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The development of multiple seepage faces along heterogeneous hillsides Rulon, Jennifer 1984

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THE DEVELOPMENT OF MULTIPLE SEEPAGE FACES ALONG HETEROGENEOUS HILLSIDES by J e n n i f e r Rulon B . S c . The U n i v e r s i t y o f Wash ington , 1979 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n THE FACULTY OF GRADUATE STUDIES (Department o f I n t e r d i s c i p l i n a r y Hydro logy) We accept t h i s t h e s i s as conforming t o the r e q u i r e d s t andard THE UNIVERSITY OF BRITISH COLUMBIA June 1984 © J e n n i f e r R u l o n , 1984 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia, I agree that the Lib r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r reference and study. I further agree that permission for extensive copying of t h i s thesis f o r s c h o l a r l y purposes may be granted by the head of my department or by h i s or her representatives. I t i s understood that copying or p u b l i c a t i o n of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of ^Qn fordi SC^JPILMAMJ cUv /otfy The Univ e r s i t y of B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date ZQ June. Iff*/ ABSTRACT A s tudy has been made to c l a r i f y the w a t e r - t a b l e c o n f i g u r a t i o n and h y d r a u l i c - h e a d d i s t r i b u t i o n i n l a y e r e d h i l l s i d e s c o n t a i n i n g m u l t i p l e seepage f a c e s . A f i n i t e - e l e m e n t model was used to s i m u l a t e t w o - d i m e n s i o n a l , s a t u r a t e d and u n s a t u r a t e d , s t e a d y - s t a t e , and t r a n s i e n t f low through l a y e r e d s l o p e s . A l a b o r a t o r y sand-tank exper iment was b u i l t to t e s t the p h y s i c a l f o u n d a t i o n o f the mathemat i ca l mode l ; the t e s t met w i t h s u c c e s s . Layered s l o p e s were found to f e a t u r e perched water t a b l e s and wedge-shaped u n s a t u r a t e d zones w h i c h , i n some c a s e s , can extend s e v e r a l k i l o m e t e r s i n t o the f low r e g i o n . The r e s u l t s demonstrate t h a t the h y d r a u l i c - h e a d d i s t r i b u t i o n and the f o r m a t i o n o f m u l t i p l e seepage faces are s t r o n g l y dependent on the p o s i t i o n o f the impeding l a y e r s , the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t , the r a i n f a l l r a t e , a n i s o t r o p y , and the s lope a n g l e . P r e d i c t i o n s o f the groundwater c o n d i t i o n s based on homogeneous, s a t u r a t e d a n a l y s e s may be s i g n i f i c a n t l y i n e r r o r when a p p l i e d to problems i n l a y e r e d s l o p e s . T h i s s tudy has i m p l i c a t i o n s w i t h r e s p e c t to s l ope s t a b i l i t y , i n f l o w s i n t o e x c a v a t i o n s , r e g i o n a l groundwater f l o w , the o c c u r r e n c e of perched f low sys tems , and h i l l s l o p e proces se s i n v o l v e d i n l andform development . i i TABLE OF CONTENTS Page LIST OF TABLES V LIST OF ILLUSTRATIONS v i ACKNOWLEDGEMENTS ' x Chapter 1. INTRODUCTION 1 Chapter 2. THE MATHEMATICAL MODEL 6 2.1 The P h y s i c a l Problem 6 2.2 The Boundary-Value Problem 8 2.3 The Numerical Method of S o l u t i o n 17 Chapter 3. MODEL VERIFICATION 25 3.1 The Experimental Design 27 S e l e c t i o n and T e s t i n g of the Medium Sand 29 S e l e c t i o n and T e s t i n g of the Fine Sand 36 E r r o r A n a l y s i s 47 Generation and Maintenence of the Boundary C o n d i t i o n s 54 3.2 The T r i a l Run 56 P r e p a r a t i o n of the Sand Tank 59 F i l l i n g the Tank With Sand 60 Re s u l t s of the T r i a l Run 61 3.3 The F i n a l Run 65 Chapter 4. STEADY-STATE SENSITIVITY ANALYSIS 74 4.1 Methodology 74 4.2 R e s u l t s 80 One-layer Flow Systems 80 Two-layer Flow Systems 86 Three-l a y e r Flow Systems 96 Anisotropy 100 Slope Angle 102 4.3 D i s c u s s i o n 104 Assumption and L i m i t a t i o n s 108 i i i TABLE OF CONTENTS (continued) Page Chapter 5. TRANSIENT ANALYSIS I l l Chapter 6. APPLICATIONS 122 6.1 S lope S t a b i l i t y 122 6.2 Other P o s s i b l e A p p l i c a t i o n s 132 Groundwater Inf lows i n t o E x c a v a t i o n s 132 R e g i o n a l Groundwater Flow 135 H i l l s l o p e Hydro logy 136 Chapter 7. SUMMARY AND CONCLUSIONS 142 BIBLIOGRAPHY 149 APPENDIX A . DEFINITION OF SYMBOLS 153 APPENDIX B. EXPERIMENTAL DATA 156 i v L I S T OF TABLES T a b l e Page 1. Range of va lue s of c o m p r e s s i b i l i t y 14 2. Cons tant -head t e s t r e s u l t s on the medium sand 31 3. Cons tant -head t e s t r e s u l t s on the f i n e sand 47 4. R e s u l t s of the e r r o r a n a l y s i s performed on Kl and K2 for a f l u i d temperature of 20 C 52 5. Summary of back c a l c u l a t i o n s 69 6. P r e d i c t e d versus observed out f low r a t e s 70 7. C l a s s i f i c a t i o n o f two- layer systems 86 8. H y d r a u l i c p r o p e r t i e s used i n the t r a n s i e n t s i m u l a t i o n s 116 9. Comparison of h y d r a u l i c - h e a d data for t ens iometer s read w i t h the S c a n i v a l v e 158 10. Comparison of h y d r a u l i c - h e a d data for t ens iometer s read w i t h manometers 159 11. P r e s s u r e head and e l e v a t i o n head data for t ens iometer s read w i t h the S c a n i v a l v e 160 12. P r e s s u r e head and e l e v a t i o n head data for t ens iometer s read w i t h manometers 161 v LIST OF ILLUSTRATIONS F i g u r e Page 1. H y p o t h e t i c a l f low r e g i o n 7 2. Examples of c h a r a c t e r i s t i c curves 11 3. The exper imenta l f low r e g i o n 28 4. Tensiometer i n s t a l l a t i o n 33 5. Measured K(\|>) data for the medium sand 35 6. Measured Q{\\>) data for the medium sand 37 7. D e s i r a b l e and u n d e s i r a b l e c h a r a c t e r i s t i c s of the exper imenta l f low r e g i o n 39 8. Three meshes used to determine the optimum t h i c k n e s s and l o c a t i o n of the impeding l a y e r 41 9. The e f f e c t of T and z on the l e n g t h over which the i n f i l t r a t i o n boundary i s ponded 42 10. The e f f e c t of of T and z on the l e n g t h of the uppermost seepage face 43 11. The e f f e c t of T and z on the d i s t a n c e the unsa tura ted wedge extends i n t o the h i l l s i d e 44 12. The optimum d e s i g n for the exper imenta l f low r e g i o n 46 13. Range of t h e o r e t i c a l p r e d i c t i o n s as K i and K 2 vary w i t h i n two s tandard d e v i a t i o n s of t h e i r means for a r a i n f a l l r a t e of 2.7 cm/min 53 14. Seepage c o l l e c t o r 57 15. L o c a t i o n and f u n c t i o n o f measurement p o r t s wi th r e s p e c t to the f i n i t e -element mesh 59 16. P r e d i c t e d r e s u l t s for the t r i a l r u n ; Ki = 2.75 x 10~ 3 m/s and K 2 = 1.21 x I O - 4 m/s 62 v i 17. Comparison of measured and p r e d i c t e d pres sure-head read ings d u r i n g the t r i a l run 63 18. Photograph of the f i n a l exper imenta l run . . . 67 19. Comparison of p r e d i c t e d and observed w a t e r - t a b l e c o n f i g u r a t i o n s for the f i n a l r u n , us ing b e s t - f i t va lues of K i and K2 68 20. F i n i t e - e l e m e n t mesh used for the s teady-s t a t e s e n s i t i v i t y a n a l y s i s 75 21. C h a r a c t e r i s t i c curves used i n the s t e a d y - s t a t e s e n s i t i v i t y a n a l y s i s 77 22. G e n e r a l i z e d boundary-value problem s o l v e d i n the s t e a d y - s t a t e s e n s i t i v i t y a n a l y s i s 79 23. S e n s i t i v i t y of the unsa tura ted wedge to the p o s i t i o n of the impeding l a y e r and to the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t 82 24. Flow reg ions e x e m p l i f y i n g the e f f e c t of the p o s i t i o n of the impeding l a y e r and the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t on the extent of the unsa tura ted wedge 83 25. S e n s i t i v i t y of the out f low to the p o s i t i o n of the impeding l a y e r and to the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t 84 26. The l e n g t h of the uppermost seepage face as a f u n c t i o n of the p o s i t i o n of the impeding l a y e r and the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t 85 27. The extent o f the unsa tura ted wedge as the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t i n c r e a s e s over s e v e r a l o rder s o f magnitude for K i = 10~7 m/s and a r a i n f a l l r a t e of 4 x 10~ 8 m/s 87 28. The e f f e c t o f the e l e v a t i o n of the impeding l a y e r s and the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t on the unsa tura ted wedges for a c o n s t a n t d i s t a n c e s e p a r a t i n g the impeding l a y e r s 89 v i i 29. The r e l a t i v e extent of the unsa tura ted wedges for K 1 / K 2 = 20 90 30. Comparison of the r e l a t i v e extent of the unsa tura ted wedges for K 1 / K 2 = 20, K i = 1.4 x 10~6 m/s , and a s teady-s t a t e r a i n f a l l r a t e of 0.3 x 1 0 " ° m/s 91 31. A comparison of the percentage of the t o t a l out f low d i s c h a r g e d acros s the upper and middle seepage faces i n two- layer systems 92 32. The e f f e c t of the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t on the percentage of the t o t a l out f low d i s c h a r g e d acros s the upper and middle seepage faces formed i n two- layer systems 94 33. The e f f e c t of i n c r e a s i n g the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t over s e v e r a l o rder s of magnitude for MeshBF, = 10~7 m/s , and a r a i n f a l l r a t e of 4 x 10~ 8 m/s 95 34. The e f f e c t of i n c r e a s i n g the number of impeding l a y e r s for K 1 / K 2 = 20, = 1.4 x 1 0 ~ ° m/s , and a r a i n f a l l r a te of 0.3 x 10~ 6 m/s 97 35. T h r e e - l a y e r systems for K 1 / K 2 = 20, K\ = 1 0 " ' m/s , and a r a i n f a l l r a t e of 4 x 10~ 8 m/s 98 36. The e f f e c t of the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t on the percentage of the t o t a l ouflow d i s c h a r g e d acros s the seepage faces formed i n t h r e e - l a y e r systems for a r a i n f a l l r a t e of 4 x 10~ 8 m/s 99 37. The e f f e c t of a n i s t r o p y for K i / K 2 = 30 and a s t e a d y - s t a t e r a i n f a l l r a t e of 4 x I O " 8 m/s 101 38. The w a t e r - t a b l e c o n f i g u r a t i o n as a f u n c t i o n o f the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t for a s l ope angle of 8 ° 103 v i i i 39. R e f r a c t i o n of groundwater acros s a l e s s -permeable l a y e r 105 40. T r a n s i e n t s i m u l a t i o n of the development o f a perched flow system, ( t , t ime i n hours) 112 41. F i n i t e - e l e m e n t mesh used i n t r a n s i e n t s i m u l a t i o n 114 42. One- l ayer flow system for the t r a n s i e n t s i m u l a t i o n 115 43. T r a n s i e n t response of the w a t e r - t a b l e for the o n e - l a y e r flow system; t = time i n hours 118 44. Two- layer flow system for the t r a n s i e n t s i m u l a t i o n 119 45. T r a n s i e n t response of the water t a b l e for the o n e - l a y e r flow system; t = time i n hours 120 46. P o s s i b l e h y d r a u l i c - h e a d d i s t r i b u t i o n s for use i n s l o p e - s t a b i l i t y a n a l y s i s 125 47. Comparison of p re s sure-head and f l u i d -pre s sure d i s t r i b u t i o n s a long a p o t e n t i a l f a i l u r e sur f ace 128 48. Geology and groundwater c o n d i t i o n s at Bender Park s i t e i n W i s c o n s i n 130 49. Groundwater c o n d i t i o n s at a highway cut i n A l b e r t a 131 50. Methods used to c o n t r o l groundwater i n f l o w s i n t o excava t ions 133 51. H y p o t h e t i c a l w a t e r - t a b l e c o n f i g u r a t i o n for an e x c a v a t i o n i n t o heterogeneous m a t e r i a l 134 52. P o s s i b l e f low paths for water to f o l l o w for a homogeneous h i l l s i d e 139 53. Index for tens iometer l o c a t i o n s i n the exper imenta l flow r e g i o n 157 i x ACKNOWLEDGMENTS I would l i k e to thank A l Freeze for h i s guidance and encouragement throughout the e n t i r e p r o j e c t and for sugges t ing the t h e s i s t o p i c . In a d d i t i o n , I would l i k e to thank him for h i s generous f i n a n c i a l s u p p o r t . A l l funding came from a grant awarded to A l Freeze by The N a t i o n a l Sc ience and E n g i n e e r i n g Research C o u n c i l of Canada. I would a l s o l i k e to thank the t a l e n t e d t e c h n i c a l s t a f f i n the Geology Department at the U n i v e r s i t y of B r i t i s h Co lumbia . Ray Rodway b u i l t most o f the equipment used i n the l a b o r a t o r y exper iment ; Ed Montgomery took photos of the l a b o r a t o r y work; John Knight assembled the e l e c t r o n i c equipment; and, Gord Hodge d r a f t e d the f i g u r e s . I a p p r e c i a t e the h e l p f u l comments of my committee members; namely, L e s l i e S m i t h , Jan d e V r i e s , B i l l Mathews, Olav S laymaker , and Peter B y r n e . I am g r a t e f u l to Nick S i t a r for a d v i c e d u r i n g the e a r l y p a r t of t h i s work and to John Nieber who p r o v i d e d v a l u a b l e guidance i n the d e s i g n of the p h y s i c a l model . F i n a l l y , s p e c i a l thanks to K e i t h Loague, Grant Garven , and Chuck Mase for t h e i r a s s i s t a n c e i n the l a b as w e l l as for h e l p f u l d i s c u s s i o n s throughout the e n t i r e p r o j e c t . x 1 Chapter 1 INTRODUCTION Groundwater c o n d i t i o n s w i t h i n a h i l l s l o p e p l a y an important r o l e i n many g e o t e c h n i c a l , h y d r o g e o l o g i c a l , and geomorpho log i ca l problems . For example, g e o t e c h n i c a l eng ineers must p r e d i c t the f l u i d - p r e s s u r e d i s t r i b u t i o n a long p o t e n t i a l f a i l u r e su r f ace s and c a l c u l a t e i n f l o w r a t e s i n t o e x c a v a t i o n s . H y d r o g e o l o g i s t s may be r e q u i r e d to assess the r o l e of groundwater d i s c h a r g e i n areas where s h o r e l i n e e r o s i o n i s a problem. Geomorphologi s t s s tudy the r o l e of groundwater i n h i l l s l o p e proces se s as they r e l a t e to landform development. In each i n s t a n c e , the f l u i d - p r e s s u r e d i s t r i b u t i o n , the l o c a t i o n of d i s c h a r g e a r e a s , and the r a t e s of d i s c h a r g e must be p r e d i c t e d . Our a b i l i t y to make these p r e d i c t i o n s i s w e l l e s t a b l i s h e d for homogeneous h i l l s i d e s . However, r e a l g e o l o g i c environments are almost always heterogeneous . T h i s t h e s i s c o n t r i b u t e s to our a b i l i t y to p r e d i c t groundwater c o n d i t i o n s i n heterogeneous h i l l s i d e s . In both homogeneous and heterogenous h i l l s i d e s , the d i s c h a r g e phenomena are a s s o c i a t e d w i t h the presence of a seepage f a c e . A seepage face i s d e f i n e d as a s a t u r a t e d out f low boundary a long which the f l u i d p re s sure i s a tmospher i c . A seepage face forms when the water t a b l e i n t e r s e c t s the s o i l su r f ace above the water l e v e l i n an ad jacent s i n k . The f a c t o r s govern ing the fo rmat ion of seepage faces are t h e r e f o r e the same as those c o n t r o l l i n g the p o s i t i o n of the water t a b l e , namely, i n t e r a c t i o n s between: a) the s a t u r a t e d and unsa tura ted flow 2 systems, b) the geology of the h i l l s i d e , and c) the h y d r o -m e t e o r o l o g i c a l c o n d i t i o n s a long the boundar ies o f the flow r e g i o n . There are two g e n e r a l approaches that can be used to ana lyze flow c o n d i t i o n s i n reg ions c o n t a i n i n g a seepage f ace . The f i r s t , known as the " f r e e - s u r f a c e " approach , assumes that f low above the water t a b l e i s n e g l i g i b l e so t h a t o n l y the s a t u r a t e d p o r t i o n of the flow r e g i o n i s a n a l y z e d . The advantage o f t h i s approach i s i t s s i m p l i c i t y ; the govern ing equa t ion of flow i s r e l a t i v e l y easy to s o l v e w i t h e x i s t i n g a n a l y t i c a l or n u m e r i c a l t e c h n i q u e s . The d i sadvantage i s tha t i t does not g e n e r a l l y a l low one to s p e c i f y a recharge r a te to the s a t u r a t e d zone. The second approach i n v o l v e s an a n a l y s i s of both the s a t u r a t e d and unsa tura ted flow r e g i o n s . T h i s approach i s more p h y s i c a l l y - b a s e d and i s g e n e r a l l y amenable to n u m e r i c a l s o l u t i o n . However, the data r e q u i r e d to c h a r a c t e r i z e the h y d r a u l i c p r o p e r t i e s o f the unsa tura ted zone are d i f f i c u l t to measure on a r o u t i n e b a s i s . In a d d i t i o n , n u m e r i c a l d i f f i c u l t i e s can prevent one from o b t a i n i n g a s o l u t i o n i n some i n s t a n c e s . Both approaches have been a p p l i e d s u c c e s s f u l l y to homogeneous h i l l s i d e s c o n t a i n i n g a s i n g l e , cont inuous seepage f a c e . However, i n n a t u r e , we observe that h i l l s i d e s may e x h i b i t s e v e r a l d i s c o n t i n u o u s seepage faces r e s u l t i n g from the complex, s a t u r a t e d - u n s a t u r a t e d flow systems that deve lop i n heterogeneous g e o l o g i c a l env i ronments . These f low systems 3 f ea ture perched water t a b l e s and wedge-shaped unsa tura ted zones l o c a t e d above the main s a t u r a t e d zone and must be s t u d i e d w i t h a s a t u r a t e d - u n s a t u r a t e d a n a l y s i s . The work presented i n t h i s t h e s i s p r o v i d e s such an a n a l y s i s of the f l u i d - p r e s s u r e d i s t r i b u t i o n and geometry of m u l t i p l e seepage faces on heterogeneous h i l l s i d e s . The s tudy i s l i m i t e d to h o r i z o n t a l l y l a y e r e d s l o p e s , as these are an important type of heterogeneous g e o l o g i c a l environment . From a t h e o r e t i c a l s t a n d p o i n t , t h i s r e s e a r c h w i l l : a) c l a r i f y the nature of the f l u i d - p r e s s u r e d i s t r i b u t i o n and h y d r a u l i c - h e a d d i s t r i b u t i o n i n l a y e r e d s lopes c o n t a i n i n g more than one seepage f a c e , b) i n v e s t i g a t e the r e l a t i v e q u a n t i t i e s of out f low from the seepage f a c e s , c) i n d i c a t e which combinat ions of h y d r o g e o l o g i c v a r i a b l e s are most l i k e l y to produce m u l t i p l e seepage f a c e s , d) show the response of these systems to t r a n s i e n t r a i n f a l l e v e n t s , and e) p r o v i d e i n s i g h t i n t o the mechanisms by which perched flow systems form. From a p r a c t i c a l s t a n d p o i n t , t h i s r e s e a r c h w i l l a i d f i e l d s t u d i e s r e q u i r i n g an unders tanding of the groundwater c o n d i t i o n s i n l a y e r e d s l o p e s . Dur ing the w r i t i n g s tages of t h i s t h e s i s , Coo ley (1983) p u b l i s h e d a s tudy i n which he developed a new technique to overcome some of the n u m e r i c a l d i f f i c u l t i e s p a r t i c u l a r to s a t u r a t e d - u n s a t u r a t e d m o d e l i n g . He present s s e v e r a l examples of the a p p l i c a b i l i t y of h i s model , i n c l u d i n g a dra inage problem i n v o l v i n g m u l t i p l e seepage f a c e s . To the a u t h o r ' s knowledge, t h i s i s the o n l y p r e v i o u s l y p u b l i s h e d attempt to model m u l t i p l e 4 seepage f a c e s . Whi le C o o l e y ' s work dea l s p r i m a r i l y wi th h i s n u m e r i c a l methodology, the emphasis of the s tudy presented here i s p l a c e d on the f a c t o r s c o n t r o l l i n g the o c c u r r e n c e of m u l t i p l e seepage f a c e s . There are four o b j e c t i v e s of t h i s s tudy . The f i r s t i s to s e l e c t a f i n i t e - e l e m e n t model that w i l l p r e d i c t the f l u i d -pre s sure d i s t r i b u t i o n and seepage-face l o c a t i o n s i n l a y e r e d , heterogeneous h i l l s i d e s . The second i s to b u i l d a l a b o r a t o r y model to v e r i f y the p h y s i c a l f ounda t ion of the s o l u t i o n s generated by the n u m e r i c a l model . The t h i r d o b j e c t i v e i s to use the n u m e r i c a l model i n s e n s i t i v i t y s t u d i e s des igned to reach q u a n t i t a t i v e c o n c l u s i o n s about the f a c t o r s govern ing the development of m u l t i p l e seepage f a c e s . The f i n a l o b j e c t i v e i s to form g e n e r a l i z e d c o n c l u s i o n s r e g a r d i n g the importance of m u l t i p l e seepage faces i n g e o t e c h n i c a l , h y d r o g e o l o g i c a l , and geomorpho log ica l problems . I t i s not the i n t e n t i o n of t h i s s tudy to p r o v i d e a model to be used r o u t i n e l y i n p r a c t i c a l , s i t e - s p e c i f i c problems; the amount of f i e l d data r e q u i r e d by the model would be p r o h i b i t i v e i n most c a s e s . I n s t e a d , the model i s best s u i t e d to g e n e r i c s t u d i e s , tha t i s , i n the a p p l i c a t i o n to r e l a t i v e l y s i m p l e , h y p o t h e t i c a l h i l l s i d e s to assess g e n e r a l mechanisms. The r e s u l t s of these g e n e r i c s t u d i e s should prove u s e f u l i n the development of a p p r o p r i a t e s t r a t e g i e s to r e s o l v e i n d i v i d u a l , p r a c t i c a l problems . For example, a g e n e r a l under s tand ing of the w a t e r - t a b l e c o n f i g u r a t i o n and the geometry o f the s a t u r a t e d and unsa tura ted flow reg ions would be u s e f u l when d e s i g n i n g data c o l l e c t i o n networks i n l a y e r e d media . 5 T h i s t h e s i s i s o r g a n i z e d i n the f o l l o w i n g manner: Chapter 2 p r o v i d e s a d e s c r i p t i o n of the mathemat ica l model , Chapter 3 d e s c r i b e s the l a b o r a t o r y work performed to v e r i f y the mathemat ica l model , Chapter 4 d e s c r i b e s the s t e a d y - s t a t e s e n s i t i v i t y s t u d y , Chapter 5 p re sent s the r e s u l t s of two p r e l i m i n a r y t r a n s i e n t s t u d i e s , Chapter 6 d i s c u s s e s a p p l i c a t i o n s of the r e s u l t s to g e o t e c h n i c a l , h y d r o g e o l o g i c a l and geomorpholog ica l problems , and Chapter 7 c o n t a i n s the summary and c o n c l u s i o n s . 6 Chapter 2 THE MATHEMATICAL MODEL Freeze (1978) o u t l i n e d four s teps i n v o l v e d i n b u i l d i n g a mathemat ica l model . The f i r s t s t ep i s to d e s c r i b e the p h y s i c a l prob lem. The second s tep i s to r e p l a c e the p h y s i c a l problem w i t h an e q u i v a l e n t boundary-va lue prob lem, n o t i n g the assumptions r e q u i r e d to e s t a b l i s h an e q u i v a l e n c e . The t h i r d s tep i s to choose a mathemat ica l t echnique to s o l v e the boundary-va lue problem. These three s teps are d i s c u s s e d i n t h i s c h a p t e r . The f i n a l s t e p , i n t e r p r e t i n g the mathemat ica l r e s u l t s i n terms of the p h y s i c a l prob lem, w i l l be d i s c u s s e d l a t e r i n the t h e s i s , f o l l o w i n g the s e n s i t i v i t y s t u d i e s . 2.1 The P h y s i c a l Problem The p h y s i c a l problem i n t r o d u c e d i n Chapter 1 i s i l l u s t r a t e d i n a s i m p l i f i e d form i n F i g u r e 1. The r e g i o n ABCDEFA r e p r e s e n t s a t w o - d i m e n s i o n a l , v e r t i c a l c ro s s s e c t i o n through a l a y e r e d h i l l s i d e . Each g e o l o g i c u n i t i s homogeneous and i s o t r o p i c w i t h r e spec t to 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 of the shaded l a y e r i s l e s s than t h a t of the s u r r o u n d i n g m a t e r i a l . The flow r e g i o n i s bounded below by impermeable s t r a t a . The boundar ies AB and CD r e p r e s e n t groundwater d i v i d e s and p r o v i d e v e r t i c a l impermeable b o u n d a r i e s . Water may i n f i l t r a t e i n t o the r e g i o n a long ED and may d i s c h a r g e i n t o a r i v e r l o c a t e d a long AF and ac ros s seepage f a c e s , as they deve lop a long F E . The p h y s i c a l problem i s to F i g u r e 1 . H y p o t h e t i c a l f l o w r e g i o n . 8 determine the s a t u r a t e d - u n s a t u r a t e d flow p a t t e r n s r e s p o n s i b l e for the development of m u l t i p l e seepage faces i n t h i s type of h y d r o g e o l o g i c env i ronment . 2.2 The Boundary-Value Problem Toth (1962, 1963) was the f i r s t h y d r o g e o l o g i s t to r e c o g n i z e tha t r e g i o n a l groundwater f low p a t t e r n s c o u l d be o b t a i n e d m a t h e m a t i c a l l y as s o l u t i o n s to boundary-va lue problems . The method has become a s tandard t h e o r e t i c a l approach for s o l v i n g both r e g i o n a l and l o c a l groundwater f low problems and w i l l be used i n the present s t u d y . To set up the boundary-va lue problem for t r a n s i e n t flow c o n d i t i o n s i n the r e g i o n shown i n F i g u r e 1, one needs a govern ing e q u a t i o n of f l o w , knowledge of the h y d r o g e o l o g i c parameters tha t c o n t r o l the f l o w , boundary c o n d i t i o n s , and i n i t i a l c o n d i t i o n s . These requirements are d i s c u s s e d i n t h i s s e c t i o n . T r a n s i e n t , s a t u r a t e d - u n s a t u r a t e d flow through a two-d i m e n s i o n a l , heterogeneous and i s o t r o p i c flow r e g i o n i s governed by the f o l l o w i n g e q u a t i o n : where = p r e s s u r e head, [ L ] , K = h y d r a u l i c c o n d u c t i v i t y , [ L / T ] , C = s p e c i f i c moi s ture c a p a c i t y , [ 1 / L ] , S s = s p e c i f i c s t o r a g e , [ 1 / L ] , 0 = v o l u m e t r i c water c o n t e n t , [ L 3 / L 3 ] , n = 2.1 p o r o s i t y , [ L ^ / L 3 ] , t = t i m e , [ T ] , and x , z = h o r i z o n t a l and 9 v e r t i c a l c o o r d i n a t e d i r e c t i o n s , [ L ] . T h i s e q u a t i o n i s a combina t ion of the s a t u r a t e d flow equa t ion developed by Jacob (1940), l a t e r c l a r i f i e d by Cooper (1966), and the unsa tura ted flow e q u a t i o n developed by R ichards (1931); these equat ions were coup led by Freeze (1971) . The h y d r o g e o l o g i c parameters tha t appear i n Equa t ion 2.1 w i l l be examined f i r s t , f o l lowed by a d i s c u s s i o n of the assumptions and l i m i t a t i o n s i m p l i c i t i n the use of E q u a t i o n 2 . 1 . The f i r s t term to be examined i s the pre s sure head, \ J J . I t appears as the dependent v a r i a b l e i n the govern ing e q u a t i o n o f f l o w . I t i s r e l a t e d to the t o t a l mechanica l energy of the f l u i d at a p o i n t by the e x p r e s s i o n : h = TJJ + z 2.2 where h = h y d r a u l i c head = mechan ica l energy per u n i t weight o f f l u i d , [ L J , and z = e l e v a t i o n head = e l e v a t i o n o f the p o i n t wi th r e s p e c t to an a r b i t r a r y datum, [ L ] . The pre s sure head i s d e f i n e d by : i|> = p/pg 2.3 where p = f l u i d p re s sure expressed i n terms o f gage p r e s s u r e , [ M T ~ 2 L - 1 ] , p = f l u i d d e n s i t y , [ M / L 3 ] , and g = g r a v i t a t i o n a l a c c e l e r a t i o n , [ L / T ^ ] . The s o l u t i o n o f E q u a t i o n 2.1 i s the d i s t r i b u t i o n o f I)J throughout the f low r e g i o n at a g i v e n time t . From t h i s f i e l d , 10 we can i d e n t i f y the p o s i t i o n of the water t a b l e as the i|> = 0 i s o b a r , and d i s t i n g u i s h between the s a t u r a t e d zone, where i|> > 0, and the unsa tura ted zone, where \p < 0. The second term to be examined i s the h y d r a u l i c c o n d u c t i v i t y , K, which i s a measure of the ease w i t h which water passes through the porous medium. I t i s a p r o p e r t y o f both the f l u i d and the porous medium and i s g i v e n by : K - ^ 2.4 where k = p e r m e a b i l i t y , [ L ^ ] , and u = dynamic v i s c o s i t y of the f l u i d , [ M L - l T - 1 ] . The h y d r a u l i c c o n d u c t i v i t y appears as the p r o p o r t i o n a l i t y cons t an t i n D a r c y ' s law, w r i t t e n here for two-d imens iona l flow through a s a t u r a t e d - u n s a t u r a t e d , homogeneous and i s o t r o p i c r e g i o n : v x = -KOp) — 2.5a 3 x v z = -K0l>) ^ .. 2.5b 9 z where v x , v z = s p e c i f i c d i s c h a r g e i n the x and z d i r e c t i o n s , [ L / T ] . Note the dependence of the c o n d u c t i v i t y on the pre s sure head . An example of a K(4>) r e l a t i o n s h i p i s i l l u s t r a t e d i n F i g u r e 2a. For t|> > \|»A, K i s a c o n s t a n t ; for I|J < \p a , K v a r i e s over many o r d e r s of magnitude. ^ a i s known as the a i r - e n t r y va lue o f a s o i l and i s the pre s sure head at which a i r f i r s t n K (m/s) - - l O " 5 T 1 1 1 1 1 1—~f-\ 1 1 1 r**" -6 -4 -2 <i>a0 2 4 a. K(^) relationship J i #<m3/m3) I I I I I I I -0.30 1 1 1 r^" -6 -4 -2 ^ A 0 2 4 b. 6((/J) relationship F i g u r e 2 . Examples o f c h a r a c t e r i s t i c c u r v e s . 12 e n t e r s the pore spaces of a s o i l as i t d e s a t u r a t e s . Note a l s o t h a t the K ) r e l a t i o n s h i p i s h y s t e r e t i c . The two bounding curves are known as the pr imary d r y i n g and pr imary wet t ing c u r v e s . There are an i n f i n i t e number of scanning curves between t h e s e ; two have been i n d i c a t e d i n F i g u r e 2a. The dependence of K on ^ i n the unsa tura ted zone i s r e l a t e d to the v a r i a t i o n o f ^ w i t h water c o n t e n t , 9 . An example o f a 9 ) r e l a t i o n s h i p i s shown i n F i g u r e 2b. For > t | J a , 0 i s a cons tant equa l to the p o r o s i t y of the s o i l , n . For i|» < 9 d e c r e a s e s , as shown. As the water content of a s o i l d e c r e a s e s , the t e n s i o n a l f o r c e s a c t i n g upon the s o i l water i n c r e a s e , so t h a t the f l u i d p r e s s u r e , and hence the p re s sure head, d e c r e a s e s . In a d d i t i o n , as the water content decrea se s , the c r o s s s e c t i o n a l area a v a i l a b l e for flow a l s o decreases and we observe a s teep d e c l i n e i n the h y d r a u l i c c o n d u c t i v i t y . The K ( ^ ) and 9 (ij>) curves are c o l l e c t i v e l y termed the c h a r a c t e r i s t i c curves o f a s o i l . A l though attempts have been made, i t appears that they cannot be d e r i v e d a n a l y t i c a l l y from fundamental p h y s i c a l l aws ; they must be e i t h e r measured or e s t imated for each s o i l . Fur ther i n f o r m a t i o n c o n c e r n i n g the c h a r a c t e r i s t i c curves can be found i n s tandard s o i l - p h y s i c s t e x t s such as H i l l e l (1980). The l a s t two terms i n E q u a t i o n 2.1 to be examined are the s p e c i f i c mo i s ture c a p a c i t y , C , and the s p e c i f i c s t o r a g e , S s ; these r e l a t e to the s torage p r o p e r t i e s of the porous medium. The s p e c i f i c moi s ture c a p a c i t y d e s c r i b e s the dominant mechanism by which water i s r e l e a s e d from storage i n the unsa tura ted 13 zone, namely, d e s a t u r a t i o n . I t i s d e f i n e d as the change i n water content per u n i t change i n pre s sure head: C(i|0 = ^ - 2.6 dip I t i s not a c o n s t a n t , but r a t h e r , i t i s a h y s t e r e t i c f u n c t i o n of 4» and i s o b t a i n e d from the s lope of the 9 (ip) c u r v e . The s p e c i f i c s t o r a g e , S s , r e f l e c t s the dominant mechanisms by which water i s r e l e a s e d from s torage i n the s a t u r a t e d zone. I t i s d e f i n e d as the volume of water r e l e a s e d from s torage per u n i t volume of the medium as a r e s u l t of a u n i t d e c l i n e i n the h y d r a u l i c head. A d e c l i n e i n the h y d r a u l i c head i s accompanied by a d e c l i n e i n the f l u i d p r e s s u r e . T h i s , i n t u r n , t r i g g e r s two mechanisms t h a t r e l e a s e water from s t o r a g e . F i r s t , the e f f e c t i v e s t r e s s i n c r e a s e s c a u s i n g the r e l e a s e of water as the porous medium compacts . Second, the f l u i d expands, thereby r e l e a s i n g a d d i t i o n a l water . The components of the s p e c i f i c s torage r e f l e c t these two mechanisms: S s = pg(a + n8) 2.7 where a = c o m p r e s s i b i l i t y o f the porous medium, [ L T 2 / M ] / a n ( j g = c o m p r e s s i b i l i t y of water [ L T 2 / M ] . The c o m p r e s s i b i l i t y of the porous medium i s determined from the s lope of the s t r e s s - s t r a i n diagram t h a t r e l a t e s the v o i d r a t i o to the e f f e c t i v e s t r e s s for a s o i l . The c o m p r e s s i b i l i t y of water , 8 ' i s assumed to be a c o n s t a n t , e q u a l to 4.4 x 1 0 - I ° m 2 / N . 14 Data r e p r e s e n t i n g K (\J;) , 8(\|>), C ) , and S g for a v a r i e t y of g e o l o g i c m a t e r i a l s are a v a i l a b l e i n the l i t e r a t u r e . Mualem (1976) compi led data on the unsa tura ted p r o p e r t i e s of over 80 s o i l s . From t h i s r e p o r t , c h a r a c t e r i s t i c curves can be chosen to r e p r e s e n t n a t u r a l l y o c c u r r i n g a g r i c u l t u r a l s o i l s . The e f f e c t s of h y s t e r e s i s w i l l not be c o n s i d e r e d i n t h i s t h e s i s ; o n l y the pr imary d r y i n g p o r t i o n of each curve w i l l be used . The va lues o f S s can be computed from Table 1, which l i s t s va lues of a for v a r i o u s types o f m a t e r i a l . Tab le 1. Range of va lues of c o m p r e s s i b i l i t y . a (m 2/N) c l a y 10~ 6 - 10~ 8 sand 1 0 - 7 - 1 0 - 9 g r a v e l 1 0 - 8 - 1 0 - 1 0 j o i n t e d rock 1 0 - 8 - 1 0 - 1 0 sound rock 1 0 - 9 - 1 0 - 1 1 (Source : Freeze and C h e r r y , 1979) The d e r i v a t i o n o f E q u a t i o n 2 . 1 , as p re sented i n Freeze and Cherry (1979), r e q u i r e s s e v e r a l a s sumpt ions . F i r s t , i t assumes that the a i r phase i n the unsa tura ted zone i s cont inuous and i s ma in ta ined at a tmospher ic p r e s s u r e . The e f f e c t s o f entrapped a i r are t h e r e f o r e n e g l e c t e d . Second, i t assumes tha t D a r c y ' s law i s v a l i d . T h i s i m p l i e s t h a t flow i s l aminar and the o n l y .sources o f energy d r i v i n g f l u i d f low are d i f f e r e n c e s i n the 15 e l e v a t i o n head and i n the pre s sure head. The model i s t h e r e f o r e r e s t r i c t e d to flow reg ions where the f o l l o w i n g proces se s are i n s i g n i f i c a n t : a) flow through macropores , b) flow caused by thermal g r a d i e n t s , and c) flow caused by the uptake o f water through p l a n t r o o t s . E qua t ion 2.1 i s developed for i s o t r o p i c porous media and i t can o n l y be a p p l i e d to r e g i o n s where flow i s p redominant ly t w o - d i m e n s i o n a l . The e f f e c t s of e v a p o t r a n s p i r a t i o n are a l s o i g n o r e d . S e v e r a l assumptions are necessary w i t h r e s p e c t to the c o m p r e s s i b i l i t y o f the porous medium. We assume that the porous medium i s compres s ib le but the i n d i v i d u a l s o i l g r a i n s are n o t . The porous medium i s assumed to be l i n e a r l y and r e v e r s i b l y e l a s t i c . The t o t a l s t r e s s i s assumed to be cons tant and a c t i n the v e r t i c a l d i r e c t i o n o n l y . We w i l l a l s o assume t h a t the va lue of S s remains cons tant i n both the s a t u r a t e d and unsa tura ted zones . T h i s i s a reasonable assumption i n the s a t u r a t e d zone, however, e f f e c t i v e s t r e s s i s a complex and i n c o m p l e t e l y understood f u n c t i o n of the negat ive f l u i d p re s sure s tha t occur i n the unsa tura ted zone. F o r t u n a t e l y : c w 11 » £ s s Si a t n 3t i n the unsa tura ted zone, so we can use a cons t an t S s va lue wi thout i n t r o d u c i n g s i g n i f i c a n t e r r o r s . In order to s o l v e the flow e q u a t i o n , the p h y s i c a l c o n d i t i o n s that e x i s t a long the boundar ies must be expressed m a t h e m a t i c a l l y i n terms of ty. The boundary c o n d i t i o n s w i l l be developed wi th r e f e r e n c e to F i g u r e 1. 16 On the impermeable b a s a l boundary, BC, there i s no flow i n the z - d i r e c t i o n so that & = 0 3z o r , i n terms of \p, ^ - -1 2.8 3z Along the v e r t i c a l impermeable boundar i e s , AB and CD, there i s no flow i n the x - d i r e c t i o n so t h a t : a* = o o r , i n terms of vp: l i = 0 2.9 3x One can view these boundar ies as p r e s c r i b e d - f l u x boundar ies on which the f l u x i s z e r o . The base o f the s t ream, A F , i s a p r e s c r i b e d - ^ boundary, a long which where zs i s the depth of water o v e r l y i n g the h o r i z o n t a l stream bottom. We w i l l assume throughout t h i s s tudy tha t the boundary c o n d i t i o n s g i v e n by Equat ions 2.8 through 2.10 remain cons tant w i t h t i m e . The i n f i l t r a t i o n boundary, ED, and the seepage face boundary, F E , may c o n t a i n both p r e s c r i b e d - f l u x and p r e s c r i b e d - ^ 17 segments. For example, p o r t i o n s of the i n f i l t r a t i o n boundary may exper ience i n c i p i e n t ponding so that \|> i s p r e s c r i b e d to be z e r o ; other p o r t i o n s may remain unsa tura ted so t h a t the f l u x r a t e i s s p e c i f i e d as the r a i n f a l l r a t e . S i m i l a r l y , those face w i l l have a p r e s c r i b e d equa l to z e r o ; those p o r t i o n s tha t remain unsa tura ted w i l l have a p r e s c r i b e d f l u x equa l to z e r o . S ince c o n d i t i o n s a long both ED and FE may vary i n response to v a r i a t i o n s i n the pres sure-head d i s t r i b u t i o n throughout the e n t i r e flow r e g i o n , these repre sent t r a n s i e n t boundary c o n d i t i o n s . T h e i r n u m e r i c a l treatment w i l l be d i s c u s s e d i n more d e t a i l i n the f o l l o w i n g s e c t i o n d e s c r i b i n g the mathemat ica l method of s o l u t i o n . In a d d i t i o n to boundary c o n d i t i o n s , one needs an i n i t i a l d i s t r i b u t i o n o f vp(x,z) at t = 0 i n order to s o l v e Equa t ion 2 . 1 . The d i s t r i b u t i o n can e i t h e r r e p r e s e n t s t a t i c c o n d i t i o n s , whereby the water t a b l e i s h o r i z o n t a l and i s at the same e l e v a t i o n as the stream s u r f a c e , o r , s t e a d y - s t a t e flow c o n d i t i o n s . In the l a t t e r c a s e , the s t e a d y - s t a t e form of the govern ing e q u a t i o n must be s o l v e d : s u b j e c t to a se t o f s t e a d y - s t a t e boundary c o n d i t i o n s s i m i l a r to those a l r e a d y d e s c r i b e d . p o r t i o n s o f the seepage face boundary tha t c o n t a i n a seepage 1 - K ( x , z , ^ ) 11 9* [_ 3x K(x,z ,40 (11 + 1) ].. 2.11 2.3 The N u m e r i c a l Method o f S o l u t i o n 18 The govern ing equa t ion of flow i s a n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n . I t i s very d i f f i c u l t to s o l v e by exact a n a l y t i c a l methods, p a r t i c u l a r l y i n heterogeneous reg ions c o n t a i n i n g complex boundary c o n d i t i o n s . Approximate n u m e r i c a l t echniques have been developed to s o l v e these types of equat ions w i t h r e l a t i v e ease . Rubin (1968) and Freeze (1971) s o l v e d s i m i l a r equat ions d e s c r i b i n g s a t u r a t e d - u n s a t u r a t e d flow wi th f i n i t e - d i f f e r e n c e models . Neuman (1972, 1973) adapted the G a l e r k i n f i n i t e - e l e m e n t method to the a n a l y s i s . The computer program named UNSAT I , w r i t t e n and documented by Neuman (1972) was s e l e c t e d and m o d i f i e d for use i n the present s t u d y . UNSAT I can be used to model t w o - d i m e n s i o n a l , t r a n s i e n t , n o n h y s t e r e t i c f low through heterogeneous , a n i s o t r o p i c , s a t u r a t e d - u n s a t u r a t e d r e g i o n s . The documentat ion presented by Neuman (1972) g i v e s a complete d e s c r i p t i o n of the d e r i v a t i o n of the f i n i t e - e l e m e n t equat ions and the use of the model . T h i s i n f o r m a t i o n w i l l not be repeated h e r e . I n s t e a d , the b a s i c idea behind the method w i l l be p r e s e n t e d . The problems p a r t i c u l a r to the s o l u t i o n of the g i v e n boundary-va lue problem w i l l be d i s c u s s e d , and the l i m i t a t i o n s i n t r o d u c e d by the f i n i t e - e l e m e n t a n a l y s i s w i l l be n o t e d . In order to app ly the f i n i t e - e l e m e n t method, the f low r e g i o n i s f i r s t d i v i d e d i n t o a se t of t r i a n g u l a r and/or q u a d r i l a t e r a l subreg ions known as e lements . Each element i s d e f i n e d by the l i n e s j o i n i n g the corner nodal p o i n t s and each element i s a s s i gned a se t of h y d r a u l i c p r o p e r t i e s . The cont inuous p a r t i a l d i f f e r e n t i a l equa t ion i s then r e p l a c e d at 19 each node by an approximate a l g e b r a i c e q u a t i o n to produce a se t o f N d i s c r e t e e q u a t i o n s , where N = number of noda l p o i n t s i n the f low r e g i o n . To do t h i s , the govern ing e q u a t i o n o f flow i s r e w r i t t e n a s : L(i|;) = 0 2.12 where L i s a d i f f e r e n t i a l opera tor g i v e n by L(i|>) = 3x K(x , z , i p ) 3x 3_ 3z K(x,z , 1 | ; ) 3z (C (Tp) + £ Sg) 3t An approximate f u n c t i o n , i | i ( x , z , t ) , which s a t i s f i e s the boundary and i n i t i a l c o n d i t i o n s i s chosen and d e f i n e d as N \p (x ,z , t ) = Z tyn(t) <Sn(x,z) n = 1.2 N n=l 2.13 where ^nt t ) = t n e exact va lue o f (t) at each nodal p o i n t , and 5 n ( x , z ) are l i n e a r l y independent f u n c t i o n s of the s p a t i a l c o o r d i n a t e s , known as b a s i s , shape, or c o o r d i n a t e f u n c t i o n s . When i s r e p l a c e d by Equa t ion 2.12 becomes: L(\|>) = R(x , z ) / 0 2.14 where R(x , z ) i s the r e s i d u a l , or e r r o r , c r e a t e d by the a p p r o x i m a t i o n . To minimize the r e s i d u a l throughout the flow r e g i o n , V , we perform the f o l l o w i n g i n t e g r a t i o n : 20 / w ( x , z ) R ( x , z ) d V = 0 V 2.15 where w(x,z) i s a we ight ing f u n c t i o n , a l s o a f u n c t i o n o n l y o f the s p a t i a l c o o r d i n a t e s . D i f f e r e n t methods can be used at t h i s p o i n t i n the f i n i t e - e l e m e n t f o r m u l a t i o n , depending upon the G a l e r k i n method i n which the b a s i s f u n c t i o n s are chosen as the w e i g h t i n g f u n c t i o n . Equa t ion 2.15 becomes: W r i t i n g R(x , z ) i n terms o f e q u a t i o n 2.14 and r e p l a c i n g vp by the e x p r e s s i o n g i v e n i n E q u a t i o n 2 .13 , we o b t a i n : T h i s i n t e g r a l i s then e v a l u a t e d at a l l nodes to produce a se t o f s imul taneous e q u a t i o n s , expressed i n mat r ix form as [A] i s the conductance , or g l o b a l s t i f f n e s s , m a t r i x . I t i s an N x N m a t r i x and i s s p a r s e , banded, and symmetr ic . [F] i s the c a p a c i t a n c e m a t r i x . I t expresses the a b i l i t y of the r e g i o n to absorb or r e l e a s e water from s torage due to a change i n p re s sure head . I t i s an N x N, d i a g o n a l m a t r i x . {ty} i s the vec tor c o n t a i n i n g the unknown va lues of ^ at i n d i v i d u a l nodes . c h o i c e of the we ight ing f u n c t i o n . Neuman (1972) uses the / 5i (x ,z) R ( x , z ) d V = 0 V 2.16 [A] {>} + [F] (^ ) = (Q) - (B) 21 {Q} i s a v e c t o r that c o n t a i n s the f l u x ac ros s the boundary nodes o f the flow r e g i o n . I t i s equa l to zero at a l l i n t e r n a l nodes that do not ac t as sources or s i n k s . {B} c o n t a i n s the f l u x at each node due to g r a v i t y a l o n e . A f u l l y i m p l i c i t time scheme i s used to e v a l u a t e the time d e r i v a t i v e , 3ip/31. The time domain i s d i s c r e t i z e d i n t o a sequence of t i m e s t e p s . At the beg inning o f each t i m e s t e p , p r e d i c t i o n s o f the new va lues of are made, based on the va lues from the p r e v i o u s t i m e s t e p . M o d i f i c a t i o n s are then made to the m a t r i x e q u a t i o n to i n c o r p o r a t e the boundary c o n d i t i o n s and the m a t r i x e q u a t i o n i s s o l v e d by Gauss ian e l i m i n a t i o n for -C\L»>- These r e s u l t s are improved by an i t e r a t i v e process u n t i l a s a t i s f a c t o r y degree of convergence i s a c h i e v e d . F u r t h e r i n f o r m a t i o n c o n c e r n i n g the d e t a i l s of f i n i t e - e l e m e n t model ing can be found i n t e x t s such as Wang and Anderson (1982), P inder and Gray (1977) , Z i e n k i e w i c z (1977), and Bathe and W i l s o n (1976) . The problems p a r t i c u l a r to the s o l u t i o n of the g i v e n boundary-va lue problem r e l a t e to the treatment of the boundary c o n d i t i o n s . Cons tant-head boundar ies and f i x e d impermeable boundar ies are e a s i l y handled by the f i n i t e - e l e m e n t method; the equat ions c o r r e s p o n d i n g to these boundary nodes are m o d i f i e d to ensure that \J> = p r e s c r i b e d value and Q = 0, r e s p e c t i v e l y . However, the t r a n s i e n t boundary c o n d i t i o n s t h a t occur a long the seepage-face boundary and the i n f i l t r a t i o n boundary r e q u i r e s p e c i a l t r e a t m e n t . 22 The seepage-face boundary i s s p e c i a l because the p o s i t i o n and l e n g t h of a seepage face may vary u n p r e d i c t a b l y as the p re s sure head v a r i e s wi th time throughout the r e g i o n . The boundary c o n d i t i o n i s t h e r e f o r e a f u n c t i o n o f the dependent v a r i a b l e and cannot be f i x e d a p r i o r i ; an i t e r a t i v e scheme i s r e q u i r e d to a r r i v e at the proper boundary c o n d i t i o n . Neuman (1972) employed the f o l l o w i n g scheme to p r e d i c t the p o s i t i o n at each t i m e s t e p . Dur ing the f i r s t i t e r a t i o n , set = 0 a long the seepage face and t r e a t t h i s segment of the boundary as a p r e s c r i b e d - i p boundary. Set Q = 0 a long the unsa tura ted p o r t i o n o f the boundary and t r e a t t h i s segment as a p r e s c r i b e d - f l u x boundary. The m a t r i x e q u a t i o n i s then s o l v e d w i t h the e x p e c t a t i o n s tha t a) the newly c a l c u l a t e d value of Q i s n e g a t i v e , i n d i c a t i n g that flow i s d i r e c t e d out of the porous medium, o n l y a long the p r e s c r i b e d - ^ segment, and b) the newly c a l c u l a t e d va lue o f i s nega t ive o n l y where Q was p r e v i o u s l y se t equa l to z e r o . I f these e x p e c t a t i o n s are not met, the boundary c o n d i t i o n s at the e r r a n t nodes are r e d e f i n e d to agree wi th the new s o l u t i o n . T h i s procedure i s repeated u n t i l the s o l u t i o n converges w i t h i n a g i v e n t i m e s t e p . The i t e r a t i v e scheme e x p l o i t s the ease w i t h which the f i n i t e - e l e m e n t method can formulate and modify boundary c o n d i t i o n s . In order to improve the convergence r a t e , Neuman (1972) des igned the i t e r a t i v e procedure so that the boundary c o n d i t i o n s are always m o d i f i e d s e q u e n t i a l l y , from node to node, beg inn ing at the base and proceed ing to the top o f the seepage-face boundary. In a d d i t i o n , i f i t becomes neces sary d u r i n g an 23 i t e r a t i o n to se t Q = 0 at any node, Q at a l l the h igher nodes on the boundary are a u t o m a t i c a l l y se t equa l to z e r o . T h i s a spect of the i t e r a t i v e scheme was removed for the present s tudy to a l l ow for the development of more than one seepage face a long the h i l l s l o p e . P r i o r to t h i s m o d i f i c a t i o n , i t was p o s s i b l e to model more than one seepage face a long a s l o p e , p r o v i d e d t h a t the lowermost node a s s o c i a t e d w i t h each seepage face c o u l d be s p e c i f i e d a p r i o r i . The m o d i f i c a t i o n removed t h i s r e s t r i c t i o n . The i n f i l t r a t i o n boundary a l s o r e q u i r e s s p e c i a l treatment because the f l u x acros s the s o i l sur face cannot be s p e c i f i e d a p r i o r i ; i t depends upon the antecedent mois ture c o n d i t i o n s and the i n f i l t r a t i o n c a p a c i t y o f the s o i l . Dur ing the f i r s t i t e r a t i o n i n each t i m e s t e p , the i n f i l t r a t i o n boundary i s t r e a t e d as a p r e s c r i b e d - f l u x boundary. Each node i s a s s igned an a r b i t r a r y f r a c t i o n of the r a i n f a l l r a t e . The mat r ix equat ions are s o l v e d and i f the computed value o f ij> i s nega t ive at some nodes a long the boundary, the p r e s c r i b e d f l u x i s i n c r e a s e d . I f a p o r t i o n of the i n f i l t r a t i o n boundary becomes ponded d u r i n g an i t e r a t i o n , then tha t p o r t i o n i s t r e a t e d as a p r e s c r i b e d - ^ boundary wi th vp = 0. T h i s proces s i s repeated u n t i l convergence occur s and each node a long the i n f i l t r a t i o n boundary i s e i t h e r unsa tura ted and t r a n s m i t t i n g the f u l l r a i n f a l l r a t e , or i s ponded and t r a n s m i t t i n g water at a r a te l e s s than the imposed r a i n f a l l r a t e . T h i s i t e r a t i v e scheme was i n t r o d u c e d by Neuman e t . a l . (1974, 1975) i n a program t i t l e d UNSAT I I . UNSAT I was m o d i f i e d for the present s tudy to i n c o r p o r a t e the a l g o r i t h m . 24 S e v e r a l l i m i t a t i o n s are i n t r o d u c e d by the f i n i t e - e l e m e n t a n a l y s i s . A node that i s l o c a t e d on the seepage-face boundary cannot be i n c l u d e d i n the i n f i l t r a t i o n boundary. T h e r e f o r e , r a i n f a l l must be r e s t r i c t e d to the f l a t , upland sur face and i n f i l t r a t i o n cannot be modeled a long the s l o p e . In a d d i t i o n , there i s no mechanism to route sur f ace r u n o f f . When ponding occur s on the i n f i l t r a t i o n s u r f a c e , or when water d i s c h a r g e s acros s a seepage f a c e , the subsequent path of the runof f i s not modeled. Fur thermore , the model does not take i n t o account the i n t e r a c t i o n s between the subsurface flow system and changes i n the stream l e v e l . The model i s a l s o l i m i t e d by the nature of the c h a r a c t e r i s t i c c u r v e s . R e c a l l from F i g u r e 2 , t h a t where the s o i l i s very d r y , the pres sure-head g r a d i e n t i s s t eep . In a d d i t i o n , the h y d r a u l i c c o n d u c t i v i t y v a r i e s i n the unsa tura ted zone over s e v e r a l o rder s of magnitude for s m a l l changes i n p re s sure head. These two f a c t o r s can combine to produce slow convergence r a t e s , o r , i n some i n s t a n c e s , n u m e r i c a l i n s t a b i l i t y . To h e l p a v o i d such problems , one can i n c r e a s e the number of nodes and space them more c l o s e l y i n the p o r t i o n s of the f low r e g i o n where dry c o n d i t i o n s are a n t i c i p a t e d . The t o t a l number of nodes , however, i s l i m i t e d by the a v a i l a b l e computer s torage c a p a c i t y . A l t e r n a t i v e l y , one may modify the shape of the c h a r a c t e r i s t i c curves to reduce t h e i r s t e p - l i k e form. I n s t a b i l i t y problems were encountered d u r i n g the s e n s i t i v i t y s t u d i e s and w i l l be d i s c u s s e d i n more d e t a i l , f o l l o w i n g the l a b o r a t o r y v e r i f i c a t i o n of the n u m e r i c a l model . 25 Chapter 3 MODEL VERIFICATION When n u m e r i c a l model ing i s used to s o l v e new types of boundary-value problems , i t i s important to v e r i f y tha t the mechanisms supported by these s o l u t i o n s are p h y s i c a l l y c o r r e c t . In many i n s t a n c e s , our unders tanding of the p h y s i c s of ground-water f low i s s u f f i c i e n t to eva lua te whether or not a s o l u t i o n i s r e a s o n a b l e . T h i s i s not the case when h y d r o g e o l o g i c f a c t o r s combine to produce more than one seepage f ace ; the c o m p l e x i t y of the boundary c o n d i t i o n s and the s a t u r a t e d - u n s a t u r a t e d flow systems p r o h i b i t an i n t u i t i v e approach to model v e r i f i c a t i o n . There are s e v e r a l ways to v e r i f y a model . The n u m e r i c a l model c o u l d be v e r i f i e d i f i t were t e s t e d a g a i n s t : a) an a n a l y t i c a l s o l u t i o n , b) f i e l d o b s e r v a t i o n s , or c) a p h y s i c a l l a b o r a t o r y model . An a n a l y t i c a l s o l u t i o n i s not a v a i l a b l e i n t h i s case because of the c o m p l e x i t y of the f low r e g i o n and boundary c o n d i t i o n s . A q u a n t i t a t i v e check i n the f i e l d would r e q u i r e knowledge of the boundar ies of the r e g i o n of f l o w , the geometry of the s o i l u n i t s , the antecedent moi s ture c o n d i t i o n s , the h y d r a u l i c p r o p e r t i e s of each s o i l u n i t , and measurements of h y d r a u l i c head, i n f l o w , and o u t f l o w . I t i s u n l i k e l y tha t these f i e l d data would be known w i t h s u f f i c i e n t accuracy to p r o v i d e u n e q u i v o c a l v e r i f i c a t i o n of the n u m e r i c a l a n a l y s i s . However, wi th a p h y s i c a l l a b o r a t o r y model , much of the u n c e r t a i n t y s u r r o u n d i n g these data requirements can be e l i m i n a t e d through proper e x p e r i m e n t a l d e s i g n and m a t e r i a l t e s t i n g . The 26 l a b o r a t o r y approach was t h e r e f o r e s e l e c t e d to be the most f e a s i b l e means of v e r i f i c a t i o n . P h y s i c a l models have been used i n t h i s manner i n s e v e r a l p r e v i o u s s t u d i e s . For example, Freeze and Banner (1970) used a s o i l column to v e r i f y t r a n s i e n t , o n e - d i m e n s i o n a l , unsa tura ted flow to a r e c h a r g i n g groundwater flow system. Nieber and Walter (1981) des igned a p h y s i c a l model to v e r i f y the t r a n s i e n t r u n o f f response of a homogeneous h i l l s i d e . In both c a s e s , a flow domain was c o n s t r u c t e d and sub jec ted to a set of boundary and i n i t i a l c o n d i t i o n s . The h y d r a u l i c - h e a d d i s t r i b u t i o n was measured and compared to that p r e d i c t e d by the n u m e r i c a l model . A f a v o r a b l e comparison then l e n t credence to the c o n c l u s i o n s drawn from the numer ica l a n a l y s i s of more complex flow r e g i o n s . In the present s t u d y , a sand-box model was b u i l t to v e r i f y t w o - d i m e n s i o n a l , s t e a d y - s t a t e , s a t u r a t e d and unsa tura ted flow through a heterogeneous h i l l s i d e i n which two seepage faces were a l lowed to d e v e l o p . The p r e d i c t e d and observed p r e s s u r e -head d i s t r i b u t i o n , seepage-face l o c a t i o n s , and d i s c h a r g e r a t e s were then compared i n order to judge the v a l i d i t y of the mathemat ica l model . D e t a i l s o f the experiment are g i v e n i n t h i s c h a p t e r . F i r s t , the e x p e r i m e n t a l d e s i g n i s d e s c r i b e d . Second, r e s u l t s of a t r i a l run are p r e s e n t e d . T h i s run was des igned to t r o u b l e s h o o t t e c h n i c a l problems and consequent ly c o n t a i n e d o n l y a sparse sampl ing o f the h y d r a u l i c - h e a d d i s t r i b u t i o n . T h i r d , r e s u l t s are p re sented from a f i n a l run i n which the h y d r a u l i c -head d i s t r i b u t i o n was moni tored e x t e n s i v e l y . 27 3.1 The E x p e r i m e n t a l Des ign The exper imenta l f low r e g i o n , i l l u s t r a t e d i n F i g u r e 3, c o n s i s t s o f a v e r t i c a l c r o s s s e c t i o n through a h i l l s i d e tha t i s 2.44 m l o n g , 1.0 m h i g h , and 0.1 m wide . I t i s composed of medium sand , w i t h i n which there i s a h o r i z o n t a l l a y e r of f i n e sand intended to impede flow and c r e a t e a seepage f a c e . Four types of boundary c o n d i t i o n s are p r e s e n t : 1. A B , BC, and CD are impermeable boundar ies 2. AF i s a p r e s c r i b e d - h e a d boundary 3. DE i s an i n f i l t r a t i o n boundary 4. EF i s a seepage-face boundary S t e a d y - s t a t e , n o n h y s t e r e t i c f low c o n d i t i o n s were monitored for a s e r i e s of d e c r e a s i n g r a i n f a l l r a t e s a p p l i e d to an i n i t i a l l y s a t u r a t e d h i l l s i d e . The d e s i g n of the p h y s i c a l model operated w i t h i n many c o n s t r a i n t s . Des ign d e c i s i o n s were s u b j e c t to the l i m i t a t i o n s i n h e r e n t i n the l a b o r a t o r y equipment and the u n c e r t a i n t i e s i n t r o d u c e d by exper imenta l e r r o r . Many of the c o n s t r a i n t s c o u l d not be p r e d i c t e d at the onset of the exper iment . As a r e s u l t , the e x p e r i m e n t a l d e s i g n was not complete u n t i l a f t e r the t r i a l r u n ; by t h i s t i m e , a l l c o n s t r a i n t s had been i d e n t i f i e d . The d e s c r i p t i o n of the exper imenta l de s i gn has been o r g a n i z e d i n the f o l l o w i n g manner. F i r s t , the s e l e c t i o n and t e s t i n g of the medium and f i n e sand i s d e s c r i b e d . Then , an 28 R A I N F A L L 1.0 H E N 0.0 0.1 m X (m) 2.44 Fine sand Medium sand F i g u r e 3. The e x p e r i m e n t a l f l o w r e g i o n . 29 a n a l y s i s of the exper imenta l e r r o r a s s o c i a t e d w i t h the d e t e r m i n a t i o n of the h y d r a u l i c p r o p e r t i e s i s p r e s e n t e d . T h i s i s f o l l o w e d by a d e s c r i p t i o n of the dev i ce s des igned to generate and m a i n t a i n the boundary c o n d i t i o n s . Equipment not purchased was b u i l t by Ray Rodway, a m a c h i n i s t i n the Department of G e o l o g i c a l S c i e n c e s , at the U n i v e r s i t y of B r i t i s h Co lumbia . H i s c ra f t smansh ip was fundamental to the success of the exper iment . SELECTION AND TESTING OF THE MEDIUM SAND The medium sand was s i e v e d from a supply of Ottawa S i l i c a Sand. The f r a c t i o n tha t was chosen passed through a 30-mesh s i e v e and was r e t a i n e d on a 40-mesh s i e v e , so tha t the g r a i n s i z e ranged from 0.4 to 0.6 mm. F o l l o w i n g the MIT or B r i t i s h Standards c l a s s i f i c a t i o n scheme ( C r a i g , 1978), t h i s r epre sen t s a medium sand. A p o o r l y - g r a d e d sand was chosen for i t s a b i l i t y to approximate a homogeneous and i s o t r o p i c porous medium. However, the more uni form the g r a i n s i z e , the more s t e p - l i k e the c h a r a c t e r i s t i c curves become. T h i s may cause i n s t a b i l i t y i n the n u m e r i c a l a n a l y s i s . The f i n i t e - e l e m e n t program was t h e r e f o r e t e s t e d wi th a se t of s t e p - l i k e c h a r a c t e r i s t i c curves and the e f f e c t was to i n c r e a s e the number o f i t e r a t i o n s r e q u i r e d for convergence from approx imate ly three to seven . T h i s was judged to be an a c c e p t a b l e t r a d e o f f for the homogeneity and i s o t r o p y ga ined through the use o f a p o o r l y -graded sand. 30 Once the medium sand was s e l e c t e d , the h y d r a u l i c c o n d u c t i v i t y , the p o r o s i t y , and the c h a r a c t e r i s t i c curves were measured. Samples were prepared for t e s t i n g by s l o w l y adding wet sand to a column of water and a l l o w i n g the sand to s e t t l e i n a s t a t e o f l o o s e s t - p a c k i n g . I t was f e l t tha t attempts to compact the sand would l e a d to a more heterogeneous sample tha t would be d i f f i c u l t to reproduce i n the sand tank . D i s t i l l e d water was used for s o i l t e s t i n g , un le s s o therwise n o t e d . The s a t u r a t e d c o n d u c t i v i t y , K, of the medium sand was determined from a cons tant -head permeameter t e s t . The procedure i n v o l v e s the a p p l i c a t i o n o f a cons tant -head d i f f e r e n t i a l to a s a t u r a t e d s o i l sample and measurement of the r e s u l t i n g v o l u m e t r i c flow r a t e . A d e t a i l e d d e s c r i p t i o n o f the s tandard l a b o r a t o r y procedure i s g iven by Lambe (1951). The p o r o s i t y , n , was c a l c u l a t e d from the r e l a t i o n s h i p : n = 1 - P K / P „ b p when P. = dry bulk d e n s i t y [M/L 3 ] and p = p a r t i c l e d e n s i t y [M/L3] . The sand was e s s e n t i a l l y pure s i l i c a so that Pp = 2.65 gm/cm 3 . The s a t u r a t e d c o n d u c t i v i t y and p o r o s i t y were determined for f i v e samples . The r e s u l t s are summarized below i n Tab le 2 and are d i s c u s s e d l a t e r i n regard to the e r r o r a n a l y s i s . 31 Table 2. Constant-head t e s t r e s u l t s on the medium sand. Sample # p o r o s i t y , n K(m/s) 1 .435 2.83 x 10-3 2 .431 2.81 x 10-3 3 .427 3.14 x 10-3 4 .431 2.99 x 10-3 5 .428 3.07 x 10" 3 There are s e v e r a l l a b o r a t o r y procedures for determining the c h a r a c t e r i s t i c curves, K (ip) and 9(4)). K l u t e (1972) provides a comprehensive review. The method used to determine K(vp) was f i r s t d e s c r i b e d by C h i l d s and C o l l i s - G e o r g e (1950). I t i s based on Darcy's law for one-dimensional, v e r t i c a l , unsaturated flow through homogeneous and i s o t r o p i c s o i l , w r i t t e n as v 2 = Q/A = - K(i|») [ S i + 1] 3.1 3z where Q = v o l u m e t r i c flow r a t e , [L-3/T], and A = c r o s s s e c t i o n a l area of the sample, [ L 2 ] . When water i s s u p p l i e d to a long, v e r t i c a l column of homogeneous s o i l such that the s p e c i f i c discharge i s l e s s than the s a t u r a t e d c o n d u c t i v i t y , s t e a d y - s t a t e flow c o n d i t i o n s are c h a r a c t e r i z e d by a t r a n s m i s s i o n zone i n which the water content and pressure head are uniform. In t h i s zone, 3 ip /9z = 0 and Equation 3.1 s i m p l i f i e s t o : v z = Q/A = K(I|>) 32 The c o n d u c t i v i t y i n the t r a n s m i s s i o n zone i s t h e r e f o r e equa l to the s p e c i f i c d i s c h a r g e . To o b t a i n a p o i n t on the K(\p) c u r v e , one needs to measure \|i i n the t r a n s m i s s i o n zone and c a l c u l a t e the s p e c i f i c d i s c h a r g e from measurements of Q and A . The e n t i r e pr imary d r y i n g curve i s o b t a i n e d by proceed ing through a s e r i e s of s t e a d y - s t a t e c o n d i t i o n s i n which the s p e c i f i c d i s c h a r g e i s s u c c e s s i v e l y reduced from a maximum value near the s a t u r a t e d c o n d u c t i v i t y . The column used for t h i s procedure was 1.3 m long w i t h an i n s i d e d i a m e t e r , ID, = 6.0 cm. Water was s u p p l i e d to the top of the column at a r a te c o n t r o l l e d by a c o n s t a n t - d i s c h a r g e pump. Outflow from the base of the column was c o l l e c t e d from an a d j u s t a b l e water r e s e r v o i r . To measure pre s sure head, t ens iometer s were i n s t a l l e d i n the column w a l l , as shown i n F i g u r e 4. The tens iometer s used i n t h i s s tudy c o n s i s t of ho l low ceramic tubes , 7 inches long w i t h an ID = 3/16 i n c h and an o u t s i d e d i amete r , OD < 5/16 i n c h . One end i s s e a l e d with g l a s s and the other end may be cu t to the r e q u i r e d l e n g t h and a t t ached to a manometer. When i n use , the tens iometer i s s a t u r a t e d and the l i n e s to the manometer are f i l l e d w i t h water . Water f lows acros s the porous ceramic w a l l s u n t i l the f l u i d p r e s s u r e i n s i d e the tens iometer has e q u i l i b r a t e d w i t h the f l u i d p re s sure o f the s o i l water . In the case o f p o s i t i v e f l u i d p r e s s u r e , water f lows from the s o i l i n t o the tens iometer u n t i l e q u i l i b r i u m i s r eached . In the case of n e g a t i v e f l u i d p r e s s u r e , water f lows from the tens iometer to the s o i l . 33 Tensiometer Plexiglass wall 1/8"-1/4" NPT brass reducing nipple Threaded brass endcap O-rings: 6 mm ID, 9 mm OD Shrink-fit tubing to seal the end of the tensiometer F i g u r e 4. Tens iometer i n s t a l l a t i o n . 34 Tensiometers cease to function when either the air-entry value of the ceramic i s exceeded, or when the water content of the s o i l becomes too low to provide s u f f i c i e n t hydraulic contact for e q u i l i b r a t i o n to occur. The air-entry value of the ceramic was reported by the manufacturer to be between 0.82 bar and 0.95 bar. This means that for \b < -830 cm, a i r may enter the tensiometer through the ceramic and invalidate future readings. The second c r i t e r i o n proved to be far more r e s t r i c t i v e as the s o i l became too dry to equilibrate at \\i = -18 cm. The design of the physical model was therefore constrained to demonstrate the development of multiple seepage faces in a saturated-unsaturated regime in which \J» > - 18 cm. This constraint was acceptable because i t includes almost the f u l l range of moisture-content values experienced by the s o i l as i t desaturated. Three experimentally determined K(\J0 curves are shown in Figure 5. Error bars indicating one standard deviation about each data point have been added to one of the curves to indicate the magnitude of the measurement errors involved in the K (4*) determination. A complete discussion of the error analysis i s presented l a t e r . An air-entry value of 4>a = -11-5 cm was measured as the distance over which the s o i l remains saturated above a s t a t i c water table. The 9(^) curve was determined using a Tempe C e l l . A Tempe C e l l consists of a 3-cm long brass ring, 5.4 cm in diameter, that contains the s o i l sample between two plexiglass endcaps. 35 -1 x 10 LEGEND • — — Run * 1 » * Run #2 Run *Z ® Air-entry value '—\—1 One standard deviat ion -1 x 10 -1 x 10 .-5 £ O D O z o o o -I D < CC a > - 1 8 - 1 6 - 1 4 - 1 2 - 1 0 PRESSURE HEAD, cm ) 1 x 10 F i g u r e 5. Measured K(\J>) d a t a f o r t h e medium s a n d . 36 The lower endcap c o n t a i n s a 1/2-bar ceramic p l a t e on which the s o i l r e s t s . There i s a s m a l l r e s e r v o i r of water beneath the ceramic p l a t e t h a t i s connected by w a t e r - f i l l e d tub ing to an out f low c o l l e c t i o n f l a s k . Negat ive f l u i d p re s sure i s a p p l i e d to the base of the ceramic p l a t e by lower ing the out f low f l a s k r e l a t i v e to the s o i l sample. To prepare the Tempe C e l l , the porous p l a t e was vacuum s a t u r a t e d and the ou t f low l i n e s and the brass sample c o n t a i n e r were f i l l e d w i t h d e - a i r e d water . Sand was p l a c e d l o o s e l y i n t o the brass c o n t a i n e r . The bulk d e n s i t y was measured to p r o v i d e a means of d e t e r m i n i n g the p o r o s i t y and i n i t i a l weight o f water c o n t a i n e d i n the sample. On l o w e r i n g the sample, water d r a i n e d from the sand u n t i l the f l u i d p re s sure had e q u i l i b r a t e d acros s the ceramic p l a t e . Twelve to twenty- four hours were a l lowed for e q u i l i b r a t i o n . The amount of water r e l e a s e d was measured g r a v i m e t r i c a l l y and used to c a l c u l a t e the water content for each decrement i n p re s sure head. The pre s sure head was measured as the v e r t i c a l d i s t a n c e from the out f low l e v e l to the center of the s o i l sample. Measured i n t h i s way, A|J r e p r e s e n t s an average over the l e n g t h of the sample. F i g u r e 6 shows two e x p e r i m e n t a l l y determined 9 (\J>) curves wi th one showing e r r o r b a r s . The method used to q u a n t i f y the e r r o r i s d i s c u s s e d l a t e r . SELECTION AND TESTING OF THE FINE SAND Numer ica l s t u d i e s were used e x t e n s i v e l y to guide the s e l e c t i o n of the f i n e sand . These s i m u l a t i o n s were used to F i g u r e 6. Measured 9 ( ) da t a for the medium s a n d . 38 p r o v i d e t h e o r e t i c a l p r e d i c t i o n s o f the response o f the p h y s i c a l model to v a r i o u s d e s i g n o p t i o n s . S p e c i f i c a l l y , n u m e r i c a l s t u d i e s he lped to determine the optimum t h i c k n e s s and l o c a t i o n of the f i n e sand l a y e r and the a s s o c i a t e d range of K v a l u e s tha t t h e o r e t i c a l l y would produce m u l t i p l e seepage f a c e s . To avo id c o n f u s i o n , the s a t u r a t e d c o n d u c t i v i t y o f the f i n e sand w i l l be termed K2 and tha t of the medium sand w i l l be termed K]_. For the n u m e r i c a l s i m u l a t i o n s , c h a r a c t e r i s t i c curves averaged from those shown i n F i g u r e s 5 and 6 were used to r epre sent the medium sand . C r i t e r i a for s e l e c t i o n were based upon c h a r a c t e r i s t i c s judged to be d e s i r a b l e for the f i n a l flow r e g i o n . F i g u r e 7a shows three such c h a r a c t e r i s t i c s . F i r s t , two seepage f a c e s , l a b e l e d BC and E F , are p r e s e n t . Second, the uppermost p o r t i o n of the water t a b l e , l a b e l e d BA, does not i n t e r s e c t the i n f i l t r a t i o n boundary to produce pond ing . T h i r d , an unsa tura ted wedge, l a b e l e d CDE, i s formed beneath the uppermost seepage f a c e . These three c h a r a c t e r i s t i c s are h e r e a f t e r i m p l i c i t i n the phrase "an a c c e p t a b l e s o l u t i o n . " Two types o f unacceptable s o l u t i o n s were a l s o i d e n t i f i e d d u r i n g p r e l i m i n a r y n u m e r i c a l s t u d i e s . The f i r s t type occur s i f the c o n d u c t i v i t y c o n t r a s t , K i / K 2 > i s so low t h a t s o l u t i o n s t y p i f i e d by F i g u r e 7b are p roduced . Two seepage faces w i l l not form, r e g a r d l e s s o f the r a i n f a l l r a t e . Note a l s o t h a t ponding has o c c u r r e d a long the i n f i l t r a t i o n boundary. T h i s i s u n d e s i r a b l e because the n u m e r i c a l model does not model sur f ace r u n o f f . A second type of u n d e s i r a b l e s o l u t i o n i s shown i n F i g u r e 7. D e s i r a b l e , and u n d e s i r a b l e c h a r a c t e r i s t i c s o f e x p e r i m e n t a l f low r e g i o n . the 40 F i g u r e 7 c . In t h i s c a s e , K 1 / K 2 i s so g rea t t h a t the f i n e l a y e r forms an almost impervious boundary, and a s i n g l e , cont inuous s a t u r a t e d zone cannot form. A flow r e g i o n o f t h i s type i s u n d e s i r a b l e because i t would not a l low o b s e r v a t i o n of the growth of the unsa tura ted wedge a long the base of the impeding l a y e r as the r a i n f a l l r a t e i s decrea sed . With these d e s i r a b l e and u n d e s i r a b l e q u a l i t i e s i n mind, a n u m e r i c a l s tudy was des igned to determine the optimum t h i c k n e s s and l o c a t i o n o f the impeding l a y e r . The t h i c k n e s s of the impeding l a y e r , T , and the e l e v a t i o n of the base of the l a y e r , z , were v a r i e d , as shown i n F i g u r e 8 . P r e l i m i n a r y s t u d i e s had i n d i c a t e d t h a t a va lue o f Kj_/K2 = 20 would a v o i d the u n d e s i r a b l e s o l u t i o n s shown i n F i g u r e 7 b and 7 c , so t h i s va lue was chosen and mainta ined cons tant throughout the a n a l y s i s . The response of each mesh to a s e r i e s of four r a i n f a l l r a t e s i s summarized i n F i g u r e s 9 , 1 0 , and 1 1 . F i g u r e 9 i s o l a t e s the e f f e c t of T and z on the l e n g t h over which the i n f i l t r a t i o n boundary i s ponded. Mesh 3 produces the l e a s t amount of ponding for any g i v e n r a i n f a l l r a t e . F i g u r e 10 shows the e f f e c t of T and z on the l e n g t h of the uppermost seepage f a c e . Meshes B and C are e q u a l l y d e s i r a b l e because they tend to produce a r e l a t i v e l y p e r s i s t e n t uppermost seepage f a c e . F i g u r e 11 shows the e f f e c t o f T and z on the d i s t a n c e the unsa tura ted wedge extends i n t o the h i l l s i d e . Mesh B i s the most d e s i r a b l e i n t h i s regard because i t produces the s m a l l e s t unsa tura ted wedge. W h i l e a l a r g e unsa tura ted wedge might be more i n t e r e s t i n g to m o n i t o r , i t i s a l s o more l i k e l y to c o n t a i n 4 1 T=20 cm Z=60 cm 2 . 4 4 F i g u r e 8 . Three meshes used to determine the optimum t h i c k n e s s and l o c a t i o n o f the impeding l a y e r . 42 < cr MESH A: T = 10 cm, 2 = 70 cm MESH B: T= 10 cm, Z - 60 cm MESH C: T - 20 cm, 2 = 60 cm 20 40 60 80 PONDED DISTANCE, ( cm ) F i g u r e 9 . The e f f e c t o f T and z on the l e n g t h over which the i n f i l t r a t i o n boundary i s ponded. 43 MESH A: T - 10 cm, Z = 70 cm MESH B: T - 10 cm, Z - 60 cm MESH C: T = 20 cm, Z =» 60 cm i 24 0 4 8 12 16 20 LENGTH OF UPPERMOST SEEPAGE FACE, ( cm ) F i g u r e 1 0 . The e f f e c t o f o f T and z on the l e n g t h o f the uppermost seepage f a c e . 44 9.0 -* 8.0 H I o x ~ 7.0 H co E LU DC < < CC 6.0 -5.0 -4.0 -3.0 B C A MESH A: T - 10 cm, Z - 70 cm MESH B: T - 10 cm, Z - 60 cm MESH C: T - 20 cm, Z = 60 cm 1.0 1.5 2.0 MAX. X-COORDINATE OF UNSATURATED WEDGE, ( m ) F i g u r e 11. The e f f e c t o f T and z on the d i s t a n c e the u n s a t u r a t e d wedge extends i n t o the h i l l s i d e 4 5 va lues of t < - 1 8 cm which cannot be measured wi th the t e n s i o m e t e r s . Based on these o b s e r v a t i o n s , Mesh B was chosen as the optimum flow r e g i o n . Next , a s tudy was made to q u a n t i f y the range of K 2 va lue s t h a t would produce an a c c e p t a b l e s o l u t i o n for a r a i n f a l l r a te of 2 . 8 cm/min, or 4 . 7 x 1 0 - 4 m/s . P r e l i m i n a r y s t u d i e s had i n d i c a t e d t h a t t h i s r a i n f a l l r a t e would produce an accep tab le s o l u t i o n and i t f e l l w i t h i n the range of the performance of the r a i n f a l l g e n e r a t o r . In the a n a l y s i s , the r a i n f a l l r a te and the va lue of Ki = 2 . 8 x 1 0 " " 3 m/s remained f i x e d ; K 2 was v a r i e d and shown to produce a c c e p t a b l e s o l u t i o n s for va lues between 8 . 3 x 1 0 ~ 5 m/s and 1 . 3 x 1 0 - 4 m/s . Va lues below t h i s range produced sur face p o n d i n g ; above t h i s range , the uppermost seepage face was l o s t . The optimum d e s i g n for the e x p e r i m e n t a l flow r e g i o n i s shown i n F i g u r e 1 2 for which K 2 = 1 . 0 x 1 0 - 4 m/s . To h e l p l o c a t e a sand having a s a t u r a t e d c o n d u c t i v i t y near 1 . 0 x 1 0 - 4 m/s , the f o l l o w i n g rough e s t imate was used : K ~ 1 0 0 D 1 0 2 3 . 2 where K i s measured i n cm/s and D ^ Q i s t n e p a r t i c l e s i z e , i n cm, such tha t 1 0 % o f the sand p a r t i c l e s are sma l l e r than that s i z e . A c c o r d i n g to E q u a t i o n 3 . 2 , a sand w i t h K ~ 1 . 0 x 1 0 - 4 m/s shou ld be a s s o c i a t e d w i t h D 1 0 ~ 0 - 1 nun. Four s i e v e s , w i t h openings rang ing between 0 . 2 mm and 0 . 0 7 5 mm, were then used to separate a mixture of f i n e Ottawa S i l i c a Sand. The f r a c t i o n w i t h a g r a i n s i z e between 0 . 1 mm and 46 Rainfall rate • 2.8 cm/min. - 2.8 x 1CT3m/s F i g u r e 1 2 . The optimum d e s i g n fo r the e x p e r i m e n t a l f low r e g i o n 47 0.075 had a s a t u r a t e d c o n d u c t i v i t y approx imate ly equa l to 1.2 x 10~ 4 m/s . T h i s was the sand tha t was s e l e c t e d to form the impeding l a y e r w i t h the e x p e c t a t i o n tha t a s o l u t i o n s i m i l a r to F i g u r e 12 c o u l d be o b t a i n e d e x p e r i m e n t a l l y . Cons tant-head permeameter t e s t s were performed on f i v e samples of the f i n e sand . The r e s u l t s are summarized below, i n Table 3, and are d i s c u s s e d l a t e r i n regard to the e r r o r a n a l y s i s . Tab le 3. Constant-head t e s t r e s u l t s on the f i n e sand. Sample n K(m/s) 1 .469 1.24 x I O " 4 2 .463 1.23 x I O " 4 3 .449 1.12 x I O " 4 4 .451 1.30 x I O " 4 5 .447 1.20 x 10~ 4 An a i r - e n t r y va lue of tya = -90 cm was measured for the f i n e sand . Because the p h y s i c a l model was des igned to operate at ^ > -18 cm, the impeding l a y e r would remain s a t u r a t e d and measurement of the c h a r a c t e r i s t i c curves was unneces sary . ERROR ANALYSIS Before one can judge whether an e x p e r i m e n t a l r e s u l t agrees w i t h a t h e o r e t i c a l p r e d i c t i o n , the accuracy o f both must be e s t i m a t e d . The e x p e r i m e n t a l r e s u l t , i n t h i s c a s e , i s the s t e a d y - s t a t e response of the p h y s i c a l model to a se t o f boundary c o n d i t i o n s . The accuracy w i t h which we can measure 4 8 the e x p e r i m e n t a l r e s u l t depends upon the accuracy of the p r e s s u r e - s e n s i n g system and the s e e p a g e - c o l l e c t i o n system. A d i s c u s s i o n of the accuracy of these systems i s i n c l u d e d i n the d i s c u s s i o n of the f i n a l exper imenta l r u n . The accuracy of the t h e o r e t i c a l p r e d i c t i o n , on the other hand, depends p r i m a r i l y upon the accuracy wi th which the boundary c o n d i t i o n s are ma in ta ined i n the l a b o r a t o r y model and the accuracy of the e x p e r i m e n t a l l y determined h y d r a u l i c p r o p e r t i e s . The former i s a t e c h n i c a l problem and i s d i s c u s s e d wi th the t r i a l r u n ; the l a t t e r w i l l now be d i s c u s s e d . O f t e n , the h y d r a u l i c p r o p e r t i e s o f a porous medium are c a l c u l a t e d from s e v e r a l i n d i v i d u a l o b s e r v a t i o n s , each of which c o n t r i b u t e s to the i n a c c u r a c y o f the computed r e s u l t . For example, the va lue of K o b t a i n e d from the cons tant -head t e s t c o n t a i n s e r r o r s from the measurement of out f low volume, t i m e , c r o s s - s e c t i o n a l a r e a , the cons tant -head d i f f e r e n t i a l , and the l e n g t h of the sample . P r o p a g a t i o n - o f - e r r o r a n a l y s i s i s des igned to q u a n t i f y the c o n t r i b u t i o n of these i n d i v i d u a l e r r o r s to the computed r e s u l t . A complete d i s c u s s i o n of the method i s c o n t a i n e d i n Young (1962); a summary i s p re sented below. Suppose a q u a n t i t y , P, i s c a l c u l a t e d from the measured q u a n t i t i e s a , b , c , . . . . The e x p r e s s i o n r e l a t i n g the v a r i a n c e of 2 the c a l c u l a t e d q u a n t i t y , a , to the v a r i a n c e s o f the measured hr q u a n t i t i e s , a a 2 , o^"2-, o c 2 , . . . , i s g iven as : 49 The f r a c t i o n a l s tandard d e v i a t i o n of the mean, Op/p, can be o b t a i n e d from the f o l l o w i n g e x p r e s s i o n : °*P ^ l ap 2 i a p 2 2 i a p 2 2 P P 3a A P 3b B P 3c c To i l l u s t r a t e the use o f Equat ions 3.3 and 3.4, c o n s i d e r the c a l c u l a t i o n of K from the cons tant-head t e s t , where: K = ^ l _ 3.5 tAAh where: Q = out f low volume t = time [T] A = c r o s s - s e c t i o n a l area [L 2 ] Ah = cons tant -head d i f f e r e n t i a l [L] I = l e n g t h of the sample [L] From E q u a t i o n 3.3, the v a r i a n c e of K i s g i v e n as : _ 2 ,3K> 2 2 3K 2 2 3K 2 2 3K 2 2 3K 2 2 K 3Q Q 3t t 3A A 3Ah Ah 34 I 3.6 I n s e r t i n g the p a r t i a l d e r i v a t i v e s i n d i c a t e d i n Equa t i on 3.6 l eads t o : t A A t t Q t 2 A A h t t A 2 A h A 50 t A ( A h ) 2 A " tAAh H From E q u a t i o n 3 . 4 , t h i s f u r t h e r s i m p l i f i e s t o : aK 2 a5 2 a t 2 a A 2 CTAh 2 °o 2 K Q t A Ah 2, Every q u a n t i t y i n E qu a t ion 3.8 can be e s t i m a t e d . Es t imates of a_, a f c, a A , a^^ and are based on the accuracy wi th which the observer makes the measurement. For example, d u r i n g the t e s t i n g of the f i r s t sample of medium sand : Q + a = 159 + 1 ml Q t + a = 1 min + 0 . 5 sec A + a = 28.27 cm 2 + 1.88 c m 2 A n ± a ^ n = 25.26 cm + 0.05 cm I +0^ = 76.3 cm + 0.5 cm A f t e r c a l c u l a t i n g the va lue o f K for the f i r s t sample from E q u a t i o n 3 . 5 , E q u a t i o n 3.8 can be s o l v e d for a K . T h i s procedure i s repeated for each sample t e s t e d . Then , u s ing the a r i t h m e t i c mean, the averge c o n d u c t i v i t y , K i s o b t a i n e d from 51 where N = number of samples . The v a r i a n c e of t h i s average i s found by a p p l y i n g E q u a t i o n 3.3 to E q u a t i o n 3.9 to o b t a i n : I f the f l u i d temperature i s known, then the p e r m e a b i l i t y , k, can be c a l c u l a t e d from Equa t i on 2.4 and the e r r o r i n k, can be s i m i l a r l y computed. Knowledge o f k + a_ i s u s e f u l because K + cr_ can then be c a l c u l a t e d for the f l u i d temperature at _ K which the exper iment i s r u n . T h i s a n a l y s i s was a p p l i e d to the cons tant -head t e s t s performed on the medium and f i n e sand. A f o u r - s t e p process was f o l l o w e d : 1. Measure K i n the l a b o r a t o r y . 2. Compute the e r r o r i n K due to i n d i v i d u a l e r r o r s i n the measurement of Q, t , A, Ah, and 3. Compute k. 4. Compute the e r r o r i n k due to i n d i v i d u a l e r r o r s i n the measurement of K and temperature . Tab le 4 summarizes the r e s u l t s from the a n a l y s i s . These were used to examine the range of t h e o r e t i c a l p r e d i c t i o n s o b t a i n e d as K]_ and K2 v a r i e d w i t h i n two s tandard d e v i a t i o n s of t h e i r means, for a g i v e n r a i n f a l l r a t e of 2.7 cm/min. 52 Table 4. R e s u l t s of the e r r o r a n a l y s i s performed on K]_ and K 2 for a f l u i d temperature of 2 0 ° C . MEDIUM SAND FINE SAND k (m2) 2.84 x 10-10 1.25 x 10-11 0.14 x 1 0 - 1 ° 0.67 x 1 0 - H K (m/s) 2.75 x 10-3 1.21 x I O " 4 o (m/s) K 0.25 x I O - 3 0.11 x 10~ 4 Three of these s o l u t i o n s are shown i n F i g u r e 13. W i t h i n t h i s range , one would be unable to d i s t i n g u i s h between the e f f e c t of e x p e r i m e n t a l e r r o r i n the measurement of K i and K 2 and an e r r o r i n the f i n i t e - e l e m e n t f o r m u l a t i o n of the prob lem. The v a r i a t i o n s shown i n F i g u r e 13 suggested t h a t the n u m e r i c a l model might need to be c a l i b r a t e d before i t s v a l i d i t y i s e v a l u a t e d . For example, suppose the mathemat ica l model p r e d i c t s the s o l u t i o n shown i n F i g u r e 13b, for which K i = K]_ and K 2 = K 2 / w h i l e the p h y s i c a l model produces the flow r e g i o n shown i n F i g u r e 13c , for which K i = K i + 2°^ and K 2 = K 2 + 2 0 K 2 * T n e mathemat ica l model would be v e r i f i e d i f new p r e d i c t i o n s u s ing K]_ = £]_ + 2a R ^ and K 2 = K 2 + 2 a K 2 m a t c n e d the e x p e r i m e n t a l response to r a i n f a l l r a t e s o ther than 2.7 cm/min. In t h i s way, the mathemat ica l model i s f i r s t c a l i b r a t e d a g a i n s t the r e s u l t s o b t a i n e d for the f i r s t i n a s e r i e s of r a i n f a l l r a t e s and then v e r i f i e d i f i t i s ab le to p r e d i c t the subsequent response of the p h y s i c a l model . 53' F i g u r e 13 . Range o f t h e o r e t i c a l p r e d i c t i o n s as and K 2 vary w i t h i n two s t a n d a r d d e v i a t i o n s o f t h e i r means fo r a r a i n f a l l r a t e o f 2.7 cm/min . 54 Superimposed on the e r r o r s i n s a t u r a t e d c o n d u c t i v i t y are the e r r o r s p re sent i n the K(ip) and 9 (\p) d a t a . P r o p a g a t i o n - o f -e r r o r a n a l y s i s was performed on the c h a r a c t e r i s t i c curve data for the medium sand . The e r r o r bars a s s o c i a t e d wi th one s tandard d e v i a t i o n appear for one o f the curves i n F i g u r e s 5 and 6. Because of the r e l a t i v e l y c l o s e f i t of the d a t a , the e f f e c t of the e r r o r s i n K(4>) and 9(\|>) was judged to be l e s s important than the e f f e c t of the e r r o r s i n K i and K2/ a l though n u m e r i c a l s t u d i e s were not performed to v e r i f y t h i s . I t was b e l i e v e d t h a t an a n a l y s i s of t h i s type c o u l d be done at a l a t e r date i f n e c e s s a r y . Subsequent r e s u l t s d i d not i n d i c a t e a need for such an a n a l y s i s . The e r r o r a n a l y s i s has not accounted for a l l the e r r o r s p re sent i n the e x p e r i m e n t a l p r o c e d u r e s . For example, because the sands were t e s t e d i n a s t a t e of l o o s e s t - p a c k i n g , some se t t l ement wi th time was i n e v i t a b l e . Dur ing the K(4>) measurements, the l e n g t h of the s o i l column decreased by up to 6 mm. T h i s r epre sented a 0.5% change i n the t o t a l l e n g t h o f the s o i l column. The e r r o r a n a l y s i s has not taken i n t o account the t r a n s i e n t e f f e c t of s e t t l ement on the measurement of the h y d r a u l i c p r o p e r t i e s . GENERATION AND MAINTENANCE OF THE BOUNDARY CONDITIONS As mentioned e a r l i e r i n r e f e r e n c e to F i g u r e 3, the p h y s i c a l model c o n t a i n s four types o f boundary c o n d i t i o n s : impermeable b o u n d a r i e s , a cons tant -head boundary, an i n f i l t r a t i o n boundary and a seepage-face boundary. Each of these must be s imula ted i n the l a b o r a t o r y . The impermeable 55 boundar ies were formed by the p l e x i g l a s s w a l l s o f the sand tank , the o n l y p r e c a u t i o n be ing tha t there be no l eaks a long the seams or tens iometer p o r t s . The other boundary c o n d i t i o n s r e q u i r e d d e v i c e s of v a r y i n g c o m p l e x i t y to r e g u l a t e and measure i n f l o w and out f low to the f low system. The cons tant -head boundary was mainta ined w i t h brass tub ing connec tor s threaded i n t o the w a l l of the tank . Outf low i n excess of the amount r e q u i r e d to m a i n t a i n the cons tant l e v e l of water s p i l l s through the tub i ng connector s and the flow ra te can be measured wi th a graduated c y l i n d e r and a s topwatch . R a i n f a l l was generated a long the i n f i l t r a t i o n boundary us ing a d e v i c e s i m i l a r to one des igned by Chow and Harbaugh (1965). The rainmaker c o n s i s t s o f a box, 79 cm by 10 cm by 2.5 cm, c o n s t r u c t e d from 3 / 8 - i n c h p l e x i g l a s s . The bottom of the box c o n t a i n s o n e - i n c h l e n g t h s of p o l y e t h y l e n e tub i ng (ID = .58 mm, OD = .965 mm) se t a long a o n e - i n c h square g r i d to produce the r a i n d r o p s . Threaded i n t o the top of the rainmaker i s a tub ing connector for the water supply and an a i r - e s c a p e v a l v e for use when f i l l i n g the rainmaker wi th water . The r a i n f a l l r a te i s c o n t r o l l e d w i t h a c o n s t a n t - d i s c h a r g e pump. The most c h a l l e n g i n g boundary c o n d i t i o n to s imula te was the seepage-face boundary. A dev i ce was r e q u i r e d that would r e s t r a i n the sand and a l l o w the measurement of seepage r a te s at each node a long the s l o p e , yet would not o therwise i n t e r f e r e w i t h f low c o n d i t i o n s . The d e v i c e that was des igned c o n s i s t s of n y l o n mesh s t r e t c h e d over an aluminum frame. Seepage 56 c o l l e c t o r s are g lued to the mesh and the e n t i r e d e v i c e i s b o l t e d i n p l a c e through the w a l l s of the p l e x i g l a s s tank. The n y l o n mesh was s e l e c t e d so tha t the s i z e of the mesh opening was s m a l l enough to prevent the movement of s o i l through i t , ye t l a r g e enough to o f f e r l i t t l e r e s i s t a n c e to f l o w . To r e s t r a i n the medium sand, which ranged i n s i z e from 0.4 mm to 0.6 mm, a mesh wi th an opening of 0.5 mm was chosen . To r e s t r a i n the f i n e sand, which ranged i n s i z e from 0.075 mm to 0.10 mm, a mesh wi th an opening of 0.088 mm was chosen . Seepage was c o l l e c t e d at the nodes a long the s lope w i t h the d e v i c e shown i n F i g u r e 14. The c o l l e c t o r s were c o n s t r u c t e d from PVC and g lued to the mesh at the nodal p o i n t l o c a t i o n s i n such a way tha t o n l y the l i p of the c o l l e c t o r i s i n c o n t a c t w i t h the s o i l . Brass tub ing c o n n e c t o r s , threaded through the w a l l of the t ank , served to route seepage away from the s lope and p r o v i d e a means of measuring out f low r a t e s . 3.2 The T r i a l Run As the e x p e r i m e n t a l d e s i g n p r o g r e s s e d , s e v e r a l que s t ions arose r e g a r d i n g the t e c h n i c a l a spect s o f the p h y s i c a l model . For example, c o u l d the sand be p l a c e d i n the tank i n a s t a t e of l o o s e s t packing? Would the d e v i c e des igned to m a i n t a i n the seepage-face boundary work? Cou ld the r a i n f a l l be ma inta ined at cons tant r a t e for the l e n g t h of time needed to reach s teady s t a te? A t r i a l run was c a r r i e d out to t r o u b l e s h o o t these types of problems . The o b j e c t i v e s were to g a i n an i n d i c a t i o n of the l i k e l i h o o d of success of the p h y s i c a l model and to make the Nylon mesh F i g u r e 14. Seepage c o l l e c t o r . 58 neces sary d e s i g n m o d i f i c a t i o n s to overcome t e c h n i c a l problems . The f i n a l r u n , d i s c u s s e d i n S e c t i o n 3.3 would then i n c o r p o r a t e the m o d i f i c a t i o n s and use a more e x t e n s i v e m o n i t o r i n g network to q u a n t i f y the response of the p h y s i c a l model . PREPARATION OF THE SAND TANK To prepare for the t r i a l r u n , the f i n i t e - e l e m e n t mesh was reproduced on the f r o n t w a l l of the tank w i t h L e t r a l i n e , a t h i n b lack adhes ive t ape . F i f t y - s i x 3 / 8 - i n c h diameter ho le s were threaded through the tank w a l l at the noda l p o i n t s where tens iometer s would be i n s t a l l e d d u r i n g the f i n a l r u n . For the purposes of the t r i a l r u n , ten of these p o r t s were used for tens iometer s and three c o n t a i n e d thermometers . Unused por t s were s e a l e d w i t h brass p l u g s . F i g u r e 15 shows the f i n i t e -element mesh, the f u n c t i o n of each p o r t , and the numbering system used to i d e n t i f y the t e n s i o m e t e r s . A c e n t r a l support mounted on the frame of the tank prevented placement of t ens iometer s a long x = 1.2 m. The s lope d e v i c e was lowered i n t o the tank and b o l t e d i n p l a c e . T w e l v e , 3 / 8 - i n c h diameter ho les were threaded i n t o the tank to route seepage away from the seepage-face nodes. The l o c a t i o n of the seepage c o l l e c t o r s and the numbering system used to i d e n t i f y the c o l l e c t o r s are i n d i c a t e d i n F i g u r e 15. Due to an o v e r s i g h t by the a u t h o r , a c o l l e c t o r was not p l a c e d at the lowermost seepage-face node. P r i o r to f i l l i n g the tank w i t h water , a l l p o r t s were s e a l e d . Clamped tub ing was a t t ached to tub ing connec tor s a long the seepage-face and cons tant -head b o u n d a r i e s . The F i g u r e 15. L o c a t i o n and f u n c t i o n of measurement p o r t s w i t h re spec t to the f i n i t e - e l e m e n t mesh. <_n KD 60 t ens iometer s and thermometers were i n s t a l l e d by the method shown i n F i g u r e 4. The tank was then f i l l e d w i t h tap water to a l e v e l z = 1.1 m; b leach was added to prevent a l g a l growth. The manometers were purged of a i r and a l lowed to e q u i l i b r a t e . FILLING THE TANK WITH SAND Two methods of p l a c i n g the sand i n the tank were t r i e d . F i r s t , wet sand was added to the water i n the tank and a l lowed to s e t t l e i n t o p l a c e i n a s t a t e of l o o s e s t p a c k i n g . T h i s method f a i l e d because the sand would not s e t t l e f l u s h aga in s t the ny lon mesh a long the s l o p e . Attempts to g e n t l y push the sand i n p l a c e a long the s lope were awkward and c r e a t e d d i f f e r e n t i a l p a c k i n g . The second method was to add wet sand to the water i n the tank and d i s t r i b u t e i t w i t h water pumped through a long meta l r o d . The pump i n t a k e was p l a c e d i n the water to the l e f t of the s lope d e v i c e . The pump out f low tub ing was a t t ached to a submerged rod and the sand c o u l d then be blown i n t o p l a c e by the j e t of water . In t h i s way, the sand was p l a c e d f l u s h a g a i n s t the n y l o n mesh. The o n l y d i sadvantage to the method was tha t the bulk d e n s i t y of the sand i n the tank was now d i f f e r e n t than the bulk d e n s i t y of the samples used to determine the h y d r a u l i c p r o p e r t i e s . The consequence o f t h i s w i l l be d i s c u s s e d when the r e s u l t s are p r e s e n t e d . A f t e r the f low r e g i o n had been p l a c e d i n the t ank , a t h i n l a y e r of p e a - s i z e d g r a v e l was p l a c e d a long the i n f i l t r a t i o n boundary so tha t ho les would not be bored i n t o the sand by the r a i n . 61 RESULTS OF THE TRIAL RUN To beg in the t r i a l r u n , the clamps s e a l i n g the c o n s t a n t -head boundary were removed and the water d r a i n e d s l o w l y from the t ank . Rain was s i m u l t a n e o u s l y a p p l i e d at a r a te of 2.7 cm/min. Approx imate ly two hours were a l lowed to ensure tha t s t e a d y - s t a t e c o n d i t i o n s were e s t a b l i s h e d . Readings of the pre s sure-head d i s t r i b u t i o n were recorded for three s u c c e s s i v e l y d e c r e a s i n g r a i n f a l l r a t e s . The thermometers i n d i c a t e d that the f l u i d temperature remained c o n s t a n t , w i t h i n a degree C e l s i u s , throughout the r u n . The p r e d i c t e d pre s sure-head d i s t r i b u t i o n for three r a i n f a l l r a t e s i s shown i n F i g u r e 16. Because the bulk d e n s i t y of the sand i n the tank was h igher than that of the samples used to determine the h y d r a u l i c p r o p e r t i e s , the va lues of K]_ and K 2 i n the tank were lower than those used i n the p r e d i c t i o n s shown i n F i g u r e 16. T h i s can be seen by comparing the p r e d i c t e d and measured pre s sure-head d i s t r i b u t i o n , as shown i n F i g u r e 17. The l i n e c o n n e c t i n g the data p o i n t s are added to a i d the comparison of the d a t a ; they do not r epre sent i n t e r p o l a t e d va lues nor any other type o f p h y s i c a l r e l a t i o n s h i p . Note that for a g i v e n r a i n f a l l r a t e , the measured v a l u e s l i e to the r i g h t of and are t h e r e f o r e g rea te r than the p r e d i c t e d v a l u e s . Subsequent n u m e r i c a l s t u d i e s showed tha t by s l i g h t l y d e c r e a s i n g the va lues of K i and K 2 r the p r e d i c t e d va lue s i n F i g u r e 17 s h i f t to the r i g h t to p r o v i d e b e t t e r agreement w i t h the e x p e r i m e n t a l r e s u l t s . Q u a l i t a t i v e l y , however, the r e s u l t s are good. In p a r t i c u l a r , the measured 4* 0 0.4 0.8 1.2 1.6 2.0 2.4 X, (m) F i g u r e 1 6 . P r e d i c t e d r e s u l t s f o r t h e t r i a l r u n ; Ki = 2 . 7 5 x 10 - 3 m/s a n d K 2 = 1 . 2 1 x I O " 4 m / s . P R E S S U R E H E A D , (cm) F i g u r e 17. Comparison of measured and p r e d i c t e d readings d u r i n g the t r i a l r u n . p r e s s u r e - h e a d 64 va lues show tha t an unsa tura ted wedge formed beneath the impeding l a y e r and became more ex tens ive as the r a i n f a l l r a t e d e c r e a s e d . The observed seepage-face l o c a t i o n s d i d not correspond w e l l wi th the p r e d i c t e d r e s u l t s . For example, for a r a i n f a l l r a te of 2.7 cm/min, out f low was measured at seepage c o l l e c t o r (SC) #4, #6, #7 and #12 w h i l e the t h e o r e t i c a l model p r e d i c t e d seepage from SC #6 and #12. T h i s d i s c r e p a n c y o c c u r r e d because i n the two days between f i l l i n g the tank wi th sand and beg inning the t r i a l r u n , the sand had s e t t l e d approx imate ly 5mm away from the n y l o n mesh a long the s l o p e . C o n s e q u e n t l y , when the water l e v e l was lowered at the beg inn ing of the t r i a l r u n , s lumping o c c u r r e d . T h i s a f f e c t e d the l o c a t i o n of the seepage faces i n two ways. F i r s t , i t d i s p l a c e d the p o s i t i o n of the f i n e l a y e r downslope, near the n y l o n mesh. The uppermost seepage face was t h e r e f o r e s h i f t e d s l i g h t l y downslope as i n d i c a t e d by the ou t f low measured at SC #7. Second, w h i l e the s lumping p r o v i d e d e x c e l l e n t c o n t a c t between the seepage c o l l e c t o r s and the sand for SC #4 through SC #12, much of the sand near the top of the s lope had moved downslope, l e a v i n g no c o n t a c t at a l l above SC #4. The d i s t a n c e from the n y l o n mesh to the sand i n c r e a s e d from a few m i l l i m e t e r s at SC #3 to s e v e r a l c e n t i m e t e r s at SC #1. T h e r e f o r e , when ponding o c c u r r e d a long the i n f i l t r a t i o n boundary for a r a i n f a l l r a t e of 2.7 cm/min, the r u n o f f f lowed a long the s lope face and was c o l l e c t e d i n SC #4. P o n d i n g , as w e l l as out f low at SC #4, ceased for lower r a i n f a l l r a t e s . To e l i m i n a t e the problems 65 caused by s l u m p i n g , i t was r e c o g n i z e d tha t the sand i n the f i n a l run would need to be compacted as i t i s p l a c e d i n the tank . The r a i n f a l l r a te was c a l c u l a t e d from c a l i b r a t i o n s made p r i o r to the t r i a l r u n . Dust and l i n t became e n t r a i n e d i n the water and e v e n t u a l l y c logged up to f i f t y percent of the c a p i l l a r y tubes produc ing r a i n d r o p s . T h i s i n d i c a t e d the need for a f i l t e r around the pump i n t a k e for the f i n a l run and suggested tha t the r a i n f a l l r a t e i s best c a l c u l a t e d from the t o t a l out f low r a t e at s teady s t a t e . 3.3 The F i n a l Run The f i n a l run was moni tored w i t h f i f t y - s i x t e n s i o m e t e r s . Ten were a t t a c h e d to manometers and the remainder were connected by 1 / 1 6 - i n c h ID n y l o n tub ing to a 48-port S c a n i v a l v e . The S c a n i v a l v e i s an e l e c t r o n i c scanning va lve tha t a l l ows the h y d r a u l i c head to be measured at each por t wi th a s i n g l e p re s sure t r a n s d u c e r . The output was read wi th a d i g i t a l vo l tmeter and recorded by hand. Two of the 48 p o r t s were used throughout the f i n a l run to c a l i b r a t e the t r a n s d u c e r . The e r r o r a s s o c i a t e d w i t h the h y d r a u l i c head read ings was approx imate ly + 3mm of water . T h i s corresponds to a 1 mm e r r o r i n the measurement of the p re s sure head and a 2 mm e r r o r i n the measurement of the e l e v a t i o n head. The sand was p l a c e d i n the t ank , as b e f o r e , and compacted by s t r i k i n g the w a l l s of the tank w i t h a rubber m a l l e t . D e s p i t e the compac t ion , the sand s e t t l e d o v e r n i g h t up to 2 mm 66 away from the n y l o n mesh. As the water l e v e l was lowered , some s lumping o c c u r r e d and the f i n e l a y e r was d i s p l a c e d s l i g h t l y downslope i n the v i c i n i t y of the n y l o n mesh. A d d i t i o n a l coarse sand was fed from the top of the s lope r e s u l t i n g i n good c o n t a c t between the f low r e g i o n and a l l seepage c o l l e c t o r s . , S t e a d y - s t a t e read ings were recorded for r a i n f a l l r a t e s of 1.8 cm/min, 1.5 cm/min, and 1.26 cm/min. An attempt was made to take read ings for a lower r a i n f a l l r a t e , but a f t e r four h o u r s , the va lue s were s t i l l f l u c t u a t i n g and the experiment was t e r m i n a t e d . The r a i n f a l l r a t e was c a l c u l a t e d from the t o t a l out f low r a t e at s teady s t a t e . F i g u r e 18 i s a photograph of the e x p e r i m e n t a l s e tup . The e x p e r i m e n t a l r e s u l t s are shown i n F i g u r e 19. The mathemat ica l model was c a l i b r a t e d a g a i n s t the r e s u l t s shown i n F i g u r e 19a i n the f o l l o w i n g manner. Because of the u n c e r t a i n t y c o n c e r n i n g the i n s i t u va lues of and K 2 , a f a l l i n g - h e a d permeameter t e s t was performed on a h i g h l y compacted sample of the medium sand . The r e s u l t i n g va lue served as a lower l i m i t for the p o s s i b l e i n s i t u va lue of K±. A s i m i l a r t e s t on the f i n e sand was not made because a l l of the a v a i l a b l e f i n e sand had been p l a c e d i n the sand tank. C a l c u l a t i o n s were then performed to determine i f a reasonable se t of K]_ and va lues c o u l d be found that would p r e d i c t the observed response of the p h y s i c a l model . The r e s u l t s of t h i s b a c k - c a l c u l a t i o n are summarized i n T a b l e 5. Note tha t a l l va lues cor re spond to a f l u i d temperature of 1 1 . 5 ° C , the temperature at which the experiment was r u n . F i g u r e 18 . Photograph o f the f i n a l e x p e r i m e n t a l a. Rainfall rate =1.8 cm/min 63 c. Rainfall rate = 1.26 cm/min X (m) F i g u r e 19 . Comparison o f p r e d i c t e d and observed w a t e r - t a b l e • c o n f i g u r a t i o n s fo r the f i n a l r u n , u s i n g b e s t - f i t va lue s o f K]_ and K 2 . 69 T a b l e 5. Summary of back c a l c u l a t i o n s . l o o s e s t p a c k i n g , 1 1 . 5 ° C K i (m/s) 2.2 x I O " 3 K 2 (m/s) 9.8 x I O " 5 densest p a c k i n g , 1 1 . 5 ° C 6.6 x I O " 4 not measured b e s t - f i t va lue by model c a l i b r a t i o n 1.4 x 10-3 5.5 x 10-5 The b e s t - f i t va lues for and K 2 r epre sent a 36% and 44% d e c r e a s e , r e s p e c t i v e l y , from the K va lues i n a s t a t e of l o o s e s t p a c k i n g . No attempt was made to measure new c h a r a c t e r i s t i c curves to account for changes i n t h e i r shape due to compact ion . The curves were s imply s c a l e d to the newly c a l c u l a t e d va lues of the s a t u r a t e d c o n d u c t i v i t y . Based on the b e s t - f i t va lues of K± and K 2 , the response of the p h y s i c a l model to the two subsequent r a i n f a l l r a t e s was p r e d i c t e d . A comparison of the p r e d i c t e d and observed w a t e r - t a b l e c o n f i g u r a t i o n s for a l l three r a i n f a l l r a t e s are shown i n F i g u r e 19. Appendix B c o n t a i n s a comparison of the p r e d i c t e d and measured h y d r a u l i c - h e a d d a t a , as w e l l as the c o r r e s p o n d i n g pres sure-head and e l e v a t i o n - h e a d d a t a . Tab le 6 summarizes the p r e d i c t e d and measured va lues o f out f low for the three r a i n f a l l r a t e s . Note tha t seepage was c o l l e c t e d at SC #7, but was not p r e d i c t e d , i n a l l c a s e s , due to the s l i g h t d i sp lacement of the f i n e sand l a y e r near the s l o p e . In a l l c a s e s , the t o t a l p r e d i c t e d out f low r a t e s were w i t h i n 11% of the t o t a l measured out f low r a t e s . The f i r s t run shows a l a r g e d i s c r e p a n c y between the p r e d i c t e d and measured out f low r a t e s from the uppermost seepage face (SC #5, 6, 7) Tab le 6. P r e d i c t e d versus observed out f low r a t e s . Run #1; R a i n f a l l r a t e = 1.80 cm/min Seepage C o l l e c t o r # P r e d i c t e d out f low Measured Outf low r a t e (cm 3/min) r a t e (cm^/min) 12 42 0 7 0 44 6 55 200 5 99 74 t o t a l p r e d i c t e d out f low = 1330 cm3/min t o t a l measured out f low = 1426 cmVmin Run #2; R a i n f a l l r a t e = 1.50 cm/min Seepage C o l l e c t o r # P r e d i c t e d out f low Measured Outf low r a t e (cmVmin) r a t e (cm^/min) 12 32 0 7 0 34 6 52 72 5 65 0 t o t a l p r e d i c t e d out f low = 1270 cm^/min t o t a l measured out f low = 1133 cm^/min Run #3: R a i n f a l l r a t e = 1.26 cm/min Seepage C o l l e c t o r # P r e d i c t e d out f low Measured Outf low r a t e ( c m 3 / m i n ) r a f c e ( C m 3 / m i n ) 7 0 13 6 25 16 t o t a l p r e d i c t e d out f low = 1095 cm^/min t o t a l measured out f low = 996 cm^/min 71 r e l a t i v e to the other r u n s . T h i s may be due to ponding t h a t o c c u r r e d over much of the i n f i l t r a t i o n sur face for a r a i n f a l l r a t e o f 1.8 cm/min. At s teady s t a t e , there was a cons tant depth o f water on the s u r f a c e . The maximum depth was approx imate ly 1 cm at the the r i g h t hand s ide o f the sur f ace and tapered o f f to zero depth before r e a c h i n g the s l o p e . T h i s c r e a t e d a head of water tha t was not p r e d i c t e d i n the t h e o r e t i c a l model . R e c a l l tha t when the i n f i l t r a t i o n sur face ponds d u r i n g the n u m e r i c a l s i m u l a t i o n , ijj i s set equa l to zero and the excess r u n o f f i s d i s r e g a r d e d ; i t i s not routed away nor i s i t conver ted i n t o an e q u i v a l e n t head of water . The m a t e r i a l used i n the p h y s i c a l model i s q u i t e porous and c o n d u c t i v e , so that a 0-1 cm cons t an t head of water a long the i n f i l t r a t i o n boundary can cause a l a r g e i n c r e a s e i n the flow r a t e through the r e g i o n , as o b s e r v e d . Given that the h y d r a u l i c head read ings were a c c u r a t e to w i t h i n + 3 mm, the e f f e c t on the head d i s t r i b u t i o n of a 0-1 cm head o f water a long the i n f i l t r a t i o n boundary i s not l i k e l y to be s i g n i f i c a n t . The r a i n f a l l r a te d u r i n g Run #2 was ad jus ted so that there was no ponding at s teady s t a t e . The t o t a l out f low r a t e and the out f low ra te s from the uppermost seepage face compare w e l l wi th the p r e d i c t e d r a t e s . Note , however, tha t d u r i n g both Run #1 and Run #2, seepage was p r e d i c t e d but not observed at SC #12. The s lope o f the water t a b l e immediate ly above the lowermost seepage face i s r e l a t i v e l y s t e e p . A s m a l l d i s c r e p a n c y i n the p o s i t i o n of the water t a b l e can t h e r e f o r e t r a n s l a t e i n t o a l a r g e r d i s c r e p a n c y i n the l e n g t h of the lowermost seepage f a c e . 72 For Run #3, the out f low r a t e s measured a long the s lope compare w e l l wi th t h e o r e t i c a l p r e d i c t i o n s . I t i s important to note that d u r i n g the f i n a l r u n , the sand d i d not d e s a t u r a t e ; tha t i s , i n the unsa tura ted zone, the p re s sure head remained g rea te r than the a i r - e n t r y va lue of the medium sand. The reason for t h i s i s as f o l l o w s . R e c a l l tha t the medium sand desa tura te s for \p < 11.5 cm, yet the lowest va lue of \J) tha t can be read by the tens iometer i s \p = -18 cm. T h e r e f o r e , o n l y a very narrow range of va lues c o r r e s p o n d i n g to de sa tura ted c o n d i t i o n s c o u l d be d e t e c t e d , namely, -18 cm < 4) < 11.5 cm. The experiment was purpose ly des igned to demonstrate the development of m u l t i p l e seepage faces for r e l a t i v e l y wet c o n d i t i o n . As i t turned o u t , the pre s sure head remained g rea te r than -9 cm and the medium sand d i d not d e s a t u r a t e . Whi le the e x p e r i m e n t a l c o n s t r a i n t s p r e c l u d e d a r i g o r o u s t e s t of the a b i l i t y o f the f i n i t e - e l e m e n t model to s imula te flow i n the de sa tura ted p o r t i o n of the unsa tura ted zone, the a b i l i t y of the model to p r e d i c t the w a t e r - t a b l e c o n f i g u r a t i o n and seepage-face l o c a t i o n s has been v e r i f i e d . In c o n c l u s i o n , the exper imenta l r e s u l t s c o n f i r m that the n u m e r i c a l model produces s o l u t i o n s that are p h y s i c a l l y c o r r e c t . The most p l a u s i b l e e x p l a n a t i o n for the minor n u m e r i c a l d i s c r e p a n c i e s tha t appear i n F i g u r e 19, Tab le 6, and Appendix B i s tha t the exper imenta l c o n d i t i o n s were not p e r f e c t l y matched i n the n u m e r i c a l runs . The n u m e r i c a l model was se t up assuming that each sand l a y e r i s homogeneous and i s o t r o p i c , when i n f a c t , i t i s probab le tha t they are n o t . The c o n d u c t i v i t y c o u l d 73 e a s i l y vary by a f a c t o r of two over a very shor t d i s t a n c e due to nonuniform compact ion . T h i s type of v a r i a t i o n r e s u l t s i n d i s c r e p a n c i e s between p r e d i c t e d and measured out f low r a t e s and head v a l u e s . However, i n l i g h t of the other a spects upon which the v e r i f i c a t i o n i s judged , these f a c t s do not seem to s e r i o u s l y d i s c r e d i t e i t h e r the n u m e r i c a l or the p h y s i c a l model . 74 Chapter 4 STEADY-STATE SENSITIVITY ANALYSIS Once the n u m e r i c a l model was v e r i f i e d , a s t e a d y - s t a t e a n a l y s i s was performed to i n v e s t i g a t e the s e n s i t i v i t y of the s o l u t i o n t o : a) the l o c a t i o n and number of impeding l a y e r s w i t h i n a h i l l s i d e , b) the magnitude of the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t between ad jacent g e o l o g i c u n i t s , and c) the r a i n f a l l r a t e . L e s s - d e t a i l e d s t u d i e s were performed to i n d i c a t e the s e n s i t i v i t y o f the s o l u t i o n to a n i s o t r o p y and the s lope a n g l e . Only a s m a l l p o r t i o n o f the range of h y d r o g e o l o g i c c o n d i t i o n s has been i n v e s t i g a t e d ; however, the cases chosen i l l u s t r a t e that the f l u i d - p r e s s u r e d i s t r i b u t i o n and the development of m u l t i p l e seepage faces are s t r o n g l y dependent upon the p o s i t i o n of the l a y e r s and t h e i r h y d r a u l i c p r o p e r t i e s . 4.1 Methodology The f i n i t e - e l e m e n t mesh used throughout much of the a n a l y s i s i s shown i n F i g u r e 20. The flow r e g i o n i s 350 m l o n g , 195 m h i g h , and i s bounded by a r e l a t i v e l y s teep s lope of approx imate ly 4 0 ° . There are s i x p o s s i b l e l o c a t i o n s for impeding l a y e r s , l a b e l e d A through F i n F i g u r e 20. Two types o f m a t e r i a l were s p e c i f i e d for each flow r e g i o n c o n s t r u c t e d w i t h t h i s mesh. M a t e r i a l No. 1 r e p r e s e n t s the dominant s o i l type and has a s a t u r a t e d h y d r a u l i c c o n d u c t i v i t y denoted by K]_. M a t e r i a l No. 2 A A 1 / "' / / ( ^ i / / c rr) \ — 0 350 X (m) F i g u r e 20. F i n i t e - e l e m e n t mesh used for the s t e a d y - s t a t e s e n s i t i v i t y a n a l y s i s . 76 r e p r e s e n t s the l e s s abundant s o i l type and has a s a t u r a t e d h y d r a u l i c c o n d u c t i v i t y denoted by K2- T h e r e f o r e , a flow r e g i o n i d e n t i f i e d as MeshAE i n d i c a t e s t h a t there are two l a y e r s o f M a t e r i a l No. 2 l o c a t e d at p o s i t i o n s A and E i n F i g u r e 20. The remainder o f the flow r e g i o n i s composed o f M a t e r i a l No. 1. S i m i l a r l y , examples of one- and t h r e e - l a y e r flow systems . i n c l u d e r e g i o n s i d e n t i f i e d as MeshD and MeshACE. Note tha t the number of l a y e r s r e f e r s to the number of K2 l a y e r s a s s igned to the h i l l s i d e . In a l l c a s e s , > K2» and the l a y e r s o f M a t e r i a l No. 2 w i l l be r e f e r r e d to as " impeding l a y e r s " . The c h a r a c t e r i s t i c curves c o r r e s p o n d i n g to M a t e r i a l No. 1 and M a t e r i a l No. 2 are d e r i v e d from those r e p r e s e n t i n g Pachapa F ine Sandy C l a y and Y o l o L i g h t C l a y , r e s p e c t i v e l y . These curves are shown i n F i g u r e 21. Data for both s o i l s were o b t a i n e d from Mualem (1976) and were o r i g i n a l l y r epor ted by Gardner (1959) and Moore (1939). The f i n i t e - e l e m e n t program r e q u i r e s the i n f o r m a t i o n i n the form o f t a b l e s o f 9 vs and 9 vs K r where K r i s the r e l a t i v e h y d r a u l i c c o n d u c t i v i t y , d e f i n e d a s : and K s i s the s a t u r a t e d h y d r a u l i c c o n d u c t i v i t y o f the s o i l . Throughout the s e n s i t i v i t y a n a l y s i s , the t a b l e s o f 9 vs and 9 vs Kr have remained unchanged; they have been d e r i v e d from the curves shown i n F i g u r e 21. T h e r e f o r e , i f the va lue o f K s i s changed the 9 (t) curve i s not mod i f i ed and the K(ty) curve i s 77 -6 -5 -4 -3 -2 -1 0 PRESSURE HEAD. f(m) -7 -9 -5 - 4 -3 -2 -1 0 PRESSURE HEAD, f(m) F i g u r e 21. C h a r a c t e r i s t i c curves used i n the s t e a d y - s t a t e s e n s i t i v i t y a n a l y s i s . 78 a u t o m a t i c a l l y s c a l e d i n accordance wi th E q u a t i o n 4 . 1 . I t shou ld be emphasized tha t n a t u r a l s o i l s do not e x h i b i t these i d e a l i z e d p r o p e r t i e s ; e(ty) curves are not independent of the va lue o f K s nor are K(^) curves r e l a t e d by the s imple s c a l i n g i m p l i e d i n E q u a t i o n 4 .2 . However, these assumptions have been made to s i m p l i f y the n u m e r i c a l p r o c e d u r e . In g e n e r a l , the s a t u r a t e d - u n s a t u r a t e d flow systems are far more s e n s i t i v e to the r e l a t i v e va lues o f K s than to the p r e c i s e nature o f the c h a r a c t e r i s t i c curves (Stephenson and F r e e z e , 1974) . For the g e n e r i c model ing e f f o r t presented h e r e , the approach embodied i n E q u a t i o n 4.1 i s j u s t i f i e d . The g e n e r a l i z e d boundary-value problem s o l v e d i n the s t e a d y - s t a t e a n a l y s i s i s shown i n F i g u r e 22a. Note tha t the r a i n f a l l r a te a p p l i e d to the i n f i l t r a t i o n boundary r e p r e s e n t s an average annual r a t e . In r e a l i t y , annual p r e c i p i t a t i o n p a t t e r n s form a time s e r i e s such as the one shown i n F i g u r e 22b. C o n s e q u e n t l y , the water t a b l e may f l u c t u a t e d u r i n g the course o f the y e a r , as i n d i c a t e d s c h e m a t i c a l l y i n F i g u r e 22c. I t i s assumed tha t by r e p l a c i n g the time s e r i e s wi th an average annual r a i n f a l l r a t e , the s t e a d y - s t a t e s o l u t i o n w i l l approximate the mean annual p o s i t i o n of the water t a b l e . The r a i n f a l l r a t e s used i n the s t e a d y - s t a t e s e n s i t i v i t y a n a l y s i s w i l l t h e r e f o r e appear to be perhaps an order o f magnitude lower than those observed for i n d i v i d u a l s torms . For example, a s teady s t a t e r a i n f a l l o f 1 0 - 7 m/s r e p r e s e n t s a r e g i o n t h a t a n n u a l l y r e c e i v e s 3 m of p r e c i p i t a t i o n , w h i l e the r a i n f a l l r a te 79 January Average Annual Rainfall Rate 2 A = 0 A V E R A G E A N N U A L R A I N F A L L R A T E i m u u i i i i ii) = 2m K 1 > K 2 2>X rrr = 0 sz Equation of Flow: dx K( <,z,^ ) ^ K<x,z ,40(!f + 1) = 0 F i g u r e 22. G e n e r a l i z e d boundary-va lue problem s o l v e d i n the s teady s t a t e s e n s i t i v i t y a n a l y s i s . 80 d u r i n g an i n d i v i d u a l storm i n tha t same r e g i o n may be on the order of I O - 6 m/s (3.6 mm/hr). 4.2 R e s u l t s The r e s u l t s of the s e n s i t i v i t y s tudy are presented i n terms of a) the nature of the f l u i d - p r e s s u r e d i s t r i b u t i o n , as c h a r a c t e r i z e d by the extent of the unsa tura ted wedge, and b) the percentage of the t o t a l out f low pas s ing through each of the seepage f a c e s . C o n c l u s i o n s are drawn from graphs summarizing the r e s u l t s of many s i m u l a t i o n s i n which the l a y e r i n g sequence, the r a i n f a l l r a t e , and the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t , K 1 / K 2 , have been v a r i e d . I t should be noted t h a t throughout the a n a l y s i s , the e f f e c t of i n c r e a s i n g K 1 / K 2 has been s t u d i e d by l e a v i n g the va lue of unchanged and d e c r e a s i n g the va lue of K 2 . The same c o n c l u s i o n s app ly i f K2 i s f i x e d and K i i s i n c r e a s e d , a l though i n d i v i d u a l s o l u t i o n s d i f f e r s l i g h t l y depending upon the way i n which K 1 / K 2 i s , i n c r e a s e d . ONE-LAYER FLOW SYSTEMS S i x flow reg ions were used to i n v e s t i g a t e the development of m u l t i p l e seepage faces i n o n e - l a y e r systems, each c o n t a i n i n g a 10 m t h i c k l a y e r of M a t e r i a l No. 2 at one o f the s i x l o c a t i o n s i n d i c a t e d i n F i g u r e 20. H y d r a u l i c c o n d u c t i v i t y c o n t r a s t s of 20 and 25 were s t u d i e d i n i t i a l l y ; Ki was ma inta ined at 1.4 x 10 - 6 m/s i n both ca se s . The range of r a i n f a l l r a t e s was v a r i e d between r e l a t i v e l y h igh va lues for which the r e g i o n was t r a n s m i t t i n g water at a maximum r a t e , and r e l a t i v e l y low va lues for which n u m e r i c a l i n s t a b i l i t y o c c u r r e d 31 due to the development of s teep ty- and K - g r a d i e n t s i n the unsa tura ted zone. These n u m e r i c a l l i m i t a t i o n s are d i s c u s s e d i n more d e t a i l l a t e r i n t h i s c h a p t e r . F i g u r e 23 summarizes the e f f e c t o f the p o s i t i o n o f the impeding l a y e r and the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t on the ex tent o f the unsa tura ted wedge. Note tha t for Mesh A , the unsa tura ted wedge extends the e n t i r e l e n g t h o f the h i l l s i d e for a l l r a i n f a l l r a t e s and a seepage face has not formed above the impeding l a y e r . Two c o n c l u s i o n s can be drawn from F i g u r e 23. F i r s t , for a g i v e n K]_/K2, an i n c r e a s e i n the e l e v a t i o n o f the impeding l a y e r i n c r e a s e s the extent of the unsa tura ted wedge. Second, for a g i v e n e l e v a t i o n o f the impeding l a y e r , an inc rea se i n Kj_/K2 i n c r e a s e s the ex tent o f the unsa tura ted wedge. These c o n c l u s i o n s are i l l u s t r a t e d by the h y d r a u l i c - h e a d d i s t r i b u t i o n s and w a t e r - t a b l e c o n f i g u r a t i o n s shown i n F i g u r e 24 for Mesh E and Mesh B. F i g u r e 25 summarizes the e f f e c t o f the p o s i t i o n of the impeding l a y e r and the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t on the percentage o f the t o t a l out f low d i s c h a r g e d acros s the uppermost seepage f a c e . For a g i v e n K 1 / K 2 , an i n c r e a s e i n the e l e v a t i o n of the impeding l a y e r decreases the percentage o f the t o t a l ou t f low ac ro s s the uppermost seepage f a c e . For a g i v e n e l e v a t i o n o f the impeding l a y e r , an inc rea se i n K 1 / K 2 i n c r e a s e s the percentage o f the t o t a l out f low acros s the uppermost seepage f a c e . C o r r e s p o n d i n g l y , the l e n g t h o f the uppermost seepage face i n c r e a s e s wi th a decrease i n e l e v a t i o n and an i n c r e a s e i n K i/K2# as shown i n F i g u r e 26. — K 1 / K 2 = 2 0 K 1 / K 2 = 2 5 • Seepage face not formed above Kg layer A EXTENT OF THE UNSATURATED WEDGE ALONG THE BASE OF THE IMPEDING LAYER (m) F i g u r e 23. S e n s i t i v i t y o f the unsa tura ted wedge to the p o s i t i o n o f the impeding l a y e r and to the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t . oo to F i g u r e 24. Flow reg ions e x e m p l i f y i n g the e f f e c t o f the p o s i t i o n of the impeding l a y e r and the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t on the ex tent o f the unsaturated wedge. 34 F i g u r e 25. S e n s i t i v i t y o f the o u t f l o w to the p o s i t i o n of the impeding l a y e r and to the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t . 85 F i g u r e 26. The l e n g t h of the uppermost seepage face as a f u n c t i o n o f the p o s i t i o n o f the impeding l a y e r and the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t . 86 The ex tent o f the unsa tura ted wedge and the percentage o f the t o t a l ouflow acros s the uppermost seepage face has been shown to be s e n s i t i v e to h y d r a u l i c c o n d u c t i v i t y c o n t r a s t s t h a t d i f f e r by l e s s than h a l f an order o f magnitude. N a t u r a l d e p o s i t s , however, may have h y d r a u l i c c o n d u c t i v i t i e s which vary over many o r d e r s of magnitude. Mesh C was s e l e c t e d to i n v e s t i g a t e the s o l u t i o n as K 1 / K 2 i s i n c r e a s e d over four o rder s of magnitude. To do t h i s , the flow r e g i o n was extended to x = 3350 m and the ex tent o f the unsa tura ted wedge was measured for K 1 / K 2 = 10, 100, 1000, 2500, 5000 and 10 ,000. In each ca se , K]_ - 10~^m/s and the r a i n f a l l r a te was 4 x l 0 _ 8 m / s . The r e s u l t s are shown i n F i g u r e 27. For K i / K 2 > 5000, the wedge extends the e n t i r e l e n g t h o f the h i l l s i d e and the s a t u r a t e d zone above the impeding l a y e r i s c o m p l e t e l y p e r c h e d . TWO-LAYER FLOW SYSTEMS The development of m u l t i p l e seepage faces i n two- layer systems was s t u d i e d wi th ten l a y e r i n g sequences , d i s t i n g u i s h e d on the b a s i s o f the d i s t a n c e s e p a r a t i n g the impeding l a y e r s , as o u t l i n e d i n Table 7. The seepage face formed above the upper impeding l a y e r w i l l be c a l l e d the upper seepage f a c e ; the seepage face a s s o c i a t e d wi th the lower impeding l a y e r w i l l be Tab le 7. C l a s s i f i c a t i o n o f two- layer systems. D i s t a n c e s e p a r a t i n g the impeding l a y e r s (m) Mesh Name 10 20 30 40 A C , BD, C E , DF AD, BE , CF A E , BF AF 37 10,000 n EXTENT OF THE UNSATURATED WEDGE, (m) F i g u r e 27. The ex tent of the u n s a t u r a t e d wedge as the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t i n c r e a s e s over s e v e r a l o r d e r s o f magnitude fo r K i = 1 0 - 7 m/s and a r a i n f a l l r a t e o f 4 x 1 0 - 8 m/s . 88 c a l l e d the middle seepage f a c e ; and, the seepage face which forms at the base of the h i l l s i d e w i l l be c a l l e d the b a s a l seepage f a c e . For a g i v e n d i s t a n c e s e p a r a t i n g the impeding l a y e r s , the c o n c l u s i o n s c o n c e r n i n g the f l u i d - p r e s s u r e d i s t r i b u t i o n are the same as those made for o n e - l a y e r systems. I f the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t i s c o n s t a n t , then an i n c r e a s e i n the e l e v a t i o n of the impeding l a y e r s i n c r e a s e s the extent of both unsa tura ted wedges; i f the e l e v a t i o n of the l a y e r s i s c o n s t a n t , then an i n c r e a s e i n K 1 / K 2 i n c r e a s e s the extent of the wedges. These c o n c l u s i o n s are e x e m p l i f i e d by the flow r e g i o n s shown i n F i g u r e 28 for the case i n which the d i s t a n c e s e p a r a t i n g the impeding l a y e r s i s 10 m. The c o n c l u s i o n s a l s o h o l d for 20, 30, and 40 m of s e p a r a t i o n . The r e l a t i v e extent of the unsa tura ted wedges depends upon the d i s t a n c e s e p a r a t i n g the impeding l a y e r s . In F i g u r e 29, the extent of both wedges i s p l o t t e d for Meshes A C , AD, A E , and A F , c o r r e s p o n d i n g to 10, 20, 30, and 40 m of s e p a r a t i o n , r e s p e c t i v e l y . In a l l c a s e s , the extent of the lower wedge i s l e s s than t h a t o f the upper wedge, except for 10 m of s e p a r a t i o n . The h y d r a u l i c - h e a d d i s t r i b u t i o n s and w a t e r - t a b l e c o n f i g u r a t i o n s i n F i g u r e 30 i l l u s t r a t e t h i s for Meshes BD, B E , and BF . The percentages of the t o t a l out f low d i s c h a r g e d acros s the upper and middle seepage f a c e s , %Qu and %Qm, are summarized i n F i g u r e 31 for Ki/K2=20. A comparison of the r e s u l t s down each column shows tha t i f the p o s i t i o n of the upper impeding l a y e r F i g u r e 28. The e f f e c t of the e l e v a t i o n of the impeding l a y e r s and the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t on the unsaturated wedges for a cons t an t d i s t a n c e s e p a r a t i n g the impeding l a y e r s . 00 10 (0 "•»"» 0.8 -E to 'o 0.7 -y -X 0.8 -UJ 1- 0.6 -< CC —1 0 .4 --1 FA 0.3 -z < 0.2 -a. Upper wedge AC AD Distance separating Mesh K 2 layers (m) AC AD AE AF 10 20 30 40 — i — 6 0 • - no seepage face formed above respective K 2 layer 100 — i — 160 — I — 2 0 0 2 6 0 ( 0 0.8 -E 0.7 -'o t -X 0.8 -TE 0.6 -< CC _l 0 .4 --I FA 0.3 -z < 0.2 -cc b. Lower wedge AF AE — i — 60 100 160 EXTENT OF THE UNSATURATED WEDGE (m) 2 0 0 2 6 0 Figure 29. The r e l a t i v e extent of the unsaturated wedges f o r K i / K 2 = 20. o 91 200 -) X (m) Figure 30. Comparison of the r e l a t i v e extent of the unsaturated wedges for K 1 / K 2 = 20, Ki = 1.4 x 10~6 m/s, and a steady-state r a i n f a l l rate of 0.3 x 10""6 m/s. 92 K 1 / K 2 - 2 0 , K 1 - 1.4 x 1 0 6 m / s OO tl» CO d o d o eo io •* « d o d o R A I N F A L L R A T E x 10 _ 6m/s F i g u r e 3 1 . A c o m p a r i s o n o f t h e p e r c e n t a g e o f t h e t o t a l o u t f l o w d i s c h a r g e d a c r o s s t h e u p p e r a n d m i d d l e s e e p a g e f a c e s i n t w o - l a y e r s y s t e m s . 93 remains f i x e d , then as the second l a y e r i s p l a c e d at s u c c e s s i v e l y lower p o s i t i o n s , %Qu decreases and %Qm i n c r e a s e s . C o n v e r s e l y , a comparison o f the r e s u l t s a long each row shows t h a t i f the p o s i t i o n of the lower impeding l a y e r remains f i x e d , then as the upper l a y e r i s p l a c e d at s u c c e s s i v e l y lower p o s i t i o n s , %Qu i n c r e a s e s and %Qm decrea se s . A comparison of the r e s u l t s a long each d i a g o n a l shows t h a t for a g i v e n d i s t a n c e s e p a r a t i n g the l a y e r s , an inc rea se i n the e l e v a t i o n o f the l a y e r s decreases both %Qu and %Qm. The r e l a t i v e p a r t i t i o n i n g of the t o t a l out f low between the seepage faces depends upon the d i s t a n c e s e p a r a t i n g the impeding l a y e r s . For 10m of s e p a r a t i o n (Meshes A C , BD, C E , DF) %Qu > %Qm; for 20m of s e p a r a t i o n (Meshes AD, BE , C F ) , a c o n s i s t e n t p a t t e r n i s not d e v e l o p e d ; for 30 and 40 m of s e p a r a t i o n (Meshes A E , BF, A F ) , %Qu < %Qm. The e f f e c t o f K 1 / K 2 on the percentage o f the t o t a l out f low from each seepage face i s summarized i n F i g u r e 32 for K 1 / K 2 = 20, 30, 100, and 1000. Because o f the r e d u c t i o n i n K 2 , the t o t a l flow through the h i l l s i d e i s decrea sed , on average , by 74% as K 1 / K 2 i s i n c r e a s e d from 20 to 1000. In a l l c a se s , %Qu i n c r e a s e s for an i n c r e a s e i n K 1 / K 2 . The percentage from the middle seepage face remains r e l a t i v e l y cons t an t u n t i l K 1 / K 2 i s i n c r e a s e d from 100 to 1000 where i t i s n o t i c e a b l y reduced . For K l / K 2 i % Q U > > % Q m a n < 3 the middle seepage face i s p re sen t o n l y i n a s s o c i a t i o n w i t h l a y e r s E and F . The h y d r a u l i c - h e a d d i s t r i b u t i o n and w a t e r - t a b l e c o n f i g u r a t i o n i s shown i n F i g u r e 33 for Mesh BF as K 1 / K 2 i s i n c r e a s e d from 20 to 1000. Rainfall rate - 4 x 1 0 8m/s, K, • 1 x 10 7m/s | | Upper seepage face Middle seepage face o o o o o o o o u tu N WO O N f f lOO " l " * 2 • >- o >- o O o o o o o o o CM CO O O CM CO o o »- o *- o Mesh: AC AD AE AF BO BE BF CE CF o o o CM CO O DF F i g u r e 32. The e f f e c t of the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t on the percentage of the t o t a l o u t f l o w d i s c h a r g e d acros s the upper and middle seepage f a c e s formed i n two-layer systems. F i g u r e 33. The e f f e c t of i n c r e a s i n g the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t over s e v e r a l o r d e r s o f magnitude f o r MeshBF, K]_ = 1 0 - 7 m/ s , and a r a i n f a l l r a t e of 4 x 1 0 - 8 m/s . 96 THREE-LAYER FLOW SYSTEMS The development of m u l t i p l e seepage faces i n t h r e e - l a y e r systems was s t u d i e d wi th four l a y e r i n g sequences : ACE, BDF, ACF, and ADF. The a d d i t i o n o f a t h i r d l a y e r reduces the extent of a l l the unsa tura ted wedges, as shown i n F i g u r e 34. The t o t a l flow through the h i l l s i d e decreases by 26% from the one-l a y e r system to the t h r e e - l a y e r system. The r e l a t i v e extent o f the unsa tura ted wedges for the t h r e e - l a y e r systems i s shown i n F i g u r e 35 for K]_/K2 = 20. A comparison of Mesh ACE and Mesh BDF aga in i l l u s t r a t e s tha t an i n c r e a s e i n the e l e v a t i o n o f the impeding l a y e r s i n c r e a s e s the ex tent o f the unsa tura ted wedges. For 10m s e p a r a t i n g two of the impeding l a y e r s , the ex tent o f the upper wedge i s l e s s than or equa l to tha t o f the lower wedge; for 20 m s e p a r a t i n g two of the impeding l a y e r s , the rever se i s t r u e . F i g u r e 36 summarizes the r e l a t i v e percentages o f the t o t a l out f low d i s c h a r g e d acros s each seepage face for K 1 / K 2 = 20, 30, 100, and 1000. Because o f the r e d u c t i o n i n K 2 / the t o t a l f low through the h i l l s i d e d e c r e a s e s , on average , by 82% as K 1 / K 2 i s i n c r e a s e d from 20 to 1000. In a l l cases shown, the percentage of the t o t a l out f low from the uppermost seepage face i n c r e a s e s as K 1 / K 2 i n c r e a s e s . For K 1 / K 2 > 100, more than 80% of the flow through the h i l l s i d e e x i t s acros s i t . The behav ior o f the seepage face formed above the middle impeding l a y e r i s l e s s c l e a r - c u t . The percentage o f the t o t a l ou t f low i n c r e a s e s as K l / K 2 i n c r e a s e s from 20 to 30 and decreases from 100 to 1000. As K i y R 2 i n c r e a s e s from 30 to 100, the response depends upon 97 0 40 80 120 160 200 240 230 320 X (m) F i g u r e 34. The e f f e c t of i n c r e a s i n g the number of impeding l a y e r s fo r K 1 / K 2 = 20, K i = 1.4 x 10** 6 m/s , and a r a i n f a l l r a t e of 0 .3 x 1 0 - 6 m/s . F i g u r e 35. T h r e e - l a y e r systems for K]_/K2 = 20, and a r a i n f a l l r a t e o f 4 x 1 0 - 8 m/s . K i = 1 0 - 7 m/s , 100 n 90 o 3 O < o I-III x 80 -70 -60 -50 UJ 4 0 -C3 < H Z LU 30 O <r UJ a. 20 1 o -K 1 - 1x 10~ 7m/s Upper seepage face H Middle seepage face Lower seepage face 1 m K 1 / K 2 : Me sh : CM CO O O O O OJ CO T- T-ACE CM CO O O O O CM co <- i -BDF 1 CM CO O O O O ACF n ; CM CO O O O O C\J CO i - y-ADF F i g u r e 36. The e f f e c t of the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t on the percentage of the t o t a l ouflow d i s c h a r g e d a c r o s s the seepage faces formed i n t h r e e - l a y e r systems f o r a r a i n f a l l r a t e of 4 x 10 ~8 m/s. 100 the p o s i t i o n o f the middle impeding l a y e r . I f the seepage face i s formed above l a y e r C , the percentage o f the t o t a l out f low i n c r e a s e s ; i f formed above l a y e r D, the percentage o f the t o t a l out f low d e c r e a s e s . The seepage face formed above the lowermost impeding l a y e r i s p re sent o n l y i n a s s o c i a t i o n wi th l a y e r F . The percentage o f the t o t a l out f low decreases as K 1 / K 2 i n c r e a s e s from 20 to 30 and the seepage face i s absent for K ] / K 2 > 100. ANISOTROPY For f l a t - l y i n g sedimentary r o c k s , i t i s common for the s a t u r a t e d h y d r a u l i c c o n d u c t i v i t y i n the h o r i z o n t a l d i r e c t i o n , K x , to be up to ten times tha t i n the v e r t i c a l d i r e c t i o n , K z , w i t h i n a g i v e n l a y e r (Freeze and C h e r r y , 1979) . I t i s a l s o p o s s i b l e tha t i n f r a c t u r e d r o c k s , K z may exceed K x . A p r e l i m i n a r y study to i n v e s t i g a t e the e f f e c t o f a n i s o t r o p y was performed for a o n e - l a y e r system. Throughout the e n t i r e a n a l y s i s , the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t , measured i n the v e r t i c a l d i r e c t i o n , was h e l d cons tant and equa l to 30. The r e s u l t s are shown i n F i g u r e 37. For the flow r e g i o n shown i n F i g u r e 37a, both M a t e r i a l No. 1 and M a t e r i a l No. 2 are i s o t r o p i c . In F i g u r e 37b, M a t e r i a l No. 1 has been made a n i s o t r o p i c by i n c r e a s i n g the va lue o f K x so that K x = 10K Z . M a t e r i a l No. 2 has not been changed and remains i s o t r o p i c . Note tha t i n comparison wi th F i g u r e 37a, the unsa tura ted wedge i s more e x t e n s i v e and there i s no longer ponding a long the i n f i l t r a t i o n s u r f a c e . Because K x has been i n c r e a s e d for M a t e r i a l No. 1, the t o t a l flow through the h i l l s i d e i n c r e a s e d 101 X (m) b. K 1 x - 1 0 " 6 m / s K 1 z = 1 0 " 7 m / s i X (m) c. K 1 x - 1 0 " 8 m / s K 1 z - 1 0 " 7 m / s F i g u r e 37. The e f f e c t of a n i s t r o p y fo r Kj_/K2 = 30 and a s t e a d y - s t a t e r a i n f a l l r a t e o f 4 x 1 0 - 8 m/s . 102 by approx imate ly 120%. In F i g u r e 37c, M a t e r i a l No. 1 has been made a n i s o t r o p i c by r e d u c i n g the va lue of K x so that K x = 1/10 K z ; M a t e r i a l No. 2 aga in remains i s o t r o p i c . Note t h a t i n comparison w i t h F i g u r e 37a, the unsa tura ted wedge i s g r e a t l y r e d u c e d . Because K x has been reduced for M a t e r i a l No. 1, the t o t a l flow through the h i l l s i d e i s decreased by approx imate ly 78% from F i g u r e 37a. F i g u r e 37 has shown tha t the w a t e r - t a b l e c o n f i g u r a t i o n i s s e n s i t i v e to a n i s o t r o p y w i t h i n M a t e r i a l No. 1. However, s i m i l a r s t u d i e s showed tha t a n i s o t r o p y w i t h i n M a t e r i a l No. 2 d i d not produce s o l u t i o n s tha t were s i g n i f i c a n t l y d i f f e r e n t from that shown i n F i g u r e 37a. I t appears tha t a n i s o t r o p y w i t h i n the impeding l a y e r i s secondary to the e f f e c t on the system of a h y d r a u l i c c o n d u c t i v i t y c o n t r a s t of 30, measured i n the v e r t i c a l d i r e c t i o n . SLOPE ANGLE F i g u r e 38 shows the w a t e r - t a b l e c o n f i g u r a t i o n as a f u n c t i o n o f the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t for a r e l a t i v e l y low s lope angle of 8 ° . M u l t i p l e seepage faces do not form u n t i l the r a t i o K 1 / K 2 exceeds 200. R e c a l l tha t one order of magnitude d i f f e r e n c e i n K\ and K2 was s u f f i c i e n t to produce m u l t i p l e seepage faces on the 4 0 ° s l ope s s t u d i e d p r e v i o u s l y . T h e r e f o r e , i f a l l e l s e i s c o n s t a n t , then a g r e a t e r h y d r a u l i c c o n d u c t i v i t y c o n t r a s t i s r e q u i r e d to produce m u l t i p l e seepage faces on g e n t l e s lopes as compared w i t h s teep s l o p e s . 416 -E N K 1 = 1 x 10 m/s — 8 Rainfall rate = 5 x 1 0 m/s - - K^K2 = 100 K ^ K 2 = 200 •— K^/K2 = 300 0 0 Vertical exaggeration x 2 X (m) 2988 F i g u r e 38. The w a t e r - t a b l e c o n f i g u r a t i o n as a f u n c t i o n of the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t for a s l o p e ang le of 8 ° . r—' O CO 104 4.3 D i s c u s s i o n The r e s u l t s presented i n the p r e v i o u s s e c t i o n have shown the h y d r a u l i c - h e a d d i s t r i b u t i o n and w a t e r - t a b l e c o n f i g u r a t i o n to be complex i n l a y e r e d s l o p e s . From these r e s u l t s , g e n e r a l i z a t i o n s can be made r e g a r d i n g the e f f e c t of the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t and the p o s i t i o n of the impeding l a y e r on the s o l u t i o n . These g e n e r a l i z a t i o n s are made wi th r e f e r e n c e to the o n e - l a y e r flow systems i n which they are p a r t i c u l a r l y c l e a r , but they are a l s o e v i d e n t i n the more complex r e s u l t s pre sented for two- and t h r e e - l a y e r systems. F i r s t , i f a l l o ther f a c t o r s are e q u a l , then an i n c r e a s e i n the r a t i o K 1 / K 2 i n c r e a s e s both the extent of the unsa tura ted wedge and the percentage of the t o t a l out f low from the seepage face formed above the impeding l a y e r . To e x p l a i n t h i s , c o n s i d e r the r e f r a c t i o n of f low tha t occurs w i t h i n a f u l l y -s a t u r a t e d flow r e g i o n c o n t a i n i n g a l e s s -permeab le s o i l l a y e r , as shown i n F i g u r e 39. When t h i s type of r e f r a c t i o n occur s i n the v i c i n i t y o f a s l o p e , water l eaves the flow r e g i o n through the more permeable m a t e r i a l above and below the impeding l a y e r . The f l u x through the l a y e r cannot match the out f low r a t e from the m a t e r i a l below i t un le s s an unsa tura ted wedge forms. I f the r a t i o K 1 / K 2 i s i n c r e a s e d by d e c r e a s i n g K 2 , the f l u x through the l a y e r decreases and a s m a l l e r h y d r a u l i c g r a d i e n t i s r e q u i r e d to d e l i v e r the reduced q u a n t i t y of water to the d i s c h a r g e area below. C o n s e q u e n t l y , the s lope of the water t a b l e beneath the impeding l a y e r decreases and the extent of the unsa tura ted wedge i n c r e a s e s . S i n c e l e s s of the t o t a l i n f l o w can pass 105 Source: Freeze and Cherry, 1979. F i g u r e 39. R e f r a c t i o n o f g r o u n d w a t e r a c r o s s a l e s s - p e r m e a b l e l a y e r . 106 through the impeding l a y e r , a l a r g e r percentage o f the t o t a l out f low e x i t s acros s the seepage face formed above the impeding l a y e r . The second g e n e r a l i z a t i o n i s tha t i f a l l o ther f a c t o r s are e q u a l , then a decrease i n the e l e v a t i o n of the impeding l a y e r decreases the extent of the unsa tura ted wedge and i n c r e a s e s the percentage o f the t o t a l ouflow from the uppermost seepage f ace . To e x p l a i n t h i s , one needs to r ecogn ize that the lower p o r t i o n of the s lope i s a d i s c h a r g e area and the major component o f flow i s d i r e c t e d h o r i z o n t a l l y towards the water t a b l e bounding the unsa tura ted wedge and towards the uppermost seepage f a c e . T h i s reduces the ex tent o f the unsa tura ted wedge and i n c r e a s e s the percentage o f the t o t a l out f low acros s the uppermost seepage f a c e . The converse i s t rue i f the impeding l a y e r i s l o c a t e d i n the upper p o r t i o n o f the s lope where the major component o f flow i s d i r e c t e d v e r t i c a l l y downwards. These g e n e r a l i z a t i o n s can be used to p r e d i c t the combina t ion o f h y d r o g e o l o g i c v a r i a b l e s most l i k e l y to produce m u l t i p l e seepage faces i n o n e - l a y e r flow systems. In g e n e r a l , m u l t i p l e seepage faces are l i k e l y to be pre sent on s teep s lope s i n which the r a t i o o f K 1 / K 2 i s at l e a s t an order o f magnitude. For K 1 / K 2 on the order o f 20 or 30, s t e a d y - s t a t e m u l t i p l e seepage faces are more l i k e l y to be pre sent i f the impeding l a y e r i s l o c a t e d i n the lower t w o - t h i r d s o f the s l o p e . Regard le s s o f the p o s i t i o n o f the l a y e r , s teep s l o p e s i n which K l / K 2 >, l u u f e a t u r e m u l t i p l e seepage faces and an unsa tura ted 107 zone beneath the l a y e r t h a t may be q u i t e e x t e n s i v e , as i n d i c a t e d i n F i g u r e 27. With regard to the e f f e c t o f the r a i n f a l l r a t e , note that i n F i g u r e s 23 and 25, K i = 1.4 x 1 0 - 6 m/s and the s t e a d y - s t a t e r a i n f a l l r a te has been v a r i e d between 1.0 x 10"~6 m/s (32 m/year) and 0.3 x 1 0 - 5 m / s (9m/year) . These s t e a d y - s t a t e r a i n f a l l r a t e s are u n r e a l i s t i c ; average annual r a i n f a l l r a t e s i n North America g e n e r a l l y vary between 8.0 x 1 0 - 8 m / s (2.5m/year) and 8 x 1 0 - 9 m / s (.25 m/year) (Barry and C h o r l e y , 1976) . However, by an a p p r o p r i a t e s c a l i n g o f the h y d r a u l i c c o n d u c t i v i t y , the same r e s u l t s shown i n F i g u r e 23 and 25 can be o b t a i n e d for s t e a d y - s t a t e r a i n f a l l r a t e s w i t h i n the range expected i n North A m e r i c a . One can invoke s i m i l i t u d e c o n s i d e r a t i o n s to show t h a t t h i s i s so . In e s sence , i f we model two flow r e g i o n s tha t have the same s i z e and geometry, the same type o f boundary c o n d i t i o n s , and the same h y d r a u l i c c o n d u c t i v i t y c o n t r a s t , then the h y d r a u l i c - h e a d d i s t r i b u t i o n s p r e d i c t e d by the f i n i t e - e l e m e n t model w i l l be i d e n t i c a l i f : ( R ) l = _(R)_2 4 ( K i ) 1 ( K i ) 2 where R i s the r a i n f a l l r a t e and the s u b s c r i p t s o u t s i d e the parentheses r e f e r to the r e s p e c t i v e flow r e g i o n . T h i s i s t rue o n l y i f the K(^) curves are s c a l e d i n accordance wi th E q u a t i o n 4 .1 and i f the 9 (vp) curves remain c o n s t a n t , as d i s c u s s e d i n S e c t i o n 4 . 1 . A p r o o f o f E q u a t i o n 4 .2 can be g i v e n for o n e - d i m e n s i o n a l , s a t u r a t e d flow through a l a y e r e d s o i l co lumn. Proof o f 108 E q u a t i o n 4 . 2 , as a p p l i e d to s a t u r a t e d - u n s a t u r a t e d flow through l a y e r e d h i l l s i d e s , would r e q u i r e an a n a l y s i s s i m i l a r to that of Verma and B r u t s a e r t (1971) i n which the boundary-va lue problem i s formula ted and so lved i n terms of d i m e n s i o n l e s s v a r i a b l e s s e l e c t e d to c h a r a c t e r i z e the flow p r o c e s s . K l i n e (1965) p r e s e n t s a d e t a i l e d d i s c u s s i o n o f the g e n e r a l methods used i n such ana ly se s and Hubbert (1937) p r o v i d e s a good i n t r o d u c t i o n to the s u b j e c t o f s i m i l i t u d e . To the a u t h o r ' s knowledge, a r i g o r o u s p r o o f o f Equa t io n 4.2 has not been performed for the boundary-value problem c o n s i d e r e d i n t h i s t h e s i s . However, the e m p i r i c a l v a l i d i t y o f E q u a t i o n 4.2 has been t e s t ed and conf i rmed by the author wi th computer s i m u l a t i o n s . In view o f E q u a t i o n 4 . 2 , the r e s u l t s p re sented i n F i g u r e s 23 and 25 would h o l d for an i n f i n i t e number of combinat ions o f R and K^, some o f which are r e a l i s t i c and some are n o t . The r e s u l t s o f the s e n s i t i v i t y s tudy apply to the range of annual r a i n f a l l r a t e s encountered i n North America for va lue s of K^ between I O - 7 m/s and I O " 8 m/s . For K i <L0 - 8 m/s , the s o l u t i o n s are i n s e n s i t i v e w i t h i n t h i s range of r a i n f a l l r a t e s because the maximum amount o f water the h i l l s i d e can t r a n s m i t i s l e s s than t h a t d e l i v e r e d to the i n f i l t r a t i o n boundary. Problems i n v o l v i n g n u m e r i c a l i n s t a b i l i t y prevented the study o f r e a l i s t i c r a i n f a l l r a t e s a p p l i e d to r eg ions i n which K i > I O - 7 m/s. LIMITATIONS AND ASSUMPTIONS Of the assumptions and l i m i t a t i o n s a s s o c i a t e d wi th the mathemat ica l mode l , the f o l l o w i n g three are the most important 109 i n terms of t h e i r p o t e n t i a l e f f e c t on the s t e a d y - s t a t e a n a l y s i s : 1 . Flow i s t w o - d i m e n s i o n a l . T h i s assumption r e s t r i c t s the a p p l i c a b i l i t y o f the r e s u l t s to flow r e g i o n s i n which the h y d r a u l i c g r a d i e n t i n the t h i r d d imension i s n e g l i g i b l e . For example, a t h r e e - d i m e n s i o n a l a n a l y s i s would be r e q u i r e d to model flow a long s lopes that are deep ly i n c i s e d by g u l l e y s or i n cases where the lower p e r m e a b i l i t y m a t e r i a l e x i s t s i n the form of l ense s as opposed to l a y e r s . In a d d i t i o n , the assumption i m p l i e s tha t i n f i l t r a t i o n and the s o i l p r o p e r t i e s are uni form i n the t h i r d d i m e n s i o n . 2 . There i s no i n f i l t r a t i o n a long the seepage-face boundary. In r e a l i t y , water may be a v a i l a b l e to i n f i l t r a t e a long the s lope as a r e s u l t of d i r e c t r a i n f a l l or sur f ace r u n o f f ; n e i t h e r has been modeled. For n e a r - v e r t i c a l s l o p e s , r e s t r i c t i o n o f i n f i l t r a t i o n to the f l a t upland sur face i s a reasonable a s sumpt ion . I t i s not r e a s o n a b l e , however, for g e n t l e s l ope s and one would expect t h a t the unsa tura ted wedge would be c o n s i d e r a b l y l e s s e x t e n s i v e i f i n f i l t r a t i o n were to be modeled a long the seepage-face boundary. For the s teep s lope s c o n s i d e r e d i n the pre sent s t u d y , t h i s assumption might i n t r o d u c e e r r o r for r e l a t i v e l y low va lues o f K\/K2- However, for K 1 / K 2 2, 1 0 0 , i t seems reasonable to b e l i e v e that the r e l a t i v e importance o f i n f i l t r a t i o n a long the s lope would decrease and e x t e n s i v e unsa tura ted wedges would p r e v a i l . 3 . A n u m e r i c a l s o l u t i o n may not be p o s s i b l e when steep g r a d i e n t s i n p re s sure head and h y d r a u l i c c o n d u c t i v i t y deve lop 110 w i t h i n the unsa tura ted zone. T h i s l i m i t a t i o n prevented the study o f l a y e r e d systems for low r a i n f a l l r a t e s . A new model developed by Coo ley (1983) t h a t uses the subdomain f i n i t e -element method appears to have overcome the n u m e r i c a l problems encountered i n the pre sent s tudy . Stephenson and Freeze (1974) found tha t by modi fy ing the shape o f the K(\j)) curve u n t i l K v a r i e d over o n l y one order o f magnitude, t h e i r i n s t a b i l i t y problems c o u l d be c o n t r o l l e d . In the pre sent s t u d y , t h i s l a t t e r approach reduced , but d i d not c u r e , the prob lem. An a l t e r n a t i v e s o l u t i o n i s to d e s i g n the f i n i t e - e l e m e n t mesh wi th c l o s e l y spaced noda l p o i n t s i n areas where the d r i e s t c o n d i t i o n s are a n t i c i p a t e d . T h i s was done for the t r a n s i e n t a n a l y s i s , but not for the s t e a d y - s t a t e s i m u l a t i o n s . For the purposes o f the s t e a d y - s t a t e a n a l y s i s , the lower l i m i t on the r a i n f a l l r a t e imposed by n u m e r i c a l i n s t a b i l i t y was s imply accepted because m u l t i p l e seepage faces g e n e r a l l y formed at h igher r a i n f a l l r a t e s for the m a j o r i t y o f g e o l o g i c m a t e r i a l s . I l l Chapter 5 TRANSIENT ANALYSIS The s e n s i t i v i t y s tudy i n the p r e v i o u s chapter p rov ided i n f o r m a t i o n about the s t e a d y - s t a t e c o n d i t i o n s under which m u l t i p l e seepage faces o c c u r . In order to examine the response of these flow systems to i n d i v i d u a l r a i n f a l l events and to examine the mechanisms by which m u l t i p l e seepage faces d e v e l o p , a t r a n s i e n t a n a l y s i s must be per formed. Freeze (1971) s t u d i e d the t r a n s i e n t development of perched water t a b l e s w i t h a f i n i t e - d i f f e r e n c e model of s a t u r a t e d -unsa tura ted f l o w . The flow r e g i o n he modeled i s shown i n F i g u r e 40a. There i s an impeding l a y e r i n the center of the h i l l s i d e and a r e g i o n o f r e l a t i v e l y h igh h y d r a u l i c c o n d u c t i v i t y at the base of the h i l l s i d e . The remainder of the flow r e g i o n i s composed of m a t e r i a l w i t h an i n t e r m e d i a t e va lue of h y d r a u l i c c o n d u c t i v i t y . The flow r e g i o n i s 37 m by 7 m. The boundary c o n d i t i o n s i n c l u d e a cons tant -head boundary, A B , and impermeable boundary, AFED, and an i n f i l t r a t i o n boundary, BCD. S t a t i c i n i t i a l c o n d i t i o n s were used for the s i m u l a t i o n . R a i n f a l l was s i m u l a t e d at a r a t e of 0 .09K o (1.5 mm/hr) where KQ i s the s a t u r a t e d h y d r a u l i c c o n d u c t i v i t y of the s o i l p re sent a long the i n f i l t r a t i o n boundary. F i g u r e 40b shows the t r a n s i e n t response o f the water t a b l e . A perched water t a b l e formed as a s m a l l l ens above the impeding l a y e r a f t e r 210 h o u r s . The l ens extended to the f a r - r i g h t impermeable boundary a f t e r 260 hours and c o n t i n u e d to b u i l d u n t i l at 460 h o u r s , the perched water 112 (Source: Freeze, 1971) F i g u r e 40. T r a n s i e n t s i m u l a t i o n of the development o f a perched f low sys tem, ( t , t ime i n h o u r s ) . 113 t a b l e j o i n e d the s l o w l y r i s i n g main water t a b l e . T h i s example p r o v i d e s i n s i g h t i n t o the mechanisms by which perched water t a b l e s form. The s i m u l a t i o n s pre sented i n t h i s chapter extend our knowledge to i n c l u d e h i l l s i d e i n which the impeding l a y e r i n t e r s e c t s the s lope and m u l t i p l e seepage faces d e v e l o p . The r e s u l t s of two t r a n s i e n t s i m u l a t i o n s are presented below, f o l l o w e d by a d i s c u s s i o n of the n u m e r i c a l d i f f i c u l t i e s t h a t were e n c o u n t e r e d . The f i n i t e - e l e m e n t mesh shown i n F i g u r e 41 was used for the t r a n s i e n t s i m u l a t i o n s . The flow r e g i o n i s 12 m by 25 m and has a s lope o f 4 5 ° . In an attempt to c i rcumvent problems w i t h n u m e r i c a l i n s t a b i l i t y , nodes were spaced c l o s e l y i n these p o r t i o n s o f the flow r e g i o n where the s t eepes t ^ - g r a d i e n t s were expected to o c c u r . Near the i n f i l t r a t i o n boundary, the noda l spac ing i n the v e r t i c a l d i r e c t i o n was 0.1 m. T h i s d i s t a n c e i n c r e a s e d to 2 m i n the b a s a l p o r t i o n o f the flow r e g i o n . I t should be noted tha t the s e n s i t i v i t y o f the s o l u t i o n to the noda l spac ing was not t e s t e d . The f i r s t t r a n s i e n t s i m u l a t i o n u s ing t h i s mesh i n v o l v e s a h i l l s i d e c o n t a i n i n g one impeding l a y e r ; the second s i m u l a t i o n c o n t a i n s two. The o n e - l a y e r flow system i s shown i n F i g u r e 42. AF i s a cons tant -head boundary a long 4* = 0.5 m, ABCD i s an impermeable boundary, DE i s an i n f i l t r a t i o n boundary, and EF i s a seepage-face boundary. A 1-m t h i c k l a y e r o f l e s s permeable m a t e r i a l i s l o c a t e d i n the upper p o r t i o n o f the h i l l s i d e ; the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t i s 25. The h y d r a u l i c p r o p e r t i e s o f the two s o i l types are l i s t e d i n Table 8. 534 nodes 486 elements / A A A A A A A A A --0 2 4 6 8 10 12 14 16 18 20 22 24 X (m) F i g u r e 41. F i n i t e - e l e m e n t mesh used i n t r a n s i e n t s i m u l a t i o n . F i g u r e 42. One- layer f low system for the t r a n s i e n t s i m u l a t i o n . 116 The va lues for the c o m p r e s s i b i l i t y o f the m a t e r i a l s were taken from Tab le 1 and are i n the middle of the range measured for a v a r i e t y o f g e o l o g i c a l m a t e r i a l s . The s p e c i f i c s torage was c a l c u l a t e d from E q u a t i o n 2 . 7 . The c h a r a c t e r i s t i c curves used i n the t r a n s i e n t a n a l y s i s have the same shape as those used i n the s t e a d y - s t a t e a n a l y s i s (F igure 21) except that they have been s c a l e d to correspond to the va lues o f the s a t u r a t e d h y d r a u l i c c o n d u c t i v i t y and p o r o s i t y shown i n Table 8. Table 8. H y d r a u l i c p r o p e r t i e s used i n the t r a n s i e n t s i m u l a t i o n s . M a t e r i a l No. 1 M a t e r i a l No. 2 s a t u r a t e d h y d r a u l i c 1 0 - 6 4 x 1 0 - 8 c o n d u c t i v i t y , K, (m/s) p o r o s i t y , n .450 .150 2 c o m p r e s s i b i l i t y , a , (^| ) 1 0 " 8 10~ 9 s p e c i f i c s t o r a g e , S s , (m) 9.994 x 1 0 - 5 1.045 x 1 0 - 5 The i n i t i a l c o n d i t i o n s were taken as the s t e a d y - s t a t e vp va lues for an average annual r a i n f a l l r a t e o f 1.2 x 1 0 - 7 m/s (3.8 m / y r ) . At t i m e , t , g r e a t e r than zero a r a i n f a l l r a t e of 9.0 x 1 0 - 7 m/s (3.24 mm/hr) was s imula ted a long the i n f i l t r a t i o n boundary. The s i z e o f the i n i t i a l t imes tep was one hour and each subsequent t imes tep was i n c r e a s e d by a f a c t o r of 1.4 to a maximum s i z e of 12 h o u r s . Convergence was g e n e r a l l y o b t a i n e d w i t h i n three to f i v e i t e r a t i o n s per t i m e s t e p ; the cumula t ive m a t e r i a l ba lance e r r o r remained w i t h i n 3% for the 14-day s torm. 117 F i g u r e 43 shows the t r a n s i e n t response of the water t a b l e . A f t e r 24 h o u r s , the perched water t a b l e i n t e r s e c t e d the i n f i l t r a t i o n boundary and c o n t i n u e d to b u i l d u n t i l at t=118 h o u r s , a seepage face formed above the impeding l a y e r . A f t e r t=118 h o u r s , the uppermost seepage face became more e x t e n s i v e . N o t e , however, tha t i n t h i s s i m u l a t i o n the main water t a b l e ad jacent to the cons tant -head boundary dropped s l i g h t l y d u r i n g the s torm. T h i s f ea ture w i l l be d i s c u s s e d f o l l o w i n g the second t r a n s i e n t example. A t r a n s i e n t s i m u l a t i o n was a l s o made wi th the two- layer f low system shown i n F i g u r e 44. The o n l y change from F i g 42 i s the i n t r o d u c t i o n of a second impeding l a y e r . The boundary c o n d i t i o n s , the h y d r a u l i c p r o p e r t i e s , the i n i t i a l c o n d i t i o n s , and the r a i n f a l l event are unchanged from the p r e v i o u s example. F i g u r e 45 shows the response of the water t a b l e . A f t e r 58 h o u r s , the uppermost perched water t a b l e i n t e r s e c t e d the i n f i l t r a t i o n boundary. A seepage face formed above the impeding l a y e r at t = 154 h o u r s . The perched water t a b l e a s s o c i a t e d w i t h the lower impeding l a y e r extended s l i g h t l y towards the s l o p e d u r i n g the 14-day s torm. Once a g a i n , the main water t a b l e dropped s l i g h t l y d u r i n g the s i m u l a t i o n . In comparison w i t h the o n e - l a y e r c a s e , approx imate ly 34 a d d i t i o n a l hours were r e q u i r e d i n the two- layer system for ponding to occur a long the i n f i l t r a t i o n sur f ace and for a seepage face to form. T h i s d i f f e r e n c e i s due to the i n i t i a l w a t e r - t a b l e c o n f i g u r a t i o n ; i n the o n e - l a y e r c a s e , the perched water t a b l e 2 o H i i i i i i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — i — | — 0 2 4 6 8 10 12 14 16 18 20 22 24 X (m) F i g u r e 43. T r a n s i e n t response of the w a t e r - t a b l e fo r the one-l a y e r flow system; t = time i n h o u r s . 12 E N 8 6 4 2 0 K 1 /K 2 = 25 / K i K 2 K, / K 2 1 | 1 | 1 1 1 1 ' 1 ' 1 ' 1 ' 1 ' 1 1 ' 1 ' 1 6 8 10 12 14 X (m) 16 18 20 22 24 F i g u r e 44. Two-layer f low system for the t r a n s i e n t s i m u l a t i o n . t -154 t =70 t =58 2-0 | i i i i i i i — i — i — i i i — i i i i i i T i -i i i r~ 0 2 4 6 8 10 12 14 16 18 20 22 24 X (m) F i g u r e 45. T r a n s i e n t response of the water t a b l e for the one-l a y e r f low system; t = time i n h o u r s . to o 121 i s c l o s e r to the s o i l sur f ace at t=0 and t h e r e f o r e responds more q u i c k l y . As noted i n both t r a n s i e n t s i m u l a t i o n s , the main water t a b l e dropped w i t h t i m e . For the cases at hand, which i n v o l v e i n f i l t r a t i o n i n t o a s t e a d y - s t a t e i n i t i a l c o n d i t i o n , one would not expect a d e c l i n e i n the main water t a b l e d u r i n g the i n f i l t r a t i o n e v e n t . A f a i l u r e to r i s e can be defended on the ground t h a t the r a t e of p r o p a g a t i o n of the w e t t i n g f r o n t i s r e t a r d e d by the l o w - p e r m e a b i l i t y l a y e r s , but a d e c l i n e , even the s l i g h t one o b s e r v e d , p robab ly p o i n t s to a minor n u m e r i c a l problem i n the. program. I n v e s t i g a t i o n s of t h i s i s sue were c a r r i e d out and i t i s now apparent tha t the d i s c r e p a n c y a r i s e s from the s l i g h t l y d i f f e r e n t n u m e r i c a l a l g o r i t h m s tha t are used to produce s t e a d y - s t a t e s o l u t i o n s i n : (a) a t rue s t e a d y - s t a t e a n a l y s i s i n which the r i g h t - h a n d s i d e of the t r a n s i e n t equa t ion i s s e t equa l to zero as i n E q u a t i o n 2 . 1 1 , and (b) a s teady-s t a t e a n a l y s i s o b t a i n e d by extending a t r a n s i e n t s i m u l a t i o n to s teady s t a t e . In the two t r a n s i e n t s i m u l a t i o n s under d i s c u s s i o n , the i n i t i a l c o n d i t i o n s were set w i t h method (a) , but the s i m u l a t i o n i s p roceed ing by method (b) . In the time p e r i o d p r i o r to the a r r i v a l of the w e t t i n g f r o n t , the heads at depth are moving towards a s l i g h t l y d i f f e r e n t s t e a d y - s t a t e c o n f i g u r a t i o n than the one i n i t i a l l y imposed. The n u m e r i c a l problem i s p r o b a b l y a minor one, but i t ought to be addressed before a f u l l t r a n s i e n t s e n s i t i v i t y a n a l y s i s i s c a r r i e d o u t . 122 Chapter 6 APPLICATIONS The i n f o r m a t i o n c o n t a i n e d i n t h i s t h e s i s has a p p l i c a t i o n to s t u d i e s tha t r e q u i r e an unders tanding o f the groundwater c o n d i t i o n s i n l a y e r e d s l o p e s . A p p l i c a t i o n s to g e o t e c h n i c a l , h y d r o g e o l o g i c a l , and geomorpho log ica l problems w i l l be d i s c u s s e d i n t h i s c h a p t e r . The f i r s t s e c t i o n d e a l s w i t h a p p l i c a t i o n s to s l o p e - s t a b i l i t y problems ; the second s e c t i o n c o n t a i n s a d e s c r i p t i v e a n a l y s i s of o ther p o s s i b l e a p p l i c a t i o n s , such as c o n t r o l l i n g groundwater in f lows i n t o e x c a v a t i o n s , p r e d i c t i n g r e g i o n a l groundwater flow p a t t e r n s , and s t u d y i n g h i l l s l o p e proces se s i n v o l v e d i n landform development . 6.1 S lope S t a b i l i t y Slope s t a b i l i t y problems a r i s e i n both manmade and n a t u r a l s l o p e s . For manmade s l o p e s , g e o t e c h n i c a l eng ineer s must ensure that the h e i g h t and s lope angle are des igned w i t h an adequate margin o f s a f e t y a g a i n s t s lope f a i l u r e s . Such p r o j e c t s i n c l u d e the d e s i g n o f highway and r a i l w a y c u t s , the d e s i g n of embankments, and the d e s i g n o f o p e n - p i t mines . The problems a s s o c i a t e d w i t h n a t u r a l s lopes g e n e r a l l y i n v o l v e the assessment o f the s t a b i l i t y of an e x i s t i n g s l o p e . E v a l u a t i o n o f the l o n g - t e r m s t a b i l i t y of a s l ope i s based on the f o l l o w i n g e x p r e s s i o n for the shear s t r e n g t h o f the s o i l , S: S = c + (a-p) tan 0 6.1 ivhere c = e f f e c t i v e c o h e s i o n , (a-p) = e f f e c t i v e s t r e s s , a = t o t a l s t r e s s , p = f l u i d p r e s s u r e , and 0 = 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 . A l l terms except the l a t t e r have u n i t s of [ M L - 1 T - 2 ] . The e f f e c t i v e c o h e s i o n and the 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 are e m p i r i c a l s o i l parameters that are determined from l a b o r a t o r y t e s t s . The t o t a l s t r e s s i s g e n e r a l l y c a l c u l a t e d from the s t a t i c s of the problem. The f l u i d p re s sure can be c a l c u l a t e d e i t h e r from d i r e c t measurement of the pres sure head a long the f a i l u r e s u r f a c e , or from the h y d r a u l i c - h e a d va lues o b t a i n e d from f lownet c o n s t r u c t i o n . R e c a l l from Chapter 2 that f l u i d p r e s s u r e , p , p re s sure head, i|>, and h y d r a u l i c head, h , are r e l a t e d through the f o l l o w i n g two e q u a t i o n s : P = pg^ h = y\> + z For unsa tura ted s o i l s , an e m p i r i c a l parameter x has been i n t r o d u c e d by B i shop and B l i g h t (1963) to r e l a t e e f f e c t i v e s t r e s s to f l u i d p re s sure as f o l l o w s : a = a - u a + X ( U a ~ Hw) where a = e f f e c t i v e s t r e s s , y a = a i r p re s sure i n the pore spaces , u w = water p r e s s u r e , and X = parameter r e l a t e d to the degree of s a t u r a t i o n of the s o i l . Recent r e s e a r c h on the s t r e n g t h of unsa tura ted s o i l s i s reviewed by F r e d l u n d , et a l . (1978) and F r e d l u n d (1979). 124 The f a c t o r o f s a f e t y a g a i n s t s lope f a i l u r e , F . S . , i s d e f i n e d as the r a t i o o f the shear s t r e n g t h a long a p o t e n t i a l f a i l u r e sur f ace to the shear s t r e s s a long t h a t s u r f a c e . For a s t a b l e s l o p e , F .S .>1 . From E q u a t i o n 6 . 1 , i t can be seen that an i n c r e a s e i n the f l u i d p re s sure decreases the shear s t r e n g t h and hence , the f a c t o r o f s a f e t y . O r , viewed another way, i f a l l o ther f a c t o r s are e q u a l , then the lower the f l u i d p r e s s u r e s , the s teeper the s t a b l e s lope a n g l e . Knowledge of the groundwater c o n d i t i o n s t h a t e x i s t w i t h i n a h i l l s i d e are t h e r e f o r e o f fundamental importance to l ong- te rm s t a b i l i t y a n a l y s e s . P r a c t i c i n g eng ineer s are w e l l aware that the geology o f a s i t e can have a profound e f f e c t on the f l u i d - p r e s s u r e d i s t r i b u t i o n . S t u d i e s tha t have emphasized the importance o f the h y d r o g e o l o g i c environment i n s t a b i l i t y a n a l y s i s i n c l u d e Pat ton and Deere (1971), Deere and Pat ton (1971), Pa t ton and Hendron (1974), and Hodge and Freeze (1977). However, c o l l e c t i o n o f h y d r o g e o l o g i c data i n the f i e l d i s an expensive and d i f f i c u l t t e c h n i c a l p rob lem. The s o p h i s t i c a t i o n of the f l u i d - p r e s s u r e d i s t r i b u t i o n used i n a s t a b i l i t y a n a l y s i s w i l l t h e r e f o r e depend upon the amount of data tha t i s a v a i l a b l e and i t s q u a l i t y . To i l l u s t r a t e t h i s , suppose a s t a b i l i t y a n a l y s i s i s to be performed for the h i l l s i d e shown i n F i g u r e 46a. One c o u l d e n v i s i o n the f o l l o w i n g four types o f groundwater c o n d i t i o n s t h a t might be invoked i n the a n a l y s i s , depending upon the a v a i l a b i l i t y o f d a t a : a. Hypothetical How region - i - . — i • i — . — r - r - r , T—W—i—.—i—.--i - , — . — r -0 2 4 • • 10 12 14 ! • t t 2 0 2 2 2 4 X (m) b. Case #1: Fully-saturated, quasi-static analysis / / h - l l h-IO ha0 d. Case #3: Heterogeneous, fully saturated analysis c. Case +2: Homogeneous, fully saturated analysis e. Case #4: Heterogeneous, saturated -unsaturated analysis F i g u r e 46. P o s s i b l e h y d r a u l i c - h e a d d i s t r i b u t i o n s fo r use i n s l o p e - s t a b i l i t y a n a l y s i s . 126 Case No. 1: In the absence o f r e l i a b l e d a t a , i t i s common to assume that the r e g i o n i s f u l l y s a t u r a t e d and the p re s sure head at any g i v e n p o i n t a long a p o t e n t i a l f a i l u r e sur f ace i s equa l to the v e r t i c a l d i s t a n c e from that p o i n t to the s o i l s u r f a c e . T h i s assumption presumes the somewhat u n r e a l i s t i c h y d r a u l i c - h e a d d i s t r i b u t i o n shown i n F i g u r e 46b. I t c o n s i s t s o f v e r t i c a l e q u i p o t e n t i a l l i n e s a long the s lope face and s t a t i c c o n d i t i o n s e l s ewhere . Case No. 2: I f the value of K]_ i s known, and K 2 i s not (or the e x i s t e n c e of the K 2 l a y e r i s not r e c o g n i z e d ) , i t might be c o n s i d e r e d s a t i s f a c t o r y to assume tha t the h i l l s i d e i s homogeneous and f u l l y s a t u r a t e d . The h y d r a u l i c - h e a d d i s t r i b u t i o n shown i n F i g u r e 46c would then be a p p r o p r i a t e for d e t e r m i n i n g the f l u i d - p r e s s u r e d i s t r i b u t i o n a long the p o t e n t i a l f a i l u r e s u r f a c e . Case No. 3: I f both K i and K 2 are known, a f u l l y s a t u r a t e d a n a l y s i s o f flow would y i e l d a d i s t r i b u t i o n o f h y d r a u l i c - h e a d l i k e tha t shown i n F i g u r e 46d. S u r p r i s i n g l y , a c a l c u l a t i o n o f the p re s sure heads for the s imula ted h y d r a u l i c -head p a t t e r n l eads to the r e g i o n o f nega t ive f l u i d p re s sure enc lo sed by the \p = 0 i s o b a r . A p p a r e n t l y , the f i n i t e - e l e m e n t program used i n t h i s s tudy does not r e j e c t negat ive, ^-va lues even though i t i s a " f u l l y - s a t u r a t e d " a n a l y s i s . However, i t must be r e c o g n i z e d tha t they are not the c o r r e c t va lue s tha t would r e s u l t from unsa tura ted flow t h e o r y . I t i s p o s s i b l e tha t many computer programs c u r r e n t l y i n c i r c u l a t i o n for the 127 p r e d i c t i o n of s a t u r a t e d f low may e x h i b i t t h i s type of per formance . Case No. 4: I f the data were a v a i l a b l e to run a s a t u r a t e d - u n s a t u r a t e d a n a l y s i s , the h y d r a u l i c - h e a d d i s t r i b u t i o n shown i n F i g u r e 46e c o u l d be used i n the s lope s t a b i l i t y a n a l y s i s . F i g u r e 47 shows a comparison of the f l u i d - p r e s s u r e d i s t r i b u t i o n s and pre s sure-head d i s t r i b u t i o n s a long a p o t e n t i a l f a i l u r e s u r f a c e for each of the h y d r a u l i c - h e a d d i s t r i b u t i o n s shown i n F i g u r e 46. Case No. 4 i s the most a c c u r a t e e s t imate of the a c t u a l c o n d i t i o n s . Case No. 1 p r o v i d e s the most c o n s e r v a t i v e e s t imate of the f l u i d - p r e s s u r e ' d i s t r i b u t i o n and r e q u i r e s no h y d r o g e o l o g i c d a t a . However, i t s use would l e a d to an o v e r d e s i g n of the s lope a n g l e ; t h i s may or may not be a c c e p t a b l e . In the case o f the d e s i g n of an open p i t mine , the co s t of an a n a l y s i s l i k e Case No. 4 might be j u s t i f i e d i n l i g h t of the s av ings c r e a t e d by reduced e x c a v a t i o n . Case No. 2, i n which the h i l l s i d e i s assumed to be homogeneous and f u l l y s a t u r a t e d , l e ads to s e r i o u s e r r o r s as the f l u i d pre s sure i s underes t imated above the impeding l a y e r at measurement p o i n t s 16 and 17. The shear s t r e n g t h , the f a c t o r of s a f e t y , and the s t a b l e s l o p e angle would t h e r e f o r e be o v e r e s t i m a t e d . Even g rea te r e r r o r would be i n t r o d u c e d by the homogeneous a n a l y s i s i f the water t a b l e was assumed to l i e beneath the ground sur f ace r a t h e r than c o i n c i d e n t w i t h i t . Case No. 3 appears to p r e d i c t the r e l a t i v e l y h igh f l u i d p re s sure s above the impeding l a y e r q u i t e w e l l . I t should be emphasized, however, tha t the 128 F i g u r e 47. Comparison o f p r e s s u r e - h e a d and f l u i d - p r e s s u r e d i s t r i b u t i o n s a long a p o t e n t i a l f a i l u r e s u r f a c e . 129 use of a " f u l l y - s a t u r a t e d " heterogeneous a n a l y s i s which a l lows nega t ive f l u i d p re s sure s i s p h y s i c a l l y i n c o r r e c t and the approach cannot be recommended. At l e a s t two case h i s t o r i e s have been r e p o r t e d i n which m u l t i p l e seepage faces have been present and the a s s o c i a t e d s a t u r a t e d - u n s a t u r a t e d flow c o n d i t i o n s have been taken i n t o c o n s i d e r a t i o n d u r i n g the s t a b i l i t y a n a l y s i s . In both examples, the f l u i d p re s sure was set equal to zero i n the unsa tura ted zone for s t a b i l i t y c a l c u l a t i o n s . S t e r r e t t and E d i l (1982) i n v e s t i g a t e d the s t a b i l i t y o f a 30m h i g h , l a y e r e d s lope i n W i s c o n s i n a long the Lake M i c h i g a n s h o r e l i n e . E r o s i o n of up to lOm/year had been observed near the top of the s l o p e . F i g u r e 48 shows the geology at the s i t e and the w a t e r - t a b l e c o n f i g u r a t i o n i n f e r r e d from f i e l d measurements of the pre s sure head and o b s e r v a t i o n o f the seepage face l o c a t i o n s . From t h e i r s t a b i l i t y a n a l y s i s , they were ab le to conclude that the uppermost perched flow system was a s i g n i f i c a n t f a c t o r i n the b l u f f - t o p e r o s i o n . E igenbrod and Morgenstern (1972) i n v e s t i g a t e d a l a n d s l i d e t h a t o c c u r r e d a long a highway cut i n a r i v e r v a l l e y near Edmonton, A l b e r t a . The s lope had been cut i n t o bedrock c o n s i s t i n g of in te rbedded mudstone, c l a y s t o n e , s ands tone , c o a l , and b e n t o n i t i c c l a y . F a i l u r e had o c c u r r e d a long the base of a h o r i z o n t a l b e n t o n i t e l a y e r , 2 to 30 cm i n t h i c k n e s s . D i r e c t l y beneath the f a i l u r e su r f ace was a p a r t i a l l y s a t u r a t e d c o a l l a y e r . F i e l d measurements r e v e a l e d the presence of two perched water t a b l e s a s s o c i a t e d w i t h seepage faces formed a long the s l o p e , as shown i n F i g u r e 49. (Source: Sterrett and Edil, 1982) F i g u r e 48. Geology and groundwater c o n d i t i o n s a t Bender Park s i t e i n W i s c o n s i n . CO 2 2 2 6 0 -i 2 2 4 0 -w 2 2 2 0 H Z - 2 2 0 0 < > 2 180 -UJ _ l UJ Spring Upper Coal Layer Lower Coal Layer ~ i i 1 i 1 1 1 1 1 1 1 1 4 0 8 0 120 160 2 0 0 2 4 0 Y CO-ORDINATE (ft.) (Source: Eigenbrod and Morgenstern, 1972) F i g u r e 49. Groundwater c o n d i t i o n s at a highway c u t i n A l b e r t a . 132 T h e i r s t a b i l i t y a n a l y s i s produced a f a c t o r of s a f e t y for the s l o p e , p r i o r to f a i l u r e , of between .82 and 1.13. In both examples , the a b i l i t y to make a c o n s i s t e n t i n t e r p r e t a t i o n hinged on c a r e f u l f i e l d o b s e r v a t i o n s and a p p r e c i a t i o n of the complex nature of the f low system present i n l a y e r e d h i l l s i d e s . 6.2 Other P o s s i b l e A p p l i c a t i o n s GROUNDWATER INFLOWS INTO EXCAVATIONS Groundwater in f lows occur when an e x c a v a t i o n i s taken below the water t a b l e . Freeze and Cher ry (1979) present an overview of the dra inage and dewater ing systems that can be used to c o n t r o l groundwater i n f l o w s ; Sharp et a l . (1977) t r e a t the s u b j e c t i n d e t a i l . Methods commonly used to lower the water t a b l e i n the v i c i n i t y of an e x c a v a t i o n i n c l u d e : a) i n s t a l l a t i o n of h o r i z o n t a l d r a i n s , b) c o n s t r u c t i o n of dra inage g a l l e r i e s , and c) i n s t a l l a t i o n of a network of pumping w e l l s . These methods are i l l u s t r a t e d i n F i g u r e 50 for an e x c a v a t i o n i n t o homogeneous m a t e r i a l . The success of a dewater ing scheme depends upon how w e l l the groundwater f low system i s under s tood . Cons ider the h y p o t h e t i c a l e x c a v a t i o n i n t o l a y e r e d m a t e r i a l shown i n F i g u r e 51. I t would be e s p e c i a l l y important to c h a r a c t e r i z e the s a t u r a t e d - u n s a t u r a t e d nature of the flow system and to p r e d i c t the t r a n s i e n t response o f the water t a b l e i n a r e g i o n such as t h i s i n order to make accura te e s t imates of i n f l o w r a t e s , and to p r o v i d e an e f f e c t i v e and e f f i c i e n t d e s i g n for the dra inage scheme. Such a d e s i g n may i n c l u d e the l o c a t i o n of pump in take 133 a. Horizontal drain pipes b. Drainage gallery c. Pumping wells f (source: Freeze and Cherry, 1979) F i g u r e 50. Methods used to c o n t r o l groundwater i n f l o w s i n t o e x c a v a t i o n s . 134 SEEPAGE I N T O A H E T E R O G E N E O U S O P E N PIT MINE F i g u r e 51. H y p o t h e t i c a l w a t e r - t a b l e c o n f i g u r a t i o n for an e x c a v a t i o n i n t o heterogeneous m a t e r i a l . 135 p o i n t s and t h e i r c a p a c i t i e s , the p o s i t i o n i n g and s i z i n g of dra inage a d i t s or g a l l e r i e s ; and the s p e c i f i c a t i o n of d r a i n p i p e l o c a t i o n and l e n g t h . REGIONAL GROUNDWATER PLOW In many groundwater s t u d i e s , i t i s e s s e n t i a l tha t the f o l l o w i n g a t t r i b u t e s of the r e g i o n a l hydrogeology be de te rmined : a) the boundar ies of the r e g i o n a l flow system, b) the l o c a t i o n s of recharge and d i s c h a r g e a r e a s , and c) the magnitude and d i r e c t i o n of groundwater f low throughout the r e g i o n . These a t t r i b u t e s are best determined when t h e o r e t i c a l s t u d i e s and f i e l d i n v e s t i g a t i o n s are used i n c o n j u n c t i o n w i t h one another . T h e o r e t i c a l s t u d i e s , such as those p r o v i d e d by mathemat ica l models , are v a l u a b l e i n the reconna i s sance stage of an i n v e s t i g a t i o n . They can g i v e the best e s t imate o f the g e n e r a l f low patterms based on the data tha t are i n i t i a l l y a v a i l a b l e and guide the i n v e s t i g a t o r to those areas where f u r t h e r data c o l l e c t i o n would be the most u s e f u l . Newly a c q u i r e d data can then be used to t e s t and r e f i n e the t h e o r e t i c a l a n a l y s i s . In order to o b t a i n a c o n s i s t e n t i n t e r p r e t a t i o n , however, the assumptions u n d e r l y i n g the t h e o r e t i c a l model must be met i n the f i e l d . In most c a s e s , i t i s s u f f i c i e n t to model s t eady-s t a t e s a t u r a t e d flow through a two-d imens iona l c r o s s s e c t i o n o r i e n t e d p a r a l l e l to the d i p o f the water t a b l e . The c l a s s i c s t u d i e s of r e g i o n a l f low systems by Toth (1963) and Freeze and Witherspoon (1967), are based on these a s sumpt ions . The r e s u l t s c o n t a i n e d i n t h i s t h e s i s , however, suggest t h a t i n 136 order to i d e n t i f y the a t t r i b u t e s of r e g i o n s c o n t a i n i n g l a y e r e d h i l l s i d e s c o r r e c t l y , we cannot assume tha t the r e g i o n i s f u l l y s a t u r a t e d ; a s a t u r a t e d - u n s a t u r a t e d a n a l y s i s must be per formed. F i g u r e 27 i n Chapter 4 demonstrated that an unsa tura ted wedge may extend for s e v e r a l k i l o m e t e r s i n t o a h i l l s i d e i f the h y d r a u l i c c o n d u c t i v i t y c o n t r a s t exceeds three o rder s of magnitude. R e c o g n i t i o n of the s a t u r a t e d - u n s a t u r a t e d nature of these systems c o u l d have important i m p l i c a t i o n s w i t h re spec t to the assessment of r e g i o n a l groundwater re sources and i n the p r e d i c t i o n of the movement of contaminant s . In such s t u d i e s , the data c o l l e c t i o n scheme should be des igned to a l l ow for the d e t e c t i o n of perched flow systems. Sampling the p re s sure head at s e v e r a l d i f f e r e n t depths i n each borehole might be an e f f i c i e n t and economica l approach to f i e l d i n s t r u m e n t a t i o n i n l a y e r e d systems. HILLSLOPE HYDROLOGY A grea t d e a l of p rogre s s has been made by geomorpholog i s t s i n d e s c r i b i n g the t h r e e - d i m e n s i o n a l form of h i l l s l o p e s and i n under s tand ing the g e o l o g i c and h y d r o l o g i c processes by which h i l l s l o p e s e v o l v e . Such i n f o r m a t i o n i s used by g e o l o g i s t s to r e c o n s t r u c t g e o l o g i c h i s t o r y , by g e o t e c h n i c a l eng ineer s to c o r r e l a t e h i l l s l o p e form w i t h the s t r e n g t h of the u n d e r l y i n g s o i l and rock mass, and by l and-use p l anner s to eva lua te how h i l l s l o p e proces se s might a f f e c t a s i t i n g of human a c t i v i t y . Most d e s c r i p t i v e models of s lope development d i s t i n g u i s h between the e f f e c t s of mass movements and sur face-water e r o s i o n . For example, Carson (1969) proposed a two-phase model 137 i n w h i c h , i n i t i a l l y , r a p i d mass movements reduce s teep s lopes to g e n t l e s l o p e s . In the second phase , sur face-water e r o s i o n dominates s l o p e development. Others suggest tha t mass movements c o n t r o l the upper , convex and s t r a i g h t segments of a h i l l s l o p e p r o f i l e and tha t sur face-water e r o s i o n c o n t r o l s the lower , o f t e n concave , p o r t i o n of the s lope (Bloom, 1978). The models of s l ope development w i l l not be reviewed i n d e t a i l i n t h i s s e c t i o n ; the i n t e r e s t e d reader i s r e f e r r e d to Carson and K i r k b y (1972), Young (1972), K i r k b y (1978), Dunne and Leopo ld (1978), and R i t t e r (1978). I n s t e a d , the f o l l o w i n g d i s c u s s i o n of mass movements and sur face-water e r o s i o n i s in tended to i n d i c a t e those areas where an unders tanding of the development of m u l t i p l e seepage faces c o u l d be important to s t u d i e s of s lope devlopment . Mass Movements Mass movements r e f e r to the proces ses by which sediment i s t r a n s p o r t e d downslope under a g r a v i t a t i o n a l s t r e s s f i e l d . They are f u r t h e r s u b d i v i d e d i n t o slow and r a p i d mass movements. Slow mass movements, or s o i l c r e e p , may be the r e s u l t of two p r o c e s s e s . The f i r s t i n v o l v e s the p l a s t i c flow of c l a y - r i c h s o i l s i n wet c l i m a t e s . The second i n v o l v e s the movement of s o i l p a r t i c l e s and s o i l aggregates due to s w e l l i n g and s e t t l e m e n t d u r i n g f reeze-thaw and w e t t i n g - d r y i n g c y c l e s . The e f f e c t i v e n e s s of s o i l c r e e p i n t r a n s p o r t i n g sediments downslope i s a f u n c t i o n o f the h i l l s l o p e g r a d i e n t , the s o i l t y p e , the water content o f the s o i l , and the c l i m a t e . The groundwater 138 c o n d i t i o n s t h a t g i v e r i s e to m u l t i p l e seepage faces would be expected to promote l o c a l i z e d areas of a c c e l e r a t e d s o i l c r e e p . Rapid mass movements are c o l l e c t i v e l y termed l a n d s l i d e s . Depending on the nature of the m a t e r i a l which f a i l e d and on the s t y l e of movement, l a n d s l i d e s are c l a s s i f i e d as e i t h e r f a l l s , s l i d e s , or f l o w s . Regard les s of the p r e c i s e c l a s s i f i c a t i o n , the g e n e r a t i o n of most r a p i d mass movements i s c o n t r o l l e d to a l a r g e extent by the f l u i d - p r e s s u r e d i s t r i b u t i o n w i t h i n the h i l l s l o p e . In e s sence , the s tudy of r a p i d mass movements reduces to the s tudy of s l ope s t a b i l i t y , d i s c u s s e d i n S e c t i o n 6 . 1 . The work presented i n t h i s t h e s i s shou ld be of i n t e r e s t , t h e r e f o r e , to those geomorpholog i s t s s t u d y i n g the r o l e of mass movements i n the development of s l ope s i n l a y e r e d g e o l o g i c envi ronments . Surface-Water E r o s i o n Sur face-water e r o s i o n r e f e r s to the entra inment and t r a n s p o r t of s o i l p a r t i c l e s downslope by sur f ace water . T h i s can o n l y occur i f h y d r o g e o l o g i c f a c t o r s combine to produce r u n o f f . The mechanisms by which runo f f i s generated are summarized by Dunne (1978). They w i l l be o u t l i n e d here w i t h r e f e r e n c e to four flow paths water might f o l l o w as i t moves d o w n h i l l ; these pathways are shown i n F i g u r e 52 for a homogeneous h i l l s i d e . I f the r a i n f a l l r a t e exceeds the i n f i l t r a b i l i t y o f the s o i l and the r a i n f a l l d u r a t i o n exceeds the time r e q u i r e d for the s o i l su r f ace to become s a t u r a t e d , then a p o r t i o n of the r a i n f a l l w i l l runo f f and f o l l o w Path No. 1 as Horton o v e r l a n d 139 (Source: Dunne, 1978) F i g u r e 52. P o s s i b l e f low paths for water to f o l l o w fo r a-homogeneous h i l l s i d e . 140 f l o w . Some of the water t h a t i n f i l t r a t e s may enter the groundwater f low system and f o l l o w Path No. 2 to the stream c h a n n e l . I f there i s a l a y e r o f permeable t o p s o i l o v e r l y i n g the l e s s permeable s u b s t r a t e , a p o r t i o n of the r a i n f a l l tha t i n f i l t r a t e s may t r a v e l a long Path No. 3 as subsur face s tormf low. And f i n a l l y , i f d u r i n g a storm the water t a b l e r i s e s and i n t e r s e c t s the s o i l s u r f a c e , then a seepage face w i l l form. The runof f produced at the seepage face i s termed s a t u r a t i o n o v e r l a n d flow and i s l a b e l e d as Path No. 4 i n F i g u r e 52. S a t u r a t i o n o v e r l a n d flow i n c l u d e s water generated from a) the emergence of subsur face s tormf low, b) d i r e c t p r e c i p i t a t i o n onto the s a t u r a t e d s o i l s u r f a c e , and c) groundwater d i s c h a r g e . Note tha t for a homogeneous h i l l s i d e , s a t u r a t i o n o v e r l a n d flow i s most l i k e l y to occur near the stream channe l where the water t a b l e i s r e l a t i v e l y c l o s e to the s o i l s u r f a c e . For l a y e r e d s l o p e s , the r e s u l t s pre sented i n t h i s t h e s i s demonstrate tha t m u l t i p l e seepage faces c o u l d produce a d d i t i o n a l areas of s a t u r a t i o n o v e r l a n d f l o w , p o s s i b l y q u i t e far from the stream c h a n n e l . Under s tand ing the o c c u r r e n c e of m u l t i p l e seepage faces i s t h e r e f o r e r e l e v a n t to s t u d i e s of sur face-water e r o s i o n . In a d d i t i o n to mass movements and sur face-water e r o s i o n , groundwater c o n d i t i o n s w i t h i n a h i l l s i d e can exer t a s t rong i n f l u e n c e on the l o c a t i o n o f s tream heads w i t h i n a watershed . The i n i t i a t i o n and headward e r o s i o n o f t r i b u t a r y streams cart be the d i r e c t r e s u l t of a form of subsur face e r o s i o n known as p i p i n g . As water f lows through the pore spaces o f a s o i l , 141 energy i s t r a n s f e r r e d from the water to the s o i l p a r t i c l e s i n the form of a f r i c t i o n a l d r a g . The f o r c e a s s o c i a t e d w i t h t h i s energy t r a n s f e r , as r e f l e c t e d by the h y d r a u l i c g r a d i e n t , may be s u f f i c i e n t to erode s o i l p a r t i c l e s at the e x i t p o i n t of the subsur face f low p a t h . T h i s e r o s i v e process i s termed p i p i n g . P r e d i c t i o n of the l o c a t i o n of seepage faces i s t h e r e f o r e b a s i c to under s tand ing the l o c a t i o n of areas where p i p i n g may o c c u r . I t i s p o s s i b l e that the two-d imens iona l a n a l y s i s presented i n t h i s t h e s i s c o u l d a i d the s tudy of the t h r e e - d i m e n s i o n a l development of dra inage networks i n l a y e r e d , heterogeneous r e g i o n s . 142 Chapter 7 SUMMARY AND CONCLUSIONS The work and c o n c l u s i o n s c o n t a i n e d i n t h i s t h e s i s w i l l be summarized i n terms of the four o b j e c t i v e s of the r e s e a r c h . The f i r s t o b j e c t i v e was to s e l e c t a f i n i t e - e l e m e n t model to p r e d i c t the f l u i d - p r e s s u r e d i s t r i b u t i o n and seepage-face l o c a t i o n s i n l a y e r e d , heterogeneous h i l l s i d e s . The f i n i t e -element model that was chosen for the s tudy was w r i t t e n by Shlomo Neuman at the U n i v e r s i t y of A r i z o n a . The program was o r i g i n a l l y t i t l e d UNSAT I and i s f u l l y documented i n Neuman (1972). In i t s o r i g i n a l form, UNSAT I can model t r a n s i e n t , t w o - d i m e n s i o n a l , s a t u r a t e d - u n s a t u r a t e d flow through heterogeneous , a n i s o t r o p i c f low r e g i o n s . UNSAT I was m o d i f i e d for use i n t h i s t h e s i s i n three ways. F i r s t , a s t e a d y - s t a t e v e r s i o n of the program was c r e a t e d . Second, the treatment o f the seepage-face boundary was m o d i f i e d to a l low the g e n e r a l i z e d development of more than one seepage face a long a g i v e n s l o p e . T h i r d , the n u m e r i c a l treatment of the i n f i l t r a t i o n process was m o d i f i e d so t h a t the f l u x e n t e r i n g the system would be determined i t e r a t i v e l y i n response to the h y d r a u l i c - h e a d d i s t r i b u t i o n . With these m o d i f i c a t i o n s , the f l u i d - p r e s s u r e d i s t r i b u t i o n and seepage-face l o c a t i o n s i n l a y e r e d s lope s c o u l d be s t u d i e d i n e i t h e r a s t e a d y - s t a t e or t r a n s i e n t mode. The second o b j e c t i v e was to b u i l d a l a b o r a t o r y model to v e r i f y the p h y s i c a l f o u n d a t i o n of the s o l u t i o n s generated by the n u m e r i c a l model ; the t e s t met w i t h s u c c e s s . A sand-tank 143 model was b u i l t to r epre sent a h i l l s i d e c o n t a i n i n g one impeding l a y e r and two seepage f a c e s . The s t e a d y - s t a t e response of the f low r e g i o n to three d i f f e r e n t r a i n f a l l r a t e s was r e c o r d e d . The n u m e r i c a l model was c a l i b r a t e d a g a i n s t the response of the p h y s i c a l model to the f i r s t r a i n f a l l r a te and v e r i f i e d by i t s a b i l i t y to s i m u l a t e the subsequent response to the two other r a i n f a l l r a t e s . The p h y s i c a l model conf i rmed the e x i s t e n c e of a wedge-shaped unsa tura ted zone s e p a r a t i n g the two seepage faces and conf i rmed t h a t the water t a b l e responded to changes i n the r a i n f a l l r a t e i n a manner p r e d i c t e d by the n u m e r i c a l model . The t h i r d o b j e c t i v e of t h i s r e sea rch was to use the n u m e r i c a l model i n s e n s i t i v i t y s t u d i e s des igned to reach q u a n t i t a t i v e c o n c l u s i o n s about the f a c t o r s govern ing the development of m u l t i p l e seepage f a c e s . T h i s o b j e c t i v e was met by us ing a d e t a i l e d s t e a d y - s t a t e a n a l y s i s and two p r e l i m i n a r y t r a n s i e n t s i m u l a t i o n s . In a l l s i m u l a t i o n s , the h y d r a u l i c c o n d u c t i v i t y o f the K 2 l a y e r was l e s s than the h y d r a u l i c c o n d u c t i v i t y of the m a t e r i a l c o m p r i s i n g the remainder of the h i l l s i d e , K]_. In a d d i t i o n , the impeding l a y e r s were h o r i z o n t a l , of uni form t h i c k n e s s , and of uni form h y d r a u l i c p r o p e r t i e s . The assumptions and l i m i t a t i o n s of the t h e o r e t i c a l a n a l y s i s are l i s t e d below. F o l l o w i n g each i s an i n d i c a t i o n of whether the i tem a p p l i e s to both the s t e a d y - s t a t e and t r a n s i e n t a n a l y s i s , or to j u s t the t r a n s i e n t a n a l y s i s : 144 1. Flow i s t w o - d i m e n s i o n a l . (both) 2. D a r c y ' s law i s v a l i d . (both) 3. The a i r phase i s cont inuous and at a tmospher ic p r e s s u r e . (both) 4. The unsa tura ted h y d r a u l i c p r o p e r t i e s are n o n h y s t e r e t i c . ( t r a n s i e n t ) 5. E v a p o t r a n s p i r a t i o n i s not modeled. (both) 6. The K(40 curves are s c a l e d i n accordance wi th Equa t ion 4 . 2 . (both) 7. The porous medium i s c o m p r e s s i b l e ; the i n d i v i d u a l s o i l g r a i n s are n o t . ( t r a n s i e n t ) 8. The porous medium i s l i n e a r l y and r e v e r s i b l y e l a s t i c , ( t r a n s i e n t ) 9. The t o t a l s t r e s s i s cons tant and ac t s i n the v e r t i c a l d i r e c t i o n o n l y . ( t r a n s i e n t ) 10. The s p e c i f i c s torage remains c o n s t a n t , r e g a r d l e s s of the degree o f s a t u r a t i o n . ( t r a n s i e n t ) 11. There i s no i n f i l t r a t i o n a long the seepage-face boundary. (both) 12. The fa te o f sur face runof f i s not modeled. (both) 13. A n u m e r i c a l s o l u t i o n may not be p o s s i b l e where s teep g r a d i e n t s i n p re s sure head and h y d r a u l i c c o n d u c t i v i t y deve lop w i t h i n the unsa tura ted zone. (both) 14. I n t e r a c t i o n s between the subsurface flow system and the stream l e v e l are not modeled. ( t r a n s i e n t ) 145 Of the assumptions r e l a t e d to the s t e a d y - s t a t e a n a l y s i s , No. 1 and No. 9 were c o n s i d e r e d to be the most i m p o r t a n t , p a r t i c u l a r l y i f one were to e x t r a p o l a t e the r e s u l t s to the o c c u r r e n c e o f m u l t i p l e seepage faces on g e n t l e s l o p e s . L i m i t a t i o n No. 11 r e s t r i c t e d the study to r e l a t i v e l y wet c o n d i t i o n s . The assumptions and l i m i t a t i o n s p e r t a i n i n g to the t r a n s i e n t a n a l y s i s cannot be eva lua ted as a f u l l t r a n s i e n t s e n s i t i v i t y s tudy was not per formed. The c o n c l u s i o n s from the s t e a d y - s t a t e a n a l y s i s of one-l a y e r systems are as f o l l o w s : 1. I f a l l e l s e i s c o n s t a n t , then for a g i v e n K 1 / K 2 r a t i o , an i n c r e a s e i n the e l e v a t i o n of the impeding l a y e r : a) i n c r e a s e s the extent of the unsa tura ted wedge, and b) decreases the percentage o f the t o t a l out f low acros s the uppermost seepage f a c e . 2. I f a l l e l s e i s c o n s t a n t , then for a g i v e n p o s i t i o n of the impeding l a y e r , an i n c r e a s e i n the K 1 / K 2 r a t i o : a) i n c r e a s e s the extent of the unsa tura ted wedge, and b) i n c r e a s e s the percentage of the t o t a l out f low acros s the uppermost seepage f a c e . 3. For the flow system modeled, K I / K 2 r a t i o s g r e a t e r than three o r d e r s of magnitude produced an unsa tura ted wedge tha t extended more than 1000m i n t o the h i l l s i d e . 4. For a g i v e n h y d r a u l i c c o n d u c t i v i t y c o n t r a s t measured i n the z - d i r e c t i o n , the w a t e r - t a b l e c o n f i g u r a t i o n i s i n s e n s i t i v e to a n i s o t r o p y w i t h i n the impeding l a y e r . 146 However, i t i s s e n s i t i v e to a n i s o t r o p y w i t h i n t h e m a t e r i a l su r round ing the impeding l a y e r . I f a l l e l s e i s c o n s t a n t , and i f the p r o p e r t i e s of the m a t e r i a l s u r r o u n d i n g the impeding l a y e r are such that K x > K z , then the ex tent of the unsa tura ted wedge i s i n c r e a s e d r e l a t i v e to i s o t r o p i c c o n d i t i o n s . C o n v e r s e l y , i f K z > K x ' t h e n t n e unsa tura ted wedge i s much l e s s e x t e n s i v e . 5. I f a l l e l s e i s c o n s t a n t , then a g rea te r h y d r a u l i c c o n d u c t i v i t y c o n t r a s t i s r e q u i r e d to produce m u l t i p l e seepage faces on g e n t l e s l ope s as compared to s teep s l o p e s . In the example presented i n Chapter 4, a h y d r a u l i c c o n d u c t i v i t y c o n t r a s t of at l e a s t two order s of magnitude was r e q u i r e d to produce m u l t i p l e seepage faces on an 8 ° s l o p e , w h i l e one order of magnitude was s u f f i c i e n t to produce m u l t i p l e seepage faces on a 4 0 ° s l o p e . The c o n c l u s i o n s from the s t e a d y - s t a t e a n a l y s i s of two-l a y e r systems are as f o l l o w s : 1. For a g i v e n K 1 / K 2 r a t i o , and a g i v e n d i s t a n c e s e p a r a t i n g the impeding l a y e r s , an i n c r e a s e i n the e l e v a t i o n o f the l a y e r s : a) i n c r e a s e s the extent of both unsa tura ted wedges, and b) decreases the percentage o f the t o t a l out f low from the uppermost seepage f a c e , %Qu, as w e l l as the percentage from the middle seepage f a c e , %Qm. 147 2. Regard les s of the d i s t a n c e s e p a r a t i n g the impeding l a y e r s , an i n c r e a s e i n the K 1 / K 2 r a t i o : a) i n c r e a s e s the extent of both unsa tura ted wedges, and b) i n c r e a s e s %Qu. The va lue of %Qm remains r e l a t i v e l y unchanged u n t i l the K 1 / K 2 r a t i o i s i n c r e a s e d above 100, a f t e r which %Qm d e c r e a s e s . 3. In g e n e r a l , the unsa tura ted wedge beneath the lower impeding l a y e r i s l e s s e x t e n s i v e than the unsa tura ted wedge formed beneath the upper impeding l a y e r . 4. I f the p o s i t i o n of the upper impeding l a y e r remains f i x e d , then as the second l a y e r i s p l a c e d at s u c c e s s i v e l y lower e l e v a t i o n s , %Qu decreases and %Qm i n c r e a s e s . C o n v e r s e l y , i f the p o s i t i o n of the lower impeding l a y e r remains f i x e d , then as the upper l a y e r i s p l a c e d at s u c c e s s i v e l y lower p o s i t i o n s , %Qu i n c r e a s e s and %Qm decrea se s . E x t e n s i o n of the s t e a d y - s t a t e a n a l y s i s to t h r e e - l a y e r systems p r i m a r i l y served to c o n f i r m many of the c o n c l u s i o n s from one- and two- layer systems and to demonstrate the c o m p l e x i t y tha t can r e s u l t as the number of impeding l a y e r s i s i n c r e a s e d . The t h r e e - l a y e r systems d i d show, however, t h a t i f a l l e l s e i s c o n s t a n t , then as the number of impeding l a y e r s w i t h i n a h i l l s i d e i s i n c r e a s e d , the unsa tura ted wedges become l e s s e x t e n s i v e . The l i m i t e d number of t r a n s i e n t runs p r e c l u d e s a d e t a i l e d se t of c o n c l u s i o n s at t h i s t i m e . 148 The f o u r t h , and f i n a l , o b j e c t i v e of t h i s t h e s i s was to form g e n e r a l i z e d c o n c l u s i o n s r e g a r d i n g the importance of m u l t i p l e seepage faces i n g e o t e c h n i c a l , h y d r o g e o l o g i c a l , and geomorpho log i ca l problems . With regard to g e o t e c h n i c a l problems , t h i s s tudy has i m p l i c a t i o n s w i t h r e spec t to p r e d i c t i o n s of the f l u i d - p r e s s u r e d i s t r i b u t i o n for s l o p e -s t a b i l i t y ana lyses and p r e d i c t i o n s of groundwater i n f l o w s i n t o e x c a v a t i o n s . In both i n s t a n c e s , i t can be conc luded that p r e d i c t i o n s based on homogeneous and s a t u r a t e d ana lyse s may be s i g n i f i c a n t l y i n e r r o r when a p p l i e d to l a y e r e d s l o p e s . With regard to h y d r o g e o l o g i c a l problems , i t has been suggested tha t because an unsa tura ted wedge can extend for grea t d i s t a n c e s i n t o a h i l l s i d e , a s a t u r a t e d - u n s a t u r a t e d a n a l y s i s should be c o n s i d e r e d i n s t u d i e s des igned to i d e n t i f y the a t t r i b u t e s o f r e g i o n a l f low systems c o n t a i n i n g l a y e r e d s l o p e s . 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C . , The F i n i t e Element Method, 3rd E d . , McGraw-H i l l , New Y o r k , 787 p p . , 1977. 153 APPENDIX A DEFINITION OF SYMBOLS Symbol D e f i n i t i o n A c r o s s - s e c t i o n a l a r e a , [L 2 ] [A] g l o b a l s t i f f n e s s mat r ix C s p e c i f i c mois ture c a p a c i t y , [1/L] [F] c a p a c i t a n c e matr ix ID i n s i d e diameter K h y d r a u l i c c o n d u c t i v i t y , [L/T] K r r e l a t i v e h y d r a u l i c c o n d u c t i v i t y K s s a t u r a t e d h y d r a u l i c c o n d u c t i v i t y [L/T] OD o u t s i d e diameter Q v o l u m e t r i c flow r a t e , [ L 3 / T ] Q out f low volume, [L 3 ] R r a i n f a l l r a t e , [L/T] S shear s t r e n g t h [ M L - l T _ 2 ] SC seepage c o l l e c t o r S s s p e c i f i c s t o r a g e , [1/L] T t h i c k n e s s of impeding l a y e r , [L] V flow r e g i o n c" e f f e c t i v e c o h e s i o n , [ML-lqi-2] cm cent imete r g g r a v i t a t i o n a l a c c e l e r a t i o n , [ L / T 2 ] h h y d r a u l i c head, [L] k p e r m e a b i l i t y , [L 2 ] 1 l e n g t h o f s o i l sample, [L] m meter n p o r o s i t y , [ L ^ / L ^ ] p gage f l u i d p ressure, [ M T ' ^ L - 1 ] t time, [ T ] v s p e c i f i c d i s c h a r g e , [ L / T ] w(x,z) weighting f u n c t i o n , [ L ] x h o r i z o n t a l c o o r d i n a t e d i r e c t i o n , [ L ] z v e r t i c a l c oordinate d i r e c t i o n , [ L ] z" e l e v a t i o n of the base of the impeding l a y e r , [ L ] z s depth of water o v e r l y i n g stream bed, [ L ] a c o m p r e s s i b i l i t y of the porous medium, [ L T 2 / M ] 3 c o m p r e s s i b i l i t y of water, [ L T 2 / M ] 6 n(x,z) b a s i s f u n c t i o n s , [ L ] Ah constant-head d i f f e r e n t i a l , [ L ] dynamic v i s c o s i t y , [ M L - 1 - T - 1 ] u u a a i r pressure i n the pore spaces, [ M T - 2 L - 1 ] water press u r e , [ M T ~ 2 L - 1 ] \p pressure head, [ L ] i j j a a i r entry value, [ L ] \j>(x,z,t) approximate f u n c t i o n , [ L ] i|j n(t) exact s o l u t i o n , [ L ] P f l u i d d e n s i t y , [ M / L 3 ] P b dry bulk d e n s i t y , [ M / L 3 ] p a r t i c l e d e n s i t y , [ M / L 3 ] t o t a l s t r e s s , [ M L - 1 L ~ 2 J P P a standard d e v i a t i o n of the measured q u a n t i t y a Si a 2 v a r i a n c e of the measured q u a n t i t y a v o l u m e t r i c w a t e r c o n t e n t , [L^/L^] e f f e c t i v e a n g l e o f i n t e r n a l f r i c t i o n e m p i r i c a l s o i l s t r e n g t h p a r a m t e r 156 APPENDIX B EXPERIMENTAL DATA In t h i s appendix , the e x p e r i m e n t a l data from the f i n a l run are g i v e n . F i g u r e 53 shows the f i n i t e - e l e m e n t d i s c r e t i z a t i o n of the e x p e r i m e n t a l flow r e g i o n and the l o c a t i o n of the nodal p o i n t s , or p o r t s , at which data were c o l l e c t e d . Por t #1 was used for c a l i b o r a t i o n and i s t h e r e f o r e not shown. Note tha t 46 of the p o r t s c o n t a i n e d tens iometer s connected to the S c a n i v a l e ; the measured and p r e d i c t e d h y d r a u l i c - h e a d va lues for these are pre sented i n Tab le 9. The remaining 10 p o r t s shown i n F i g u r e 53 c o n t a i n e d tens iometer s connected to manometers; the c o r r e s p o n d i n g measured and p r e d i c t e d h y d r a u l i c - h e a d va lues are g i v e n i n Table 10. The e v a l u a t i o n head and pre s sure head data are presented i n Tab le 11 and Table 12. The p r e d i c t e d e l e v a t i o n head data can be o b t a i n e d from F i g u r e 53; The p r e d i c t e d p re s sure head data can be c a l c u l a t e d from the p r e d i c t e d h y d r a u l i c head and e l e v a t i o n head d a t a . X (m) ' i ' j i r e 53. Index Eor t e n s i o m e t e r l o c a t i o n s i n the e x p e r imentn 1 f l o w r e g i o n . — i 158 T a b l e 9. Comparison of h y d r a u l i c - h e a d data cor te n s i o m e t e r s read with the S c a n i v a l v e . h m = measured h y d r a u l i c head; hp = p r e d i c t e d h y d r a u l i c head ( a l l v a l u e s i n cm) RUN #1 RUN #2 RUN #3 PORT hm hp hm _ hp hm hp hm~ hp nm hp nnT • hP 2 44. 5 34.0 10.5 43.3 34.0 9.3 43. 5 33.9 9 . 6 3 38. 5 39.1 -.6 39.1 39.0 0.1 39. 4 38.7 0. 7 4 40. 4 43.2 -2.3 40.1 43.1 -3.0 40. 1 42.5 -2. 3 5 43. 0 45.9 -2.9 42.0 45.7 -3.7 41. 2 45.0 -3. 8 6 52. 0 48.3 3.7 49.9 48.0 1.9 49. 4 47 .1 2. 3 7 47. 0 49.0 -2.0 47.9 48.8 -0.9 45. 6 47.3 -2. 2 8 50. 9 50.5 0.4 49.7 50.2 -0.5 49. 0 49.1 -0. 1 9 52. 5 52.6 -0.1 51.6 52.3 -0.7 50 . 9 51.1 -0. 2 10 53. 1 53.0 0.1 52.2 52.7 -0.5 51. 2 • 51.5 -0. 3 11 53. 7 53.3 -0.1 52.6 53.5 -0.9 51. 5 52.1 -0. 6 12 56. 5 55.6 0.9 54.1 55.2 -1.1 53. 7 53.9 -0. 2 13 57. 6 57.2 0.4 56.2 56.8 -0.6 54. 1 55.3 -1. 2 14 61. 1 61.0 0.1 59.4 60.5 -1.1 56. 1 53.6 -2. 5 15 62. 3 61.7 0.6 60.6 61.2 -0.6 57. 1 59.3 -2. 2 16 81. 8 32.4 -0.6 79.0 81.5 -2.5 76. 2 78.5 -2. 3 17 61. 2 61.1 0.1 59.4 60.6 -1.2 57 . 4 58.7 -1. 3 18 62. 0 62.1 -0.1 60.2 61.6 -1.4 58. 2 59.6 -1. 4 19 62. 6 62.7 -0.1 60.7 62.2 -1.5 58. 6 60.1 -1. 5 20 85. 7 34.0 1.7 81.6 82.9 -1.3 78 . 3 79.5 -1. 2 21 63. 9 64.3 -0 . 4 61.9 63.7 -1.3 59 . 7 61.5 -1. 3 22 65. 5 65.0 0.5 63.3 64.5 -1.2 61. 0 62.2 -1. 2 23 88. 6 86.4 2.2 83.6 35.1 -1.5 79. 3 81.2 -1. 4 24 39. 3 87.3 2.5 84.4 86.0 -1.6 80. 5 82.0 -1. 5 25 91. 4 88.4 3.0 85.3 37 .0 -1.7 81. 3 32.8 -1. 5 26 65. 2 65.7 -0.5 63.0 65.1 -2.1 60. 6 62.7 -2. 1 27 91. 3 38 . 6 2.7 85.5 87.2 -1.7 81. 4 82.9 -1. S 28 92. 5 39.5 3.0 86.4 88.1 -1.7 82. 2 83.7 -1. 5 29 94. 4 90.6 3.8 87.3 89.1 -1.3 33. 1 84.5 -1. 4 30 65. 3 65.3 -0.5 63.1 65.2 -2.1 60. 9 62.8 -1. 9 31 66 . 5 67.0 -0.5 64.1 66.4 -2.3 61. 7 63.8 -2. 1 32 67. 2 67 .8 -0.6 64.4 67.1 -2.7 63. 0 64.6 -1. 6 33 93. 7 • 90.6 3.1 37.5 89.1 -1.6 83. 1 34. 4 -1. 3 34 97. 1 92.9 4.2 89 .8 91.2 -1.4 85. 1 86.3 -1. 2 35 67. 6 63.1 -0.5 65.1 67.5 -2.4 62. 6 64.8 -2. 2 36 69. 1 69.0 0.1 66.4 63.3 -1.9 63. 7 65.6 -1. 9 37 96. 6 93.5 3.1 90.4 91.3 -1.4 85 . 4 86.3 -1. 4 38 98. 5 94.5 4.0 91.7 93.1 -1.4 86 . 7 87.9 -1. 2 39 70. 6 70.0 0.6 68.0 69.2 -1.2 65. 2 66 . 4 -1. 2 40 99 . 2 96.4 2.3 93.0 94.7 -1.7 87. 9 39.2 -1. 3 41 70. 7 70.7 0.0 67.9 70.0 ^2.1 65. 1 67.1 -2. 0 42 69. 3 69.8 -0.5 66.7 69.1 -2.4 64. 0 66 .3 -2. 3 43 71. 2 71.4 -0.2 68 .3 70 .7 -2.4 55 . 5 67.7 -2. 2 44 68. 6 69.1 -0.5 66.1 63.4 -2.3 63. 5 65.7 -2. 2 45 72. 0 71.3 0.2 69 .3 71.2 -1.9 66 . 2 68.1 -1. 9 46 100 . 2 98 . 6 l . S 95 . 4 97.7' -2.3 89 . 9 91.3 -1. Q 47 72. 2 72.2 0.0 69.3 71.5 -2.2 66 . 3 68.4 -2. 1 159 T a b l e 10. Comparison of dat a £or t e n s i o m e t e r s read with manometers. hm = measured h y d r a u l i c head; hp = p r e d i c t e d h y d r a u l i c head ( a l l v a l u e s i n cm) RUN #1 RON #2 RUN #3 PORT hm hp nm - np h m hp h m - h p 'nm hp hm" •hp Ml 53.1 51.0 2.1 51.9 50. 7 1.2 50.3 49 . 7 1. 1 M2 55.2 53.3 1.9 54.1 53.0 1.1 52.7 51.3 0. 9 M3 56.8 55.1 1.7 55.8 54.3 1.0 54.3 53.4 0. 9 M4 75.1 74.6 0.5 74.6 74. 5 0.1 73.5 73.6 -0. 1 M5 59.6 57.8 1.8 58.5 57.4 1.1 56.7 55.9 0. 3 M6 81.8 81.4 0.4 79.3 80.6 -1.3 76.5 77.7 -1. 2 M7 36.6 84.9 1.7 82.5 33.3 -1.3 78.8 30.3 -1. 5 M8 64.2 63.4 0.8 62.3 62.9 -0.6 60.3 60.8 -0. 5 M9 67.0 66.5 0.5 64.5 65.9 -1.4 62.1 63.4 -1. 3 M10 94.3 91.6 3.2 88.5 90.1 -1.6 33.9 35.3 -1. 4 160 T a b l e 11. P r e s s u r e head and e l e v a t i o n head data for t e n s i o m e t e r s read with the S c a n i v a l v e . Z = measured e l e v a t i o n head, cm ijj ™ = measured p r e s s u r e head, cm RUN f l RUN #2 RUN #3 P o r t Zm 'fm 2 9.9 34.6 33.9 33.5 3 29.7 3.3 9.4 ' 9.7 4 39.7 0.7 0.4 0.4 5 39.6 3.4 2.4 1.6 6 39.8 12.2 10.1 9.6 7 50.0 -3.0 -2.1 -4.4 a 39.7 11.2 10.0 9.3 9 40.0 12.5 11.6 10.9 10 50.0 3.1 2.2 1.2 11 9.6 44.1 43.0 41. 9 12 60.0 -3.5 -5.9 -6.3 13 • 50.0 7.6 6.2 4.1 14 49.9 11.2 9.5 6.2 15 60.0 2.3 0.6 -2.9 16 79.8 2.0 -0.8 -3.6 17 10.0 51.2 49.4 47 . 4 18 39.9 22.1 20.3 13.3 19 49.9 12.7 10.8 3.7 20 69.8 15.9 11.8 8.5 21 50.0 13.9 11.9 9.7 22 60.1 5.4 3.2 0.9 23 69.7 18.9 13.9 10.1 24 79.7 10.1 4.7 0 . 8 25 90.0 1.4 -4.7 -8.7 26 49.8 15.4 13.2 10.8 27 69.9 21.4 15.6 11.5 28 80.0 12.5 6.4 2.2 29 90.1 4.3 -2.8 -7.0 30 29.9 35.4 33.2 31.0 31 49 .9 16.6 14.2 11.8 32 60.1 7.1 4.3 2.9 33 70 .0 23.7 17.5 13.1 34 90.0 7.1 -0.2 -4.9 35 49.8 17.8 -15.3 12.3 36 60.0 9.1 6.4 3.7 37 79.9 16.7 10. 5 5.5 38 89.9 8.6 1.8 -3.2 39 60.0 10.6 3.0 5.2 40 89.9 9.3 3.1 -2.0 41 59.9 10.8 ' 8.0 5.2 42 39.9 29. 4 26.8 24.1 43 59.9 11.3 8.4 5 . 6 44 9.3 58.3 56.3 53. 7 45 60.0 12.0 9.3 5 . 2 46 90.0 10 . 2 5 . 4 -0 .1 47 50.0 12.2 9 . 3 6 . 3 1 6 1 T a b l e 12. P r e s s u r e head and e l e v a t i o n head data f o r t e n s i o m e t e r s read with manometers. 2 = measured e l e v a t i o n head, cm <l> m = measured p r e s s u r e head, cm RUN #1 RUN #2 RUN *3 P o r t Z m m m m Ml 50 . 0 3. .1 1.9 0.3 M2 59. .9 -4. ,7 -5.3 -7.2 M3 50. ,0 6. .3 5.3 4.3 M4 70. .0 5. ,1 4.6 3.5 M5 60. .0 -0. ,4 -1.5 -3.3 M6 70. .1 11. . 7 9.2 6.4 M7 79. .7 6. .9 2.8 -0.9 M8 60. ,1 4. ,1 2.2 0.2 M9 60. .1 6. .9 4.4 2.0 M10 79. ,9 14. ,9 8.6 4.0 

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