 5 . The value assigned to VERF depends on the f a c i l i t i e s of the computer, the number of data p o i n t s , the r e q u i r e d accuracy, and the geology and geometry. The smaller the value of V E R F , the c l o s e r the approach of the computer f i l t e r to the a n a l y t i c a l impulse response. However the number of COLUMNS NOTES 1-7 " 1 - 3 INTEGER - no decimal point 1-7 " 1 - 1 0 " 1 - 1 0 -1-7 " 1 - 1 0 -points i n the f i l t e r must be considerably less than the number of input points since the difference between the two (plus one) i s the number of output points. Symbolically NWP = NDP - N + 1 where NWP i s the number of well points (output) N i s the number of f i l t e r points (including t a i l ) . The computer time required i s greatly reduced i f the storage requirement i s limited to the core storage of the computer. This l a s t factor places an upper l i m i t on NDP and N. On the present University of B r i t i s h Columbia computer, an IBM 7040, the available core storage l i m i t s the number of f i l t e r points or data points to 130 unless use i s made of the u t i l i t y storage through the use of CHAIN or other features. This appears to be s u f f i c i e n t for the present cases but might not be suitable for an unconfined aquifer s i t u a t i o n . The following gives a rough guide for choosing VERF under conditions of NDP = 196 (49 hours of data with sampling at 15 minute intervals) X about 300 feet S = .001 T 6.5 5.5 3.0 1.7 VERF .10 .10 .15 .20 -36-T 6 •• 4 •• a •• o . With an increase i n T the t a i l tends to decrease more r a p i d l y and thus the necessary amount of re c t a n g u l a r t a i l decreases. Therefore VERF can be decreased. The program may output N GREATER THAN 130 or NWP GREATER THAN 130 In the f i r s t case VERF should be l a r g e r and i n the second VERF must be made smaller. I f NDP i s decreased, VERF must be increased i n order to get enough output. Thus i f the f i l t e r i s to be longer (a b e t t e r approximation to the a n a l y t -i c a l s o l u t i o n ) , VERF must be smaller and v i c e versa. This emphasis on the choice of VERF i s necessary i n that VERF c o n t r o l s the degree to which the a n a l y t i c a l s o l u t i o n i s approached. Output The output from the program has been p l o t t e d (Figure 12) and compared with the a c t u a l w e l l data. Problems with c a l c u l a t i o n s f o r w e l l 1 were increased due to the v a r i a b l e value of T. A mean of the three values given f o r w e l l 1 was used; t h i s value may be s l i g h t l y high. -37-The t i m i n g of the c a l c u l a t e d output was remarkably cl o s e to that of the a c t u a l data. In most cases the times of peaks, troughs, and mean sea l e v e l were w i t h i n a data i n t e r v a l of the observed times. The most n o t i c e a b l e o v e r a l l d e v i a t i o n of the c a l c u l a t e d values from the a c t u a l values was with respect to the magnitudes of the peaks and the troughs. In a l l cases the p r e d i c t e d high water was s l i g h t l y greater than the a c t u a l and the p r e d i c t e d low water was much lower than the a c t u a l value. There are s e v e r a l p o s s i b l e explanations f o r t h i s d e v i a t i o n of the c a l c u l a t e d r e s u l t s from the w e l l observa-t i o n s . Since t h i s i s a n a t u r a l system i t i s l i k e l y to be non-l i n e a r which f a c t immediately puts an upper l i m i t on the extent to which the computer output, based on l i n e a r i z a -t i o n s , can approach the observations. The values of S,T, and x are probably not constant. The ground i s f a r from being homogeneous and the water l e v e l i s continuously changing. There i s the p o s s i b i l i t y of recharge e f f e c t s which, although d i f f i c u l t to measure might set a c e r t a i n minimum water l e v e l f o r the a q u i f e r . Coupled with t h i s there might be an overflow system which would l i m i t the high water by an outflow from the a q u i f e r a f t e r the water reaches a c e r t a i n FIGURE 1 2 A : CALCULATED VALUES FOR WELL 1 FOR JULY 10, 1965 io«-oaoo 1 0 0 0 l a o o i<v,oox »4>po 1004 1 1 —I H i 1 3 8 + *4 x X I ' » * ^  « * * "< Wo.-te.r- L e v e l Time PST FIGURE 1 2 B : CALCULATED VALUES FOR WELL 3 FOR JULY 10, 1965 i<*4 100-984-x K O S 0 0 I O O O 1 2 0 0 Kp<i * *. 1 < * < X Wo^er LeveV T i m e > PST FIGURE 12C: CALCULATED VALUES FOR WELL 1 FOR JULY 25,26, 1965 10+ IOSL' * M A A X x 0800 IOOO laoo H O O 98' 3 t W c i t r Level T i m e — PST »c=oo IQOO zooo •2.ZOO Z*t oo FIGURE 1 2 D : CALCULATED VALUES FOR WELL 2 FOR JULY 25,26, 1965 10+ \\oz 9 8 96 1 1 1 1 1 1 1 1 1 1 1 i ° s o o \ooo i^oo i<voo i t o o 18 oo fcooo zzoo z+oo o a o o MOO O&OO I * ^ Water- L e v e l Icr*: \ ^oot T i me \ C n \ \ \ Wo or FIGURE 12E: CALCULATED VALUES FOR WELL 3 FOR JULY 2 5 , 2 6 , 1965 x x * , I ' 0 * ' x ' * X H98 Water- Level TTme, * PST » r —1 1 1 ; 1 1 1 1 1 1 1 1 • 08OO 100O 1100 WOO (feOO 1 8 0 0 2 0 0 0 2*oo O S L O O 0 * 0 0 o « , o o -38-l e v e l . The recharge idea i s very l i k e l y to be v a l i d . This would help to reduce the deep troughs of the c a l c u l a t e d output. The amplitude of the c a l c u l a t e d trough might be f u r t h e r reduced by the c o n s i d e r a t i o n that the p h y s i c a l distance from the surface water to the w e l l changes much more r a p i d l y w i t h water l e v e l s below mean sea l e v e l due to the very low slope of the mud f l a t s at e l e v a t i o n s below mean sea l e v e l . Thus the e f f e c t of the deep t i d a l lows would be decreased. Fundamental to the n o n - l i n e a r i t y of the system i s the f a c t that the r i s i n g water l e v e l has an e f f e c t i v e l y i n f i n i t e source i n the r i s i n g t i d e but the f a l l i n g water l e v e l has a very f i n i t e source (the water contained i n the pores of the a q u i f e r ) a l b e i t an i n f i n i t e s i n k. F i e l d observations i n v o l v i n g pumping and recharging w e l l s note a d i f f e r e n c e i n the r a t e at which water can be made to leave the w e l l with respect to the r a t e at which water enters the w e l l , the l a t t e r being at a decidedly greater r a t e . Accordingly i t i s to be expected that the w e l l records would not show as much c o r r e l a t i o n f o r low water l e v e l s with a theory based on a l i n e a r , homogeneous model. F i n a l l y the problem of a p o s s i b l e f i n i t e source length was considered. The theory f o r the computations had been developed from the assumption of an i n f i n i t e , s t r a i g h t source. The e f f e c t of a source subtending an angle of 135 - 3 9 -degrees at the w e l l was computed (non-dimensionally) and p l o t t e d against the corresponding Impulse response (Figure 1 3 ). In t h i s case the non-dimensional parameter was ( T t / S x 2 ) . Up to a value of one, the two curves are i d e n t i c a l and f o r higher values of the parameter the f i n i t e source curve drops o f f q u i c k e r . This w i l l e f f e c t the t i m i n g s l i g h t l y as the area under the new curve i s l e s s . Neverthe-l e s s the i n f i n i t e source model i s a good approximation. FIGURE 13: F INITE LENGTH CORRECTION TO IMPULSE RESPONSE -40-CONCLUSIONS By the use of t h i s a n a l y t i c a l model one can p r e d i c t , to a f i r s t approximation, the response of the water l e v e l s i n the w e l l s . Since the model used i s s e n s i t i v e to changes i n T i t may p o s s i b l y be used as a prospecting t o o l . Accordingly by i n p u t t i n g source data f o r a p r e l i m i n a r y w e l l with various values of T and comparing the outputs with the w e l l data, a f i r s t approximation of T could be found to w i t h i n about ±1. A d e c i s i o n on the appropriate case (confined or unconfined) must be made to give S f o r the above a n a l y s i s . ( I t i s of i n t e r e s t to note that f o r the unconfined case the data used h e r e i n would i n d i c a t e a value of x of the order of 50 feet . ) I f the value of S i s not obvious, a s i n g l e run of the above program with a rough value of T w i l l q u i c k l y show which case i s i n v o l v e d . When comparing the output to the w e l l data the p o s i t i o n s on the time s c a l e of the peaks and of mean sea l e v e l are the best i n d i c a t i o n of the closeness of the approximation i n v o l v e d . A rough s i m i l a r i t y of a c t u a l water heights i s a l l that can be expected from t h i s l i n e a r model. For an approximate "impulse" input a p l o t of l o g h versus l o g t could be used i n conjunction with the non--41-dimensional plot (Figure 9 ) to give an estimate of the r a t i o of the geological parameters. The easiest c a l c u l a t i o n would be to evaluate T/S from the displacement of the zero of log (£~ 2) which i s equivalent to log (4T/Sx 2) (Figure 10). It would be of d e f i n i t e interest to apply these hypothe-sized "tools" i n the f i e l d and to have more data of the kind used i n t h i s thesis to check the approximations and inferred problems. However "time and tide wait for no man." FIGURE 14: IMPULSE RESPONSES FOR THE THREE WELLS TAU *» 15 — IVell 1 — We.ll 2. - W e l l 3 mnuitu T I M E » t = n *. TAVJ Cn) -42-BIBLIOGRAPHY A s e l t i n e , J.A., 1958. Transform Method in Linear System Analysis. McGraw-Hill, New York: 178-184. Carslaw, H.S., and J.C. Jaeger, 1959. Conduction of Heat in S o l i d s . Oxford U n i v e r s i t y Press, London: 50-62, 92-105, 273-277, 370-371. Cooper, H.C., 1966. The Equation of Groundwater Flow in Fixed and Deforming Coordinates. Jour. Geophys. Res., 71 (20): 4785-4790. Defant, A., 1958. Ebb and Flow. U n i v e r s i t y of Michigan Press. DeWiest, R.J.M., 1965. Geohydrology. John Wiley and Sons, In c . , New York. DeWiest, R.J.M., 1966. On the Storage C o e f f i c i e n t and Equations of Ground Water Flow. Jour. Geophys. Res., 71 ( 4 ) : 1117-1122. D i c k e r , D., and W.A. Sevian, 1965. Transient Flow Through Porous Mediums. Jour. Geophys. Res., 7.0 (20): 5043-5054. Doodson, A.T., and H.D. Warburg, 1941. Admiralty Manual of Tides. Hydrographic Department, Admiralty, London. Eyres, N.R., D.R. Hartree, J . Ingham, R. Jackson, R.J. S a r j a n t , and J.B. Wagstaff,. 1948. The C a l c u l a t i o n of Variable Heat Flows in S o l i d s . P h i l . Trans. Roy. S o c , A. 240_: 1-57. F e r r i s , J.G., D.B. Knowles, R.H. Brown, and R.W. Stallman, 1962. Theory of Aquifer Tests. Geol. Survey Water-Supply Paper 1536-E> U.S. Government P r i n t i n g O f f i c e , Washington. Jacob, C.E., 1950. Flow of Ground Water. Engineering H y d r a u l i c s , H. Rouse (Ed.), John Wiley and Sons, Inc., New York: 321-386. -43-Polubarinova-Kochina, P.Ya., 1962. Theory of Ground Water Movement. T r a n s l a t i o n from Russian by R.J.M. DeWiest. P r i n c e t o n U n i v e r s i t y Press, New Jersey: 1-50. Robinson, E.A., 1964. Econometric Model Building. H.O.A. Wold (Ed.). Niorth-Holland P u b l i s h i n g Co., Amsterdam: 37-168. Schureman, P., 1941. Manual of Harmonic Analysis and P r e d i c t i o n of Tides. U.S. Government P r i n t i n g O f f i c e , Washington. Todd, D.K., 1959. Ground Water Hydrology. John Wiley and Sons, Inc., New York. Werner, P.W. and D. Noren, 1951. Progressive Waves in Non-Artesian Aquifers. American Geophys. Union, 32 ( 2 ) : 238-244. -44-SYMBOL TABLE a constant i n the d i f f u s i o n equation a' n speed number of t i d a l component n a n constant A(t) input s i g n a l i n d i g i t a l form b a q u i f e r t h i c k n e s s constants d impulse response of a system E astronomical parameter as s o c i a t e d with t i d a l forces f long p e r i o d t i d a l c o r r e c t i o n f a c t o r F ( x , t ) impulse response of system g F o u r i e r transform and i t s inverse a c c e l e r a t i o n due to g r a v i t y & n phase l a g f o r t i d a l component n h water height h» d e v i a t i o n from mean water l e v e l h o mean sea l e v e l h+ h e i g h t - d u r a t i o n ( i n foot-seconds) H stage of the t i d e j J 2 - -1 k i n t r i n s i c p e r m e a b i l i t y K e f f e c t i v e p e r m e a b i l i t y or transmis s i o n c o e f f i c i e n t K ,K ,K tran s m i s s i o n c o e f f i c i e n t s i n x,y,z, d i r e c t i o n s L d i s t a n c e , i n Darcy's Law -45-LjL"" 1 Laplace transform and Its inverse m,n integers MSL mean sea l e v e l N number of f i l t e r points i n computer analysis NDP number of data points input into computer NWP number of well points output from computer p pressure SH mean sea l e v e l input into computer S c o e f f i c i e n t of storage S s p e c i f i c y i e l d S_ s p e c i f i c retention r t time . T c o e f f i c i e n t of t r n s m i s s i b i l i t y TAU data i n t e r v a l u p a r t i c l e v e l o c i t y v1 s p e c i f i c discharge or seepage v e l o c i t y vi>V 2 v e l o c i t i e s at points 1 and 2 VERF constant used i n approximation of f i l t e r response by computer V volume ¥ volume of water w w volume of water that can be drained by gravity y X,x distance from well to surface water (or to MSL) y output signal z' elevation of the head of water i n Darcy's Law z,z(t) input signal -46-a v e r t i c a l c o m p r e s s i b i l i t y B c o m p r e s s i b i l i t y B c o m p r e s s i b i l i t y o f w a t e r 6 ( t ) i m p u l s e Y w e i g h t o f a u n i t v o l u m e o f w a t e r 6 p o r o s i t y A d i m e n s i o n a l c o r r e c t i o n t e r m f o r n o n - d i m e n s i o n a l p l o t u d y n a m i c v i s c o s i t y 5 n o n - d i m e n s i o n a l p a r a m e t e r u s e d i n i m p u l s e r e s p o n s e p d e n s i t y a c o m p r e s s i v e s t r e s s i n v e r t i c a l d i r e c t i o n T d a t a i n t e r v a l tj) v e l o c i t y p o t e n t i a l ij* c o n s t a n t p a r t o f t h e p h a s e o f t h e t i d e OJ f r e q u e n c y i n r a d i a n s p e r s e c o n d 

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