UBC Theses and Dissertations

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UBC Theses and Dissertations

Long-term transient regional groundwater flow in a heterogeneous mature basin with large hydraulic conductivity… England, Lindy Alison 1986

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LONG-TERM TRANSIENT REGIONAL GROUNDWATER FLOW IN A HETEROGENEOUS MATURE BASIN WITH LARGE HYDRAULIC CONDUCTIVITY CONTRASTS By LINDY ALISON ENGLAND B . S c , The U n i v e r s i t y of W a t e r l o o , 1983 THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE i n THE FACULTY OF GRADUATE STUDIES Department of G e o l o g i c a l S c i e n c e s We a c c e p t t h i s t h e s i s as co n f o r m i n g to the r e q u i r e d s t a n d a r d THE UNIVERSITY OF BRITISH COLUMBIA A p r i l 1986 © L i n d y A. Engla n d , 1986 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f t h e r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y o f B r i t i s h C o l u m b i a , I agree t h a t t h e L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y purposes may be g r a n t e d by t h e head o f my department o r by h i s o r h e r r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . v Department o f Geological Sciences The U n i v e r s i t y o f B r i t i s h Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date April 16, 1986 DE -6 (3/81) A b s t r a c t A f i n i t e e l e m e n t m odel i s d e v e l o p e d t o s o l v e t h e t w o - d i m e n s i o n a l , t r a n s i e n t , g r o u n d w a t e r f l o w e q u a t i o n f o r t h e p u r p o s e o f d e s c r i b i n g t h e d e v e l o p m e n t o f r e g i o n a l f l o w o v e r g e o l o g i c t i m e , i n r e s p o n s e t o e r o s i o n a l m o d i f i c a t i o n s o f t h e l a n d s u r f a c e , i n a m a t u r e b a s i n w i t h l a r g e 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 . A s i m p l i f i e d m o d e l o f t h e Red E a r t h r e g i o n , A l b e r t a , C a n a d a , r e f e r r e d t o a s t h e ' s t a n d a r d ' m o d e l , i s t h e w o r k i n g e x a m p l e . T h e r e a r e s e v e r a l n u m e r i c a l p r o b l e m s i n h e r e n t i n t r a n s i e n t f l o w m o d e l s . O s c i l l a t i o n s o c c u r a t n odes a t e a r l y t i m e as t h e y a d j u s t t o new, s t r e s s f u l b o u n d a r y c o n d i t i o n s ; t h e s e o s c i l l a t i o n s d i e o u t a t l a t e r t i m e s . O s c i l l a t i o n s a l s o o c c u r a t c e r t a i n n o d e s , a s h e a d v a l u e s r e a c h s t e a d y s t a t e , i f t h e l e n g t h t o d e p t h r a t i o s o f t h e e l e m e n t s a r e t o o l a r g e ; a r e f i n e m e n t o f t h e mesh r e d u c e s t h e s e o s c i l l a t i o n s t o i n s i g n i f i c a n t a m o u n t s . To s i m u l a t e a m o d el i n w h i c h s e v e r a l m i l l i o n y e a r s a r e r e q u i r e d t o r e a c h s t e a d y s t a t e , i t i s n e c e s s a r y t o use a c o n t i n u o u s l y v a r y i n g t i m e s t e p s u c h t h a t t h e p e r c e n t c h a n g e a t any node w i t h i n t h e f l o w r e g i o n d o e s n o t e x c e e d a s p e c i f i e d t o l e r a n c e l e v e l ; t h r e e p e r c e n t i s u s e d f o r most c a s e s i n t h i s s t u d y . The f i n i t e e l e m e n t m odel i s v e r i f i e d by c o m p a r i n g s o l u t i o n s t o s e v e r a l p r o b l e m s w i t h t h o s e o f b o t h 1-D and 2-D s t e a d y s t a t e a n a l y t i c a l s o l u t i o n s and 1-D t r a n s i e n t a n a l y t i c a l s o l u t i o n s . C o m p a r i s o n s a r e e x c e l l e n t i n a l l c a s e s . A s e n s i t i v i t y a n a l y s i s i s p e r f o r m e d on a l a y e r e d s y s t e m i n w h i c h t h e v a l u e s o f f i v e p a r a m e t e r s , h y d r a u l i c c o n d u c t i v i t y , s p e c i f i c s t o r a g e , a n i s o t r o p y , b a s i n s i z e , and w a t e r t a b l e c o n f i g u r a t i o n , a r e s l i g h t l y p e r t u r b e d . R e s u l t s i n d i c a t e t h a t t h e m o d el i s most s e n s i t i v e t o c h a n g e s i n : a) v a l u e s o f h y d r a u l i c c o n d u c t i v i t y o f low-K l a y e r s , b) t h e z - d i r e c t i o n c omponent o f h y d r a u l i c c o n d u c t i v i t y i n a n i s o t r o p i c low-K l a y e r s , c) t h e g e o m e t r y o f t h e b a s i n by c h a n g i n g t h e l a y e r t h i c k n e s s e s , and d) v a l u e s o f s p e c i f i c s t o r a g e i n h i g h - K l a y e r s . H y p o t h e t i c a l b a s i n s w i t h a v a r i e t y o f s t r a t i g r a p h i c and s t r u c t u r a l c o n f i g u r a t i o n s and a r e a l i s t i c m o d e l o f t h e Red E a r t h b a s i n a r e s i m u l a t e d . F l o w p a t t e r n s a r e a n a l y z e d and c ompared i n o r d e r t o d e v e l o p an u n d e r s t a n d i n g o f t h e t i m e - d e p e n d e n t n a t u r e o f h y d r a u l i c head a d j u s t m e n t s w i t h i n a h e t e r o g e n e o u s , m a t u r e b a s i n . The s i g n i f i c a n c e o f m a k i n g p o t e n t i a l e r r o r s i n a s s u m i n g s t e a d y s t a t e f l o w i n a m a t u r e b a s i n when t r a n s i e n t c o n d i t i o n s a c t u a l l y e x i s t i s d i s c u s s e d . A p p l i c a t i o n s o f t h i s t y p e o f t r a n s i e n t a n a l y s i s i n c l u d e : a) e x p l o r i n g f o r h y d r o c a r b o n s o r m i n e r a l d e p o s i t s , b) i s o l a t i n g r a d i o a c t i v e w a s t e s , c) d e t e r m i n i n g s a f e d e v e l o p m e n t r a t e s o f g r o u n d w a t e r r e s o u r c e s , and d) d a t i n g g e o l o g i c a l e v e n t s t h a t r e s u l t i n c h a n g e s i n b o u n d a r y c o n d i t i o n s o f r e g i o n a l g r o u n d w a t e r f l o w s y s t e m s . T a b l e o f C o n t e n t s Page ABSTRACT i i L I S T OF TABLES v i L I S T OF FIGURES v i i ACKNOWLEDGEMENT X NOMENCLATURE x i CHAPTER 1. INTRODUCTION 1 2. PREVIOUS WORK: A BRIEF REVIEW OF VARIOUS APPROACHES TO MODELING GROUNDWATER FLOW IN REGIONAL BASINS 5 I n t r o d u c t i o n 5 A n a l y s i s o f S t e a d y S t a t e F l o w i n a R e g i o n a l B a s i n 5 A n a l y s i s o f T r a n s i e n t F l o w i n a R e g i o n a l B a s i n . . . 7 3. THE F I N I T E ELEMENT MODEL 10 T h e o r e t i c a l D e v e l o p m e n t 10 P r o g r a m i n g T e c h n i q u e s 22 M o d e l V e r i f i c a t i o n 28 N u m e r i c a l P e r f o r m a n c e 42 4. S E N S I T I V I T Y ANALYSIS 51 I n t r o d u c t i o n 51 The M o d e l ' s S e n s i t i v i t y t o Ch a n g e s i n H y d r a u l i c C o n d u c t i v i t y 55 The M o d e l ' s S e n s i t i v i t y t o Changes i n A n i s o t r o p y 62 The M o d e l ' s S e n s i t i v i t y t o Changes i n S p e c i f i c S t o r a g e 67 The M o d e l ' s S e n s i t i v i t y t o Ch a n g e s i n Wa t e r T a b l e C o n f i g u r a t i o n 69 The M o d e l ' s S e n s i t i v i t y t o Changes i n B a s i n S i z e 71 C o n c l u s i o n s 74 i v T a b l e o f C o n t e n t s ( c o n t . ) Page CHAPTER 5. TRANSIENT FLOW IN REGIONAL BASINS WITH VARIOUS STRATIGRAPHIC AND STRUCTURAL CONFIGURATIONS 79 2-D F l o w i n t h e Red E a r t h R e g i o n 79 F l o w i n a B a s i n i n w h i c h t h e Low~K L a y e r s P i n c h Out n e a r t h e S h a l l o w Edge 91 F l o w i n a B a s i n w i t h a N a t u r a l C o n d u i t 102 P o t e n t i a l E r r o r s I n v o l v e d when A s s u m i n g S t e a d y S t a t e F l o w C o n d i t i o n s 114 6. DISCUSSION 118 A p p l i c a t i o n s 118 L i m i t a t i o n s 120 C o n c l u s i o n s 121 REFERENCES 127 APPENDIX 129 v L i s t o f T a b l e s Page T a b l e I . H y d r a u l i c p a r a m e t e r s o f h y d r o g e o l o g i c u n i t s o f t h e Red E a r t h r e g i o n , A l b e r t a , Canada us e d f o r a o n e - d i m e n s i o n a l f l o w a n a l y s i s 40 T a b l e I I . C o m p a r i s o n o f r e l a t i v e h ead a d j u s t m e n t s b e t w e e n t h e a n a l y t i c a l and f i n i t e e l e m e n t s o l u t i o n s o f t h e 1-D t r a n s i e n t g r o u n d w a t e r f l o w e q u a t i o n 42 T a b l e I I I . P a r a m e t e r s u s e d f o r t h e s t a n d a r d b a s i n m o d el b a s e d on t h o s e o f t h e Red E a r t h r e g i o n , A l b e r t a , Canada 53 T a b l e I V . P a r a m e t e r s i n p u t t o m o del A, t h e c o m p l e x Red E a r t h b a s i n m o del 81 T a b l e V. R c o m p a r i s o n b e t w e e n t h e 2-D f i n i t e e l e m e n t (FE) m o del and b o t h t h e 1-D FE m o d el and t h e 1~D a n a l y t i c a l (AN) model 91 v i L i s t o f F i g u r e s Page F i g u r e 1. I n d e x map s h o w i n g t h e l o c a t i o n o f t h e Red E a r t h r e g i o n 4 F i g u r e 2. An e x a m p l e o f a d i s c r e t i z e d f l o w r e g i o n 11 F i g u r e 3. An e x a m p l e o f a l i n e a r t r i a n g u l a r e l e m e n t 11 F i g u r e 4. An e x a m p l e o f a f i n i t e e l e m e n t mesh g e n e r a t e d by t h e c o m p u t e r p r o g r a m MESH 24 F i g u r e 5. A n a l y t i c a l v e r s u s f i n i t e e l e m e n t s o l u t i o n s o f t h e 1-D s t e a d y - s t a t e g r o u n d w a t e r f l o w e q u a t i o n f o r h e t e r o g e n e o u s p o r o u s m e d i a 31 F i g u r e 6. A n a l y t i c a l v e r s u s f i n i t e e l e m e n t s o l u t i o n s o f t h e 2-D s t e a d y - s t a t e g r o u n d w a t e r f l o w e q u a t i o n f o r homogeneous p o r o u s m e d i a 33 F i g u r e 7. A n a l y t i c a l v e r s u s f i n i t e e l e m e n t s o l u t i o n s o f t h e 1~D t r a n s i e n t g r o u n d w a t e r f l o w e q u a t i o n f o r v e r t i c a l f l o w i n homogeneous p o r o u s m e d i a 37 F i g u r e 8. A n a l y t i c a l v e r s u s f i n i t e e l e m e n t s o l u t i o n s o f t h e 1-D t r a n s i e n t g r o u n d w a t e r f l o w e q u a t i o n f o r h o r i z o n t a l f l o w i n homogeneous p o r o u s m e d i a 38 F i g u r e 9. The t r a n s i e n t b e h a v i o u r o f h y d r a u l i c h e a d s i n a t w o - l a y e r v e r t i c a l s e c t i o n w i t h a h i g h - K l a y e r o v e r l y i n g a low-K l a y e r 44 F i g u r e 10. The t r a n s i e n t b e h a v i o u r o f h y d r a u l i c h e a d s i n a t w o - l a y e r v e r t i c a l s e c t i o n w i t h a low-K l a y e r o v e r l y i n g a h i g h - K l a y e r 46 F i g u r e 1 1. The t r a n s i e n t b e h a v i o u r o f h y d r a u l i c h e a d s i n a t h r e e - l a y e r v e r t i c a l s e c t i o n w i t h a h i g h - K l a y e r b e t w e e n two low-K l a y e r s 47 F i g u r e 12. The t r a n s i e n t b e h a v i o u r o f h y d r a u l i c h e a d s i n a t h r e e - l a y e r v e r t i c a l s e c t i o n w i t h a low-K l a y e r b e t w e e n two h i g h - K l a y e r s 48 F i g u r e 13. The s t a n d a r d b a s i n model s h o w i n g a l l node p o s i t i o n s , l a y e r p o s i t i o n s , and t h e l o c a t i o n s o f nodes I D , 2M, and 3S 54 F i g u r e 14. H y d r a u l i c head v e r s u s l o g t i m e p l o t s f o r p e r t u r b a t i o n s o f K i n h i g h - K l a y e r s 57 v i i L i s t of F i g u r e s (cont.) Page F i g u r e 15. H y d r a u l i c head v e r s u s l o g time p l o t s f o r p e r t u r b a t i o n s of K i n low-K l a y e r s 60 F i g u r e 16. H y d r a u l i c head v e r s u s l o g time p l o t s f o r K z > K x i n low-K l a y e r s 64 F i g u r e 17. H y d r a u l i c head v e r s u s l o g time p l o t s f o r K > K_ i n high-K l a y e r s 66 F i g u r e 18. H y d r a u l i c head v e r s u s l o g time p l o t s f o r p e r t u r b a t i o n s of s p e c i f i c s t o r a g e 68 F i g u r e 19. H y d r a u l i c head v e r s u s l o g time p l o t s f o r v a r i a b l e d e p t h : l e n g t h r a t i o s 73 F i g u r e 20. H y d r a u l i c head v e r s u s l o g time p l o t s f o r v a r i a b l e l e n g t h : d e p t h r a t i o s 75 F i g u r e 21. Mesh diagram o f model A, the complex Red E a r t h b a s i n 80 F i g u r e 22. Steady s t a t e e q u i p o t e n t i a l p l o t of model A.... 82 F i g u r e 23. Steady s t a t e v e l o c i t y p l o t of model A 83 F i g u r e 24. T r a n s i e n t e q u i p o t e n t i a l p l o t s o f h y d r a u l i c head change of model A 85 F i g u r e 25. Mesh diagram o f model B, the ' p i n c h - o u t ' b a s i n 93 F i g u r e 26. H y d r a u l i c head v e r s u s l o g time p l o t s comparing model B w i t h the s t a n d a r d model at nodes i n the high-K l a y e r s of the b a s i n s . . . 94 F i g u r e 27. H y d r a u l i c head v e r s u s l o g time p l o t s comparing model B w i t h the s t a n d a r d model at nodes i n the low-K l a y e r s of the b a s i n s . . . . 95 F i g u r e 28. T r a n s i e n t e q u i p o t e n t i a l p l o t s o f model B 98 F i g u r e 29. Steady s t a t e e q u i p o t e n t i a l p l o t o f the s t a n d a r d model 103 F i g u r e 30. Steady s t a t e v e l o c i t y p l o t of the s t a n d a r d model 104 F i g u r e 31. Steady s t a t e e q u i p o t e n t i a l p l o t of model B....105 F i g u r e 32. Steady s t a t e v e l o c i t y p l o t o f model B 106 v i i i L i s t of F i g u r e s (cont.) Page F i g u r e 33. Mesh diagram of model C, the ' c o n d u i t ' b a s i n 108 F i g u r e 34. Steady s t a t e e q u i p o t e n t i a l p l o t of model C....110 F i g u r e 35. Steady s t a t e v e l o c i t y p l o t o f model C I l l F i g u r e 36. H y d r a u l i c head v e r s u s l o g time p l o t s comparing model C w i t h the s t a n d a r d model at v a r i o u s nodes w i t h i n the b a s i n s 113 F i g u r e 37. Log time v e r s u s h y d r a u l i c head, v e l o c i t y , and t r a v e l time to a d i s c h a r g e p o i n t f o r node 2M i n l a y e r IH, i n d i c a t i n g p o t e n t i a l e r r o r i n v o l v e d when u s i n g a s t e a d y s t a t e model v e r s u s a t r a n s i e n t model 115 i x A c k n o w l e d g e m e n t The a u t h o r a c k n o w l e d g e s w i t h t h a n k s and a p p r e c i a t i o n t h e s u p p o r t and a d v i c e o f D r . R. A. F r e e z e t h r o u g h o u t t h e e n t i r e t y o f t h i s r e s e a r c h . V a l u a b l e g u i d a n c e f r o m D r . J . L. S m i t h and C. Mase i s a l s o a c k n o w l e d g e d w i t h t h a n k s . R e v i e w s o f t h e t h e s i s by t h e f o l l o w i n g s c i e n t i s t s a r e a c k n o w l e d g e d w i t h a p p r e c i a t i o n : D r . R. A. F r e e z e , D r . J . L. S m i t h , and D r . T. H. Brown ( D e p a r t m e n t o f G e o l o g i c a l S c i e n c e s , U n i v e r s i t y o f B r i t i s h C o l u m b i a ) , and D r . J . T o t h ( D e p a r t m e n t o f G e o l o g y , U n i v e r s i t y o f A l b e r t a ) . Deep and s i n c e r e t h a n k s a r e e x t e n d e d t o T. D. J . E n g l a n d f o r d r a f t i n g o f f i g u r e s , a s s i s t a n c e i n m i c r o c o m p u t e r p r o g r a m m i n g , e d i t i n g o f t h e t h e s i s , a n d , most o f a l l , h i s v a l u e d s u p p o r t t h r o u g h o u t t h e d u r a t i o n o f t h i s w o r k . x N o m e n c l a t u r e A i j = s P a t i a l g l o b a l s t i f f n e s s m a t r i x a ^ j = t e r m o f t h e s p a t i a l m a t r i x A . j AN = a n a l y t i c a l A r = a r e a o f t r i a n g u l a r e l e m e n t B j = v e c t o r o f b o u n d a r y c o n d i t i o n s D = b a s i n d e p t h (km) Do = a r e a o f f l o w d o m a i n E ^ j = s p a t i a l e l e m e n t s t i f f n e s s m a t r i x e i j = t e r m o f t h e s p a t i a l m a t r i x E — FE = f i n i t e e l e m e n t h = h y d r a u l i c head (km) n = a p p r o x i m a t e h y d r a u l i c head h j = h y d r a u l i c head a t node j i , j , k = nodes K = h y d r a u l i c c o n d u c t i v i t y (m/s) K4L = K o f l a y e r number 4L K = K i n t h e x - d i r e c t i o n K z = K i n t h e z - d i r e c t i o n L = b a s i n l e n g t h (km) L ( ) = l i n e a r o p e r a t o r N_. = b a s i s f u n c t i o n a t node j n = number o f n o d e s R = r e s i d u a l R g n = r e l a t i v e h y d r a u l i c head a d j u s t m e n t S = b o u n d a r y o f d o m a i n , Do S^ .. = t e m p o r a l g l o b a l s t i f f n e s s m a t r i x S g = s p e c i f i c s t o r a g e (m 1 ) x i t e m p o r a l e l e m e n t s t i f f n e s s m a t r i x t i m e ( y e a r s ) w e i g h t i n g f o r node i l a t e r a l d i s t a n c e (km) v e r t i c a l d i s t a n c e o r e l e v a t i o n a b o v e datum (km) w e i g h t i n g f a c t o r f o r t i m e d e r i v a t i v e a p p r o x i m a t i o n o n e - d i m e n s i o n a l x i i CHAPTER 1 INTRODUCTION The n a t u r e and r a t e o f c h a n g e o f t r a n s i e n t g r o u n d w a t e r c o n d i t i o n s i n a g e o l o g i c a l l y m a t u r e b a s i n , i . e . one t h a t i s n o t b e i n g i n f l u e n c e d by t e c t o n i s m , c a n be o f s i g n i f i c a n c e f o r h y d r o g e o l o g i c s t u d i e s c o n c e r n e d w i t h d e t e r m i n i n g t h e n a t u r e o f l o n g - t e r m e v e n t s o c c u r r i n g on a r e g i o n a l s c a l e . An a n a l y s i s o f t r a n s i e n t c o n d i t i o n s s h o u l d be a p r i m e c o n s i d e r a t i o n when: a) e x p l o r i n g f o r t r a n s p o r t e d h y d r o c a r b o n s o r m i n e r a l d e p o s i t s ; b) i s o l a t i n g r a d i o a c t i v e w a s t e s ; c) d e t e r m i n i n g s a f e d e v e l o p m e n t r a t e s o f g r o u n d w a t e r r e s o u r c e s i n a r i d r e g i o n s ; and d) d a t i n g g e o l o g i c a l e v e n t s t h a t r e s u l t i n c h a n g e s o f b o u n d a r y c o n d i t i o n s o f s u b s u r f a c e f l o w f i e l d s ( T 6 t h and M i l l a r , 1 9 8 3 ) . P r e v i o u s s t u d i e s o f t h e f l o w o f g r o u n d w a t e r i n d e e p , g e o l o g i c a l l y m a t u r e , r e g i o n a l b a s i n s , h a v e g e n e r a l l y assumed t h e f l o w t o be i n a s t a t e o f d y n a m i c e q u i l i b r i u m , i n w h i c h c a s e , s t e a d y - s t a t e a n a l y s e s c a n be i n v o k e d . R e c e n t l y , s e v e r a l i n v e s t i g a t o r s have s u g g e s t e d t h a t f l o w p a t t e r n s , when o b s e r v e d f r o m t h e p e r s p e c t i v e o f g e o l o g i c t i m e , may a c t u a l l y r e p r e s e n t one t r a n s i e n t s t a g e i n t h e a d j u s t m e n t o f t h e f l o w r e g i m e . I n some b a s i n s , t h i s a d j u s t m e n t i s s u f f i c i e n t l y f a s t t h a t t h e a s s u m p t i o n o f s t e a d y - s t a t e f l o w may be d e f e n s i b l e ; i n o t h e r b a s i n s , t h e r a t e o f a d j u s t m e n t i s so s l o w t h a t t h e s t e a d y - s t a t e a s s u m p t i o n i s n o t v a l i d . The o b j e c t i v e o f t h e t h e s i s i s t o d e v e l o p an u n d e r s t a n d i n g o f t h e n a t u r e o f t r a n s i e n t f l o w i n a h e t e r o g e n e o u s , a n i s o t r o p i c , g e o l o g i c a l l y m a t u r e b a s i n t h r o u g h t h e use o f a t w o - d i m e n s i o n a l 2 f i n i t e e l e m e n t m o d e l . M o d e l r e s u l t s w i l l be u s e d t o : 1) d e t e r m i n e t h e l e n g t h s o f t i m e r e q u i r e d f o r r e g i o n a l f l o w f i e l d s t o a d j u s t t o new b o u n d a r y c o n d i t i o n s ; 2) e v a l u a t e t h e e f f e c t s o f g e o l o g i c e v e n t s on t h e f l o w p a t t e r n s o f s u b s u r f a c e f l u i d s ; 3) u n d e r s t a n d t h e t i m e - d e p e n d e n t n a t u r e o f p e r m e a b i l i t y b a r r i e r s ; and 4) r e l a t e t h e s e i s s u e s t o r e l e v a n t p r o b l e m s i n h y d r o g e o l o g y . The s t a t e o f t r a n s i e n t f l o w o f g r o u n d w a t e r i s n o r m a l l y i n d i c a t e d by t h e p r e s e n c e o f a b n o r m a l p o r e p r e s s u r e s , t h e o r i g i n o f w h i c h have been e x p l a i n e d by v a r i o u s i n v e s t i g a t o r s u s i n g a v a r i e t y o f m e c h a n i s m s (T6th and M i l l a r , 1983) i n c l u d i n g : a) o s m o s i s , p h a s e c h a n g e s o f m i n e r a l s , and f l u i d v o l u m e c h a n g e s w i t h i n t h e b a s i n ; b) d e f o r m a t i o n o f t h e r o c k f r a m e w o r k c a u s e d by c o m p a c t i o n , c o m p r e s s i o n , o r e l a s t i c r e b o u n d ; and c) c h a n g e s i n w a t e r t a b l e e l e v a t i o n s a t t h e o u t c r o p a r e a s o f c o n f i n e d a q u i f e r s . T o t h and M i l l a r (1983) h a v e more r e c e n t l y s u g g e s t e d t h a t e r o s i o n a l m o d i f i c a t i o n s o f t h e l a n d s u r f a c e may a c c o u n t f o r t h e e x i s t e n c e o f a n o m a l o u s p r e s s u r e s a t d e p t h . R e g i o n a l g r o u n d w a t e r f l o w s y s t e m s i n a r i g i d r o c k f r a m e w o r k a r e g e n e r a t e d by e l e v a t i o n d i f f e r e n c e s on t h e w a t e r t a b l e , whose c o n f i g u r a t i o n g e n e r a l l y f o l l o w s t h a t o f t h e l a n d s u r f a c e . They a r e m o d i f i e d by d i f f e r e n c e s i n p e r m e a b i l i t y o f t h e g e o l o g i c f o r m a t i o n s . A c h a n g e i n t h e s u r f a c e t o p o g r a p h y o v e r g e o l o g i c t i m e w i l l c a u s e a r e a d j u s t m e n t o f t h e f l o w p a t t e r n , a t a r a t e w h i c h i s a f u n c t i o n o f t h e p e r m e a b i l i t y and c o m p r e s s i b i l i t y ( e l a s t i c i t y ) o f t h e p o r o u s m e d i a . I n some h e t e r o g e n e o u s b a s i n s , t h e r e a d j u s t m e n t c a n be e x t r e m e l y s l o w , on t h e o r d e r o f h u n d r e d s t o m i l l i o n s o f y e a r s , c a u s i n g p a r t s o f t h e f l o w f i e l d t o be o u t - o f - p h a s e w i t h r e s p e c t t o o t h e r p a r t s o f t h e f l o w r e g i m e 3 d u r i n g a g i v e n t i m e p e r i o d . I t i s t h i s a s p e c t o f t h e t r a n s i e n t e v e n t t h a t c o u l d be o f i m p o r t a n c e when e x p l o r i n g f o r h y d r o c a r b o n s o r m o d e l i n g f l o w s y s t e m s f o r p r e d i c t i o n s o f r a d i o a c t i v e w a s t e t r a n s p o r t . The f i r s t p a r t o f t h i s t h e s i s c o m p r i s e s a d i s c u s s i o n o f p r e v i o u s work i n b o t h s t e a d y - s t a t e and t r a n s i e n t n u m e r i c a l m o d e l i n g o f r e g i o n a l g r o u n d w a t e r f l o w , an o u t l i n e o f t h e t h e o r y o f t h e f i n i t e e l e m e n t m e t h o d , d e v e l o p m e n t o f t h e F o r t r a n - I V c o m p u t e r p r o g r a m , v e r i f i c a t i o n o f t h e m o d e l , and a d i s c u s s i o n o f i t s n u m e r i c a l p e r f o r m a n c e . The s e c o n d p a r t o f t h e t h e s i s c o n s i s t s o f an a n a l y s i s o f t h e m o d e l ' s s e n s i t i v i t y t o a v a r i e t y o f h y d r o g e o l o g i c a l p a r a m e t e r s , i n c l u d i n g h y d r a u l i c c o n d u c t i v i t y , a n i s o t r o p y , w a t e r t a b l e c o n f i g u r a t i o n , b a s i n s i z e , and s p e c i f i c s t o r a g e ; a d i s c u s s i o n o f s i m u l a t i o n s o f f l o w i n b a s i n s w i t h v a r i o u s s t r a t i g r a p h i c and s t r u c t u r a l c o n f i g u r a t i o n s ; and an o u t l i n e o f t h e p r a c t i c a l a p p l i c a t i o n s o f t h e s t u d y . The Red E a r t h r e g i o n o f n o r t h e r n A l b e r t a , Canada ( F i g u r e 1) was c h o s e n a s a w o r k i n g e x a m p l e b e c a u s e an e x t e n s i v e amount o f f i e l d work h a s been p e r f o r m e d i n t h i s a r e a and t h e h y d r a u l i c p a r a m e t e r s o f t h e f i v e h y d r o g e o l o g i c u n i t s i n t h e b a s i n a r e f a i r l y a c c u r a t e l y known ( T o t h , 1978, 1 9 7 9 ) . The g e o m e t r y o f t h e l a y e r s h a s b een s i m p l i f i e d i n t h e m o d el t o a i d i n i n t e r p r e t a t i o n o f t h e d a t 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 s o f up t o e i g h t o r d e r s o f m a g n i t u d e w ere m o d e l e d . Figure 1. Index map showing the location of the Red Earth region. Adapted from Tdth and Millar (1983, Fig. 1). 5 CHAPTER 2 PREVIOUS WORK: A BRIEF REVIEW OF VARIOUS APPROACHES  TO MODELING GROUNDWATER FLOW IN REGIONAL BASINS 2.1. I n t r o d u c t i o n The purpose of t h i s c h a p t e r i s to b r i e f l y r e v i e w p r e v i o u s work d e s c r i b i n g the n a t u r e of s t e a d y s t a t e f l o w i n a r e g i o n a l b a s i n , and to document a n a l y s e s of l o n g - t e r m t r a n s i e n t f l o w i n mature b a s i n s . Attempts a t e s t i m a t i n g the r a t e and d u r a t i o n of t r a n s i e n t events w i t h i n r e g i o n a l b a s i n s are a l s o d i s c u s s e d . 2.2. A n a l y s e s of Steady S t a t e Flow i n a R e g i o n a l B a s i n E a r l y c o n t r i b u t o r s toward u n d e r s t a n d i n g the n a t u r e of s t e a d y s t a t e r e g i o n a l groundwater f l o w a re Hubbert (1940), Toth (1962, 1963), and Freeze and Withe r s p o o n (1966, 1967, 1968). These a u t h o r s a l l p r e s e n t t h e o r e t i c a l a n a l y s e s of r e g i o n a l groundwater f l o w . Hubbert (1940), from f i r s t p r i n c i p l e s , re-examined the fundamentals of the p o t e n t i a l f u n c t i o n g o v e r n i n g groundwater f l o w . He r e - d e f i n e d t h i s f u n c t i o n so t h a t i t conformed w i t h both the p r i n c i p l e of the c o n s e r v a t i o n o f matter and the laws of thermodynamics. The main c o n t r i b u t i o n s of Hubbert's (1940) paper were: 1) to e x p r e s s r e s u l t s of h y d r a u l i c head measurements such t h a t they can be e a s i l y t r a n s f o r m e d i n t o a g r a p h i c a l a p p r o x i m a t i o n of the groundwater f l o w p a t t e r n ; and 2) to f o r m u l a t e the b a s i c e q u a t i o n s such t h a t they are r e a d i l y a d a p t a b l e to more p r e c i s e a n a l y t i c a l and n u m e r i c a l s o l u t i o n methods. 6 T o t h ( 1 9 6 2 , 1963) a n a l y z e d g r o u n d w a t e r f l o w i n s m a l l d r a i n a g e b a s i n s u s i n g a 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 a n a l y t i c a l m o d e l . He c o n c l u d e d , f r o m r e s u l t s o f t h e m o d e l , t h a t : 1) t h e p i e z o m e t r i c s u r f a c e f o r c o n d i t i o n s a t d e p t h i s g e n e r a l l y a r e f l e c t i o n o f t h e s u r f a c e t o p o g r a p h y ; 2) a r e g i o n a l f l o w s y s t e m d r i v e n by a l i n e a r 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 y m m e t r i c a l r e l a t i v e t o t h e m i d - l i n e b e t w e e n a v a l l e y b o t t o m and a g r o u n d w a t e r d i v i d e ; 3) i n t h i s c a s e , t h e d o w n s l o p e h a l f o f t h e v a l l e y s i d e i s t h e d i s c h a r g e a r e a ; and 4) m i n o r t o p o g r a p h i c i r r e g u l a r i t i e s h ave t h e i r own a s s o c i a t e d f l o w s y s t e m s and a c o m p l e x p a t t e r n o f r e c h a r g e and d i s c h a r g e a r e a s . I n s u c h c a s e s s e v e r a l f l o w s y s t e m s may be s u p e r i m p o s e d on one a n o t h e r . F r e e z e and W i t h e r s p o o n (1966) d e v e l o p e d b o t h a n a l y t i c a l and n u m e r i c a l s o l u t i o n s t o t h e r e g i o n a l f l o w m o d e l . The p u r p o s e o f t h e i r work was t o e x t e n d T o t h ' s method i n o r d e r t o a p p l y i t t o t h e more g e n e r a l c a s e o f f l o w i n a t h r e e - d i m e n s i o n a l n onhomogeneous, a n i s o t r o p i c b a s i n . The a u t h o r s s o l v e d t h e g e n e r a l c a s e u s i n g a f i n i t e d i f f e r e n c e m o d e l . R e s u l t s o f n u m e r i c a l s o l u t i o n s t o a w i d e v a r i e t y o f g e o l o g i c a l and g e o g r a p h i c a l s i t u a t i o n s show q u a l i t a t i v e and q u a n t i t a t i v e e f f e c t s on t h e f l o w p a t t e r n s o f b o t h t h e c o n f i g u r a t i o n o f t h e w a t e r t a b l e and t h e v a r i a t i o n i n p e r m e a b i l i t y and c o n f i g u r a t i o n o f s t r a t a . C o n c l u s i o n s a r i s i n g f r o m t h e s e r e s u l t s a r e : 1) t h e f a c t o r s a f f e c t i n g s t e a d y s t a t e r e g i o n a l f l o w p a t t e r n s i n c l u d e t h e r a t i o o f d e p t h t o l a t e r a l e x t e n t o f t h e b a s i n , t h e w a t e r t a b l e c o n f i g u r a t i o n , and t h e s t r a t i g r a p h y and v a r i a t i o n s i n p e r m e a b i l i t y o f s t r a t a i n t h e b a s i n ; 2) d i s c h a r g e f r o m t h e b a s i n i s c o n c e n t r a t e d i n m a j o r v a l l e y s ; 3) r e c h a r g e a r e a s a r e l a r g e r 7 t h a n d i s c h a r g e a r e a s ; and 4) s t r a t i g r a p h i c d i s c o n t i n u i t i e s c a u s e d i s t r i b u t i o n s o f r e c h a r g e and d i s c h a r g e a r e a s t h a t a r e d i f f i c u l t t o p r e d i c t . The n a t u r e o f s t e a d y s t a t e f l o w i n a r e g i o n a l b a s i n i s r e a s o n a b l y w e l l u n d e r s t o o d , b u t t h e i m p o r t a n c e o f l o n g - t e r m t r a n s i e n t e v e n t s o c c u r r i n g o v e r t h o u s a n d s t o m i l l i o n s o f y e a r s , h a s r e c e i v e d l i t t l e a t t e n t i o n , and c o n s e q u e n t l y , i s p o o r l y u n d e r s t o o d . 2.3. A n a l y s e s o f T r a n s i e n t F l o w i n a R e g i o n a l B a s i n Few i n v e s t i g a t o r s h ave s t u d i e d t h e n a t u r e o f l o n g - t e r m t r a n s i e n t f l o w , p r o b a b l y due t o t h e l o n g p e r i o d o f t i m e o v e r w h i c h c h a n g e s o c c u r and t o t h e e x t r e m e l y s m a l l v a r i a t i o n s o f h y d r a u l i c head t h a t o c c u r w i t h i n t h e s u b s u r f a c e r e l a t i v e t o t h e human t i m e s c a l e . S c i e n t i s t s who have s i m u l a t e d l o n g - t e r m t r a n s i e n t e v e n t s h a v e r a r e l y a t t e m p t e d a q u a n t i t a t i v e a n a l y s i s . B r e d e h o e f t and Hanshaw (1968) p r e s e n t e d o n e - d i m e n s i o n a l a n a l y t i c a l s o l u t i o n s t o t h e p r o b l e m o f t r a n s i e n t f l o w i n t h i c k , h omogeneous, s e d i m e n t a r y s e q u e n c e s . They c o n c l u d e d t h a t h y d r a u l i c head c h a n g e s i n a t h i c k s e d i m e n t a r y s e q u e n c e w i t h a l o w h y d r a u l i c c o n d u c t i v i t y c a n p e r s i s t f o r g e o l o g i c a l l y s i g n i f i c a n t p e r i o d s o f t i m e . The a u t h o r s a l s o o b s e r v e d t h a t c r e a t i o n o f a b n o r m a l f o r m a t i o n f l u i d p r e s s u r e s d e p e n d s m a i n l y on t h e h y d r a u l i c c o n d u c t i v i t y a n d , t o a l e s s e r d e g r e e , on t h e s p e c i f i c s t o r a g e o f c l a y l a y e r s w i t h i n t h e s y s t e m . A q u a s i - t r a n s i e n t r e s i s t a n c e n e t w o r k a n a l o g u e m o del was d e v e l o p e d by L l o y d and F a r a g (1978) w h i c h a p p r o x i m a t e s t h e t i m e v a r i a n c e o f g r o u n d w a t e r f l o w i n r e g i o n a l s e d i m e n t a r y b a s i n s by a 8 s e r i e s o f s t e a d y s t a t e s i m u l a t i o n s . The a u t h o r s c o n c l u d e d t h a t a b n o r m a l h y d r a u l i c head g r a d i e n t s c a n e x i s t i n r e g i o n a l b a s i n s f o r l o n g p e r i o d s o f t i m e and t h e i r i m p l i c a t i o n s s h o u l d be c o n s i d e r e d when a s s e s s i n g l o n g - t e r m g r o u n d w a t e r d e v e l o p m e n t p r o p o s a l s . T o t h (1980) p r e s e n t e d a g e n e r a l i z e d h y d r a u l i c t h e o r y o f p e t r o l e u m m i g r a t i o n , w h i c h i s b a s e d on two i m p o r t a n t c o n c e p t s : 1) t h e r o c k f r a m e w o r k o f a g e o l o g i c a l l y m a t u r e b a s i n i s h y d r a u l i c a l l y c o n t i n u o u s ; and 2) t h e p r i n c i p a l m o t i v e f o r c e o f s u b s u r f a c e f l u i d s i s g r a v i t y . The t h e o r y i s v e r i f i e d w i t h s e v e r a l f i e l d e x a m p l e s w h i c h compare r e g i o n a l o b s e r v a t i o n s t o d i s t r i b u t i o n p a t t e r n s o f h y d r a u l i c h e a d , f l o w , v e r t i c a l p r e s s u r e g r a d i e n t , and d y n a m i c p r e s s u r e i n c r e m e n t , f o r h y d r a u l i c a l l y c o n t i n u o u s h y p o t h e t i c a l f l o w r e g i o n s . T o t h c o n c l u d e d t h a t : " l a r g e d r a i n a g e b a s i n s do f u n c t i o n as h y d r a u l i c a l l y c o n t i n u o u s r e g i o n s i n w h i c h g r o u n d w a t e r f l o w - d i s t r i b u t i o n i s d e t e r m i n e d by t h e c o n f i g u r a t i o n o f t h e w a t e r t a b l e . . . < a n d a l s o t h a t > . . . s i i g h t l y p e r m e a b l e e x t e n s i v e f o r m a t i o n s m o d i f y t h e f l o w d i s t r i b u t i o n and a l s o may d e l a y t h e t r a n s m i s s i o n o f c h a n g e s i n f l u i d p o t e n t i a l f r o m t h e b a s i n ' s s u r f a c e t o l o w e r z o n e s . " ( T o t h , 1980, p.165) I n a s u b s e q u e n t p a p e r , T o t h and M i l l a r (1983) p r e s e n t e d a r i g o r o u s a n a l y t i c a l a n a l y s i s o f r e g i o n a l t r a n s i e n t f l o w i n a g e o l o g i c a l l y m a t u r e b a s i n u s i n g a o n e - d i m e n s i o n a l s o l u t i o n o f t h e d i f f u s i o n e q u a t i o n . The p r i m a r y p u r p o s e o f t h i s p a p e r was t o d e t e r m i n e t h e t i m e s c a l e o v e r w h i c h f o r m a t i o n f l u i d p r e s s u r e s a t d e p t h a r e a l t e r e d by e r o s i o n a l m o d i f i c a t i o n o f t h e l a n d s u r f a c e . T o t h and M i l l a r (1983) c o n c l u d e d f r o m t h e i r work t h a t : 1) e r o s i o n a l m o d i f i c a t i o n s o f t h e l a n d s u r f a c e c a n g e n e r a t e p o r e 9 p r e s s u r e changes at g r e a t depths i n s e d i m e n t a r y b a s i n s ; 2) p r e v i o u s l y e s t a b l i s h e d p r e s s u r e p a t t e r n s are a b l e to s u r v i v e changes i n the boundary c o n d i t i o n s and w i l l appear anomalous, w i t h r e s p e c t to the c u r r e n t t o p o g r a p h i c r e l i e f , f o r hundreds to m i l l i o n s of y e a r s ; 3) the d u r a t i o n and r a t e of e r o s i o n - i n d u c e d pore p r e s s u r e changes are c o n t r o l l e d by those beds w i t h the g r e a t e s t h y d r a u l i c r e s i s t a n c e ; 4) a o n e-order-of-magnitude change i n the v a l u e of h y d r a u l i c c o n d u c t i v i t y of the more r e s i s t a n t beds r e s u l t s i n a o n e - o r d e r - o f - m a g n i t u d e change i n the adjustment t i m e ; and 5) the t r a n s i e n t n a t u r e of the p r o c e s s of p r e s s u r e adjustment to changed boundary c o n d i t i o n s can o n l y be p e r c e i v e d on the proper time s c a l e . The r e s u l t s of Toth and M i l l a r ' s a n a l y s i s are f u r t h e r d i s c u s s e d i n Chapter t h r e e . To the a u t h o r ' s knowledge, t h e r e have been no two- or t h r e e - d i m e n s i o n a l s i m u l a t i o n s of l o n g - t e r m t r a n s i e n t b e h a v i o u r o f r e g i o n a l groundwater f l o w i n g e o l o g i c a l l y mature b a s i n s u s i n g n u m e r i c a l t e c h n i q u e s . A n a l y t i c a l s o l u t i o n s have been l i m i t e d to assessment of o n e - d i m e n s i o n a l f l o w o n l y . The f o l l o w i n g c h a p t e r i s an o u t l i n e of the f i n i t e element model developed f o r a n a l y z i n g the n a t u r e of t w o - d i m e n s i o n a l t r a n s i e n t groundwater f l o w i n a mature b a s i n . The model i s a p p l i e d to g e n e r i c and a c t u a l b a s i n s w i t h h y d r a u l i c c o n d u c t i v i t y c o n s t r a s t s of up to e i g h t o r d e r s - o f - m a g n i t u d e between f o r m a t i o n s . 10 CHAPTER 3 THE FINITE ELEMENT MODEL T h i s c h a p t e r c o n s i s t s of f o u r s e c t i o n s i n c l u d i n g t h e o r e t i c a l development of the f i n i t e element method, development o f the F o r t r a n IV computer program, v e r i f i c a t i o n of the model, and n u m e r i c a l performance of the model. Those r e a d e r s who are a l r e a d y f a m i l i a r w i t h the f i n i t e element method may wish t o s k i p s e c t i o n 3.1. 3.1. T h e o r e t i c a l Development 3.1.1. I n t r o d u c t i o n The f i n i t e element model p r o v i d e s a method f o r s o l v i n g the s e t of 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 s t h a t d e s c r i b e the f l o w of groundwater. The g o v e r n i n g e q u a t i o n f o r t r a n s i e n t groundwater f l o w i s a second-order 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 n o r d e r to s o l v e t h i s e q u a t i o n u n i q u e l y , a s e t o f c o n d i t i o n s on a l l b o u n d a r i e s o f the f l o w r e g i o n and v a l u e s o f i n i t i a l h y d r a u l i c head a t a l l p o i n t s w i t h i n the f l o w r e g i o n must be d e f i n e d . The f i n i t e element method r e q u i r e s d i s c r e t i z a t i o n o f the f l o w regime i n t o s m a l l elements (see F i g u r e 2) which may be o f s e v e r a l d i f f e r e n t g e o m e t r i c a l forms. L i n e a r , t r i a n g u l a r elements w i t h a nodal p o i n t l o c a t e d a t each apex of the t r i a n g l e (see F i g u r e 3) were s e l e c t e d f o r t h i s s t u d y . A l l parameters are c o n s t a n t w i t h i n each element but may v a r y from element to element. To s o l v e the f l o w e q u a t i o n , l i n e a r b a s i s (or i n t e r p o l a t i o n ) f u n c t i o n s a re d e f i n e d over each element and a c o n s i s t e n t form of the c o e f f i c i e n t m a t r i x i s developed through the a p p l i c a t i o n of Figure 2. An example of a discretized flow region. k t x k l z k j Figure 3. An example of a linear triangular element. 12 G a l e r k i n ' s method ( P i n d e r and Gray, 1977) and the d i r e c t s u b s t i t u t i o n of the a p p r o x i m a t i n g e q u a t i o n . The C r a n k - N i c o l s o n or c e n t e r e d f i n i t e d i f f e r e n c e procedure (Remson e_t a _ l . , 1971) i s a p p l i e d to the temporal term of the e q u a t i o n . The r e s u l t i s a s e t of a l g e b r a i c e q u a t i o n s , i n m a t r i x form, i n which the unknowns are the h y d a u l i c head v a l u e s a t each node w i t h i n the f l o w regime a t a s p e c i f i e d t i m e . 3.1.2. G a l e r k i n ' s A p p r o x i m a t i o n of the Flow E q u a t i o n The e q u a t i o n of t w o - d i m e n s i o n a l t r a n s i e n t f l u i d f l o w i n a non-homogeneous, a n i s o t r o p i c porous medium i s the second-order p a r a b o l i c , 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 : & ( K x $ + & ( K z f f > - s s It ^ where: h = h y d r a u l i c head K = h y d r a u l i c c o n d u c t i v i t y S s = s p e c i f i c s t o r a g e t = time x = h o r i z o n t a l d i s t a n c e z = e l e v a t i o n above datum E l a s t i c d e f o r m a t i o n of the rock framework i s a p o s s i b l e r e s u l t o f e r o s i o n a l u n l o a d i n g o f sediment a t the l a n d s u r f a c e . The e f f e c t s of t h i s on h y d r a u l i c heads are n e g l e c t e d f o r a l l s i m u l a t i o n s p r e s e n t e d i n t h i s s t u d y because: a) changes i n h y d r a u l i c head due e x c l u s i v e l y to changes i n the topography can be i s o l a t e d and e v a l u a t e d ; and b) pore p r e s s u r e s i n the Red E a r t h r e g i o n are c l e a r l y c o n t r o l l e d by the t o p o g r a p h i c c o n f i g u r a t i o n o f the l a n d s u r f a c e and show no s i g n s of e l a s t i c 13 d e f o r m a t i o n ( T o t h and M i l l a r , 1 9 8 3 ) . C h a nges i n t h e f l u i d p r o p e r t i e s w i t h d e p t h due t o i n c r e a s i n g t e m p e r a t u r e a r e n o t c o n s i d e r e d i n t h i s s t u d y . A d e c r e a s e i n f l u i d v i s c o s i t y due t o i n c r e a s i n g t e m p e r a t u r e w i t h d e p t h c a u s e s an i n c r e a s e i n h y d r a u l i c c o n d u c t i v i t y r e s u l t i n g i n d e e p e r c i r c u l a t i o n o f f l u i d . A d e c r e a s e i n f l u i d d e n s i t y due t o i n c r e a s i n g t e m p e r a t u r e w i t h d e p t h c o n t r i b u t e s t o w a r d b u o y a n t f l u i d f l o w . The e f f e c t o f t h e s e two f a c t o r s on c a l c u l a t e d h y d r a u l i c head a d j u s t m e n t t i m e s w o u l d l i k e l y be t o d e c r e a s e t h e c a l c u l a t e d t i m e f o r c o m p l e t e head a d j u s t m e n t a t t h e b a s e o f t h e s y s t e m . C o n d i t i o n s on t h e b o u n d a r i e s o f t h e f l o w r e g i o n f o r t h e b o u n d a r y v a l u e p r o b l e m (bvp) s o l v e d i n t h i s s t u d y a r e shown i n F i g u r e 2. The l o w e r and two v e r t i c a l b o u n d a r i e s a r e a l l n o - f l o w b o u n d a r i e s and a r e r e p r e s e n t e d m a t h e m a t i c a l l y as f o l l o w s : Ml = 0 8x ( v e r t i c a l ) = 0 ciz ( l o w e r ) The u p p e r b o u n d a r y i s a s l o p i n g w a t e r t a b l e and i s m a t h e m a t i c a l l y e x p r e s s e d a s : h ( x ) = z ( x ) where z i s t h e e l e v a t i o n o f t h e w a t e r t a b l e . The i n i t i a l c o n d i t i o n s f o r a l l o f t h e m o d e l s p r e s e n t e d t h r o u g h o u t t h e t h e s i s a r e no f l o w , o r s t a t i c , c o n d i t i o n s . A d e c r e a s e i n h y d r a u l i c h ead v a l u e s a l o n g t h e u p p e r b o u n d a r y a t time=0 r e p r e s e n t s i n s t a n t a n e o u s e r o s i o n o f t h e l a n d s u r f a c e , w h i c h i n i t i a t e s t h e t r a n s i e n t e v e n t . E q u a t i o n (1) d e s c r i b e s t h e r e s u l t i n g c h a n g e s 14 of head i n both space and t i m e . To s o l v e t h i s type of e q u a t i o n , a s e t of a p p r o x i m a t i n g i n t e g r a l e q u a t i o n s are f o r m u l a t e d by a p p l y i n g e i t h e r the v a r i a t i o n a l t e c h n i q u e or the method of w e i g h t e d r e s i d u a l s . The R a y l e i g h - R i t z p r o c e d u r e ( S t r a n g and F i x , 1973), based on the c a l c u l u s of v a r i a t i o n s , has been used e x t e n s i v e l y i n the p a s t . However, G a l e r k i n ' s f o r m u l a t i o n , a s p e c i a l case of the method of weighted r e s i d u a l s , i s more g e n e r a l i n a p p l i c a t i o n and w i l l converge f o r a broader spectrum of d i f f e r e n t i a l o p e r a t o r s than the v a r i a t i o n a l p r o c e d u r e ( P i n d e r and Gray, 1977). For t h e s e r e a s o n s , e q u a t i o n (1) was t r a n s f o r m e d u s i n g G a l e r k i n ' s f o r m u l a t i o n . The f l o w e q u a t i o n , ( 1 ) , can be w r i t t e n as: L<h> " 0 " £ <Kx f|f> + JI <Kz ?F> - Ss It <2> where L() i s a l i n e a r o p e r a t o r . The e x a c t v a l u e of h y d r a u l i c head i s r e p l a c e d by the t r i a l f u n c t i o n , d e f i n e d as: n h(x,z) = i h.N,(x,z) (3) j=l J J where: h.. = h y d r a u l i c head a t node j N.. = b a s i s f u n c t i o n a t node j n = number of nodes T h i s f u n c t i o n l i n e a r l y a p p r o x i m a t e s the head, h, p i e c e w i s e a c r o s s each element, i n a way d e f i n e d by the b a s i s f u n c t i o n s , N j , a t each node. The f o l l o w i n g diagram i l l u s t r a t e s t h i s o n e - d i m e n s i o n a l l y . 15 element node The d i f f e r e n c e b e t w e e n t h e e x a c t s o l u t i o n and t h e a p p r o x i m a t e s o l u t i o n , l a b e l e d t h e r e s i d u a l , R, i s d e f i n e d as R(h) = L(h) - L(h) T 8X 8X ; 8Z ^ N Z dZ> " ° S 3 t (4) To o b t a i n an e x a c t s o l u t i o n o v e r t h e f l o w d o m a i n t h e r e s i d u a l , R, must v a n i s h . T h i s i s a c c o m p l i s h e d by c h o o s i n g w e i g h t s , w., w h i c h r e d u c e t h e w e i g h t e d i n t e g r a l s o f t h e r e s i d u a l t o z e r o : / / R(h)w-dxdz = 0 i=l,n D 1 (5) where D i s t h e a r e a o f t h e f l o w d o m a i n . G a l e r k i n ' s f o r m u l a t i o n s e l e c t s t h e b a s i s f u n c t i o n s , N i , a s t h e w e i g h t i n g f u n c t i o n s . H e nce, II R(h)N.dxdz = 0 i=l,n D 1 ( 6 ) S u b s t i t u t i n g i n t h e e x p r e s s i o n f o r R, e q u a t i o n (4), g i v e s : »Hi{Tx (Kx & < I h J N J » + Jz ( K z TI (shjN.,)) - Ss A (EhjNj)}dxdz=0 (7) The a p p l i c a t i o n o f G a l e r k i n ' s method p r o d u c e s banded 16 c o e f f i c i e n t m a t r i c e s i n t h e r e s u l t i n g m a t r i x e q u a t i o n . The c o e f f i c i e n t m a t r i x w h i c h m u l t i p l i e s t h e t i m e d e r i v a t i v e v e c t o r i s t r i d i a g o n a l and i s known as t h e c o n s i s t e n t mass m a t r i x . 3.1.3. E l e m e n t s and B a s i s F u n c t i o n s T r i a n g u l a r e l e m e n t s were s e l e c t e d b e c a u s e t h e y a r e s i m p l e , t h e y c a n be a p p l i e d t o p r o b l e m s w i t h i r r e g u l a r g e o m e t r i c a l b o u n d a r i e s , and t h e y f a c i l i t a t e c o n c e n t r a t i o n o f t h e n o d e s i n t h o s e r e g i o n s o f t h e f l o w d o m a i n where a r a p i d l y v a r y i n g s o l u t i o n i s a n t i c i p a t e d ( P i n d e r and G r a y , 1 9 7 7 ) . B a s i s f u n c t i o n s a r e d e f i n e d o v e r e a c h l i n e a r t r i a n g u l a r e l e m e n t and d e s c r i b e a p l a n e o v e r t h e t r i a n g l e . The e x p r e s s i o n f o r s u c h a p l a n e i n g l o b a l x-z c o o r d i n a t e s i s t h e g e n e r a l p l a n a r p o l y n o m i a l : where A, B, and C a r e c o n s t a n t s . To g e n e r a t e b a s i s f u n c t i o n s , c o n s t r a i n t s w h i c h must be i m p o s e d a r e t h a t t h e n o d e s o f e a c h t r i a n g u l a r e l e m e n t must be numbered s e q u e n t i a l l y i n a c o u n t e r c l o c k w i s e d i r e c t i o n and t h e b a s i s f u n c t i o n s f o r e a c h node must be u n i t y a t t h e node f o r w h i c h t h e y a r e d e f i n e d and z e r o a t a l l o t h e r n o d e s . The p o l y n o m i a l e q u a t i o n (8) i s s o l v e d f o r e a c h n o d e , i , j , and k, o f t h e e l e m e n t shown i n t h e d i a g r a m b e l o w , i n o r d e r t o a c h i e v e e x a c t i n t e r p o l a t i o n a t t h e n o d e s . The e q u a t i o n s a r e w r i t t e n a s : h ( x , z ) = A x + B z + C (8) z h. = h. = Ax- + Bz- + C Ax k + Bz k + C x 17 S o l v i n g f o r A, B, and C, i n t r o d u c i n g new v a r i a b l e s a ^ , b^, and c^ i n t o t h e s e e q u a t i o n s , and s u b s t i t u t i n g b a c k i n t o e q u a t i o n (8) p r o d u c e s t h e f o l l o w i n g s e t o f e q u a t i o n s w h i c h d e s c r i b e t h e b a s i s f u n c t i o n s : N. = ( a j x + b L z + c L ) / 2 * A r a i = z j " z k b. = x k - X j c i = x j z k " x k z j Nj = (3jX + b j Z + Cj) / 2 * A r a. = 3 b. = 3 c . = z k " z i x i " x k x k z i x i z k N. = ( a t x + b k z + c k ) / 2 * A r a k - z • — z • i 3 b k = x j " x i c. = X . z • -k i D 1 (9) w h e r e : A r = a r e a o f t r i a n g u l a r e l e m e n t b ^ j ) / 2 ( a . b j z = m x = m z - c o o r d i n a t e o f node m, m = i , j , k x - c o o r d i n a t e o f node m, m = i , j , k 3.1.4. A p p l i c a t i o n o f G r e e n ' s Theorem I n t e g r a t i o n by p a r t s o f a t w o - d i m e n s i o n a l e q u a t i o n , known as G r e e n ' s t h e o r e m ( B o y c e and D i P r i m a , 1 9 7 7 , p . 5 5 7 ) , i s a p p l i e d t o e q u a t i o n (7) i n o r d e r t o e l i m i n a t e t h e s e c o n d d e r i v a t i v e s . E q u a t i o n (7) t h e n becomes: 18 3 N i • -a 9 N i a " 15T ( Kx & <*W> + i f ( Kz h ( E W } + N i S s h ( z h j V d x d z (10) 3.1.5. E l e m e n t S t i f f n e s s M a t r i x The i n f o r m a t i o n o b t a i n e d f r o m i n t e g r a t i o n o f e q u a t i o n (10) o v e r one t r i a n g u l a r e l e m e n t i s i n i t i a l l y s t o r e d i n an e l e m e n t s t i f f n e s s ( o r c o e f f i c i e n t ) m a t r i x o f o r d e r t h r e e . The g e n e r a l f o r m o f t h e e l e m e n t s t i f f n e s s m a t r i x i s : at \ = o ( i i ) w h e r e : E ^ = s p a t i a l m a t r i x T ^ j = t e m p o r a l m a t r i x B j = v e c t o r o f b o u n d a r y c o n d i t i o n s . E a c h o f t h e s e t e r m s i s e x p l a i n e d i n d e t a i l i n t h e r e m a i n d e r o f t h i s s u b s e c t i o n . 3.1.5.1. S p a t i a l m a t r i x The s p a t i a l m a t r i x i s d e t e r m i n e d by d i f f e r e n t i a t i o n and i n t e g r a t i o n o f t h e l e f t - h a n d s i d e o f e q u a t i o n (10) e x c l u d i n g t h e l a s t t e r m c o n t a i n i n g t h e t i m e d e r i v a t i v e : 3N 'D' {~w (K X 3 x L ^ W ) + ^ ( ^ ( E W ) } D X D Z (12) F o r node i , d i f f e r e n t i a t i o n o f t h e b a s i s f u n c t i o n , , g i v e s : 3N i 2-Ar 3N. ~3z~ 2-Ar (13) 19 S u b s t i t u t i n g e q u a t i o n (13) i n t o e q u a t i o n (12) g i v e s : a. Ea.h. b. Eb.h. .'„' {2^AT Kx (-2^) + 2TAT Kz h i W } d x d z (14) M o v i n g t h e c o n s t a n t s o u t s i d e t h e i n t e g r a l and n o t i n g t h a t : / / K (Ea.h.) dxdz = K (Ea.h.) (Ar) Q A J J A J J g i v e s t h e r e s u l t i n g t e r m : a.K b.K 4 ^ ( ^ h d ) + ^ ( . b j h j ) ( 1 5 ) E a c h t e r m i n t h e m a t r i x i s o f t h e f o l l o w i n g f o r m : e i j = ( K x a i a j + K z b i b j ) / 4 * A r (16) 3.1.5.2. T e m p o r a l m a t r i x The t e m p o r a l m a t r i x i s f o r m u l a t e d f r o m d i f f e r e n t i a t i o n and i n t e g r a t i o n o f t h e l a s t t e r m on t h e l e f t - h a n d s i d e o f e q u a t i o n (10), w r i t t e n a s : ssinr'D/Ni ( Z W d x d z (iv) F o r node i , i n t e g r a t i o n o f t h e b a s i s f u n c t i o n i s d e f i n e d a s : / / N.N. dxdz = 4r i l 6 / / N.N. dxdz = , j*i J l c (18) T h i s l e a d s t o t h e f i n a l f o r m o f t h e e q u a t i o n : S l 6 at 12 a t ' ( 1 9 ) The d i a g o n a l t e r m s o f t h e c o e f f i c i e n t m a t r i x a r e (S * A r ) / 6 20 and the o f f - d i a g o n a l terms are ( S s * Ar) / 12. 3.1.5.3. V e c t o r of boundary c o n d i t i o n s The r i g h t - h a n d s i d e of e q u a t i o n (10) forms the v e c t o r of boundary c o n d i t i o n s . These i n t e g r a l s need not be e v a l u a t e d m a t h e m a t i c a l l y . I t can be shown (see, f o r example, Wang and Anderson, 1982, p.126-128) t h a t : i ) the i n t e g r a l reduces to zero f o r a l l i n t e r i o r nodes and nodes on a n o - f l o w boundary; and i i ) f o r a c o n s t a n t head boundary node, n, f o r which the head i s s p e c i f i e d , the nth e q u a t i o n o f the m a t r i x , r e p r e s e n t i n g the s o l u t i o n to node n, can be e l i m i n a t e d , s i n c e the s o l u t i o n i s known, and the remainder of the e q u a t i o n s m o d i f i e d . T h i s t e c h n i q u e , of p a r t i t i o n i n g the m a t r i x , i s d i s c u s s e d i n d e t a i l i n S e c t i o n 3.2.3. 3.1.6. G l o b a l S t i f f n e s s M a t r i x The g l o b a l c o e f f i c i e n t m a t r i x i s o b t a i n e d by summing, f o r each node, the c o n t r i b u t i o n s to t h a t node from each element c o e f f i c i e n t m a t r i x . Each c o n t r i b u t i o n must be p l a c e d i n the row and column o f the g l o b a l m a t r i x c o r r e s p o n d i n g to the same node numbers as those of the element m a t r i x . The g e n e r a l e q u a t i o n f o r the g l o b a l s t i f f n e s s m a t r i x i s of the form: (20) For example, a^j i s the e n t r y i n row i , column j of the g l o b a l m a t r i x and i s f o r m u l a t e d as f o l l o w s : 21 13 1 N 1 0 D 2 <4 E S 3 4 3 1 2 N O D E S 2 1 3 1 !13 :33 1 3 N 0 D E S N O D E S 2 4 3 2 2 2 e l l e 1 2 e 1 3 2 2 2 !21 e 2 2 e 2 3 2 !31 2 2 !32 e 3 3 a i j = a 2 3 1 = e 2 3 + 2 e 1 3 a 3 2 = e 3 2 + •§1 a 3 3 1 " e 3 3 + e 3 3 + 3 e l l 5 3 1 The same p r o c e d u r e i s f o l l o w e d t o c o n s t r u c t t h e m a t r i x S To c o m p l e t e t h e g l o b a l m a t r i x , t h e t i m e d e r i v a t i v e must be e v a l u a t e d . A c c o r d i n g t o P i n d e r and G r a y (1977) n u m e r i c a l e x p e r i m e n t s have shown t h a t t h e c h o i c e among f i n i t e d i f f e r e n c e , f i n i t e e l e m e n t , and l a g g e d f i n i t e e l e m e n t a p p r o x i m a t i o n s g e n e r a l l y d e p e n d s upon t h e n a t u r e o f t h e p r o b l e m and t h e t i m e - s t e p p i n g p r o c e d u r e . None o f t h e methods t e s t e d by t h e a u t h o r s p e r f o r m e d s i g n i f i c a n t l y b e t t e r t h a n t h e c e n t e r e d f i n i t e d i f f e r e n c e p r o c e d u r e . A l s o , t h e f i n i t e d i f f e r e n c e method i s t h e s i m p l e s t p r o c e d u r e i n b o t h a p p l i c a t i o n and t h e o r e t i c a l d e v e l o p m e n t . The a u t h o r s c o n c l u d e d t h a t " t h e f i n i t e d i f f e r e n c e scheme i n t i m e i s t h e b e s t o v e r a l l c h o i c e i n t h e m a j o r i t y o f t r a n s i e n t f i n i t e e l e m e n t a n a l y s e s . " 3.1.7. F i n i t e D i f f e r e n c e A p p r o x i m a t i o n o f t h e Time Der i v a t i v e The c e n t e r e d - i n - t i m e o r C r a n k - N i c o l s o n f i n i t e d i f f e r e n c e a p p r o x i m a t i o n was c h o s e n o v e r t h e f o r w a r d o r b a c k w a r d f i n i t e / 22 d i f f e r e n c e a p p r o x i m a t i o n s b e c a u s e i t i s u n c o n d i t i o n a l l y s t a b l e and c o n v e r g e n t and t h e most a c c u r a t e o f t h e t h r e e m e t h o d s . The e x p l i c i t o r f o r w a r d t i m e a p p r o x i m a t i o n r e q u i r e s t h e l e a s t c o m p u t a t i o n a l e f f o r t b u t i s o n l y c o n d i t i o n a l l y s t a b l e ( P i n d e r and G r a y , 1 9 7 7 , p . 4 8 - 4 9 ) . The i m p l i c i t o r b a c k w a r d t i m e a p p r o x i m a t i o n i s u n c o n d i t i o n a l l y s t a b l e , r e q u i r e s l e s s c o m p u t a t i o n a l e f f o r t b u t h a s a l o w e r o r d e r o f a c c u r a c y . I n e q u a t i o n (20) t h e t i m e d e r i v a t i v e , d h / d t , i s r e p l a c e d by a w e i g h t e d f i n i t e d i f f e r e n c e a p p r o x i m a t i o n , r e s u l t i n g i n t h e g e n e r a l e q u a t i o n : A, t+At T. . "1J ( ». h ) t+At + ( l - e ) ) B. / = 0 t+At J t (21) F o r t h e C r a n k - N i c o l s o n m e t h o d , e = 1/2. The f i n a l f o r m o f t h e g l o b a l m a t r i x e q u a t i o n i s : Xi 1J J _ "At 1J ) h ( t+At ( At 1J A. . 1J B. (22) The v e c t o r h^ i s known a t t i m e t and h e n c e i t r e m a i n s t o s o l v e t h e e q u a t i o n f o r t h e v e c t o r h^ a t t i m e t + A t . 3.2. P r o g r a m i n g T e c h n i q u e s 3.2.1. I n t r o d u c t i o n The f i n i t e e l e m e n t model d e v e l o p e d i n t h e p r e v i o u s s e c t i o n was p r o g r a m e d , by t h e a u t h o r , i n F o r t r a n I V c o m p u t e r l a n g u a g e ( s e e A p p e n d i x ) . The m a i n p r o g r a m , named 2DTF, i s s u p p o r t e d by 23 f i v e s u b r o u t i n e s ; s u b p r o g r a m , MESH i s an a u t o m a t i c mesh g e n e r a t o r ; s u b p r o g r a m s DBAND and SBAND a r e r o u t i n e s f o r s o l v i n g a s e t o f l i n e a r e q u a t i o n s , i n t h e f o r m o f e q u a t i o n ( 1 8 ) ; s u b p r o g r a m s HPLOT and VPLOT a r e r o u t i n e s f o r p l o t t i n g c o n t o u r s o f h y d r a u l i c head v a l u e s and f o r p l o t t i n g v e l o c i t y a r r o w s s h o w i n g m a g n i t u d e and d i r e c t i o n o f f l o w , r e s p e c t i v e l y . The B a s i c p r o g r a m , named XYPLOTS, g e n e r a t e s t h e p l o t s o f h y d r a u l i c h e a d v e r s u s l o g t i m e on a m i c r o - c o m p u t e r . 3.2.2. F i n i t e E l e m e n t Mesh S u b r o u t i n e MESH g e n e r a t e s a t w o - d i m e n s i o n a l mesh f o r a r e c t a n g u l a r f l o w r e g i o n w i t h a s l o p i n g l i n e a r , c o n c a v e , o r c o n v e x w a t e r t a b l e ( t h e u p p e r b o u n d a r y ) and s l o p i n g o r h o r i z o n t a l i m p e r m e a b l e b a sement r o c k ( t h e l o w e r b o u n d a r y ) . Node s p a c i n g i n b o t h t h e x ( l a t e r a l ) and z ( v e r t i c a l ) d i r e c t i o n s c a n be v a r i a b l e o r c o n s t a n t . N u m b e r i n g o f nodes and e l e m e n t s b e g i n s i n t h e l o w e r l e f t c o r n e r o f t h e mesh and i n c r e a s e s s e q u e n t i a l l y i n e i t h e r t h e x o r z d i r e c t i o n . Nodes a t t h e a p i c e s o f e a c h e l e m e n t a r e a u t o m a t i c a l l y t a g g e d i n a c o u n t e r c l o c k w i s e d i r e c t i o n and a s s i g n e d t o t h a t e l e m e n t . B o t h x and z c o o r d i n a t e s a r e c a l c u l a t e d f o r e a c h node. E a c h node i s a l s o t a g g e d as e i t h e r a c o n s t a n t - h e a d node o r an unknown-head node. The s u b r o u t i n e a s s i g n s an h y d r a u l i c c o n d u c t i v i t y t a g t o e a c h e l e m e n t . B o u n d a r i e s b e t w e e n e a c h l a y e r o f e l e m e n t s must be l i n e a r and o f c o n s t a n t s l o p e . An e x a m p l e mesh d i a g r a m i s shown i n F i g u r e 4. N o t e t h e l a r g e v e r t i c a l e x a g g e r a t i o n i n t h i s and a l l s u b s e q u e n t mesh d i a g r a m s , c r e a t i n g h i g h e l e m e n t a s p e c t r a t i o s . Figure 4. An example of a finite element mesh generated by the computer program MESH. 25 3.2.3. P a r t i t i o n i n g of the M a t r i x Constant-head boundary nodes have known v a l u e s of h y d r a u l i c head which do not change t h r o u g h t i m e ; hence, t h e s e nodes can be removed from the m a t r i c e s of e q u a t i o n ( 1 8 ) . By p a r t i t i o n i n g the m a t r i x and removing the rows c o r r e s p o n d i n g to the c o n s t a n t head nodes, computer s t o r a g e r e q u i r e m e n t s are m i n i m i z e d and fewer a r i t h m e t i c o p e r a t i o n s w i l l be r e q u i r e d f o r s o l u t i o n of the m a t r i c e s ( P i n d e r and F r i n d , 1972). In o r d e r to p r e s e r v e the squareness of the m a t r i x , a l l of the r e m a i n i n g e q u a t i o n s must be m o d i f i e d and the columns of c o n s t a n t head nodes e l i m i n a t e d . T h i s i s a c c o m p l i s h e d by m u l t i p l y i n g each e n t r y i n the m a t r i x under a column which i s to be e l i m i n a t e d by the c o n s t a n t h y d r a u l i c head v a l u e of the node c o r r e s p o n d i n g to t h a t column number ( S e g e r l i n d , 1976, p.109-112). T h i s p r o d u c t i s then s u b t r a c t e d from the r i g h t - h a n d s i d e of the e q u a t i o n . The f o l l o w i n g i s an example of p a r t i t i o n i n g of a 3x3 c o e f f i c i e n t m a t r i x , i f node 3 i s a c o n s t a n t head node: K l l K 1 2 K 2 1 K 2 2 K l l K 1 2 1 2 L I 2 2 K 3 1 K 3 2 K 13 23_ 33_ F l F 2 " K 2 3 h 3 1 V F 3 . K 1 3 h 3 3.2.4. S o l u t i o n of the M a t r i x E q u a t i o n S o l v i n g the m a t r i x e q u a t i o n r e q u i r e s a l a r g e amount of s t o r a g e c a p a c i t y and i n c u r s h i g h c o m p u t a t i o n a l c o s t s . Cost i s m i n i m i z e d by s e l e c t i n g the most e f f i c i e n t s o l u t i o n s t r a t e g y f o r the problem. The banded, symmetric, and p o s i t i v e - d e f i n i t e 26 s y s t e m o f l i n e a r e q u a t i o n s g e n e r a t e d i n t h e p r e v i o u s s e c t i o n i s b e s t s o l v e d by C h o l e s k y d e c o m p o s i t i o n ( M e y e r , 1 973, 1974; a s  c i t e d by F r i n d and V e r g e , 1 9 7 8 ) , a common method o f s o l u t i o n f o r t w o - d i m e n s i o n a l g r o u n d w a t e r s y s t e m s . The s y m m e t r i c a l n a t u r e o f t h e c o e f f i c i e n t m a t r i x a l l o w s s t o r a g e o f o n l y h a l f o f t h e m a t r i x c o e f f i c i e n t s , t h e u p p e r o r l o w e r t r i a n g l e , i n c l u d i n g t h e d i a g o n a l . To r e d u c e s t o r a g e r e q u i r e m e n t s , t h e n o n - z e r o m a t r i x d i a g o n a l s a r e s t o r e d as c o l u m n s , h e n c e t h e number o f c o l u m n s e q u a l s h a l f t h e b a n d w i d t h o f t h e m a t r i x . T h e r e a r e two s t e p s t o t h e C h o l e s k y d e c o m p o s i t i o n m e t h o d . The f i r s t s t e p i n v o l v e s d e c o m p o s i t i o n o f t h e m a t r i x i n t o t h e p r o d u c t o f a l o w e r t r i a n g u l a r m a t r i x and an u p p e r t r i a n g u l a r m a t r i x . D e c o m p o s i t i o n i s a c c o m p l i s h e d u s i n g t h e s u b r o u t i n e DBAND and need o n l y be p e r f o r m e d once i f t h e s i z e o f t h e t i m e s t e p i s c o n s t a n t . The s e c o n d s t e p i n v o l v e s f o r w a r d and b a c k w a r d s u b s t i t u t i o n . T h i s i s a c c o m p l i s h e d i n t h e s u b r o u t i n e SBAND and must be r e p e a t e d f o r e a c h t i m e s t e p . B o t h o f t h e s u b r o u t i n e s , DBAND and SBAND, were a d a p t e d f r o m t h e p r o g r a m STEADY, w r i t t e n by D r . E m i l 0. F r i n d o f t h e U n i v e r s i t y o f W a t e r l o o ( F r i n d , 1 9 8 2 , p e r s . comm.). 3.2.5. S e l e c t i o n o f t h e Time S t e p L a r g e c o n t r a s t s i n h y d r a u l i c c o n d u c t i v i t y a c r o s s b o u n d a r i e s b e t w e e n l a y e r s o f d i f f e r i n g m a t e r i a l p o s e a p r o b l e m f o r t i m e s t e p p i n g . F o r p r o b l e m s i n v o l v i n g h e t e r o g e n e o u s h y d r o g e o l o g i c a l c o n f i g u r a t i o n s , i t i s n o t p o s s i b l e t o s e l e c t a s i n g l e t i m e s t e p f o r t h e e n t i r e a d j u s t m e n t p e r i o d d u r i n g w h i c h t h e s o l u t i o n a p p r o a c h e s s t e a d y s t a t e . T h e r e f o r e , t h e t i m e s t e p must c h a n g e 27 c o n t i n u o u s l y by v a r y i n g amounts d e p e n d i n g upon w h i c h a r e a o f t h e f l o w r e g i o n i s a d j u s t i n g a t a p a r t i c u l a r t i m e t o c h a n g e s i m p o s e d on t h e u p p e r b o u n d a r y . To a c c o m p l i s h t h i s an i n i t i a l t i m e s t e p i s s p e c i f i e d and c o n s e q u e n t t i m e s t e p s a r e a u t o m a t i c a l l y s e l e c t e d by t h e c o m p u t e r , a s d e s c r i b e d b e l o w . An optimum i n i t i a l t i m e s t e p i s c h o s e n s u c h t h a t t h e r e i s no c h a n g e i n h y d r a u l i c head a l o n g t h e f i r s t row o f n o d e s b e n e a t h t h e w a t e r t a b l e f o l l o w i n g t h e f i r s t t i m e s t e p . C h a n g e s i n head v a l u e s f a l l j u s t s h o r t o f r e a c h i n g t h e s e n o d e s . The f i r s t t i m e s t e p i s c a l c u l a t e d b y : t = S s * A z 2 / 4 * K (23) w h e r e : S s = s p e c i f i c s t o r a g e o f t o p l a y e r Az = s m a l l e s t mesh d i s c r e t i z a t i o n o f t o p l a y e r K = h y d r a u l i c c o n d u c t i v i t y o f t o p l a y e r . C h a n g e s i n h y d r a u l i c head a r e n o t m o n i t o r e d f o r t h e f i r s t few t i m e s t e p s . D u r i n g t h e s e e a r l y t i m e s t h e t i m e s t e p i s i n c r e a s e d by a f a c t o r o f 1.25. The s o l u t i o n o s c i l l a t e s a t e a r l y t i m e s b e c a u s e o f t h e l a r g e c h a n g e i n h y d r a u l i c head a t t h e u p p e r b o u n d a r y r e p r e s e n t i n g i n s t a n t a n e o u s e r o s i o n o f t h e l a n d s u r f a c e . A f t e r t h e f i r s t few s t e p s t h r o u g h t i m e , t h e p e r c e n t a g e c h a n g e , o r d e c a y , i n h y d r a u l i c head i s c a l c u l a t e d a t e a c h node i n t h e s y s t e m a t e a c h t i m e s t e p . The p e r c e n t c h a n g e i n h y d r a u l i c head a t any node i s n o t a l l o w e d t o e x c e e d a s p e c i f i e d t o l e r a n c e l i m i t - t h r e e p e r c e n t i n most c a s e s . The t i m e i n c r e m e n t i n c r e a s e s o r d e c r e a s e s w i t h e a c h s t e p t h r o u g h t i m e d e p e n d i n g on t h e l a r g e s t p e r c e n t c h a n g e c a l c u l a t e d a t any node i n t h e f l o w r e g i o n a c c o r d i n g t o t h e f o r m u l a : 28 d t 2 = dtj^ * f a c t o r = dtx * (1 + ( t o l - %change) / t o l ) (24) where: f a c t o r = 1 i f %change = t o l f a c t o r > 1 i f %change < t o l f a c t o r < 1 i f %change > t o l The v a l u e of ' f a c t o r * cannot be l e s s than 0.1. I f the perc e n t a g e change i n head a t any node exceeds the t o l e r a n c e , the time increment used f o r t h a t c a l c u l a t i o n i s d e c r e a s e d and the c a l c u l a t i o n i s r e p e a t e d . The program a l l o w s o n l y two i t e r a t i o n s per time s t e p . In most c a s e s , no i t e r a t i o n s a re r e q u i r e d to keep the p e r c e n t a g e change i n head below the t o l e r a n c e . U s i n g t h i s method of c a l c u l a t i n g the time s t e p , the increment i n time i s a l l o w e d to grow a t a r a t e c o r r e s p o n d i n g to the r a t e of change of h y d r a u l i c head v a l u e s i n the f l o w r e g i o n . The main drawback to t h i s method i s t h a t the c o e f f i c i e n t m a t r i x must be decomposed f o r each t i m e s t e p . T h i s i s o f f s e t by the f a c t t h a t the s o l u t i o n w i l l reach s t e a d y s t a t e i n many fewer time s t e p s than i f the time s t e p were h e l d c o n s t a n t . By r e s t r i c t i n g the p e r c e n t a g e change i n head a t a l l nodes f o r e v e r y time s t e p , a l l s t a g e s i n h y d r a u l i c head adjustment can be observed i n both high-K and low-K u n i t s . 3.3. Model V e r i f i c a t i o n 3.3.1. I n t r o d u c t i o n The purpose of the f i n i t e element model i s to p r e d i c t the v a l u e of h y d r a u l i c head a t each node w i t h i n the r e g i o n of f l o w over a s p e c i f i e d p e r i o d of t i m e . B e f o r e a p p l y i n g the model to h y p o t h e t i c a l or s i t e - s p e c i f i c s i t u a t i o n s , i t s a c c u r a c y must be 29 v e r i f i e d . The approach taken here i s to compare r e s u l t s g e n e r a t e d by the f i n i t e element model w i t h r e s u l t s from a n a l y t i c a l s o l u t i o n s o b t a i n e d from v a r i o u s s o u r c e s . The s t e a d y - s t a t e a n a l y s i s was v e r i f i e d i n both one and two d i m e n s i o n s . The t r a n s i e n t s o l u t i o n was v e r i f i e d i n one d i m e n s i o n o n l y . 3.3.2. V e r i f i c a t i o n of Steady S t a t e S o l u t i o n s T h i s s e c t i o n d e s c r i b e s the s o l u t i o n s used to v e r i f y o n e - d i m e n s i o n a l s t e a d y - s t a t e f l o w i n homogeneous and heterogeneous porous.media and 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 f l o w i n homogeneous porous media. R e s u l t s of both the n u m e r i c a l and a n a l y t i c a l a n a l y s e s are p r e s e n t e d and compared. C o n s t a n t mesh s p a c i n g i s used f o r a l l s t e a d y - s t a t e a n a l y s e s . 3.3.2.1. One-dimensional Flow, Homogeneous Porous Media The a c c c u r a c y o f the o n e - d i m e n s i o n a l homogeneous s t e a d y - s t a t e model can be e a s i l y v e r i f i e d by i n s p e c t i o n , a c c o r d i n g to Darcy's law. The r e g i o n of f l o w , a h o r i z o n t a l l i n e , i s modeled w i t h the f o l l o w i n g boundary c o n d i t i o n s : 1 V p L x=0 x=l h L = 0.0 A c o n s t a n t g r a d i e n t i s e s t a b l i s h e d from x=0 to x = l . The model c o r r e c t l y p r e d i c t e d t h i s r e s u l t . 3.3.2.2. One-dimensional Flow, Heterogeneous Porous Media The c o r r e s p o n d i n g heterogeneous case i s more complex and r e q u i r e s an a n a l y t i c a l s o l u t i o n t o v e r i f y the r e s u l t s . The f l o w r e g i o n and boundary c o n d i t i o n s f o r t h i s s o l u t i o n are i d e n t i c a l 30 to those for the homogeneous case. The hydraulic conductivity at each point, x, within the region of flow i s represented by the following equation: K(x) = (1 + x ) 3 (25) The a n a l y t i c a l solution to Darcy's law in one dimension with K replaced by equation (25) i s : ( h , - h 0 ) (1+L)2 ( ( l + x ) 2 - l ) h(x) = h n + L U ? T -U (1+L)2-1 (1+x) (26) (Mase, 1984, pers. comm.) Figure 5 shows the results of both the a n a l y t i c a l (AN) solution and the f i n i t e element (FE) solution plotted as hydraulic head versus distance. The f i n i t e element model was run three times, decreasing the mesh d i s c r e t i z a t i o n size for each run. The smallest mesh d i s c r e t i z a t i o n , dx = .033, produced results almost i d e n t i c a l to those of the an a l y t i c a l solution. A l l hydraulic heads and distances are normalized values. 3.3.2.3. Two-dimensional Flow, Homogeneous Porous Media V e r i f i c a t i o n of the two-dimensional steady-state model involves comparison with an an a l y t i c a l solution developed for regional groundwater flow (T6th, 1962, 1963; a_s cited by Freeze and Cherry, 1979). The solution i s for a v e r t i c a l cross-section bounded on top by a sloping water table. It i s not possible to obtain an a n a l y t i c a l solution for a trapezoidal region, hence the region of flow, shown below, i s approximated by a rectangular region with head values on the actual water table projected onto the upper boundary of the rectangular region. Legend Q AN solution FE solutions: A Ax = .200 -\~ Ax = .067 X Ax = .033 ~B) ffi B BE BP EE) E3 0.8 0.9 NORMALIZED DISTANCE Figure 5. Analytical versus finite element solutions of the 1-D steady state groundwater flow equation for heterogeneous porous media. 32 .6 C *-x=0 0 x=S T h i s s o l u t i o n i s o n l y v a l i d f o r s m a l l v a l u e s o f a ( i . e . , g e n t l e s l o p e s ) . The b o u n d a r y c o n d i t i o n s f o r t h e f l o w r e g i o n a r e : h ( x ' z 0 ) = z 0 ~ t a n x f£(0 ,z) = = i^f-- 0 where - t a n a i s t h e s l o p e o f t h e w a t e r t a b l e . T 6 t h s o l v e d t h e f l o w e q u a t i o n w i t h t h e s e b o u n d a r y c o n d i t i o n s , u s i n g t h e s e p a r a t i o n o f v a r i a b l e s t e c h n i q u e . The s o l u t i o n i s : h(x z) = z - t a n o t ' s + 4tancrS £ cos((2m+l)Trx/S) # cosh((2m+l)Trz/S) 0 2 I T 2 m=0 (2m+l)2 cosh((2m+l)Trz0/S) (27) F i g u r e 6 A and B shows t h e r e s u l t s o f b o t h t h e a n a l y t i c a l and t h e f i n i t e e l e m e n t a n a l y s e s f o r a l i n e o f n o d e s l o c a t e d a t z=0.5, w h i c h i s a p p r o x i m a t e l y t h e c e n t e r o f t h e f l o w r e g i o n , and f o r two d i f f e r e n t v a l u e s o f s l o p e o f t h e w a t e r t a b l e , n a m e l y -.005 and - . 0 5 . N o t e t h a t t h e s c a l e o f t h e y - a x i s i n F i g u r e 6A i s much l a r g e r t h a n t h a t i n F i g u r e 6B. The f i n i t e e l e m e n t model was r u n w i t h b o t h a f i n e and a c o a r s e mesh. The f i n e mesh d i d n o t i m p r o v e t h e s o l u t i o n . The f i n i t e e l e m e n t s o l u t i o n s a p p r o x i m a t e t h e a n a l y t i c a l s o l u t i o n much more c l o s e l y f o r t h e c a s e w i t h t h e more g e n t l y d i p p i n g w a t e r t a b l e . T h i s i s l i k e l y a r e s u l t o f e r r o r i n t h e 33 <x>~! I I I | I I | r~ ' 1 I 0.0 0.! 0.2 0.3 0.4 0.5 0.6 0.7 0.B 0.9 1.0 NORMAL IZED DISTANCE Figure 6. Analytical versus finite element solutions of the 2-D steady state groundwater flow equation for homogeneous porous media. A. Slope = -.005 NORMALIZED HYDRAULIC HERD (XlCr 1 ) 35 a n a l y t i c a l s o l u t i o n f o r s t e e p l y d i p p i n g w a t e r t a b l e s . A n o t h e r s o u r c e o f e r r o r i n t h e a n a l y t i c a l s o l u t i o n i s i n c u r r e d d u r i n g t h e s u m m a t i o n o f t h e i n f i n i t e s e r i e s ( s e e e q u a t i o n ( 2 7 ) ) . H o w ever, i t i s q u i t e e v i d e n t t h a t t h e two m o d e l s a r e i n ag r e e m e n t . 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 f l o w i n a h e t e r o g e n e o u s medium was n o t v e r i f i e d d i r e c t l y f o r two r e a s o n s : 1) t h e a n a l y t i c a l s o l u t i o n i s l o n g and c o m p l e x ; and 2) t h e h e t e r o g e n e o u s p o r t i o n o f t h e m o d e l was v e r i f i e d i n t h e o n e - d i m e n s i o n a l c a s e . 3.3.3. V e r i f i c a t i o n o f T r a n s i e n t S o l u t i o n s A n a l y t i c a l s o l u t i o n s a r e a v a i l a b l e f o r v e r i f i c a t i o n o f o n e - d i m e n s i o n a l t r a n s i e n t f l o w i n homogeneous and h e t e r o g e n e o u s p o r o u s m e d i a . I t i s assumed t h a t v e r i f i c a t i o n o f t w o - d i m e n s i o n a l f l o w i n t h e s t e a d y - s t a t e a n a l y s i s s u f f i c e s f o r t h a t i n t h e t r a n s i e n t a n a l y s i s a l s o , p r o v i d i n g t h e t r a n s i e n t p o r t i o n o f t h e m o del c a n be v e r i f i e d i n t h e o n e - d i m e n s i o n a l s i t u a t i o n . 3.3.3.1. O n e - d i m e n s i o n a l F l o w , Homogeneous P o r o u s M e d i a Two a n a l y t i c a l s o l u t i o n s were p r o g r a m e d w h i c h d e s c r i b e o n e - d i m e n s i o n a l , homogeneous, t r a n s i e n t f l o w . The f i r s t , a d a p t e d f r o m C a r s l a w and J a e g e r ( 1 9 5 9 , p . 5 8 - 6 2 ) , m o d e l s t h e f o l l o w i n g f l o w r e g i o n , b o u n d a r y and i n i t i a l c o n d i t i o n s : h n — r — z = 0 h ( 0 , t ) = 0.0 h. (~,t) = 1.0 hi-i-z=«> h. ( z , 0 ) = 1.0 The s o l u t i o n i s : 36 h(z,t) = hn - (hn-h.)-erf ( __) (2.{(t.K)/Ss}i^) ( 2 8 ) For the f i n i t e element model, the upper one kilometer of the flow region i s f i n e l y d i s c r e t i z e d ; s p a t i a l increments increase exponentially from one kilometer to several hundreds of kilometers. If the decay of hydraulic head were to reach the lower boundary of the system, equation (28) would no longer be v a l i d . For t h i s reason, only the upper one kilometer of the region i s examined and i t i s ensured that decay of hydraulic head never reaches the lowest node of the mesh. The results of both the a n a l y t i c a l and f i n i t e element models are plotted in Figure 7. The match i s excellent. A second one-dimensional a n a l y t i c a l solution, a form of the Terzaghi solution (Scott, 1963), can also be used to v e r i f y the f i n i t e element model. The flow region, boundary and i n i t i a l conditions for t h i s problem are: 0 \ h 0 (0,t) = 0.0 x=0 x=l h L ( l , t ) = 1.0 h. (x,0) = 0.0 The solution equation i s long and i s not shown here. Figure 8 i s a graph of hydraulic head versus distance plotted at several times for both solutions. Again, the two models show an excellent match. 3.3.3.2. One-dimensional Flow, Heterogeneous Porous Media A mathematical solution for one-dimensional flow in a multilayer column was developed by Tdth and M i l l a r (1983). They 37 Legend NORMAL I ZED HYDRAULIC HEAD Figure 7. Analytical versus finite element solutions of the 1-D transient groundwater flow equation for vertical flow in homogeneous porous media. Figure.8. Analytical versus finite element solutions of the 1-D transient groundwater flow equation for horizontal flow in homogeneous porous media. 39 computed t r a n s i e n t h y d r a u l i c heads a t the base o f a 5 - l a y e r column u s i n g an adapted v e r s i o n of the o n e - d i m e n s i o n a l heat f l o w e q u a t i o n f o r a m u l t i l a y e r column p r e s e n t e d by C a r s l a w and Jae g e r (1959). Development of the a n a l y t i c a l s o l u t i o n i s shown i n the appendix of Tdth and M i l l a r (1983). The a n a l y t i c a l s o l u t i o n i s d e s i g n e d s p e c i f i c a l l y f o r p r e d i c t i o n o f t r a n s i e n t h y d r a u l i c head v a l u e s a t the base of the Devonian I a q u i f e r (Tdth, 1978, p.808), which c o m p r i s e s the Keg R i v e r , E r n e s t i n a - C h i n c h a g a and G r a n i t e Wash F o r m a t i o n s of the Red E a r t h r e g i o n , n o r t h e r n A l b e r t a . The parameters i n p u t to the model are l i s t e d i n Table I . The r e g i o n of f l o w , d i m e n s i o n s of the column, boundary, and i n i t i a l c o n d i t i o n s a r e : ZQ=1363m Unit 5 137m h 0 ( z 0 ) , t ) = 1363 m 518m Unit 4 L 3: h L ( z L , t) / 3 z = 0 183m Unit 3 h. l (z,0) = 1863 m 480m Unit 2 Unit 1 40 UNIT NUMBER 5 4 3 2 THICKNESS (M) 137 518 183 480 45 K (M/S) 1x10 * to l x l 0 ~ J 1 l x l 0 I a 1x10 * to 1X10-} 4 4 x l 0 _ / (M & 1) 5 X 1 0 " A l x l 0 ~ 4 l x l 0 ~ 4 l x l 0 ~ 4 l x l 0 " 4 Table I: Hydraulic parameters of hydrogeologic units used in one-dimensional flow computations in the Red Earth region. Adapted from Toth and M i l l a r (1983). Results of the a n a l y t i c a l solution are presented as the time required for hydraulic head values to att a i n a spe c i f i e d r e l a t i v e hydraulic head adjustment, R , at the base of the 9 n column: Ah an an Ah T h0 - h n Ah-r (29) h = h0 = T = where: Ah^ = ultimate head change at time = T transient hydraulic head at base of column at time = t i n i t i a l hydraulic head (sta t i c conditions) time to reach steady-state Since the hydraulic conductivities, K, of the two aquitards (layers 2 and 4) are not well-known, the analysis was repeated for combinations of several values within the determined ranges of K. The conductivity values for the other layers were kept constant. Results are shown in T6th and M i l l a r (1983, Table 2). Three sets of values of K were chosen to compare to the f i n i t e element model: a) K 2 = l x 10~ 9, = 1 x 10~ 9; b) K 2 = 1 x 10 1 0 , K 4 = 1 x 10" 1 1; and c) K 2 = 1 x 10~ 1 4, = 1 10 -11 41 m/s. P a r a m e t e r s , boundary and i n i t i a l c o n d i t i o n s , and f l o w r e g i o n d i m e n s i o n s i d e n t i c a l to those used by Toth and M i l l a r (1983) were i n p u t to the f i n i t e element model. The r e s u l t a n t h y d r a u l i c head v a l u e s were c o n v e r t e d to r e l a t i v e h y d r a u l i c head adjustment (R _ ) . Due to the n a t u r e of the f i n i t e element a l l model, time to reach a s p e c i f i e d R a n , f o r example 10%, can not be c a l c u l a t e d e x a c t l y . T h e r e f o r e , i n s t e a d o f comparing t i m e s , the r e l a t i v e head adju s t m e n t s c a l c u l a t e d by the f i n i t e element model were compared to t h o s e of the a n a l y t i c a l s o l u t i o n at the t i m e s c a l c u l a t e d by Toth and M i l l a r (1983). The r e s u l t s are p r e s e n t e d i n Table I I . The s o l u t i o n s agree to w i t h i n 2% of the f o r a l l cases c o n s i d e r e d . a n 3.3.4. C o n c l u s i o n s S o l u t i o n s g e n e r a t e d by the f i n i t e element model f o r both s t e a d y - s t a t e and t r a n s i e n t f l o w problems i n homogeneous and heterogeneous porous media are compared to s e v e r a l a n a l y t i c a l s o l u t i o n s . In a l l c a s e s , the f i n i t e element model s o l u t i o n s match w e l l w i t h the v a r i o u s a n a l y t i c a l s o l u t i o n s ; no d i s c r e p a n c i e s were noted. The a u t h o r c o n c l u d e s , from the r e s u l t s p r e s e n t e d i n t h i s s e c t i o n , t h a t the f i n i t e element model programed f o r t h i s r e s e a r c h p r o v i d e s a c c u r a t e r e s u l t s f o r both one and 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 and t r a n s i e n t f l o w i n h e t e r o g e n e o u s , l a y e r e d , porous media. 42 C a s e No. K2 (m/s) 1 x 10 K4 (m/s) 1 x 10 -9 -9 +2 Time f o r R g n ( y r s ) 2 x 10 R f o r AN s o l . .000 R a„ f o r FE s o l . .004 a n Time f o r R ( y r s ) 5.1 x 10 R f o r AN s o l . .100 R a„ f o r FE s o l . .095 a n +2 Time f o r R ( y r s ) 8.4 x 10 R f o r AN s o l . .250 R a n f o r FE s o l . .243 a H Time f o r R ( y r s ) 1.53 x 10 R f o r AN s o l . .500 R a„ f o r FE s o l . .502 Time f o r R ( y r s ) 2.73 x 10 R f o r AN s o l . .750 R a„ f o r FE s o l . .768 a n Time f o r R ( y r s ) 4.18 x 10 R__ f o r AN s o l . .900 f o r FE s o l . .906 a n +2 +3 +3 +3 Time f o r R ( y r s ) 1 x 10 R f o r AN s o l . 1.00 R a f o r FE s o l . .998 a n +4 -10 -11 +4 1 x 10 1 x 10 1 x 10 . 000 . 004 3.28 x 10 .100 .111 +4 5.8 x 10 . 250 .249 +4 1.22 x 10 . 500 .514 2.25 x 10 .750 .760 3.77 x 10 .900 .918 +5 +5 +5 1 x 10 1.00 .999 +6 -14 -11 +6 1 x 10 1 x 10 4 x 10 .000 .002 1.1 x 10 .100 .094 +7 1.85 x 10 .250 .250 +7 3.3 x 10 .500 .502 5.8 x 10 .750 .763 9.2 x 10 . 900 .911 +7 +7 +7 2 x 10 1.00 .997 +8 T a b l e I I : C o m p a r i s o n o f r e l a t i v e head a d j u s t m e n t s b e t w e e n t h e a n a l y t i c a l s o l u t i o n and t h e f i n i t e e l e m e n t s o l u t i o n f o r o n e - d i m e n s i o n a l h e t e r o g e n e o u s t r a n s i e n t f l o w i n t h e Red E a r t h r e g i o n . 3.4. N u m e r i c a l P e r f o r m a n c e The p u r p o s e o f t h i s s e c t i o n i s t o show how t h e model r e s p o n d s t o t h e t i m e - s t e p p i n g p r o c e d u r e d e v e l o p e d i n t h e c o m p u t e r p r o g r a m and t o d i s c u s s t h e r e s p o n s e o f h y d r a u l i c head a t n o d e s n e a r b o u n d a r i e s b e t w e e n l a y e r s o f c o n t r a s t i n g h y d r a u l i c c o n d u c t i v i t y . A l s o i n c l u d e d h e r e a r e t h e r e s u l t s o f t h e f i n i t e e l e m e n t mesh s e n s i t i v i t y a n a l y s i s . 43 3.4.1. Time s t e p p e r f o r m a n c e A l l t r a n s i e n t s i m u l a t i o n s b e g i n w i t h a s m a l l i n i t i a l t i m e s t e p o f l e s s t h a n one d a y . D u r i n g t h e f i r s t few t i m e s t e p s t h e s o l u t i o n o s c i l l a t e s a t n o d e s n e a r t h e u p p e r b o u n d a r y i n r e s p o n s e t o t h e s u d d e n d e c r e a s e i n h y d r a u l i c head t h e r e . Once t h e o s c i l l a t i o n s have s u b s i d e d , t h e l a r g e s t c h a n g e i n h y d r a u l i c head a t t h e n o d e s i s d e t e r m i n e d f o r e a c h t i m e s t e p . I f t h e c h a n g e a t any one node e x c e e d s t h r e e p e r c e n t , t h e c a l c u l a t i o n s f o r t h a t t i m e s t e p a r e r e p e a t e d u s i n g a s m a l l e r t i m e i n c r e m e n t . The h y d r a u l i c head c h a n g e s r e m a i n w e l l b e l o w t h r e e p e r c e n t f o r most o f t h e t i m e s t e p s . F o u r c a s e s were c o n s i d e r e d i n o r d e r t o e v a l u a t e t h e r e s p o n s e o f t i m e s t e p s t o c h a n g e s i n h y d r a u l i c c o n d u c t i v i t y ( K ) : 1) a h i g h - K l a y e r o v e r l y i n g a low-K l a y e r ; 2) a low-K l a y e r o v e r l y i n g a h i g h - K l a y e r ; 3) a h i g h - K l a y e r b e t w e e n two low-K l a y e r s ; and 4) a low-K l a y e r b e t w e e n two h i g h - K l a y e r s . A l l l a y e r s a r e h o r i z o n t a l . F i g u r e 9 i l l u s t r a t e s t h e t r a n s i e n t b e h a v i o u r o f h y d r a u l i c h e a d s a l o n g a v e r t i c a l s e c t i o n f o r t h e f i r s t c a s e . The h i g h - K l a y e r i s 0.8 km t h i c k and o v e r l i e s a 0.2 km t h i c k l a y e r w h i c h h a s an h y d r a u l i c c o n d u c t i v i t y t h r e e o r d e r s - o f - m a g n i t u d e l o w e r . The h y d r a u l i c head i s r e d u c e d f r o m 1.5 km t o 1.0 km a t t=0. The g r a p h i n d i c a t e s t h a t a l l h y d r a u l i c h e a d v a l u e s i n t h e h i g h - K l a y e r h a v e d e c a y e d by more t h a n 7 5 % a f t e r f i v e d a y s ; h e a d v a l u e s i n t h e l o w -K l a y e r a r e j u s t b e g i n n i n g t o d e c a y a t t h i s t i m e . Once h y d r a u l i c head v a l u e s n e a r t h e b o u n d a r y b e t w e e n t h e two l a y e r s b e g i n t o c h a n g e , o s c i l l a t i o n s o c c u r a t n o d e s d i r e c t l y b e l o w t h i s b o u n d a r y ; h y d r a u l i c head v a l u e s a b o v e t h e b o u n d a r y HYDRRUL1C HERO (KM) Figure 9. The transient behaviour of hydraulic heads in a two-layer vertical section with a high-K layer overlying a low-K layer. 45 c o n t i n u e to decay. A f t e r t h r e e or f o u r days, v a l u e s of h y d r a u l i c head i n the low-K l a y e r b e g i n to decay, but a t a much slo w e r r a t e . Both l a y e r s show a g r a d u a l decay o f h y d r a u l i c head t h r o u g h t i m e . The e f f e c t on h y d r a u l i c head changes of a low-K l a y e r o v e r l y i n g a high-K l a y e r i s i l l u s t r a t e d i n F i g u r e 10. Both l a y e r s have the same t h i c k n e s s e s as i n case 1. The high-K l a y e r responds d i f f e r e n t l y i n t h i s c a s e . V e r t i c a l changes of h y d r a u l i c head are i n s t a n t a n e o u s , u n l i k e the g r a d u a l decay i n case 1. T h i s i s due to the l a r g e time s t e p s , which are a r e s u l t of the o v e r l y i n g low-K l a y e r . The slow r a t e of decay i n the low-K l a y e r a l s o s lows down the r a t e of decay i n the high-K l a y e r below. F i g u r e s 11 and 12 i l l u s t r a t e h y d r a u l i c head changes t h r o u g h time i n a t h r e e - l a y e r b a s i n of which 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 i d d l e l a y e r c o n t r a s t s w i t h t h a t of the a d j a c e n t l a y e r s by t h r e e o r d e r s - o f ~ m a g n i t u d e . The graphs behave s i m i l a r l y to those f o r the t w o - l a y e r c a s e . In F i g u r e 12, the top high-K l a y e r almost c o m p l e t e l y a d j u s t s to the new boundary c o n d i t i o n b e f o r e the low-K l a y e r below has begun to respond to the change. In F i g u r e 11, the bottom low-K l a y e r b e g i n s to a d j u s t i m m e d i a t e l y a f t e r the high-K l a y e r above b e g i n s to a d j u s t , due to the l a r g e time s t e p s . T h i s e f f e c t i s o n l y observed i f t h e r e i s a low-K l a y e r o v e r l y i n g a high-K l a y e r and i s not always observed f o r t h i s s i t u a t i o n . For the f i v e - l a y e r b a s i n s i m u l a t i o n , the time s t e p i s c o n t i n u a l l y i n c r e a s i n g a t a r a t e d e termined by the low h y d r a u l i c c o n d u c t i v i t y l a y e r s . The s i z e of the f i n a l time s t e p and the F i g u r e 10. The t r a n s i e n t b e h a v i o u r o f h y d r a u l i c heads i n a t w o - l a y e r v e r t i c a l s e c t i o n w i t h a low-K l a y e r o v e r l y i n g a high - K l a y e r . HYDRAULIC HERD (KM) Figure 1 1 . The transient behaviour of hydraulic heads in a three-layer vertical section with a high-K layer between two low-K layers. Figure 12. The transient behaviour of hydraulic heads in a three-layer vertical section with a 1OWL-K layer between two high-K layers. 49 t i m e when s t e a d y s t a t e i s r e a c h e d a r e o f t h e same o r d e r - o f - m a g n i t u d e - m i l l i o n s o f y e a r s . 3.4.2. O s c i l l a t i o n s a t t h e b o u n d a r i e s As d i s c u s s e d i n s e c t i o n 3.2, o s c i l l a t i o n o f t h e s o l u t i o n o c c u r s a l o n g t h e u p p e r b o u n d a r y due t o t h e i n s t a n t a n e o u s d e c r e a s e i n he a d v a l u e a t t h e s e n o d e s . A f t e r t h e f i r s t few t i m e s t e p s , t h e o s c i l l a t i o n s dampen as t h e y move down t h r o u g h t h e b a s i n and he a d v a l u e s n e a r t h e s u r f a c e b e g i n t o d e c a y . O s c i l l a t i o n s a r e a l s o o b s e r v e d d u r i n g t h e t r a n s i e n t e v e n t when h y d r a u l i c head c h a n g e s r e a c h t h e b o u n d a r y b e t w e e n an o v e r l y i n g h i g h - K l a y e r and an u n d e r l y i n g low-K l a y e r . These a l s o damp o u t a s t i m e p r o g r e s s e s . T h e r e a r e no o s c i l l a t i o n s o b s e r v e d a t t h e b o u n d a r y when t h e low-K l a y e r o v e r l i e s t h e h i g h - K l a y e r , b e c a u s e head c h a n g e s o c c u r v e r y r a p i d l y i n t h e h i g h - K l a y e r . 3.4.3. Mesh s e n s i t i v i t y a n a l y s i s The s e n s i t i v i t y o f h y d r a u l i c head v a l u e s t o c h a n g e s i n t h e d i s c r e t i z a t i o n o f t h e f l o w r e g i o n was d e t e r m i n e d i n o r d e r t o s e l e c t t h e b e s t f i n i t e e l e m e n t mesh f o r c o m p u t i n g a c c u r a t e head v a l u e s . The mesh d e s i g n e d f o r t h i s m o d e l h a s v a r y i n g i n c r e m e n t s i n b o t h t h e x- and z - d i r e c t i o n s . I n t h e x - d i r e c t i o n , t h e x i n c r e m e n t d e c r e a s e s t o w a r d s t h e s h a l l o w end o f t h e b a s i n , b e c a u s e t h i s i s where h y d r a u l i c head c h a n g e s o c c u r f i r s t . I n c r e m e n t s i n t h e z - d i r e c t i o n a r e s m a l l e r i n low-K l a y e r s where c h a n g e s i n h y d r a u l i c head a r e much s l o w e r . D e l t a z was h e l d c o n s t a n t f o r a n a l y s i s o f t h e h o r i z o n t a l d i s c r e t i z a t i o n and v i c e v e r s a . 50 Most o f t h e c h a n g e i n h y d r a u l i c head o c c u r s v e r t i c a l l y , and t h e r e f o r e , c h a n g e s i n t h e s i z e o f t h e x i n c r e m e n t d i d n o t r e f i n e t h e s o l u t i o n s by a s i g n i f i c a n t amount. The c o a r s e s t mesh u s e d Ax r a n g i n g f r o m 38 t o 150 km. The s o l u t i o n s o b t a i n e d u s i n g t h i s mesh were r e f i n e d by i n c r e a s i n g t h e number o f n o d e s by 4 f r o m 9 t o 13. T h i s r e s u l t e d i n AX v a l u e s r a n g i n g f r o m 25 t o 100 km. F u r t h e r r e f i n i n g o f t h e mesh i n t h e x - d i r e c t i o n d i d n o t i n c r e a s e t h e p r e c i s i o n o f t h e s o l u t i o n . The c o a r s e s t mesh o f 19 nodes u s e d i n t h e z - d i r e c t i o n , w i t h Az r a n g i n g f r o m 0.1 km i n t h e low-K l a y e r s t o 0.5 km i n t h e h i g h - K l a y e r s , r e s u l t e d i n a f a i r l y p r e c i s e s o l u t i o n a t most o f t h e n o d e s . R e f i n i n g o f t h e mesh by h a l v i n g Az i n a l l l a y e r s , r e s u l t e d i n o n l y a few n o d e s s h o w i n g a s i g n i f i c a n t c h a n g e o f g r e a t e r t h a n one p e r c e n t i n t h e h y d r a u l i c h e a d v a l u e . I t was c o n c l u d e d t h a t a s l i g h t l y r e f i n e d v e r s i o n o f t h e c o a r s e r mesh was s u f f i c i e n t f o r p r e c i s e s o l u t i o n s and no f i n e r mesh s i z e s were a n a l y z e d . The r e s u l t i n g mesh u s e d f o r s i m u l a t i o n s c o n s i s t s o f 13 n o d e s i n t h e x - d i r e c t i o n and 22 nodes i n t h e z - d i r e c t i o n H y d r a u l i c head v a l u e s o f n o d e s i n t h e t o p r i g h t c o r n e r o f t h e f l o w r e g i o n , where t h e e l e m e n t s become s m a l l , w i l l o s c i l l a t e , o n c e s t e a d y s t a t e i s r e a c h e d a t t h e s e n o d e s , i f t h e a s p e c t r a t i o s o f t h e e l e m e n t s become t o o l a r g e . O s c i l l a t i o n s a r e o b s e r v e d i n s o l u t i o n s u s i n g t h e f i n e r A\z mesh o f 37 n o d e s . D e c r e a s i n g t h e number o f nodes i n t h e z - d i r e c t i o n a l s o d e c r e a s e s t h e a s p e c t r a t i o s o f t h e e l e m e n t s . No o s c i l l a t i o n s a r e o b s e r v e d a f t e r s t e a d y s t a t e i s r e a c h e d i n s o l u t i o n s i n w h i c h t h e mesh i s 22 n o d e s by 13 n o d e s . 51 CHAPTER 4 SENSITIVITY ANALYSIS 4.1. I n t r o d u c t i o n The s e n s i t i v i t y a n a l y s i s documented i n t h i s c h a p t e r i s a c o m b i n a t i o n of both a g e n e r i c and s i t e - s p e c i f i c a n a l y s i s . The g e n e r i c model used f o r the s e n s i t i v i t y s t u d y v e r y c l o s e l y resembles the Red E a r t h r e g i o n of A l b e r t a , Canada, but i s s i m p l i f i e d to a l l o w c a l c u l a t i o n s and comparisons to be c a r r i e d out more e a s i l y . The v a l u e s of the h y d r o g e o l o g i c parameters t h a t are used as a s t a r t i n g p o i n t f o r the s i m u l a t i o n s are based on those r e p o r t e d by Toth and M i l l a r (1983). T h i s i s the ' s t a n d a r d ' data s e t . The e f f e c t s of s m a l l p e r t u r b a t i o n s of the s t a n d a r d data s e t on the time r e q u i r e d f o r complete adjustment and on the r a t e o f decay of h y d r a u l i c head v a l u e s w i t h i n the f l o w system, a r e p r e s e n t e d and d i s c u s s e d . The purpose of t h i s a n a l y s i s i s to determine the s e n s i t i v i t y o f the f i n i t e element model to s m a l l changes i n the n u m e r i c a l v a l u e s of s e v e r a l parameters. Those parameters t e s t e d a r e : h y d r a u l i c c o n d u c t i v i t y , a n i s o t r o p y , s p e c i f i c s t o r a g e , water t a b l e c o n f i g u r a t i o n , and b a s i n s i z e . For each s i m u l a t i o n , a l l c o n d i t i o n s remain the same except f o r the v a l u e of the one parameter b e i n g a n a l y z e d . The model used f o r the s e n s i t i v i t y a n a l y s i s i s a s i m p l i f i e d g e o l o g i c c r o s s - s e c t i o n o f the Red E a r t h r e g i o n (see F i g u r e 1, A-A'), r e f e r r e d to as the ' s t a n d a r d ' model. The model b a s i n i s 600 km i n l e n g t h and 3 km t h i c k a t the s o u t h w e s t e r n boundary. The s t r a t i g r a p h i c column i s d i v i d e d i n t o f i v e h y d r o g e o l o g i c 52 u n i t s , e a c h o f w h i c h e x t e n d a c r o s s t h e e n t i r e l e n g t h o f t h e b a s i n and t h i n t o t h e n o r t h - e a s t . T hese u n i t s a r e u n d e r l a i n by an i m p e r m e a b l e b a s e . E a c h u n i t i s homogeneous and i s o t r o p i c b u t a l l f i v e u n i t s h a v e c o n t r a s t i n g v a l u e s o f h y d r a u l i c c o n d u c t i v i t y . The s o u t h w e s t e r n and n o r t h e a s t e r n v e r t i c a l b o u n d a r i e s a r e g r o u n d w a t e r d i v i d e s a n d , h e n c e , i m p e r m e a b l e . The u p p e r b o u n d a r y i s a l i n e a r l y s l o p i n g w a t e r t a b l e ; l o c a l v a r i a t i o n s i n t o p o g r a p h y a r e c o n s i d e r e d n e g l i g i b l e f o r t h i s l a r g e s c a l e a n a l y s i s . V a l u e s o f a l l p a r a m e t e r s o f t h e model b a s i n a r e l i s t e d i n T a b l e I I I . I n a l l s i m u l a t i o n s , t h e i n i t i a l h ead a t time=0 i s assumed t o be 3.5 km f o r t h e e n t i r e b a s i n . The c h a n g e i n head a l o n g t h e u p p e r b o u n d a r y r a n g e s f r o m 0.5 km a t t h e deep edge o f t h e b a s i n t o 0.74 km a t t h e s h a l l o w edge. I n i t i a l c o n d i t i o n s w e r e s e l e c t e d on t h e b a s i s o f t h e e v o l u t i o n a r y s t a g e s o f t h e Red E a r t h r e g i o n a s d e t e r m i n e d by T o t h ( 1 9 7 8 ) . I n i t i a l l y , a h o r i z o n t a l l a n d s u r f a c e e x i s t s c r e a t i n g h y d r o s t a t i c f l u i d c o n d i t i o n s . G r a d u a l e r o s i o n a l d e e p e n i n g o f t h e v a l l e y s t o d e p t h s o f b e t w e e n 0.5 and 0.74 km i s r e p r e s e n t e d i n t h e m o d e l by an i n s t a n t a n e o u s d r o p i n head a t t h e w a t e r t a b l e . P r e s e n t r a t e s o f e r o s i o n o f t h e l a n d s u r f a c e a r e g r e a t e r t h a n c a l c u l a t e d r a t e s o f h y d r a u l i c head a d j u s t m e n t i n t h e low-K l a y e r s o f t h e b a s i n . I t i s q u i t e p r o b a b l e t h a t e r o s i o n r a t e s w ere e v e n h i g h e r i n t h e p a s t . T h e r e f o r e , i t w o u l d seem r e a s o n a b l e t o m o d el e r o s i o n as an i n s t a n t a n e o u s e v e n t . The n a t u r e o f t h e f i n i t e e l e m e n t c o m p u t e r p r o g r a m makes i t i m p r a c t i c a l t o m o d el e r o s i o n as a t r a n s i e n t e v e n t . 53 LAYER HYDRAULIC CONDUCTIVITY SPECIFIC LAYER THICKNESS (M/S) STORAGE NO. (KM) X-DIRECTION Z-DIRECTION (M ) IH 0.5 6 x 10~?. 6 x 10"7. 1 X 10"^ 2L 0.3 1 x 10 1 x 1 0 " i 1 x 1 0 ~ 4 3H 1.2 1 x 10~° 1 x 10"° 1 x 1 0 ~ 4 4L 0.8 1 x 10"*^ 1 x 10 " " J i 1 x 1 0 ~ 4 5H 0.2 1 x 1 0 ~ 5 1 x 1 0 ~ 5 5 x 1 0 ~ 4 B a s i n Length: 600. km B a s i n Depth (at SW boundary): 3. km S l o p e of Water T a b l e : .0004 Slope of Basement Rock: .004 Table I I I : Parameters used f o r the s t a n d a r d b a s i n model based on those of the Red E a r t h r e g i o n , A l b e r t a , Canada. F i g u r e 13 i l l u s t r a t e s the model b a s i n used f o r most of the s i m u l a t i o n s , i n d i c a t i n g the p o s i t i o n s of those nodes which were a n a l y z e d i n d e t a i l . Because the b a s i n i s v e r y l o n g and t h i n i t i s n e c e s s a r y to exaggerate the v e r t i c a l s c a l e f o r a l l b a s i n diagrams. The model b a s i n s used f o r the water t a b l e c o n f i g u r a t i o n and b a s i n s i z e s e n s i t i v i t y a n a l y s e s are v e r y s i m i l a r and t h e r e f o r e not i l l u s t r a t e d . F i f t e e n nodes were a n a l y z e d i n d e t a i l , t h r e e nodes i n each l a y e r . They are s i t u a t e d as c l o s e to the bottom of the l a y e r as p o s s i b l e w i t h o u t b e i n g on e i t h e r an e x t e r n a l boundary or a boundary between l a y e r s , and are l o c a t e d i n the r e c h a r g e , c e n t e r , and d i s c h a r g e a r e a s o f the f l o w system. The nodes are tagged ID (deep edge of b a s i n ) , 2M ( m i d - b a s i n ) , and 3S ( s h a l l o w edge of b a s i n ) i n each l a y e r . The l a y e r s are numbered from 1 ( b a s a l ) to 5 (top) and tagged H (high-K) or L ( l o w - K ) . With such a c o n f i g u r a t i o n , e f f e c t s on a l l p a r t s of the f l o w system can be s t u d i e d w i t h o u t l o o k i n g a t the e n t i r e d a t a s e t i n d e t a i l . s w N E l O O 2 0 0 3 0 0 4 0 0 5 0 0 BOO K M Figure 13. The standard basin model showing all node positions, layer positions, and the locations of nodes ID, 2M, and 3S. 55 The e f f e c t s o f c h a n g i n g t h e v a l u e o f e a c h p a r a m e t e r a r e d i s c u s s e d b e l o w . P l o t s o f h y d r a u l i c h e a d v e r s u s l o g t i m e were c o n s t r u c t e d f o r a l l f i f t e e n n o d e s i n e a c h o f t h e a n a l y s e s , b u t o n l y t h e most i n t e r e s t i n g p l o t s a r e i l l u s t r a t e d . A d j u s t m e n t t i m e s t o " s t e a d y s t a t e " o r t o " c o m p l e t e a d j u s t m e n t o f h y d r a u l i c h e a d " r e f e r t o t h o s e t i m e s a t w h i c h 99.7 p e r c e n t o f t h e t o t a l h e ad a d j u s t m e n t h a s o c c u r r e d a t a p a r t i c u l a r node i n t h e s y s t e m . T h i s s t r i c t d e f i n i t i o n o f t h e s t e a d y s t a t e l i m i t s i s r e q u i r e d b e c a u s e s t e a d y s t a t e i s a p p r o a c h e d a s y m p t o t i c a l l y b u t n e v e r a c t u a l l y r e a c h e d . 4.2. The M o d e l ' s S e n s i t i v i t y t o C h a n g e s i n H y d r a u l i c C o n d u c t i v i t y To d e t e r m i n e t h e m o d e l ' s s e n s i t i v i t y t o s m a l l c h a n g e s i n h y d r a u l i c c o n d u c t i v i t y , K, t h e v a l u e o f t h i s p a r a m e t e r was v a r i e d i n e a c h l a y e r , s e p a r a t e l y . S i m u l a t i o n s w e r e r u n w i t h v a l u e s o f K b o t h one o r d e r - o f - m a g n i t u d e h i g h e r and one o r d e r - o f m a g n i t u d e l o w e r t h a n t h e s t a n d a r d v a l u e l i s t e d i n T a b l e I I I . The f o l l o w i n g d i s c u s s i o n d e a l s f i r s t w i t h c h a n g e s o f K1H, K3H, and K5H i n t h e h i g h ~ K l a y e r s 1H, 3H, and 5H, r e s p e c t i v e l y , and s e c o n d l y , w i t h c h a n g e s o f K2L and K4L i n t h e low-K l a y e r s 2L and 4 L , r e s p e c t i v e l y . 4.2.1. L a y e r s w i t h h i g h v a l u e s o f h y d r a u l i c c o n d u c t i v i t y P e r t u r b a t i o n s o f t h e v a l u e o f K1H a f f e c t t h e a d j u s t m e n t t i m e s o f h y d r a u l i c h e a d s i n l a y e r s 1H and 2 L , b u t h a v e no e f f e c t on any o t h e r p a r t o f t h e f l o w s y s t e m . S i m i l a r l y , c h a n g i n g t h e v a l u e o f K3H a f f e c t s h y d r a u l i c head a d j u s t m e n t s i n l a y e r s 3H and 4L o n l y . S i m i l a r p a t t e r n s o f h y d r a u l i c head d e c a y o v e r t i m e a r e 56 o b s e r v e d i n b o t h t h e p e r t u r b e d h i g h - K l a y e r ( p - l a y e r ) and t h e o v e r l y i n g l o w - K l a y e r f o r t h e s i m u l a t i o n s i n w h i c h K1H i s v a r i e d (K1H s i m u l a t i o n s ) and t h e s i m u l a t i o n s i n w h i c h K3H i s v a r i e d (K3H s i m u l a t i o n s ) . Nodes l o c a t e d a t t h e s h a l l o w edge o f t h e b a s i n a p p e a r t o be most s e n s i t i v e t o c h a n g e s i n h y d r a u l i c c o n d u c t i v i t y . T h e s e r e s u l t s i n d i c a t e t h a t t h e l o w e r t h e v a l u e o f h y d r a u l i c c o n d u c t i v i t y i n t h e p e r t u r b e d l a y e r , t h e f a s t e r t h e r a t e o f h y d r a u l i c head a d j u s t m e n t d u r i n g e a r l y t i m e s ( s e e F i g u r e 14 A t o D ) . No d i f f e r e n c e i n t h e amount o f t i m e r e q u i r e d f o r h y d r a u l i c h e a d s t o r e a c h s t e a d y s t a t e i s a p p a r e n t f r o m t h e K1H s i m u l a t i o n s , i n w h i c h t h e b a s a l l a y e r i s p e r t u r b e d . H owever, t h e K3H s i m u l a t i o n s i n d i c a t e t h a t f o r a one o r d e r - o f - m a g n i t u d e d e c r e a s e f r o m t h e s t a n d a r d v a l u e o f K3H, t h e t i m e r e q u i r e d f o r c o m p l e t e a d j u s t m e n t o f h y d r a u l i c h e a d s a t n o d e s 3S i n l a y e r s 3H and 4L d e c r e a s e s by 3 t o 5 t i m e s ( i . e . , f r o m a p p r o x i m a t e l y .69 Ma t o b e t w e e n .14 and .21 M a ) . An i n c r e a s e f r o m t h e s t a n d a r d v a l u e o f K3H h a s no s i g n i f i c a n t a f f e c t on t h i s a d j u s t m e n t t i m e . I n t u i t i v e l y , one m i g h t e x p e c t a l o w e r i n g o f t h e v a l u e s o f K1H and K3H t o c a u s e a d e c r e a s e i n t h e r a t e o f h y d r a u l i c head a d j u s t m e n t ; q u i t e t h e o p p o s i t e i s o b s e r v e d h e r e . F a s t e r r a t e s o f head d e c a y a t e a r l y t i m e s a r e o b s e r v e d f o r l o w e r v a l u e s o f K i n l a y e r s IH and 3H p o s s i b l y b e c a u s e t h e l o w e r K v a l u e s d e c r e a s e t h e c o n t r a s t i n K b e t w e e n t h e p - l a y e r and t h e s u p e r j a c e n t low-K l a y e r . As h y d r a u l i c head d e c a y moves t h r o u g h t h e low-K l a y e r and a c r o s s t h e b o u n d a r y a t t h e s h a l l o w edge o f t h e b a s i n i n t o t h e h i g h - K l a y e r , l e s s r e s i s t a n c e i s e n c o u n t e r e d f o r l o w e r v a l u e s o f K i n t h e h i g h - K l a y e r and h e n c e d e c a y c a n o c c u r a t a Node 3S Layer 1H Shallow edge 9 0 1 3 1 1 - Node 3S —1 1 B Layer 2L Shallow edge 1 s / P-lo>.er:lH 102 103 10' 105 106 1 1 . Node 3S 1 1 1 c Layer 3H Shallow edge - 1/ 7 U P-l 0yer:3H 1 1 . Node 3S 1 I 1 D Layer 4 L Shallow edge 1/7 / 1/ / 3 Figure 14. Hydraulic head versus log time plots for perturbations of K in high-K layers. A,B: l i s KlH=6xlO-8; 2 is KlH=6xlO"7; 3 is KlH=6xlO"6.m/s. G,D,E: 1 is K3H=10-7; 2 is K3H=10"6; 3 is K3H=10"5 m/s. Curve legend 1 - low K 2 - standard K 3 - high K 58 g r e a t e r r a t e . The e f f e c t i s o b s e r v e d most s t r o n g l y a t n odes n e a r t h e s h a l l o w edge o f t h e b a s i n b e c a u s e o n c e a d j u s t m e n t has begun i n t h e h i g h - K l a y e r , h e a d v a l u e s b e g i n t o d e c a y l a t e r a l l y i n t h a t l a y e r , r e d u c i n g t h e s t r e s s a l o n g t h e b o u n d a r y i n t h e c e n t r a l and d e e p p a r t s o f t h e b a s i n b e f o r e head d e c a y f r o m above r e a c h e s i t . N o t e t h a t f o r n o d e s 3S i n l a y e r s 2L and 4L ( F i g u r e 14 B and D) h y d r a u l i c h e a d v a l u e s a r e a l l c h a n g i n g a t t h e same r a t e f o r a l l v a l u e s o f K1H and K3H u n t i l a d j u s t m e n t b e g i n s i n t h e r e s p e c t i v e l y s u b j a c e n t h i g h - K l a y e r s ( F i g u r e 14 A and C ) , c a u s i n g t h e r a t e s o f d e c a y a t t h e two n o d e s t o v a r y f r o m t h i s t i m e o n w a r d . H y d r a u l i c head v a l u e s a t b a s i n c e n t r a l n o des 2M i n l a y e r s 1H and 2L f o r K1H s i m u l a t i o n s , and b a s i n c e n t r a l nodes 2M i n l a y e r s 3H and 4L f o r K3H s i m u l a t i o n s , d e c a y a t a f a s t e r r a t e d u r i n g e a r l y t i m e s f o r l o w e r v a l u e s o f K i n t h e p e r t u r b e d l a y e r . A f t e r h y d r a u l i c head v a l u e s h a v e d e c a y e d by a p p r o x i m a t e l y 50%, t h e t r e n d r e v e r s e s . Node K i s shown a s an e x a m p l e i n F i g u r e 14 E. F o r e a c h o f t h e f o u r n o d e s , h y d r a u l i c head v a l u e s r e a c h s t e a d y s t a t e a t a p p r o x i m a t e l y t h e same t i m e i n a l l t h r e e c a s e s . H y d r a u l i c head v a l u e s n e a r t h e deep edge o f t h e b a s i n show s i m i l a r r a t e s o f a d j u s t m e n t and r e q u i r e a p p r o x i m a t e l y t h e same l e n g t h o f t i m e t o r e a c h s t e a d y s t a t e . P e r t u r b a t i o n s o f K5H o n l y a f f e c t h y d r a u l i c head d e c a y i n l a y e r 5H. A one o r d e r - o f - m a g n i t u d e i n c r e a s e i n t h e v a l u e o f K5H r e s u l t s i n a p p r o x i m a t e l y a one o r d e r - o f - m a g n i t u d e d e c r e a s e i n t h e h y d r a u l i c head a d j u s t m e n t t i m e . 59 4.2.2. L a y e r s w i t h l o w v a l u e s o f h y d r a u l i c c o n d u c t i v i t y The f l o w s y s t e m a p p e a r s t o be much more s e n s i t i v e t o p e r t u r b a t i o n s o f h y d r a u l i c c o n d u c t i v i t y i n t h e l e s s p e r m e a b l e l a y e r s , 2L and 4 L , t h a n i n t h e h i g h - K l a y e r s I H , 3H, and 5H. S i n c e t h e r e s u l t s o f t h e s i m u l a t i o n s i n w h i c h t h e v a l u e o f K2L i s v a r i e d a r e v e r y s i m i l a r t o t h o s e i n w h i c h t h e v a l u e o f K4L i s v a r i e d , t h e f o l l o w i n g d i s c u s s i o n a p p l i e s t o b o t h s e t s o f r e s u l t s . The most s e n s i t i v e l a y e r s t o v a r i a t i o n s o f t h e v a l u e o f K2L a r e l a y e r s IH and 2L; t h e most s e n s i t i v e t o v a r i a t i o n s o f K4L a r e l a y e r s 3H and 4L. P e r t u r b a t i o n s o f t h e v a l u e o f K4L a l s o h a ve s m a l l e f f e c t s on n o d e s i n l a y e r s IH and 2L; t h i s w i l l be d i s c u s s e d l a t e r . C h a n g i n g t h e v a l u e o f K a f f e c t s b o t h t h e r a t e o f d e c a y and t h e t i m e r e q u i r e d f o r c o m p l e t i o n o f a d j u s t m e n t o f h y d r a u l i c h e a d s i n b o t h t h e p e r t u r b e d l a y e r and t h e s u b j a c e n t h i g h - K l a y e r . A one o r d e r - o f - m a g n i t u d e d e c r e a s e i n t h e v a l u e o f K2L r e s u l t s i n an i n c r e a s e o f up t o one o r d e r - o f - m a g n i t u d e i n t h e t i m e r e q u i r e d f o r h y d r a u l i c h e a d s t o r e a c h s t e a d y s t a t e ( e g . an i n c r e a s e f r o m a p p r o x i m a t e l y 7 Ma f o r K2L = l x l 0 ~ * 1 3 m/s t o a p p r o x i m a t e l y 69 Ma f o r K2L = l x l 0 - 1 4 m/s); a one o r d e r - o f - m a g n i t u d e d e c r e a s e i n K4L r e s u l t s i n an i n c r e a s e o f b e t w e e n one and two o r d e r s - o f - m a g n i t u d e , d e p e n d i n g on t h e n o d a l l o c a t i o n w i t h i n t h e l a y e r , i n t h e t i m e r e q u i r e d f o r c o m p l e t e head a d j u s t m e n t ( e g . an i n c r e a s e f r o m a p p r o x i m a t e l y 6,900 y e a r s f o r K4L = l x l 0 " 1 0 m/s t o a p p r o x i m a t e l y 137,000 y e a r s f o r K4L = l x l 0 _ 1 1 m/s a t node ID i n l a y e r 4L) (see F i g u r e 15 A t o D ) . The f o l l o w i n g g e n e r a l t r e n d s a r e a l s o n o t e d : a) f o r K2L s i m u l a t i o n s , t h e r a t e o f d e c a y o f h y d r a u l i c h e a d s i s much s l o w e r Figure 15. Hydraulic head versus log time plots for Curve legend perturbations of K in low-K layers. 1 - low K A,B,C: 1 is K2L=10"15; 2 is K2L=10"14; 2 - standard K 3 is K2L=10"13 m/s. 3 " h l * g h K D,E: 1 is K4L=10"12; 2 is K4L=10_11; 3 is K4L=10"10 m/s. 61 a t the s h a l l o w edge o f the b a s i n than a t the deeper edge, e s p e c i a l l y a t e a r l y t i m e s (compare F i g u r e 15 A w i t h 15 B ) ; b) h y d r a u l i c head decay r a t e s decrease w i t h a decrease i n v a l u e s of K2L and K4L, e s p e c i a l l y d u r i n g the e a r l y p e r i o d s of head adjustment and near the s h a l l o w e r edge of the b a s i n (see F i g u r e 15 B and D); and c) f o r K2L s i m u l a t i o n s , h y d r a u l i c head decay r a t e s a re much g r e a t e r i n the low-K l a y e r ( l a y e r 2L) than i n the high-K l a y e r ( l a y e r IH) a t e a r l y times (compare F i g u r e 15 C w i t h 15 B) . The d e c rease i n the r a t e of adjustment of h y d r a u l i c heads a t node 3S, l a y e r 2L, f o r cases 1 and 2 o f the K2L s i m u l a t i o n ( F i g u r e 15 C) o c c u r s a t a p p r o x i m a t e l y the same time t h a t h y d r a u l i c head decay b e g i n s i n l a y e r IH below ( F i g u r e 15 B ) . The o n s e t o f head decay i n l a y e r IH appears to cause a decrease i n the r a t e of head decay i n l a y e r 2L a t node 3S. T h i s e f f e c t o n l y o c c u r s near the s h a l l o w edge o f the b a s i n where h y d r a u l i c head adjustment f i r s t b e g i n s w i t h i n the l a y e r ; i t does not oc c u r at nodes ID and 2M o f l a y e r 2L because head decay a t s u b j a c e n t nodes i n l a y e r IH has e i t h e r begun p r e v i o u s to or a t a p p r o x i m a t e l y the same time as head decay a t nodes ID and 2M, due to the f a s t e r r a t e of h o r i z o n t a l decay i n the high-K l a y e r IH. T h i s e f f e c t does not occur i n K4L s i m u l a t i o n s p o s s i b l y because h y d r a u l i c head decay r a t e s i n the u n d e r l y i n g high-K l a y e r 3H a r e much g r e a t e r than those i n l a y e r IH. When the v a l u e of K4L becomes v e r y low ( i . e . , l e s s than the s t a n d a r d v a l u e o f the model) i t r e s u l t s i n an i n c r e a s e i n the time r e q u i r e d f o r complete adjustment o f h y d r a u l i c heads i n both l a y e r s 3H and 4L. T h i s , i n t u r n , causes a s l i g h t decrease i n 62 the i n i t i a l decay r a t e of h y d r a u l i c heads i n both l a y e r s IH and 2L below. However, the h y d r a u l i c head decay r a t e r a p i d l y i n c r e a s e s w i t h t i m e , c a u s i n g no change i n the time r e q u i r e d f o r complete head adjustment i n l a y e r s IH and 2L. An example i s shown i n F i g u r e 15 E. T h i s e f f e c t c o u l d become s i g n i f i c a n t f o r v a l u e s of K4L lower than those s i m u l a t e d h e r e . 4.3. The Model's S e n s i t i v i t y t o Changes i n A n i s o t r o p y I s o t r o p i c c o n d i t i o n s are assumed to e x i s t i n a l l f i v e l a y e r s o f the s t a n d a r d model. To d etermine the e f f e c t s of v a r y i n g degrees of a n i s o t r o p y on the r a t e of h y d r a u l i c head decay and the time r e q u i r e d f o r complete adjustment of h y d r a u l i c heads, v a l u e s of h y d r a u l i c c o n d u c t i v i t y were v a r i e d f i r s t i n the x d i r e c t i o n and then i n the z d i r e c t i o n i n the two low-K l a y e r s t o g e t h e r and the t h r e e high-K l a y e r s t o g e t h e r . For each s i m u l a 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 was v a r i e d over t h r e e o r d e r s - o f - m a g n i t u d e . A n i s o t r o p y v a l u e s of up to 100:1 are r e a l i s t i c on a r e g i o n a l s c a l e , e s p e c i a l l y i n l a y e r e d heterogeneous systems (Freeze and C h e r r y , 1979, p. 3 4 ) . V a l u e s of a n i s o t r o p y of 1000:1 may be u n r e a l i s t i c but h e l p to c o n c e p t u a l i z e and i l l u s t r a t e e f f e c t s . 4.3.1. L a y e r s w i t h low v a l u e s of h y d r a u l i c c o n d u c t i v i t y V a r y i n g the v a l u e of K x (the h y d r a u l i c c o n d u c t i v i t y i n the x - d i r e c t i o n ) i n the low-K l a y e r s , 2L and 4L, has a b s o l u t e l y no e f f e c t on the r e s u l t i n g h y d r a u l i c head adjustment t i m e s . T h i s can be e x p l a i n e d by the dominance of v e r t i c a l f l o w i n the low-K l a y e r s and the d o m i n a n t l y v e r t i c a l changes i n h y d r a u l i c head i n the f l o w system. S m a l l p e r t u r b a t i o n s of the v a l u e of K are not 63 e xpected to a f f e c t these v e r t i c a l f o r c e s s i g n i f i c a n t l y . V a r y i n g the v a l u e of K (the h y d r a u l i c c o n d u c t i v i t y i n the z - d i r e c t i o n ) i n l a y e r s 2L and 4L a f f e c t s the h y d r a u l i c head adjustment t i m e s t h r o u g h o u t the e n t i r e , f l o w system. For a l l f i f t e e n nodes e v a l u a t e d , the l a r g e r the r a t i o K :K , the sooner s t e a d y s t a t e f l o w i s a t t a i n e d . In g e n e r a l , f o r a one o r d e r - o f - m a g n i t u d e i n c r e a s e i n K :K , the time r e q u i r e d f o r z x h y d r a u l i c head v a l u e s to reach s t e a d y s t a t e d e c r e a s e s by one o r d e r - o f - m a g n i t u d e o r more (see F i g u r e 16 A to D). For example, s t e a d y s t a t e i s reached a f t e r a p p r o x i m a t e l y 3,500 y e a r s f o r K = 100 K and a f t e r a p p r o x i m a t e l y 270 y e a r s f o r K = 1000 K at A. X node 2M i n l a y e r 3H. I n c r e a s i n g the v a l u e o f K i s e q u i v a l e n t to d e c r e a s i n g the l a y e r t h i c k n e s s f o r a c o r r e s p o n d i n g i s o t r o p i c s e c t i o n , and f o r a t h i n n e r l a y e r one would expect the time r e q u i r e d f o r h y d r a u l i c head adjustment to be l e s s . There are s e v e r a l p a t t e r n s to note i n the diagrams o f F i g u r e 16: a) f o r some nodes, such as 3S i n l a y e r s IH and 2L i n F i g u r e 16 A and B, r e s p e c t i v e l y , t h e r e i s a g e n e r a l d e c r e a s e i n the r a t e of decay of h y d r a u l i c heads w i t h a decrease i n the r a t i o K :K , e s p e c i a l l y d u r i n g e a r l y p e r i o d s of head a d j u s t m e n t ; Z X b) h y d r a u l i c head v a l u e s a t nodes i n high-K l a y e r s g e n e r a l l y decay a t a g r e a t e r r a t e than those a t nodes i n low-K l a y e r s , e s p e c i a l l y a t l a t e r t i m e s (compare nodes 3S i n l a y e r s IH and 2L w i t h nodes 2M i n l a y e r s 3H and 4L i n F i g u r e 16); and c) i n some i n s t a n c e s , as i n case 1 of F i g u r e 16 B, the onset of h y d r a u l i c head adjustment a t a node i n a high-K l a y e r (node 3S, l a y e r IH, case 1, F i g u r e 16 A) causes a d e l a y i n the head adjustment of nodes d i r e c t l y above i n a low-K l a y e r (node 3S, l a y e r 2L, case 10J 10 6 1 I Node 3S layer 2L Shallow edge I — i i - i r r . Node 2M Layer 3H Mid-Basin 2 / \ / C Figure 16. Hydraulic head versus log time plots for K >K in low-K layers. z x J 1 is Kz=Kx 4 is K =1000K 2 is K =10K 3 is K =100K ; Z A Curve legend 1 2 3 4 isotropic increasing K 65 1, F i g u r e 16 B ) . T h i s seems t o o c c u r o n l y when t h e i n i t i a l r a t e o f head d e c a y i n t h e h i g h - K l a y e r i s v e r y s l o w . 4.3.2. L a y e r s w i t h h i g h v a l u e s o f h y d r a u l i c c o n d u c t i v i t y The b a s i n m o d el i s much l e s s s e n s i t i v e t o c h a n g e s o f a n i s o t r o p y i n t h e h i g h - K l a y e r s , 1H, 3H, and 5H, t h a n i n t h e low-K l a y e r s d i s c u s s e d p r e v i o u s l y . V a r y i n g t h e v a l u e o f K i n t h e h i g h - K l a y e r s c a u s e s c h a n g e s i n t h e r a t e o f d e c a y o f h y d r a u l i c h e a d s i n l a y e r s 3H and 4 L , b u t n o t i n t h e t i m e r e q u i r e d f o r h y d r a u l i c h e a d v a l u e s t o r e a c h s t e a d y s t a t e ; i t h a s no e f f e c t on l a y e r s 1H, 2L, and 5H. Nodes 3S i n l a y e r s 3H and 4 L , l o c a t e d a t t h e s h a l l o w edge o f t h e b a s i n , show t h e g r e a t e s t s e n s i t i v i t y t o c h a n g e s o f K X : K Z i n t h e h i g h - K l a y e r s ( F i g u r e 17 A and B ) . The h i g h e r t h e r a t i o K :K_, t h e s l o w e r t h e i n i t i a l r a t e o f a d j u s t m e n t o f h y d r a u l i c h e a d s . A f t e r a p p r o x i m a t e l y 25% h y d r a u l i c head a d j u s t m e n t , t h e t r e n d r e v e r s e s and s t e a d y s t a t e i s r e a c h e d a t a p p r o x i m a t e l y t h e same t i m e i n a l l f o u r c a s e s . N o t e t h a t f o r node 3S, i n t h e low-K l a y e r 4 L , h y d r a u l i c head v a l u e s a r e a l l c h a n g i n g a t t h e same r a t e f o r a l l c a s e s u n t i l h ead a d j u s t m e n t b e g i n s i n t h e h i g h - K l a y e r 3H b e l o w , w h i c h c a u s e s t h e r a t e s o f head d e c a y i n l a y e r 4L t o v a r y , d e p e n d i n g on t h e v a l u e o f K •K i n l a y e r 3H. H y d r a u l i c head a d j u s t m e n t s a t b a s i n c e n t r a l n o d e s 2M i n l a y e r s 3H and 4L b o t h show a s i m i l a r p a t t e r n i n w h i c h t h e l i n e s c r o s s e a c h o t h e r i n d i c a t i n g d i f f e r e n c e s i n t h e r a t e s o f h y d r a u l i c head c h a n g e s b e t w e e n t h e f o u r c a s e s ( s e e F i g u r e 17 C ) . H y d r a u l i c head v a l u e s r e a c h s t e a d y s t a t e a t a p p r o x i m a t e l y t h e same t i m e f o r a l l f o u r c a s e s . H y d r a u l i c head v a l u e s i n l a y e r 5H a r e s e n s i t i v e t o 66 Years 102 103 104 105 106 \Q7 1 1 1 1 1 1— 2.7 _ Node 3S 1 :—i 1 1 1 r 2.7 _ Node 2 M 2.8 _ Layer 3H Mid-Basin -^-^ =1^ — F igure 17. Hydrau l i c head versus log time p l o t s f o r K x>K z i n h igh-K l a y e r s . Curve legend 1 - i s o t r o p i c 1 i s K x=K z 3 i s K =100K A £ 2 i s K =10K: A L. 4 i s K =1000K . A L. 2 3 4 i n c r ea s i ng K. 67 p e r t u r b a t i o n s of the v a l u e of K i n the high-K l a y e r s . A g a i n , f o r a one o r d e r - o f ~ m a g n i t u d e i n c r e a s e i n the r a t i o K :K , the Z X time r e q u i r e d f o r h y d r a u l i c head v a l u e s to reach s t e a d y s t a t e d e c r e a s e s by a p p r o x i m a t e l y one o r d e r - o f - m a g n i t u d e . L a y e r s IH and 3H a r e u n a f f e c t e d by changes of K :K i n these l a y e r s , Z X p r o b a b l y due to the presence of low-K l a y e r s above, which c o n t r o l the r a t e s of h y d r a u l i c head adjustment and the time r e q u i r e d f o r heads to reach s t e a d y s t a t e i n t h e s e l a y e r s . 4.4. The Model's S e n s i t i v i t y to Changes i n S p e c i f i c S t o r a g e V a l u e s of s p e c i f i c s t o r a g e ( S s ) are v a r i e d i n t h r e e o f the l a y e r s o f the s t a n d a r d model i n o r d e r to determine the model's s e n s i t i v i t y to t h i s parameter. The l a y e r s t e s t e d a r e : l a y e r 3H, a high-K l a y e r ; l a y e r 4L, a low-K l a y e r ; and l a y e r 5H, a h i g h - K , u n c o n f i n e d l a y e r . The s t a n d a r d v a l u e o f S s of l a y e r s 3H -4 -1 and 4L, 1x10 m as g i v e n by Toth and M i l l a r (1983) appears h i g h , and, t h e r e f o r e , o n l y one o t h e r v a l u e o f S s, l x l 0 ~ ^ m-"*", i s s i m u l a t e d . The s t a n d a r d s p e c i f i c s t o r a g e v a l u e of l a y e r 5H, -4 -1 5x10 m , i s an average v a l u e , and, t h e r e f o r e , two o t h e r —5 —3 —1 v a l u e s of S , 5x10 and 1.25x10 m , are s i m u l a t e d . A l l s' v a l u e s of S s a r e w i t h i n the range of those r e p o r t e d elsewhere (Freeze and C h e r r y , 1979, p.60-61). R e s u l t s o f the a n a l y s i s are shown i n F i g u r e 18. Changes i n the S g v a l u e of l a y e r 3H a f f e c t h y d r a u l i c head adjustment t i m e s i n both l a y e r s 3H and 4L; the o t h e r l a y e r s remain unchanged. H y d r a u l i c heads a t nodes ID i n l a y e r 3H and 2M i n l a y e r 4L b e g i n to decay a t a p p r o x i m a t e l y the same time f o r both c a s e s 1 and 2, but f o r case 1, t h a t of the lower v a l u e of Figure 18. Hydraulic head versus log time plots for Curve legend perturbations of specific storage. ^ _ -|ow ^ 2 - standard S< 3 - high S$ s S_5=1.25xl0"3 m"1. A ,B: 1 is S 3=10" s •6. 2 is Ss3=10"4 m"1 C,D: 1 is Ss4=10" •6. » 2 is Ss4=10"4 E: 1 is Ss5=5xl0" •5. 2 is Ss5=5xl0' -4. 5 69 ; S s , the heads decay a t a f a s t e r r a t e , and reach s t e a d y s t a t e 2 to 4 t i m e s sooner (see F i g u r e 18 A and B ) . For example, a t node 2M i n l a y e r 4L, s t e a d y s t a t e i s a c h i e v e d a f t e r a p p r o x i m a t e l y .22 Ma f o r case 1 and a p p r o x i m a t e l y .86 Ma f o r case 2. Changes to the S g v a l u e of l a y e r 4L, a low-K l a y e r , a f f e c t r e s u l t i n g h y d r a u l i c head v a l u e s i n both l a y e r s 3H and 4L. S i m i l a r p a t t e r n s are observed f o r a l l nodes i n these two l a y e r s , two of which are i l l u s t r a t e d i n F i g u r e 18 C and D. Lowering the v a l u e o f S s causes h y d r a u l i c head a d j u s t m e n t s to b e g i n e a r l i e r and proceed a t s l o w e r r a t e s ; s t e a d y s t a t e i s reached a t the same time f o r both cases 1 and 2. D u r i n g the e a r l y p e r i o d of h y d r a u l i c head adjustment a t nodes ID, 2M, and 3S i n l a y e r 4L, t h e r e i s a sharp decrease i n the decay r a t e , f o l l o w e d by a sharp i n c r e a s e . The i n c r e a s e a t node ID i n l a y e r 4L c o r r e s p o n d s to the b e g i n n i n g of h y d r a u l i c head adjustment a t node ID i n l a y e r 3H below ( F i g u r e 18 C and D). S i m i l a r c o n d i t i o n s e x i s t f o r nodes 2M i n l a y e r s 4L and 3H and nodes 3S i n l a y e r s 4L and 3H. Changes i n the v a l u e of S g of l a y e r 5H, the u n c o n f i n e d l a y e r , o n l y a f f e c t h y d r a u l i c head decay a t nodes ID, 2M, and 3S i n l a y e r 5H. H y d r a u l i c head adjustment r a t e s are s i m i l a r f o r a l l t h r e e nodes, however, the lower the v a l u e of S g , the sooner h y d r a u l i c head decay b e g i n s and ends. A p l o t o f h y d r a u l i c head v e r s u s l o g time f o r node ID i n l a y e r 5H i s shown i n F i g u r e 18 E. 4.5. The Model's S e n s i t i v i t y to Changes i n Water T a b l e C o n f i g u r a t i o n To determine the e f f e c t of the shape of the r e g i o n a l water t a b l e on the h y d r a u l i c head adjustment times f o r the f l o w system, t h r e e g e n e r a l water t a b l e c o n f i g u r a t i o n s were a n a l y z e d : 70 a l i n e a r l y s l o p i n g water t a b l e ; a concave water t a b l e ; and a convex water t a b l e . Most n a t u r a l l y o c c u r r i n g water t a b l e s can be r e p r e s e n t e d by one or a c o m b i n a t i o n o f these t h r e e c o n f i g u r a t i o n s . For these s i m u l a t i o n s , the f i n i t e element mesh was a l t e r e d s l i g h t l y i n o r d e r to model the concave and convex water t a b l e s . To m i n i m i z e the changes i n t h i c k n e s s e s o f the l a y e r s , the water t a b l e b e g i n s and ends a t the same e l e v a t i o n a t each end of the b a s i n f o r a l l t h r e e c a s e s , and changes shape i n between. I t i s i m p o s s i b l e t o change the shape of the water t a b l e w i t h o u t changing the t h i c k n e s s of the top l a y e r i n some p l a c e s , and t h i s w i l l a f f e c t the r e s u l t s i n l a y e r 5H. However, the top l a y e r i s not v e r y i m p o r t a n t because i t i s a high-K l a y e r and t h e r e f o r e h y d r a u l i c head adjustment r a t e s w i t h i n the l a y e r do not i n f l u e n c e head adjustment r a t e s w i t h i n the remainder of the f l o w system. R e s u l t s o f t h i s a n a l y s i s suggest t h a t changing the shape of the r e g i o n a l water t a b l e from concave t o l i n e a r l y s l o p i n g to convex, has v e r y l i t t l e e f f e c t on the h y d r a u l i c head adjustment t i m e s o f the system as a whole. As e x p e c t e d , the major changes o c c u r i n the top l a y e r . However, s i n c e the t h i c k n e s s o f the top l a y e r v a r i e s due to the n a t u r e of the a n a l y s i s , i t i s l i k e l y t h a t the changes are a r e s u l t of d i f f e r e n c e s i n t h i c k n e s s o f the l a y e r , r a t h e r than of the change i n shape o f the water t a b l e . H y d r a u l i c head v a l u e s a t nodes 2M and 3S i n l a y e r 5H reach s t e a d y s t a t e l e s s than one o r d e r - o f - m a g n i t u d e e a r l i e r f o r the concave water t a b l e , and l e s s than one o r d e r - o f - m a g n i t u d e l a t e r f o r t he convex water t a b l e . Node ID, a t the deep edge of the 71 b a s i n i n l a y e r 5H, shows no d i f f e r e n c e i n the time r e q u i r e d f o r h y d r a u l i c head v a l u e s to reach s t e a d y s t a t e i n any o f the t h r e e c a s e s . 4.6. The Model's S e n s i t i v i t y to Changes i n B a s i n S i z e To a n a l y z e the model's s e n s i t i v i t y to the s i z e of the b a s i n , f o u r s i m u l a t i o n s were run: a) the depth remains f i x e d and the l e n g t h i s reduced to one h a l f o f the s t a n d a r d model's l e n g t h ; b) the depth remains f i x e d and the l e n g t h i s reduced to one q u a r t e r of the s t a n d a r d model's l e n g t h ; c) the l e n g t h remains f i x e d and the depth i s t w i c e the s t a n d a r d model's d e p t h ; and d) the l e n g t h remains f i x e d and the depth i s f o u r t i m e s the s t a n d a r d model's d e p t h . 4.6.1. S i m u l a t i o n s i n which the depth remains f i x e d w h i l e the l e n g t h v a r i e s The d e p t h : l e n g t h (D:L) r a t i o of the s t a n d a r d model i s 1:200. T h i s i s a s m a l l r a t i o and t h e r e f o r e o n l y l a r g e r D:L r a t i o s of 1:100 and 1:50 were s i m u l a t e d . The r e s u l t s o f t h e s e s i m u l a t i o n s i n d i c a t e t h a t v a r y i n g the l e n g t h o f the b a s i n has v e r y l i t t l e e f f e c t on the r a t e of decay of and the time r e q u i r e d f o r complete adjustment of h y d r a u l i c head v a l u e s . In g e n e r a l , h y d r a u l i c head v a l u e s a t nodes l o c a t e d near the deep and c e n t r a l p a r t s o f the b a s i n b e g i n to decay a l i t t l e sooner f o r a s h o r t e r b a s i n and those a t nodes l o c a t e d near the c e n t e r o f the b a s i n a d j u s t a t a f a s t e r r a t e f o r a l o n g e r b a s i n . T h i s may be due to the d i f f e r e n c e s i n the s l o p e of the water t a b l e , which i s a r e s u l t o f changing the b a s i n l e n g t h . In a l l c a s e s , h y d r a u l i c head v a l u e s r e a c h a s t e a d y s t a t e c o n d i t i o n a t a p p r o x i m a t e l y the 72 same t i m e . The o n l y n o d e s n e a r t h e s h a l l o w edge o f t h e b a s i n w h i c h show a s i g n i f i c a n t d i f f e r e n c e i n t h e r a t e s o f a d j u s t m e n t o f h y d r a u l i c head v a l u e s a r e n o d e s 3S, l o c a t e d a t t h e b a s e o f l a y e r s 3H and 4L. As shown i n F i g u r e 19 A and B, t h e r a t e o f h e a d a d j u s t m e n t i n c r e a s e s w i t h an i n c r e a s e i n t h e b a s i n l e n g t h . N o t e i n F i g u r e 19 t h a t h y d r a u l i c head a d j u s t m e n t s a t node 3S i n l a y e r 4L a r e t h e same i n a l l t h r e e c a s e s up u n t i l t h e t i m e t h a t t h e h y d r a u l i c h e a d v a l u e s a t node 3S i n l a y e r 3H, l o c a t e d v e r t i c a l l y b e l o w , b e g i n t o d e c a y . A t t h i s p o i n t i n t i m e , t h e d e c a y r a t e s a t node 3S i n l a y e r 4L a r e d i f f e r e n t f o r e a c h c a s e , d e p e n d i n g on t h e D:L r a t i o . 4.6.2. S i m u l a t i o n s i n w h i c h t h e l e n g t h r e m a i n s f i x e d w h i l e t h e d e p t h v a r i e s The s t a n d a r d m o d el b a s i n i s s h a l l o w w i t h a l e n g t h : d e p t h (L:D) r a t i o o f 1:.005. Two s i m u l a t i o n s w e re r u n w i t h d e p t h s o f 2 and 4 t i m e s t h e s t a n d a r d d e p t h o f t h e m o d e l , o r L:D r a t i o s o f 1:.01 and 1:.02, r e s p e c t i v e l y . The r e s u l t s o f t h i s a n a l y s i s , as shown i n F i g u r e 20, i n d i c a t e t h a t a c h a n g e i n t h e t h i c k n e s s o f t h e b a s i n , due t o a c h a n g e i n t h e t h i c k n e s s o f e a c h l a y e r , h a s a s i g n i f i c a n t e f f e c t on t h e h y d r a u l i c head a d j u s t m e n t t i m e s . I n g e n e r a l , d o u b l i n g t h e t h i c k n e s s o f a l l l a y e r s i n c r e a s e s t h e t i m e r e q u i r e d f o r c o m p l e t e h y d r a u l i c head a d j u s t m e n t by 3 t o 5 t i m e s . F o r e x a m p l e , a t node 3S i n l a y e r 1H, a d j u s t m e n t t i m e i n c r e a s e s f r o m a p p r o x i m a t e l y 69 Ma t o a p p r o x i m a t e l y 218 Ma f o r an i n c r e a s e i n t o t a l t h i c k n e s s o f t h e l a y e r s f r o m 3 km t o 6 km. R a t e s o f d e c a y o f and t h e amount o f t i m e r e q u i r e d f o r c o m p l e t e a d j u s t m e n t o f h y d r a u l i c h e a d s a r e s i m i l a r f o r a l l f i f t e e n nodes e x a m i n e d . 73 Figure 19. Hydraulic head versus log time Curve legend plots for variable depth:length 1 ratios. ^ 1 is D:L=1:50; 2 is D:L=1:100; \ ~ r " Q S ' " ' T ^ ' 3 is D:L=l:200v 3 " s t a n d a r d b a s i n size increasing basin length 74 S i n c e h y d r a u l i c head v e r s u s l o g time p l o t s are a l s o s i m i l a r f o r t h e s e s i m u l a t i o n s , o n l y p l o t s f o r one node from each l a y e r are shown i n F i g u r e 20 A to E. A g a i n , note the d e c r ease i n the r a t e of a d j u s t m e n t of h y d r a u l i c heads a t node 3S i n l a y e r 2L ( F i g u r e 20 B) at the p o i n t i n time at which the nodes v e r t i c a l l y below i t i n the high-K l a y e r IH f i r s t b e g i n to decay ( F i g u r e 20 A ) . T h i s o n l y o c c u r s at nodes near the s h a l l o w edge of the b a s i n , where h y d r a u l i c head decay f i r s t b e g i n s i n each l a y e r . I t s h o u l d be noted t h a t the v e r t i c a l s c a l e o f F i g u r e 20 o n l y r e f e r s to the s t a n d a r d b a s i n . For the c o r r e c t s c a l e s of c a s e s 2 and 3, the h y d r a u l i c head v a l u e s s h o u l d be i n c r e a s e d by 3 and 9 km, r e s p e c t i v e l y . The p l o t was c o n s t r u c t e d i n t h i s manner to a l l o w easy comparison of the h y d r a u l i c head adjustment r a t e s f o r each s i m u l a t i o n . 4.7. C o n c l u s i o n s The r e s u l t s i n d i c a t e t h a t the model i s most s e n s i t i v e to changes i n (a) the v a l u e s of h y d r a u l i c c o n d u c t i v i t y of the low-K l a y e r s , (b) the z - d i r e c t i o n component of h y d r a u l i c c o n d u c t i v i t y i n the low-K l a y e r s i n those c a s e s where such l a y e r s are a n i s o t r o p i c , (c) the geometry of the b a s i n by changing l a y e r t h i c k n e s s e s , and (d) the v a l u e s of s p e c i f i c s t o r a g e i n the high-K l a y e r s . The major p o i n t s and t r e n d s are summarized below f o r a l l of the parameters a n a l y z e d . 1) The f l o w system i s much more s e n s i t i v e to p e r t u r b a t i o n s of the v a l u e s of h y d r a u l i c c o n d u c t i v i t y i n low-K l a y e r s than i n high-K l a y e r s . Figure 20. Hydraulic head versus log time plots for variable length:depth ratios. 1 is L:D=1:.005; 2 is L:D=1:.01; 3 is L:D=1:.02. Curve legend 1 - standard basin size 2 - increasing 3 - Ibasin depth 76 2) H y d r a u l i c head v a l u e s a t nodes near the s h a l l o w edge o f the b a s i n , where h y d r a u l i c head decay f i r s t b e g i n s w i t h i n a l a y e r , a re s e n s i t i v e to p e r t u r b a t i o n s of the v a l u e s of K i n i s o t r o p i c high-K l a y e r s . The lower the v a l u e of K i n the p e r t u r b e d l a y e r , the f a s t e r the r a t e of h y d r a u l i c head adjustment i n both the p e r t u r b e d l a y e r and the s u p e r j a c e n t low-K l a y e r a t e a r l y t i m e s . 3) Changing the v a l u e o f K i n an i s o t r o p i c low-K l a y e r a f f e c t s t h a t l a y e r and the s u b j a c e n t high-K l a y e r , c a u s i n g changes i n both the r a t e of decay and the time r e q u i r e d f o r complete adjustment of h y d r a u l i c heads. A one o r d e r - o f - m a g n i t u d e decrease i n the v a l u e o f K r e s u l t s i n an i n c r e a s e of up to two o r d e r s - o f ~ m a g n i t u d e i n the time r e q u i r e d f o r h y d r a u l i c heads to reach s t e a d y s t a t e . Decay r a t e s of h y d r a u l i c heads g e n e r a l l y d e c r ease w i t h an i n c r e a s e i n the v a l u e of K, e s p e c i a l l y d u r i n g e a r l y t i m e s and near the s h a l l o w edge of the b a s i n . 4) The f l o w system i s much more s e n s i t i v e to p e r t u r b a t i o n s of the v a l u e s of i n a n i s o t r o p i c low-K l a y e r s than i n a n i s o t r o p i c high-K l a y e r s . For a one o r d e r - o f - m a g n i t u d e i n c r e a s e i n K :K i n a low-K l a y e r , the time r e q u i r e d f o r h y d r a u l i c head v a l u e s to reach s t e a d y s t a t e d e c r e a s e s by one o r d e r - o f - m a g n i t u d e or more a t a l l nodes w i t h i n the e n t i r e f l o w system. 5) V a r y i n g the v a l u e s of K^ i n a n i s o t r o p i c low-K l a y e r s has no e f f e c t on the system. V a r y i n g the v a l u e s of K^ i n a n i s o t r o p i c high-K l a y e r s o n l y a f f e c t s the r a t e of decay of h y d r a u l i c heads i n c e r t a i n l a y e r s and has no e f f e c t on the time 77 r e q u i r e d f o r head v a l u e s t o reach s t e a d y s t a t e . 6) V a r y i n g the v a l u e o f S s i n a high-K l a y e r a f f e c t s both t h a t l a y e r and the s u p e r j a c e n t low-K l a y e r . A d e c r e a s e of two o r d e r s - o f - m a g n i t u d e i n the v a l u e o f S g r e s u l t s i n an i n c r e a s e i n the r a t e of decay of h y d r a u l i c head v a l u e s and an i n c r e a s e of 2 to 4 t i m e s i n the time r e q u i r e d f o r heads to reach s t e a d y s t a t e . 7) D e c r e a s i n g the v a l u e of i n a low-K l a y e r d e c r e a s e s the r a t e of h y d r a u l i c head decay i n t h a t l a y e r and i n the s u b j a c e n t high-K l a y e r , but has no e f f e c t on the time r e q u i r e d f o r h y d r a u l i c head v a l u e s to reach s t e a d y s t a t e . 8) Changing the g e n e r a l shape of the r e g i o n a l water t a b l e has l i t t l e to no e f f e c t on the t i m i n g o f adjustment o f h y d r a u l i c head v a l u e s . 9) V a r y i n g the l e n g t h of the b a s i n has v e r y l i t t l e e f f e c t on the adjustment t i m e s of h y d r a u l i c head v a l u e s w i t h i n the b a s i n . T h i s s u g g e s t s t h a t h o r i z o n t a l decay of h y d r a u l i c head v a l u e s i n the b a s i n i s much l e s s s i g n i f i c a n t than v e r t i c a l decay. 10) The model i s e x t r e m e l y s e n s i t i v e to c e r t a i n changes i n the b a s i n geometry caused by changes i n l a y e r t h i c k n e s s e s . D o u b l i n g the t h i c k n e s s o f a l l l a y e r s w i t h i n the b a s i n r e s u l t s i n an i n c r e a s e of 3 to 5 t i m e s i n the time r e q u i r e d f o r complete h y d r a u l i c head a d j u s t m e n t . 11) Two major t r e n d s appear i n the h y d r a u l i c head v e r s u s l o g - t i m e p l o t s : a) h y d r a u l i c head v a l u e s a t nodes l o c a t e d near the s h a l l o w edge of the b a s i n i n low-K l a y e r s b e g i n changing a t the same time and r a t e f o r a l l c a s e s , u n t i l h y d r a u l i c head v a l u e s a t nodes i n the s u b j a c e n t high-K l a y e r b e g i n to a d j u s t , 78 c a u s i n g the r a t e s of head decay i n the low-K l a y e r to d i f f e r f o r each case ( t h i s t r e n d i s observed i n the h i g h - K - l a y e r s i m u l a t i o n s , the K^ s i m u l a t i o n s i n the high-K l a y e r s , and the b a s i n l e n g t h s i m u l a t i o n s ) ; and b) a sh a r p decrease i n the r a t e of decay of h y d r a u l i c heads i n low-K l a y e r s o c c u r s when adjustment b e g i n s a t nodes i n the s u b j a c e n t high-K l a y e r . T h i s o c c u r s o n l y a t nodes l o c a t e d near the s h a l l o w edge o f the b a s i n and o n l y when h y d r a u l i c head decay r a t e s i n the high-K l a y e r are slow ( t h i s t r e n d i s observed i n the l o w - K - l a y e r s i m u l a t i o n s , the K s i m u l a t i o n s i n the low-K l a y e r s , the s p e c i f i c s t o r a g e s i m u l a t i o n s i n the low-K l a y e r s , and the b a s i n depth s i m u l a t i o n s ) . 79 CHAPTER 5 TRANSIENT FLOW IN REGIONAL BASINS WITH VARIOUS  STRATIGRAPHIC AND STRUCTURAL CONFIGURATIONS 5.1. 2-D Flow i n the Red E a r t h Region 5.1.1. I n t r o d u c t i o n The purpose of s i m u l a t i n g t r a n s i e n t groundwater f l o w i n the Red E a r t h r e g i o n , u s i n g a more a c c u r a t e model, r e f e r r e d to as model A, than the s t a n d a r d model employed i n Chapter 4, i s t w o - f o l d : f i r s t , to compare the 2-D r e s u l t s w i t h both the 1-D f i n i t e element model r e s u l t s and Toth and M i l l a r ' s (1983) 1-D a n a l y t i c a l s o l u t i o n ; and second, to i l l u s t r a t e the n a t u r e of t r a n s i e n t groundwater f l o w i n a r e a l i s t i c mature b a s i n w i t h l a r g e 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 . The mesh diagram f o r t h i s s i m u l a t i o n i s shown i n F i g u r e 21. The g e o g r a p h i c a l l o c a t i o n of the s e c t i o n i s shown i n F i g u r e 1 as A-A'. Note t h a t due to the l o n g and s h a l l o w n a t u r e of the b a s i n , t h e r e i s a l a r g e v e r t i c a l e x a g g e r a t i o n i n t h i s and a l l subsequent b a s i n diagrams i n t h i s c h a p t e r . To s i m p l i f y the computer r u n , the mesh g e n e r a t o r i s the same as t h a t used f o r the s t a n d a r d model. The l a y e r s are d e f i n e d such t h a t t h e i r geometry c l o s e l y resembles t h a t o f the Red E a r t h r e g i o n . The upper boundary i s the water t a b l e and r e p r e s e n t s the t o p o g r a p h i c a l c o n f i g u r a t i o n of the l a n d s u r f a c e i n P l i o c e n e t i m e s ; the r e m a i n i n g b o u n d a r i e s are a l l n o - f l o w b o u n d a r i e s . The s l o p e of the basement rock and the c o n f i g u r a t i o n o f the water t a b l e are a l s o somewhat s i m p l i f i e d . For the t r a n s i e n t s i m u l a t i o n , s t a t i c c o n d i t i o n s are assumed to p r e v a i l i n i t i a l l y , Figure 21. Mesh diagram of Model A , the complex Red Earth basin. 00 o 81 w i t h h y d r a u l i c head v a l u e s of 3.657 km throughout the b a s i n . At time=0, an i n s t a n t a n e o u s e r o s i o n a l event causes a d e c rease i n head v a l u e s a l o n g the upper boundary r a n g i n g from .53 to 1.1 km. These i n i t i a l c o n d i t i o n s were chosen to c l o s e l y resemble the i n i t i a l s t a g e i n the e v o l u t i o n of the Red E a r t h b a s i n (T6th, 1978). A l t h o u g h o n l y the P l i o c e n e topography i s modeled h e r e , a second e r o s i o n a l m o d i f i c a t i o n of the l a n d s u r f a c e o c c u r r e d at some time d u r i n g the P l e i s t o c e n e , c r e a t i n g the p r e s e n t - d a y topography. The e f f e c t of t h i s i s c o n s i d e r e d i n S e c t i o n 5.3. The parameters i n p u t to the model are l i s t e d i n T able IV. LAYER HYDRAULIC CONDUCTIVITY SPECIFIC LAYER THICKNESS (M/S) STORAGE NO. (KM) X-DIRECTION Z-DIRECTION (M - 1) 1 0.500 4 x 10~7 4 x 10-?. 1 x 1 0 ~ 4 2 0.275 1 x 1 0 " i 1 x 10~£ 1 x 10_J 3 1.189 1 x 10 ^ 1 x 1 0 ~ T , 1 x 10 7 4 0.854 1 x 10~£ 1 x 10_f 1 x 10 4 5 0.305 1 x 10 b 1 x 1 0 5 x 10 4 B a s i n Length: 644 km B a s i n Depth (at SW boundary): 3.123 km S l o p e of Water T a b l e : .000426 Slope of Basement Rock: .00362 Ta b l e IV: Parameters i n p u t to model A. 5.1.2. The n a t u r e of t r a n s i e n t f l o w i n a 5 - l a y e r b a s i n In o r d e r to v i s u a l i z e the f l o w r e g i o n towards which the t r a n s i e n t h y d r a u l i c heads are e q u i l i b r a t i n g , s t e a d y s t a t e p o t e n t i a l and v e l o c i t y diagrams are p r e s e n t e d i n F i g u r e s 22 and 23. F i g u r e 22 shows e q u i p o t e n t i a l l i n e s w i t h i n the f l o w regime. The c o n t o u r s are g e n e r a l l y v e r t i c a l i n the high-K l a y e r s and d i a g o n a l to h o r i z o n t a l i n the low-K l a y e r s . H y d r a u l i c g r a d i e n t s are q u i t e h i g h i n l a y e r 2, the bottom-most low-K l a y e r . F i g u r e R E D E A R T H Figure 22. Steady state equipotential plot of model A. R F D F O R T H MAXIMUM VELOCITY = 2.46E -5 ro T : 1 r 1 1 i i 0.0 10.737 21.473 32.21 42.947 53.683 64.42 KILOMETERS (X 101 ) Figure 23. Steady state velocity plot of model A . 84 23 i n d i c a t e s d i r e c t i o n and magnitude of f l o w w i t h i n the r e g i o n . V e l o c i t i e s are c a l c u l a t e d w i t h i n each element and are c o n s t a n t a c r o s s the element. C a l c u l a t i n g v e l o c i t i e s i n t h i s way may r e s u l t i n a c e r t a i n amount of e r r o r f o r l a r g e elements; however, the f i g u r e shows the g e n e r a l p a t t e r n o f f l o w w i t h i n the b a s i n . The l a r g e v e l o c i t y c o n t r a s t s between high-K and low-K l a y e r s does not a l l o w p l o t t i n g of v e l o c i t y magnitudes as a b s o l u t e v a l u e s ; t hey are s c a l e d l o g a r i t h m i c a l l y i n o r d e r to show f l o w d i r e c t i o n i n both the high-K and low-K l a y e r s s i m u l t a n e o u s l y . — 1 7 S m a l l c r o s s e s i n d i c a t e a v e l o c i t y of l e s s than 3x10 m/day —17 (3x10 m/s). Flow i s g e n e r a l l y v e r t i c a l i n the low-K l a y e r s and h o r i z o n t a l i n the high-K l a y e r s . F i g u r e 24 A to E shows c o n t o u r e d v a l u e s of the change i n h y d r a u l i c head a t s p e c i f i c t i m e s throughout the t r a n s i e n t e v ent. The f o l l o w i n g p a t t e r n s can be noted from t h i s f i g u r e , which a p p l y g e n e r a l l y to most of the t r a n s i e n t s i m u l a t i o n s d i s c u s s e d i n the t h e s i s : 1) v e r t i c a l adjustment i n high-K l a y e r s s u b j a c e n t to low-K l a y e r s i s i n s t a n t a n e o u s t h r o u g h o u t any v e r t i c a l column i n the high-K l a y e r ; 2) decay of h y d r a u l i c heads a t nodes i n both low-K and high-K l a y e r s i s a f f e c t e d by v e r t i c a l changes above the nodes much more than by h o r i z o n t a l changes from n e i g h b o u r i n g nodes; 3) g e n e r a l l y , the r a t e of a djustment of ' h y d r a u l i c heads i n a high-K l a y e r u n d e r l y i n g a low-K l a y e r i s c o n t r o l l e d by the r a t e of adjustment i n the low-K l a y e r , as shown i n F i g u r e 24 C; however, h y d r a u l i c heads at nodes i n high-K l a y e r s do, i n some c a s e s , b e g i n a d j u s t i n g sooner than s u p e r j a c e n t nodes i n low-K l a y e r s but the r a t e of a djustment i s slow u n t i l h y d r a u l i c heads i n the low-K l a y e r have R E D E A R T H m 0.0 10.737 21.473 32.21 42.947 53.683 64 KILOMETERS (X101 ) Figure 24. Transient equipotential plot of hydraulic head change (km) at nodes in model A. A. Time = 10 days. R E D E A R T H Figure 24. Transient equipotential plot of hydraulic head change (km) at nodes in model A. B. Time = 2,740 years. oo s R E D E A R T H Figure 24. Transient equipotential plot of hydraulic head change (km) at nodes in model A. C. Time - .137 Ma. 00 R E D E A R T H Figure 24. Transient equipotential plot of hydraulic head change (km) at nodes in model A. D. Time =2.74 Ma. R E D E A R T H Figure 24. Transient equipotential plot of hydraulic head change (km) at nodes in model A. " E. Time = 27.4 Ma. 90 c a u g h t up. T h i s o c c u r s i n l a y e r 1 b u t i s n o t v i s i b l e i n F i g u r e 24 due t o t h e v e r y s m a l l c o n t o u r i n t e r v a l r e q u i r e d t o show i t . How e v e r , a s i m i l a r p a t t e r n c a n be s e e n i n F i g u r e 28 D i n t h e f o l l o w i n g s e c t i o n , 5.2. T h i s o b s e r v e d phenomenon i s due t o t h e m o n o t o n i c d e c r e a s e o f h y d r a u l i c h e a d s f r o m t h e i r i n i t i a l v a l u e s a t time=0 t o t h e i r f i n a l s t e a d y s t a t e v a l u e s , a s c a n be s e e n i n a l l o f t h e h y d r a u l i c head v e r s u s l o g t i m e p l o t s p r e s e n t e d i n C h a p t e r 4. B e c a u s e h e a d v a l u e s t h r o u g h o u t t h e f l o w r e g i m e a r e c o n s t a n t l y d e c r e a s i n g t o w a r d s s t e a d y s t a t e , a d j u s t m e n t i n h i g h - K l a y e r s c a n n e v e r be c o m p l e t e d p r i o r t o a d j u s t m e n t i n s u p e r j a c e n t low-K l a y e r s ; and 4) h y d r a u l i c head v a l u e s i n low-K l a y e r s a d j u s t much more s l o w l y t h a n t h o s e i n h i g h - K l a y e r s . 5.1.3. C o m p a r i s o n w i t h ID Red E a r t h m o d e l s I n C h a p t e r 3, t h e f i n i t e e l e m e n t s o l u t i o n t o t h e 1-D t r a n s i e n t f l o w e q u a t i o n f o r a h e t e r o g e n e o u s medium i s compared t o t h e 1-D a n a l y t i c a l s o l u t i o n o f T o t h and M i l l a r (1983) b y c o m p a r i n g r e l a t i v e h y d r a u l i c head a d j u s t m e n t s , R g n a t s p e c i f i e d t i m e s d u r i n g t h e t r a n s i e n t e v e n t . V a l u e s o f R were d e t e r m i n e d 3 n a t t h e s e same t i m e s f o r t h r e e n o d e s , A, B, and C, l o c a t e d i n t h e b a s a l u n i t o f model A, i n t h e d e e p , c e n t r a l , and s h a l l o w r e g i o n s o f t h e b a s i n . The r e s u l t s a r e shown i n T a b l e V a l o n g w i t h t h o s e o f t h e 1-D m o d e l s . R e s u l t s f o r o n l y one s e t o f h y d r a u l i c c o n d u c t i v i t y v a l u e s were a n a l y z e d , n a m e l y , t h o s e l i s t e d i n T a b l e I V . 91 RELATIVE HYDRAULIC HEAD ADJUSTMENT TIME 1-D AN 1-D FE 2-D FE (Ma) A B C 4 . 000 .002 .076 .101 .149 11 .100 .094 . 263 . 285 . 323 18.5 . 250 .250 .441 .458 . 486 33 . 500 . 502 .679 .689 . 705 58 .750 .763 .881 .884 . 890 92 . 900 .911 .972 . 973 . 974 137 - - .997 .997 .997 200 1.000 .997 1. 000 1. 000 1.000 T a b l e V: R c o m p a r i s o n b e t w e e n t h e 2-D f i n i t e e l e m e n t (FE) model and b o t h t h e 1-D FE model and t h e 1-D a n a l y t i c a l (AN) m o d e l . I t i s c l e a r f r o m T a b l e V t h a t h y d r a u l i c head v a l u e s i n t h e b a s a l l a y e r have a g r e a t e r R a t a l l t i m e s f o r t h e 2-D model a n t h a n f o r e i t h e r o f t h e 1-D m o d e l s . A t 25% h y d r a u l i c head a d j u s t m e n t f o r t h e 1-D m o d e l s , h y d r a u l i c head v a l u e s h a v e r e a c h e d b e t w e e n 44% and 49% a d j u s t m e n t f o r t h e 2-D m o d e l . H owever, t h i s gap n a r r o w s s i g n i f i c a n t l y a t l a t e r t i m e s and f o r t h e 1-D m o d e l , h y d r a u l i c h e a d v a l u e s r e q u i r e a p p r o x i m a t e l y 63 Ma l o n g e r t o r e a c h s t e a d y s t a t e t h a n t h e 2-D m o d e l . T h i s d i f f e r e n c e i n t i m e b e t w e e n t h e 2-D model and t h e 1-D m o d e l s c a n be a c c o u n t e d f o r by t h e d i f f e r e n c e s i n l a y e r t h i c k n e s s a l o n g t h e b a s i n c r o s s - s e c t i o n i n t h e 2-D m o d e l , and t h e e f f e c t o f l a t e r a l d e c a y i n t h e h i g h - K l a y e r s o f t h e 2-D m o d e l . 5.2. F l o w i n a B a s i n i n w h i c h t h e Low-K L a y e r s P i n c h Out n e a r t h e S h a l l o w Edge 5.2.1. I n t r o d u c t i o n The p u r p o s e o f t h i s a n a l y s i s i s t o d e t e r m i n e t h e i m p o r t a n c e o f l a t e r a l c h a n g e s i n h y d r a u l i c head i n t h e h i g h - K l a y e r s o f t h e 92 s t a n d a r d model and the e f f e c t of these changes on the o v e r a l l t i m i n g of h y d r a u l i c head adjustments w i t h i n the e n t i r e f l o w system. To c r e a t e model B, the s t a n d a r d model was m o d i f i e d such t h a t the two low~K l a y e r s p i n c h out a t a p p r o x i m a t e l y 50 km to the SW o f the NE boundary of the r e g i o n of f l o w (see F i g u r e 2 5 ) . A l l o t h e r parameters and c o n d i t i o n s remain i d e n t i c a l to those d e s c r i b e d f o r the s t a n d a r d model i n Chapter 4. 5.2.2. Comparison w i t h the s t a n d a r d b a s i n model Cau s i n g the two low-K l a y e r s to p i n c h out near the s h a l l o w edge of the b a s i n has a s i g n i f i c a n t e f f e c t on the t i m i n g of h y d r a u l i c head adjustment a t nodes i n the two b a s a l l a y e r s , IH and 2L, and a t nodes l o c a t e d near the s h a l l o w edge of the b a s i n i n a l l l a y e r s (nodes l a b e l l e d 3 S ) . F i g u r e s 26 and 27 show the h y d r a u l i c head v e r s u s l o g time p l o t s f o r some of the s e nodes. At a l l t h r e e m o n i t o r e d nodal l o c a t i o n s i n l a y e r IH ( F i g u r e 26 A to C) h y d r a u l i c head v a l u e s reach s t e a d y s t a t e sooner f o r model B than f o r the s t a n d a r d model. No changes are observed a t o t h e r nodes i n the high-K l a y e r s , except f o r nodes 3S l o c a t e d near the s h a l l o w edge of the b a s i n i n l a y e r s 3H and 5H. A g a i n , adjustment i s completed much sooner f o r model B. H y d r a u l i c head v a l u e s a t nodes ID and 2M i n l a y e r IH r e q u i r e a p p r o x i m a t e l y a f i v e f o l d i n c r e a s e i n time to reach s t e a d y s t a t e f o r the s t a n d a r d model than f o r model B, the p i n c h - o u t model ( i . e . , s t e a d y s t a t e i s reached a f t e r a p p r o x i m a t e l y 13.7 Ma f o r model B, and a f t e r a p p r o x i m a t e l y 68.5 Ma f o r the s t a n d a r d model). The f a s t e r adjustment a t th e s e nodes f o r model B can be accounted f o r by the i n c r e a s e d area of h i g h v a l u e s of K i n the b a s i n , e s p e c i a l l y t h r o u g h o u t the s h a l l o w 0 . 0 !0 .0 2 0 . 0 30 .0 K I L O M E T E R S ( X 1 0 4 0 . 0 50 .0 6 0 . 0 Figure 25. Mesh diagram of model B, the 'pinch-out' basin. 1^3 co Figure 26. Hydraulic head versus log time plots comparing model B (case 1) with the standard model (case 2) at nodes in the high-K layers of the basins. 95 2 7 2 . 8 2 . 9 £ 3 . 0 3 . 1 O <D I 3 . 2 3 . 3 3.4 3 . 5 io- 10 6 1 — Years 10 7 10£ N o d e ID L a y e r 2L D e e p edge T 10V 10 10 Figure 27. Hydraulic head versus log time plots comparing model B (case 1) with the standard model (case 2) at nodes in the low-K layers of the basins. 96 p o r t i o n . T h i s a l l o w s h y d r a u l i c heads i n a l l of the t h r e e high-K l a y e r s to a d j u s t sooner. Changes occur f i r s t t hroughout the s h a l l o w edge of the b a s i n and through l a y e r 5H, then b e g i n to move l a t e r a l l y a l o n g the high-K l a y e r s and v e r t i c a l l y down through the upper-most low-K l a y e r 4L, and f i n a l l y move both v e r t i c a l l y up and down from the high-K l a y e r s i n t o the low-K l a y e r s . H y d r a u l i c head v a l u e s a t nodes i n high-K l a y e r s never reach s t e a d y s t a t e b e f o r e s u p e r j a c e n t nodes i n low-K l a y e r s , because of the monotonic p a t t e r n by which adjustment t a k e s p l a c e . Node 3S a t the s h a l l o w edge o f the b a s i n i n l a y e r 1H i n d i c a t e s t h a t the time r e q u i r e d f o r h y d r a u l i c head v a l u e s to reach s t e a d y s t a t e f o r the s t a n d a r d model i s a p p r o x i m a t e l y 69 Ma, seven to e i g h t o r d e r s - o f - m a g n i t u d e more than the 1.4 y e a r s r e q u i r e d by model B (see F i g u r e 26 C ) . Node 3S i n l a y e r 3H shows a s i m i l a r r e s u l t . T h i s i s o b v i o u s l y due to the l a c k of low-K l a y e r s s u p e r j a c e n t to these nodes i n model B and, t h e r e f o r e , adjustment i s not impeded as i t i s by t h e s e l a y e r s i n the s t a n d a r d model. Node 3S i n l a y e r 5H, however, shows the r e v e r s e t r e n d ; h y d r a u l i c heads r e q u i r e a p p r o x i m a t e l y one o r d e r - o f - m a g n i t u d e l o n g e r to reach s t e a d y s t a t e f o r the p i n c h - o u t model. One would expect to see no d i f f e r e n c e i n head adjustment between the two models a t t h i s p a r t i c u l a r node. The author i s unable to e x p l a i n t h i s s e e m i n g l y e r r a t i c r e s u l t . H y d r a u l i c head v e r s u s l o g time p l o t s f o r nodes i n the low-K l a y e r 2L are shown i n F i g u r e 27. Nodes ID and 2M, l o c a t e d near the deep edge and c e n t e r of the b a s i n , r e s p e c t i v e l y , i n l a y e r 4L are not a f f e c t e d by the p i n c h - o u t of the low-K l a y e r s and 97 t h e r e f o r e are not i l l u s t r a t e d . However, nodes ID and 2M i n l a y e r 2L ( F i g u r e 27 A and B) show a d e c r e a s e o f one to two o r d e r s - o f - m a g n i t u d e i n the time r e q u i r e d f o r h y d r a u l i c heads to reach s t e a d y s t a t e f o r model B. Nodes 3S i n l a y e r s 2L and 4L of the s t a n d a r d model are l o c a t e d i n high-K l a y e r s i n model B and t h e r e f o r e are not compared. Nodes i n l a y e r 4L remain u n a f f e c t e d by the p i n c h - o u t s because not enough time has e l a p s e d f o r l a t e r a l adjustment i n the s u b j a c e n t high-K l a y e r 3H to reach the c e n t r a l and deep p a r t s of the b a s i n p r i o r to the onset of adjustment a t b a s a l nodes i n l a y e r 4L. T h i s i s due to the f a c t t h a t the b a s i n i s 600 km lo n g and l a y e r 4L i s o n l y 0.8 km t h i c k a t i t s deepest edge. However, l a t e r a l adjustment i n l a y e r 1H at nodes ID and 2M o c c u r s j u s t p r i o r to the onset of adjustment a t b a s a l nodes 1H and 2M i n the s u p e r j a c e n t low-K l a y e r 2L. T h i s r e s u l t s i n adjustment time i n l a y e r 2L to be l e s s f o r model B as compared to the s t a n d a r d model. Contoured v a l u e s o f h y d r a u l i c head at f o u r d i f f e r e n t t i m e s d u r i n g the t r a n s i e n t event i n the p i n c h o u t b a s i n are shown i n F i g u r e 28 A to D. H y d r a u l i c head v a l u e s a t the s h a l l o w edge o f the b a s i n , where the h y d r a u l i c c o n d u c t i v i t y i s h i g h t h r o u g h o u t the b a s i n , are the f i r s t to reach s t e a d y s t a t e . I t i s shown i n F i g u r e 28 C t h a t h y d r a u l i c head decay i n the high-K l a y e r 3H does not decay h o r i z o n t a l l y f a s t e r than the r a t e of v e r t i c a l decay i n l a y e r 4L due to the time f a c t o r and the l o n g h o r i z o n t a l d i s t a n c e , as d i s c u s s e d p r e v i o u s l y . In l a y e r 1H, h o r i z o n t a l decay does occur f a s t e r than v e r t i c a l decay i n the s u p e r j a c e n t low-K l a y e r 2L (see F i g u r e 28 D ) , f o r reasons a l s o d i s c u s s e d p r e v i o u s l y . P I N C H 0 U T m 9 0.0 10.0 20.0 30.0 40.0 50.0 60 KILOMETERS -(X101 ) Figure 28. Transient equipotential plot of hydraulic head values at nodes in model B. A. Time = 1,000 days P I N C H o u r 0.0 10.0 20 .0 30 .0 4 0 . 0 KILOMETERS (X101 ) 50 .0 Figure 28. Transient equipotential plot of hydraulic head values at nodes in model B. B. Time = 137 years P I N C H 0 U T m q Figure 28. Transient equipotential plot of hydraulic head values at nodes in model B. C. Time = 2,740 years. i—• o o P I N C H 0 U T Figure 28. Transient equipotential plot of hydraulic head val nodes in model B. D. Time = .137 Ma. 102 The s t e a d y s t a t e p o t e n t i a l and v e l o c i t y p l o t s o f b o t h t h e s t a n d a r d b a s i n model and t h e p i n c h - o u t b a s i n model a r e shown i n F i g u r e s 29 t o 32. T hese f o u r f i g u r e s a l l have a l a r g e v e r t i c a l e x a g g e r a t i o n w h i c h d i s t o r t s t h e e q u i p o t e n t i a l p a t t e r n s . A c o m p a r i s o n o f t h e two p o t e n t i a l p l o t s ( F i g u r e s 29 and 3 1 ) , i n d i c a t e s t h a t t h e e q u i p o t e n t i a l l i n e s a t t h e d e e p and c e n t r a l p a r t s o f t h e b a s i n a r e s i m i l a r , e x c e p t f o r a h i g h e r v e r t i c a l g r a d i e n t i n l a y e r 2 o f t h e p i n c h - o u t m o d e l ( F i g u r e 3 1 ) . N e a r t h e s h a l l o w edge o f t h e p i n c h - o u t b a s i n , an a r e a o f d o m i n a n t l y h o r i z o n t a l f l o w t h r o u g h o u t t h e e n t i r e d e p t h o f t h e b a s i n i s c r e a t e d due t o t h e low-K l a y e r s p i n c h i n g o u t t h e r e . The v e l o c i t y p l o t s a l s o l o o k s i m i l a r f o r t h e two m o d e l s n e a r t h e deep and c e n t r a l p a r t s o f t h e b a s i n . N e a r t h e s h a l l o w e d g e , v e l o c i t i e s a r e g r e a t e r f o r t h e p i n c h - o u t m o d e l ( F i g u r e 32) and i n d i c a t e a h o r i z o n t a l d i r e c t i o n o f f l o w o n l y . A l t h o u g h t h e f l o w a r r o w s on F i g u r e s 30 and 32 a r e drawn a c r o s s t h e NE b o u n d a r y , t h i s i s a c t u a l l y a n o - f l o w b o u n d a r y . The l o n g a r r o w s a r e m e r e l y a f u n c t i o n o f t h e m a g n i t u d e o f t h e v e l o c i t y a c r o s s t h e e l e m e n t i n w h i c h t h e a r r o w o r i g i n a t e s . P o s s i b l e s o u r c e s o f e r r o r i n t h e s e d i a g r a m s a r e t h e o m i s s i o n o f r e c h a r g e and i n f i l t r a t i o n r a t e s , and t h e a s s u m p t i o n s t h a t t h e t o p o g r a p h i c b o u n d a r y c a n be r e p r e s e n t e d by t h e w a t e r t a b l e , and t h a t t h e l o c a l t o p o g r a p h y i s n e g l i g i b l e . 5.3. F l o w i n a B a s i n w i t h a N a t u r a l C o n d u i t 5.3.1. I n t r o d u c t i o n The p h a s e s o f h y d r o d y n a m i c e v o l u t i o n , i n t h e Red E a r t h r e g i o n , w h i c h a r e i n f e r r e d f r o m t h e p r e s e n t d i s t r i b u t i o n o f Figure 29. Steady state equipotential plot of the 'standard' model. S T A N D A R D S S MAXIMUM VELOCITY = 1.14E -7 0 .0 10.0 20 .0 30 .0 4 0 . 0 5 0 . 0 60 .0 KILOMETERS (X101 ) Figure 30. Steady state velocity plot of the 'standard' model. o Figure 31. Steady state equipotential plot of model B. P I N C H O U T S S MAXIMUM VELOCITY = 1.14E -7 o 0.0 10.0 20 .0 30 .0 40 .0 50 .0 60 .0 KILOMETERS (X101 ) Figure 32. Steady state velocity plot of model B. o 107 observed p r e s s u r e s are d e s c r i b e d i n d e t a i l by Toth (1978). He s t a t e s t h a t the seven phases p r e s e n t e d t h e r e are p u r e l y s p e c u l a t i v e and based on types of f o r c e s b e l i e v e d to dominate d u r i n g the e v o l u t i o n of a s e d i m e n t a r y b a s i n . The f i n a l s t a g e of the e v o l u t i o n of the Red E a r t h b a s i n i n v o l v e s a major change i n the hydrodynamics o f the r e g i o n i n which a m i d d l e zone of minimum p o t e n t i a l a t t r a c t s f l o w from both the s u b j a c e n t and s u p e r j a c e n t w a t e r - b e a r i n g l a y e r s . The low p o t e n t i a l of t h i s zone i s caused by i t s exposure a t low t o p o g r a p h i c l e v e l s due to an e r o s i o n a l event which took p l a c e d u r i n g or a f t e r the p r e v i o u s e v o l u t i o n a r y phase. Toth c o n c l u d e s t h a t f l o w i n the system below the exposed l o w - p o t e n t i a l zone i s m a i n t a i n e d by r e l i c t energy d i f f e r e n c e s and i s i n the t r a n s i e n t p r o c e s s of e q u a l i z a t i o n . To d e t e r m i n e the importance of t h i s type of p r o c e s s i n d e l a y i n g the h y d r a u l i c head a d j u s t m e n t s near the base of a mature b a s i n , model C, the ' c o n d u i t ' model, was d e v e l o p e d . I n i t i a l and boundary c o n d i t i o n s are the same as those used f o r the s t a n d a r d model, however at time = 0.274 Ma the upper boundary i s m o d i f i e d by i n s t a n t a n e o u s e r o s i o n of a s e c t i o n of rock near the s h a l l o w edge of the b a s i n . The e r o s i o n a l event causes the m i d d l e high-K l a y e r 3H to become exposed at the s u r f a c e . A l l o t h e r parameters and c o n d i t i o n s remain i d e n t i c a l •to t h o s e o f the s t a n d a r d model. F i g u r e 33 shows the mesh diagram a f t e r the e r o s i o n a l event a t time = 0.274 Ma. Figure 33. Mesh diagram of model C, the 'conduit' basin. 109 5.3.2. E f f e c t of c o n d u i t on the t r a n s i e n t h y d r a u l i c head adj ustments F i g u r e s 34 and 35 show s t e a d y s t a t e e q u i p o t e n t i a l and v e l o c i t y p l o t s , r e s p e c t i v e l y , f o r model C. When compared to s i m i l a r p l o t s f o r the s t a n d a r d model ( F i g u r e s 29 and 3 0 ) , q u i t e a d i f f e r e n c e can be seen i n both the h y d r a u l i c head g r a d i e n t s and the magnitude and d i r e c t i o n o f f l o w i n a l l p a r t s o f the system. E r o s i o n o f the l a n d s u r f a c e i n the c o n d u i t model a t the n o r t h e a s t e r n boundary of the f l o w r e g i o n causes g r e a t e r h o r i z o n t a l head g r a d i e n t s i n high-K l a y e r 3 and g r e a t e r v e r t i c a l head g r a d i e n t s i n low-K l a y e r s 2 and 4, e s p e c i a l l y near the s h a l l o w edge of the b a s i n where e r o s i o n has o c c u r r e d (compare F i g u r e s 29 and 3 4 ) . These h i g h e r g r a d i e n t s are m a i n l y a r e s u l t of c r e a t i n g a g r e a t e r t o p o g r a p h i c g r a d i e n t , i n d u c i n g g r e a t e r f l o w . The v e l o c i t y p l o t s ( F i g u r e s 30 and 35) i n d i c a t e a maximum v e l o c i t y o f g r e a t e r than two o r d e r s - o f - m a g n i t u d e more f o r the c o n d u i t model than f o r the s t a n d a r d model. The major d i f f e r e n c e s to be noted i n F i g u r e 35 as compared to F i g u r e 30 a r e : a) an i n c r e a s e i n s i z e of the r e c h a r g e a r e a ; b) c o n c e n t r a t i o n o f the f l o w towards high-K l a y e r 3H, the n a t u r a l c o n d u i t ; and c) the l a c k of f l o w i n the deep and c e n t r a l p a r t s o f the b a s i n i n low-K l a y e r 2L. These r e s u l t s are under the assumption t h a t the water t a b l e s t i l l m i r r o r s the topography and t h a t p r e c i p i t a t i o n i s s u f f i c i e n t to s u p p l y the i n c r e a s e d volume of groundwater f l o w . I t i s more l i k e l y t h a t , i n t h i s s i t u a t i o n , t h e r e would not be s u f f i c i e n t r e c h a r g e to e s t a b l i s h t h i s type of e q u i p o t e n t i a l p a t t e r n and t h e r e f o r e the water t a b l e may be l owered to r e - e s t a b l i s h the o l d g r a d i e n t s . However, f o r the p r e s e n t e v a l u a t i o n , r e c h a r g e r a t e s are n e g l e c t e d and the water N R T U R R L C O N D U I T Figure 34. Steady state equipotential plot of model C. Figure 3 5 . Steady state velocity plot of model C. 112 t a b l e i s assumed t o m i r r o r t h e t o p o g r a p h y . I n o r d e r t o compare t h e e f f e c t o f t h e n a t u r a l c o n d u i t on t h e t r a n s i e n t d e c a y o f h y d r a u l i c head v a l u e s t h r o u g h o u t t h e s y s t e m , h y d r a u l i c h e a d v e r s u s l o g t i m e p l o t s were c o n s t r u c t e d f o r n o des I D , 2M, and 3S i n a l l f i v e l a y e r s ( s e e F i g u r e 1 3 ) . P l o t s f o r t h o s e n odes w h i c h show s i g n i f i c a n t d i f f e r e n c e s b e t w e e n model C and t h e s t a n d a r d m o d el a r e i l l u s t r a t e d i n F i g u r e 36 A t o D. H y d r a u l i c h e a d v a l u e s a t n o des i n t h e t o p h i g h - K l a y e r 5H a r e u n a f f e c t e d by t h e c h a n g e i n b o u n d a r y c o n d i t i o n s b e c a u s e t h e y a r e a l r e a d y a t s t e a d y s t a t e a t t h e t i m e o f t h e s e c o n d e r o s i o n a l e v e n t . R e a d j u s t m e n t o f t h e i r e q u i l i b r i u m v a l u e s t o t h e new u p p e r b o u n d a r y o c c u r s i n s t a n t a n e o u s l y , g e o l o g i c a l l y s p e a k i n g . Nodes i n l a y e r s 3H and 4L show s i m i l a r r e s u l t s ; s t e a d y s t a t e h y d r a u l i c h e a d v a l u e s a r e l o w e r f o r t h e c o n d u i t m o d e l b u t t h e r e i s no c h a n g e i n t h e t i m i n g o f h e a d a d j u s t m e n t s ( s e e F i g u r e 36 C and D ) . Nodes i n l a y e r s 1H and 2L show s i m i l a r r e s u l t s ; s t e a d y s t a t e h y d r a u l i c head v a l u e s a r e l o w e r f o r t h e c o n d u i t m o d el and t h e r a t e o f d e c a y o f h e a d s i s s l i g h t l y g r e a t e r . However, h y d r a u l i c head v a l u e s a l l r e a c h s t e a d y s t a t e a t a p p r o x i m a t e l y t h e same t i m e f o r b o t h m o d e l s . I t i s c o n c l u d e d f r o m t h i s a n a l y s i s t h a t t h e e x p o s u r e o f a h i g h - K l a y e r a t t h e s u r f a c e a t some t i m e d u r i n g t h e t r a n s i e n t d e c a y o f h e a d s may a l t e r t h e s t e a d y s t a t e f l o w p a t t e r n w i t h i n t h e b a s i n b u t h a s l i t t l e e f f e c t on t h e t i m i n g o f h y d r a u l i c head a d j u s t m e n t s . 2.7 2.8 2.9 J 3.0 -D 3.1 Years 106 10' r . Node T ID — r T r "• r -Layer IH Deep edge i / / lh A 10' -T— Node ID Layer 3H Deep edge 10" 10' —|— Figure 36. Hydraulic head versus log time plots comparing model C (case 1) with the standard model (case 2) at various nodes within the basin. 114 5.4. P o t e n t i a l E r r o r s I n v o l v e d when Assuming Steady S t a t e Flow C o n d i t i o n s The purpose of t h i s s e c t i o n i s to p r o v i d e a rough e s t i m a t e of the magnitude of e r r o r which c o u l d be i n v o l v e d i n hydrodynamic c a l c u l a t i o n s i f s t e a d y s t a t e c o n d i t i o n s were assumed to p r e v a i l i n the s t a n d a r d b a s i n model, when i n a c t u a l f a c t the b a s i n i s s l o w l y a d j u s t i n g t o boundary c o n d i t i o n s imposed upon i t some time p r i o r to the p r e s e n t day. The example d i s c u s s e d here a p p l i e s o n l y to a r e g i o n a l b a s i n which has undergone i n s t a n t a n e o u s e r o s i o n of a p p r o x i m a t e l y 0.5 km. Hence, the e r r o r s i n v o l v e d do not n e c e s s a r i l y a p p l y to any o t h e r r e g i o n a l b a s i n ; each case has to be c o n s i d e r e d s e p a r a t e l y . E r r o r s may range from b e i n g n e g l i g i b l e - f o r example, a b a s i n e r o d i n g s l o w l y over time - to b e i n g v e r y s i g n i f i c a n t - f o r example, a b a s i n which has e x p e r i e n c e d i c e rebound. Node 2M, l o c a t e d a t m i d - b a s i n i n l a y e r 1H, the b a s a l l a y e r , i s used as an example f o r d i s c u s s i n g p o t e n t i a l e r r o r s . Graphs of l o g time v e r s u s h y d r a u l i c head, v e l o c i t y , and t r a v e l time to a d i s c h a r g e p o i n t , are p r e s e n t e d i n one diagram (see F i g u r e 37) i n o r d e r to a l l o w q u i c k comparison o f e s t i m a t e s f o r the t h r e e parameters a t a chosen t i m e . V e l o c i t y e s t i m a t e s f o r node 2M i n l a y e r 1H a r e s i m p l y an average o f the v e l o c i t i e s a c r o s s the s i x elements s u r r o u n d i n g the node; a l l elements are l o c a t e d i n l a y e r 1H. T r a v e l times a re determined f o r a p a r t i c l e t r a v e l l i n g w i t h the groundwater, h o r i z o n t a l l y from node 2M along the base o f the system i n l a y e r 1H and v e r t i c a l l y upwards th r o u g h the f i v e l a y e r s to the p o i n t o f d i s c h a r g e at the water t a b l e . Average l a y e r v e l o c i t i e s a r e used f o r the t r a v e l time e s t i m a t e s . S t e a d y S t a t e H y d r a u l i c H e a d (km) 0 4 8 12 16 20 V e l o c i t y (m/s x 1 0 - 1 2 ) -r-18 22 -T-2 6 3 0 T r a v e l T i m e ( y e a r s x 1 0 8 ) Figure 37. Log time versus hydraulic head, velocity, and travel time to a discharge point for node 2M in layer 1H, indicating potential error involved when using a steady state model versus a transient model. 116 The l o g time v e r s u s h y d r a u l i c head p l o t i s v e r y s i m i l a r to those seen i n p r e v i o u s diagrams and i n d i c a t e s a monotonic d e c l i n e i n h y d r a u l i c head towards the s t e a d y s t a t e v a l u e . The l o g time v e r s u s v e l o c i t y p l o t has a d i f f e r e n t shape; v e l o c i t y i s zero a t i n i t i a l t i m e , reaches a maximum which i s much g r e a t e r than the s t e a d y s t a t e v a l u e , and then s l o w l y decays to i t s s t e a d y s t a t e v a l u e . The i n i t i a l i n c r e a s e i n v e l o c i t y i s due to the h i g h head g r a d i e n t s c r e a t e d by the adjustment of h y d r a u l i c heads i n l a y e r 1 p r i o r to those i n l a y e r 2. The l o g time v e r s u s t r a v e l time to a d i s c h a r g e p o i n t shows a sharp decrease i n t r a v e l time i n i t i a l l y which then l e v e l s o f f , f o l l o w e d by a sharp i n c r e a s e i n t r a v e l time towards the s t e a d y s t a t e v a l u e . I t i s shown i n F i g u r e 37 t h a t , f o r t h i s p a r t i c u l a r c a s e , s i g n i f i c a n t e r r o r s c o u l d r e s u l t from assuming a s t e a d y s t a t e f l o w system i f the b a s i n was a t some s t a g e i n the t r a n s i e n t a d j u s t m e n t . A s t e a d y s t a t e a n a l y s i s would u n d e r e s t i m a t e head and v e l o c i t y v a l u e s and o v e r e s t i m a t e the t r a v e l t i m e , i n t h i s example. In o r d e r to e v a l u a t e whether or not a p a r t i c u l a r b a s i n i s at s t e a d y s t a t e or r e p r e s e n t s a c e r t a i n stage i n the t r a n s i e n t development of the f l o w system one must know: a) b a s i n p a r a m e t e r s , such as h y d r a u l i c c o n d u c t i v i t y and s p e c i f i c s t o r a g e ; b) the g e o l o g i c h i s t o r y and p a l e o t o p o g r a p h y of the b a s i n ; and c) the p r e s e n t day 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 b a s i n . The p r e s e n t day p r e s s u r e d i s t r i b u t i o n i s compared to a p r e s e n t day s t e a d y s t a t e model of the b a s i n . I f the observed p r e s s u r e systems cannot be e x p l a i n e d by c r o s s - f o r m a t i o n a l f l o w due to p r e s e n t day topography then those p r e s s u r e systems may i n c l u d e 117 m a n i f e s t a t i o n s o f r e l i c t t o p o g r a p h y , a n d / o r d e f o r m a t i o n a l s t r e s s e s o f t h e r o c k f r a m e w o r k . 118 CHAPTER 6 DISCUSSION 6.1. A p p l i c a t i o n s There e x i s t a wide range of problems i n v o l v i n g r e g i o n a l groundwater f l o w i n mature b a s i n s , many of which are i n v e s t i g a t e d under the assumption t h a t s t e a d y s t a t e c o n d i t i o n s p r e v a i l . A l t h o u g h t h i s assumption may be q u i t e v a l i d i n many s i t u a t i o n s , the p o s s i b l e e x i s t e n c e o f h y d r a u l i c head g r a d i e n t s and f l o w p a t t e r n s t h a t a re i n a t r a n s i e n t s t a t e due to p a l e o t o p o g r a p h y , may be o f importance f o r a n a l y s e s i n some mature b a s i n s . Toth and M i l l a r suggest the f o l l o w i n g examples i n which the e f f e c t of anachronous f o r m a t i o n p r e s s u r e s or h y d r a u l i c heads may be s i g n i f i c a n t : "...(1) s a f e development r a t e s of r e g i o n a l groundwater r e s o u r c e s i n a r i d r e g i o n s where e x i s t i n g h y d r a u l i c g r a d i e n t s may be r e l i c t f e a t u r e s i n s t e a d of i n d i c a t i n g c u r r e n t r e c h a r g e , (2) e x p l o r a t i o n f o r hydrocarbons or c e r t a i n m e t a l l i c m i n e r a l s where a c c u m u l a t i o n s may have been brought about by p a l e o s y s t e m s o f groundwater f l o w , (3) s a f e i s o l a t i o n o f r a d i o a c t i v e waste p r o d u c t s which might be m o b i l i z e d i n the f u t u r e by changes i n boundary c o n d i t i o n s , and (4) the d a t i n g o f g e o l o g i c a l e v e n t s t h a t r e s u l t i n changes of boundary c o n d i t i o n s o f s u b s u r f a c e f l o w f i e l d s a s , f o r i n s t a n c e : e r o s i o n a l d o w n c u t t i n g , f a u l t i n g , s e a l i n g by b u r i a l , d i a g e n e s i s , or c e m e n t a t i o n , e l a s t i c rebound upon u n l o a d i n g by d e n u d a t i o n o r m e l t i n g i c e , and so on." (Toth and M i l l a r , 1983, p.1585-86) C a l c u l a t i o n s over s e v e r a l t e n s to hundreds of y e a r s of s a f e development r a t e s of r e g i o n a l groundwater r e s o u r c e s i n any r e g i o n may be too h i g h i f they are based on h y d r a u l i c g r a d i e n t s which r e f l e c t a p r e v i o u s l y e x i s t i n g topography and which are i n the p r o c e s s o f s l o w l y a d j u s t i n g and d e c a y i n g to e q u i l i b r a t e w i t h 119 the p r e s e n t - d a y topography. T h i s becomes e s p e c i a l l y i m p o r t a n t i n a r i d r e g i o n s where p r e s e n t - d a y r e c h a r g e to the groundwater f l o w system i s v e r y l i m i t e d . The e x i s t e n c e of t r a n s i e n t h y d r a u l i c heads may a l s o a f f e c t the a c c u r a c y of c a l c u l a t i o n s of r e g i o n a l b a s i n water budgets. A c c u m u l a t i o n and s p a t i a l d i s t r i b u t i o n of h y d r o c a r b o n s may be c o n t i n u o u s l y m o d i f i e d as the t o p o g r a p h i c r e l i e f i s a l t e r e d due to e r o s i o n and/or d e p o s i t i o n . Depending on the magnitude of the s u r f a c e changes, i t i s p o s s i b l e t h a t r e l i c t h y d rocarbon a c c u m u l a t i o n s , which have developed under the i n f l u e n c e of a p a l e o t o p o g r a p h y u n l i k e t h a t e x i s t i n g a t p r e s e n t , may e x i s t i n l a r g e , mature b a s i n s . In the hydrodynamic approach to p e t r o l e u m e x p l o r a t i o n (Hubbert, 1953; T o t h , 1980) such h y d r o c a r b o n a c c u m u l a t i o n s may not be i d e n t i f i e d i f a s t e a d y s t a t e f l o w a n a l y s i s i s used. The t r a n s p o r t of c e r t a i n m e t a l l i c m i n e r a l s may a l s o be r e l a t e d to r e l i c t t o p o g r a p h i c a l f e a t u r e s . T6th and M i l l a r (1983, p.1593) s u g g e s t , f o r example, t h a t t r a n s i e n t a n a l y s e s may be r e q u i r e d f o r " . . . r e s e a r c h i n the g e n e s i s . . . o f g e o l o g i c p r o c e s s e s and phenomena i n v o l v i n g s u b s u r f a c e h y d r a u l i c t r a n s p o r t of m a t t e r and/or time v a r i a t i o n s of p o r e - p r e s s u r e s on a r e g i o n a l s c a l e such a s , f o r example, the o r i g i n of c e r t a i n t y p e s of uranium and s u l f i d e ore d e p o s i t s , t h e r m a l a n o m a l i e s , and so on." R a d i o a c t i v e waste d i s p o s a l i s a major c o n c e r n , p r e s e n t l y , t h r o u g h o u t the w o r l d . Many p r o s p e c t i v e r e p o s i t o r i e s are l o c a t e d w i t h i n a r e g i o n a l groundwater f l o w system and p r e d i c t i o n s c o n c e r n i n g t r a v e l time of wastes to a p o i n t o f d i s c h a r g e sometimes i n v o l v e thousands to m i l l i o n s of y e a r s . There are two 120 ways to view the p o s s i b l e a p p l i c a t i o n of a t r a n s i e n t groundwater f l o w a n a l y s i s of a p o t e n t i a l d i s p o s a l s i t e . On the one hand, one might c o n s i d e r the p o s s i b i l i t y of f u t u r e changes i n boundary c o n d i t i o n s d i s t u r b i n g p r e s e n t h y d r a u l i c head p a t t e r n s i n the b a s i n , c a u s i n g m o b i l i z a t i o n of the waste. A more l i k e l y a p p l i c a t i o n , on the o t h e r hand, i s the use of such a model i n p r e d i c t i n g t r a v e l time to the s u r f a c e , s h o u l d the waste e n t e r the r e g i o n a l groundwater f l o w system. In some c a s e s , a more a c c u r a t e p r e d i c t i o n of t r a v e l t i m e s and f l o w r a t e s may be o b t a i n e d w i t h a t r a n s i e n t a n a l y s i s , s h o u l d the r e g i o n a l b a s i n be s t i l l u ndergoing h y d r a u l i c head adju s t m e n t s due to a p r e v i o u s change i n the upper boundary c o n d i t i o n . 6.2. L i m i t a t i o n s There are s e v e r a l l i m i t a t i o n s and assumptions i n h e r e n t i n the f i n i t e element model, which are summarized below. 1) The model i s s p a t i a l l y t w o - d i m e n s i o n a l and assumes t h a t none of the parameters v a r y i n the t h i r d d i m e n s i o n . 2) The e q u a t i o n o f f l o w assumes t h a t changes i n f l u i d d e n s i t y and v i s c o s i t y i n space are n e g l i g i b l e ( i . e . the e f f e c t s of i n c r e a s e temperature and s a l i n i t y a t d e p t h , r e s u l t i n g i n a d e c r e a s e i n both f l u i d d e n s i t y and v i s c o s i t y are n e g l e c t e d ) . 3) The e q u a t i o n of f l o w n e g l e c t s changes i n the s t r e s s / s t r a i n p a t t e r n over t i m e . 4) The b a s i n i s f u l l y s a t u r a t e d . 5) The s i m p l i f i e d geometry of the model i s assumed to be a r e a l i s t i c r e p r e s e n t a t i o n o f the Red E a r t h r e g i o n . 6) Groundwater v e l o c i t y i s l i n e a r . 121 7) Only 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 i n the s i m u l a t i o n s and, hence, h y d r a u l i c head adjustment times f o r b a s i n s h a v i n g 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 s due to a p r e - e x i s t i n g t o p o g r a p h i c c o n f i g u r a t i o n are not c o n s i d e r e d . 8) The a c c u r a c y of the r e s u l t s p r e s e n t e d f o r the Red E a r t h r e g i o n , model A, i s l i m i t e d p a r t l y by the a c c u r a c y of parameter v a l u e s i n p u t to the model. 9) R ates of i n f i l t r a t i o n and r e c h a r g e from the s u r f a c e are not c o n s i d e r e d . 10) E r o s i o n which o c c u r s over thousands to m i l l i o n s of y e a r s i s assumed to be i n s t a n t a n e o u s . 6.3. C o n c l u s i o n s T h i s s t u d y i s concerned w i t h t r a n s i e n t c o n d i t i o n s of r e g i o n a l groundwater f l o w i n g e o l o g i c a l l y mature, heterogeneous b a s i n s w i t h 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 up to e i g h t o r d e r s - o f - m a g n i t u d e . Changes i n the f l o w regime, caused by changes i n the c o n f i g u r a t i o n of the water t a b l e due to e r o s i o n of the l a n d s u r f a c e , are n u m e r i c a l l y s i m u l a t e d u s i n g a 2-D, t r a n s i e n t f i n i t e element model of a s i m p l i f i e d v e r s i o n of the Red E a r t h r e g i o n , A l b e r t a , Canada. The model i s v e r i f i e d by comparing i t s r e s u l t s w i t h t h o s e o f s e v e r a l 1-D and 2-D s t e a d y s t a t e and 1-D t r a n s i e n t a n a l y t i c a l s o l u t i o n s . S e v e r a l n u m e r i c a l problems were encountered d u r i n g the development of the f i n i t e element computer program. The immense s t r e s s p l a c e d on the upper boundary, due to i n s t a n t e r o s i o n of a p p r o x i m a t e l y 0.5 km of r o c k , and s t r e s s e s on i n t e r i o r b o u n d a r i e s between l a y e r s o f v a r y i n g h y d r a u l i c c o n d u c t i v i t i e s , 122 due to the l a r g e c o n t r a s t i n the v a l u e s of K, cause o s c i l l a t i o n s o f h y d r a u l i c head v a l u e s a t nodes when they f i r s t a d j u s t to t h i s new upper boundary c o n d i t i o n . E v e n t u a l l y , the o s c i l l a t i o n s dampen and h y d r a u l i c head v a l u e s b e g i n to decay. O s c i l l a t i o n s do not oc c u r i f 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 a re s m a l l . For 1- D heterogeneous s o l u t i o n s , o s c i l l a t i o n s , a l t h o u g h t r o u b l e s o m e , do not appear to s i g n i f i c a n t l y a l t e r the r e s u l t s , as shown by the agreement between the 1-D f i n i t e element s o l u t i o n and Toth and M i l l a r ' s (1983) 1-D a n a l y t i c a l s o l u t i o n (see Chapter 3 ) . T h e r e f o r e , the a u t h o r c o n c l u d e s t h a t o s c i l l a t i n g h y d r a u l i c head v a l u e s a l s o have no s i g n i f i c a n t e f f e c t on the a c c u r a c y of the 2- D f i n i t e element s o l u t i o n s . H y d r a u l i c head v a l u e s a t c e r t a i n nodes w i t h i n the b a s i n a l s o o s c i l l a t e a f t e r r e a c h i n g s t e a d y s t a t e . The c o n t r o l l i n g f a c t o r here appears to be the l e n g t h to depth r a t i o s of the elements. I f t h e s e r a t i o s are too l a r g e , the s o l u t i o n i s u n s t a b l e and tends to o s c i l l a t e around the s t e a d y s t a t e v a l u e , w i t h the o s c i l l a t i o n s i n c r e a s i n g w i t h t i m e . These are reduced to an i n s i g n i f i c a n t l y s m a l l v a l u e by d e c r e a s i n g l e n g t h to depth r a t i o s of the elements. The major o b j e c t i v e o f the s t u d y i s to d e v e l o p an u n d e r s t a n d i n g o f the time-dependent n a t u r e o f h y d r a u l i c head a d j u s t m e n t s w i t h i n a heterogeneous mature b a s i n . T h i s i s a c c o m p l i s h e d by i n v e s t i g a t i n g the s e n s i t i v i t y of the f i n i t e element model to s e v e r a l b a s i n p a r a m e t e r s , namely h y d r a u l i c c o n d u c t i v i t y , a n i s o t r o p y , s p e c i f i c s t o r a g e , water t a b l e c o n f i g u r a t i o n , and b a s i n s i z e , and by modeling and a n a l y z i n g f l o w p a t t e r n s i n h y p o t h e t i c a l b a s i n s of v a r y i n g s t r u c t u r a l and 123 s t r a t i g r a p h i c c o n f i g u r a t i o n s . The r e s u l t s o f the s e n s i t i v i t y a n a l y s i s can be summarized as f o l l o w s : 1) The model i s f a r more s e n s i t i v e to changes i n the v a l u e s of h y d r a u l i c c o n d u c t i v i t y o f i s o t r o p i c low-K l a y e r s than of i s o t r o p i c high-K l a y e r s . A one o r d e r - o f - m a g n i t u d e decrease i n the v a l u e of K i n an i s o t r o p i c low-K l a y e r r e s u l t s i n an i n c r e a s e of up to two o r d e r s - o f - m a g n i t u d e i n the time r e q u i r e d f o r h y d r a u l i c head v a l u e s to reach s t e a d y s t a t e i n the p e r t u r b e d l a y e r and the s u b j a c e n t high-K l a y e r . 2) The f l o w system i s more s e n s i t i v e to p e r t u r b a t i o n s of the z - d i r e c t i o n component of K than i t i s to the x - d i r e c t i o n component of K i n a n i s o t r o p i c l a y e r s . T h i s e f f e c t i s more pronounced i n low-K l a y e r s than i t i s i n high-K l a y e r s . A one o r d e r - o f - m a g n i t u d e i n c r e a s e i n the r a t i o K :K i n a n i s o t r o p i c Z X low-K l a y e r s r e s u l t s i n a d e c r e a s e of one o r d e r - o f - m a g n i t u d e o r more f o r the time r e q u i r e d f o r h y d r a u l i c head v a l u e s to reach s t e a d y s t a t e a t a l l nodes w i t h i n the b a s i n . 3) A decrease i n the v a l u e of s p e c i f i c s t o r a g e of two o r d e r s - o f - m a g n i t u d e i n a high-K l a y e r a f f e c t s both the p e r t u r b e d l a y e r and the s u p e r j a c e n t low-K l a y e r by i n c r e a s i n g the time r e q u i r e d f o r h y d r a u l i c heads to reach s t e a d y s t a t e by two to f o u r t i m e s . V a r y i n g the v a l u e of S g i n low-K l a y e r s has l i t t l e e f f e c t on the system. 4) The model i s much more s e n s i t i v e to v a r i a t i o n s i n t h i c k n e s s e s o f the l a y e r s than i t i s to v a r i a t i o n s i n l e n g t h . D o u b l i n g the t h i c k n e s s of a l l l a y e r s i n the b a s i n r e s u l t s i n an i n c r e a s e o f t h r e e to f i v e t i m e s i n the h y d r a u l i c head adjustment 124 t i m e . 5) Changing the g e n e r a l shape of the r e g i o n a l water t a b l e has l i t t l e to no e f f e c t on the t i m i n g of h y d r a u l i c head adjustment w i t h i n the b a s i n . The f o l l o w i n g b a s i n s were modeled i n o r d e r to f u r t h e r u n d erstand the f a c t o r s which c o n t r o l t r a n s i e n t decay of h y d r a u l i c head v a l u e s i n heterogeneous mature b a s i n s : i ) model A, the Red E a r t h r e g i o n , A l b e r t a , Canada - a f i v e - l a y e r b a s i n w i t h l a r g e 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 between l a y e r s ; i i ) model B, a h y p o t h e t i c a l b a s i n i n which the low-K l a y e r s p i n c h out near the s h a l l o w edge; and i i i ) model C, a h y p o t h e t i c a l b a s i n i n which a n a t u r a l c o n d u i t d e v e l o p s d u r i n g the t r a n s i e n t adjustment o f heads, due to the e r o s i o n of rock near the s h a l l o w edge o f the b a s i n . The remainder o f t h i s s e c t i o n d i s c u s s e s c o n c l u s i o n s drawn as a r e s u l t of these s i m u l a t i o n s . M o d e l i n g the Red E a r t h b a s i n ensures t h a t a r e a l i s t i c s e t of v a l u e s are used as i n p u t to the model and a r e a l i s t i c s e t of r e s u l t s are o b t a i n e d from the s i m u l a t i o n . S e v e r a l i m p o r t a n t c o n c l u s i o n s can be drawn from t h i s s i m u l a t i o n r e g a r d i n g the g e n e r a l n a t u r e o f t r a n s i e n t groundwater f l o w i n he t e r o g e n e o u s , mature b a s i n s w i t h l a r g e 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 : 1) h y d r a u l i c head v a l u e s i n low-K l a y e r s a d j u s t a t a much s l o w e r r a t e than those i n high-K l a y e r s ; 2) v e r t i c a l adjustment of h y d r a u l i c heads i n high-K l a y e r s s u b j a c e n t to low-K l a y e r s i s e f f e c t i v e l y i n s t a n t a n e o u s throughout any v e r t i c a l column i n the high-K l a y e r s ; 3) decay of h y d r a u l i c heads i n the b a s i n i s a f f e c t e d by v e r t i c a l changes from above much more than by h o r i z o n t a l changes; and 4) i n some i n s t a n c e s , nodes i n high-K 125 l a y e r s c a n b e g i n i n i t i a l a d j u s t m e n t s o o n e r t h a n nodes l o c a t e d d i r e c t l y a bove i n a low-K l a y e r ; h o w e v e r , t h e r a t e o f d e c a y a t no d e s i n t h e s u p e r j a c e n t l a y e r s o o n o v e r t a k e s t h a t o f t h e h i g h - K l a y e r . I n g e n e r a l , t h e r a t e o f a d j u s t m e n t i n h i g h - K l a y e r s a p p e a r s t o be c o n t r o l l e d by t h e r a t e o f a d j u s t m e n t i n s u p e r j a c e n t low-K l a y e r s and h y d r a u l i c h e a d s a r e n o t a b l e t o d e c a y much f a s t e r i n t h e h i g h - K l a y e r s . By c o m p a r i n g m odel A, t h e 2-D Red E a r t h m o d e l , w i t h t h e 1-D m o d e l s o f C h a p t e r 3, i t i s e v i d e n t t h a t t h e 2-D model p r e d i c t s t h a t a l e s s e r amount o f t i m e i s r e q u i r e d f o r h y d r a u l i c head v a l u e s t o c o m p l e t e l y a d j u s t t o p r e s e n t b o u n d a r y c o n d i t i o n s . Head v a l u e s i n t h e b a s a l h i g h - K l a y e r o f t h e 2-D model b e g i n t o a d j u s t s o o n e r t h a n i n t h e 1-D m o d e l s . However, t h e r a t e o f a d j u s t m e n t d e c r e a s e s a t l a t e r t i m e s , p o s s i b l y due t o t h e s u p e r j a c e n t low-K l a y e r . The 1-D model p r e d i c t s t h e amount o f t i m e r e q u i r e d f o r c o m p l e t e a d j u s t m e n t o f h y d r a u l i c h e a d v a l u e s t o be a p p r o x i m a t e l y one and a h a l f t i m e s g r e a t e r t h a n t h a t p r e d i c t e d b y t h e 2-D m o d e l . T h i s c a n be a s i g n i f i c a n t d i f f e r e n c e on a g e o l o g i c t i m e s c a l e . The 2-D model g i v e s a much more a c c u r a t e p i c t u r e o f t h e t r a n s i e n t r a t e o f c h a n g e o f h y d r a u l i c head v a l u e s . * S i m u l a t i o n s i n w h i c h low-K l a y e r s p i n c h o u t , model B, i n d i c a t e t h a t t h i s c o n f i g u r a t i o n c a n have a s i g n i f i c a n t e f f e c t on t h e t i m e r e q u i r e d f o r c o m p l e t e head a d j u s t m e n t . M o d e l B o n l y r e q u i r e s o n e - f i f t h o f t h e t i m e n e e d e d by t h e s t a n d a r d model f o r nodes i n t h e b a s a l l a y e r t o a d j u s t , and o n e - t e n t h t o o n e - h u n d r e d t h o f t h e t i m e r e q u i r e d f o r nodes i n t h e s u p e r j a c e n t low-K l a y e r t o a d j u s t . A l l nodes n e a r t h e s h a l l o w edge o f t h e 126 basin require six to eight orders-of-magnitude longer to adjust for the standard model than for model B. For model C, a natural conduit i s created by introducing an erosional event part of the way through the transient decay of hydraulic head values in the standard model; t h i s causes the high-K layer 3H to be exposed at the surface. This has a very i n s i g n i f i c a n t e f f e c t on the timing of adjustment of hydraulic heads within the basin. It appears to al t e r the steady state equipotential plot and flow net, but the transient e f f e c t i s neglig i b l e . It i s evident that much more work can be performed using hypothetical basins; the p o s s i b i l i t i e s , in fact, are endless. However, the author feels that analysis of the examples presented in th i s thesis provides a s i g n i f i c a n t advance in the understanding of transient regional flow in mature basins with large hydraulic conductivity contrasts. 127 REFERENCES CITED B o y c e , W. E., and R. C. D i P r i m a . 1977. E l e m e n t a r y D i f f e r e n t i a l  E q u a t i o n s and B o u n d a r y V a l u e P r o b l e m s . 3 r d . e d . , J o h n W i l e y and S o n s , I n c . , New Y o r k , N.Y.. B r e d e h o e f t , J . D., and B. B. Hanshaw, 1968, On t h e m a i n t e n a n c e o f a n o m a l o u s f l u i d p r e s s u r e s : I . T h i c k s e d i m e n t a r y s e q u e n c e s : G e o l . S o c . Am. B u l l . , v . 79, p. 1 0 9 7-1106. C a r s l a w , H. S., and J . C. J a e g e r . 1959. 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T o t h , J . , 1978, G r a v i t y - i n d u c e d c r o s s - f o r m a t i o n a l f l o w o f f o r m a t i o n f l u i d s , Red E a r t h r e g i o n , A l b e r t a * C a n a d a : A n a l y s i s , p a t t e r n s , and e v o l u t i o n : W a t e r R e s . R e s . , v . 14, no. 5, p. 8 0 5 - 8 4 3 . T o t h , J . , 1979, P a t t e r n s o f d y n a m i c p r e s s u r e i n c r e m e n t o f f o r m a t i o n - f l u i d f l o w i n l a r g e d r a i n a g e b a s i n s , e x e m p l i f i e d by t h e Red E a r t h r e g i o n , A l b e r t a , C a n a d a : B u l l . Can. P e t . G e o l . , v . 2 7 , no. 1, p. 63-86. T o t h , J . , 1980, C r o s s - f o r m a t i o n a l g r a v i t y - f l o w o f g r o u n d w a t e r : A m e c h a n i s m o f t h e t r a n s p o r t and a c c u m u l a t i o n o f p e t r o l e u m ( t h e g e n e r a l i z e d h y d r a u l i c t h e o r y o f p e t r o l e u m m i g r a t i o n ) ; i n W. H. R o b e r t s , I I I , and R. J . C o r d e l l , e d s . , P r o b l e m s o f P e t r o l e u m M i g r a t i o n : T u l s a , A.A.P.G. S t u d i e s i n G e o l o g y No. 10, p. 121-167. T o t h , J . , and R. F. M i l l a r , 1 9 8 3 , P o s s i b l e e f f e c t s o f e r o s i o n a l c h a n g e s o f t h e t o p o g r a p h i c r e l i e f on p o r e p r e s s u r e s a t d e p t h : W a t e r R e s . R e s . , v . 19, n o . 6, p. 1 5 8 5 - 1 5 9 7 . Wang, H. F., and M. P. A n d e r s o n . 1982. I n t r o d u c t i o n t o G r o u n d w a t e r M o d e l i n g . W. H. Freeman and Co., San F r a n c i s c o . 129 APPENDIX L i s t i n g o f 2 D T F a t 0 9 : 5 0 : 0 0 o n MAR 2 5 , 1 9 8 6 f o r C C 1 d = F A U L P a g e 1 1 2 3 4 5 6 7 8 9 1 0 1 1 12 13 14 15 16 17 18 19 2 0 21 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 31 3 2 3 3 3 4 3 5 3 6 3 7 3 8 3 9 4 0 4 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 5 0 51 5 2 5 3 5 4 5 5 5 6 5 7 5 8 * * * * * * * * * * * * * * * * * * * * T W O - D I M E N S OR S T E A D Y W I T H U P TO B E T W E E N L A * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * I O N A L F I N I T E E L E M E N T P R O G R A M TO D E T E R M I N E T H E T R A N S I E N T FLOW R E G I M E OF A G E O L O G I C A L L Y M A T U R E , H E T E R O G E N E O U S B A S I N , S I X O R D E R S OF M A G N I T U D E C O N T R A S T I N H Y D R A U L I C C O N D U C T I V I T Y Y E R S . D E L T A T , T H E T I M E I N C R E M E N T , I S V A R I A B L E . W R I T T E N BY L I N D Y A . E N G L A N D * * * * * * * * * * * * * * '***************************• ***************************** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * L I S T O F V A R I A B L E S : AB A K N P AR A T B T C O N I N T C O N Z DT DX D Z E C M ECMT F T S GCM GCMR H C X H C Z IBW I C O N I FLOW I H E A D I L A B I M E S H I N I P E I P L I P R I N I P R O P I Q P L O T = I T A G = I T I T L E = I T Y P E = I V P L O T = J T A G = K A S E = S L O P E OF B A S E M E N T ROCK H E A D V A L U E S OF K N P N O D E S A R E A O F E L E M E N T S L O P E OF WATER T A B L E Z - C O O R D OF WATER T A B L E AT X = 0 . C O N T O U R I N T E R V A L H E A D V A L U E O F L O W E S T C O N T O U R T I M E I N C R E M E N T D I S C R E T I Z A T I O N I N X D I R E C T I O N D I S C R E T I Z A T I O N I N Z D I R E C T I O N E L E M E N T C O E F F I C I E N T M A T R I X T I M E E L E M E N T C O E F F I C I E N T M A T R I X F I R S T T I M E S T E P TO C A L C U L A T E %CHANGE G L O B A L C O E F F I C I E N T M A T R I X G L O B A L C O E F F I C I E N T M A T R I X OF T H E RHS H Y D R A U L I C C O N D U C T I V I T Y OF L A Y E R I N X-H Y D R A U L I C C O N D U C T I V I T Y OF L A Y E R I N Z-F U L L B A N D W I D T H O F M A T R I X N O . OF H E A D C O N T O U R S 1 : S T E A D Y F L O W ; 2 T R A N S I E N T FLOW I N I T I A L H E A D C O N D I T I O N S ; 1 : S T A T I C ; O : D O N ' T L A B E L C O N T O U R S ON H P L O T ; 1 0 : D O N ' T P L O T M E S H ; 1 : P L 0 T M E S H E L E M E N T I N C I D E N C E S N O . O F N O D E S L Y I N G ON B O U N D A R I E S NODE N O . OF N O D E S L Y I N G ON B O U N D A R I E S 1 : P R I N T M E S H I N F O R M A T I O N ; 2 : D 0 N ' T P R I N T M E S H I N F O 1 : P R I N T R E S U L T S AT A L L N O D E S A S C H A N G E I N H E A D ; 2 : P R I N T R E S U L T S AT S P E C I F I E D N O D E S ; 3 : P R I N T R E S U L T S AT A L L N O D E S AS A B S O L U T E H E A D ; 4 : D O B O T H 2 AND 3 ; 0 : N 0 P L O T ; 1 : P L 0 T M E S H O N L Y ; 2 : P L 0 T M E S H 3 : P L 0 T O N L Y ; H E A D V A L U E S R E A D FROM F I L E ; P A R A M E T E R T A G F O R E A C H E L E M E N T T I T L E OF P L O T T Y P E O F N O D E : 1 = C O N S T A N T H E A D ; 0=UNKNOWN H E A D O : D O N ' T P L O T V E L O C I T Y A R R O W S ; 1 : P L O T V E L O C I T Y ARROWS H Y D R A U L I C C O N D . T A G FOR R E G I O N S D E F I N E D BY K T A G U N C O N F I N E D A Q U I F E R 1 : W T = C O N S T A N T H E A D , O T H E R S = N O FLOW 2 : W T AND LOWER BNDR = C O N S T A N T H E A D , O T H E R S =N0 FLOW D I R E C T I O N ( K M / D A Y ) D I R E C T I O N ( K M / D A Y ) 2 , 0 : S T E A D Y S T A T E : L A B E L C O N T O U R S H E A D A N D / O R V E L O C I T Y 4 : P L O T N O D E S O N L Y O L i s t i n g o f 2 D T F a t 0 9 : 5 0 : 0 0 o n MAR 2 5 , 1 9 8 6 f o r C C i d = F A U L P a g e 2 5 9 C 6 0 c 61 c 6 2 c 6 3 c 6 4 c 6 5 c 6 6 c 6 7 c 6 8 c 6 9 c 7 0 c 71 c 72 c 7 3 c 7 4 c 7 5 c 7 6 c 7 7 c 7 8 c 7 9 c 8 0 c 8 1 c 8 2 c 8 3 c 8 4 c 8 5 c 8 6 c 8 7 c 8 8 c 8 9 c 9 0 c 9 1 c 9 2 c 9 3 c 9 4 c 9 5 c 9 6 c 9 7 c 9 8 c 9 9 c 1 0 0 c 101 c 102 c 1 0 3 c 104 c 105 c 1 0 6 c 107 c 108 c 1 0 9 c 1 1 0 c 111 c 112 c 113 c 114 c 1 15 c 116 c 3 : C 0 N F I N E D A Q U I F E R , END B N D R S = C O N S T A N T H E A D , 0 T H E R S = N 0 FLOW AND H E T E R O G E N E O U S 4 : K A S E 3 B U T HOMOGENEOUS 5 : K A S E 1 W I T H S L O P I N G WT B E G I N N I N G AT X 6 : K A S E 1 W I T H WATER T A B L E O T H E R T H A N L I N E A R 7 : K A S E 1 W I T H A C H A N G E I N BOUNDARY C O N D I T I O N S AT T I M E = T C H KNP - NODE N O S . L Y I N G ON C O N S T A N T HEAD BOUNDARY K T A G = E L E M E N T N O . AT WHICH C O N D U C T I V I T Y C H A N G E S L B = H A L F BAND W I D T H ( I N C L U D I N G D I A G O N A L ) L C » P A R A M E T E R . FOR P A R T I T I O N I N G M A T R I X M T Y P E = T Y P E OF M E S H : 1 - V A R I A B L E S P A C I N G , NODES NUMBERED I N X - D I R E C T I O N 2 = V A R I A B L E S P A C I N G , NODES N U M B E R E D I N Z - O I R E C T I O N , K A S E 2 ONLY 3 = C 0 N S T A N T S P A C I N G , NODES NUMBERED I N X - D I R E C T I O N 4 = C 0 N S T A N T S P A C I N G , NODES NUMBERED I N Z - D I R E C T I O N , K A S E 2 ONLY NC = NUMBER OF C O N S T A N T H E A D N O D E S N C H = N O . OF A D D I T I O N A L N O D E S A S S I G N E D C O N S T A N T H E A D O T H E R T H A N WT NCHN = NODE N O . OF C O N S T A N T HEAD N O D E S O T H E R T H A N WT ND = N O . OF D I V I S I O N S A L O N G X - A X I S OF V A R Y I N G D E L T A X NE = NUMBER OF E L E M E N T S NEX = NUMBER OF E L E M E N T S I N X - D I R E C T I O N A L O N G ROW NF = NUMBER OF D E G R E E S OF F R E E D O M (UNKNOWN H E A D S ) N L = NUMBER OF L A Y E R S OF V A R Y I N G M A T E R I A L T Y P E NN - NUMBER OF N O D E S NNPR = NUMBER OF N O D E S AT WHICH R E S U L T S P R I N T E D NODE = NODE N O . OF NODES AT WHICH R E S U L T S P R I N T E D N S L = NODE N O . , A L O N G LOWER B N D R Y . AT W H I C H WT S L O P E C H A N G E S ( K A S E 5 O N L Y ) N T A G = NUMBER OF T A G S FOR C H A N G E S I N P A R A M E T E R S ( E L E M E N T N O . ) N T P R = N O . OF T I M E S A T WHICH TO P R I N T R E S U L T S 1 : P R I N T FOR A L L T I M E S ; > 1 : P R I N T FOR S P E C I F I E D T I M E S N T P L = NUMBER OF T I M E S TO P L O T R E S U L T S N T S = T O T A L NUMBER OF T I M E S T E P S NX = NUMBER OF N O D E S A L O N G X - A X I S ( L E N G T H ) NXD = N O . OF N O D E S I N X - D I R E C T I O N OF E A C H D I V I S I O N , ND ( I N C L U D I N G O N L Y 1 BNDRY NODE FOR E A C H D I V I S I O N ) NZ = NUMBER OF N O D E S A L O N G Z - A X I S ( D E P T H ) N Z L - N O . OF N O D E S I N Z - D I R E C T I O N OF E A C H L A Y E R ( I N C L U D I N G ONLY ONE BOUNDARY NODE FOR E A C H L A Y E R ) POR = P O R O S I T Y OF M A T E R I A L SC = S C A L I N G F A C T O R FOR D E T E R M I N I N G Z T A G S S / S S L = S P E C I F I C S T O R A G E ( 1 / K M ) S T P H = S T O R E D "/.HEAD FOR P R I N T I N G AT S P E C I F I E D N O D E S S T H = S T O R E D H E A D FOR P R I N T I N G AT S P E C I F I E D N O D E S T = A B S O L U T E T I M E ( D A Y S ) T C H « T I M E AT W H I C H BOUNDARY C O N D I T I O N S C H A N G E T H I = T H I C K N E S S OF S E C T I O N ( K M ) T O L = % H E A D C H A N G E ALLOWED D U R I N G T I M E S T E P T P L = T I M E S AT W H I C H TO P L O T HEAD AND V E L O C I T Y N E T S TPR = T I M E S AT W H I C H TO P R I N T R E S U L T S V B = V E C T O R OF BOUNDARY C O N D I T I O N S VC = V E C T O R OF C O N S T A N T H E A D S ( K M ) V E L X = V E L O C I T Y OF R E C T . E L E M E N T I N X - D I R E C T I O N ( K M / D A Y ) V E L Z = V E L O C I T Y OF R E C T . E L E M E N T I N Z - D I R E C T I O N ( K M / D A Y ) V I = I N I T I A L H E A D AT T I M E ZERO V S L = NEW V A L U E OF S L O P E FOR K A S E 5 VT = V E C T O R OF T O T A L HEAD C H A N G E FROM T I M E Z E R O TO S T E A D Y S T A T E VU = V E C T O R OF KNOWN AND UNKNOWN H E A D S ( K M ) L i s t i n g of 2DTF at 09:50:00 on MAR 25, 1986 for CCid=FAUL P a g e 3 1 16.2 C VUI = INITIAL HEADS FOR STATIC INITIAL CONDITIONS 117 C X = X COORDINATE OF NODE 1 18 C XCEN = X-COORD OF CENTER OF RECTANGULAR ELEMENT 1 19 C XLEN = LENGTH OF SECTION (KM) 120 C XMAX = MAXIMUM X COORDINATE 121 C XSIZE = REAL SIZE OF PLOT IN X-DIRECTION 122 C XT = SINGLE PRECISION VALUE OF TIME FOR PLOTTING 123 C XX = X-COORO AT WHICH NODE SPACING CHANGES ALONG X-AXIS 124 C YH = SINGLE PRECISION VALUE OF HEAD FOR PLOTTING 125 C Z = Z COORDINATE OF NODE 126 C ZCEN = Z-COORD OF CENTER OF RECTANGULAR ELEMENT 127 C ZMAX = MAXIMUM Z COORDINATE 128 C ZMIN = DATUM 129 C ZSIZE = REAL SIZE OF PLOT IN Z-DIRECTION 130 C ZZ = Z-COORD OF TOP OF LAYER AT X=0 131 C ZZTAG/ZTAG = Z-COORD OF TOP OF EACH LAYER AT X 132 C 133 C 134 IMPLICIT REAL*8 (A-H.O-Z) 135 REAL*4 XT(60), YH(60). YH2 (60) 136 DIMENSION ITAG(600), HCXL(5), HCX(600), HCZL(5), HCZ(600), 137 1IN(600,3), X(340), Z(340). VC(70), VI(340). VU(340,2), ECM(3,3). 138 2GCM(340, 15) , GCMR(340,30) , A(3), B(3). C O ) , AR( 600) , ITYPE ( 340) . 139 3LC(340), ZZ(5). KTAG(5), JTAG(5), SS(600), SSL(5), TPR(60). 140 4VB(340), ECMT(3,3), NZL(5), VELX(600), VELZ(600), VT(340), 141 5NCHN(70), IPL(80), TPL(5), N0DE(15). STH(15,60), STPH(15,60), 142 6P0RL(5), POR(600), XX(5), NXD(5) 143 L0GICAL*1 ITITLE(20), JTITLE(60) 144 C 145 C READ IN PARAMETER VALUES 146 c 147 90 READ (5,9) (JTITLE(I),1=1,60), THI. XLEN, NL, MTYPE, IPROP, 148 1IHEAD. ND, (XX(I),NXD(I),I=1,ND) 149 READ (5.10) (HCXL(K),HCZL(K),PORL(K),SSL(K),ZZ(K),NZL(K),K=1,NL) 150 READ (5,11) AT, BT, AB, IPRIN, KASE, NTAG, (KTAG(K), JTAG(K), 151 1K=1,NTAG) 152 READ (5,30) NTS, NTPR, DT, VUI, TOL, FTS, (TPR(I), I=1,NTPR) 153 READ (5,33) NNPR, (NODE(I). I=1,NNPR) 154 IF (KASE.E0.5) READ (5,45) NSL.VSL 155 IF (KASE.E0.6) READ (5,46) NCH, (NCHN(J), d=1,NCH) 156 IF (KASE.EQ.7) READ (5,47) TCH, NCH, (NCHN(J), d=1,NCH) 157 C 158 C READ IN VALUES FOR PLOTTING ROUTINE 159 C 160 95 READ (5,27) IOPLOT 161 READ (5,28) (ITITLE(I ) .I=1,20). NTPL, (TPL(I),I = 1,NTPL) 162 IF (IQPLOT.EQ.O.OR.IQPLOT.EQ.1.OR.IQPLOT.EQ.4) GO TO 100 163 READ (5,29) ICON, CONZ, CONINT, IMESH, IVPLOT, I LAB 164 C 165 C CALCULATE CONSTANTS 166 C 167 100 IFLOW = 1 168 . NZ = 1 169 DO 105 1=1,NL 170 105 NZ = NZ + NZL(I) 171 NX = 1 172 DO 110 1 = 1, ND 173 1 10 NX = NX + NXD( I ) L i s t i n g of 2DTF at 09:50:00 on MAR 25, 1986 for CCid=FAUL 174 NN = NZ*NX 175 NE = 2*(NZ-1)*(NX-1) 176 C 177 C GENERATE FINITE ELEMENT MESH 178 C 179 CALL MESH (NN, NE , NX. NZ, NL, AT, BT , AB , KTAG, JTAG, THI , 180 1XLEN, IN, X, Z, ITAG. ITYPE, KASE, ZZ. NZL, MTYPE, NSL, 181 2VSL, XX, NXD, ND) 182 C 183 C DEFINE INITIAL HEADS FOR STATIC INITIAL CONDITIONS 184 C 185 IF (IHEAD.NE.1) GO TO 116 186 00 115 1=1,NN 187 VI(I) = VUI 188 115 VU(I,1) = VUI 189 C 190 C ASSIGN HYDRAULIC CONDUCTIVITY, SPECIFIC STORAGE. AND POROSITY TO ELEMENTS 191 C 192 116 DO 120 K=1,NE 193 KK = ITAG(K) 194 POR(K) = PORL(KK) 195 SS(K) = SSL(KK) 196 HCX(K) = HCXL(KK) 197 120 HCZ(K) = HCZL(KK) 198 C 199 C PRINT OUT INITIAL INFORMATION 200 C 201 IF (IHEAD.EQ.O) WRITE (6,41) 202 WRITE (6,12) (JTITLE(I), 1=1,60) 203 WRITE (6,13) THI, XLEN, NZ, NX, NL, NN, NE, AT, AB, DT, TOL 204 WRITE (6.14) (HCXL(I),HCZL(I),SSL(I).PORL(I),ZZ(I),NZL(I), 1=1,NL) 205 WRITE (6,34) (KTAG(I), JTAG(I), 1=1,NTAG) 206 IF (IHEAD.NE.1) GO TO 121 207 WRITE (6,31) VUI 208 121 IF (IPRIN.E0.2) GO TO 125 209 WRITE (6,41) 210 WRITE (6,15) 211 WRITE (6.16) (I, ( IN(I ,J) , J=1.3), ITAG(I), 1=1,NE) 212 WRITE (6,41) 213 WRITE (6,17) 214 WRITE (6,18) (K, X(K), Z(K), ITYPE(K). K=1,NN) 215 C 216 C CALCULATE FULL AND HALF BANDWIDTH OF MATRIX 217 C 218 125 LB = IN(1,3)-IN(1,1)+1 219 IF (IN(1,2).GT.IN(1,3)) LB = IN(1,2)-IN(1,1)+1 220 IBW = (LB-1 )*.2+1 221 C 222 C CONDENSATION (PARTITION) OF MATRIX TO REMOVE KNOWN HEAD NODES 223 C 224 IF (KASE.NE.6) GO TO 129 225 DO 128 1=1,NCH 226 128 ITYPE(NCHN(I)) = 1 227 129 LC(1)=0 228 IF (ITYPE(1).EQ.1) LC(1)=1 229 DO 130 I = 2,NN 230 K=0 231 IF (ITYPE(I).EQ.1) K=1 L i s t i n g of 2DTF at 09:50:00 on MAR 25, 1986 for CC i d=FAUL Page 5 232 130 LC(I)=LC(I-1)+K 233 NC=LC(NN) 234 NF=NN-NC 235 WRITE (6,19) NC.NF 236 C 237 C SKIP TO PLOTTING ROUTINES IF PLOTTING ONLY 238 C 239 IF (IQPLOT.EQ.1.OR.IQPL0T.EQ.4) GO TO 170 240 IF (IQPL0T.NE.3) GO TO 131 241 GO TO 170 242 C 243 C , ASSIGN POTENTIALS TO CONSTANT HEAD NODES 244 C. 245 131 IF (KASE.EO.3.OR.KASE.EO.4) GO TO 155 246 IF (KASE.E0.2) GO TO 140 247 00 135 I=1,NN 248 IF (ITYPE(I).EO.O) GO TO 135 249 VC(LC(I)) = Z(I) 250 135 CONTINUE 251 GO TO 170 252 140 DO 150 I=1,NN 253 IF (ITYPE(I).EO.O) GO TO 150 254 RNUM = X(I)*AB 255 IF (Z(I).EO.RNUM) GO TO 145 256 VC(LC(I)) = VU(I,1) 257 GO TO 150 258 145 VC(LC(I)) = O.DO 259 150 CONTINUE 260 GO TO 170 261 ' 155 DO 165 1 = 1,NN 262 IF (ITYPE(I).EQ.O) GO TO 165 263 IF (X(I) .EO.O.DO) GO TO 160 264 VC(LC(I)) = 1.D0 265 GO TO 165 266 160 VC(LC(I)) = O.DO 267 165 CONTINUE 268 C 269 C BEGIN THE TIME LOOP 270 C 271 170 KOUNT = 0 272 KCOUNT = 0 273 KTPR = 1 274 KTPL = 1 275 T1 = O.DO 276 DO 455 LL=1,NTS 277 IF ( IQPLOT.EQ. 1 .OR. IQPLOT.EQ.4 ) GO TO 355 278 IF (IQPLOT.EQ.3) GO TO 350 279 IF (IFLOW.EQ.1) GO TO 180 280 175 T = T1 + DT 281 IF (KTPR.EQ.O) GO TO 180 282 IF (KOUNT.GT.O) GO TO 180 283 C 284 C ALTER TIME AND TIMESTEP IF CLOSE TO A PRINTING 285 C 286 IF (T.GE.TPR(KTPR)) GO TO 176 287 PC = (TPR(KTPR)-T)/TPR(KTPR) 288 IF (PC.LE.0.2D0) GO TO 176 289 GO TO 180 co L i s t i n g of 2DTF at 09:50:00 on MAR 25, 1986 for CCid=FAUL 290 176 DT = DT - T + TPR(KTPR) 291 T = TPR(KTPR) 292 C 293 C IF BNDRY CONDITIONS CHANGE, REDUCE TIMESTEP AND REDO MATRIX PARTITION 294 C 295 IF (KASE.NE.7) GO TO 180 296 IF (T.NE.TCH) GO TO 180 297 IF (KCOUNT.NE.0) GO TO 174 298 DT = DT - 1.DO 299 T = T - 1.DO 300 KCOUNT = 1 301 GO TO 180 302 174 DO 177 1=1,NCH 303 177 ITYPE(NCHN(I)) =1 304 LC( 1 ) = O 305 IF (ITYPE(1).EO.1) LC(1)=1 306 DO 178 I=2,NN 307 K = O 308 IF (ITYPE(I).EO.1) K=1 309 178 LC(I) = LC(I-1) + K 3 10 , NC = LC(NN) 311 NF = NN - NC 312 DO 179 1=1,NN 313 IF (ITYPE(I).EO.O) GO TO 179 314 VC(LC(I)) = Z(I) 315 179 CONTINUE 316 C 317 C INITIALIZE MATRICES AND VECTORS 318 C 319 180 DO 185 1=1,NF 320 VB(I) = O.DO 321 VU(I,2) = O.DO 322 DO 185 d=i,LB 323 185 GCM(I.d) = O.DO 324 DO 190 1=1,NN 325 DO 190 J=1,IBW 326 190 GCMR(I.d) = O.DO 327 C 328 C FORM THE ELEMENT STIFFNESS MATRIX 329 C 330 LB 1 = O 331 DO 240 L=1,NE 332 A(1) = Z(IN(L,2))-Z(IN(L,3)) 333 A(2) = Z(IN(L.3))-Z(IN(L,1)) 334 A(3) = Z(IN(L,1))-Z(IN(L,2)) 335 B(1) = X(IN(L.3))-X(IN(L,2)) 336 B(2) = X(IN(L,1))-X(IN(L,3)) 337 B(3) = X(IN(L,2))-X(IN(L,1)) 338 C(1) = X(IN(L,2))*Z(IN(L,3))-X(IN(L,3))*Z(IN(L,2)) 339 C(2) = X(IN(L,3))*Z(IN(L,1))-X(IN(L.1))*Z(IN(L,3)) 340 C(3) = X(IN(L,1 ))*Z(IN(L.2))-X(IN(L,2))»Z(IN(L, 1)) 341 AR(L) = DABS ((A(1)*B(2)-B(1)*A(2))/2.DO) 342 DO 200 1=1,3 343 DO 200 J=1 ,3 344 ECM(I.d) = ((HCX(L)*A(I)*A(d))+(HCZ(L)*B(I)*B(d)))/ 345 1 (4D0»AR(L)) 346 IF (IFL0W.E0.1) GO TO 200 347 IF (I.EO.J) GO TO 195 L i s t i n g of 2DTF at 09:50:00 on MAR 25, 1986 for CCid=FAUL P a g e 7 348 349 350 195 351 200 352 C 353 C i 354 c 355 356 357 358 359 360 361 362 363 364 365 205 366 367 368 369 370 371 372 210 373 374 375 215 376 377 220 378 379 380 225 381 230 382 235 383 240 384 C 385 C F 386 C 387 388 389 390 391 392 245 393 C 394 C ( 395 C 396 397 398 399 400 250 401 C 402 C / 403 C 404 405 ECMT(I.J) = 0.5DO»AR(L)*SS(L)/6.DO GO TO 200 ECMT(I.J) = AR(L)*SS(L)/6.D0 CONTINUE ASSEMBLE THE GLOBAL STIFFNESS MATRIX DO 235 1=1,3 KI = IN(L,I) IF (ITYPE(KI).EQ.1) GO TO 235 II = KI-LC(KI) DO 230 J=1,3 KJ = IN(L,J) IF (IFLOW.EQ.1) GO TO 205 KK = KJ-KI+LB IF (KK.LT.1) GO TO 205 GCMR(II.KK) = GCMR(II.KK) - ECM(I,J)*0.5D0 + ECMT(I,J)/DT IF (ITYPE(KU).EQ.1) GO TO 220 dd = KJ-LC(Kd)-II+1 IF (dd.LT.1 ) GO TO 230 IF (dd.GT.LB1) LB1=dd IF (dd.LE.LB) GO TO 210 WRITE (6.20) L,dd GO TO 230 IF (IFLOW.EQ.1) GO TO 215 GCM(II.dd) = GCM(II.Jd) + ECM(I,J)*0.5D0 + ECMT(I.J)/DT GO TO 230 GCNI(II,JJ) = GCM(II.JJ) + ECM(I.J) GO TO 230 IF (IFLOW.EQ.1) GO TO 225 VB(II) = VB(II) - (ECM(I,d)"0.5D0+ECMT(I,d)/DT)*VC(LC(KJ)) GO TO 230 VU(II,2) = VU(II,2) - ECM(I.d) * VC(LC(Kd)) CONTINUE CONTINUE CONTINUE PRINT OUT FINAL BANDWIDTH OF MATRIX IF (LL GT.1) GO TO 245 WRITE (6.21) LB 1 IF (LB1.LE LB) GO TO 245 WRITE (6,22) GO TO 490 CONTINUE CALL SUBROUTINE TO DECOMPOSE GLOBAL MATRIX CALL DBAND (NF,LB 1,GCM.I EX) IF (I EX.EQ.O) GO TO 250 WRITE (6,23) GO TO 490 CONTINUE ASSEMBLE THE RHS VECTOR IF (IFLOW.EQ.1) GO TO 275 DO 270 I=1,NF C O L i s t i n g of 2DTF at 09:50:00 on MAR 25, 198G for CCid=FAUL Page 8 406 DO 265 J=1,IBW 407 IF (GCMR(I.J).EQ.O.DO) GO TO 265 408 DO 255 K=1,NN 409 II = I+LC(K) 410 IF (II.EQ.K) GO TO 260 411 255 CONTINUE 412 260 JJ = J-LB+II 413 VU(I,2) = VU(I,2) + GCMR(I,J)*VU(M 414 265 CONTINUE 415 VU(I,2) = VU(I,2) + VB(I) 416 270 CONTINUE 417 C 418 C CALL SUBROUTINE TO SOLVE THE GLOBAL MATRIX 419 C 420 275 CALL SBAND (NF,LB 1,GCM,VU) 421 C 422 C EXPAND HEAD VECTOR 423 C 424 00 285 II=1,NN 425 I = NN-II+1 426 IF (I TYPE(I).EO. 1) GO TO 280 427 K = I-LC(I ) 428 VU(I,2) = VU(K,2) 429 GO TO 285 430 280 VU(I,2) = VC(LC(I)) 431 285 CONTINUE 432 C 433 C DEFINE INITIAL HEADS FOR STEADY STATE INITIAL ( 434 C 435 IF (IHEAD.NE.2) GO TO 288 436 DO 287 1=1,NN 437 VI(I) = VU(I.2) 438 287 VU(I.1) « VU(I,2) 439 GO TO 390 440 C 44 1 C CALCULATE PERCENT CHANGE IN HEAD 442 C 443 288 IF (IFLOW.EQ.1) GO TO 335 444 CH1 = O.DO 445 IF (T.LT.FTS) GO TO 320 446 00 290 1=1,NN 447 IF (I TYPE (I).EQ.1 ) GO TO 290 448 CH2 = DABS((VU(I,1)-VU(1,2))/VU(1,1)) 449 IF (CH2.GT.CH1) CH1=CH2 450 290 CONTINUE 451 IF (IPR0P.NE.2.0R.T.EQ.TPR(NTPR)) WRITE 452 C 453 C CALCULATE NEXT DELTA T 454 C 455 IF (KASE.EQ.7.AND.T.EQ.TCH) GO TO 294 456 IF (T.EQ.TPR(KTPR)) GO TO 295 457 294 IF (CH1.LE.TOL) DT1 = DT 458 IF (KOUNT.GT.O) GO TO 305 459 295 KOUNT = 0 460 300 FACT = (T0L-CH1)/TOL 461 IF (FACT.LE.-1.DO) FACT=-.9D0 462 IF (KOUNT.EQ.1) FACT = -0.5D0 463 DT = 0T1 * (1.DO + FACT) LL, DT, CO L i s t i n g of 2DTF at 09:50:00 on MAR 25, 1986 for CC1d=FAUL P a g e 9 464 IF (KOUNT.NE.1) GO TO 305 465 KOUNT = 2 466 GO TO 175 467 305 IF (CH1.LE.T0L) GO TO 310 468 IF (KOUNT.EQ.2) GO TO 315 469 IF (KOUNT. E0.1) GO TO 300 470 KOUNT = 1 471 GO TO 175 472 310 IF (KOUNT.EO.1.OR.KOUNT.EQ.2) GO TO 295 473 GO TO 325 474 315 WRITE (6,32) 475 GO TO 490 476 320 DT1 = DT 477 DT = DT1 * 1.25DO 478 C 479 C REASSIGN INITIAL HEAD VECTOR 480 C 481 325 T1 = T 482 DO 330 1=1,NN 483 330 VU(I,1) = VU(I,2) 484 C 485 C SAVE HEAD VALUES FOR PLOTTING, IF DESIRED 486 C 487 IF (KTPL.GT.NTPL) GO TO 365 488 IF (IQPLOT.NE.O) GO TO 345 489 IF (T.NE.TPL(KTPL)) GO TO 365 t 490 KTPL = KTPL + 1 491 WRITE (11,43) (VU(I,2), I=1,NN) 492 C 493 C PLOT THE POTENTIALS 494 C 495 GO TO 345 496 335 DO 340 J= 1 , NN 497 340 VT(d) = VI(J) - VU(d,2) 498 345 IF (IQPLOT.EQ.0) GO TO 365 499 IF (IFLOW.EQ.1) GO TO 355 500 IF (T.NE.TPL(KTPL)) GO TO 365 501 IF (IQPL0T.NE.3) GO TO 355 502 350 READ (11,43) (VU(I,2), 1=1,NN) 503 355 CALL HPLOT (VU,NC,NN,NE,NX,NZ,ICON, CONZ.CONINT 504 1 IMESH,ITITLE,IN.X.Z,IPL,IPE,IOPLOT, I TYPE,I LAB) 505 IF (IVPLOT.EQ.O) GO TO 364 506 C 507 C CALCULATE AND PLOT THE VELOCITIES 508 C 509 IF (IQPLOT.EQ.1.OR.IQPLOT.EQ.4) GO TO 490 510 IF (IQPLOT.EQ.O) GO TO 365 511 VNUM = 0..' 512 VAV = 0 513 II = 1 514 WRITE (6,48) 515 DO 360 1=1,NE 516 Al = IN(1,1) 517 AK = IN(I,3) 518 Ad = IN(I,2) 519 IF (VU(AI,2).GT.VI(AI)) VU(AI,2) = VI(AI) 520 IF (VU(Ad,2).GT.VI(Ad)) VU(Ad,2) = VI(Ad) 521 IF (VU(AK,2).GT,.VI(AK)) VU(AK,2) = VI(AK) L i s t i n g of 2DTF at 09:50:00 on MAR 25, 1986 for CC1d=FAUL P a g e 10 522 523 524 525 526 527 528 529 530 359 531 532 533 534 535 536 537 538 539 540 541 542 361 543 544 545 546 547 360 548 549 550 364 551 552 553 554 C 555 C 1 556 C 557 365 558 559 560 561 366 562 563 564 565 370 566 375 567 568 376 569 570 571 572 573 380 574 575 385 576 577 578 390 579 IF (IOPLOT.NE.3) GO TO 359 A(1) = Z(Ad)-Z(AK) A(2) = Z(AK)-Z(AI) A(3) = Z(AI)-Z(Ad) B( 1 ) = X(AK)-X(Ad) B(2) = X(AI)-X(AK) B(3) = X(Ad)-X(AI) AR(I) = DABS ((A(1 )*B(2)-B(1)*A(2))/2 DO) AA = ((A(1)*VU(Al,2))+(A(2)*VU(Ad,2))+(A(3)*VU(AK,2)))/ (2*AR(I)) BB = ((B(1)*VU(AI,2))+(B(2)*VU(Ad,2))+(B(3)*VU(AK,2)))/ (2*AR(I)) VELX(I) = - (HCX(I)/POR(I)) * AA VELZ(I) = - (HCZ(I)/POR(I)) * BB VMAG = (VELX(I)**2 + VELZ(I)**2)* *.5 WRITE (6,49) VELX(I), VELZ(I), VMAG VAV = VAV + VMAG VNUM = VNUM + 1 IF (I.EO.KTAG(II)) GO TO 361 GO TO 360 VAV = VAV/VNUM WRITE (6,50) II, VAV VAV = 0 VNUM = 0 1 1 = 1 1 + 1 CONTINUE CALL VPLOT (NN, NE, NX, NZ, IPE, IPL, VELX. VELZ, XLEN, THI, X, Z, IN, ITITLE, I TYPE) IF ( IOPLOT. NE. 3) GO TO 365 IF (KTPL.EO.NTPL) GO TO 490 KTPL = KTPL + 1 GO TO 350 PRINT RESULTS: ALL NODES IF (I FLOW.EO.1.AND.IPROP.EQ.1) GO TO 366 IF (IFL0W.E0.1) GO TO 390 IF (LL.EQ.2) WRITE (6,41) IF (IPROP.NE.1) GO TO 376 DO 375 1=1,NN IF (VU(I,1).GE.VI(I) ) GO TO 370 VU(I,2) = VI(I)-VU(I,1) GO TO 375 VU(I,2) = 0.DO CONTINUE IF (IFLOW.EO.1) GO TO 390 IF (LL.EO.NTS) GO TO 385 IF (KTPR.EQ.O) GO TO 385 IF (NTPR.GT.1) GO TO 380 KTPR = 0 GO TO 385 IF (T.NE.TPR(KTPR)) GO TO 455 KTPR = KTPR + 1 IF (IPROP.EQ.2) GO TO 435 WRITE (6,24) T GO TO 395 WRITE (6,41) ' IF (IHEAD.EQ.2) GO TO 393 l-j C O L i s t i n g of 2DTF at 09:50:00 on MAR 25, 1986 for CC id = FAUL P a g e 1 1 580 WRITE (6,35) 581 GO TO 395 582 393 WRITE (6,51) 583 395 IF (X(2).NE.O.DO) GO TO 410 584 IMIN = N2 585 IMAX = NZ*8 586 400 IF (NN.LT.IMAX) IMAX = NN 587 DO 405 d=1,NZ 588 WRITE (6,25) (VU(I, 2), I=IMIN,IMAX.NZ) 589 IMIN = IMIN - 1 590 405 IMAX = IMAX - 1 591 WRITE (6.26) 592 IMIN = IMAX + NZ*2 593 IMAX = IMIN + NZ*7 594 IF (NN.GE.IMIN) GO TO I 400 595 IF (IHEAD.E0.2) GO TO I 460 596 IF (IPR0P.EQ.2) GO TO I 435 597 GO TO 434 598 4 10 IMIN = NX * (NZ-1) + 1 599 IMAX = IMIN + 7 600 415 IF (NN.LT.IMAX) IMAX = NN 601 JMAX = NZ 602 IF (KASE.EQ.2) JMAX = NZL(1 ) + 1 603 DO 420 J=1,JMAX 604 WRITE (6,25) (VU(I, 2), I=IMIN,IMAX) 605 IMIN = IMIN - NX 606 420 IMAX = IMAX - NX 607 WRITE (6,26) 608 IF (KASE.NE.2) GO TO 430 609 IMAX = 3 * NX 610 IMIN = IMAX - NX + 1 611 DO 425 J=1 ,3 612 WRITE (6,25) (VU(I, 2), I=IMIN.IMAX) 613 IMIN = IMIN - NX 614 425 IMAX = IMAX - NX 615 GO TO 434 616 430 IMIN = IMIN + NN + 8 617 IMAX = IMIN + 7 618 IF (NN.GE.IMIN) GO TO 415 619 IF (IHEAD.EQ.2) GO TO 460 620 434 IF (IPROP.EQ.1.OR.IPR0P.EQ.3) GO TO 450 621 C 622 C CALCULATE AND STORE PERCENT CHANGE IN HEAD AT SPECIFIED NODES 623 C 624 435 DO 445 1=1,NNPR 625 IF (IFLOW.EQ.1) GO TO 440 626 VU(NODE( I ) , 2) = ( VI ( NODE ( I ) )-VU(NODE( I ) , 1 ) )/VT(NODE( '. 627 IF (VU(NODE(I),2).LE.O.002D0) VU(NODE(I),2)=0.DO 628 IF (VU('N0DE(I),2).GE.O.998DO) VU(NODE (I ) , 2 ) = 1 . DO 629 STH(I.KTPR) = VU(NODE(I),1) 630 STPH(I.KTPR) = VU(N0DE(I),2) 631 GO TO 445 632 440 STH(1,1) = VU(NODE(I),2) 633 STPH(I,1) = 1.00 634 445 CONTINUE 635 C 636 C END THE TIME LOOP 637 C 4== O L i s t i n g of 2DTF at 09:50:00 on MAR 25, 1986 for CC1d = FAUL Page 12 638 450 639 640 * 455 641 C 642 C I 643 C 644 460 645 646 647 648 649 650 651 652 653 465 654 655 470 656 C 657 C ! 658 C 659 660 661 662 663 664 665 666 667 668 480 669 670 485 671 C 672 C I 673 C 674 9 675 10 676 11 677 12 678 13 679 680 681 682 683 684 685 14 686 687 688 689 15 690 691 16 692 17 693 694 18 695 19 IF (KTPR.GT.NTPR) GO TO 460 IFLOW = 2 CONTINUE PRINT RESULTS: SPECIFIED NODES IF (IHEAD.EQ.2) GO TO 90 IF (IPROP.EO.1.OR.IPROP.EO.3) GO TO 490 WRITE (6,41) WRITE (6,37) DO 470 1=1,NNPR WRITE (6,38) NODE(I),X(NODE(I)),Z(NODE(I)),STH(I,1),STPH(I.1) DO 465 K=1,NTPR IF (STPH(I,K+1).LE.0.DO.0R.STPH(I,K+1).EQ.1.DO) GO TO 465 WRITE (6,39) TPR(K), STH(I,K+1), STPH(I,K+1) CONTINUE WRITE (6,42) VT(NODE(I)) CONTINUE STORE VALUES FOR PLOT OF HEAD VS TIME AT SPECIFIED NODES DO 485 d=1,NNPR I = O DO 480 K=1,NTPR IF (TPR(K) .LT. 1 .DO) GO TO 480 IF (STPH(d,K+1).LT.O.002.OR.STPH(J,K+1).GT.0.998) GO TO 480 1 = 1 + 1 YH(I) = STPH(d,K+1) YH2(I) = STH (d,K+1) XT(I) = DL0G10(TPR(K)) CONTINUE WRITE (12,44) I, (XT(L), YH(L), YH2(L), L=1.I) CONTINUE FORMAT STATEMENTS FORMAT (60A1/2F10.5,5I5/(F10.5,15)) FORMAT ((2E10.4,3F10.5,15)) FORMAT (E12.5,F10.5,E12.5,3I5/(2I5)) FORMAT (/5X.60A1//) FORMAT (//5X,'THICKNESS OF SECTION (KM):',12X,E10.5/5X,'LENGTH OF 1SECTI0N (KM) : ' , 15X , E 10. 5/5X , 'NO. OF NODES IN Z-DIRECTION: '., 16X , 14/ 25X,'N0. OF NODES IN X-DIRECTI ON: ' , 16X,14/5X, 'NO. OF MATERIAL TYPES 3:'.23X.I3/5X.'TOTAL NO. OF NODES:',25X,I4/5X,'TOTAL NO. OF ELEMENT 4S:',22X,I4/5X,'SLOPE OF WATER TABLE:',15X,E12.5/5X, 5'SLOPE OF BASEMENT:',20X,E10.5/5X.'TIME INCREMENT:'.23X,E10.5/ 65X,'TOLERANCE:',31X,F7.2) FORMAT (//5X,'HYDRAULIC CONDUCTIVITY, SPECIFIC STORAGE, POROSITY, 1 AND ELEVATION OF EACH LAYER:'/5X,'X-DIRECTION', 10X,'Z-DIRECTION' , 213X,'STORAGE',12X,'POROSITY',12X,'ELEVATION',10X,'NO. OF NODES'// 3(5X,E11.5, 10X,E11 .5, 10X,F10.5. 10X,F10.5, 10X,F10.5. 15X,15)) FORMAT (//5X,'ELEMENT',10X,'INCIDENCE',12X,'MATERIAL TYPE'/5X, 17(1H-),10X,9(1H-),12X,8(1H-),1X,4(1H-)//18X. 'I ' ,6X, 'd ' ,6X. 'K'//) FORMAT (5X,I4.4X.3I7,12X,12) FORMAT (//5X,'NODE',20X,'COORDINATES',15X.'NODE TYPE'/5X, 14(1H-),20X,11(1H-),15X,4(1H-).1X,4(1H-)//29X,'X'.12X.'Z'//) FORMAT (5X,14,14X,E10.5,4X.E10.5,10X,14) FORMAT (//5X,'NUMBER OF CONSTANT HEAD NODES',I5/5X, L i s t i n g of 2DTF at 09:50:00 on MAR 25, 1986 for CCid=FAUL Page 13 696 1'NUMBER OF DEGREES OF FREEDOM',16) 697 20 FORMAT (//5X.'ELEMENT',14,2X,'REQUIRES BANDWIDTH OF',14) 698 21 FORMAT (//5X, 'FINAL BANDWIDTH',15X,14) 699 22 FORMAT (//5X, 'EXECUTION TERMINATED-BANDWIDTH TOO SMALL') 700 23 FORMAT (//5X,'DECOMPOSITION FAILED - EXECUTION TERMINATED') 701 24 FORMAT (////10X,'TIME:',15X.E15.5/10X,4(1H-)///10X, 702 1'TRANSIENT HEAD DISTRIBUTION'/10X,9(1H-),1X.4(1H-),1X,12(1H-)//) 703 25 FORMAT (8E15.5) 704 26 FORMAT (////) 705 27 FORMAT (15) 706 28 FORMAT (20A1,I5/(5E10.4)) 707 29 FORMAT (I5,2F10.2,3I5) 708 30 FORMAT (2I5,4F10.4/(5E10.4)) 709 31 FORMAT (//5X,'INITIAL HEAD AT EACH NODE:'//(5X,E10.5)) 710 32 FORMAT (//5X,'EXECUTION TERMINATED - TIME STEP TOO LARGE') 711 33 FORMAT (1015) 712 34 FORMAT (//5X,'TAGS FOR IDENTIFYING LAYER POSITIONS:'./5X, 713 1'ELEMENT NO.' , 10X,'CONDUCT IVITY TAG'//(9X,14,22X,14 ) ) 714 35 FORMAT (////10X, 'STEADY STATE HEAD DISTRIBUTION'/10X,6(1H-) , 715 11X,5(1H-),1X,4(1H-),1X,12(1H-)//) 716 36 FORMAT (//5X,'VELOCITIES; X AND Z: '/(5X,2E20 . 5 ) ) 717 37 FORMAT (///'NODE NO.',5X,'X-COORD'.11X,'Z-COORD',12X,'TIME (D) ' . 718 19X,'HEAD (KM) ' ,9X, '"/. HEAD'/) 7 19 38 FORMAT (/I5,2E18.5,8X,'STEADY STATE',3X,2E15.5) 720 39 FORMAT (46X,E15.5,3X,2E15.5) 721 40 FORMAT (//5X,'GREATEST 54CHANGE ' , 5X , ' TIMESTEP ' , 10X , 'DT ' , 15X , 722 1'TIME'/5X,F10.5,12X,I5,7X,E10.5,10X,E10.5) 723 41 FORMAT ( ' 1 ' ) 724 42 FORMAT (/'TOTAL HEAD CHANGE: ' .5X.E15.5) 725 43 FORMAT (5E12.7) 726 44 FORMAT (I5/(3E15.5)) 727 .45 FORMAT (I5.E12.5) 728 46 FORMAT (1215) 729 47 FORMAT (E10.4/(1215)) 730 48 FORMAT (//'ELEMENT AND AVERAGE LAYER VELOCITIES'//'X-VELOCITY', 731 15X.'Z-VELOCITY',10X,'VELOCITY MAGNITUDE'///) 732 49 FORMAT (E10.5,5X,E10.5,18X.E10.5) 733 50 FORMAT (//'AVERAGE VELOCITY IN LAYER ' , 1 2 , ' = ' .E10.5//) 734 51 FORMAT (////10X,'INITIAL STEADY STATE HEAD DISTRIBUTION'/ 735 110X,7(1H-),1X,6(1H-),1X,5(1H-).1X.4(1H-),1X,12(1H-)//) 736 490 . STOP 737 END L i s t i n g of MESH at 09:49:42 on MAR 25, 1986 for CCid=FAUL P a g e 1 1 SUBROUTINE MESH (NN, NE, NX, NZ, NL, AT. BT, AB, KTAG, UTAG, 2 1THI, XLEN, IN, X, Z, ITAG, I TYPE, KASE, ZZ, NZL, MTYPE, NSL, 3 2VSL, XX, NXD, ND) 4 5 6 C 7 C SUBPROGRAM MESH 8 C 9 c WRITTEN BY LINDY A. ENGLAND 10 c 11 12 13 c 14 c THIS SUBPROGRAM GENERATES AN AUTOMATIC MESH FOR A RECTANGULAR 15 c FLOW REGION WITH SLOPING WATER TABLE (UPPER BNDRY) AND SLOPING 16 c BASEMENT (LOWER BNDRY). MESH SPACING CAN BE CONSTANT OR VARIABLE. 17 c NODES CAN BE NUMBERED IN THE X OR Z DIRECTION. 18 c 19 c 20 c LIST OF VARIABLES: 21 c 22 c SEE MAIN PROGRAM - 2DTF 23 c 24 IMPLICIT REAL*8 (A-H.O-Z) 25 DIMENSION X(340), Z(340), IN(600,3), ITAG(600), ITYPE(340), 26 1KTAG(5), JTAG(5), ZTAG(5). ZZTAG(5). SC(5), ZZ(5), NZL(5), 27 2XX(5), NXD(5), XXTAG(5) 28 c 29 C ASSIGN X AND Z COORDS AND NODE TYPES TO EACH NODE 30 C 31 IF (MTYPE.EO.3.OR.MTYPE.EO.4) GO TO 100 32 DO 90 1=1,ND 33 90 XXTAG(I) = XX(I) - .00O05D0 34 X( 1 ) = 0.DO 35 JMAX = NL - 1 36 GO TO 105 37 100 DX = XLEN/DFLOAT(NX-1) 38 X( 1) = O.DO 39 105 IF (MTYPE.EQ.1.OR.MTYPE.EO.3) GO TO 110 40 MAX = NN 41 INC = NZ 42 GO TO 115 43 110 MAX = NX 44 INC = 1 45 LL = 1 46 1 15 DO 260 K=1,MAX,INC 47 IF (MTYPE.EO.1 OR MTYPE.EO.3) GO TO 116 48 KMAX = K + NZ - 1 49 GO TO 117 SO 1 16 KMAX = K + NX * (NZ-1) 51 117 IF (KASE.NE.3.AND.KASE.NE.4) GO TO 125 52 IF (K.EO.1 OR.KMAX.EO.NN) GO TO 120 53 ITYPE(KMAX) = 0 54 ITYPE(K) = 0 55 GO TO 135 56 120 I TYPE(KMAX) = 1 57 ITYPE(K) = 1 58 GO TO 135 -Pi co L i s t i n g of MESH at 09:49:42 on MAR 25, 1986 for CC1d=FAUL 59 125 ITYPE(KMAX) = 1 60 IF (KASE.LE.1.OR.KASE.GE.5) GO TO 130 61 ITYPE(K) = 1 62 GO TO 135 63 130 ITYPE(K) = 0 64 135 X(KMAX) = X(K) 65 IF (KASE.E0.2) GO TO 231 66 Z(K) = AB*X(K) 67 IF (KASE.NE.5) GO TO 137 68 IF (K.LE.NSL) GO TO 137 69 VIN = THI - VSL * X(NSL) 70 Z(KMAX) = VSL * X(K) + VIN 71 GO TO 138 72 137 Z(KMAX) = AT*X(K) + BT 73 138 IF (MTYPE.EO.3.OR.MTYPE.EO.4) GO TO 155 74 IF (NL.EO.1) GO TO 150 75 DO 145 J=1,JMAX 76 IF (K.EO.1) GO TO 140 77 SC(J) = THI/ZZ(J) 78 ZTAG(J) = ((Z(KMAX)-Z(K))/SC(J)) + Z(K) 79 GO TO 145 80 140 ZTAG(J) = ZZ(J) 81 145 ZZTAG(J) = ZTAG(d) - .00005D0 82 150 ZTAG(NL) = Z(KMAX) 83 ZZTAG(NL) = Z(KMAX) 84 ZMAX = ZZTAGd) 85 KT = 1 86 NUM = 0 87 GO TO 160 88 155 DZ = (Z(KMAX)-Z(K))/(DFL0AT(NZ-1)) 89 160 IF (MTYPE.EO.1 OR.MTYPE.EO.3) GO TO 165 90 KKMAX = KMAX - 2 91 KINC = 1 92 GO TO 170 93 165 KKMAX = KMAX - 2 * NX 94 KINC = NX 95 170 DO 230 I=K,KKMAX,KINC 96 IF (MTYPE.EO.1.OR.MTYPE.EO.3) GO TO 180 97 ITYPEO + 1) = ITYPE(I) 98 GO TO 195 99 180 ITYPE(I+NX) = ITYPE(I) 100 195 IF (MTYPE.EO.3.OR.MTYPE.EQ.4) GO TO 215 101 IF (Z(I).GT.ZMAX) GO TO 210 102 IF (NUM.EO.1) GO TO 215 103 200 IF (KT.NE.1) GO TO 205 104 DZ = (ZTAG(KT)-Z(K))/DFLOAT(NZL(KT)) 105 NUM = 1 106 GO TO 215 107 205 DZ = (ZTAG(KT)-ZTAG(KT-1))/DFL0AT(NZL(KT)) 108 GO TO 215 109 210 KT = KT + 1 110 ZMAX = ZZTAG(KT) 1 1 1 GO TO 200 112 215 IF (MTYPE.EQ.1.OR.MTYPE.EQ.3) GO TO 225 113 Z(1 + 1 ) = Z(I) + DZ 1 14 X(I+1) = X(K) 1 15 GO TO 230 1 16 225 Z(I+NX) = Z(I) + DZ L i s t i n g of MESH at 09:49:42 on MAR 25, 1986 for CC i d=FAUL Page 3 117 X(I+NX) = X(K) 118 230 CONTINUE 1 19 GO TO 234 120 231 Z(KMAX) = O.DO 121 DZ = THI/OFLOAT(NZL(1)) 122 IMAX - NZL(1) 123 DO 232 1=1,IMAX 124 d = KMAX - NX * I 125 Z(d) = DZ * I 126 IF (d.EO.K) GO TO 232 127 ITYPE(J) = 0 128 X(J) = X(K) 129 232 CONTINUE 130 IF (NL.E0.1) GO TO 234 131 IMIN = IMAX + 2 132 DO 233 I=IMIN,NZ 133 DZ = DZ * 1.05 134 d = KMAX - NX * (1-1) 135 Z(d) = Z(d+NX) + DZ 136 IF (d.EQ.K) GO TO 233 137 ITYPE(d) = O 138 X(d) = X(K) 139 233 CONTINUE 140 234 IF (KMAX.EO.NN) GO TO 260 141 IF (MTYPE.EQ.3.0R.MTYPE.EQ.4) GO TO 245 142 235 IF (X(K).GT.XXTAG(LL)) GO TO 240 143 IF (LL.EO.1) GO TO 237 144 OX = (XX(LL) - XX(LL-D) / (DFLOAT (NXD(LL))) 145 GO TO 245 146 237 OX = XX(1) / (DFLOAT (NXD(1))) 147 GO TO 245 148 240 LL = LL + 1 149 GO TO 235 150 245 IF (MTYPE.EQ.1.OR.MTYPE.EQ.3) GO TO 255 151 X(KMAX+1) = X(K) + DX 152 GO TO 260 153 255 X(K+1) = X(K) + DX 154 260 CONTINUE 155 C 156 C ASSIGN ELEMENT INCIDENCES 157 C 158 NEX = (NX-1)*2 159 NODE = 1 160 DO 280 K= 1 , NE , NEX 161 KOUNT = 1 162 UMAX = K + NEX - 1 163 DO 270 d = K,dMAX,2 164 IN(d,1) = NODE 165 IF (MTYPE.EO.1.OR.MTYPE.EO.3) GO TO 265 166 IN(d,2) = IN(d,1) + NZ 167 IN(d,3) = IN(J,1) + 1 168 IN(d+1,3) = IN(d,2) 169 IN(d+1,1) = IN(d+1,3) + 1 170 IMJ+1.2) = IN(d,3) 171 NODE = NZ*KOUNT + IN(K,1) 172 KOUNT = KOUNT + 1 173 GO TO 270 174 265 IN(J,2) = IN(d,1) + 1 L i s t i n g of MESH at 09:49:42 on MAR 25. 1986 for CCid=FAUL Page 4 175 IN(J.3) = IN(J.1) + NX 176 IN(J+1,2) = IN(d,3) 177 IN(d+1,1) = IN(d+1,2) + 1 178 IN(J+1.3) = IN(d,2) 179 NODE = NODE + 1 180 270 CONTINUE 181 IF (MTYPE.EQ.1.OR.MTYPE.EQ.3) GO TO 275 182 NODE - IN(K,1) + 1 183 GO TO 280 184 275 NODE - IN(K,1) +• NX 185 280 CONTINUE 186 C 187 C ASSIGN AN HYDRAULIC CONDUCTIVITY TAG TO EACH ELEMENT 188 C 189 IF (KASE.EQ.3) GO TO 300 190 KKTAG =.KTAG(1) 191 KOUNT = 1 192 DO 295 L=1,NE 193 IF (U.GT.KKTAG) GO TO 290 194 285 ITAG(L) = JTAG(KOUNT) 195 GO TO 295 196 290 KOUNT = KOUNT + 1 197 KKTAG = KTAG(KOUNT) 198 GO TO 285 199 295 CONTINUE 200 GO TO 315 201 300 KKTAG = NEX 202 ITAGO) = 1 203 ITAG(2) = 1 204 DO 310 L=3,NE,2 205 IF (L.GT.KKTAG) GO TO 305 206 ITAG(L) = ITAG(L-1) + 1 207 ITAG(L+1) = ITAG(L) 208 GO TO 310 209 305 ITAG(L) = 1 210 ITAG(L+1) = ITAG(L) 211 KKTAG = KKTAG + NEX 212 310 CONTINUE 213 315 RETURN 214 END L i s t i n g of DBAND at 09:49:35 on MAR 25, 1986 for CC1d=FAUL 1 SUBROUTINE DBAND (NF,LB,GCM,I EX) 2 cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc 3 cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc 4 c 5 c 6 -j c Q SUBPROGRAM TO DECOMPOSE A MATRIX USING CHOLESKY LU DECOMPOSITION 8 c (TAKEN FROM PROGRAM "STEADY" BY EMIL 0. FRIND, U. OF WATERLOO) 9 c 10 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCGCCCC 1 1 cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc 12 c 13 c 14 IMPLICIT REAL*8 (A-H.O-Z) 15 DIMENSION GCM(340,15) 16 IEX = 0 17 DO 100 I=1,NF 18 IP = NF-I+1 19 IF (LB.LT.IP) IP=LB 20 DO 100 d=1,IP 21 10 = LB-J 22 IF ( ( I - D . L T . I O ) 10=1-1 23 SUM = GCM(I.J) 24 IF (10.LT.1) GO TO 110 25 DO 120 K=1,IQ 26 II = I-K 27 dZ = J+K 28 120 SUM = SUM - GCM(II,K+1)*GCM(II, dZ) 29 1 10 IF (d.NE.1) GO TO 130 30 IF (SUM.LE.O.DO) GO TO 140 31 TEMP = 1,DO/DSQRT(SUM) 32 GCM(I.d) = TEMP 33 GO TO 100 34 140 WRITE (6,10) I 35 IEX = 1 36 RETURN 37 130 GCM(I.d) = SUM*TEMP 38 100 CONTINUE 39 10 FORMAT (10X,'DBAND FAILS AT ROW ' ,14) 40 RETURN 41 END L i s t i n g of SBAND at 09:49:10 on MAR 25, 1986 for CC1d=FAUL Page 1 1 SUBROUTINE SBAND (NF,LB,GCM,VU) 2 c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c 3 ^ c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c 5 \J c 6 c SUBPROGRAM TO SOLVE A MATRIX USING CHOLESKY L-U DECOMPOSITION 8 c (TAKEN FROM PROGRAM "STEADY" BY EMIL 0. FRIND, U. OF WATERLOO) 9 c 10 c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c 1 1 c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c c 12 c 13 c 14 IMPLICIT REAL*8 (A-H.O-Z) 15 DIMENSION GCM(340,15), VU(340,2) 16 DO 100 I=1,NF 17 J = I-LB+1 18 IF ( ( I + D.LE.LB) J=1 19 SUM = VU(I,2) 20 K1 = 1-1 21 IF (J.GT.K1) GO TO 110 22 DO 120 K=J,K1 23 I I = I-K+1 24 120 SUM = SUM - GCM(K,II)*VU(K,2) 25 110 VU(1,2) = SUM » GCM(1,1) 26 100 CONTINUE 27 DO 130 II=1,NF 28 I = NF-II+1 29 J = I+LB-1 30 IF (J.GT.NF) d=NF 31 SUM = VU(I,2) 32 K2 = 1+1 33 IF (K2.GT.J) GO TO 140 34 DO 150 K=K2,J 35 KK = K-I+1 36 150 SUM = SUM - GCM(I,KK)*VU(K,2) 37 140 VU(I,2) = SUM*GCM(I,1) 38 130 CONTINUE 39 RETURN 40 END 00 L i s t i n g of PLOTH at 09:50:08 on MAR 25, 198S for CC i d = FAUL Page 1 1 SUBROUTINE HPLOT (VU, NC, NN, NE, NX, NZ, ICON, CONZ, CONINT, 2 1XMAX.ZMAX,IMESH,ITITLE,NODE,X,Z,IPL.IPE,IOPLOT,I TYPE,I LAB) 3 C 4 C 5 C PLOTTING PACKAGE 6 C 7 C 8 C 9 C SUBROUTINE HPLOT CONTOURS HYDRAULIC HEAD DATA FOR 10 C THE FINITE ELEMENT MESH,PLOTS THE SCALE,AND DRAWS 11 C THE PERIMETER OF THE MESH 12 C 13 C LIST OF VARIABLES: 14 C 15 C SEE MAIN PROGRAM - 2DTF 16 C 17 IMPLICIT REAL*8 (A-H.O-Z) 18 DIMENSION VU(340,2), IPL(80), N0DE(6OO,3). X(340), Z(340), 19 1KNP(30). ITYPE(340) 20 L0GICAL*1 ITITLE(20) 21 C 22 C CALCULATE CONSTANTS 23 C 24 IF (NZ.GT.NX) GO TO 100 25 FACT = NZ 26 • GO TO 105 27 100 FACT = NX 28 105 NCOL = NX - 1 29 XSIZE = 6.0D0 30 ZSIZE = 3.0DO 31 IF (IQPLOT.EO.1) GO TO 146 32 KOUNT = 2 33 IJTAG = 0 34 IPE = 0 35 DO 145 1=1,NN 36 IF (IJTAG.EQ.1) GO TO 130 37 IF (IJTAG.EQ.2) GO TO 140 38 IF (I.GT.FACT) GO TO 120 39 IPL(I) = I 40 IPE = IPE + 1 41 GO TO 145 42 120 IPL(I) = FACT*KOUNT 43 IPE = IPE + 1 44 KOUNT = KOUNT + 1 45 IF (IPL(I).EQ.NN.OR.ITYPE(IPL(I)).EQ.1) GO TO 125 46 GO TO 145 47 125 IJTAG = 1 48 KNP(1) = IPL(I) 49 K1 = 1 50 GO TO 145 51 130 NCK = IPL(1-1 ) - NX 52 IF (IPL(1-2).EQ.NCK) GO TO 132 53 IF (ITYPE(NCK).EQ.1) GO TO 131 54 , GO TO 132 55 131 IPL(I) = NCK 56 K1 = K1 + 1 57 KNP(K1) = IPL(I) 58 GO TO 133 L i s t i n g of PLOTH at 09:50:08 on MAR 25, 1986 for CCid=FAUL Page 2 59 132 NCK = IPL(I-1) - 1 60 N1 = NCK + NX 61 IF (N1.GT.NN) GO TO 131 62 IF (I TYPE(NCK).EO.1) GO TO 131 63 NCK = IPL(I-1) - 1 + NX 64 IF (ITYPE(NCK).EO.1) GO TO 131 65 NCK = IPL(I-1) + NX 66 IF (ITYPE(NCK).EO.1) GO TO 131 67 WRITE (6,10) 68 10 FORMAT (//'ERROR IN WATER TABLE PLOTTING'//) 69 STOP 70 133 IPE = IPE + 1 71 IF (X(IPL (D).EO.O.DO) GO TO 135 72 GO TO 145 73 135 IJTAG =2 74 GO TO 145 75 140 IPL(I) = IPL(I-I) - FACT 76 IF (Z(IPL (D).EO.O.DO) GO TO 146 77 IPE = IPE + 1 78 145 CONTINUE 79 146 HSCALE = XSIZE / XMAX 80 VSCALE = ZSIZE / ZMAX 81 HSI = XMAX / XSIZE 82 VSI = ZMAX / ZSIZE 83 S2 = ZMAX « VSCALE + 0.6D0 84 11 = 1 85 X1 = X(I1) * HSCALE 86 Z1 = Z(I1) * VSCALE 87 C 88 C PLOT FINITE ELEMENT MESH 89 C 90 IF (IOPLOT.EO.1.OR.IOPLOT.EO.4) IMESH = 1 91 IF (IMESH.EQ.O) GO TO 158 92 A = XSIZE + 2.DO 93 CALL PL0T(2.0, 0.0, -3) 94 CALL PL0T(X1, Z1, 3) 95 CALL AXIS(X1, Z1, 'KILOMETERS', -10, XSIZE, OJ 96 CALL AXIS(X1, Z1, 'KILOMETERS', 10. ZSIZE, 90.1 97 AZ = XMAX * HSCALE - 3.ODO 98 CALL SYMBOHAZ, S2, 0.25, 'MESH', 0.0, 4) 99 IF (IQPLOT.EQ.4) GO TO 155 100 DO 150 L = 1. NE 101 IF (ITYPE(NODE(L,1)).EQ. 1.AND.I TYPE(NODE(L,2 102 I ITYPE(NODE(L,3)).E0.1) GO TO 150 103 XI = X(N0DE(L,1)) * HSCALE 104 XO = X(N0DE(L,2)) * HSCALE 105 XK = X(N0DE(L,3)) * HSCALE 106 ZI = Z(N0DE(L,1)) * VSCALE 107 ZJ = Z(N0DE(L,2)) * VSCALE 108 ZK = Z(N0DE(L,3)) * VSCALE 109 CALL PLOT(XI, ZI, 3) 1 10 CALL PLOT(Xd, Zd, 2) 111 CALL PLOT(XK. ZK. 2) 112 CALL PLOT(XI, ZI, 2) 113 150 CONTINUE 1 14 CALL PLOT(A, 0.0, -3) 1 15 IF (IQPLOT.NE.1) GO TO 158 1 16 GO TO 245 . 0.0, HSI) o L i s t i n g of PLOTH at 09:50:08 on MAR 25, 1986 for CC1d=FAUL Page 3 1 17 155 ZCMAX = Z(NN) * VSCALE 1 18 ZCMIN = Z(NX) * VSCALE 1 19 IF (MTYPE.EO.2.OR.MTYPE.E0.4) ZCMIN = Z(NN-NX+1) 120 XCOORD = XMAX»HSCALE 121 DO 156 L=1,NN 122 XC = X(L)*HSCALE 123 ZC = Z(L)*VSCALE 124 CALL SYMBOL ( XC , ZC , 0 . 04 , 3 . 0. 0, - 1 ) 125 156 CONTINUE 126 CALL PLOT (0 .0 ,0 .0 ,3) . 127 CALL PLOT (XCOORD,ZCMIN,2) 128 CALL PLOT (XCOORD,ZCMAX,2) 129 CALL PLOT (0.0.ZMAX,2) 130 CALL PLOT (A,0 .0 , -3) 131 GO TO 245 132 C 133 C PRINT TITLE OF PLOT 134 C 135 158 CALL PLOT(1.5, 0.0, -3) 136 DO 160 IZ = 1, 20 137 AZ = 0.2D0 » DFLOAT(IZ - 1) + (XMAX*HSCALE - 3 138 160 CALL SYMBOLfAZ, S2, 0.14, ITITLE(IZ), 0.0, 1) 139 C 140 C PLOT AXES AND PERIMETER ON DRAWING 141 C 142 CALL PL0T(X1, Z1, 3) 143 CALL AXIS(X1, Z1, 'KILOMETERS', -10, XSIZE, 0.0, 144 CALL AXIS(X1, Z1, 'KILOMETERS', 10, ZSIZE, 90.0, 145 DO 165 M = 2, IPE 146 11 = IPL(M) 147 X1 = X(I1) » HSCALE 148 Z1 = Z(I1) * VSCALE 149 165 CALL PL0T(X1, Z1, 2) 150 C 151 C CONTOUR HYDRAULIC HEAD DATA 152 C 153 DO 220 L = 1, NE 154 IF (ITYPE(NODE(L,1)).EO.1.AND.ITYPE(NODE(L,2)) 155 1 ITYPE(N0DE(L.3)).E0.1) GO TO 220 156 ICO = 0 157 CN = CONZ 158 170 DO 175 d « 1 , 3 159 IM = NODE(L.J) 160 IF (VU(IM,2) .GT. CN) GO TO 175 161 GO TO 180 162 175 CONTINUE 163 GO TO 215 164 180 DO 185 I = 1, 3 165 IP = NOOE(L,I ) 166 IF (VU(IP,2) .LT. CN) GO TO 185 167 GO TO 190 168 185 CONTINUE 169 GO TO 215 170 190 IPO = 0 171 DO 210 N = 1, 3 172 K = N + 1 173 IF (N EO. 3) K = 1 174 dl = NODE(L.N) * VSCALE 0.0, VSI) L i s t i n g of PLOTH at 09:50:08 on MAR 25, 1986 for CC id=FAUL 175 J2 = NODE(L,K) 176 IF (CN .GE. VU(d1,2) .AND. CN .GE. VU(d2,2)) GO TO 210 177 IF (CN .LT. VU(d1.2) .AND. CN . LT . VU(d2,2)) GO TO 210 178 DELH = VU(J1,2) - VU(d2,2) 179 IF (DABS(DELH).GT.O.00001DO) GO TO 195 180 XA = X(d1 ) * HSCALE 181 ZA = Z(J1) * VSCALE 182 XB = X(d2) * HSCALE 183 ZB = Z(J2) * VSCALE 184 CALL PLOT(XA, ZA, 3) 185 CALL PLOT(XB, ZB, 2) 186 GO TO 215 187 195 CH = CN - VU(d1,2) 188 XA = X(d1) + (CH*(X(J1) - X(J2))/DELH) 189 XB = XA * HSCALE 190 ZA = Z(J1) + (CH*(Z(J1) - Z(J2))/DELH) 191 ZB = ZA * VSCALE 192 IPO = IPO + 1 193 GO TO (200, 205), IPO 194 200 CALL PLOT(XB, ZB. 3) 195 GO TO 210 196 205 CALL PLOT(XB, ZB, 2) 197 GO TO 215 198 210 CONTINUE 199 215 ICO = ICO + 1 200 IF (ICO .GE. ICON) GO TO 220 201 CN = CN + CONINT 202 GO TO 170 203 220 CONTINUE 204 C 205 C LABEL CONTOURS AT WATER TABLE, IF DESIRED 206 C 207 IF (ILAB.EQ.O) GO TO 240 208 ICO = 0 209 CN = CONZ 210 JMAX = K1 - 1 211 225 DO 235 d = 1. dMAX 212 N = d + 1 213 d1 = KNP(d) 214 d2 = KNP(N) 215 IF (Z(J1) .GE. CN .AND. CN .GT. Z(d2)) GO TO'230 216 IF (Z(d2) .GT. CN .AND. CN . GE . Z (dD) GO TO 231 217 GO TO 235 218 230 DELZ = Z(d1) - Z(d2) 219 ZH = CN - Z(d1) 220 XA = X(d1) + (ZH*(X(d1) - X(J2))/DELZ) 221 XB = XA * HSCALE 222 ZA = Z(J1) + (ZH*(Z(d1) - Z(d2))/DELZ) 223 ZB = ZA * VSCALE + 0.10DO 224 GO TO 232 225 231 DELZ = Z(d2) - Z(d1) 226 ZH = CN - Z(d2) 227 XA = X(J2) + (ZH*(X(J2) - X(d1))/DELZ) 228 XB = XA * HSCALE 229 ZA = Z(d2) + (ZH*(Z(J2) - Z(J1))/DELZ) 230 ZB = ZA * VSCALE + 0.10D0 231 232 CALL NUMBER(XB, ZB, 0.10, CN, 45.0, 2) 232 235 CONTINUE L i s t i n g o f P L O T H a t 0 9 : 5 0 : 0 8 o n MAR 2 5 , 1 9 8 6 f o r C C i d = F A U L Page 5 2 3 3 I C O = I C O + 1 2 3 4 I F ( I C O . G E . I C O N ) GO TO 2 4 0 2 3 5 C N = C N + C O N I N T 2 3 6 GO TO 2 2 5 2 3 7 2 4 0 A = X S I Z E + 2 . D O 2 3 8 C A L L P L O T ( A , 0 . 0 , - 3 ) 2 3 9 2 4 5 R E T U R N 2 4 0 END cn co L i s t i n g of PLOTV at 09:50:22 on MAR 25, 1986 for CC1d=FAUL 1 SUBROUTINE VPLOT (NN, NE, NX, NZ, IPE, IPL, XVEL, ZVEL, 2 1XMAX, ZMAX, X, Z. NODE. ITITLE, ITYPE) 3 C 4 C ************************************************ 5 C PLOTTING PACKAGE 6 C ***************************************************************** 7 C 8 C 9 C SUBROUTINE VPLOT DRAWS VELOCITY ARROWS SHOWING MAGNITUDE AND 10 C DIRECTION OF FLOW IN EACH TRIANGULAR ELEMENT 11 C LARGEST VELOCITY ARROW IS 0.55 INCHES 12 C 13 C LIST OF VARIABLES: 14 C 15 C SEE MAIN PROGRAM - 2DTF 16 C 17 IMPLICIT REAL'S (A-H.O-Z) 18 REAL*4 VI. V2, SCTEMP 19 DIMENSION XCEN(600), ZCEN(600), XVEL(600). ZVEL(600), 20 1N0DE(6OO,3), X(340), Z(340), IPL(80), ITYPE(340) 21 L0GICAL»1 ITITLE(20) 22 C 23 C CALCULATE CONSTANTS 24 C 25 ZSIZE = 3.0D0 26 XSIZE = 6.0D0 27 HSCALE=XSIZE/XMAX 28 VSCALE=ZSIZE/ZMAX 29 HSI = XMAX / XSIZE 30 VSI = ZMAX / ZSIZE 31 S2 = ZMAX * VSCALE + 0.600 32 S3 = S2 - 0.2 33 S4 = S3 - 0.2 34 C 35 C SCAN FOR PLOTTING SCALE, VELOCITY ARROWS 36 C 37 TEST1=0.D0 38 TEST2=1000.D0 39 DO 100 1=1,NE 40 VMAG = (XVEL(I)**2 + ZVEL(I)* *2)**.5 41 IF (VMAG.LT.3.E-15) GO TO 100 42 IF (VMAG.GT.TEST 1) TEST 1=VMAG 43 IF (VMAG.LT.TEST2) TEST2=VMAG 44 100 CONTINUE 45 SCTEMP = DL0G10(TEST2) 46 SCTEMP = FLOAT(- IFIX(SCTEMP)) + 1. 47 SC1 = 10.DO ** SCTEMP 48 SC2 = .5D0 / DL0G10 (TEST 1 * SC1) 49 C 50 C PRINT TITLE OF PLOT 51 C 52 CALL PL0T(1.5, 0.0, -3) 53 DO 115 IZ = 1, 20 54 AZ = 0.2D0 * DFLOAT(IZ - 1) + (XMAX"HSCALE - 3.ODO) 55 115 CALL SYMBOL(AZ, S2, 0.14, ITITLE(IZ), 0.0, 1) 56 AZ = XMAX*HSCALE - 3.DO 57 V1 = DL0G1O(TEST1) 58 V2 = FLOAT(IFIX(V1)) L i s t i n g of PLOTV at 09:50:22 on MAR 25, 1986 for CC1d=FAUL Page 2 59 IF (V1.LT.0.) V2 = V2 - 1. 60 V1 = 10**(V1-V2) 61 CALL SYMBOL (AZ,S3,O. 14, 'MAX I MUM VELOCITY = ' ,0 .0 ,19) 62 A1 = AZ + 2.3DO 63 CALL NUMBER (A1,S3,0.14,V1,0.0.2) 64 A2 = A1 + O.5D0 65 CALL SYMBOL (A2 ,S3 ,0 .14 , 'E ' ,0 .0 ,1) 66 A3 = A2 + 0.2D0 67 CALL NUMBER (A3,S3,0.14,V2,O.O,-2) 68 C 69 C PLOT BOUNDARIES OF REGION 70 C 71 1 1 = IPL(1) 72 X1 = X(I1) * HSCALE 73 Z1 = Z(11) * VSCALE 74' CALL PL0T(X1, Z I . 3) 75 CALL AXIS(X1, Z I , 'KILOMETERS', -10, XSIZE. 0.0. 0 .0 , HSI) 76 CALL AXIS(X1, Z I , 'KILOMETERS'. 10. ZSIZE, 90.0. 0.0, VSI) 77 DO 120 M = 2, IPE 78 11 = IPL(M) 79 X1 = X(I 1) * HSCALE 80 Z1 = Z(I1) * VSCALE 81 120 CALL PL0T(X1, Z I , 2) 82 C 83 C COMPUTE CENTERS FOR TRIANGULAR ELEMENTS 84 C 85 . DO 130 LL=1,NE 86 IA=NODE(LL,1) 87 IB=N0DE(LL,2) 88 IC=N0DE(LL,3) 89 XCEN(LL) = ((X(IA)+X(IB) + X(IC))/3)«HSCALE 90 130 ZCEN(LL) = ((Z(IA) + Z(IB ) + Z(IC) )/3)*VSCALE 91 C 92 C DETERMINE LENGTH OF VELOCITY ARROW 93 C DETERMINE DIRECTION OF ARROWHEAD 94 C THETA EQUAL O IS VERTICALLY UP FOR UBC PLOTTING ROUTINES 95 C 96 NUM = NX - 1 97 KOUNT = 0 98 LL = O 99 IMAX = (((NE/2)-NX+1)/2)-3 100 DO 150 1 = 1, IMAX 101 LL = LL + 2 102 KOUNT = KOUNT + 1 103 IF (KOUNT.GT.NUM) GO TO 146 104 140 IF (ITYPE(NODE(LL,1)).EQ.1.AND.ITYPE(N0DE(LL,2)).EQ.1.ANO. 105 . 1 ITYPE(N0DE(LL,3)).EQ.1) GO TO 150 106 CALL PL0T(XCEN(LL),ZCEN(LL),3) 107 IF(XVEL(LL).EQ.O.ODO) GO TO 150 108 IF(XVEL(LL) LT.O.ODO) GO TO 160 109 IF(ZVEL(LL).LT O.ODO) GO TO 170 110 THETA=DATAN(ZVEL(LL)/XVEL(LL)) 111 THETA=270.0DO+THETA*180.0DO/3.1416D0 112 GO TO 180 113 170' THETA=DATAN(DABS(ZVEL(LL))/XVEL(LL)) 1 14 THETA = 270.0DO-THETA*18O.0DO/3.1426D0 115 GO TO 180 116 160 IF(ZVEL(LL) LT.O.ODO) GO TO 190 L i s t i n g of PLOTV at 09:50:22 on MAR 25, 1986 for CCid=FAUL Page 3 117 THETA=DATAN(ZVEL(LL)/DABS(XVEL ( LL))) 1 18 THETA=90.0D0-THETA*180.0D0/3.1426D0 119 GO TO 180 120 190 THETA=DATAN(DABS(ZVEL(LL))/DABS(XVEL(LL))) 121 THETA=90.0DO+THETA*180.DO/3.1426D0 122 180 VMAG=(XVEL(LL)**2 + ZVEL(LL)**2)»* .5 123 IF (VMAG.LT.3.E-15) GO TO 145 124 VMAG1 = DL0G10(VMAG*SC1) * 5C2 125 IF (VMAG1.LT.0.05DO) GO TO 145 126 VELX = XVEL(LL)*VMAG1/VMAG 127 VELZ = ZVEL(LL)*VMAG1/VMAG 128 XVX=XCEN(LL)+VELX 129 ZVZ=ZCEN(LL)+VELZ 130 CALL PL0T(XVX,ZVZ,2) 131 CALL SYMBOL(XVX.ZVZ,0.10,6,THETA,- 1) 132 GO TO 150 133 145 CALL SYMBOL (XCEN(LL).ZCEN(LL).0.04.4,0.0,- 1) 134 GO TO 150 135 146 KOUNT = 1 136 LL = LL + NUM«2 137 GO TO 140 138 150 CONTINUE 139 A=XSIZE+2.0D0 140 CALL PL0T(A,O.O.-3) 141 RETURN 142 END 

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