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Regional geology, groundwater flow systems and slope stability 1976

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REGIONAL GEOLOGY, GROUNDWATER FLOW SYSTEMS AND SLOPE STABILITY by ROBERT A.L. HODGE B.A.Sc, U n i v e r s i t y of B r i t i s h Columbia, 1972 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of G e o l o g i c a l Sciences The U n i v e r s i t y of B r i t i s h Columbia We accept t h i s t h e s i s as conforming to the r e q u i r e d standard THE UNIVERSITY OF BRITISH COLUMBIA June, 1976 0 Robert A.L. Hodge, 1976 In presenting th i s thes is in pa r t i a l fu l f i lment of the requirements for an advanced degree at the Un ivers i ty of B r i t i s h Columbia, I agree that the L ibrary sha l l make it f ree ly ava i l ab le for reference and study! I fur ther agree that permission for extensive copying of th is thesis for scho lar ly purposes may be granted by the Head of my Department or by his representat ives. It is understood that copying or pub l i ca t ion of th is thes is fo r f i nanc ia l gain sha l l not be allowed without my wr i t ten permission. Depa rtment The Univers i ty of B r i t i s h Columbia 2075 Wesbrook P l a c e Vancouver, Canada V6T 1W5 Date i i ABSTRACT The purpose of t h i s t h e s i s i s to show, using computer s i m u l a t i o n of flow systems i n a v a r i e t y of h y p o t h e t i c a l s l o p e s , how d i f f e r e n t g e o l o g i c a l environments a f f e c t the groundwater fl o w regime, which i n t u r n i s funda- mental to the s t a b i l i t y of a slope. G a l e r k i n ' s method i s used to d e r i v e a f i n i t e element program to model two dimensional, s a t u r a t e d , steady s t a t e flow through a n i s o t r o p i c and heterogeneous r i g i d porous media. An understanding of the r e g i o n a l geology i s r e q u i r e d i n order to un- derstand the r e g i o n a l flow system. The f o l l o w i n g p o i n t s are i l l u s t r a t e d . a. In a n i s o t r o p i c media, the most adverse groundwater c o n d i t i o n f o r slope s t a b i l i t y occurs when the major a x i s of c o n d u c t i v i - t y l i e s down the d i p of the sl o p e . b. Depending on t h e i r c h a r a c t e r i s t i c s , f a u l t s , contacts and dykes can be e i t h e r d e t r i m e n t a l or favourable i n t h e i r e f - f e c t on the flow system. C a r e f u l f i e l d i n v e s t i g a t i o n i s r e q u i r e d to e s t a b l i s h that e f f e c t . c. Deep weathering commonly causes a c o n f i n i n g zone of low c o n d u c t i v i t y , a s i t u a t i o n very d e t r i m e n t a l to s t a b i l i t y . d. Stress r e l i e f f r a c t u r e s on v a l l e y w a l l s can adversely i n - fl u e n c e the e f f e c t of groundwater on s t a b i l i t y . e. A r e g i o n a l a q u i f e r can cause hig h pore pressure development beneath a v a l l e y . f. F l u c t u a t i o n s i n the r e g i o n a l groundwater system can cause i n s t a b i l i t y i n P l e i s t o c e n e t e r r a c e s . g. The presence of an u n d e r l y i n g l e s s conductive zone or u n i t i i i can have an adverse e f f e c t on the flow system. C o n d u c t i v i t y c o n t r a s t s of l e s s than two orders of magnitude can cause pore pressure development c r i t i c a l to s t a b i l i t y . Three other p o i n t s are demonstrated which have d i r e c t a p p l i c a t i o n to slope s t a b i l i t y a n a l y s i s and c o n t r o l . 1. The pressure head d i s t r i b u t i o n on rock wedges can be non- l i n e a r r a t h e r than the commonly assumed l i n e a r d i s t r i b u - t i o n . 2. The i n t r o d u c t i o n of a r e s e r v o i r at the toe of a slope can i n f l u e n c e the groundwater regime w e l l above the r e s e r v o i r s u r f a c e ; even a low r e s e r v o i r can cause, the change r e q u i r e d to cause i n s t a b i l i t y . 3. P i e z o m e t r i c measurements and drainage systems must penetrate through any l e s s conductive u n i t that might be a c t i n g as a s l i d e plane. i v CONTENTS Page CHAPTER ONE: INTRODUCTION 1 CHAPTER TWO: A REVIEW OF SLOPE STABILITY ANALYSES 3 I n t r o d u c t i o n 3 Categories of Slope S t a b i l i t y A n a l y s i s 5 L i m i t E q u i l i b r i u m Methods 5 E l a s t i c S o l u t i o n s of the F i n i t e Element Type 9 D i s c r e t e P a r t i c l e A n a l y s i s 10 H i s t o r i c Development of L i m i t E q u i l i b r i u m Methods 11 The Concept of E f f e c t i v e S t r e s s 30 Groundwater and L i m i t E q u i l i b r i u m Methods 31 CHAPTER THREE: GROUNDWATER FLOW, GALERKIN'S METHOD AND THE FINITE ELEMENT METHOD 34 I n t r o d u c t i o n 34 Theory 35 Example 42 M a t r i x S o l u t i o n 47 Heterogeneity and Aniso t r o p y . 48 CHAPTER FOUR: LIMITING ASSUMPTIONS AND HYDRAULIC CONDUCTIVITY 51 L i m i t i n g Assumptions 51 H y d r a u l i c C o n d u c t i v i t y 57 CHAPTER FIVE: MODEL RESULTS AND DISCUSSION 60 I n t r o d u c t i o n 60 The E f f e c t s of Ani s o t r o p y 64 Thrusts and Interbedded Sedimentary Rocks 66 F a u l t s , Contacts, Dykes and Weathering P r o f i l e s 69 Layered Colluvium and Buried Weathering P r o f i l e s 71 F l a t L y i n g Weak Rocks, F r a c t u r e s Due to Str e s s R e l i e f and the E f f e c t s of a Regional A q u i f e r 74 P l e i s t o c e n e Terraces 75 Deformed Metamorphic Rocks and the E f f e c t of a Rese r v o i r on a Deep Rock S l i d e 79 V CONTENTS (Cont'd) Page CHAPTER SIX: SUMMARY AND CONCLUSIONS 85 Slope S t a b i l i t y Analyses 85 L i m i t i n g Assumptions 86 Model R e s u l t s 87 REFERENCES 91 APPENDIX I : COMPUTER PROGRAM APPENDIX I I : FIGURES FOR CHAPTER FIVE TABLES Page Table 2-1 Categories of Slope S t a b i l i t y A n a l y s i s Table 5-1 Weathering p r o f i l e f o r igneous and metamorphic rocks . 72 Table 5-2 R e s u l t s of s t a b i l i t y a n a l y s i s of a P l e i s t o c e n e t e r r a c e 79 Table 5-3 R e s u l t s of s t a b i l i t y a n a l y s i s of a p o t e n t i a l rock s l i d e i n metamorphic t e r r a i n 62 v i i ILLUSTRATIONS Page Figure 2-1 Schematic stress-strain curves for ideal and real materials 7 Figure 2-2 Fellenius' method of slices for a curved sliding surface through the toe of a slope 14 Figure 2-3 Force diagram 14 Figure 2-1- The ^ - c i r c l e method for toe failures 16 Figure 2-5 Rendulic's log-spiral method 17 Figure 2-6 Taylor's ^ - c i r c l e method for complete sub- mergence 18 Figure 2-7 Taylor's ^ - c i r c l e analysis for capillary saturation 20 Figure 2-8 Forces used by Janbu in his Generalized Proceedure of Slices 21 Figure 2-9 Forces used in Bishop's Method 23 Figure 2-10 Forces used in the Morgenstern-Price Method 25 Figure 3-1 Descritization of region into f i n i t e triangu- lar elements 33 Figure 3-2 Region of flow 43 Figure 3-3 Matrix form of equation (3.19) 47 Figure 3-4 Trigonometry used in defining nodal coordin- ates i n terms of a local coordinate system 49 Figure 4-1 Ranges of hydraulic conductivity 58 Figure 5-1 Summary of Models 62 Figure 5-2 Hydraulic head, elevation head, and pressure head 63 v i i i ILLUSTRATIONS (Cont'd) Page F i g u r e 5-3 The e f f e c t s o f a n i s o t r o p y * F i g u r e 5-4 Summary of p i e z o m e t r i c l i n e s from F i g u r e 5-3 * F i g u r e 5-5 T h r u s t s and i n t e r b e d d e d s e d i m e n t r y r o c k s * F i g u r e 5-6 F a u l t s , c o n t a c t s , dykes and w e a t h e r i n g p r o f i l e s * F i g u r e 5-7 Summary of p i e z o m e t r i c l i n e s f o r F i g u r e 5-6 * F i g u r e 5-8 L a y e r e d c o l l u v i u m and b u r i e d w e a t h e r i n g p r o f i l e s * F i g u r e 5-9 F l a t l y i n g weak r o c k s , f r a c t u r e s due t o s t r e s s r e l i e f , and t h e e f f e c t s of a r e - g i o n a l a q u i f e r * F i g u r e 5-10 P l e i s t o c e n e T e r r a c e s * F i g u r e 5-11 Deformed Metamorphic Rocks and t h e e f - f e c t s o f a r e s e r v o i r o r a deep r o c k s l i d e * F i g u r e 5-12 Summary o f p i e z o m e t r i c l i e s from F i g u r e 5-11 * * APPENDIX i i - I S , M rttap Cabinet^ ix ACKNOWLEDGMENTS A number of people have given much needed help and advice throughout this project. Katsayuki Fujinawa, v i s i t i n g graduate student from Kyoto University, Japan, spent many hours with me patiently discussing the nu- merical and computer techniques. Peter Byrne,: Department of C i v i l Engi- neering, U.B.C. reviewed my work on slope s t a b i l i t y analyses. Hardy Bunn, C.B.A. Engineering Ltd., and Julie Wytrwal, Department of Mathematics, U.B.C, managed to wade through my hieroglyphic writing and produce a neat- ly typed manuscript. Two people in particular have influenced and guided me during the past few years: F.D. Patton, Consultant Engineering Geologist, and my supervi- sor, R.A. Freeze, Department of Geological Sciences, U.B.C. Without the technical and moral assistance they have provided, this thesis would not have been possible. 1. CHAPTER ONE: INTRODUCTION The c r i t i c a l r o l e that groundwater plays i n the s t a b i l i t y of slopes has long been recognized. However, for many years the mechanism that related porewater pressures to the strength of the s o i l or rock was not understood. In 1923 Terzaghi provided that l i n k with h is e f f e c t i v e stress law. Using the concept of ef f e c t i v e stress, a number of tech- niques have been developed to quantitatively analyze the stresses i n a slope to provide an estimate of s t a b i l i t y . A l l of these methods assume that porewater pressures are a known quantity. In practice', these are usually obtained by f i e l d measurement with piezometers. Unfortunately, such measurements are both d i f f i c u l t to obtain and sometimes inaccurate. In 1940, Hubbert proposed the exact mathematical equations that govern steady-state groundwater flow. However, i t was not u n t i l Toth (1962, 1963) introduced the method of mathematically modelling a groundwater flow system by solving a formal boundary value problem, that i t became possible to obtain an estimate of the groundwater flow system by a means other than with d i r e c t measurement of the pore pressure d i s - t r i b u t i o n . With an understanding of conductivity contrasts and a know- ledge of the water table configuration, flow systems could be rea d i l y modelled and an estimate of the complete pore pressure d i s t r i b u t i o n obtained. The power of thi s technique was demonstrated by Freeze and Witherspoon (1966, 1967) who discussed i n d e t a i l the t h e o r e t i c a l analysis of regional groundwater flow systems. I t i s now possible to model three- dimensional, transient groundwater flow through a porous media that i n - cludes both saturated and unsaturated zones (Freeze, 1971a). 2. Mathematical models can provide an independent check on measured pore pressures. When anomalies a r i s e , models can p o i n t to f i e l d c o n d i t i o n s not p r e v i o u s l y expected and m o d i f i c a t i o n s to f i e l d i n v e s t i g a t i o n s can r e s u l t . Deere and Patton (1967, 1971) and Patton and Hendron (1974) have discussed the i m p l i c a t i o n s to slope s t a b i l i t y of groundwater flow i n a number of d i f f e r e n t slope environments. Despite the r e c o g n i t i o n of these i m p l i c a t i o n s , the v e r s a t i l e m o delling techniques now a v a i l a b l e have not been a p p l i e d to a c c u r a t e l y i l l u s t r a t i n g flow systems i n the complex g e o l o g i c environments o f t e n a s s o c i a t e d w i t h unstable s l o p e s . The purpose of t h i s t h e s i s i s to show, usi n g computer simu- l a t i o n s of flow c o n d i t i o n s i n a v a r i e t y of h y p o t h e t i c a l s l o p e s , how d i f f e r e n t g e o l o g i c environments a f f e c t the groundwater flow regime which i n t urn i s fundamental to the s t a b i l i t y of a slope. I t w i l l be shown that to understand the d i s t r i b u t i o n of pore pressures i n a given s l o p e , i t i s necessary to understand the r e g i o n a l flow system i n which the slope i s l o c a t e d . To f a m i l i a r i z e myself w i t h the mechanics of slope s t a b i l i t y analyses and to e s t a b l i s h how the e f f e c t s of groundwater were f a c t o r e d i n t o the a n a l y s e s , a review was undertaken of a l l the a v a i l a b l e l i t e r a - t u r e p e r t a i n i n g to the techniques of slope s t a b i l i t y a n a l y s i s . The r e s u l t s of t h i s review are reported i n Chapter Two. In Chapter Three, the theory u n d e r l y i n g the computerized mathematical models i s given i n - d e t a i l and a s i m p l i f i e d example i s presented which i l l u s t r a t e s the numerical technique. I n any p r a c t i c a l a p p l i c a t i o n of mathematical models i t i s c r i t i c a l that u n d e r l y i n g assumptions are understood. These are presented i n Chapter Four. The r e s u l t s of the modelling and d i s - c ussion of t h e i r i m p l i c a t i o n s i s found i n Chapter F i v e and i n Chapter S i x the p r o j e c t i s summarized. CHAPTER TWO: A REVIEW OF SLOPE STABILITY ANALYSES I n t r o d u c t i o n To completely document a l l the developments which have l e d to -the p r e s e n t l y used techniques of a n a l y z i n g slope s t a b i l i t y would r e q u i r e a lengthy volume. The purpose of t h i s review i s t w o f o l d . F i r s t l y , a b r i e f d e s c r i p t i o n i s given of the three c a t e g o r i e s of a n a l y s i s : methods .based on l i m i t e q u i l i b r i u m , e l a s t i c s o l u t i o n s of the f i n i t e element type, and Cundall's method based on d i s c r e t e p a r t i c l e mechanics. F o l - lowing t h i s i n i t i a l d i s c u s s i o n , a more d e t a i l e d account i s presented of the v a r i o u s steps which have l e d to the present-day form of the l i m i t e q u i l i b r i u m method. Three d i s t i n c t aspects of t h i s development can be i s o l a t e d : i d e n t i f y i n g a l l the s t r e s s e s a c t i n g , understanding the mecha- n i c s which r e l a t e the s t r e s s e s , and determining the n a t u r a l p r o p e r t i e s which govern the m a t e r i a l behaviour when the s t r e s s e s are a c t i n g . The i n i t i a l i n t e n t of t h i s review was to gain i n s i g h t i n t o the ^assumptions that, v a r i o u s i n v e s t i g a t o r s have made regarding the r o l e of groundwater. This u n d e r l y i n g m o t i v a t i o n remains. For t h i s reason, the ^chapter concludes w i t h a b r i e f d i s c u s s i o n of the concept of e f f e c t i v e s t r e s s f o l l o w e d by a summary of d i f f e r e n t ways of ha n d l i n g groundwater i n l i m i t e q u i l i b r i u m techniques. 5. Categories of Slope Stability Analysis The three categories of slope s t a b i l i t y analysis are summarized in Table 2-1. As well as descriptions, also lis t e d are advantages, dis- advantages, and the a b i l i t y of each to account for groundwater. Limit Equilibrium Methods Limit equilibrium methods assume a failure plane, consider forces acting on the f a i l i n g mass, and further assume that a c r i t i c a l limit occurs after which resisting forces are no longer greater than forces causing i n s t a b i l i t y . Failure then occurs simultaneously every- where along the failure surface. For a given slope, a factor of safety i s calculated by considering the ratio of the materials shear strength (forces resisting failure) to the shear stress (forces causing fa i l u r e ) . The c r i t i c a l limit referred to above is equivalent to a factor of safety of one. In relating shear strength to normal stress, materials are assumed to follow either Coulomb's Law (Coulomb, 1776): T = c + ô tannS (2.1) or the Revised Coulomb Equation (Terzaghi, 1923, 1936): x = c' + (a - u)tan<j>' (2.2) where T = shear strength c and c' = cohesion and effective cohesion = total normal stress u = porewater pressure CATEGORY OF ANALYSIS ADVANTAGES DISADVANTAGES GROUNDWATER LIMIT EQUILIBRIUM The equilibrium condition of a rigid body or a system of rigid bodies is considered. A factor of safety is cal- culated by comparing the shear strength to the shear stress. ELASTIC SOLUTIONS OF THE FINITE ELEMENT TYPE The stress-strain condition of a continuous body is developed, based on the theory of elasticity. DISCRETE PARTICLE ANALYSIS A slope is mod collection of rigid blocks, librium of eac considered. U explicit time by considering static equilib (quasi-dynamic calculated as time. elled as a discrete The equi- h block is ses an integration a series of rium analyses ). Motion is a function of Results in a single factor cf safety (within confidence limits) which is easily understandable. Readily handles heterogeneous material properties. Readily handles any porewater pressure distribution. When a pre-existing, known discon- tinuity controls the failure, exact modelling is possible. Results In a complete stress picture for the body. Stresses, strains, and displacements can be obtained. A failure plane need not be assumed. A l l the laws of mechanics are satis- fied: equilibrium, compatibility, and boundary conditions. Fairly easy to handle a quasi-dynamic situation. Results *n a dynamic illustration of the slope which stands i f stable or f a l l s i f unstable. Does not require the preselection of a failure surface. Systems of fractures can be modelled exactly. Allows for unlimited displacements and rotations. Can describe a progressive failure. 1. If the failure surface has not been located, the worst case must be found by t r i a l and error. 2. No measure of strain on displacement is obtained. 3 . Because only static equilibrium con- ditions are considered, dynamic situations such as a progressive failure cannot be modelled. 4 . Is indeterminate due to an inadequate understanding of the normal stress distribution on the base and the normal and shear stress distribution on the sides of slices. Depending on the method, different assumptions are made to render the problem determinate. 1. Limited to small strain by the theory of elasticity. (This does not pre- clude large displacements.) 2. Because the body must remain a con- tinuum, discontinuities cannot be truly modelled although approxima- tions can be made by using highly elastic joint elements. 3 . The correct stress-strain relationship for each element must be known. 4 . Cannot handle a progressive failure. 1. Is written in terms of total stresses, not effective stresses. 2. Applies only to elopes which can be modelled as systems of rigid blocks. Any groundwater flow system can be r e a d i l y handled. Groundwater can be handled in terms cf porewater pressures at nodes or seepage forces. However, the effect of groundwater on a materials modulus of elasticity is not generally agreed upon. Cannot at present handle porewater pressures. Table 2-1. Categories of Slope S t a b i l i t y Analysis 7. <(> and <J>' = angle and effective angle of internal f r i c t i o n Depending on which law Is used, analyses are said to be i n terms of total or effective stresses. Most s t a b i l i t y analyses involve an experimental calculation of c and $ or c' and <J>' These values are then used in the analysis, i n essence replacing the real by an ideal material. Terzaghi and Peck (1967, p. 104) point out that this replacement involves the assumption that both c and <f> or c' and <{>' are independent of strain and only when the shearing stress at every point along the continuous failure surface overcomes the shear strength is., a failure generated. In fact, with the exception of ideal plastic materials, strength i s dependent on strain. After an i n i t i a l peak is overcome, strength decreases with further strain u n t i l a constant residual value i s attained. Figure 2-1 shows schematically the stress-strain curves for an ideal plastic material (peak strength not strain dependent) and a'more real- i s t i c material (strength i s strain dependent). | (a) i l e a l strain Figure 2-1. Schematic stress-strain curves for ideal and real materials 8. For any n a t u r a l m a t e r i a l c h a r a c t e r i z e d by curve (b) i n F i g u r e 2-1, the assumption of a simultaneous f a i l u r e w i l l be v i o l a t e d because s t r a i n s along a p o t e n t i a l f a i l u r e s u rface w i l l not be uniform. As a con- sequence, m a t e r i a l along p a r t s of the f a i l u r e s u r f a c e w i l l be e x e r t i n g peak str e n g t h and the remainder, something l e s s . F a i l u r e w i l l then s t a r t at one p o i n t and propagate along the f a i l u r e s u r f a c e causing a "progressive f a i l u r e " . P r o g r e s s i v e f a i l u r e s i n v a l i d a t e the assumptions on which l i m i t e q u i l i b r i u m techniques are based. A review of a l l the ideas p e r t a i n i n g to p r o g r e s s i v e f a i l u r e i s beyond the scope of t h i s p r o j e c t . However, f o r f u r t h e r background, the reader i s r e f e r r e d to Bishop (1967, 1971), Bjerrum (1967), C h r i s t i a n and Whitman (1969), H a e f e l i (1965), Lo (1972), Lo and Lee (1973), Lutton (1971), Romani et a l (1972), T u r n b u l l and Hvorslev (1967), and M a n f r e d i n i et a l (1975). Despite the above l i m i t a t i o n s , l i m i t e q u i l i b r i u m techniques remain a powerful t o o l . Heterogeneous m a t e r i a l p r o p e r t i e s cause no d i f f i c u l t y . Any porewater pressure d i s t r i b u t i o n can e a s i l y be i n c l u d e d . The geometry of f a i l u r e planes c o n t r o l l e d by d i s c o n t i n u i t i e s can be modelled e x a c t l y . And f i n a l l y , provided one i s aware of the l i m i t a - t i o n s and assumptions u n d e r l y i n g i t s c a l c u l a t i o n , the r e s u l t i n g f a c t o r of s a f e t y i s understandable and u s e f u l f o r both design purposes and decision-making. Because of an inadequate understanding of the normal s t r e s s d i s t r i b u t i o n along the base of the s l i d e and the normal and shear s t r e s s d i s t r i b u t i o n on the s i d e s of s l i c e s ( i n the case of methods i n v o l v i n g s l i c e s ) , l i m i t e q u i l i b r i u m methods are indeterminate. To render the problem determinate, an assumption must be made regarding e i t h e r the s i d e or base f o r c e s . The v a r i o u s techniques of l i m i t e q u i l i b r i u m a n a l y s i s are d i f f e r e n t i a t e d by the assumption used to overcome t h i s mechanics problem. E l a s t i c S o l u t i o n s of the F i n i t e Element Type An a l t e r n a t e approach to the l i m i t e q u i l i b r i u m methods i s to use the s t r e s s - s t r a i n r e l a t i o n s h i p s of the. t h e o r i e s of e l a s t i c i t y and model a slope as a continuous e l a s t i c medium. Commonly, such analyses are solved n u m e r i c a l l y u s i n g the f i n i t e element technique. The r e s u l t i s an understanding of the s t r e s s - s t r a i n c o n d i t i o n of the s l o p e . No f a c t o r of s a f e t y i s generated but the method i s i d e a l f o r i d e n t i f y i n g zones of s t r e s s c o n c e n t r a t i o n . An u n d e r l y i n g and l i m i t i n g assumption r e q u i r e d by the theory of e l a s t i c i t y i s that s t r a i n s are s m a l l , l e s s than one percent. To overcome t h i s d i f f i c u l t y and to approximate d i s - c o n t i n u i t i e s , Goodman et a l (1968) proposed the use of j o i n t elements w i t h p r o p e r t i e s much d i f f e r e n t than the surrounding rock. The medium i s s t i l l , however, modelled as a continuum. Another l i m i t a t i o n i s that most s t r e s s - s t r a i n analyses are i n terms of t o t a l s t r e s s e s . Although porewater pressures can be introduced as an e x t e r n a l f o r c e on any node, i t i s not c l e a r what e f f e c t the water has on the s t r e s s - s t r a i n r e l a t i o n s h i p s which must be assumed f o r every element. (P. Byrne, personal communication). 10. Nevertheless, t h i s approach i s used f o r both s o i l and rock and i s p a r t i c u l a r l y u s e f u l i n developing the s t r e s s - s t r a i n s i t u a t i o n that e x i s t s around excavations such as open p i t s and underground openings. One advantage i s that i n o b t a i n i n g the s t r e s s - s t r a i n p i c - t u r e , a c t u a l displacements can be measured and p r e d i c t e d . Examples i n the l i t e r a t u r e are numerous but f o r a s t a r t the reader i s r e f e r r e d to Desai (1972). D i s c r e t e P a r t i c l e A n a l y s i s Cundall (1971, 1974) and Cundall et a l (1975) de s c r i b e another approach to slope s t a b i l i t y u t i l i z i n g the concepts of d i s c r e t e p a r t i c l e mechanics. A computer program has been developed which models a rock slope as an assemblage of d i s c r e t e and r i g i d b l o c k s . Presumably i f the computer c a p a c i t y and data were a v a i l a b l e , s m a l l e r and smaller rock blocks could be considered u n t i l , i n f a c t , a s o i l was being modelled. The program i s s i m i l a r to l i m i t e q u i l i b r i u m analyses i n that a l l displacements occur along j o i n t s and d i s c o n t i n u i t i e s , but d i f f e r e n t i n that displacements, r o t a t i o n s , and i n t e r a c t i o n s of the blocks are c a l c u l a t e d as a f u n c t i o n of time w i t h f a i l u r e surfaces being generated as i n s t a b i l i t i e s develop. The program d i f f e r s from f i n i t e element a n a l y s i s i n that as an i n s t a b i l i t y develops the rock b l o c k s are f r e e to undergo l a r g e displacements and r o t a t i o n s . The a n a l y s i s i s not t r u l y dynamic: an e x p l i c i t time i n t e g r a - t i o n i s used by c o n s i d e r i n g a s e r i e s of s t a t i c e q u i l i b r i u m analyses. In each step the e q u i l i b r i u m of each b l o c k i s considered. Resultant 11. f o r c e s a r e used w i t h Newton's second law to c a l c u l a t e an a c c e l e r a t i o n . Thus, motion i s c a l c u l a t e d as a f u n c t i o n o f time. I n t e r a c t i o n w i t h the program i s v i a a g r a p h i c s t e r m i n a l . The geometry o f a s l o p e , complete w i t h d i s c o n t i n u i t i e s , i s drawn on the t e r m i n a l and, w i t h program e x e c u t i o n , i n s t a b i l i t i e s can be seen to d e v e l o p and f a i l u r e s o c c u r . U n f o r t u n a t e l y , C u n d a l l ' s work i s e n t i r e l y i n terms o f t o t a l s t r e s s e s . Because the e f f e c t s o f water a r e almost always c r i t i c a l to s l o p e s t a b i l i t y i n s o i l o r r o c k , i t i s apparent t h a t t h i s method has s e v e r e l i m i t a t i o n s u n t i l i t can be m o d i f i e d t o work i n terms o f e f f e c - t i v e s t r e s s . H i s t o r i c Development o f L i m i t E q u i l i b r i u m Methods Without q u e s t i o n , t h e f i r s t s i g n i f i c a n t c o n t r i b u t i o n was Coulomb's (1776) memoir p r e s e n t e d t o the Academy o f S c i e n c e i n P a r i s . Heyman (1972) g i v e s n o t o n l y a copy o f the o r i g i n a l memoir but a l s o an E n g l i s h t r a n s l a t i o n w i t h d i s c u s s i o n o f the p r i n c i p a l f e a t u r e s . I n h i s paper, Coulomb f i r s t p r e s e n t e d the law t h a t now b e a r s h i s name (2 . 1 ) . A l t h o u g h s i n c e r e f i n e d , t h i s b a s i c e m p i r i c a l law remains the backbone o f many a s p e c t s o f A p p l i e d S c i e n c e . M o d i f i e d t o a l l o w f o r e f f e c t i v e s t r e s s ( 2 . 2 ) , i t u n d e r l i e s almost a l l p r e s e n t l y used t e c h n i q u e s f o r a n a l y z i n g s l o p e s t a b i l i t y . I n c o n s i d e r i n g p o t e n t i a l f a i l u r e s b e h i n d r e t a i n i n g w a l l s , Coulomb used, f o r the " s i m p l i c i t y o f the r e s u l t s o b t a i n e d , t h e ease o f t h e i r a p p l i c a t i o n i n p r a c t i c e , and the w i s h t o be u s e f u l to and unders 12. by workmen" a p l a n a r f a i l u r e s u r f a c e . However, he d i d go on to b r i e f l y d e s c r i b e a method to h a n d l e c u r v e d s u r f a c e s . I n t h i s t e c h n i q u e he broke h i s r e g i o n i n t o v e r t i c a l s l i c e s . The c r e d i t f o r f i r s t u s i n g c u r v e d f a i l u r e s u r f a c e s and s l i c e s i s u s u a l l y g i v e n to much more r e c e n t i n v e s t i g a t o r s . Coulomb a l s o r e c o g n i z e d the c r i t i c a l r o l e o f water. He s u g g e s t e d t h a t the p r e s s u r e o f water reduced the a n g l e of i n t e r n a l f r i c t i o n . H i s r e a s o n i n g was perhaps i n c o m p l e t e ; he a t t r i b u t e d the problem to " r e p l a c i n g s o i l f o r c e s by the f r i c t i o n l e s s p r e s s u r e of a f l u i d " (Heyman, 1972^ p. 5 7 ) , but h i s a s t u t e o b s e r v a t i o n i s n e v e r t h e - l e s s worth n o t i n g . L e g g e t t (1962, p. 430) d e s c r i b e s the c o n t r i b u t i o n o f A l e x a n d r a C o l l i n , a l i t t l e known F r e n c h e n g i n e e r . C o l l i n ' s book (1846) d e s c r i b e s the f i r s t s h e a r t e s t s on c l a y . These were completed to d e t e r m i n e the m e c h a n i c a l p r o p e r t i e s o f the s o i l w hich he might then a p p l y i n the m a t h e m a t i c a l a n a l y s i s o f the s t a b i l i t y o f an u n s u p p o r t e d s l o p e . C o l l i n a l s o r e c o g n i z e d the dependence o f a l l t h e o r e t i c a l s t u d i e s of s l o p e s t a b i l i t y upon the l o c a l g e o l o g i c s t r u c t u r e and c o n d i t i o n s . In 1860, Rankine put f o r w a r d h i s t h e o r y o f e a r t h p r e s s u r e . L i k e Coulomb, he d e a l t w i t h p o t e n t i a l f a i l u r e s and f o r c e s b e h i n d r e t a i n i n g w a l l s . Both Coulomb and Rankine c o n s i d e r e d " a c t i v e and p a s s i v e p r e s s u r e s and t h r u s t s caused by a c o h e s i o n l e s s s o i l n o t s u b j e c t to seepage f o r c e s " ( T a y l o r , 1948, p. 488). Culmann (1866) d e s c r i b e d p l a n e f a i l u r e t hrough the toe o f a s l o p e . F o r c e s a c t i n g on the f a i l i n g wedge i n c l u d e d o n l y the w eight o f 13. the wedge and a resist ing force resulting from a cohesion that was constant along the fa i lure surface. During the early 1 9 0 0 ' s much work was conducted i n Sweden because of numerous slope fai lures that had occurred along the Swedish rai lroads . A major commission on slope s t a b i l i t y was established and several important contributions resulted. From observing many fa i lure surfaces that approximated circular areas, several "c i rcular arc" methods were developed, including the method of s l i c e s , cj>-circle method and the log s p i r a l method. The method of s l ices was developed by Fellenius ( 1 9 2 7 , 1936). The mass above an assumed fa i lure surface i s divided into v e r t i c a l s l i ces (Figure 2 - 2 ) . The forces acting on each s l i c e include the weight "W", the cohesion "c" and a third force "P" due to the materials angle of internal f r i c t i o n . Figure 2 - 3 shows how c and P are direct ly related to Coulomb's law. Letting c be the resis t ing force due to cohesion, the total resis t ing force is F R = c + WNtan<j> (2.3) The f r i c t i o n a l component of the resis t ing force i s W tand> and can N be represented vectorial ly by + P where P i s inclined at an. angle <{> to the normal component of the weight (w^). Because only moments are being considered, the use of P i s equivalent to the use of W t̂an<j> because W_T i t s e l f causes no moment. N 14. F i g u r e 2-3. Force diagram showing a weight W a c t i n g on a slope i n c l i n e d at an angle a 15. F o r each s l i c e , P i s tangent to a s m a l l c i r c l e w i t h c e n t r e the same as t h a t of the s l i d i n g s u r f a c e and r a d i u s rsincj) ( F i g u r e 2- 2 ) . The b a s i c e q u i l i b r i u m e q u a t i o n used i n the d e r i v a t i o n o f the f a c t o r of s a f e t y i s the summation o f moments about the c e n t r e o f r o t a t i o n . I n t e r s l i c e f o r c e s are assumed to a c t p a r a l l e l to the base o f the s l i d e and t h e r e f o r e c a n c e l out i n t h e summation. The worse case i s found by t r i a l and e r r o r a f t e r a l a r g e number o f f a i l u r e s u r f a c e s have been c o n s i d e r e d . The <|>-circle method was d e s c r i b e d by T a y l o r (1937) a l t h o u g h he a t t r i b u t e d an e a r l i e r o r i g i n t o P r o f e s s o r s Glennon G i l b o y and A r t h u r Casagrande. T a y l o r c o n s i d e r e d b o t h f a i l u r e s p a s s i n g through and below the t o e . The r e g i o n above the f a i l u r e s u r f a c e i s assumed to be a s i n g l e f r e e body ( F i g u r e 2-4). The f o r c e s a c t i n g on t h e mass a r e the w eight "W", a c o h e s i o n " c " a c t i n g p a r a l l e l to t h e c h o r d j o i n i n g the two ends o f the f a i l u r e s u r f a c e , and "P" a " r e s u l t a n t f o r c e t r a n s - m i t t e d from g r a i n to g r a i n of the s o i l a c r o s s " the f a i l u r e s u r f a c e ( T a y l o r , 1937, p. 345). "P", as b e f o r e , i s assumed to a c t a t an a n g l e <j> to t h e f a i l u r e c i r c l e , making i t t a n g e n t to a s m a l l c i r c l e ( t h e <f>-circle) w i t h c e n t r e a t the o r i g i n o f the f a i l u r e c i r c l e . Moments o f the mass about the o r i g i n a r e c o n s i d e r e d . A s p e c i a l c ase f o r <j> = 0 r e s u l t s when the m a t e r i a l i n q u e s t i o n ( f o r example, s a t u - r a t e d u n d r a i n e d c l a y ) can be c o n s i d e r e d p u r e l y c o h e s i v e . T h i s case i s t h o r o u g h l y r e v i e w e d by Skempton (1948). As w i t h F e l l e n i u s ' method of s l i c e s , a l a r g e number of assumed f a i l u r e s u r f a c e s must be c o n s i d e r e d i n o r d e r to f i n d the most c r i t i c a l c a s e . 16. rsin<)> F i g u r e 2-4. The ^ - c i r c l e method f o r t o e f a i l u r e s ( T a y l o r , 1937, p. 344) R e n d u l i c (1935) p r o p o s e d a s l i g h t v a r i a t i o n f r o m the <j>~circle method based on assuming the f a i l u r e c i r c l e t o be a l o g s p i r a l . U s i n g t h i s assumption he a v o i d e d the b a s i c assumption o f the <j>-circle method as a l l r a d i u s v e c t o r s can be shown to i n t e r s e c t the f a i l u r e c u r v e a t an a n g l e o f <fj ( F i g u r e 2-5) . A g a i n , t r i a l and e r r o r must be used to f i n d the c r i t i c a l f a i l u r e s u r f a c e . A l l o f t h e t h r e e " c i r c u l a r a r c " methods i n i t i a l l y assumed d r y , homogeneous and i s o t r o p i c m a t e r i a l s . A f t e r the i n t r o d u c t i o n o f the con- ce p t o f e f f e c t i v e s t r e s s they were extended t o ac c o u n t f o r pore w a t e r p r e s s u r e s . 17. l o g s p i r a l F i g u r e 2 - 5 . R e n d u l i c ' s l o g - s p i r a l method T e r z a g h i ' s development o f t h i s concept s t a n d s w i t h Coulomb's law i n importance t o the u n d e r s t a n d i n g o f s l o p e s t a b i l i t y . A l t h o u g h f i r s t s u g g e s t e d i n h i s work o f 1923, i t was n o t u n t i l he p u b l i s h e d h i s 1936 paper, i n E n g l i s h , t h a t the f u l l impact o f h i s e m p i r i c a l law c o u l d be f e l t . H i s e f f e c t i v e s t r e s s law i s 0 = a - u (2.4) e t where o" = e f f e c t i v e s t r e s s e a ••= t o t a l s t r e s s u = porewater p r e s s u r e Coulomb's law (2.1) c o u l d t h e n be r e s t a t e d i n terms o f e f f e c t i v e s t r e s s ( 2 . 2 ) . 18. Although commonly used i n p r a c t i c e , the. v a l i d i t y of Terzaghi's e m p i r i c a l e f f e c t i v e s t r e s s law has been the s u b j e c t of some controversy. Some of the debated ideas w i l l be presented i n a l a t e r s e c t i o n . F o l l o w i n g the i n t r o d u c t i o n of h i s e f f e c t i v e s t r e s s law, Terzaghi (1936(c)) went on to show how F e l l e n i u s ' and Rendulic's methods could be m o d i f i e d f o r water pressures i f p a r t of the f a i l i n g mass was below a s t a t i c water t a b l e . For the submerged p o r t i o n he simply used buoyant u n i t weights. He a l s o noted that " p e r c o l a t i n g water" could be represented by flow n e t s . Using the porewater pressures he obtained from h i s flow net, he was able to use F e l l e n i u s ' method f o r the case of steady seepage. T a y l o r (1937) expanded Terzaghi"s d i s c u s s i o n i n c o n s i d e r i n g four cases using the ^ - c i r c l e method: complete submergence (Figure. 2-6), sudden drawdown, steady seepage, and c a p i l l a r y s a t u r a t i o n (Figure 2-7). Figure 2--6. T a y l o r ' s (J)-circle. a n a l y s i s f o r complete submergence 1 9 . F o r complete submergence ( F i g u r e 2-6) , T a y l o r used the " e f f e c - t i v e " w e i g h t , "Wg, d e f i n e d by W S = ( Y T " Y w a t e r ) * V ° W ( 2 ' 5 ) where Y T = t o t a l u n i t w e i g h t Y = u n i t w eight o f water 'water b U s i n g Wg causes a r e d e f i n i t i o n o f P because o f t h e i r i n t e r - dependence. Two a d d i t i o n a l water f o r c e s a r e i n t r o d u c e d , E , t h e r e s u l - t a n t o f water f o r c e s a c t i n g everywhere normal to the f a i l u r e c i r c l e ; and E '2, the r e s u l t i n g " s u p p o r t i n g " f o r c e due to the water above the f a i l i n g mass. E^ w i l l n o t cause a moment. The moments o f the e n t i r e mass a r e then c o n s i d e r e d as b e f o r e . The sudden drawdown c a s e was s i m p l y c a l c u l a t e d by removing t h e f o r c e E 2 • F o r s t e a d y seepage, T a y l o r commented t h a t , a l t h o u g h f l o w n e t s c o u l d be s k e t c h e d and used t o o b t a i n s o l u t i o n s , the work i n v o l v e d was l o n g and t e d i o u s . The t r u e s o l u t i o n f o r s t e a d y seepage would l i e some- where i n between complete submergence and sudden drawdown. He recom- mended t h a t the sudden drawdown c a l c u l a t i o n be used f o r the s t e a d y seepage c a s e . A l t h o u g h h i s approach was c o n s e r v a t i v e , he doubted t h a t " t h e a c c u r a c y which can be a t t a c h e d t o s t a b i l i t y computations was g r e a t enough to warrant the drawing o f a f l o w n e t and the i n v o l v e d p r o c e d u r e t h a t must f o l l o w " ( T a y l o r , 1937, p. 373). T a y l o r c o n s i d e r e d one o t h e r c a s e , c a p i l l a r y s a t u r a t i o n ( F i g u r e 2-7). 20. F i g u r e 2-7. T a y l o r ' s ^ - c i r c l e a n a l y s i s f o r c a p i l l a r y s a t u r a t i o n F o r t h i s case he used a t o t a l w e i g h t which c o n s i s t e d o f the weight o f the s o i l p l u s the weight o f water r e t a i n e d by c a p i l l a r y a c t i o n . T h i s c o n d i t i o n , he e x p l a i n e d , c o u l d be a t t a i n e d by s h u t t i n g o f f the s u p p l y o f seepage water and thus e l i m i n a t i n g seepage f o r c e s . E s s e n t i a l l y , t h i s would r e q u i r e d r a i n i n g t o lower t h e water t a b l e below t h e f a i l i n g s l o p e . The v a r i o u s c i r c u l a r a r c methods d i s c u s s e d so f a r a r e t h e o r e - t i c a l l y i n c o r r e c t , s a t i s f y i n g o n l y one o f the three, a s p e c t s of s t a t i c e q u i l i b r i u m , moment e q u i l i b r i u m ; l e a v i n g u n s a t i s f i e d the q u e s t i o n o f h o r i z o n t a l and v e r t i c a l f o r c e e q u i l i b r i u m . I n 1954, Janbu (1954, 1957, 1973) f i r s t p r e s e n t e d h i s G e n e r a l - i z e d P r o c e d u r e o f S l i c e s . H i s work i n c l u d e d the f i r s t attempt to c o n - s i d e r a l l the t h r e e a s p e c t s o f e q u i l i b r i u m as w e l l as p r o v i d i n g a method t h a t c o u l d h a n d l e f a i l u r e s u r f a c e s o f any shape. The f o r c e s he con- s i d e r e d a r e shown i n F i g u r e 2-8, where dP = e x t e r n a l l o a d dW = s l i c e weight 21. E and E + dE T and T + dT dS dN h o r i z o n t a l i n t e r s l i c e f o r c e s v e r t i c a l i n t e r s l i c e f o r c e s r e s i s t i n g f o r c e s along the base normal f o r c e along the base. E + dEl T + dT dP dW Fi g u r e 2-8. Forces used by Janbu i n h i s Generalized Procedure of S l i c e s For each s l i c e he considered v e r t i c a l , h o r i z o n t a l , and moment e q u i l i b r i u m . I n t e g r a t i n g the v e r t i c a l and h o r i z o n t a l f o r c e s f o r the e n t i r e body, he was able to consider v e r t i c a l and h o r i z o n t a l e q u i l i b r i u m f o r the mass as a whole. For s t a b i l i t y c r i t e r i a he used the h o r i z o n t a l c o n d i t i o n of e q u i l i b r i u m . The r e s u l t i n g expression f o r s a f e t y f a c t o r was indeterminate without assuming a l i n e of t h r u s t f o r the i n t e r s l i c e f o r c e s . These were then obtained by s u c c e s s i v e approximations and r e - introduced i n t o h i s s a f e t y f a c t o r expression. Terzaghi and Peck (1967, p. 253) suggest that w i t h Janbu's method, moment e q u i l i b r i u m f o r i n d i - v i d u a l s l i c e s i s not s a t i s f i e d by the f o r c e s d e r i v e d from the s o l u t i o n . However, more recent summaries ( f o r example, Wright et a l , 1973; Duncan. 1975) i n c l u d e Janbu's method w i t h those that s a t i s f y a l l c o n d i t i o n s of e q u i l i b r i u m . Bishop (1955) put F e l l e n i u s ' c i r c u l a r arc method of s l i c e s on a more c o r r e c t foundation from a s t a t i s t i c s p o i n t of view. Figure 2-9 shows the f o r c e s he considered, where w = weight of s l i c e X ,X ,, = v e r t i c a l shear fo r c e s n n+1 E ,E ,, = r e s u l t a n t h o r i z o n t a l i n t e r s l i c e f o r c e s n n+1 S = shear f o r c e a c t i n g on the s l i c e base P = normal f o r c e a c t i n g on the s l i c e base By c o n s i d e r i n g the moments about the o r i g i n "0" he developed an expression f o r the f a c t o r of s a f e t y which i n c l u d e d the i n t e r s l i c e f o r c e s . The terms i n v o l v i n g these i n t e r s l i c e f o r c e s had been p r e v i o u s l y shown by Krey (1926) and Terzaghi (1929) to cause only minor changes i f neglected. The r e s u l t i n g simple expression meant an easy c a l c u l a t i o n of the s a f e t y f a c t o r but was i n general c o n s e r v a t i v e and could lead to unenconomical design. F o r . t h i s reason, Bishop went on to d e s c r i b e a 23. method that by successive approximations ensured that the v e r t i c a l shear forces should sum to zero. L i k e Janbu, he had to assume a reasonable l i n e of t h r u s t . He a l s o gave an expression that i f s a t i s f i e d would ensure that h o r i z o n t a l i n t e r s l i c e f o r c e s would be i n e q u i l i b r i u m as w e l l . F i g u r e 2-9. Forces used i n Bishop's method However, he added t h a t , although the gain i n accuracy obtained by ensuring v e r t i c a l shear f o r c e s summed to zero was s i g n i f i c a n t , the same could not be s a i d of ensuring h o r i z o n t a l i n t e r s l i c e f o r c e e q u i l i b r i u m . Bishop's s i m p l i f i e d method which i n v o l v e s s a t i s f y i n g o v e r a l l moment e q u i l i b r i u m and v e r t i c a l shear f o r c e e q u i l i b r i u m but not h o r i z o n t a l i n t e r s l i c e f o r c e e q u i l i b r i u m i s commonly used and has been shown 24. (Duncan, 1975; M o r g e n s t e r n and P r i c e , 1965; Wright e t a l , 1973) to g i v e a c c u r a t e r e s u l t s f o r most c a s e s . Lowe and K a r a f i a t h (1960) p r o p o s e d a g r a p h i c a l , m o d i f i e d method o f s l i c e s which s a t i s f i e d h o r i z o n t a l and v e r t i c a l f o r c e e q u i l i - b r ium but not moment e q u i l i b r i u m . A f t e r c a l c u l a t i n g the magnitude and d i r e c t i o n o f weight and water f o r c e s , they assumed a " r e a s o n a b l e " d i r e c - t i o n f o r l a t e r a l e a r t h f o r c e s . F o r a t r i a l f a c t o r o f s a f e t y , "a s e r i e s o f f o r c e p o l y g o n s were c o n s t r u c t e d , one f o r each s l i c e . By p l o t t i n g t h e f o r c e p o l y gons c o n t i g u o u s w i t h each o t h e r and i n sequence, a c l o s u r e p o l y g o n i s o b t a i n e d " (Lowe and K a r a f i a t h , 1960, p. 542). I f c l o s u r e i s not o b t a i n e d , a new f a c t o r o f s a f e t y i s assumed and the e x e r c i s e i s r e p e a t e d u n t i l the a p p r o p r i a t e f a c t o r o f s a f e t y i s o b t a i n e d . The i n t r o d u c t i o n o f the computer i n t o common usage i n the 1950's r e v o l u t i o n i z e d p r o c e d u r e s f o r a n a l y z i n g s l o p e s t a b i l i t y . C l e a r l y many time-consuming and r e p e t i t i v e o p e r a t i o n s became si m p l e and q u i c k . L i t t l e and P r i c e (1958), Horn (1960), and Whitman and B a i l e y (1967) d i s c u s s the a p p l i c a t i o n o f the computer t o s l o p e s t a b i l i t y . U n t i l the i n t r o d u c t i o n o f the computer, t e c h n i q u e s were d e v e l o p e d t o g i v e f a s t , r e a s o n a b l y a c c u r a t e answers even though Laws o f Mechanics were n o t always s a t i s f i e d . Because o f the i n d e t e r m i n a t e n a t u r e o f L i m i t E q u i l i b r i u m methods, t e c h n i q u e s which s a t i s f i e d the Laws o f Mechanics commonly r e q u i r e d s u c c e s s i v e a p p r o x i m a t i o n s to r e a c h a s o l u t i o n . Without the computer such s o l u t i o n s were unwieldy and t i m e - consuming. However, w i t h the computer, t h i s d i f f i c u l t y was removed and i t e r a t i v e methods which ensured a l l the Laws of Mechanics were s a t i s - f i e d , r e p l a c e d the e a r l i e r - a p p r o x i m a t e methods. Such a method was proposed by Morgenstern and P r i c e (1965). The f o r c e s they considered are shown i n Figure 2-10 where x, x + dx = v e r t i c a l i n t e r s l i c e shear f o r c e s E', E' + dE' = l a t e r a l e f f e c t i v e s i d e t h r u s t s P„, P„ + dP„ = r e s u l t a n t s i d e water pressure W w w dP^ = water pressure a c t i n g on the base dN 1 = e f f e c t i v e normal pressure dS = shear f o r c e a c t i n g on the base a = i n c l i n a t i o n of the base dW = weight of the s l i c e F igure 2-10. Forces used i n the Morgenstern-Price Method By t a k i n g moments about the centre of the base of the s l i c e , summing forces perpendicular and p a r a l l e l to the f a i l u r e s u rface and subse- quently s i m p l i f y i n g , two governing d i f f e r e n t i a l equations i n three 26. unknowns remain, a s t a t i c a l l y i n d e t e r m i n a t e problem. The r e m a i n i n g un- knowns a r e the l a t e r a l s i d e t h r u s t E ', i t s p o s i t i o n o f a c t i o n , and the v e r t i c a l i n t e r s l i c e shear f o r c e x. A number of assumptions a r e p o s s i b l e t o reduce the p roblem t o a s o l v a b l e case i n c l u d i n g assuming the p o s i t i o n o f the l i n e o f t h r u s t o r assuming a r e l a t i o n s h i p between E and x. M orgenstern and P r i c e use a form o f t h i s l a t t e r assumption by r e l a t i n g t h e t o t a l h o r i z o n t a l s t r e s s E (= E' + P ) and x by w x = X f ( z ) E (2.6) where X i s a c o n s t a n t which depends on r e a s o n a b l e boundary c o n d i t i o n s and f ( z ) i s an a r b i t r a r y f u n c t i o n o f z, the v e r t i c a l c o o r d i n a t e . f ( z ) can be e s t i m a t e d from a v a i l a b l e e l a s t i c i t y s o l u t i o n s o r ( M o r g e n s t e r n and P r i c e , 1965, pp. 87 and 8 8 ) : S p e c i f i e d on t h e b a s i s o f the i n t u i t i v e assumption t h a t f o r most c r o s s - s e c t i o n s the h i g h e r the r a t e o f c u r v a t u r e o f the s l i p s u r f a c e , the g r e a t e r the r a t i o between the s h e a r and h o r i z o n t a l f o r c e s a t the s l i p i n t e r f a c e ... U l t i m a t e l y r e l i a b l e f i e l d measurements of i n t e r n a l s t r e s s w i l l p r o v i d e the b e s t g u i d e . S t a r t i n g w i t h guessed v a l u e s o f A and f ( z ) they i n t e g r a t e a c r o s s a l l the s l i c e s t o o b t a i n v a l u e s of E , the t o t a l h o r i z o n t a l s t r e s s and M , the moment; which i n g e n e r a l w i l l not b o t h be z e r o . By m o d i f y i n g A and f ( z ) and s y s t e m a t i c a l l y i t e r a t i n g , v a l u e s a r e f i n a l l y found f o r which E n and M a r e z e r o . M o r g e n s t e r n and P r i c e (1965) i n t r o d u c e p o r e w a t e r p r e s s u r e d a t a i n the form o f a c o e f f i c i e n t " ^ u " as d e f i n e d by B i s h o p and M o r g e n s t e r n (1960): 27. where u = pore pressure Y = s p e c i f i c weight of the s l i d e m a t e r i a l h = depth below s l i d e s u r f a c e Although can be v a r i e d from s l i c e to s l i c e , i t would seem more reason- able to develop a flow net s o l u t i o n and in t r o d u c e d i r e c t l y the a p p r o p r i a t e values of the pore pressure. (This approach i s used by Fredlund (1974).) The use of r w i l l be discussed more thoroughly i n a l a t e r s e c t i o n . Since Morgenstern and P r i c e p u b l i s h e d t h e i r paper, s e v e r a l others have described s o l u t i o n s which depend on d i f f e r e n t assumptions used to e l i m i n a t e the indeterminacy of the problem. Spencer (1967) assumed p a r a l l e l i n t e r s l i c e f o r c e s ; i . e . , a constant i n t e r s l i c e f o r c e i n c l i n a t i o n . B e l l (1968) made an assumption about the v a r i a t i o n of the normal s t r e s s a c t i n g on the p o t e n t i a l s l i p s u r f a c e . I n t e r s l i c e f o r c e s , however, d i d not enter i n t o h i s equation of e q u i l i b r i u m as he considered them i n t e r n a l to the f r e e body being examined. Janbu (1973) r e s t a t e d h i s method which assumes a l i n e of t h r u s t of i n t e r s l i c e f o r c e s . The various methods are summarized by Duncan (1975). Thus f a r i n t h i s d i s c u s s i o n , no comment has been given on the r e l a t i o n s h i p between analyses f o r rock slope s t a b i l i t y and s o i l s lope s t a b i l i t y . The f i r s t concepts and methods f o r a n a l y z i n g slope s t a b i l i t y o r i g i n a t e d w i t h i n what i s now considered the science of S o i l Mechanics. The c l o s e l y r e l a t e d science of Rock Mechanics developed some time l a t e r but i s based on many of the same p r i n c i p l e s . L i m i t e q u i l i b r i u m tech- niques are used f o r the a n a l y s i s of both s o i l and rock slope s t a b i l i t y . 28. I f a rock slope i s so h i g h l y f r a c t u r e d or weathered that i t can be considered to act l i k e a s o i l , then the techniques of s o i l mechanics are a p p l i c a b l e . More commonly, the s t r e n g t h of a rock slope i s c o n t r o l l e d by d i s c o n t i n u i t i e s , f a u l t s , and f r a c t u r e s , and i s b e t t e r modelled as a system of rock blo c k s and wedges. In some cases, i n d i - v i d u a l blocks and wedges must be considered. Techniques f o r h a n d l i n g these cases are d e s c r i b e d by Goodman and Taylor (1967), John (1968) , Londe et a l (1969, 1970), and Hendron et a l (1971). Obviously, s c a l e and the nature and spacing of d i s c o n t i n u i t i e s are c r i t i c a l i n d e c i d i n g on the most appropriate model. In summarizing the development of l i m i t e q u i l i b r i u m tech- niques, three d i s t i n c t aspects can be i s o l a t e d : i d e n t i f y i n g the d i f - f erent s t r e s s e s a c t i n g ; understanding the mechanics which r e l a t e the s t r e s s e s ; and determining the n a t u r a l p r o p e r t i e s that govern the mate- r i a l behaviour when the s t r e s s e s are a c t i n g . The r e c o g n i t i o n of d i f f e r e n t s t r e s s e s , one due to overburden load and the other due to f l u i d p r e s s u r e s , l e d Terzaghi to h i s e f f e c t i v e s t r e s s law. His r e c o g n i t i o n of t h i s concept was a major step forward. A f u r t h e r s t r e s s problem i s the r o l e of r e s i d u a l t e c t o n i c s t r e s s e s due to past and present g e o l o g i c processes. These can be grouped i n t o l o c a l or r e g i o n a l r e s i d u a l s t r e s s e s . L o c a l s t r e s s e s are o f t e n due to s t r e s s r e l i e f and rebound a f t e r removal of m a t e r i a l s . This removal can be a r e s u l t of n a t u r a l processes such as r i v e r s and g l a c i e r s or man-made excavations. Regional s t r e s s e s are r e l a t e d to l a r g e g e o l o g i c s t r u c t u r e s r e s u l t i n g from r e g i o n a l t e c t o n i c a c t i v i t y . The exact r o l e of r e s i d u a l t e c t o n i c s t r e s s e s i s not•completely understood and c e r t a i n l y at the 29. present time i s not in c l u d e d i n l i m i t e q u i l i b r i u m slope s t a b i l i t y c a l - c u l a t i o n s . Although the magnitude of these s t r e s s e s i s probably s m a l l , i n some cases i t may be s i g n i f i c a n t . A second aspect of the development of l i m i t e q u i l i b r i u m t e c h - niques was a r r i v i n g at a f u l l understanding of the mechanics which r e l a t e the s t r e s s e s being considered. Given the problem i n terms of e f f e c t i v e s t r e s s e s and a computer to e l i m i n a t e r e p e t i t i v e and time- consuming o p e r a t i o n s , the task was then to p r o p e r l y r e l a t e a l l the forces a c t i n g . This was achieved by a p p l i c a t i o n of a l l the laws of s t a t i c e q u i l i b r i u m . The problem, however, i s s t i l l indeterminate and the v a r i o u s assumptions used f o r s o l u t i o n s , r e s u l t i n the d i f f e r e n t methods. The f i n a l aspect i s a l a r g e subject i n i t s e l f and a thorough treatment i s w e l l beyond the scope of t h i s review. However, f o r com- pleteness I mention i t here. The usefulness of any method depends on the understanding and the accurate measurement of the n a t u r a l p r o p e r t i e s used i n the v a r i o u s equations. Such p r o p e r t i e s i n c l u d e cohesion, i n t e r n a l f r i c t i o n angle, s p e c i f i c weight, p e r m e a b i l i t y , p o r o s i t y , and c o m p r e s s i b i l i t y . To o b t a i n a meaningful value or d i s t r i b u t i o n of values which d e s c r i b e each of these p r o p e r t i e s i s a d i f f i c u l t task. Laboratory measurements, f i e l d measurements, and back c a l c u l a t i o n s can a l l be used but i t remains a d i f f i c u l t problem. I t i s t h i s aspect, more than the others, which l i m i t s the accuracy and usefulness of the techniques themselves. 30. The. Concept of E f f e c t i v e Stress The concept of e f f e c t i v e s t r e s s may not be as simple as sug- gested by Terzaghi's e m p i r i c a l law (2-3). In f a c t , Terzaghi (1923) o r i g i n a l l y proposed the. lav/ i n a s l i g h t l y d i f f e r e n t form: a = a - au (2.8) e t where a i s a parameter, the exact value of which i s c o n t r o v e r s i a l . Terzaghi (1923) suggested that a should equal the p o r o s i t y but found experimentally that a - 1. These conclusions were repeated i n h i s 1936(a) paper where he c a l l e d a a r e d u c t i o n f a c t o r . Hubbert and Rubey (1959, 1960) attempted to show t h e o r e t i c a l l y that tx = 1 but the v a l i d i t y of t h e i r proof has been questioned by Laubscher (1960). Skempton (1961) proposed that more c o r r e c t expressions f o r e f f e c t i v e s t r e s s i n f u l l y s aturated m a t e r i a l are f o r shear s t r e n g t h a b e f o r volume change a £ where a = the area of contact between p a r t i c l e s per u n i t gross area of the m a t e r i a l U> = i n t r i n s i c angle of i n t e r n a l f r i c t i o n <(>' = e f f e c t i v e angle of i n t e r n a l f r i c t i o n c g = c o m p r e s s i b i l i t y of the s o l i d p a r t i c l e s c = c o m p r e s s i b i l i t y of the o v e r a l l m a t e r i a l and the other symbols are as defined p r e v i o u s l y . For f u l l y s a t u r a t e d s o i l s , at pressures normally encountered i n engineering p r a c t i c e , , both "a" and " c o / c " are very s m a l l and = a - (1 (1 atanif\ c — ) u (2.9) (2.10) 3 1 . (2.9) and (2.10) w i l l degenerate to Terzaghi's e m p i r i a l r e l a t i o n s h i p (2.3). However f o r saturated rock and concrete r e l a t i o n s h i p s (2.9) and (2.10) should be used. Nur and Byerlee. (1971) give both experimental evidence and a, t h e o r e t i c a l proof of (2.10). However, they note that "when the compres- s i b i l i t y of the aggregate i s s u f f i c i e n t l y g reater than that of the g r a i n s , " Terzaghi's e m p i r i c a l law i s an e x c e l l e n t approximation regardless of p o r o s i t y . For mechanically i s o t r o p i c , e l a s t i c a l l y l i n e a r aggregates, the law (2.10) hold s . They f u r t h e r suggest that one can determine " a c c u r a t e l y the. s t r a i n i n a porous s o l i d w i t h pore pressure from the measured e l a s t i c m o d u l i i of the m a t e r i a l without pore p r e s - sure." In p r a c t i c e , Terzaghi's e m p i r i c a l law i s commonly used. I t would appear from the above d i s c u s s i o n that under c e r t a i n circumstances Terzaghi's law gives an adequate approximation but i n other cases more accurate r e l a t i o n s h i p s should be used. The problem of e f f e c t i v e s t r e s s i n unsaturated m a t e r i a l s i s 1 discussed by Jennings (1960), B l i g h t (1967), and Skempton (1961), who p o i n t out that above the s a t u r a t e d zone Terzaghi's e m p i r i c a l law does not hold. They proposed a l t e r n a t e forms of the e f f e c t i v e s t r e s s law and the reader i s r e f e r r e d to those papers f o r d e t a i l s . Groundwater and L i m i t E q u i l i b r i u m Methods Fredlund (1974) summarized the d i f f e r e n t procedures f o r handling groundwater i n l i m i t e q u i l i b r i u m methods. He l i s t e d the f o l - lowing f i v e a l t e r n a t i v e s : 32. I. total stress analysis, no porewater pressure 2o use of the coefficient r u 3. pore pressures are expressed as a ratio of overburden •pressure A . pore pressures are expressed as x and y coordinates -of a peizometric line J>. pore pressures are evaluated by interpolation between .grid of known pore pressures ^Except under very special circumstances (for example clays with ^strength due only to cohesion), a total stress analysis i n which strength ;sLs not considered to be a function of pore pressures, i s not appropriate .-asrfor ?slope s t a b i l i t y analysis. In fact, the presence of water i s commonly iMiie. cause of s t a b i l i t y problems and an effective stress analysis i s best. .;As.previously mentioned, Bishop and Morgenstern (I960) intro- ..'u^cluced-the porewater pressure coefficient r^ defined by (2.7) or •*u r u y h (2.11) ?Xhe -pore pressure can be seen by (2.11) to be expressed as a simple f u n c t i o n of the depth h . Although r u can be made to vary from s l i c e 'jgto s l i c e , i t results i n an approximation that can be markedly different Mrom the actual case. In re a l i t y , porewater pressures are rarely a simple SLinear function of depth. The exception i s the case of a saturated static **water table where r u would simply indicate the ratio of the specific ^weight of water to that of the overburden. Such a condition i s not com- ^anon i n nature. Because techniques are now available to accurately model ; f i e l d conditions with relative ease, i t no longer makes sense to use an ^^approximation l i k e r u and i t s use should be discouraged. 33. Expressing pore pressures as a r a t i o of overburden pressure was suggested by H i l f (1948) as a means of e s t i m a t i n g c o n s t r u c t i o n pore pressures i n r o l l e d e a r t h dams. As w i t h the use of r u , means are now a v a i l a b l e to a c c u r a t e l y e s t a b l i s h c o n d i t i o n s by numerical modelling and the use of t h i s approximation i s a l s o no longer j u s t i f i e d . In a s i t u a t i o n where f i e l d measurements have e s t a b l i s h e d the p i e z o m e t r i c s u r f a c e , expressing pore pressures as x and y coordinates o a p i e z o m e t r i c l i n e may be the best way to handle groundwater. E v a l u a t i n g pore pressures by i n t e r p o l a t i o n between a g r i d of known pore pressures which have been obtained by f i e l d measurement and/or mathematical modelling i s the best of the l i s t e d a l t e r n a t i v e s . F i n i t e d i f f e r e n c e and f i n i t e element techniques are now commonly a v a i l - able to model complex groundwater flow regimes. Provided the boundary co n d i t i o n s have been c o r r e c t l y i n t e r p r e t e d and reasonable estimates hav been obtained of n a t u r a l p r o p e r t i e s , these methods give accurate s o l u - t i o n s of the groundwater flow problem. 34. CHAPTER THREE: GROUNDWATER FLOW, GHERKIN'S FETTIOD AND THE FINITE ELFENT fOIOD I n t r o d u c t i o n F l u i d f l o w through porous media can be t r e a t e d m a t h e m a t i c a l l y as a boundary v a l u e problem. For r e g i o n s t h a t a r e g e o m e t r i c a l l y s i m p l e and t h a t c o n s i s t o f homogeneous i s o t r o p i c m a t e r i a l s , one can o b t a i n an exac t a n a l y t i c , s o l u t i o n . F o r more c o m p l i c a t e d r e g i o n s and m a t e r i a l s , e x a c t s o l u t i o n s a r e n o t p o s s i b l e and n u m e r i c a l a p p r o x i m a t i o n s a r e n e c e s s a r y . G a l e r k i n ' s method i s one of a number o f p r o c e d u r e s a v a i l a b l e f o r f i n d i n g an approximate n u m e r i c a l s o l u t i o n t o a d i f f e r e n t i a l e q u a t i o n . Others i n c l u d e t h e method o f K a n t o r o v i c h , R a l e i g h - R i t z , and E u l e r ' s F i n i t e D i f f e r e n c e t e c h n i q u e ( C r a n d a l l , 1956; F o r r a y , 1968; K a n t o r o v i c h and K r y l o v , 1964). A l l of th e s e methods u t i l i z e an a p p r o x i m a t i n g f u n c t i o n f o r the unknown q u a n t i t y and i n v o l v e m i n i m i z i n g the r e s u l t i n g e r r o r . The v a r i o u s methods a r e d i f f e r e n t i a t e d by t h e i r degree o f g e n e r a l i z a t i o n and te c h n i q u e o f e r r o r m i n i m i z a t i o n . In some c a s e s , one method can be shown t o reduce to a n o t h e r . F o r example, C r a n d a l l (1956, p 233), K a n t o r o v i c h and K r y l o v (1964, pp 262-264) and F o r r a y (1968, p 193) d i s c u s s the e q u i v a l e n c e under c e r t a i n c o n d i t i o n , o f the R a l e i g h - R i t z and G a l e r k i n methods. The f o l l o w i n g d i s c u s s i o n i s l i m i t e d t o s o l v i n g the e q u a t i o n of f l o w f o r two d i m e n s i o n a l , s a t u r a t e d , s t e a d y s t a t e f l o w t h r o u g h a n i s o t r o p i c 35. and heterogeneous porous media. G a l e r k i n ' s approach and the f i n i t e element technique are used to develop a set of s o l v a b l e l i n e a r equations. The works of Pinder and F r i n d (1972) and of Remson, Hornberger and Molz (1971) were p a r t i c u l a r l y u s e f u l i n s y n t h e s i z i n g concepts. Theory Consider a region R i n which we wish to compute the h y d r a u l i c head, cj>(x,y), at c e r t a i n p o i n t s or nodes. Let these nodes form the v e r t i c e s of small but f i n i t e t r i a n g u l a r elements (Figure 3-1). Figure 3-1 . D e s c r i t i z a t i o n of r e g i o n i n t o f i n i t e t r i a n g u l a r elements. Other more, complex (higher order) elemental shapes are p o s s i b l e and are d e s i r a b l e f o r more accurate s o l u t i o n s to g e o m e t r i c a l l y complex reg i o n s . For example, Zi e n k i e w i c z (1971) and Pinder and F r i n d (1972) d i s c u s s the curved i s o p a r a m e t r i c q u a d r i l a t e r a l . However, f o r s i m p l i c i t y t h i s d i s c u s s i o n 36. w i l l be l i m i t e d to that of the t r i a n g l e . ' The s o l u t i o n a)(x,y) i n R can be viewed as a su r f a c e w i t h j • x ± ± ^ J, . th . t h . . th . -coordinates §. , <j>. , <f> at the i , j , and k nodes r e s p e c t i v e l y i 3 K The simplest approximation of t h i s s o l u t i o n s urface i s a l i n e a r plane which f o r each element i s of the form: <j) = Ax + By + C ( 3.1) We want to sol v e f o r the nodal values of cf>(x,y) t h e r e f o r e we so l v e f o r A, B and C knowing the plane used i n approximating the element i , j , k must pass through a l l three p o i n t s , thus: d>. = Ax. + By. + C l i l <j>. = Ax. + By. + C ( 3.2) 3 3 3 + k = A x k + B y k + C Equations ( 3.2) can be used to so l v e f o r A, B and C . The r e s u l t i n g expressions f o r A, B and C can then be s u b s t i t u t e d i n t o equation ( 3.1) so (J)', the approximation w i t h i n the t r i a n g u l a r element, can be w r i t t e n i n terms of the nodel values of <j> at the v e r t i c e s and three c o e f f i c i e n t s which are f u n c t i o n s of the node p o s i t i o n s : <f>' = N.caS. + N.cJ). + N (j), ( 3 . 3 ) i i 2 3 k k where N i • [ ( x / k - + ( y j " y k ) x 4 ( x k - X j ) y ] / 2 D N j = C ( x k y i " x i y k ) + ( y k " y i ) x + ( x i " \ ) y ] / 2 T ) \ = t ( x i y j " x j y k } 4 ( y ± " y j ) x + ( x j " x i ) y ] / 2 D and 2D = 2 x area of the elemental t r i a n g l e i , j , k = determinant 1 x. y ± 1 x. y J J I1 \ y k Equation (3.3) can be r e - w r i t t e n as n * = +* = I N .c j ) , i = l (3.4) (3.5) where i l a b e l s the nodes c o n t r i b u t i n g to that element, n i s the t o t a l number of nodes c o n t r i b u t i n g , <f>.. are the heads at the nodes and N. l l are defined by equation (3.4) and are c a l l e d "shape" or " b a s i s " f u n c t i o n s . For more complicated elements, the b a s i s f u n c t i o n s are l i n e a r , q u a d r a t i c or cubic polynomials, depending on the element shape (Pinder and F r i n d , 1972, p 110). Equation (3.5) expresses the l i n e a r approximation to the r e a l cj> v a l u e s . The b a s i s f u n c t i o n s , N^, are chosen so t h a t the boundary co n d i t i o n s imposed on the governing p a r t i a l d i f f e r e n t i a l equation, (the equation of f l o w ) , are s a t i s f i e d . Where Darcy's Law i s assumed to be c o r r e c t , the governing equation f o r two-dimensional s t e a d y - s t a t e flow through a n i s o t r o p i c porous media i s : 38. - ^ (K v f j ) + / (K . l i ) = 0 dx xx 3x dy yy 8y (3.6) where L i s the d i f f e r e n t i a l operator and K and K are the xx yy h y d r a u l i c c o n d u c t i v i t i e s i n the x and y d i r e c t i o n s . The x and y axes are chosen to correspond l o c a l l y ( i . e , f o r each element) w i t h the p r i n c i p l e d i r e c t i o n s of the h y d r a u l i c c o n d u c t i v i t y e l l i p s e . Equation (3.5) w i l l be the exact s o l u t i o n to (3.6) i f L(<J>') = 0 (3.7) This c o n d i t i o n i s e q u i v a l e n t to the requirement that L(aS') be orthogonal to a l l the shape f u n c t i o n s (Kantorovich and K r y l o v , 1964 p. 262) o r , L(t}) ' ) N dx dy = 0 R and s u b s t i t u t i n g i n equation (3.5) : (3.8) n L( I N.cJ>.) N. dx dy = 0 R  i = l 1 1 J j = 1 . . . n (3.9) Equation (3.9) r e s u l t s i n n equations i n n unknowns. Expanding equation (3.9) : 3x xx ^- R — + -—-K —-) ( 7 N. (j). ) ] N. dx dy = 0 (3.10) ^ Now, f o r a given t r i a n g u l a r element, three equations w i l l be generated, one f o r each node of that t r i a n g l e . Assuming K and K xx yy are constant w i t h i n the element i n q u e s t i o n , equation (3.10) can be r e - w r i t t e n 39. R i = l K xx 3?'N. + K 3 x 2 y y 3 N. 3y 2 J N dx dy = 0 (3.11) This equation i n c l u d e s second d e r i v a t i v e s which are d i f f i c u l t to handle nu m e r i c a l l y . To e l i m i n a t e then, Green's Theorem i s a p p l i e d i n the f o l l o w i n g form (Weinstock, 1952, p 13): R * + * r | dx dy 3x y 9x 9x 9y 9y i 11 j * 9 ^ d s (3.12) 8 d> where c represents the boundary of R and — i s the outward normal 9n d e r i v a t i v e of the f u n c t i o n tj>(x,y): 96 3d) „ . 3<J) .. 3n 9x x 9y y £ and £ are d i r e c t i o n c o s i n e s , x y Comparing (3.11) and (3.12) we can w r i t e <Jj = N. i and r e w r i t e Green's Theorem as 3 2N. N. 3x — + N 2 + N j 9 2N. 9y 2J dx dy = - 9N. 9N. 9N. R + 3N. J. [9x 3x 9y 9y J dx dy 3N. N. - r - i ds J 3n (3.13) W ± 8N ± 9N ± where - r — = ~ — £ + - r — & 3n 3x x 3y j 40. Combining t h i s r e s u l t w i t h e q u a t i o n (3.11) we can w r i t e the g o v e r n i n g d i f f e r e n t i a l e q u a t i o n a s : R i = l 3N. 9N. K ^ ' T - 1 + K 3N. 3N. i . 1 [ xx 3x 3x yy 3y 3y J dx dy + N. n 3N. (3.14) 3N. 3N. 3N. where I + K 3n "xx 3x ~x ' "yy 3y ~y E q u a t i o n (3.14) r e p r e s e n t s the f l o w o v e r a r e g i o n which i n our case i s a t r i a n g u l a r element. At any g i v e n node t h e r e w i l l be c o n t r i b u t i o n s from the s u r r o u n d i n g e l e m e n t s . We a r e i n t e r e s t e d i n t h e v a l u e o f the head a t each n o d a l p o i n t . To w r i t e the e q u a t i o n f o r a g i v e n node we must sum the c o n t r i b u t i o n s from each of t h e a d j a c e n t e l e m e n t s . I f t h e r e a d j a c e n t elements, we can w r i t e the e q u a t i o n f o r a g i v e n node a s : E I e=l 1=1 3N? 3N e 3N e 3N? e — i . . 1 yy 9y 3y J dx dy n 3N. J i ' l 8 n 1 ds = 0 (3.15) j = 1 . . . n where i l a b e l s t h e node i n q u e s t i o n and j i d e n t i f i e s the nodes o f the c o r n e r s o f the c o n t r i b u t i n g e l e m e n t s . The l a s t term i n e q u a t i o n (3.15) i n c o r p o r a t e s the Neuman boundary c o n d i t i o n ( c o n s t a n t f l u x ) : K l i = 3n (3.16) where q i s the f l u x o f water i n t o the element p e r u n i t l e n g t h o f boundary. I t i s formed o n l y when q i s non ze r o i n which case i t t a k e s the form ( P i n d e r and F r i n d , 1972, p 112), 41. N. q ds x (3.17) According to Z i e n k i e w i c z and Cheung (1968, p 154) i t reduces to J3_A 2 (3.18) where I i s the l e n g t h of the element on the boundary. At nodes where a D i r i c h l e t boundary c o n d i t i o n (constant head) i s encountered, equation (3.15) i s not generated as <j> i s a known quan t i t y at such nodes. Combining (3.15) and (3.18) we can r e w r i t e the equation f o r a given node as: E I e = l n K 9N? 9N e 9N? 9N 6 + K [ xx 9x 9x yy 9y 9y.J dx dy q I 2 3 = 1 or i n matrix form: (3.19) [P]{<|>} + {F} = 0 (3.20) where i»3 E I e=l I K 9N? 9N. 3N? l 9N. xx 9x 9x yy 9y 3y dx dy (3.21) and 42. [P] and {F} can be evaluated, and u s i n g standard techniques of m a t r i x algebra vze can s o l v e f o r {<)>} . The m a t r i x of c o e f f i c i e n t s [ P ] , which i s often c a l l e d the s t i f f n e s s m a t r i x by analogy to s t r u c t u r a l mechanics, w i l l be mostly zeros. This r e s u l t s from c o n t r i b u t i o n s to a given nodal equation being only from those elements surrounding the node f o r which the equation i s being w r i t t e n . [P] can be described as sparse and banded. The programmed s o l u t i o n , l i s t e d and documented i n Appendix I , sweeps through the r e g i o n element by element. For each element, three q u a n t i t i e s are c a l c u l a t e d corresponding to the c o n t r i b u t i o n s of that element to each of the equations of the three v e r t i c e s . During program execution the equation f o r each node i s generated as the computer sweeps a l l the c o n t r i b u t i n g elements surrounding that node. A b r i e f o u t l i n e of the s o l u t i o n used i n my program f o r the r e s u l t i n g matrix equation i s given i n a l a t t e r s e c t i o n of t h i s chapter. Example To i l l u s t r a t e the numerical technique, an example f o l l o w s . For s i m p l i c i t y of computation an i s o t r o p i c m a t e r i a l w i l l be considered. We can then set K = K = K and f a c t o r a c c o r d i n g l y . The flow r e g i o n - xx yy ° J ° i s bounded by constant head boundaries and i s i l l u s t r a t e d i n Figure 3 _2 . Figure 3-2 . Region of flow. Consider node 1 i n Figure 3-2 . The c o n t r i b u t i o n from element e to the equation f o r node 1 i s : K 3N _1 3x 3N: 3N e + 3x 3N 3x 3N„ 3y 3yJ e 3N, 3x 3N, 3x 3N: + 3N® 3N!T — — + — -3x 3y 3y 3N! 3y 3N 3yJ (3.23) From element e + 1 we can w r i t e : e+1 K 3N e+1 3N e+1 3N e+1 3N e+1 3x 3y + 3y 3y + (j,. 3N e+1 3N e+1 3N e+1 3N e+1 3N 3x e+1 3x 3x 3y 3y 3N e+1 3N e+1 3N e+1 3x + 3y e+1 A = 0 3y (3.24) From element e + 2 we can w r i t e : 44. K e+2 9N e+2 3N e+2 3N .e+2 9N e+2 9N e+2 9N e+2 3N e+2 3N e+2 3x 3x + 3y 3y + A 3x 3x + 3y 9y + 4>, 3N e+2 3N e+2 3N e+2 3N e+2 3x 9x 9y 3y J A e+2 A =0 (3.25) The equation f o r node 1 i s obtained by summing the three c o n t r i b u t i o n s from the three surrounding elements. Combining ap p r o p r i a t e terms, the f o l l o w i n g equation r e s u l t s : 3N: 3x 3N!T 3N^ 3x 3N" + + e e K A e e K A 3N, 9JT' 3N, 3x 9y 3y e+2 e+2 + K A + K 6 + 1 A e + 1 3N e+1 9x ~„e+l ..,e+l 9N 9N _ _ L + _ _ L _ 9x 9y 3N® 3N^ 3x 3N̂ ' 3N!T 3x 3N e+2 3N e+2 9N e+2 3x 3x 3N, 3y 3y 3N, 9y 9y + K e + 2 A e + 2 + K e + 1 A e + 1 3N 9x e+1 9N 9x 9N e+2 9y e+2 3N 9y e+2 9N .e+1 Te+2 1 2 - + 9x 9N e+1 3y e+1 3N 9x + 9y 3y 3N e+2 9N 9y e+1 3y + <>. K e + l A e + l 3N 6 + 1 3 N f 1 anf 1 4 > 1 + 4 3x * 3x 3N e+1 3y 3y e+2 e+2 + K A 9N e+2 3x 3 N f 2 3 N f 2 1 + — i ~ 3x 3y 3N e+2 3y = 0 (3.26) 45. For node 2 : K V 3N: 3x 3N" 9x K E A E ax" 8y 9y + K 6 + 2 A E + 2 3N^ 3N!J + 3x 3N e+2 3N e+2 3N e+2 3x 3 N , 3y 3y e+2 e+? + K A 3N e+2 3x 3N e+2 3x + 3 N f 2 3Nf2 ~3T~ + 3y ay 3N e+2. 3y K E A E 3N, ax 3N^ 3N® + 6 3x 3 N , 3y 3y e+2 e+2 K. A 3N e+2 _4 3x 3N e+2 3N e+2 3x 3y 3N e+2. 3y = 0 ( 3 . 2 7 ) For node 3 : K V 3 N , 3x + <f>2i K E A 6 3N^ 3N^ + 3x 3x 3y 3y e+1 e+1 + K A 3N e+1 3x 3 N ! 37" 3 N , + 3N; ay 3y 3N e+1 3N e+1 3x a y 3N e+1 3y + <f>, K E A 6 3N! ax 3 N ! 37" 3N; + 8N: 3y 3y e+1 e+1 + K A 3N e+1 3x 3N e+1 3N Te+1 3x + 3y 3y + <f>. ..e+1 e+1 K A 3N e+1 3N e+1 3N e+1 aN e+1 ax ax ay 3y ( 3 . 2 8 ) 46. For node 4 K e + l A e + l 9N e+1 9N e+1 9N e+1 9N e+1 3x 3x + 9y 9y + K e + 2 A 6 4 - 2 9N e+2 9x + <J>, K 6 + 2 A e + 2 K e + l A e + l 9N 2 9N e+2 9N e+2 9x 9N„ 9x 9y 9N e+1 9N e+1 9x 9x + 9y + e+1 e+1 K A 9N e+1 9x 9 N f + 1 3 N f + 1 4 + 4 9x 9y + K 6 + 2 A e + 2 9N e+2 9N e+2 9x 9y 9N e+2 9y 9N e+1 9y 9N e+1 9y 9N e+2 9N e+2 9N e+2 9x 9x 9y = 0 9N e+2 9y 9N e+2 9y (3.29) The four equations, one f o r each node can be put i n m a t r i x form as f o l l o w s : e, e + 1, e + 2 e, e + 2 e, e + 1 e + 1 , e + 2 e, e + 2 e, e + 2 e e + 2 e, e + 1 e + 1 , e + 2 • l 0 e e + 2 *2 0 e, e + 1 e + 1 *3 0 e + 1 e + 1, e + 2 *4 0 (3.30) 47. where the e, e + 1 , e + 2 represent a c o n t r i b u t i o n from the element of the same l a b e l . The i ^ 1 row can be seen to represent the equation f o r the i ^ node and the j t ' r i column contains e n t r i e s corresponding to the c o e f f i c i e n t of the j t * 1 head value. M a t r i x S o l u t i o n Figure 3-3 i l l u s t r a t e s the mat r i x form of equation (3.19). n x n [ P 1 : j ] U.} {F.} {0} F i g u r e 3-3 M a t r i x form of equation (3.19) [P ] , the m a t r i x of c o e f f i c i e n t s i s dimensioned m x m where m i s the t o t a l number of nodes i n the r e g i o n . {<}>} i s a v e c t o r of mostly unknown head values and {F} i s a v e c t o r of constants which depend on the boundary c o n d i t i o n s . [P] can be p a r t i t i o n e d i n t o two p a r t s of which one c o n s i s t s of a s m a l l e r n x n m a t r i x where n i s the t o a l number of unknowns. In numbering the.nodes, the known values are numbered l a s t . 48. I t i s then a simple matter of m u l t i p l y i n g the known head values by the appropriate c o e f f i c i e n t s i n the. p a r t of [P^ ̂  ] where 0 < i <_ n and n < j < m . For each row "i" the r e s u l t i n g products can be summed, added to F^ and t r a n s f e r r e d to the r i g h t hand s i d e of the equation. The r e s u l t i n g equation i s of the form [A] {$} = {B} (3.31) Where [A] i s an n x n m a t r i x of c o e f f i c i e n t s , {cj>} i s a v e c t o r of n unknown head values and {?,} i s a v e c t o r of known constants. Standard computer programs can be used to s o l v e f o r {a)} . Such programs are commonly based on Cholesky Decomposition or Gaussian E l i m i n a t i o n . The subroutine c a l l e d by the program used i n t h i s study i s based on Gaussian e l i m i n a t i o n ( B i r d , 1975, p. 42). K r e y s i g (1972) d i s c u s s e s the theory of the v a r i o u s numerical techniques. Heterogeneity and Anisotropy W i t h i n any element, p h y s i c a l p r o p e r t i e s are considered homogeneous ( i s o t r o p i c or a n i s o t r o p i c ) . However, from element to element, p r o p e r t i e s can vary w i d e l y a l l o w i n g one to model hete r o g e n e i t y . If the p r i n c i p l e d i r e c t i o n s of any given a n i s o t r o p i c element c o i n c i d e w i t h the g l o b a l c o o r d i n a t e s , equation (3.19) h o l d s . I f not, a coordinate t r a n s f o r m a t i o n i s r e q u i r e d . Z i e n k i e w i c z (1971, p. 301) p o i n t out that an important d i f f e r e n c e a r i s e s here from the s t r u c t u r a l s i t u a t i o n . As the matrix of c o e f f i c i e n t s [P] d e f i n e s r e l a t i o n s h i p s between s c a l a r 49. q u a n t i t i e s , i t i s e q u a l l y v a l i d whatever the o r i e n t a t i o n of the l o c a l axes. Thus, use of l o c a l axes does not r e q u i r e a matr i x t r a n s f o r m a t i o n or a change i n the assembly technique. For any element, a l o c a l coordinate system can be d e f i n e d , skewed a p p r o p r i a t e l y from the g l o b a l coordinate system. W i t h i n that element, equation (3.19) i s v a l i d r e l a t i v e to the l o c a l coordinate system and c o n t r i b u t i o n s to the nodal equations can be c a l c u l a t e d and added i n as before. Figure 3-4 i l l u s t r a t e s the simple trigonometry used to d e f i n e nodal coordinates r e l a t i v e to axes c o i n c i d e n t w i t h the l o c a l p r i n c i p a l d i r e c t i o n s of h y d r a u l i c c o n d u c t i v i t y . y x / y. X . 1 , Figure 3 4 Trigonometry used i n d e f i n i n g nodal coordinates i n terms of a l o c a l coordinate system. 50. The o r i g i n at (0,0) i s kept f i x e d and the l o c a l coordinate axes are skewed an angle of 0 from the g l o b a l coordinates. The f o l l o w i n g r e l a t i o n s h i p s h o l d : -1 y i a + 9 •= tan — (3.32) i -1 y l o = tan — - 0. (3.33) X l 1 = vx± + y i cos a (3.34) 2 2 y i = ^ x i + y i s 7 n a (3.35) Equation (3.19) i s solved f o r the transformed element i n terms of x|, x!, x/ and y!, y!, y' where the s u b s c r i p t s i , j , and k l a b e l the three v e r t i c e s . Such a tr a n s f o r m a t i o n imposes the r e s t r i c t i o n on data that x and y coordinates must be greater than zero. 51. C H A P T E R F O U R : L I M I T I N G A S S U M P T I O N S A N D H Y D R A U L I C C O N D U C T I V I T Y Limiting Assumptions A number of assumptions underlie the theory presented i n the previous chapter. In creating models or groundwater flow systems for studies of engineering significance, i t i s es s e n t i a l to understand these l i m i t i n g assumptions. In the following paragraphs, eight assumptions are stated, underlined, and b r i e f l y discussed. The models are f u l l y saturated with the top flow boundary corresponding to the water table. Freeze and Witherspoon (1966) point out that i n the rigorous approach the. entire saturated-unsaturated system should be considered as continuous and the ground surface should be used as the upper boundary of flow. Although possible for s o i l s (Freeze, 1971a, 1971b), the approach requires knowing the rather complex relationship between permeability and s o i l moisture content i n the unsaturated zone. Such data are d i f f i c u l t to obtain and not commonly available.. Further, I know of no work that has investigated flow i n unsaturated fractured rock. In any case, the assumption i s often quite reasonable for regional studies because the thickness of the unsaturated zone i s small compared to the t o t a l thickness of a groundwater basin and therefore the effects of the unsaturated flow are probably small. The po s i t i o n of the water table must be known and i n most cases i s chosen as the ground surface. From an engineering point of 52. view, choosing the ground surface as the water t a b l e i s v a l i d because i t represents the worst case. I n f a c t , f o r many s i t u a t i o n s , t h i s c o n d i t i o n may a l s o be the most reasonable one to assume as s t a b i l i t y problems often occur a f t e r heavy r a i n s or during snow-melt when s a t u r a t i o n would extend to the ground s u r f a c e . The models are s t e a d y - s t a t e . This means that boundary con- d i t i o n s and t h e r e f o r e the p o t e n t i a l d i s t r i b u t i o n are assumed to be time- independent. Freeze and Witherspoon (1966) r a t i o n a l i z e the a p p r o x i - mation of t r a n s i e n t by steady-state c o n d i t i o n s on the f o l l o w i n g b a s i s : 1. For the r e g i o n a l s c a l e of most i n v e s t i g a t i o n s , the d i f - ferences of a few f e e t between high water and low water p o s i t i o n s of the water t a b l e w i l l have l i t t l e e f f e c t on flow p a t t e r n s . 2. The r e l a t i v e c o n f i g u r a t i o n of the water t a b l e u s u a l l y remains the same throughout the c y c l e of f l u c t u a t i o n s ; t h a t i s , h i g h p o i n t s remain the highest and low p o i n t s remain the lowest. I t may be that f o r some slope s t a b i l i t y problems, t r a n s i e n t e f f e c t s of water t a b l e f l u c t u a t i o n s are important, but again, by c o n s i d e r i n g the steady, f u l l y s a t u r a t e d c o n d i t i o n , the worst case i s covered. The models are two-dimensional. This assumption i s e q u i v a l e n t t o assuming a n e g l i g i b l e gradient i n the t h i r d dimension. For the homogeneous i s o t r o p i c case t h i s requirement leads to the r e s u l t that flow i n the t h i r d dimension i s a l s o zero. However, f o r a n i s o t r o p i c media, having the gradient zero i n the t h i r d dimension does not neces- s a r i l y mean that flow i s zero i n the t h i r d dimension. Flow i n the t h i r d dimension w i l l vanish only i f the plane of the model c o i n c i d e s w i t h two of the p r i n c i p a l axes. This r e s t r i c t i o n i s discussed i n d e t a i l by Bear (1972 p. 142). I n complex-geologies i t w i l l not always be p o s s i b l e to 53. f u l f i l l t h i s r e s t r i c t i o n and some e r r o r w i l l be introduced i n t o the models. I know of no s e n s i t i v i t y a n a l y s i s which has e s t a b l i s h e d the magnitude of the e r r o r . The best that can be done at present i s to minimize the e r r o r by t a k i n g the two-dimensional s e c t i o n p e r p e n d i c u l a r to the contours of the water t a b l e s u r f a c e . The model boundaries are e i t h e r &_ s p e c i f i e d constant head ( D i r i c h l e t c o n d i t i o n ) or constant f l u x (Nejuman c o n d i t i o n ) . A s p e c i a l case of the Neuman c o n d i t i o n i s when the f l u x equals zero, corresponding to an impermeable boundary. The top flow boundary i s e i t h e r the assumed constant head water t a b l e or assumed f l u x boundary. I t can be chosen to reasonably approximate f i e l d c o n d i t i o n s . By constant head, i t i s not meant that the head on the water t a b l e i s everywhere the same but r a t h e r the head at any p o i n t on the water t a b l e i s kept constant w i t h time. The s i d e and bottom model boundaries cannot be observed i n the f i e l d and t h e i r choice r e q u i r e s some e x p l a n a t i o n . The s i d e s are u s u a l l y chosen as v e r t i c a l impermeable bounda- r i e s corresponding to groundwater d i v i d e s . This i s an a p p l i c a t i o n of image theory which i s v a l i d provided the groundwater d i v i d e i s indeed s t r a i g h t and v e r t i c a l . This suggests a symmetry r a r e l y met by f i e l d c o n d i t i o n s , p a r t i c u l a r l y i n regions of complex geology. In f a c t , the assumption holds only under mountains or v a l l e y s e x h i b i t i n g symmetric topography and u n d e r l a i n by a homogeneous, i s o t r o p i c , porous media. A number of examples of i r r e g u l a r and perhaps more r e a l i s t i c groundwater d i v i d e s can be drawn f o r the models de s c r i b e d i n the next chapter (Figures 5-3 to 5-11). 54. The d i f f i c u l t y can be overcome i n two ways. F i r s t l y a p e r s o n e x p e r i e n c e d i n m o d e l l i n g r e a l c a s e s may, w i t h a knowledge of g e o l o g i c c o n d i t i o n s and r e s u l t i n g h y d r a u l i c p r o p e r t i e s , be a b l e t o e s t i m a t e a r e a s o n a b l e p o s i t i o n f o r a groundwater d i v i d e . The r e s u l t i n g i r r e g u l a r boundary can be e a s i l y h a n d l e d by the f i n i t e element program. However, i n some cases such a guess c o u l d be i n a c c u r a t e . A second way, used i n t h i s p r o j e c t , i s t o model a l a r g e enough r e g i o n so t h a t i n c o r r e c t boundary e f f e c t s w i l l n o t be f e l t i n the l o c a l a r e a of i n t e r e s t . The c h o i c e o f a h o r i z o n t a l impermeable boundary at depth i s d i s c u s s e d by F r e e z e and W i t h e r s p o o n (1967). At some depth, p e r m e a b i l i - t i e s w i l l be lower than i n the near s u r f a c e u n i t s and a t t h i s p o i n t e q u i p o t e n t i a l s become v e r t i c a l . The lower boundary of the h i g h e r p e r - m e a b i l i t y u n i t a c t s l i k e an impermeable boundary and the a d d i t i o n o r d e l e t i c a o f t h e l o w e r , l e s s permeable u n i t has a n e g l i g i b l e e f f e c t on the p o t e n t i a l p a t t e r n . In d e s i g n i n g a model f o r a g i v e n b a s i n , t h e y suggest t h a t p r e l i m i n a r y s t u d i e s s h o u l d p r o b a b l y b e g i n w i t h a g r e a t e r b a s i n depth than would seem n e c e s s a r y ... i f one f i n d s v e r t i c a l e q u i - p o t e n t i a l s as s u g g e s t e d above, t h e b a s i n depth can be l i m i t e d a c c o r d i n g l y . Many i n v e s t i g a t o r s ( f o r example, D a v i s , 1969) have n o t e d the apparent d e c r e a s e i n p e r m e a b i l i t y o f r o c k w i t h d e p t h . T h i s o b s e r v a t i o n c e r t a i n l y s u b s t a n t i a t e s the above r e a s o n i n g . However, assuming a d e c r e a s e i n p e r m e a b i l i t y o r h y d r a u l i c c o n d u c t i v i t y w i t h depth may not always be a r e a s o n a b l e assumption. Snow (1968) and Brace and M a r t i n (1968), amongst o t h e r s , d i s c u s s the dependence of p e r m e a b i l i t y on e f f e c - t i v e s t r e s s , e s p e c i a l l y In r o c k . Because e f f e c t i v e s t r e s s does not 5 5 . always d e c r e a s e w i t h depth, t h e r e may be some, perhaps many g e o l o g i c s i t u a t i o n s where p e r m e a b i l i t y i n c r e a s e s r a t h e r than d e c r e a s e s w i t h depth. Such a s i t u a t i o n , however, r e q u i r e s a d e f o r m i n g media. The models used i n t h i s study a r e based on the assumption t h a t the con- t i n u o u s porous media i s r i g i d . Of a l l the a s s u m p t i o n s , t h i s may be the most d i f f i c u l t t o d e f e n d . The models d e s c r i b e d i n the next c h a p t e r i l l u s t r a t e a number o f cases t h a t c o u l d l e a d to l o c a l d e v e l o p m e n t s ' o f h i g h pore p r e s s u r e s and subsequent s l o p e f a i l u r e s . I n f a c t , what may a c t u a l l y happen i s t h a t i n c r e a s e s i n pore p r e s s u r e may be l i m i t e d t o the p o i n t a t which t h e accompanying d e c r e a s e i n e f f e c t i v e s t r e s s i s enough t o cause a s m a l l l o c a l f a i l u r e . An i n s t a n t a n e o u s i n c r e a s e i n permea- b i l i t y r e s u l t s , p r e s s u r e s a r e r e l i e v e d and e f f e c t i v e s t r e s s i s i n c r e a s e d , which i - i t u r n causes an i n c r e a s e i n s t a b i l i t y . Snow (1968) d i s c u s s e s the e l a s t i c i t y o f f r a c t u r e d media i n r e s p o n s e to f l u i d p r e s s u r e changes and g i v e s e v i d e n c e f o r " f r a c t u r e b r e a t h i n g " . T h i s p o s s i b i l i t y does not i n v a l i d a t e t h e r e s u l t s o f m o d e l l i n g r i g i d media, b u t i t may be cause f o r m o d i f i c a t i o n o f some of the r e s u l t s . The t o p i c o f f l o w i n deforrnable media i s p r e s e n t l y an a c t i v e a r e a o f r e s e a r c h ( f o r example, see G a l e , 1975). With g r e a t e r u n d e r s t a n d i n g i t may be p o s s i b l e t o m o d i f y r e g i o n a l groundwater models a c c o r d i n g l y . F o r the p r e s e n t , one can r a t i o n a l i z e t h e r i g i d media assumption, p a r t i c u l a r l y f o r e n g i n e e r i n g s t u d i e s , by s t a t i n g t h a t the worst s i t u a t i o n i s b e i n g m o d e l l e d and i n any case the problem a r e a s a r e b e i n g i d e n t i f i e d . One o f the most fundamental assumptions i s t h a t groundwater f l o w through porous media oan be d e s c r i b e d by D a r c y ' s Law. Bear (1972, 56. p. 127) d i s t i n g u i s h e s t h r e e t y p e s o f f l u i d f l o w : l i n e a r l a m i n a r f l o w , n o n l i n e a r l a m i n a r f l o w , and n o n l i n e a r t u r b u l e n t f l o w . D a r c y ' s Law h o l d s o n l y f o r low v e l o c i t i e s and head g r a d i e n t s when f l o w of the f i r s t type e x i s t s . F o r groundwater f l o w systems a s s o c i a t e d w i t h most g e o l o g i c s i t u a t i o n s , t h i s i s a r e a s o n a b l e assumption. F o r n o n l i n e a r l a m i n a r and t u r b u l e n t f l o w , a number of power-law r e l a t i o n s h i p s have been p r o p o s e d . These a r e a l s o d i s c u s s e d by Bear (1972, p. 182). One f i n a l and i m p o r t a n t assumption remains to be d i s c u s s e d : the g e o l o g i c case b e i n g m o d e l l e d can be r e a s o n a b l y approximated by an e q u i v a l e n t , c o n t i n u o u s porous media. I t i s c l e a r t h a t examples e x i s t f o r which an e q u i v a l e n t porous media cannot be assumed and i n m o d e l l i n g groundwater f l o w systems f o r s l o p e s t a b i l i t y a n a l y s e s t h e s e must be r e c o g n i z e d . C e r t a i n l y models o f l a r g e r e g i o n s which i n c l u d e h i g h l y f r a c t u r e d o r o t h e r w i s e porous g e o l o g i c u n i t s can be r e a s o n a b l y m o d e l l e d as a c o n t i n u o u s porous media, w h i l e a s m a l l s l o p e of sound r o c k cut by a few f r a c t u r e s cannot. L o u i s i s quoted by Londe (1971, p. 3) as sug- g e s t i n g t h a t f o r a g i v e n c r o s s - s e c t i o n of r o c k , an e q u i v a l e n t porous media can be assumed o n l y i f the number o f f i s s u r e s c u t by the c r o s s - s e c t i o n i s o f the o r d e r 10,000. A l t h o u g h p r i n c i p l e s o f f l o w t h r o u g h porous media a r e w e l l e s t a b l i s h e d , t h e p r i n c i p l e s g o v e r n i n g f l o w t h r o u g h a f r a c t u r e d media t h a t cannot be approximated by an e q u i v a l e n t porous media a r e n o t . The problem i s d i s c u s s e d by many i n v e s t i g a t o r s i n c l u d i n g C a s t i l l o (1972), G a l e (1975), Londe (1971), L o u i s (1969), L o u i s and P e r n o t (1972), Sharp and M a i n i (1972), Snow (1968, 1969, 1972), W i t t k e (1971), W i t t k e and L o u i s (1966) and W i t t k e e t a l (1972). I t i s n o t p o s s i b l e a t the p r e s e n t time to make g e n e r a l i z a t i o n s about the v a l i d i t y 57. of t h i s a ssumption. R a t h e r , each model must be judged i n d e p e n d e n t l y on the b a s i s o f t h e g e o l o g i c c o n d i t i o n s e n c o u n t e r e d . H y d r a u l i c C o n d u c t i v i t y Under the s e t o f assumptions d e s c r i b e d above, the o n l y mate- r i a l p r o p e r t y t h a t r e q u i r e s s p e c i f i c a t i o n i s the h y d r a u l i c c o n d u c t i v i t y . The h y d r a u l i c c o n d u c t i v i t y "K" i s the c o e f f i c i e n t o f p r o p o r t i o n a l i t y which appears i n D a r c y ' s Law. I t r e f l e c t s b o t h p r o p e r t i e s o f the f l u i d and the s o l i d m a t r i x and i s not t o be c o n f u s e d w i t h the i n t r i n s i c p e r - m e a b i l i t y , "k", which i s a p r o p e r t y o f the m a t r i x o n l y . The two a r e r e l a t e d by K = k 1 (4.1) y where y i s the s p e c i f i c weight o f the f l u i d and u i s the v i s c o s i t y . C o n f u s i o n has a r i s e n i n the p a s t because the term " c o e f f i c i e n t o f p e r - m e a b i l i t y " has been a p p l i e d t o the h y d r a u l i c c o n d u c t i v i t y . F i g u r e 4-1 g i v e s a pproximate ranges o f h y d r a u l i c c o n d u c t i v i t y f o r d i f f e r e n t r o c k t y p e s and u n c o n s o l i d a t e d m a t e r i a l s . T h i s f i g u r e was used t o o b t a i n a f i r s t e s t i m a t e o f r e l a t i v e c o n d u c t i v i t i e s between d i f f e r e n t g e o l o g i c u n i t s . However, i n some c a s e s , h y d r a u l i c c o n d u c t i - v i t i e s were a s s i g n e d on a b a s i s o t h e r t h a n l i t h o l o g y . D a v i s (1969) has p o i n t e d out t h a t i n many c a s e s w e a t h e r i n g h i s t o r y and f r a c t u r i n g a r e more s i g n i f i c a n t t o h y d r a u l i c c h a r a c t e r i s t i c s than l i t h o l o g i e s . I n t e r e s t i n g l y , i t i s n o t t h e a b s o l u t e v a l u e s o f h y d r a u l i c c o n d u c t i v i t y t h a t c o n t r o l the form of the p o t e n t i a l n e t s d e s c r i b e d i n 58. r o c k «rff sediments 1 ", ™* — • • -J* Hydraulic Conductivity cm./sec. ft./sec. ,2 3 r-10 h i o r - l -10 •10 h i o r - io - l - A hio -5 •10 -10 h i o h l O 10 •10 •11 Figure 4-1. Approximate ranges of hydraulic conductivity (Freeze, 1975, personal communication). r-l 0 J h i o r- i h i o h l O -1 -2 - 4 h i o h i o 10 h i o h i o ' - l O •10 59. the next chapter, but r a t h e r the r e l a t i v e c o n d u c t i v i t i e s between u n i t s . For p r a c t i c a l a p p l i c a t i o n , i t may be more accurate to estimate or d e r - o f - magnitude d i f f e r e n c e s i n h y d r a u l i c c o n d u c t i v i t y r a t h e r than exact values and be able to produce a reasonable model of the flow system even though exact h y d r a u l i c c o n d u c t i v i t i e s are not known. 60. C H A P T E R F I V E : M O D E L R E S U L T S AMD D I S C U S S I O N I n t r o d u c t i o n . To model the complex geologies of i n t e r e s t , f a i r l y l a r g e models were r e q u i r e d . However, because of the time r e q u i r e d i n s e t t i n g up element g r i d s and t a b u l a t i n g element and nodal data, the number of elements were l i m i t e d to 500 and the number of nodes to 275. This r e s u l t e d i n a storage requirement of 162 k. Execution times were between 6 and 13 seconds, depending on the number of i t e r a t i o n s r e q u i r e d to achieve the d e s i r e d t o l e r a n c e of .001 i n c a l c u l a t i n g the nodal poten- t i a l s . A l l models were run on the IBM 370/168 at the U n i v e r s i t y of B r i t i s h Columbia. Gi l l h a m and Farvolden (1974) reported s o l u t i o n i n s t a b i l i t y i n t h e i r f i n i t e element flow program usi n g s i n g l e p r e c i s i o n v a r i a b l e s f o r c o n d u c t i v i t y c o n t r a s t s of over 500. This r e q u i r e d t h e i r use of double p r e c i s i o n v a r i a b l e s . They a l s o found that as the number of nodes i n a c r o s s - s e c t i o n was i n c r e a s e d , the s o l u t i o n became unstable at lower con- d u c t i v i t y r a t i o s . Using the matrix s o l u t i o n SLIMP, programmed by the UBC Computing Centre, I encountered no such d i f f i c u l t i e s . S i n g l e p r e c i - s i o n v a r i a b l e s were used w i t h v a r i a t i o n s i n c o n d u c t i v i t y of up to s i x orders of magnitude. For s m a l l e r computers the storage requirement of 162 k may at present cause d i f f i c u l t i e s but storage c a p a b i l i t i e s are r a p i d l y i n c r e a s i n g , rendering t h i s a temporary problem, i f one at a l l . 61. Six d i f f e r e n t topographic arrangements were modelled, each with a number of d i f f e r e n t geologies exhibiting d i f f e r e n t hydraulic conductiv- i t y contrasts. In a l l , over 75 models were run. Figure 5-1 summarizes the models and the various p r i n c i p l e s to be i l l u s t r a t e d . Some of these hydrogeologic settings were suggested to me by the discussion of Deere and Patton (1967). To show the effect of d i f f e r e n t flow systems on slope s t a b i l i t y , use is.made of piezometric l i n e s which .indicate hydraulic head at the base of an a r b i t r a r y plane, perhaps a s l i d e surface. Water would r i s e to the piezometric l i n e , i n an open standpipe connected only to the s l i d e surface. The hydraulic head "<j>" consists of two parts: <j> = z + ¥ (5.1) where z i s the elevation head and l i s the pressure head. Figure 5-2 i l l u s t r a t e s the d i f f e r e n t parts of the hydraulic head. The pore pressure "u" found i n Terzaghi's e f f e c t i v e stress law, (2.4), i s related to the pressure head by u = YY (5.2) where y i s the s p e c i f i c weight of water. Also shown on Figure 5-2 i s a plot of pressure head at the base of the s l i d e plane. In most slope s t a b i l i t y analyses the s l i d e i s broken into v e r t i c a l s l i c e s and the pressure head i s used to calculate the pore pressure acting on each s l i c e . In practice, i t i s not necessary to plot a separate pressure head diagram, as a measurement of the height of the piezometric l i n e above the s l i d e plane can be taken d i r e c t l y from the numerically simulated hydraulic 62. The e f f e c t s o f a n i s o t r o p y . F i g u r e 5-3 T h r u s t s and i n t e r b e d d e d s e d i m e n t r y r o c k s . F i g u r e 5-5 F a u l t s , c o n t a c t s , dykes and weather- i n g p r o f i l e s . F i g u r e 5-6 L a y e r e d c o l l u v i u m and w e a t h e r i n g p r o f i l e s . F i g u r e 5-8 F l a t l y i n g weak r o c k s , f r a c t u r e s due to s t r e s s r e l i e f , and the e f f e c t s o f a r e g i o n a l a q u i f e r . F i g u r e 5-9 P l e i s t o c e n e T e r r a c e s . F i g u r e 5-10 Deformed metamorphic r o c k s , and the e f f e c t o f a r e s e r v o i r on a deep r o c k s l i d e . F i g u r e 5-11 F i g u r e 5-1. Summary o f Models 63. Figure 5-2. Hydraulic head, pressure head and el e v a t i o n head. 64. head net. Slope s t a b i l i t y formulae can be adapted to process t h i s data d i r e c t l y . The remaining f i g u r e s discussed i n t h i s chapter are found i n APPENDIX I I . For reader convenience they should be unfolded and kept handy w h i l e reading. The v a r i o u s f i g u r e s can be considered dimensionless and s c a l e i s presented as a m u l t i p l e of an a r b i t r a r y constant. Such models can e q u a l l y w e l l represent flow patterns f o r systems covering only a few acres or f o r those extending over many hundreds of square mil e s (Freeze and Witherspoon, 1967). The E f f e c t s of Anisotropy H y d r a u l i c c h a r a c t e r i s t i c s of both s o i l and rock are more commonly a n i s o t r o p i c than i s o t r o p i c . Maasland (1957) gives an extensive d i s c u s s i o n of ani s o t r o p y . He p o i n t s out that a n i s o t r o p y i n s o i l s can be caused by s t r a t i f i c a t i o n . An apparent d i r e c t i o n a l h y d r a u l i c c o n d u c t i v i t y i s created i n a s t r a t i f i e d medium which i s i d e n t i c a l to th a t of an a n i s o t r o p i c medium. The s t r a t i f i c a t i o n may be a r e s u l t of the shape of mine r a l g r a i n s or a l t e r n a t i n g l a y e r s of d i f f e r e n t t e x t u r e . In gene r a l , h o r i z o n t a l c o n d u c t i v i t y i s greater than v e r t i c a l c o n d u c t i v i t y . He al s o notes that i n many sedimentary rocks there may be a di s s e m i n a t i o n of very t h i n shale or micaceous l a m i n a t i o n through the porous bed. In common co r i n g techniques these laminations would cause f r a c t u r e s between pieces of core. T e s t i n g of the core pieces would miss the t h i n l a y e r s and an erroneous measurement of anisotropy would be obtained. He c i t e s l a b o r a t o r y 65. and f i e l d e v i d e n c e o f a n i s o t r o p y i n s o i l s w i t h h o r i z o n t a l t o v e r t i c a l c o n d u c t i v i t y r a t i o s r a n g i n g from 1 t o 42. A n i s o t r o p y r e s u l t i n g from f r a c t u r e s i n r o c k i s d i s c u s s e d by Snow (1968). He n o t e s t h a t " a n i s o t r o p y i s the r u l e ; i s o t r o p y i n f r a c - t u r e d r o c k s i s a s p e c i a l c i r c u m s t a n c e o f p r e s s u r e and f r a c t u r e geometry." F i g u r e 5-3 i l l u s t r a t e s the e f f e c t s of a n i s o t r o p y on the ground- water regime. As w i l l be seen i n the next s e c t i o n , the asymmetric t o p o g r a p h i c p r o f i l e c o u l d r e p r e s e n t a v a r i e t y o f cases of d i p p i n g - g e o - l o g i c u n i t s which have been eroded t o form f l a t i r o n s . The homogeneous, i s o t r o p i c case i s shown i n F i g u r e 5-3a, w h i l e F i g u r e s 5-3b t h r o u g h 5-3e i l l u s t r a t e an a n i s o t r o p y o f one o r d e r o f mag- n i t u d e skewed a t v a r i o u s a n g l e s from the g l o b a l c o o r d i n a t e s . To s i m u l a t e a d e c r e a s e o f c o n d u c t i v i t y w i t h d e p th, two i s o t r o p i c u n i t s u n d e r l i e the a n i s o t r o p i c r e g i o n . Note t h a t t h e e q u i p o t e n t i a l s tend t o f o l l o w the d i r e c t i o n o f maximum h y d r a u l i c c o n d u c t i v i t y . To c o n s i d e r the e f f e c t o f the d i f f e r e n t o r i e n t a t i o n s o f a n i s o - t r o p y on s l o p e s t a b i l i t y , an a r b i t r a r y f a i l u r e s u r f a c e was drawn beneath the r i g h t - h a n d v a l l e y w a l l . F o r each f i g u r e , the p i e z o m e t r i c l i n e r e p r e s e n t i n g t h e h y d r a u l i c head a t t h e base o f the s l i d e was drawn. These l i n e s a r e p l o t t e d t o g e t h e r on F i g u r e 5-4. I t can be seen t h a t the h i g h e s t h y d r a u l i c and p r e s s u r e heads o c c u r when the maximum c o n d u c t i v i t y i s p a r a l l e l t o the d i p of the s l o p e , w h i l e the lowest o c c u r when the a n i s o t r o p y i s f l a t l y i n g . I f the d i f f e r e n c e i n e l e v a t i o n between v a l l e y f l o o r and the h i g h e r , r i g h t - h a n d peak i s 1,000 f e e t , the maximum d i f f e r e n c e i n p r e s s u r e 66. head of the two extreme cases (5-3b and 5-3d) i s approximately 100 feet. This represents a pore pressure difference of over 6,000 pounds per foot length of s l i d e surface, assuming a one-foot-thick section! The example v i v i d l y i l l u s t r a t e s that d i f f e r e n t orientations of anisotropy can cause very d i f f e r e n t pore pressure d i s t r i b u t i o n s i n a slope. I t shows, as w i l l the other examples to be discussed, the impor- tance of understanding the geology and hydraulic c h a r a c t e r i s t i c s of a large area around a s l i d e , not just the s l i d e i t s e l f . I n t e r e s t i n g l y , a dip slope, as w e l l as causing the most adverse groundwater conditions, i s also the weakest from a strength of reck perspective. Thrusts and Interbedded Sedimentary Rocks A common geologic s i t u a t i o n occurs when thick sedimentary units are folded or thrust up from thei r o r i g i n a l horizontal p o s i t i o n . With time s the less resistant units are eroded to form v a l l e y s , while the more resistant beds form ridges or f l a t i r o n s . Figure 5-5 (modified from Deere and Patton, 1967, Figure 5) shows s i x d i f f e r e n t models of i n t e r - bedded units with a variety of conductivity contrasts. As discussed i n the previous example (Figure 5-3), the dip slope i s underlain by the most adverse head d i s t r i b u t i o n . To compare the effects of d i f f e r e n t conduc- t i v i t y contrasts, piezometric l i n e s have been plotted i n d i c a t i n g the hydraulic head at the base of Unit A beneath the dip slope on the r i g h t - hand side of the v a l l e y . Figure 5-5a shows what could be interbedded sandstones and shales with the more resistant sandstone occupying the ridges. The 67. sandstone has a. c o n d u c t i v i t y two orders of magnitude greater than the shale. I t i s p o s s i b l e that a s i t u a t i o n could e x i s t where the more r e s i s t a n t capping u n i t i s a l s o the l e s s conductive u n i t . For example, such a s i t u a t i o n i s b e l i e v e d to e x i s t on S a l t s p r i n g I s l a n d i n the S t r a i t of Georgia where the shale occupying the v a l l e y s has developed a secon- dary f r a c t u r e p e r m e a b i l i t y , making i t more conductive than the sandstone. The groundwater regime f o r such an example i s shown i n Figure 5-5b. For comparison, the p i e z o m e t r i c l i n e from 5-5a i s p l o t t e d on 5-5b. In both examples, hig h h y d r a u l i c heads e x i s t beneath the dip s l o p e , although i n 5-5b the p i e z o m e t r i c l e v e l i s somewhat higher because of the t h i c k e r c o n f i n i n g u n i t . With a greater c o n d u c t i v i t y c o n t r a s t the r e s u l t s are more s t a r t l i n g . This i s seen on F i g u r e 5-5c where Unit A has been assigned a c o n d u c t i v i t y of 10 ^, four orders of magnitude lower than the u n i t s on e i t h e r s i d e . The r e s u l t i n g r i s e i n the p i e z o m e t r i c l i n e i s very great. U n i t A may represent a d i s t i n c t l i t h o l o g y w i t h low c o n d u c t i v i t y or i t could represent a t h r u s t s u r f a c e . The t h r u s t i n g , through produc- t i o n of a f a u l t gorge could have caused the lower c o n d u c t i v i t y ; or a u n i t of lower c o n d u c t i v i t y could have caused the high pore pressures which i n t u r n l e d to the t h r u s t f a u l t i n g . This l a t t e r p o s s i b i l i t y was proposed by Hubbert and Ruby (1959). In r e a l examples, both mechanisms might be at work and i t may not be p o s s i b l e to i s o l a t e cause and e f f e c t . Figure 5-5c does show, however, the h i g h f l u i d p o t e n t i a l s that could develop beneath a t h r u s t s u r f a c e . I t appears that the s t a b i l i t y of enormous t h r u s t blocks i s based on the same p r i n c i p l e s as the s t a b i l i t y of much sm a l l e r slopes. 6 8 . No method has yet been developed f o r s u c c e s s f u l l y measuring a n i s o t r o p i c c o n d u c t i v i t i e s i n a l l g e o l o g i c s i t u a t i o n s , although Louis and Pernot (1972) describe a d e t a i l e d f i e l d i n v e s t i g a t i o n they used to e s t a - b l i s h a n i s o t r o p i c h y d r a u l i c c h a r a c t e r i s t i c s i n metamorphic rocks at a p o t e n t i a l damsite. However, such d e t a i l e d work i s expensive and not always p o s s i b l e . For that reason i t i s o f t e n necessary to match model r e s u l t s w i t h a r a t h e r scanty amount of f i e l d i n f o r m a t i o n . I f a f i t i s not obtained, the model i s adjusted u n t i l a f i t i s found. U n f o r t u n a t e l y , i n complex g e o l o g i e s , more.than one combination of a n i s o t r o p i c and hetero- geneous h y d r a u l i c c o n d u c t i v i t i e s can r e s u l t i n the same p o t e n t i a l net. This problem i s i l l u s t r a t e d by Figure 5-5d, where Unit A has been assigned an a n i s o t r o p i c c o n d u c t i v i t y w i t h the major a x i s (x') 10 3 and the minor a x i s ( y ' ) , 10 6 . Even w i t h the low c o n d u c t i v i t y across the u n i t , the r e s u l t i s a s i m i l a r but s l i g h t l y lower p i e z o m e t r i c l i n e than i n 5-5a, where. U n i t A has an i s o t r o p i c c o n d u c t i v i t y of 10 4 . With minor a d j u s t - ments to the a n i s o t r o p i c c o n d u c t i v i t i e s the two could be made the same. This r e s u l t s from the f a c t that any homogeneous a n i s o t r o p i c medium can be transformed i n t o a f i c t i t i o u s i s o t r o p i c medium (Maasland, 1957, p. 238). A general example of the t r a n s f o r m a t i o n of an a n i s o t r o p i c two-layer system i s described by Stevens (1936). The reader i s r e f e r r e d to these papers f o r d e t a i l s . The l e s s o n to be learned i s that great care must be taken i n a d j u s t i n g models on the b a s i s of scanty f i e l d data. Figures 5-5e and 5-5f show an a n i s o t r o p i c c o n d u c t i v i t y i n U n i t A w i t h a c o n t r a s t of two orders of magnitude. I n F i g u r e 5-5f, however, ponded water has been introduced i n the v a l l e y . Both p i e z o m e t r i c 69. l i n e s a r e p l o t t e d on 5-5f. The ponded water causes a s i g n i f i c a n t r i s e i n t h e p i e z o m e t r i c l i n e . I t i s p a r t i c u l a r l y worth n o t i n g t h a t the e f f e c t o f t h e ponded water i s f e l t w e l l above the l e v e l o f t h e water on the v a l l e y s i d e s . F a u l t s , C o n t a c t s , Dykes and Weathering P r o f i l e s Where t h e r e i s e x t e n s i v e s u b s i d i a r y f r a c t u r i n g , f a u l t s may t a k e t h e form o f h i g h p e r m e a b i l i t y c o n d u i t s . C o n v e r s e l y , movement a l o n g a f a u l t may cause a gouge zone o f v e r y low c o n d u c t i v i t y . C o n t a c t s can a l s o cause a wide v a r i e t y o f e f f e c t s on the p o t e n t i a l d i s t r i b u t i o n depending on the d i f f e r e n t l i t h o l o g i e s i n c o n t a c t , t h e i r r e s p e c t i v e s t r u c t u r e s , and t h e n a t u r e o f t h e c o n t a c t i t s e l f . S i m i l a r l y a dyke, a l t h o u g h i n i t i a l l y much l e s s permeable than the s u r r o u n d i n g c o u n t r y r o c k , may be much more s u s c e p t i b l e t o w e a t h e r i n g and w i t h time become more permeable than the s u r r o u n d i n g r o c k . D e f o r m a t i o n of an a r e a i n t r u d e d by l e s s d u c t i l e dyke r o c k s can cause the dykes to be more permeable through a h i g h e r i n t e n s i t y o f f r a c t u r i n g . C l e a r l y , c a r e f u l f i e l d i n v e s t i g a t i o n i s r e q u i r e d t o e s t a b l i s h the s i g n i f i c a n c e o f any one o f t h e s e f e a t u r e s . F i g u r e 5-6 i l l u s t r a t e s some o f the d i v e r s e e f f e c t s d e s c r i b e d above. F o r comparison, a p l a n e has been drawn on the s e c t i o n beneath the l e f t - h a n d v a l l e y w a l l and the p i e z o m e t r i c l i n e s r e p r e s e n t i n g the h y d r a u l i c head on t h i s p l a n e have been drawn f o r each o f 5-6a through 5- 6 f . A l l the p i e z o m e t r i c l i n e s a r e drawn t o g e t h e r on F i g u r e 5-7. 70. The homogeneous i s o t r o p i c case i s shown i n Figure 5-6a. In Figure 5-6b a l e s s conductive u n i t i s seen o v e r l y i n g another. The contact i s simple. The d i f f e r e n c e i s not great but the p i e z o m e t r i c l i n e of 5-6b i s higher than 5-6a. Presumably i f the c o n d u c t i v i t y c o n t r a s t was grea t e r or the o r i e n t a t i o n of the contact d i f f e r e n t , a more dramatic d i f f e r e n c e would be apparent. In Figure 5-6c a f a u l t zone i s shown w i t h a c o n d u c t i v i t y s i g n i f i c a n t l y higher than the surrounding rock u n i t s . Such a f a u l t acts as a d r a i n and no problems r e l a t e d to slope s t a b i l i t y are apparent. In Figures 5-6d and 5-6e the f a u l t zone has been assigned a lower con- d u c v i t i t y than the surrounding rock. The two d i f f e r only i n that opposite c o n d u c t i v i t i e s have been assigned the two major rock u n i t s . In both Figures 5--6d and 5-6e the f a u l t zone causes a higher gradient beneath the toe of the slope i n the u n d e r l y i n g rock u n i t . A s i m i l a r s i t u a t i o n could r e s u l t from the i n t r u s i o n of a l e s s permeable dyke along the contact. Figure 5-6f shows a s i t u a t i o n which combines some of the feat u r e s of the previous four f i g u r e s . As w e l l , a weathering p r o f i l e has been added and the o v e r l y i n g u n i t has.been assigned an a n i s o t r o p i c c o n d u c t i v i t y . The f i g u r e could represent d i p p i n g sedimentary rocks t h r u s t against or l y i n g unconformably on a massive i n t r u s i v e rock of much lower c o n d u c t i v i t y . A deep weathering p r o f i l e has developed extending much f u r t h e r down the f a u l t zone or contact between the two u n i t s . The s i t u a t i o n i s reasonable f o r many no n - g l a c i a t e d areas of the world. 71. An e x t e n s i v e r e v i e w o f w e a t h e r i n g p r o f i l e s and t h e i r e f f e c t on f l o w systems and s l o p e s t a b i l i t y i s g i v e n by Deere and P a t t o n (1971). T a b l e 5-1 g i v e s t h e i r d e s c r i p t i o n o f the p a r t s o f a w e a t h e r i n g p r o f i l e f o r i gneous and metamorphic r o c k s . I n many s e d i m e n t a r y r o c k s t h e w e a t h e r i n g p r o f i l e i s s i m i l a r b u t i n c a r b o n a t e s the s a p r o l i t e (I-C) may be absent and the p a r t l y weathered r o c k (II--B) may n o t be p o s s i b l e t o i d e n t i f y . The r e l a t i v e p e r m e a b i l i t i e s and s t r e n g t h s a r e a l s o l i s t e d i n T a b l e 5-1. Note t h a t the l o w - s t r e n g t h r e s i d u a l s o i l s a r e u n d e r l a i n by a zone of much h i g h e r p e r m e a b i l i t y . I n e v i t a b l y t h i s c o n f i n i n g s i t u a t i o n w i l l l e a d t o s t a b i l i t y p roblems. The r e l a t i v e p e r m e a b i l i t i e s s u g g e s t e d by Deere and P a t t o n were used as a b a s i s f o r a s s i g n i n g c o n d u c t i v i t y v a l u e s t o the v a r i o u s zones shown i n F i g u r e 5 - . 6 f . The toe of b o t h s l o p e s i n the v a l l e y shows h y d r a u l i c heads d e t r i m e n t a l to s l o p e s t a b i - l i t y . T h i s can be seen i n F i g u r e 5-7 by the h i g h p i e z o m e t r i c l e v e l o f 5-6f r e l a t i v e t o the o t h e r c a s e s . L a y e r e d C o l l u v i u m and B u r i e d W e a t h e r i n g P r o f i l e s The r e c u r r e n c e o f l a n d s l i d e s i n a r e a s o f o l d s l i d e d e b r i s o r c o l l u v i u m i s a f a i r l y common problem. In t r o p i c a l a r e a s the problem i s i n t e n s i f i e d by the development o f w e a t h e r i n g p r o f i l e s b o t h on o r i g i n a l ground s u r f a c e and i n subsequent l a y e r s o f c o l l u v i u m . The p r o b l e m o f s l o p e s t a b i l i t y i n l a y e r e d c o l l u v i u m i s d i s c u s s e d i n d e t a i l by Deere and P a t t o n (1971) who p o i n t out t h a t d e s i c c a t i o n o r c o n s o l i d a t i o n o f the c o l l u v i u m causes complex l a y e r i n g and t h a t b u r i e d w e a t h e r i n g p r o f i l e s can cause examples of t h e " c l a s s i c c o n f i n e d a r t e s i a n a q u i f e r s i t u a t i o n . 72. Zone D e s c r i p t i o n R e l a t i v e P e r m e a b i l i t y R e l a t i v e S t r e n g t h I RESIDUAL SOIL IA T o p s o i l , r o o t s , o r g a n i c m a t e r i a l zone o f l e a c h - i n g and a l u v i a t i o n . May be por o u s . medium t o h i g h low t o medium IB C h a r a c t e r i s t i c a l l y c l a y e n r i c h e d w i t h accumula- t i o n s o f Fe, A l , S i . May be cemented. No r e l i c t s t r u c t u r e s p r e s e n t . low commonly low, h i g h i f cemented IC S a p r o l i t e . R e l i c t r o c k s t r u c t u r e s r e t a i n e d . S i l t y t o sandy m a t e r i a l w i t h l e s s t h a n 10% c o r e s t o n e s . O f t e n micaceous. medium low t o medium; r e l i c t s t r u c - t u r e s v e r y s i g n i f i c a n t I I WEATHERED ROCK I I A T r a n s i t i o n from s a p r o - l i t e t o p a r t l y weath- e r e d r o c k . H i g h l y v a r i a b l e , s o i l - l i k e t o r o c k - l i k e . 10% to 95% c o r e s t o n e s w i t h f i n e s commonly f i n e t o c o a r s e sand. S p h e r o i d a l w e a t h e r i n g common. h i g h medium t o low where weak s t r u c t u r e s and r e l i c t s t r u c - t u r e s a r e p r e s e n t I I B P a r t l y weathered r o c k . S o f t t o h a r d r o c k . J o i n t s s t a i n e d t o a l t e r e d . Some a l t e r a - t i o n of f e l d s p a r s and mi c a s . medium t o h i g h medium t o to h i g h * I I I UNWEATHERED ROCK I I I No i r o n s t a i n s a r e appar e n t a l o n g j o i n t s . No w e a t h e r i n g o f f e l d - s p a r s and m i c a s . low t o medium v e r y h i g h * * C o n s i d e r i n g o n l y i n t a c t r o c k masses w i t h no a d v e r s e l y o r i e n t e d g e o l o g i c s t r u c t u r e s . T a b l e 5-1. Weathering P r o f i l e f o r Igneous and Metamorphic Rocks a f t e r Deere and P a t t o n (1971, p. 92) 73. F i g u r e 5-8 i s an attempt t o i l l u s t r a t e the groundwater f l o w system i n such an environment. The b a s i s o f the models i s F i g u r e 8 o f Deere and P a t t o n (1971, p. 108). F i g u r e 5-8a i s the homogeneous i s o - t r o p i c c a s e . A s i n g l e l a y e r o f c o l l u v i u m o v e r a d i p p i n g a n i s o t r o p i c r o c k u n i t i s shown i n F i g u r e 5-8b. The c o l l u v i u m c o v e r s a s i m p l i f i e d w e a t h e r i n g p r o f i l e . I n the w e a t h e r i n g p r o f i l e , c o n d u c t i v i t i e s have been a d j u s t e d where a p p r o p r i a t e , but the a n i s o t r o p y has been m a i n t a i n e d . I n F i g u r e 5-8c the s i n g l e l a y e r o f c o l l u v i u m has been s p l i t by U n i t A, which r e p r e s e n t s a p a r t i a l w e a t h e r i n g p r o f i l e d e v e l o p e d on a lower l a y e r o f c o l l u v i u m . I t s e f f e c t i s a zone of lower c o n d u c t i v i t y . U n i t A con- f i n e s t h e l ower c o l l u v i u m l a y e r and h i g h pore p r e s s u r e s have d e v e l o p e d . F o r c omparison, the p i e z o m e t r i c l i n e r e p r e s e n t i n g t h e h y d r a u l i c head a t the base of U n i t A has been p l o t t e d f o r each model. A l s o p l o t t e d i s a dashed l i n e r e p r e s e n t i n g the w eight o f o v e r b u r d e n above the base of U n i t A. A s p e c i f i c w e ight o f 130 pounds p e r c u b i c f o o t has-been assumed and the l i n e p l o t t e d i n terms o f an e q u i v a l e n t h e i g h t o f water. I f o v e r b u r d e n weight becomes l e s s than p o r e p r e s s u r e s the dashed l i n e w i l l d i p below the p i e z o m e t r i c l i n e . An example o f t h i s i s shown i n F i g u r e 5- 8c and i s l a b e l l e d an a r e a o f i n s t a b i l i t y . I n r e a l c a s e s , f a i l u r e would p r o b a b l y o c c u r b e f o r e such a l a r g e d i s c r e p a n c y between the two l i n e s d e v e l o p e d . However, the e f f e c t o f the b u r i e d w e a t h e r i n g p r o f i l e has been c l e a r l y demonstrated. 74. Flatlying Weak Rocks, Fractures Due to Stress Relief and the Effects of a Regional Aquifer Matheson and Thomson (1973) have investigated the occurrence in central Alberta of valley rebound due to stress r e l i e f . The rebound was accompanied by interbed s l i p that would give rise to gouge zones in valley walls. Ferguson (1967, 1974) describes similar rebound phenomena in the Allegheny Plateau region where he found compression faults i n the valley bottoms and tension fractures in the valley walls. Patton and Hendron (1974) discuss the above phenomena and further point out the effect of high f l u i d pressures on valleys in groundwater discharge areas. Figure 5-9 shows three models which i l l u s t r a t e the groundwater regime for an area which exhibits some of these features. The basis for the models i s Figure 5 of Patton and Hendron (1974, p. 10). The models are of a valley cut into f l a t l y i n g sandstones and shales. The position of the water table below the ground surface has been assumed. For comparison between models, the pressure head d i s t r i - bution on a wedge in the valley wall i s shown. The isotropic homogeneous case is shown in Figure 5-9a. In 5- 9b the sandstone and shale units have been introduced with a conduc- t i v i t y contrast of two orders of magnitude. Rebound beneath the valley due to stress r e l i e f i s indicated by a slight upwarping of the units. Lower conductivity zones, parallel to bedding and somewhat exaggerated from the mylonitic zones described by Patton and Hendron (1974) or the gouge zones of Matheson and Thomson (1973) are seen in the valley walls. Beneath the valley floor are faults due to heaving. Clearly the flow 75. regime i s d e t r i m e n t a l to the s t a b i l i t y of the v a l l e y w a l l s , p a r t i c u l a r l y i n the c i r c l e d areas above the h o r i z o n t a l gouge zones which act to t r a i n the groundwater i n t o the v a l l e y . The groundwater regime i s even more d e t r i m e n t a l to s t a b i l i t y i n Figure 5-9c where an increased head has been assigned to the l e f t boundary of the t h i n n e r sandstone u n i t , g i v i n g i t the r o l e of a r e g i o n a l a q u i f e r . Beneath the v a l l e y , the i n c r e a s e of h y d r a u l i c head w i t h depth i s much grea t e r than i n the two previous f i g u r e s . Because of t h e i r o r i e n t a t i o n p erpendicular to the e q u i p o t e n t i a l s , the v e r t i c a l f a u l t s i n the base of the v a l l e y have l i t t l e e f f e c t on the flow system. However, the e f f e c t of the h o r i z o n t a l gouge zones i s pronounced. Comparison of the pressure head d i s t r i b u t i o n on the wedges from the three models c l e a r l y i l l u s t r a t e s the i n f l u e n c e on slope s t a b i l i t y that the v a r i o u s groundwater regimes would have. A l s o , note the non l i n e a r d i s t r i b u t i o n of pressure head on the wedge, d i f f e r e n t from the commonly assumed l i n e a r d i s t r i b u t i o n s . - • : P l e i s t o c e n e Terraces In many p a r t s of c e n t r a l B r i t i s h Columbia d i s t i n c t t e r r a c e s are found on the s i d e s of present-day r i v e r v a l l e y s . They are u s u a l l y a r e s u l t of g l a c i o f l u v i a l , g l a c i p l a c u s t r i n e , or a l l u v i a l processes and c o n s i s t of a v a r y i n g amount of c l a y , s i l t , sand and g r a v e l , depend- i n g on t h e i r o r i g i n . Good examples are found i n the Fr a s e r V a l l e y , 7 6 . Thomson Valley and throughout the Okanagan. If water i s available for i r r i g a t i o n , they provide i d e a l farming s i t e s . Roads and railways take advantage of the terraces when possible, but i n many places i t i s necessary for them to traverse a slope beneath a terrace. S t a b i l i t y problems i n the unconsolidated terrace sediments are f a i r l y common. Deere and Patton (1967) discuss the s t a b i l i t y of varved clay terraces. The models presented i n Figure 5-10 approximate an area adjacent to Trout Creek i n Summerland, Southern Okanagan, B.C., where a s l i d e was i n i t i a t e d about 60 years ago and i s s t i l l moving. This "per- petual landslide" has been studied i n d e t a i l by R i g l i n (1976). For Figure 5-10, pre-slide topography was estimated. Although the d e t a i l s of the geology are from the Trout Creek area, the intent i s to i l l u s t r a t e a geologic environment and associated groundwater flow system that i s simi- l a r to many places. The i s o t r o p i c , homogeneous case i s shown i n Figure 5-10a. Simplified versions of the geology are shown i n Figures 5-10b and 5-10c. Volcanic rocks (Unit A) o v e r l i e less conductive granodiorite (Unit B). The terrace sediments consist of a lowermost group of Tertiary sediments (Unit C), a f l a t l y i n g layer of less conductive t i l l or s i l t (Unit D), and an upper zone of sands and gravels (Unit E). In Figure 5-10c the volcanics have been assigned an anisotropic conductivity. In Figure 5-10d, Units C and D have also been given aniso- tropic conductivities, the water table i n the sands and gravels has been lowered to a more r e a l i s t i c l e v e l , and a less conductive unit has been added representing the possible effects of a buried weathering p r o f i l e on the top of the granodiorite. 77. This same case i s shown i n Figure 5-10e except that i r r i g a t i o n of a s e c t i o n of the t e r r a c e has been simulated by r a i s i n g the water t a b l e back to the ground s u r f a c e . In Figure 5-10f the u n i t r e p r e s e n t i n g the weathering p r o f i l e has been assigned a c o n d u c t i v i t y higher than the surrounding r o c k s , perhaps r e p r e s e n t i n g a grus developed on the grano- d i o r i t e . I n the same f i g u r e , p a r t of U n i t C, the T e r t i a r y sediments have a l s o been given a higher c o n d u c t i v i t y . To compare the e f f e c t s of the d i f f e r e n t flow systems, an a r b i t r a r y , i n i t i a l s l i d e s u r f a c e was drawn, p i e z o m e t r i c l i n e s e s t a - b l i s h e d , and s t a b i l i t y analyses undertaken f o r the models shown i n Figures 5-10a, 5-10c, 5-10d and 5-10f. No attempt was made to f i n d the most c r i t i c a l s l i d e s urface although the f a i l u r e s u rface used i s s i m i l a r i n form to the s l i d e s urface that has developed at the Trout Creek l o c a l i t y . Figures 5-10b and 5-10e are e x a c t l y the same, i n the area of the s l i d e , as 5-10c and 5-10f ' r e s p e c t i v e l y , rendering analyses unnecessary. The technique of a n a l y s i s used i s a s i m p l i f i e d l i m i t e q u i - l i b r i u m method described by Patton and Hendron (1974). The s l i d e i s broken i n t o s l i c e s and f o r each s l i c e shearing f o r c e s p a r a l l e l to the b?.se and r e s i s t i n g "forces p a r a l l e l to the base are c a l c u l a t e d . For the e n t i r e s l i d e , r e s i s t i n g f o r c e s are summed and compared t o shearing f o r c e s to c a l c u l a t e a f a c t o r of s a f e t y . I n t e r s l i c e f o r c e s are con- s i d e r e d to sum to zero and moments are not considered. The method i s appropriate f o r a f i r s t estimate of the s t a b i l i t y . The absolute values of the f a c t o r s of s a f e t y may not be exact but the r e l a t i v e values are u s e f u l f o r comparison. 78. In c a l c u l a t i n g f a c t o r s of s a f e t y , i t i s f i r s t necessary to decide on a base model, a s s i g n i t a f a c t o r of s a f e t y of one and back- c a l c u l a t e f o r the angle of i n t e r n a l f r i c t i o n , p h i . This p h i value i s then used i n the remaining f a c t o r of s a f e t y c a l c u l a t i o n s . Another way of comparison i s to c a l c u l a t e i n each case the p h i r e q u i r e d f o r a f a c t o r of s a f e t y of one. These p h i values can then be compared to each other, to l a b o r a t o r y data, or to values reported i n the l i t e r a t u r e . In t h i s way, both the v a l i d i t y of the model can be checked and the s i g n i f i c a n c e of the d i f f e r e n t flow systems can be judged. R e s u l t s of the c a l c u l a t i o n s are given i n Table 5-2. For the lower water t a b l e shown i n Figures 5-10e and 5-10f, a p h i value of only 10.9 - 11.3° i s r e q u i r e d , w h i l e the s a t u r a t e d cases of Figures 5-10a and 5-10c r e q u i r e p h i values of 15.5 - 17.3° f o r s t a b i l i t y . Lab t e s t i n g by R i g l i n (1976) of the remoulded m a t e r i a l from the s l i d e plane of Trout Creek r e s u l t e d i n a p h i value of 18°. This value i n d i c a t e s that a more c r i t i c a l s l i d e surface probably e x i s t s f o r which a higher p h i value i s re q u i r e d f o r s t a b i l i t y than suggested by my c a l c u l a t i o n s . N e v e r t h e l e s s , 18° i s c l o s e and w i t h changes i n the flow system causing a 55% range i n th-s f a c t o r of s a f e t y (.89 to 1.44), i t i s reasonable to conclude that changes i n the pressure head due to flow system v a r i a t i o n s could cause the t e r r a c e to f a i l . Many s i m i l a r s i t u a t i o n s probably e x i s t i n B r i t i s h Columbia and other p a r t s of the world. In a l l cases, the nature of the r e g i o n a l flow system and the p o s i t i o n of the t e r r a c e i n the r e g i o n a l system are c r i t i - c a l to understanding and improving s t a b i l i t y . 79. F a c t o r o f S a f e t y PHI f o r F a c t o r Case f o r PHI = 1 5 . 5 ° o f S a f e t y = 1 . 0 F i g u r e 5-10a 1.0 15.5° F i g u r e 5--10c .89 17.3° F i g u r e 5-10d 1.39 11.3° F i g u r e 5-10f 1.44 10.9° T a b l e 5-2. R e s u l t s o f S t a b i l i t y A n a l y s i s o f a P l e i s t o c e n e T e r r a c e ( F i g u r e 5-10). S l i c e geometry i s shown i n the f i g u r e . A s p e c i f i c weight o f 130 pounds per c u b i c f o o t was used. Deformed Metamorphic Rocks and t h e E f f e c t o f a R e s e r v o i r on a Deep Rock S l i d e The h y d r a u l i c c h a r a c t e r i s t i c s o f metamorphic r o c k s have n o t , t o my knowledge, been s t u d i e d i n g r e a t d e t a i l . P o l y p h a s e d e f o r m a t i o n can cause complex f o l d p a t t e r n s and d i f f e r e n t l i t h o l o g i e s w i t h v e r y d i f f e r e n t c o n d u c t i v i t i e s may be a r r a n g e d i n p a t t e r n s o n l y u n d e r s t a n d a b l e a f t e r d e t a i l e d l o c a l and r e g i o n a l mapping. I n some cases i t may not be p o s s i b l e t o r e a s o n a b l y model a l l the c o m p l e x i t i e s . On the o t h e r hand, metamorphic r o c k s o f t e n e x h i b i t a s u r p r i s i n g l y c o n s i s t e n t f o l i a t i o n d e f i n e d by m i c r o f a u l t s o r f r a c t u r e s , p r e f e r r e d o r i e n t a t i o n o f i n e q u a n t m i n e r a l s , l a m i n a r m i n e r a l a g g r e g a t e s o r a c o m b i n a t i o n o f t h e s e m i c r o - s t r u c t u r e s (Hobbs e t a l , 1976). In many cases the f o l i a t i o n i s i n c l i n e d t o o r i g i n a l b e d d i n g . Such a c o n s i s t e n t f o l i a t i o n may be much more c r i t i c a l t o t h e groundwater regime than l i t h o l o g i c d i f f e r e n c e s and i t s 80. p r e s e n c e may make p o s s i b l e r e a s o n a b l e m o d e l l i n g o f a s e e m i n g l y complex r e g i o n . The models shown i n F i g u r e 5-11 a r e i n t e n d e d to i l l u s t r a t e a metamorphic t e r r a i n w i t h a c o n s i s t e n t f o l i a t i o n t h a t has i t s e l f been g e n t l y f o l d e d i n a b road a n t i f o r m . The groundwater regime of the d i f - f e r e n t models w i l l be compared by c o n s i d e r i n g the p i e z o m e t r i c l i n e r e p r e s e n t i n g the p r e s s u r e head at the base o f a d e e p - s e a t e d p o t e n t i a l r o c k s l i d e on t h e d i p s l o p e of the v a l l e y . Each d i f f e r e n t g e o l o g i c s i t u a t i o n i s shown w i t h and w i t h o u t a r e s e r v o i r c o v e r i n g a s m a l l p o r t i o n o f the t o e . Each f i g u r e i l l u s t r a t i n g the r e s e r v o i r case a l s o i n c l u d e s the p i e z o m e t r i c l i n e from the n o n - r e s e r v o i r c a s e . F i g u r e 5-12 i s a c o m p i l a t i o n o f a l l the p i e z o m e t r i c l i n e s . F i g u r e 5 - l l a i s t h e homogeneous i s o t r o p i c c a s e w i t h no r e s e r - v o i r , w h i l e F i g u r e 5 - l l b i s the m a tching r e s e r v o i r c a s e . I n F i g u r e 5- 11c an a n i s o t r o p y has been i n t r o d u c e d to r e p r e s e n t the f o l i a t i o n . D a v i s (1969) n o t e s t h a t r e l i a b l e f i e l d measurement o f a n i s o t r o p y i n dense r o c k s i s v i r t u a l l y l a c k i n g but goes on to d e s c r i b e a case where p e r - m e a b i l i t y was a p p r o x i m a t e l y t w i c e as l a r g e i n the d i r e c t i o n o f s c h i s t o - c i t y as i t was p e r p e n d i c u l a r . L a c k i n g b e t t e r i n f o r m a t i o n , tb:is a n i s o t r o p y was used. The i n t r o d u c t i o n of even t h i s s l i g h t a n i s o t r o p y causes a n o t i c e a b l e r i s e i n the p i e z o m e t r i c l i n e . The e f f e c t o f the r e s e r v o i r , F i g u r e 5 - l l d , i s more pronounced than i n the p r e v i o u s p a i r . I n F i g u r e 5 - l l e and 5 - l l f a more c o n d u c t i v e zone has been i n t r o d u c e d n e a r the s u r f a c e , r e p r e s e n t i n g a h i g h e r f r a c t u r e f r e q u e n c y due t o s t r e s s r e l i e f and w e a t h e r i n g . I t has a m o d e r a t i n g i n f l u e n c e on the p i e z o m e t r i c l i n e . 81. A through-going, less permeable zone i s seen i n Figure 5 - l l g . Compared to the surrounding rock., conductivity across the f o l i a t i o n has been reduced by ten, while along the f o l i a t i o n only by four. This zone represents a through-going, more schistose layer, perhaps a thrust surface, that could provide the necessary s l i d e plane for i n s t a b i l i t y to occur. The re s u l t i n g r i s e i n piezometric surface i n both 5 - l l g and 5 - l l h , the matching reservoir case i s dramatic. - To show the s e n s i t i v i t y of the groundwater flow system to conductivity contrasts, another model was run l i k e 5 - l l g but with a reduction by ten of the conductivity i n the schistose zone. I t i s shown i n Figure 5 - l l i . The re s u l t i n g piezometric l i n e i s almost twice the height of the previous p a i r , above the s l i d e plane. The thrust surface and the more conductive zone representing fractures and weathering near the ground surface are combined i n Figures 5 - l l j and 5 - l l k . Again, the more conductive surface zone has a moderating influence on the piezometric l i n e . A zone three times higher i n conductivity than the country rock has been added beneath the thrust surface i n Figures 5-111 and 5-llm. Such a zone may be due to subsidiary fracturing or perhaps a di f f e r e n t l i t h o l o g y . By comparing the piezometric l i n e of 5 - l l J i to that of 5 - l l j (non-reservoir cases) and 5-llm to that of 5 - l l k (reservoir cases), on Figure 5-12 i t can be seen that the confined, more conductive zone causes a small but measurable r i s e i n the piezometric .surface. 82. To f u r t h e r i l l u s t r a t e the e f f e c t o f the d i f f e r e n t groundwater f l o w regimes on the p o t e n t i a l r o c k s l i d e , a s t a b i l i t y a n a l y s i s was c a r r i e d out f o r each c a s e . The p r o c e d u r e s o u t l i n e d i n the p r e v i o u s s e c t i o n were used. The r e s u l t s a r e summarized i n T a b l e 5-3. Case F a c t o r o f S a f e t y f o r PHI •= 31.9° PHI f o r F a c t o r o f S a f e t y = 1.0 5 - l l a 1.0 31.9 5 - l l b .99 . 32.1 5 - l l c .97 32.8 5 - l i d .94 33.6 5 - l l e 1.04 30.9 5 - l l f 1.02 31.4 5 - l l g .79 38.1 5 - l l h .74 40.0 ( 5 - l l i ) (.26) (67.3) 5 - l l j .88 35.3 5 - l l k .85 36.1 5 - l L c .78 38.7 5 - l l m .76 39.2 T a b l e 5-3. R e s u l t s of s t a b i l i t y a n a l y s i s of a p o t e n t i a l r o c k s l i d e i n metamorphic t e r r a i n . S l i c e geometry i s shown i n F i g u r e 5-12. A s p e c i f i c weight of 170 pounds p e r c u b i c f o o t was used. 83. With the exception of 5 - l l i , f a c t o r s of s a f e t y vary from .74 to 1.04, a range of 33%. For the same models, c o n d u c t i v i t y c o n t r a s t s were l e s s than two orders of magnitude. These c o n t r a s t s were chosen to be as r e a l i s t i c as p o s s i b l e based on data i n the l i t e r a t u r e and my l i m i t e d experience. The model shown i n Figure 5 - l l i has a c o n d u c t i v i t y contrast of about three orders of magnitude. The r e s u l t s of the s t a b i - l i t y a n a l y s i s f o r t h i s case are shown bracketed i n Table 5-3 because the s i t u a t i o n i s completely u n r e a l i s t i c , i n order f o r that slope to stand, the s p e c i f i c weight of the s l i d e m a t e r i a l would have to be w e l l over 300 pounds per cubic f o o t . C o n d u c t i v i t y c o n t r a s t s of at l e a s t twelve orders of magnitude e x i s t i n the r e a l w o r l d but i t can be concluded from the above c a l c u - l a t i o n s that v a r i a t i o n s i n geology causing c o n d u c t i v i t y c o n t r a s t s of only two orders of magnitude can have a c r i t i c a l e f f e c t on the ground- water flow regime of a s l o p e . A f a u l t gouge or g e o l o g i c u n i t need not be impermeable or even of exceedingly l e s s c o n d u c t i v i t y than the s u r - rounding m a t e r i a l to adversely a f f e c t the flow regime i n such a way as to cause i n s t a b i l i t y . Two important p r a c t i c a l p o i n t s are demonstrated i n these models. F i r s t l y , they show that to get u s e f u l and accurate f i e l d measurements of pressure head i n a slope i n v e s t i g a t i o n , i t i s c r i t i c a l t h at the piezometers penetrate through any l e s s conductive u n i t t h a t might be a c t i n g as a s l i d e plane. The second p o i n t i s that i f a l e s s per- meable zone u n d e r l i e s a s l i d e , f o r drainage to be an e f f e c t i v e remedial measure, d r a i n holes must penetrate beneath t h i s l e s s permeable zone. 84. In a l l reservoir cases i l l u s t r a t e d i n Figure 5-11, the effects of the ponded water are f e l t i n the v a l l e y slope w e l l above the l e v e l of the water surface. As small as the reservoir i s , i t causes a reduction i n the factor of safety of from 1% to 5%. Although t h i s reduction may seem i n s i g n i f i c a n t , i f a slope i s already close to f a i l u r e , the change i n groundwater regime caused by the introduction of even a small reser- v o i r may be. a l l that i s required to cause i n s t a b i l i t y . However, accord- ing to Patton (Personal communication, 1976) the detrimental effect on s t a b i l i t y shown by my modelling, i s not general and cases can be found for which the introduction of a reserovir causes an increase i n a slope's s t a b i l i t y . The phi's calculated for a factor of safety of one should be considered to establish the reasonableness of the models. The strength parameters compiled by Deere and Patton (1971, p. 142) include a number of phi values for gneisses, s c h i s t s , p h y l l i t e and granites, weathered to varying degrees, that bracket the 30.9° - 40° range l i s t e d i n Table 5-3. From th i s comparison the models appear r e a l i s t i c . 85. C H A P T E R S I X : SUMMARY A N D C O N C L U S I O N S Slope S t a b i l i t y Analyses 1. There are three d i s t i n c t aspects i n the a n a l y s i s of the s t a b i - l i t y of a s l o p e : i d e n t i f y i n g a l l the s t r e s s e s a c t i n g , understanding the mechanics which r e l a t e the s t r e s s e s , and determining the n a t u r a l proper- t i e s which govern the m a t e r i a l behaviour when, the s t r e s s e s are a c t i n g . 2. The three c a t e g o r i e s of a n a l y s i s , l i m i t e q u i l i b r i u m , e l a s t i c - f i n i t e element, and Cundall's d i s c r e t e p a r t i c l e method each have t h e i r own advantages and l i m i t a t i o n s . For any given s i t u a t i o n , l o c a l c o n d i - t i o n s and the purpose of the a n a l y s i s w i l l govern the choice of category. 3. L i m i t e q u i l i b r i u m techniques have reached the stage where a l l the laws of s t a t i c e q u i l i b r i u m can be s a t i s f i e d . However, the r e s u l t i n g equations are indeterminate without some s i m p l i f y i n g assumption. D i f - f e r e n t l i m i t e q u i l i b r i u m methods u t i l i z e d i f f e r e n t assumptions. 4. Terzaghi's e m p i r i c a l e f f e c t i v e s t r e s s law i s v a l i d f o r s a t u - rated s o i l s but, i n some cases of saturate d rock, Skempton's modified form may be more ap p r o p r i a t e . 86. 5. To undertake a slope s t a b i l i t y analysis i n terms of eff e c t i v e stress, pore pressures must be known. In practice, these are approximated as a percentage of overburden pres- sure, approximated with an assumed c o e f f i c i e n t , obtained by f i e l d measure- ment or obtained from mathematical models. 6. D i f f i c u l t i e s remain i n both accurate f i e l d measurement and mathematical modelling of flow systems i n complex geological environ- ments. The best understanding of the pore pressure d i s t r i b u t i o n i n a slope w i l l be obtained i f f i e l d measurement and modelling are used together. Use of the pore pressure c o e f f i c i e n t or the assumption of pore pressures as a percentage of overburden pressure are no longer necessary methods of approximation. Limiting Assumptions The following assumptions underlie the theory used to produce the mathematical models developed i n t h i s project: 1. The models are f u l l y saturated with the top flow boundary corresponding to the water table. 2. The position of the water table must be known and i n most cases i s chosen as the ground surface. 87. 3. The models are steady-state. 4. The models are two-dimensional. 5. The model boundaries are either a specified constant head (D i r i c h l e t condition) or constant f l u x (Neuman condition). . 6. . The continuous porous media i s r i g i d . 7. Darcy's Law accurately describes the groundwater flow. 8. The geologic case being modelled can be reasonably approxi- mated by an equivalent continuous porous media. Model Results 1. The models developed i n t h i s study serve to rigorously v e r i f y many of the possible implications of d i f f e r e n t flow systems on slope s t a b i l i t y , p a r t i c u l a r l y those previously suggested by Deere and Patton (1967, 1971) and Patton and Hendron (1974). 2. Anisotropic hydraulic c h a r a c t e r i s t i c s are common i n both s o i l and rock. The models completed for t h i s study suggest that for slope s t a b i l i t y the worst case i s when the p r i n c i p a l axis of hydraulic conduc- t i v i t y dips down the slope, while the most stable case i s when the p r i n c i p a l axis i s horizontal. 88. 3. Thick interbedded sedimentary units that have been folded or thrust up, eroded, and now stand as ridges and valleys often have as- sociated complex flow systems. As i n the simple anisotropic case, the flow system has a maximum adverse effect on the dip slope. 4. Faults, contacts, and dykes can cause a variety of effects on the groundwater flow system. Careful f i e l d investiagtion i s required to establish the significance of any one of these features. 5. Weathering p r o f i l e s can commonly result i n a less conductive zone confining another with the r e s u l t i n g flow system extremely d e t r i - mental to s t a b i l i t y . 6. Layers of colluvium or old landslide debris, p a r t i c u l a r l y i f buried weathering p r o f i l e s are present, can cause complex groundwater flow patterns. Less permeable units are often present and can act as confining layers causing zones of high pore pressures which can lead to i n s t a b i l i t y . 7 . Stress r e l i e f fractures on v a l l e y w a l l s , i f accompanied by the production of gouge can adversely influence the effect of the ground- water flow regime on s t a b i l i t y . 8. The presence of a regional aquifer beneath a v a l l e y can cause anomalously high pore pressure development which can have a detrimental effect on the s t a b i l i t y of v a l l e y walls. 89. 9. The pressure head d i s t r i b u t i o n on rock wedges can be non- lin e a r and quite different from the commonly assumed l i n e a r d i s t r i b u t i o n . 10. The s t a b i l i t y of the Pleistocene terraces found throughout B r i t i s h Columbia i s i n part dependent on the regional groundwater flow system i n which the terraces are located. Fluctuations i n the ground- water regime causing anomalous increases i n the water table could cause a terrace to f a i l . l l e Modelling of flow systems i n highly deformed metamorphic rocks may not be possible because of the complexities of the geology. However, i n many cases a through-going f o l i a t i o n e xists which may control the flow system and allow reasonable modelling. 12. A more permeable zone near the surface, due. to a higher f r a c - ture frequency or weathering, has a moderating influence on the ground- water flow regime from a slope s t a b i l i t y perspective. 13. The presence of a l i t h o l o g i c unit or thrust surface of lower conductivity than the surrounding rocks can have a major effect on the flow system. Conductivity contrasts of less than two orders of magni- tude can cause pore pressure developments c r i t i c a l to s t a b i l i t y . 90. 14. The i n t r o d u c t i o n of a r e s e r v o i r at the toe of the slo p e can i n f l u e n c e the groundwater regime w e l l up the slo p e from the r e s e r v o i r s u r f a c e . I f a slope was already c l o s e to being u n s t a b l e , even a low r e s e r v o i r could provide the change r e q u i r e d to cause i n s t a b i l i t y . 15. Piezometers should penetrate through any l e s s conductive u n i t that might be a c t i n g as a s l i d e plane. For drainage to be an e f f e c t i v e remedial measure, d r a i n h o l e s must a l s o penetrate beneath such a l e s s . permeable zone. 16. A l l of the r e s u l t s of the mod e l l i n g i l l u s t r a t e the b a s i c theme of t h i s t h e s i s : to understand the flo w system, and t h e r e f o r e pore pres- sures, that are fundamental to the s t a b i l i t y a n a l y s i s of a s l o p e , i t i s necessary to understand the r e g i o n a l flow system i n which the slope i s l o c a t e d . This understanding can only be obtained w i t h a thorough know- ledge of the r e g i o n a l geology. 91. REFERENCES Bear, J . , 1972. Dynamics of F l u i d s i n Porous Media, American E l s e v i e r P ubl. Co. Inc., New York, 764 p. B e l l , J.M., 1968. General slope s t a b i l i t y a n a l y s i s , Jour. S o i l Mech. and Fdn. Div., A.S.C.E., V. 94, SM 6, 1253-1270. B i r d , C., 1975. UBC M a t r i x , A Guide to S o l v i n g M a t r i x Problems, U n i v e r s i t y of B r i t i s h Columbia Computing Centre, Vancouver, Canada, 115 p. Bishop, A.W., 1955. 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West Vancouver, B.C. 99 A P P E N D I X I C O M P U T E R PROGRAM APPENDIX I : COMPUTER PROGRAM 100 Language-: F o r t r a n IV Purpose: To s o l v e , u s i n g the f i n i t e element method, the steady- s t a t e equation of flow f o r two-dimensional, s a t u r a t e d , heterogeneous, a n i s o t r o p i c porous media. L i s t i n g : A complete l i s t i n g of the program, sample output, and sample data f i l e , i s found at the end of t h i s appendix. Flow Chart: See Figure 1-1. Subroutines: 1. NOTRAN Purpose: To c a l c u l a t e c o n t r i b u t i o n s to nodal equa- t i o n s from elements not r e q u i r i n g a c o o o r d i - nate t r a n s f o r m a t i o n . A v a i l a b i l i t y : S e l f - c o n t a i n e d . 2. TRANS Purpose: To c a l c u l a t e c o n t r i b u t i o n s to nodal equa- t i o n s from elements r e q u i r i n g a coordinate t r a n s f o r m a t i o n . A f t e r t r a n s f o r m a t i o n , a p p r o p r i a t e c o n t r i b u t i o n s are c a l c u l a t e d . A v a i l a b i l i t y : S e l f - c o n t a i n e d . 101 3. S L I M P Purpose: To s o l v e and improve i t e r a t i v e l y a system of l i n e a r equations of the form [A] • {x} = {B} A v a i l a b i l i t y : General l i b r a r y , Computing Centre, U n i v e r s i t y of B r i t i s h Columbia. 4. GRAPH 1 Purpose: To set up data a p p r o p r i a t e l y and c a l l SCATCN f o r p l o t t i n g . A v a i l a b i l i t y : S e l f - c o n t a i n e d . 5. GRAPH 2 Purpose: To set up a data f i l e f o r MPLOT A v a i l a b i l i t y : S e l f - c o n t a i n e d . 6. SCATCN Purpose: To produce a contour map from a set of s c a t t e r e d data p o i n t s . A v a i l a b i l i t y : General l i b r a r y , Computing Centre, U n i v e r s i t y of B r i t i s h Columbia. 7. • MPLOT Purpose: To produce an e q u i p o t e n t i a l map from the r e s u l t s of a f i n i t e element flow program. A v a i l a b i l i t y : Department of C i v i l E n g i n e e r i n g , Univer- s i t y of B r i t i s h Columbia. 102 Input: Parameter N0DT0T N LTOT MCON MPLOT IND NO NIB SIZE SCA • D e s c r i p t i o n T o t a l number of nodes T o t a l number of unknown p o t e n t i a l values T o t a l number of elements Number of contours to be drawn I f MPLOT = 0, no p l o t i s generated I f MPLOT = 1, GRAPH 1 i s c a l l e d and a p l o t generated w i t h SCATCN I f MPLOT = 2, GRAPH 2 i s , c a l l e d and a p l o t generated w i t h MPLOT Used by SCATCN. I n d i c a t e s type of p l o t d e s i r e d . I f IND = 0, s c a t t e r e d data p o i n t s w i l l be p l o t t e d i f there are l e s s than 25. I f IND = 1, no data p o i n t s w i l l be p l o t t e d , and i f IND = 2, a l l data p o i n t s w i l l be p l o t t e d . Number of boundary nodes Number of the f i r s t boundary node Si z e i n inches of the f i n a l p l o t made by SCATCN Scale f a c t o r used by MPLOT so y ^ 9 J max 103 Parameter LDAT ( I , J ) XC O ( I ) , YCO(I) PERMX(NEL), PERMY(NEL) Q(NEL) EL(NEL) THETA(NEL) CN(I) D e s c r i p t i o n Element d a t a . A 2-D a r r a y c o n t a i n i n g the i d e n t i t y o f the v e r t i c e s o f the t r i a n g u l a r e l e m e n t s . F o r the j t h element, LDAT (1, J ) c o n t a i n s the n o d a l number of the f i r s t c o r n e r LDAT (2, J ) c o n t a i n s the n o d a l number o f the second c o r n e r LDAT (3, J) c o n t a i n s the n o d a l number o f the t h i r d c o r n e r O n e - d i m e n s i o n a l a r r a y s c o n t a i n i n g the x ( h o r i z o n t a l ) and y ( v e r t i c a l ) c o o r d i n a t e o f th e I nodes. O n e - d i m e n s i o n a l a r r a y s c o n t a i n i n g the x ( h o r i z o n t a l ) and y ( v e r t i c a l ) h y d r a u l i c con- d u c t i v i t i e s f o r each element. O n e - d i m e n s i o n a l a r r a y c o n t a i n i n g the f l u x i n t o each element p e r u n i t l e n g t h o f element boundary. O n e - d i m e n s i o n a l a r r a y c o n t a i n i n g t h e l e n g t h o f element boundary t h r o u g h which Q(NEL) f l o w s . O n e - d i m e n s i o n a l a r r a y c o n t a i n i n g the a n g l e which the p r i n c i p a l a x i s o f h y d r a u l i c conduc- t i v i t y f o r any g i v e n element i s skewed from the g l o b a l c o o r d i n a t e s system. O n e - d i m e n s i o n a l a r r a y c o n t a i n i n g the v a l u e s of t h e MCON c o n t o u r s t o be drawn. 1 COMMON LDAT !3t500},XCC(275),YCO(275), PHI (275) , G{ 275)» 2 IT(275,2 75),X(275),B(275),I PERM(550),RZ(275),AR(500),C(275), 3 2P(275,275),0(500),EL(500),F(500),THETA(500),ALPHA(275),XC0T(275) 4 3YC0T(2 75),RT{275),PERNX(500),PERMY(500),BX(5 00 ),BY(500),CN{50), 5 40ATA(3,275 ) ,TITLE(20),JCCPCl),I,J,K,NEL,NODTOT,N,NP1,MCON, 6 ~t 5SIZE,IND,NG,N1B,SCA,LTOT i 8 L READ(5,99) TITLE 9 99 F0RMAT(20A4) 10 READ(5,1) NODTOT,N,LTOT,MCON,MPLOT,IND,NO,NIB,SIZ5,SCA 11 1 F0RMAT(8I6,2F10„1) 12 READ(5,2)((LDAT(I ,J),1=1,3),J = 1,LT0T) 13 2 F0RMATU8I4) 14 READ(5,4)(YCO II),XCOlI),1=1,NODTOT) 15 4 FORMAT(10F8.1) 16 READ( 5, 8 ) { PERMX(NEL) ,PER MY (NEL) »NEL=i , LTOT ) 17 8 FORMAT(8F10.6) 18 READ(5,18)(Q(NEL),NEL=1,LT0T) 19 18 F0RMAT113F6.1J 20 READ (5,29) (EL(NSL) ,NEL=1,LTOT> 21 29 FORMAT(10F8o2) 22 READ(5,2 0)(THETA(NEL),NEL = 1,LTOT) 23 20 F0RMAT(16F5ol) 24 READ(5,80) (CN(.I), I = 1,MC0N) 25 80 FORMAT ( 10F8, 1) 26 C 27 C ••COMPLETION OF DATA READ IN,WRITE OUT INPUT** 28 C 29 WRITE(6,98) TITLE 30 98 FORMAT(1H1,20A4) 3i WRITE (6, 19) NODTOT., LTOT, N 32 19 FORMAT(1HO,'NODE TOTAL IS» , I 5,5X, •ELEMENT TOTAL IS', 33 115,;X,'UNKNOWNS•,15) 34 W fi I T S { 6, 7) 35 7 FORMAT(1H0,3X,'ELEMENT NO.»,4X,•AREA«,11X, * PERMX',5X, • PERMY',9X, 36 11THETA',8X »'Q• ,9X,•£L»,1 OX , »I' , 5X , • J • , 5X,»K » / / ) 37 C 33 r **SET TOP FLOW BOUNDARY TO THE GROUND SURFACE** 39 c 40 NP1=N+1 41 00 6 J = NPl " f NQDT0T 42 P H I ( J ) = Y C 0 { J ) 43 6 CONTINUE 44 C 45 C • • I N I T I A L I Z E PARAMETERS** 46 C 47 DO 10 1=1,275 43 DO 10 J = l , 2 7 5 49 P ( I , J ) = 0 . 0 0 0 50 T I I , J ) = 0 . 0 0 0 51 10 CONTINUE 52 DO 5 J = l , 2 7 5 5 3 X{J J = Oo 000 54 B ( J ) = 0 . 0 0 0 55 RZC J ) = 0 o 000 56 5 CONTINUE 57 DO. 3 J = l , 5 5 0 53 I P E R M ( J ) = 0 . 0 0 0 59 3 CONTINUE 60 DO 9 1=1,500 61 F ( I ) =OoO 62 9 CONTINUE 63 C 64 C * *CONVERT COORDINATE ANGLE TO RADIANS** 65 . C 66 DO 31 N £ L = 1 , L T 0 T 67 T H E T A ( N E L ) = ( T H E T A ( N E L ) / 3 6 0 . 0 ) * 2 . 0 * 3 . 1 4 1 5 9 68 31 CONTINUE 69 C 70 C * * C A L C U L A T E MATRIX C O E F F I C I E N T S * * 71 C 72 DO 11 NE L=1 ,LTOT 73 I = L D A T ( 1 , N E L ) 74 J = LDAT ( 2 » N E D 75 K = L D A T ( 3 , N E D • 76 c 77 A R ( N E L ) = ( { ( X C O { J ) * Y C O ( K ) ) - { X C Q { K ) * Y C Q ( J ) i i-t{XCOU>*YCO(K) 78 1 - { X C 0 ( K ) * Y C 0 ( ! > ) ) + ( U C O ( I ) * Y C O ( J ) J - i X C • ( J J * Y C O ( I ) ) ) ) * 0 . 5 79 IF(AR(NEL)«LT.O.O) AR { NEL) = -AR (NEL) 80 C 81 WRITE(6, 51) NEL,AR(NEL),PERMX(NEL),PfcRMY{NEL),THETA(NEL), 82 IQ(NEL),EL(NEL),I,J,K 83 51 FORMAT(1H ,5X,I 5,4X,F10o1,5X,2F10o6,5X,F7a4,5X,F6o2,5X,F6.2, 84 15X.3I6) 85 C 86 BX(NEL)=PERMX(NEL)/(4»0*AR(NEL)) ' 87 BY(NEL) = PERMY(NEL)/14«>0*AR(NEL) ) 88 C 89 C **CALL APPROPRIATE SUBROUTINE** 90 C 91 IF(THETA(NELJ.EQoOoO) GO TO 42 92 CALL TRANS 93 GO TO 11 94 42 CALL NQTRAN 95 C 96 11 CONTINUE 97 C 98 C **SET SUBROUTINE PARAMETERS AND CALL SLIMP** 99 C 100 M=275 101 NDIMAT=M 102 ITMAX=14 103 EPS=5oE-3 104 NRHS=1 105 C 106 DO 14 1 = 1,N 107 DO 14 J=NP1,NODTOT 108 PII,J)=P(I,J)*PHI(J) 109 BlI) = B( I)-P( I,J)+F(I) 110 14 CONTINUE 111 C 112 DO 12 1=1,N 113 DO 12 J = NP 1,NODTOT 114 P(I,J)=0e000 115 12 CONTINUE 116 DO 39 I=NP1,NODTOT 117 DO 3 9 J = l , NODTOT •p(I,.J)=0.000 J 39 CONTINUE ' " ° C CALL S L I « P . P - T , B , X , R Z , I P E R H . H , N 0 I H 4 T . £ P S . N R H S , n M A X > 121 122 C 123 C 124 C 125 126 22 ••WRITE OUT RESULTS** WRITEC6.22 ) FORMAT(1H1,6X,«N0DE N O . ' » 1 OX» 'XCQORD*,9X. ' YCOORD', 15X ,»PH I«// ) 127 DO 16 1=1,N 128 WRITE(6,15) I, XCQ{ I >, YCOU ), X (1) 129 15 FORMAT!iH ,5X,I 5 , 10X ,F10 .1 ,5X ,F10 .1 ,1 OX,F10•2) 130 16 CONTINUE 131 DO 17 I=NP1,N0DT0T WRITE(6,15} I , XCO( I)tYCO ( I ) f PH I { I ) 1 4 i 144 145 146 147 1 3 2 1 7 CONTINUE 133 134 C 135 C ••CA-LL PLOTTER IF REQUIRED** 136 C 137 IF(MPLOToEQ.O) GO TO 21 138 IF(MPL0ToEGo2) GO TO 23 139 CALL GRAPH1 140 GO TO 21 141 23 CALL GRAPH2 142 21 CONTINUE STOP END SUBROUTINE NOTRAN COMMON LDAT(3 i 500 ) ,XC0<275 ) fYC0 (275> tPH l<275 ) ,G (275 ) , IT( 27 5,2 75) , X t 2 7 5 ) , B ( 2 75 ) , I PERM{53 0 ) , R Z ( 2 7 5 ) , A R 1 5 0 0 ) , C ( 2 7 5 ) , 148 2P ( 275,275} »Ql 500) , EL( 500) ,F (500) , TH'cT A { 500 ) , ALPHA {2 75) ,XC0T{275 ), 149 3YC0T(275),RT(275),PERMX{500),PERMY( 500),3X( 500 ) , 8Y< 500) ,CNC50) . 150 4DATA(3,2 75) »T ITLE(20) , JCCP ( l ) , I , J ,K ,NEL ,NJQTOT,N,NP1 ,MCON, 151 5S IZE, IND,NO, NIB,SCA.LTOT 152 C 153 C •••••CALCULATION OF MATRIX COEFFICIENTS FOR ELEMENTS NOT 154 C REQUIRING COORDINATE TRANSFORMATION***** M 155 C . o I S A P {I , I ) = P( I , I ).+ { 8 X ( NEL ) * H YCO{ J ) -YCO(K) )**2 ) + 157 1BY(N5U*( ( X C O ( K ) - X C O < J ) )**2) ) 158 P ( I f J ) = P « I , J ) + { B X ( N E L ) * ( ( Y C O ( J ) - Y C Q ( K ) ) * C Y C O ( K ) - Y C O ( I ) ) ) 159 1+BY(NEL)*(IXCO(K}-XCO(J))*(XCO{I)-XCO(K)})) 160 P(I,K ) = P(I,K ) + ( B X(NEL)*{(YCO{J)-YCO(K) J *CY CO(I)-YCO(J))) 161 1+BY(NEL)*( {XCO ( K )-XCO { J ) ) * ( XCO ( J } - XCG ( 1) ) ) ) 162 C 163 P C J , I ) = P ( J , I ) + I B X ( N E L ) * C I Y C 0 1 K ) - Y C 0 ( I ) ) * { Y C O ( J ) - Y C O ( K ) ) ) 164 • 1 + BY(NEL)*{ (XCO{ I)-XCO(K))*{XCOIK}-XCG I J ) ) ) 3 165 P ( J r J ) = P ( J ,J)+(BX(NEL)*((YCOIK)-YCO{I ) )**2)+ 166 18Y(NEL)*llXC0m-XC0(K))**2)} 167 P(J,K)=P( J,K)MBX(NEL)*UYCQ(K}-YCG(I ) )*(YCO{ I)-YCO(J) ) } 168 1 + B Y(NEL)*( (XCO< I)-XCO(K) )*(XCO(J)-XCJU)) ) ) 169 C 170 P ( K , I ) = P ( K , I ) + C B X ( N E L ) * ( ( Y C O I I } - Y C O ( J ) ) * ( Y C O ( J ) - Y C O ( K } ) ) 171 1 + BY(N£L)*UXC0{ J )-XCO( I ) ) * ( XCO ( K)-XCO { J) ) ) ) 172 P(K,J ) = P(K,J)+(BX(NEL)*((YCOlI)-YCO(J)J*(YCO(K)-YCO(Ij)) 173 1+BY(NEL)*((XCO(JJ-XCOU))*{XCO{ I)-XCG(K)))) 174 P(K,K)=P(K,K)+(3X{NEL>*( (YCO( D-YCOU ) )**2)+ 175 lBY(NEL)*{{XCOIJ)-XCO(I))**2)J 176 C 177 F{I)=F{ I)+(0.5*Q(NEL)*EL(NEL)) 178 F{J) = F(J J + I0.5*Q(NEU*EL(NED ) 179 F(K)=F(K)+{0.5*Q{NEL)*EL(NEU) 130 C 131 RETURN 182 END 183 SUBROUTINE TRANS 184 COMMON LDAT(3,500),XCO(275),YCO(275},PHI(275),G(275), 135 17(275,2 75) ,X(275),B(275),I PERM(550),RZ(275),AR(500),C( 275), 186 2P(275,275),Q(500),EL(500),F(500),THETA(500),ALPHA{275),XCOT{275), 137 3YC0T127 5),RT1275) ,PERMX(500)>PERMY{50 0),BX(500),8Y(500),CN(50), 188 4DATA(3,2 75) ,T ITLE{20),JCCP(1) ,I , J,K,He L,NODTOT,N,NP1,MCON, 189 5SIZE,IND,NO,NIB » SCA,LTOT 190 C 191 C ****COORDINATE TRANSFORMATION AND CALCULATION OF MATRIX g 192 C COEFFICIENTS FOR ELEMENTS HAVING ANISOTROPY SKEWED 0 ° 193 C FROM GLOBAL COORDINATES***** 194 C 195 ALP HA(I) = ATAN(YCO( I )/XCO(I))-THETA(Nc L) R T { K ) = S Q R T ( ( X C 0 { K ) * * 2 ) ' + { Y C 0 ( K ) * * 2 } } X C O T ( I ) = C O S < A L P H A ( I ) ) * R T ( I ) X C O T l J ) = C O S ( A L P H A ( J ) ) * R T ( J ) X C O T ( K ) = C O S ( A L P H A < K ) ) * R T ( K ) Y C O T ( I ) = S I N ( A L P H A ( I ) ) * R T ( I ) Y C O T ( J ) = S I N { A L P H A { J ) ) * R T ( J ) Y C O T i K } = S I N ( A L P H A ( K ) ) * R T ( K ) P ( I , I ) = P ( I , I ) + ( B X ( N £ D * ( ( Y C O T C J ) - Y C O T ( K ) } « * 2 ) + 1 B Y ( N E L ) * ( { X C O T ( K ) - X C O T l J ) ) * * 2 ) ) 1 9 6 A L P H A ( J ) = A T A N ( Y C O ( J ) / X C O ( J ) ) - T H E T A ( N c L ) 1 9 7 A L P H A I K )•= A T A N C Y C O l K J / X C O I K i ) - T H E T A ( N S i L ) 1 9 8 C 1 9 9 R T ( I ) = S Q R T ( { X C O ( I ) * * 2 ) - M Y C C U ) * * 2 I ) 2 0 0 R T ( J ) = S Q R T U X C 0 ( J ) * * 2 ) + < Y C 0 U ) * * 2 ) ) 2 0 2 C 2 0 3 2 0 4 2 0 5 2 0 6 C 2 0 7 2 0 8 2 0 9 2 1 0 C 2 1 1 2 1 3 P ( I , J ) = P ( I , J ) + ( B X I N E L ) * ( ( Y C 0 T ( J ) - Y C O T I K ) ) * { Y C 0 T ( K ) - Y C O T t I ) ) ) 2 1 4 1 + B Y ( N E L ) * ( ( X C O T ( K ) - X C O T ( J ) ) * ( X C O T I I ) - X C 0 T ( K ) ) ) ) 2 1 5 P ( I , K ) = P ( I , K ) + { 3 X I N E L ) * ( ( Y C 0 T ( J ) - Y C O T ( K ) ) * ( Y C 0 T ( I ) - Y C O T I J ) ) ) 2 1 6 1 + B Y ( N E L ) * ( ( X C O T I K ) - X C O T ( J ) ) * ( X C O T ( J ) - X C 0 T { I ) ) ) ) 2 1 7 C • 2 1 8 P ( J , I ) = P ( J , I ) + ( B X ( N £ L ) * ( ( Y C 0 T ( K ) - Y C 0 T ( I ) ) * I Y C O T l J ) - Y C O T ( K ) ) ) 2 1 9 1 + B Y { N E L ) * I ( X C O T { I ) - X C 0 T ( K > ) * ( X C O T ( K ) - X C O T ( J ) ) ) ) 2 2 0 P ( J , J } = P U , J ) + ( B X { N E L I * ( { Y C O T i K ) - Y C O T ( I ) ) * * 2 ) + 2 2 1 1 B Y ( N E L ) * ( I X C 0 T { I ) - X C 0 T ( K ) ) * * 2 ) ) 2 2 2 P ( J , K ) = P ( J , K ) + i B X ( N £ L ) * { ( Y C O T ( K ) - Y C O T ( I ) } * ( Y C O T t I ) - Y C O T { J ) ) ) 2 2 3 1 + B Y ( N E L ) * ( ( X C O T ( I ) - X C O T l K ) ) * ( X C O T ( J ) - X C O T i I ) ) ) ) 2 2 4 C 2 2 5 P { K » I ) = P ( K » I ) + ( 8 X ( N E L ) H { Y C 0 T ( I ) - Y C O T ( J ) ) * ( Y C 0 T ( J ) - Y C O T ( K > ) ) 2 2 6 l + 8 Y ( N E L ) * l ( X C O T ( J ) - X C O T ( I ) ) * ( X C O T ( K ) - X C O T { J ) ) ) ) 2 2 7 P ( K , J ) = P ( K , J i + ( B X ( N E L ) * . ( Y C O T ( I ) - Y C O T ( J ) ) * 1 Y C O T ( K ) - Y C O T ( I ) ) ) 2 2 8 1 + BY •' N E L ) * ( ( X C O T ( J 5 - X C O T i I ) ) £( X C O T ( I i - X C O T ( K ) ) ) ) 2 2 9 P { K , K ) = P ( K , K ) + ( B X ( N E L ) * ( ( Y C O T l I i - Y C O T ( J ) ) * * 2 ) * 2 3 0 1 B Y ( N = L ) * U X C O T ( J J - X C O T l I ) ) * * 2 ) ) o 2 3 1 C " ° 2 3 2 F ( I ) = F ( I ) + I 0 . 5 * Q ( N E L ) * E L { N E L ) ) 2 3 3 F ( J ) = F U ) + ( 0 * 5 * 0 ( N E L ) * ? =L<NF.L ) ) 2 3 4 F ( K ) = F ( K ) + ( 0 . 5 * 0 ( N E L ) * E L ( N E D ) 2 3 5 C 2 3 6 R E T U R N 2 3 7 E N D 2 3 8 S U B R O U T I N E G R A P H 1 2 3 9 C O M M O N L D A T ( 3 , 5 0 0 ) , X C O < 2 7 5 ) , Y C O i 2 7 5 ) , P H I ( 2 7 5 ) , G ( 2 7 5 ) , 2 4 0 1 T ( 2 7 5 , 2 7 5 ) , X ( 2 7 5 ) , B ( 2 7 5 ) , I P E R M ( 5 5 0 ) , R Z 1 2 7 5 ) , A R ( 5 0 0 ) , C { 2 7 5 ) , 2 4 1 2 P ( 2 7 5 , 2 7 5 ) , Q ( 5 0 0 ) , E L ( 5 0 0 ) , F ( 500) , T H s T A (.500) , A L P H A { 2 7 5 ) , X C O T { 2 7 5 ) , 2 4 2 * 3 Y C 0 7 ( 2 7 5 ) , R T ( 2 7 5 ) , P E R M X ( 5 0 0 ) , P 5 R M Y ( 5 0 0 ) , B X ( 5 0 0 ) , B Y ( 5 0 0 ) , C N ( 5 0 ) , 2 4 3 4 D A T A ( 3 , 2 7 5 ) , T I T L E ( 2 0 ) , J C C P C 1 ) , I , J , K , N E L , N O D T O T , N , N P 1 , M C O N , 2 4 4 5 S I Z E , I N D , N 0 , N 1 B , S C A , L T 0 T 2 4 5 C 2 4 6 C * * * * P L O T O F E Q U I P O T E N T I A L S * * * * 2 4 7 C 2 4 3 DO 7 0 1 = 1 , N 2 4 9 D A T A { 1 , I ) = X C O ( I ) 2 5 0 D A T A ( 2 , I ) = Y C O ( I ) 2 5 1 D A T A ( 3 , I ) = X ( I ) 2 5 2 7 0 C O N T I N U E 2 5 3 DO 7 1 I = N P 1 , N 0 D T 0 T 2 5 4 D A T A ( 1 , I ) = X C O ( I ) 2 5 5 D A T A ( 2 » I ) = Y C O ( I ) 2 5 6 D A T A ( 3 , I ) = P H I ( 1 ) 2 5 7 7 1 C O N T I N U E 2 5 8 C 2 5 9 C A L L S C A T C N ( D A T A , N O D T Q T . C N , M C O N , S I Z E , I N D ) 2 6 0 C A L L P L Q T N D 2 6 1 R E T U R N 2 6 2 E N D 2 6 3 S U B R O U T I N E G R A P H 2 2 6 4 C O M M O N L D A T ( 3 , 5 0 0 ) , X C O ( 2 7 5 ) , Y C O ( 2 7 5 ) , P H K 2 7 5 ) , G ( 2 7 5 ) , 2 6 5 I T { 2 7 5 , 2 7 5 ) , X ( 2 7 5 ) , B ( 2 7 5 ) , I P E R M ( 5 5 0 ) , R Z ( 2 7 5 ) , A R ( 5 0 0 ) , C ( 2 7 5 ) , 2 6 6 2 P ( 2 7 5 , 2 7 5 ) , 0 ( 5 0 0 ) , E L ( 5 0 0 ) , F ( 5 0 0 ) , T H E T A ( 5 0 0 ) , A L P H A ( 2 7 5 ) , X C 0 T ( 2 7 5 ) , 2 6 7 3 Y C 0 7 ( 2 7 5 ) , R 7 ( 2 7 5 ) , P E R M X I 5 0 0 > , P E R M Y { 5 0 0 ) , 8 X 1 5 0 0 ) , B Y ( 5 0 0 ) , C N { 5 0 ) , 2 6 8 4 D A T A ( 3 , 2 7 5 ) , T I T L E ( 2 0 ) , J C C P (1 ) » I , J , K , N E L , N U D T O T , N , N P 1 , M C Q N , 2 6 9 5 S I Z E , I N D , N 0 , N 1 B , S C A , L T 0 T 2 7 C C 2 7 1 C * * * * W R I T E D A T A I N T E M P O R A R Y F I L E I N F O R M A T R G Q U I R E D B Y M P L O T * * * * P 2 7 2 C • 2 7 3 N E 1 = 1 274 NE2=LT0T 275 NN = NODTOT 276 NS= 1 277 NA=10 278 XM=1„0 279 YM = lo 0 280 IFL0W=0 231 JCCP l 1)=1 282 C 233 WRITE(8,8 I ) TITLE 284 81 FORMAT(20A4) 285 W,RITE(8, 82 ) LTQT,MC0N,N£1,NE2,NN, N0,.M5,NA, SCA, XM,YM,IFL0W 286 82 F 0 R M A T * 8 I 6 , 3 F 3 » 2 , I 4 ) 287 W R I T E ( 8 , 8 2 ) ( J C C P(I), I = 1,NS) 283 WRITE(8,83 ) (CN( I ) , I=1,MCQN) 289 83 FORMAT16F12.0) 290 DO 84 N E L =' 1, L TO T 291 I=LDAT(1,NEL) 292 J = LDAT(2 »NEL) 293 K=LDAT(3,NEL) 294 WRITE (8 ,85 ) I , J , K , X C O ( I ) , X C O t J ) , XCQ IK ) ,YC0( I ) »YCO( J ) , YCQ{ K) 295 85 F0RMAT(3 I6,6F8. 1) 296 84 CONTINUE 297 DO 86 I=1,N 298 G ( I ) =X ( I ) 299 86 CONTINUE 300 DO 90 I=NP1,N0DT0T 301 G ( I ) = P H I ( I ) 302 90 CONTINUE 303 WRITE(8,89)tGCI),1=1,NODTOT) 304 89 FORMAK 6F12o3) 305 C 306 DO 88 I=NIB,NODTOT 307 W R I T E ( 8 , 8 7 ) XCOtI),YCO<I) 308 87 F0RMAT{2F12o2) 309 88 CONTINUE 310 C 311 RETURN 312 END 1 S L O P E GROUNDWATER 6 2 253 215 4 4 6 11 2 3 215 216 1 1 216 217 217 218 4 15 220 29 29 220 221 221 29 5 223 53 224 224 53 54 54 224 6 81 226 80 80 226 227 227 30 7 103 228 229 229 103 230 230 103 8 15 2 16 16 15 29 29 3.6 9 54 41 55 55 54 62 62 54 10 214 2 3 3 2 16 16 3 11 41 18 30 30 41 42 42 41 12 4 3 7 7 3 5 5 7 13 19 17 18 18 19 21 21 18 14 33 30 42 42 33 35 35 42 15 48 43 55 55 48 57 57 55 16 64 62 61 61 64 65 65 61 17 82 72 81 81 82 83 83 81 18 90 95 93 93 90 91 101 230 19 100 107 111 l i l 107 232 232 111 20 109 108 233 233 109 118 118 233 21 117 234 235 235 117 119 119 117 22 236 129 130 130 129 119 130 236 23 131 130 128' 132 237 138 138 237 24 140 238 137 137 238 239 239 137 25 148 239 240 240 148 152 152 148 26 164 152 151 164 241 159 159 241 27 160 242 158 158 242 243 243 158 28 172 170 171 171 170 169 169 170 29 181 244 179 179 244 2 45 245 179 30 4 212 6 6 4 7 7 6 31 11 10 12 12 11 19 19 12 32 23 22 24 24 23 31 31 24 33 35 34 36 36 35 46 46 35 34 49 45 48 48 49 59 59 43 35 66 56 63 63 66 67 67 63 36 77 65 75 75 77 7 6 76 75 37 35 82 83 83 85 36 86 8 3 3 8 96 95 93 93 96 94 94 93 39 104 103 101 101 104 102 102 101 2 58 196 21.0 2400.0 1 1 218 15 15 218 219 219 220 15 40 40 221 222 222 40 53 53 222 223 71 71 224 225 225 71 226 226 71 81 90 90 227 228 228 90 91 91 228 103 101 214 215 1 i 214 2 2 1 15 41 41 29 40 40 41 53 53 41 54 61 61 54 71 71 61 72 72 71 81 5 5 16 17 17 16 18 18 16 41 43 43 41 55 213 214 3 3 213 4 9 9 5 17 17 9 11 11 17 19 23 23 18 30 30 23 31 31 30 33 44 44 42 43 43 44 45 45 43 48 56 56 55 62 62 56 63 63 62 64 75 75 61 72 72 75 73 73 72 82 84 84 81 80 30 84 95 95 80 90 231 231 101 100 100 231 107 107 231 232 108 108 23 2 233 108 111 110 110 108 109 234 109 113 122 118 234 117 117 118 122 120 120 117 122 119 235 129 129 235 236 123 128 236 237 237 128 132 132 128 131 238 238 138 140 140 138 139 139 138 132 148 148 137 149 149 137 140 149 148 150 150 152 240 151 151 240 241 241 151 164 242 242 159 160 160 159 162 162 1 59 164 174 174 158 160 174 243 170 170 174 172 243 243 244 169 169 244 181 181 171 169 178 173 245 246 246 178 247 212 213 4 8 8 i 7 9 9 8 10 10 9 11 20 20 19 21 21 20 22 22 21 23 32 32 31 33 33 32 34 34 33 35 44 44 46 47 47 44 45 45 47 49 57 57 59 53 58 57 56 56 58 66 64 64 67 63 60 64 65 65 68 77 73 73 76 74 74 73 82 82 74 85 84 34 86 • 87 87 84 95 95 37 96 91 91 94 92 92 91 103 103 92 104 100 100 102 112 112 100 111. 111 112 113 40 113 111 110 110 113 114 114 110 109 109 114 124 124 109 122 122 124 123 41 123 122 120 120 123 121 121 120 119 119 121 133 133 119 130 130 133 134 42 134 130 131 131 134 135 135 131 132 132 135 143 143 132 139 139 143 142 A3 142 139 140 140 142 141 141 140 149 149 141 155 155 149 150 150 155 154 44 154 150 152 152 154 153 153 152 164 164 153 165 165, 164 162 162 165 163 45 163 162 160 160 163 161 161 160 174 174 161 175 175 174 172 172 175 173 46 173 172 171 171 173 183 183 171 181 181 183 182 182 181 179 179 182 180 47 180 179 178 178 180 187 137 178 247 247 187 248 211 212 6 6 211 8 48 8 211 13 13 8 10 10 13 12 12 13 14 14 12 20 20 14 25 49 25 20 22 22 25 24 24 25 37 37 24 32 32 37 34 34 37 38 50 38 34 36 36 38 46 46 38 50 50 46 47 -47 50 49 49 50 60 .51 60 49 59 59 60 58 58 60 69 69 58 66 66 67 69 69 67 68 52 68 69 78 78 68 77 77 78 76 76 78 74 74 78 38 88 74 85 53 85 86 88 88 86 87 87 88 97 97 87 96 96 97 94 94 97 92 . 54 92. 97 105 105 92 104 104 105 102 102 105 112 112 105 115 115 112 113 55 113 115 114 114 115 125 125 114 124 124 125 123 123 125 121 121 125 136 56 136 121 133 133 136 134 134 136 135 135 136 144 144 135 143 143 144 142 57 142 144 141 141 144 156 156 141 155 155 156 154 154 156 153 153 156 166 58 166 153 165 165 166 163 163 166 176 176 163 161 161 176 175 175 176 173 59 173 176 184 184 173 183 183 184 182 182 184 185 185 182 180 180 185 188 60 138 180 187 187 183 248 248 188 249 249 138 189 189 188 185 189 249 250 61 250 139 193 193 250 251 251 193 195 195 251 252 252 195 196 196 252 253 62 211 210 13 13 210 14 14 210 209 209 14 26 26. 14 25 25 26. 39 63 39 25 37 37 39 38 38 39 51 51 38 50 50 51 60 60 51 70 64 70 60 69 69 70 78 73 70 89 89 78 83 88 89 97 97 89 106 65 106 97 105 105 106 115 115 106 126 126 115 125 125 126 136 136 126 145 66 145 136 144 144 145 156 156 145 167 167 156 166 166 167 176 176 167 186 67 186 176 184 184 136 185 185 186 189 189 186 194 194 i 8 9 193 193 194 197 68 197 193 195 195 197 196 209 26 27 27 26 39 39 27 52 52 39 51 69 51 52 79 79 51 70 70 79 89 89 79 98 98 89 106 106 98 127 70 127 106 126 126 127 146 146 126 145 145 146 157 157 145 167 167 157 177 71 177 167 186 186 177 190 190 186 194 194 190 198 198 194 197 208 209 27 72 27 208 28 23 27 52 52 28 206 206 52 205 205 52 79 79 205 99 73 99 79 98 98 99 116 116 98 127 127 116 147 147 127 146 146 147 157 74 147 157 168 163 157 177 177 16.8 191 191 177 190 190 191 199 199 190 198 75 207 208 28 28 207 206 205 99 204 204 99 116 116 204 203 203 116 147 76 147 203 202 202 147 168 168 202 192 192 168 191 195- 192 200 200 191 199 77 202 192 201 201 192 200 78 19800.C 3200.0 18000.0 4000.0 I6400o0 3200.0 15600.0 2400.0 15600.0 4000.0 ) J H I - (-J H H H H r 1 r 1 K l J H I - 1 r " H h H H H , J H H O O O O O O O O O O J ) v O O > O v O < ( 3 > O i O > 0 ' 0 ( » C o a ) ( ) ) 0 ) 0 ) ( I ) ( » C S | I ) > l I X J r - ' r - ' t - ' r - ' l - ' r - ' i — ' r - ' l - ' c r o c o c o r o c r o o r o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 0 0 0 0 0 * 0 0 0 0 0 0 0 o o o o o o o o o o o o o o t—• t—• »—> I—• »—• H ' r - t - ' M !—* iV—» t—• I—• O ' O C C ^ C O O H O I - ' ^ O N O O O o o o o o o c n o c o u i o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o v 0 C 0 a - ^ U 1 l s J ^ H H a ) O ^ - s l ^ U l 4 > l v ) N I J C » v 0 ^ ^ U l U l N ^ I - ' O t » > 0 C 0 ( > ^ O O U O ( M » O O ( » ^ ( » 0 W 6 0 > 0 + N O 0 ( M M i l 0 ) N O W O 0 0 l > 0 O ( M V ) o o m o o o o o o o o o o o r u o o o o u i o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o „ 0 0 0 0 0 0 0 » 0 0 9 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 o o o o o o o o o a o o o o o o o o o o o o o o o o o o •4> •*> o o o 0 o o o o o o o o ^ o o o o o « a c o ro o o o a O 1—' 1—• >—• —• c o o o o r o v o u i c o o - o r \ ) C o o - r > o e o | N o CD CT -si ^ ro o o o o o o o o o o o o o o o o o o o o o o Q O O O O O Q O O o o o o o o o o o |-i r — ' r—• (-• r~» r-" I-* 1—' r-* CO-JvOCK -J Ul Ul W Ul o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O O O ^ ^ u i ^ w ^ r u ^ f u u J r - ' r o - . j r u - f ' W u i 4 > O 4 > C 0 O O O O O O O O O O O O O O O o o O O O o o o C O O C u i O O O U J O O O O O O O O O O 0 0 * 0 0 0 0 0 9 O O O O O O O O O o o r v ) c o o o o o j o u i o > - ' c o o - 4 > u i o r o o u i j - o o i N > O O O ' J l O O O O O O O U I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o O O O O O O O O O O O O O O O O O O O O O O O O ^ f—i Ul W N H ^ r— O O O O o o o o o o o o o o o o o o vD CO Cf- O Ul Ul -*N ro ro 4 s o o o o o o o o o o o o o o o o o o o o o a o o o o o o o o o o ro . . O v O O v O C O C O v O C O C O - g O O O O O O O O O O O O O O O O o- o o (--4 f_l CO Cr* ->l ro co cr O O O O Cr Ui o o O O O O O O O O o- 4> o o o o o O  o 9 O 0 0 0 * o o o 0 O O O j co -P* ro Ul O o o o o o o 0 0 0 0 Ul CO o o 0 o o o o o o o I—• •—• t—• i—• I—' t—' —• »•—* t—» H-» »—• I—" I—* *—' I—« o o o o o o o o o o o r o o o o o o o o o o o o o o u i o o o o O O O O O O O O O O O O O O o o o o o o o o o o o o o o UJ o o o 0 o o o 0 o W W O J W U l U i W W W ^ N N W W N M W r J N i - ' H H r - ' r ' r - H I - ' CC 0 ' N U l ̂  C l t : H O C O ) 0 - - J U ' i + N W U ; C O O » l f f ' U l U ! W N t-f fri\)(»aOWCOvOC^UlO'^4>OI\ICDO*OOWCON*'CO^O' o o u i o o o o o o o r o o o o o o u i o o o o o o o o o o o o o o o o o o o o o o o o o 0 * 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 9 0 0 O 0 0 O o o o o o o c o c 0 o O O vO C O W J > O N O N O o o o o o o o o o o 0 0 0 0 0 o o o o o o o 0 o o- a u> u> o o o ro o o o o o o o o e o o o o o o o ro o o o e o N j t D C O O v O O C O v D O J C O O 1 ^ ^ ^ G o co o ro ro o 4> ro o cr cr o o- o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 0 0 0 o 0 0 0 0 0 0 0 0 O O O O O O O O o 0 o o o o o o •—4 ^ ' I—' ( 1 l—* I - * O 0 0 - C r u i 4 ^ 4 > 0 i ^ rocoororocococr -^ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o I—• (-< r-» I " 4 »•* (-* k̂ J J—• |u4' V-J k—' f- O f f l O N ^ O P ^ C O W O O ' O - O ^ O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O o o o o o o o o o o o o o o o O O O O O O O O O O O O O O O ^ ^ OJ U W U W U W W Ui N N IS) W N M N N i\) H H H H H H H H H H H -f- t - * i£> O O O O O - O O O O O O o o 0 0 0 0 o o o o ^ 4 > c r r o r o t - < o » £ > c o - v l c r 4 > v r i U J U ) o O W O M C O W ^ O ^ H N U C t l C M I l C D o o o o o o o o o o o o o o o o o o o o o o o o o o o e o o o o o o o o o o o o o o o o o o o o o c o v o c o ~ i u i u i U ) U > o r u o c o c r O + N O 0 - O i 0 O + N C 0 O O < J ' 0 ' o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 o 0 o o 0 o o o o o o o o o o o o o 0 0 o o -si O ro o o o o o 0 0 o o co cr o o o o a « o o 1̂ o o o ro (-* O O O o O o o o o o o o o o o o O v 0 O 0 0 « 0 C 0 C 0 - £ " o ro o cr o o o o o o o o o o o o o o o o o o r~4 (—• t-J r-« (-4 r—' (—• f-» r—• r—' K• - j . - j - ^ J U l C r U i O J > U l - P - U i - f > O N O C O W 0 4 % 0 0 0 + N 0 ' t M > 0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o U l W O W I - ' W I - ' N c o N U H W i O crcr04^4>crru-f- O O O O O O O O O O O O O O O O o o o o o o o o O O O O O O O O CO Cr O O O O O O 0 0 9 O ro Cr O O O O O O 0 0 9 O O O O o o o o O 0 o + N N O ^ ^ ^ U l W W O O O ' f l - ~ l 0 > U I U l U ) U J H I - ' C D i 0 0 - 0 a + s + N O O - O O - O C O N f f U l U I C O O ^ C D O O N O O ^ ^ N O i T O C O O O O O O O O O O U I U I O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 0 0 0 0 0 0 0 0 0 0 0 9 0 0 0 0 0 0 0 0 0 0 o o o o o o o o o o o o o o o o o o o o o o 0 0 9 0 o o o o o r-1 vO >0 CO CO o ro ro co o o o o o o o o o 0 0 9 0 o o o o o o o o Ul Ul U) ro co o cr o o o o o o o o o o o o o o o o t n 115 o o O O O o o o o o o o o o o o o o o 0 9 0 0 0 0 0 0 0 o o o o o o o o o o O O O o o o o o rH rH rH rH rH rH rH rH o o O O O o o o o o o o o o o o o o o o O O O CO o OJ OJ o o o o o o o o o v f OJ oj r - i n o i n rH r - o o o o o o o o o rH OJ OJ ro. ro v t 0 0 0 o 9 0 9 9 o o O O O o o o o o 0 0 0 0 0 • 0 0 0 0 o o O O O o o o o o o o o o o o o o o o o o r - o o s O i n o o o o o o o o o o o O vO rH vO s 0 ro i n i n OJ OJ OJ OJ OJ OJ OJ OJ 0 0 oj o o r - i n r - CT-o o o o o o o o o o rH OJ OJ rH rH rH rH OJ OJ o o o o o o o o o o o o o o o o o o o o o o o o o o 0 0 o 9 9 o 9 9 0 0 • 0 0 0 0 0 0 • o o o o o o o o o o o o O O O i n o o o o o o O 0 3 O v f o o o o o o o o o o o o »—1 CM OJ IS; v f o <r o v O o o o o o o o o rH OJ OJ m ro ro rH r - l I-l r - l rH r-l rH rH o o o o o o o o o o O O O o o o o o o o o o o o o o 0 o 0 0 0 0 0 0 9 0 o o o o o o o o o o O O O o o o o o 0 0 0 0 9 9 9 0 o o o o ro o o o i n o o o vO i n o rH o i n ro OJ a* r - o r - m r - o o o rH OJ rH rH rH rH OJ OJ o o o o o o o o o o o o o o o o o o o o a o o o o o o OJ OJ oj OJ OJ OJ OJ OJ 0 0 9 0 0 9 9 <» 0 0 0 o o o o o o o o o a O O O o o o o o o o o o o o o o o o o O O O o o o o o o o o o o o o o o o o O O fsl o rn OJ oo - 0 o 0 9 9 9 0 0 V 9 OJ .-NJ v f H o rr> < f CO < t rH OJ OJ OJ ro ro v f o a O O O o o o o o o o o o o o o o o 0 o 0 0 0 o 0 9 o 0 0 o o o o o o o o o o O O O o o o o o o rH rH rH rH rH r - l rH rH o o O O O o o o o rH i n o o o o o o o o o o O O O o oo r- v f ro v f o o o o o o o o oo vO vO r-l 0 > vO o Q 0 o o rH o o o o o o o o rH OJ rH i-H r-l rH OJ OJ OJ 0 0 0 9 9 9 0 a o o O O O o o o o o o 0 a 9 0 0 o 0 9 0 0 0 o o O O O o o o o o o o o o o o o o o o o O O O o o o i n o o o o o o o o o o o o <r o v f o OJ rH OJ OJ OJ OJ OJ OJ OJ OJ OJ o OJ OJ to 0 0 OJ co co CO ro o o o o o o o o rn r-l OJ OJ ro ro v f o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 0 0 9 0 0 0 a 0 0 O O O a 0 a 0 0 o o O O O o o o o o o o o O O O o o o o 0 0 i n o o v Q N O m v f ro o ro v t o o o o o o o o OJ OJ i n r-l o o o 0 0 o a rH o o o o o o o o r-l OJ OJ rH r-l rH OJ OJ OJ r-l rH r H rH rH rH rH rH o o o o o o o o o o O O O o o o o o o o o o o o o o o 0 0 0 9 0 0 0 0 0 o a o o o o o o o o o o O O O o o o o o o 0 0 a 9 9 9 0 a o o O O O o >o o o o o o o O O - 0 o OJ CO v f oo v t v f 0 0 OJ OJ 0 0 r~ rH vO 0 4 r - rH v t rH OJ OJ ro ro v f o o o o o o o o o o o o o o o o o o O O O o o o o o o OJ OJ OJ OJ OJ OJ OJ OJ 0 a a o o o 0 0 0 o 0 o o o o o o o o o o O O O o o o o o o o o o o o o o o o o o o r - i n o o ro o rH o o o o o o o o o o o rn r - o o r ~ <r o 0 a 9 a 9 9 9 9 OJ v f r H O vO o co 0> o rH OJ OJ rH rH rH rH OJ OJ oo o> O rH 0-1 ro v f i n vO f - co CP o rH OJ ro v f i n vO rH rH OJ OJ OJ OJ OJ OJ OJ OJ OJ OJ ro ro ro ro m ro ro ' r H r-l r-l r-l . H rH r-t r-l rH r-l rH rH rH rH rH rH rH rH o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o rH rH r-l rH rH rH r-l rH rH rH rH rH rH rH rH rH rH rH r H rH o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 0 9 9 9 9 0 9 9 9 9 9 9 0 9 0 9 9 9 9 o . o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o OJ OJ ro OJ OJ OJ OJ OJ OJ oo OJ OJ oo OJ OJ OJ OJ OJ oo OJ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 0 9 9 « 0 9 9 9 9 9 9 9 9 9 9 0 9 a 9 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o rH r-l r-l rH rH rH rH rH rH rH r-l rH rH H rH rH rH rH rH rH o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 0 a 9 0 9 0 9 9 • 9 9 9 9 0 9 4 0 9 a 0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o OJ OO OJ OJ OJ OJ OJ OJ OJ OJ c\J OJ OO OJ OJ OJ OJ OJ OJ OJ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o a o o o o o o o o o o o o a o o o o o o o o 9 0 a 9 0 9 9 9 a 9 9 a 9 0 9 9 0 9 9 9 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o rH rH rH rH rH r-l rH rH rH rH r-l rH rH rH rH rH rH rH rH o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o a 0 9 0 0 9 0 9 0 o 0 0 9 9 9 9 9 9 9 9 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o OJ oo OJ OJ OJ oo OJ OJ oo OJ OJ oo oo OJ OJ OJ OJ OJ OJ oj o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 9 0 9 0 9 0 0 9 s 9 0 0 9 0 a 9 9 9 9 9 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o rH rH rH rH r-l rH r-| rH r-l rH rH rH rH rH rH /—1 rH r-i rH o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o a o o o o o o o o o o o o o o o o o o o o 9 0 9 9 9 0 0 9 0 0 0 9 9 9 9 9 9 9 9 0 o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o OJ OJ OJ OJ OJ O-l OJ OJ OJ O-l OJ oo OJ oo oo OJ O-l OJ OJ OJ o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O o o o o 9 9 0 0 9 9 0 0 9 a 9 0 9 9 0 9 9 9 9 9 r - co o r-l OJ ro v t i n o r - CO cr> o rH OJ ro v t i n ro r o ro v f v t v t v t v t v t v f v t v t v t m i n i n i n i n m i n rH rH rH rH rH rH rH rH .-1 rH rH r-l rH r-l rH rH r H r-l rH rH 116 o o o o o o o o o O o O r-4 r-l r H r H r—1 i-H r-i F - l r H r H r H r H r H r H r H I--I r H r H r H r H r H r-H r H r H r H i - H r H o o o o o o o o o o o o O o o o o o o o o o o o o o o o o O o o o o o o o o o • H r H r - l r H r-H r H r H r - l r H r - l r H r H o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o. p o o o o o p o p o o o.. o •o o o o o o o o o • 0 e 0 0 e 9 e e 0 0 e 0 0 a 0 o 0 0 0 0 e 0 0 c 0 e •o 0* d ' . 0 0 0 o 0 0 0 e 0 o o o o o O O O o O o O in in in in in in in in in in in in in in in in in in in in in in in in in m m o o o o o o o o o o o O o o O o o o o o o a O o o a O o o o o a o o o o o o o CM CM CM CM CM CM CM CM CM CM CM CM o o o a o o o O o o o o o o O o a o o o o o o o o o O o o O O O O O O O O O O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o O O O O O O a O O o o o o o o o o o o o o o o o o o o o a o o o o o o o o o o O o O O o O o O O o o o o a o a o o o a o o o o o a o o c o o o o o o o • 0 « 9 a 0 0 9 » 0 e 0 0 0 e e 0 0 9 0 0 a 0 0 0 0 « 0 9 9 0 9 0 0 9 0 0 0 0 o o o o o O o O o o o o r H r H r H r H r H r H r-H r H r-H r H r H r H r H r— 1 r - l r H r H r H r - l r-4 r-l r-H r-l r-4 r H r H r - l o o o o o o o O o o o o o o o o o o o o o o o o o o o o o o o o o o o a o o o r H r H r H r-4 r H r - l r - l r H r-l r-4 a o o o o o o o o o c o o o o o o o o o o o o o o o o O O o o o O o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o a o o o o o o o o o o o o o o o o a o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o 3 o o o o o o o o o o o o o o o o o o o o o o o o o o o e 9 0 e 9 e 9 9 0 9 9 0 • 0 c 9 o 9 9 0 9 c e 0 0 9 9 o 9 0 o 0 0 9 9 9 9 0 o o o o o o o o a o o o m in in in in in in in in in in m in in in in in in in in in m tn m m m m o o o o o o o o o o a o o a o a o o o o o o o a o a o o o o o o o a o o o o o CM <M rM CM CM CM c\l CM CM CM CM CM o o o o o a o o o o a o o a o o o o o o o o o a o o o o o o o o o o o o o o o o o o o o o o o o a o o a o o o o o o o o o o o a o o o o a o o a o a o o o o o o o o o o o a o a -O a o o o a o o o o o o a o o o o o a o o o o o o o o o o o o o o o o o o o o o o o a o o o o o o o o o o o o o 0 9 9 9 0 e o 0 9 9 9 9 9 9 9 9 9 9 9 9 0 9 0 e 0 0 9 9 9 9 0 9 9 0 9 9 9 9 0 o o o o o o o o o o a o r-H r H r-H r-H r-H r-H r-H "rH) r H r H r H r H r H r-H r-H r-H r-H r H r H r H —H r H r H r H r H r H r H o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o —1 r H r H r H r H r H i - H r H r H r H r H r H o o o o o o o o o o o o o o o o o o o o o a o o o o o o o o o o o o o o o o o o o o a o o o o o o o o o o o o o o o o. o o o o o o o o o o o o o a o o o o a o c o o o o o o o o o o o o o o o a o o o o o o o o o o a o o o o o a o o o o o o o o o b o o o o <5 o o o o o o o o o o o 9 0 9 9 0 o e 0 9 9 0 9 0 9 9 9 9 9 9 9 0 9 0 0 9 9 9 9 0 0 9 9 9 0 o 0 0 0 o o o o o o o o o o o o o in in in in in in in in in in in in m in in in in in in m in in in in in in in o o a o o o o o o o o o o o o Q o o o o o o o o o o o o o o o o o o a o o o o CM CM CM CM CM CM CM CM CM CM CM CM o o o 6 o o o o o o o o a o o o o o o o o o o o o o o o o o o o o o o o o o o o a o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o Q o o o o o o o o o o o o o o c 9 9 9 9 0 9 9 0 9 o 9 9 9 9 9 0 9 9 9 9 0 0 9 a « 0 9 0 0 9 9 9 9 0 0 o 0 9 0 o o o o o o o o o o o o o rH t-H r-H rH r - l rH r - l r - l rH r-H rH I-t r - l rH rH r H I - l r-H rH rH r - l rH r - l rH r-l o o o o o o o o o o o o o o o o o o o o o o o •o o o o o o o o o o o o o o o o rH r-H r - l —t r-H rH r-H rH r - l r-H rH r - l o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o c o o o o o o o o o o o o o o o o o o o o o a o o o o o o o o o a o o o o o o o o o o o o o o o o o o o o o o o 9 0 9 9 e 9 9 0 9 0 9 9 9 9 9 0 9 9 9 9 0 0 0 o o 9 9 9 0 9 9 9 ' 9 0 9 0 0 9 o o o o o o o o o o o o o o in tn in in in in in in in in in in in m in in in in in in tn in in in in in o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o CM CM CM CM CM CM CM CM CM CM CM CM CM o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o c o o o c o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o a o o o o o o o o o o o o o o o o o o o o o o o 9 9 0 9 0 9 9 0 9 9 0 9 0 9 9 0 9 0 0 9 9 9 9 0 9 0 9 9 0 9 9 9 9 0 9 9 0 9 o r - 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O 16 .0 16.0 1 6 , 0 - 4 0 . 0 - 4 0 . 0 - 4 0 . 0 - 4 0 . 0 338 - 4 0 . 0 - 4 0 . 0 - 2 7 . 0 - 2 7 . 0 - 2 7 . 0 - 2 7 . 0 - 27 .0 -12 . 0 -12. 0 - 12 . 0 - 12 . 0 0.0 0.0 0. 0 0.0 0 . 0 339 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 1 6 . 0 340 16. 0 16. 0 16.0 16. 0 16.0 16.0 16.0 16. 0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 341 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16 .0 16.0 16.0 16.0 16.0 16.0 16.0 342 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 343 16.0 1 6 . 0 - 3 0 . 0 - 3 0 . 0 - 3 0 . 0 - 3 0 . 0 - 1 5 . 0 - 1 5 . 0 - 1 0 . 0 - 1 0 . 0 0.0 0.0 10.0 16.0 16.0 16.0 .344 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 !345 16.0 16.0 16.0 16.0 16.0 16.0 16.0 1 6 . 0 - 1 5 . 0 - 1 5 . 0 - 8 . 0 - 8 . 0 7.0 7.0 16.0 16.0 ;346 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 -2 5.0-2 5 .0 -13 .0 347 - 8 . 0 0.0 7.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 348 - 2 5 . 0 - 1 0 . 0 10.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 16.0 349 16000.0 16500.0 17000.0 17500.0 18000.0 13500.0 19000.0 19500.0 20000.0 20500 .0 ro .350 21000.0 ° 121 S t a r t Read in d a t a W r i t e o u t i n p u t Set t o p f l o w b o u n d a r y t o t h e ground s u r f a c e I n i t i a l i z e p a r a m e t e r s t o z e r o I C o n v e r t t h e t a t o r a d i a n s N e l = 0 N e l = N e l + 1 I C a l c u l a t e element a r e a Yes r e q u i r e d ? C a l l TRANS. C a l c u l a t e [ c o n t r i b u t i o n t o n o d a l equation) C a l l NOTRAN. C a l c u l a t e [ c o n t r i b u t i o n t o n o d a l e q u a t i o n J!o_ C a l l SLIMP t o s o l v e m a t r i x e q u a t i o n F i g u r e 1-1 S t o p F l o w c h a r t . 122 APPENDIX II FIGURES FOR CHAPTER FIVE

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