UBC Theses and Dissertations

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UBC Theses and Dissertations

Numerical modeling of horizontal drain drainage in an open pit slope Ge, Shemin 1985

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N U M E R I C A L M O D E L I N G O F H O R I Z O N T A L D R A I N D R A I N A G E I N A N O P E N P I T S L O P E by S H E M I N G E B . A . S c . , WUHAN I N S T I T U T E O F B U I L D I N G M A T E R I A L S , C H I N A , 1 9 8 2 A T H E S I S S U B M I T T E D 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 T H E D E G R E E O F M A S T E R O F A P P L I E D S C I E N C E i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S T h e D e p a r t m e n t o f M i n i n g a n d M i n e r a l P r o c e s s i n g E n g i n e e r i n g We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F BRITI^tKcOLUMBIA ^ A p r i l , 1 9 8 5 © S h e m i n G e , A p r i l , 1 9 8 5 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 a d v a n c e d d e g r e e a t t h e 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 a g r e e 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 a n d s t u d y . I f u r t h e r a g r e e 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 p u r p o s e s may be g r a n t e d by t h e h e a d o f my d e p a r t m e n t o r b y 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 . D e p a r t m e n t o f W '^'^W ho^^ ^f^J^J T h e 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 1956 Main Ma l l V a n c o u v e r , C a n a d a V6T 1Y3 D a t e £ ^ >JK~ D E - 6 (3/81) ft ABSTRACT A s t u d y h a s b e e n made t o e v a l u a t e t h e e f f e c t s o f h o r i z o n t a l d r a i n d r a i n a g e on t h e w a t e r t a b l e drawdown i n o p e n p i t s l o p e s . Two m a j o r p a r a m e t e r s o f a h o r i z o n t a l d r a i n d r a i n a g e s y s t e m , l e n g t h a n d s p a c i n g , were s t u d i e d . A two d i m e n s i o n a l f i n i t e e l e m e n t c o m p u t e r m o d e l was c o n s t r u c t e d t o s i m u l a t e t h e w a t e r f l o w i n t o d r a i n s i n r o c k s l o p e s . W a t e r f l o w i n t h e s a t u r a t e d z o n e was a s s u m e d . T h e c o m p u t e r m o d e l was t e s t e d by t h e f i e l d d a t a o b t a i n e d f r o m t h e LORNEX M i n e i n B r i t i s h C o l u m b i a a n d t h e d a t a t a k e n f r o m INTRODUCTION TO GROUNDWATER MODELING (Wang & A n d e r s o n , 1 9 8 2 ) . S a t i s f a c t o r y a g r e e m e n t s were o b t a i n e d . A s t h e r e s u l t o f c o m p u t e r s i m u l a t i o n s , a s e r i e s o f g r a p h s were p l o t t e d . T h e s e g r a p h s show t h e r e l a t i o n s h i p b e t w e e n h y d r a u l i c h e a d d i s t r i b u t i o n v s . d r a i n s p a c i n g a n d l e n g t h . T h e y c o u l d be u s e d i n h o r i z o n t a l d r a i n d e s i g n a s an a i d t o d e t e r m i n e t h e s p a c i n g a n d l e n g t h o f a d r a i n s y s t e m . T h e c o m p u t e r s i m u l a t i o n s were a l s o made t o s t u d y t h e d r a i n a g e c h a r a c t e r i s t i c s o f a n i s o t r o p i c r o c k s l o p e s . The r e s u l t s i n d i c a t e d t h e i n f l u e n c e o f s u c h r o c k c o n d i t i o n s on t h e d r a i n a g e e f f e c t . A n o t h e r f e a t u r e o f m i n i n g s l o p e s i s t h a t t h e i r h e i g h t v a r i e s a s t h e m i n i n g o p e r a t i o n p r o g r e s s e s . T h e r e f o r e , t h e s u i t a b l e v e r t i c a l s p a c i n g b e t w e e n d r a i n rows was i n v e s t i g a t e d by c o m p u t e r s i m u l a t i o n . A c o m p a r i s o n o f t h e d r a i n a g e e f f e c t s o f d i f f e r e n t d r a i n p a t t e r n s , p a r a l l e l d r a i n a n d f a n n e d d r a i n l a y o u t s , was a l s o m a d e . T A B L E OF CONTENTS Page ABSTRACT i i T A B L E OF CONTENTS i i i L I S T OF F IGURES v L I S T OF T A B L E S v i i i ACKNOWLEDGEMENT i x C h a p t e r 1. INTRODUCTION 1 C h a p t e r 2 . MATHEMATICAL MODELING 6 2.1 T h e P h y s i c a l P r o b l e m 7 2 . 2 T h e M a t h e m a t i c l P r o b l e m 9 2 . 3 F i n i t e E l e m e n t S o l u t i o n 13 B a s i c T h e o r y 13 F i n i t e E l e m e n t F o r m u l a t i o n 15 B o u n d a r y C o n d i t i o n s 18 2 . 4 C o m p u t e r P r o g r a m a n d V e r i f i c a t i o n 20 C h a p t e r 3 . COMPUTER A N A L Y S I S 28 3.1 H o r i z o n t a l M o d e l A n a l y s i s 28 3 . 2 V e r t i c a l M o d e l A n a l y s i s . . 4 7 3 . 3 A n i s o t r o p y E f f e c t s on D r a i n a g e 56 3 . 4 T h e C o m p a r i s i o n o f D i f f e r e n t D r a i n P a t t e r n L a y o u t s . . . 5 8 C h a p t e r 4 . A p p l i c a t i o n s 65 4.1 H y p o t h e t i c a l E x a m p l e 65 4 . 2 T h e A p p l i c a t i o n f o r LORNEX M i n e 72 C h a p t e r 5 . SUMMARY AND CONCLUSIONS 77 5.1 C o n c l u s i o n s 77 5 . 2 Summary 78 i i i T A B L E OF CONTENTS ( c o n t i n u e d ) P a g e B IBLIOGRAPHY 80 A p p e n d i x A . F i n i t e E l e m e n t F o r m u l a t i o n 83 A p p e n d i x B . C o m p u t e r P r o g r a m O r g a n i z a t i o n 89 i v L I S T OF F IGURES F i g u r e P a g e 1. A H o r i z o n t a l D r a i n D r a i n a g e S y s t e m 2 2 . H y p o t h e t i c a l S l o p e w i t h D r a i n s . 8 3 . The W a t e r T a b l e P o s i t i o n i n Two S e c t i o n s o f a S l o p e .10 4 . A s s o c i a t e d N o d e s a n d E l e m e n t s 16 5 . B o u n d a r y C o n d i t i o n A p p l i e d 19 6 . The P l a n V i e w o f LORNEX Open P i t 24 7 . M o d e l S e c t i o n o f LORNEX M i n e a n d t h e H y d r a u l i c C o n d u c t i v i t y D i s t r i b u t i o n 25 8 . The C o m p a r i s o n o f C o m p u t e r O u t p u t a n d t h e F i e l d D a t a o f LORNEX M i n e 26 9 . F i n i t e E l e m e n t M e s h f o r H o r i z o n t a l M o d e l 29 10 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 150m, D r a i n L e n g t h 90m 32 11 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 150m, D r a i n L e n g t h 80m 33 12 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 150m, D r a i n L e n g t h 70m 34 13 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 150m, D r a i n L e n g t h 60m 35 14 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 150m, D r a i n L e n g t h 50m 36 15 . D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 175m, D r a i n L e n g t h 100m 37 16 . D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 175m, D r a i n L e n g t h 90m 38 17 . D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 175m, D r a i n L e n g t h 80m 39 18 . D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 175m, D r a i n L e n g t h 70m 40 19 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 175m, D r a i n L e n g t h 60m 41 v L I S T OF F IGURES ( " c o n t i n u e d ) F i g u r e Page 2 0 . D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 200m, D r a i n L e n g t h 100m 42 2 1 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 200m, D r a i n L e n g t h 90m 43 2 2 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 200m, D r a i n L e n g t h 80m 44 2 3 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 200m, D r a i n L e n g t h 70m 45 2 4 . D i m e n s i o n l e s s H y d r a u l i c Head D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 200m, D r a i n L e n g t h 60m 46 2 5 . F i n i t e E l e m e n t M e s h f o r V e r t i c a l M o d e l 48 2 6 . W a t e r T a b l e P o s i t i o n i n a S l o p e 100m H e i g h t 50 2 7 . W a t e r T a b l e P o s i t i o n i n a S l o p e 120m H e i g h t 51 2 8 . W a t e r T a b l e P o s i t i o n i n a S l o p e 140m H e i g h t . . . . . . . . 5 2 2 9 . W a t e r T a b l e P o s i t i o n i n a S l o p e 160m H e i g h t 53 3 0 . W a t e r T a b l e P o s i t i o n i n a S l o p e 180m H e i g h t 54 3 1 . W a t e r T a b l e P o s i t i o n i n a S l o p e 200m H e i g h t 55 3 2 . T h e D r a i n a g e E f f e c t s o f A n i s o t r o p i c M a t e r i a l s 57 3 3 . P a r a l l e l a n d F a n n e d D r a i n P a t t e r n s 59 3 4 . F i n i t e E l e m e n t M e s h f o r a F a n n e d D r a i n P a t t e r n 60 3 5 . T h e C o m p a r i s o n o f F a n n e d a n d P a r a l l e l D r a i n P a t t e r n s , S l o p e H e i g h t 150m 62 3 6 . T h e C o m p a r i s o n o f F a n n e d a n d P a r a l l e l D r a i n P a t t e r n s , S l o p e H e i g h t 175m 63 3 7 . T h e C o m p a r i s o n o f F a n n e d a n d P a r a l l e l D r a i n P a t t e r n s , S l o p e H e i g h t 200m 64 3 8 . T h e S l o p e S e c t i o n o f t h e E x a m p l e 66 v i L I S T OF F IGURES ( " c o n t i n u e d ) F i g u r e Page 3 9 . E s t i m a t e d W a t e r T a b l e P o s i t i o n i n t h e S a m p l e S l o p e 70 4 0 . E s t i m a t e d W a t e r T a b l e P o s i t i o n i n LORNEX M i n e 75 A . 1. T h e G l o b a l M a t r i x A s s e m b l y f r o m I n d i v i d u a l E l e m e n t s 88 B . 1. F l o w C h a r t f o r t h e C o m p u t e r P r o g r a m FESHDMS 91 B . 2 . An E x a m p l e t o Show t h e C o m p u t e r I n p u t D a t a 98 v i i L I S T OF T A B L E S T a b l e s • P a g e I. T h e C o m p a r i s o n o f Two S e t s o f R e s u l t 27 I I . C a l c u l a t i o n T a b l e 68 I I I . E x a m p l e C a l c u l a t i o n T a b l e 69 I V . C a l c u l a t i o n T a b l e f o r LORNEX M i n e 74 v i i i ACKNOWLEDGEMENT I w o u l d l i k e t o e x p r e s s my t h a n k s t o P r o f e s s o r C . 0 . B r a w n e r f o r h i s c o n t i n u o u s g u i d a n c e a n d e n c o u r a g e m e n t d u r i n g t h i s r e s e a r c h p r o g r a m . I a p p r e c i a t e t h e comments o f my c o m m i t t e e m e m b e r s ; D r . G . W. P o l i n g a n d D r . H . D. S . M i l l e r . I am g r a t e f u l t o D r . A . J . R e e d a n d D r . J . L . S m i t h f o r h e l p f u l a d v i c e . I w o u l d a l s o l i k e t o t h a n k D r . A . R. F r e e z e , D r . J . R u l o n a n d D r . D . G o l d m a n . T h e d i s c u s s i o n s w i t h them h a v e p r o v e n t o be v a l u a b l e . D i s c u s s i o n s w i t h g r a d u a t e s t u d e n t s , e n c o u r a g e m e n t f r o m t h e f a c u l t y members a n d s t a f f i n t h e D e p a r t m e n t o f M i n i n g a n d M i n e r a l P r o c e s s i n g E n g i n e e r i n g , a n d t h e e n c o u r a g e m e n t f r o m Xu C h a n g y u a n d L u N i n g i n C h i n a a r e a l l a p p r e c i a t e d . 1 C h a p t e r 1 INTRODUCTION G r o u n d w a t e r c o n d i t i o n s a r e one o f t h e m a j o r c o n c e r n s i n o p e n p i t m i n e s l o p e s t a b i l i t y . A h i g h w a t e r t a b l e i n a s l o p e g r e a t l y r e d u c e s t h e s t a b i l i t y o f t h e s l o p e . T h e m e c h a n i s m i s t h a t t h e w a t e r p r e s s u r e r e d u c e s t h e e f f e c t i v e s h e a r s t r e n g t h o f r o c k , w a t e r p r e s s u r e i n t e n s i o n c r a c k s i n c r e a s e s s h e a r s t r e s s , h y d r o d y n a m i c s h o c k may be i n d u c e d due t o b l a s t i n g o r o t h e r v i b r a t i o n s , a n d w a t e r p r e s s u r e d e v e l o p s s e e p a g e f o r c e s a c t i n g i n t h e d i r e c t i o n o f movement t o r e d u c e s l o p e s t a b i l i t y . ( B r a w n e r , 1983) T o i m p r o v e s l o p e s t a b i l i t y , i t i s n e c e s s a r y t o l o w e r t h e w a t e r t a b l e i n t h e s l o p e . T h e r e a r e s e v e r a l d r a i n a g e s y s t e m s u s e d i n p r a c t i c e a c c o r d i n g t o d i f f e r e n t g e o l o g i c c o n d i t i o n s , d r a i n a g e r e q u i r e m e n t s , e q u i p m e n t a v a i l a b i l i t y a n d c o s t e f f e c t i v e f a c t o r s . T h e y a r e h o r i z o n t a l d r a i n d r a i n a g e , p u m p i n g w e l l s , d r a i n a g e g a l l e r i e s , d r a i n a g e t r e n c h e s a n d h o r i z o n t a l d r a i n s u n d e r v a c u u m . (CANMET, 1980) Among t h e s e m e t h o d s , h o r i z o n t a l d r a i n d r a i n a g e h a s some a d v a n t a g e s o v e r o t h e r s i n many c i r c u m s t a n c e s . T h e y a r e q u i c k a n d e a s y t o i n s t a l l , work by g r a v i t y , n e e d l i t t l e u p k e e p , p r o v i d e l o n g l i f e a n d a r e low c o s t c o m p a r e d t o o t h e r s y s t e m s . A r e c e n t d e v e l o p m e n t i n c o m b i n i n g h o r i z o n t a l d r a i n s w i t h a vacuum h a s shown i t s g r e a t a d v a n t a g e s i n d r a i n i n g s l o p e s w i t h l o w e r c o n d u c t i v i t y m a t e r i a l . A t y p i c a l h o r i z o n t a l d r a i n d r a i n a g e s y s t e m i s shown i n F i g u r e 1. 2 F i g u r e 1.- A H o r i z o n t a l D r a i n Drainage System 3 ( S e e g m i l l e r , 1979) In t h e g e o t e c h n i c a l l i t e r a t u r e , t h e r e a r e no r i g o r o u s d e s i g n g u i d e s f o r a h o r i z o n t a l d r a i n d r a i n a g e s y s t e m . ( W i l l i a m s , 1982) In m i n i n g , p a r t i c u l a r l y , e v e n l e s s r e s e a r c h h a s b e e n d o n e i n t h i s a s p e c t . T h e d e t e r m i n a t i o n o f d r a i n p a r a m e t e r s s u c h a s l e n g t h a n d s p a c i n g i s commonly b a s e d on e x p e r i e n c e o r t r i a l a n d e r r o r i n s i t u t e s t i n g . D u r i n g t h e p a s t two y e a r s W i l l i a m s ( 1 9 8 2 ) a n d T e s a r i k ( 1 9 8 4 ) r e s p e c t i v e l y p u b l i s h e d a s t u d y r e l a t e d t o h o r i z o n t a l d r a i n d e s i g n i n e m b a n k m e n t s . T o t h e a u t h o r ' s k n o w l e d g e , t h e s e a r e t h e o n l y two p r e v i o u s l y p u b l i s h e d s t u d i e s a t t e m p t i n g t o u s e t h e f i n i t e e l e m e n t a n d f i n i t e d i f f e r e n c e m e t h o d s f o r h o r i z o n t a l d r a i n d e s i g n . W h i l e t h e i r work d e a l t p r i m a r i l y w i t h t h e d e s i g n o f e m b a n k m e n t s , t h e e m p h a s i s o f t h i s s t u d y p r e s e n t e d i n t h e t h e s i s was on t h e d e s i g n f o r o p e n p i t m i n i n g s l o p e s w h i c h d i f f e r f r o m embankments i n some ways s u c h a s t h e h e t e r o g e n e i t y a n d a n i s o t r o p y o f t h e s l o p e m a t e r i a l s a n d t h e v a r y i n g h e i g h t o f s l o p e s . T h e r e a r e f o u r o b j e c t i v e s i n t h i s s t u d y . 1. T o e s t i m a t e t h e h y d r a u l i c h e a d d i s t r i b u t i o n on t h e d r a i n l e v e l a f t e r t h e h o r i z o n t a l d r a i n s a r e i n s t a l l e d i n s l o p e s w i t h d i f f e r e n t l e n g t h s a n d s p a c i n g s . 2 . T o e s t i m a t e t h e s u i t a b l e v e r t i c a l d i s t a n c e b e t w e e n rows o f d r a i n s a s t h e p i t e x t e n d s d e e p e r . 3 . T o i n v e s t i g a t e t h e i n f l u e n c e of a n i s o t r o p y o f s l o p e m a t e r i a l s on t h e d r a i n a g e r e s u l t s . 4 4 . T o c o m p a r e t h e d r a i n a g e r e s u l t s o f d i f f e r e n t d r a i n l a y o u t s , f a n n e d a n d p a r a l l e l d r a i n p a t t e r n s . T o a c h i e v e t h e s e o b j e c t i v e s , f o u r p r o c e d u r e s w e r e f o l l o w e d . F i r s t , t o d e v e l o p a c o m p u t e r m o d e l w h i c h s i m u l a t e s 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 , s a t u r a t e d w a t e r f l o w i n t h e r o c k s l o p e b y t h e f i n i t e e l e m e n t m e t h o d . S e c o n d , t o v e r i f y t h e v a l i d i t y o f t h e c o m p u t e r m o d e l w i t h f i e l d d a t a f r o m t h e L O R N E X M i n e a n d a n e x a m p l e t a k e n f r o m t h e b o o k , 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 a n g & A n d e r s o n , 1 9 8 2 ) . T h e t h i r d s t e p i s t o u s e t h e p r o g r a m t o s i m u l a t e t h e d r a i n a g e p e r f o r m a n c e s f o r a h o r i z o n t a l d r a i n d r a i n a g e s y s t e m w i t h d i f f e r e n t c o m b i n a t i o n s o f d r a i n l e n g t h s a n d s p a c i n g s , d i f f e r e n t d r a i n p a t t e r n l a y o u t s a n d v a r y i n g s l o p e h e i g h t s . T h e f i n a l s t e p i s t o i n t e r p r e t r e s u l t s i n a g r a p h i c a l f o r m . T h e d e s i g n o f a h o r i z o n t a l d r a i n s y s t e m i s n o t a s i m p l e t a s k . I n p r a c t i c e s e l e c t i o n o f d r a i n p a r a m e t e r s d e p e n d s o n t h e r e l i a b i l i t y o f g e o l o g i c i n f o r m a t i o n a n d t h e g o o d j u d g e m e n t o f e x p e r i e n c e d e n g i n e e r s . T h e i n t e n t i o n o f t h i s s t u d y i s t o p r o v i d e a b a s i c e v a l u a t i o n o f t h e d r a i n p a r a m e t e r s f o r d e s i g n . M o r e r e s e a r c h i s n e e d e d i n o r d e r t o d e v e l o p a m e t h o d o l o g y f o r t h e h o r i z o n t a l d r a i n d e s i g n . C o m b i n i n g t h e c o m p u t e r s i m u l a t i o n t e c h n i q u e a n d f i e l d e x p e r i e n c e s s h o w s g r e a t p o t e n t i a l i n t h i s r e s e a r c h a r e a . I n t h e t h e s i s o r g a n i z a t i o n , t h e d e s c r i p t i o n o f t h e c o m p u t e r m o d e l d e v e l o p m e n t a n d m o d e l v e r i f i c a t i o n a r e p r e s e n t e d f i r s t , i n C h a p t e r 2 . T h e c o m p u t e r s i m u l a t i o n s 5 w h i c h i n v o l v e t h e c o m p u t e r a n a l y s i s on d i f f e r e n t m o d e l s and t h e s i m u l a t i o n i n t e r p r e t a t i o n a r e i n C h a p t e r 3. Two e x a m p l e s a r e p r e s e n t e d i n C h a p t e r 4 t o show t h e a p p l i c a t i o n o f t h i s s t u d y . C h a p t e r 5 c o n t a i n s t h e summary a n d c o n c l u s i o n s . The d e t a i l s o f t h e f i n i t e e l e m e n t f o r m u l a t i o n a n d c o m p u t e r p r o g r a m a r e p r e s e n t e d i n A p p e n d i x A a n d B , r e s p e c t i v e l y . 6 Chapter 2 MATHEMATICAL MODEL Groundwater in mining slopes is represented by the flow of fluid through porous media. Earlier research in this area of fluid flow through porous media was based on analytical methods. These mathematical methods, originally, were developed for treatment of problems of heat flow, elect r i c i t y and magnetic field s . With the development of the digital computer and its widespread availability, many of the important recent advances in the analysis of groundwater systems have been based on much different mathematical approaches generally known as numerical methods. The fin i t e element method is one of the numerical methods. It is the most widely used in research work and is employed in this study. (Freeze & Cherry, 1979) To solve groundwater problems by numerical methods involves four steps. The f i r s t step is to examine the physical problem encountered in practice. The second is to replace the physical problem by an equivalent mathematical problem, which, in more specific terms, is refered to as a boundary value problem. The third is to solve the boundary value problem for unknowns by computer techniques. The final step is to interpret the numerical solution in terms of the physical problem. The emphasis of this study was placed on the final step. The details of that part, therefore, were arranged in a separate chapter while the other three parts 7 a r e i n c l u d e d i n t h i s c h a p t e r . 2.1 The P h y s i c a l P r o b l e m The p h y s i c a l p r o b l e m u n d e r s t u d y i s a common p r o b l e m e n c o u n t e r e d i n o p e n p i t s l o p e d r a i n a g e . A s i l l u s t r a t e d i n F i g u r e 2 , a v e r t i c a l s e c t i o n o f an open p i t s l o p e and a h o r i z o n t a l s e c t i o n on t h e d r a i n l e v e l show t h e p r o b l e m i n t h r e e d i m e n s i o n s . The s l o p e w i l l l i k e l y c o m p r i s e h e t e r o g e n e o u s a n d a n i s o t r o p i c m a t e r i a l s . T o s i m u l a t e t h e w a t e r f l o w i n t o d r a i n s , s o a s t o o b t a i n t h e w a t e r p r e s s u r e d i s t r i b u t i o n on t h e d r a i n l e v e l u n d e r d i f f e r e n t c o n d i t i o n s , a n a l y s e s were p e r f o r m e d on h o r i z o n t a l s e c t i o n s . P a r t i c u l a r l y i n t e r e s t i n g i s t h e p r e s s u r e d i s t r i b u t i o n a l o n g t h e l i n e ( B F ) w h i c h i s midway b e t w e e n two d r a i n s . A l o n g B F , t h e p r e s s u r e w o u l d be t h e h i g h e s t c o m p a r e d w i t h o t h e r p a r a l l e l l i n e s on t h e same p l a n e , t h e r e f o r e , t h i s h a s most s i g n i f i c a n c e i n p r o v i d i n g t h e i n f o r m a t i o n f o r a s t a b i l i t y a n a l y s i s . The r i g h t b o u n d a r y EG was c o n s i d e r e d t o be f a r e n o u g h f r o m t h e s l o p e t o e t h a t t h e h y d r a u l i c h e a d on EG i s d e t e r m i n e d by r e g i o n a l g r o u n d w a t e r c o n d i t i o n s and i s a s s u m e d n o t t o be a f f e c t e d by d r a i n a g e . The l e f t b o u n d a r y i s t h e t o e o f t h e s l o p e a n d h a s z e r o p r e s s u r e a l o n g i t . As t h e h o r i z o n t a l d r a i n s ( C D a n d AH i n F i g u r e 2) a r e i n s t a l l e d , w a t e r f l o w s o u t o f t h e r e g i o n n o t o n l y a l o n g A C , b u t a l s o a l o n g t h e s e d r a i n s . 8 Free Surface after Drainage Free Surface before Drainage drain H (a). Vertical section (b). Horizontal section Figure 2.-Hypothetical Slope with Drains 9 T o e s t i m a t e t h e v e r t i c a l d i s t a n c e b e t w e e n d r a i n r o w s , v a r y i n g h e i g h t s o f t h e s l o p e must be c o n s i d e r e d . In t h i s c a s e i t i s more c o n v e n i e n t t o work on t h e v e r t i c a l s e c t i o n m o d e l w h i c h i s shown i n F i g u r e 2 ( a ) . 2 . 2 M a t h e m a t i c a l P r o b l e m W a t e r f l o w i n g i n t o d r a i n s i n a s l o p e i s a t h r e e d i m e n s i o n a l p r o b l e m . H o w e v e r , f o r a t h r e e d i m e n s i o n a l f i n i t e e l e m e n t m o d e l , t h e e l e m e n t mesh c o n s t r u c t i o n r e q u i r e s c o n s i d e r a b l e t i m e . T h e c o m p u t e r p r o g r a m o f t e n r e q u i r e s much c o m p u t e r t i m e a n d l a r g e s t o r a g e due t o t h e l a r g e number o f unknowns w h i c h c o u l d r e s u l t i n a v e r y h i g h c o s t o f c o m p u t e r r u n . T e s a r i k ' s s t u d y ( T e s a r i k , 1 9 8 4 ) on t h e s u b j e c t o f h o r i z o n t a l d r a i n d e s i g n f o r embankments i n d i c a t e d t h a t two d i m e n s i o n a l a n d t h r e e d i m e n s i o n a l c o m p u t e r c o d e s p r o d u c e d n e a r l y t h e same r e s u l t s f o r p h r e a t i c s u r f a c e l o c a t i o n s b e t w e e n d r a i n s . T h e r e f o r e , two two d i m e n s i o n a l m o d e l s , one on a h o r i z o n t a l s e c t i o n , a n o t h e r on a v e r t i c a l s e c t i o n , were u s e d i n t h i s s t u d y . T h e h o r i z o n t a l m o d e l d o e s n o t i n c l u d e t h e w a t e r f l o w c o m p o n e n t i n t h e v e r t i c a l d i r e c t i o n . T h i s i s t h e D u p u i t a s s u m p t i o n . The v a l i d i t y o f t h i s a s s u m p t i o n h a s been e v a l u a t e d a n a l y t i c a l l y by M u r r a y a n d Monkmeyer ( M u r r a y , 1 9 7 3 ) . In g e n e r a l , t h e b e s t r e s u l t s u s i n g t h e D u p u i t a s s u m p t i o n a r e a c h i e v e d f o r t h e s i t u a t i o n s w h e r e t h e s l o p e o f t h e p h r e a t i c s u r f a c e i s r e l a t i v e l y f l a t . drain » t w 1 I T i i 1 1 A — (a). Plan View H - w - f -(b). Section A - A h - " — H (c). Section B - B Figure 3 . -Water Table Position along Slope 11 In v e r t i c a l s e c t i o n s i m u l a t i o n , t h e d r a i n i s t r e a t e d a s a b l a n k e t d r a i n i n t h r e e d i m e n s i o n s , w h i c h i s n o t r e a l i n m i n i n g p r a c t i c e . As shown i n F i g u r e 3 , i n d i v i d u a l d r a i n s a r e u s u a l l y u s e d i n s l o p e d r a i n a g e . The h e i g h t o f t h e w a t e r t a b l e i n t h e s e c t i o n midway b e t w e e n two d r a i n s , s e e F i g u r e 3 ( b ) , w o u l d be t h e maximum. In t h e s e c t i o n a l o n g t h e d r a i n s , F i g u r e 3 ( c ) , t h e w a t e r s u r f a c e i s f u r t h e r b a c k i n t o t h e s l o p e . In F i g u r e 3 ( b ) , w , t h e d i s t a n c e f r o m t h e s l o p e t o e t o t h e p o i n t a t w h i c h t h e w a t e r s u r f a c e r e a c h e s t h e d r a i n l e v e l , i s a f u n c t i o n o f d r a i n s p a c i n g a n d l e n g t h . I t i s o b v i o u s t o s e e t h a t w i s a l w a y s s m a l l e r t h a n t h e a c t u a l d r a i n l e n g t h . In any s p e c i f i c c a s e , w c o u l d be e s t i m a t e d by s i m u l a t i o n on a h o r i z o n t a l m o d e l a n d f i n d i n g t h e i n t e r s e c t i o n o f t h e w a t e r t a b l e w i t h t h e p l a n e on t h e d r a i n l e v e l . U s i n g t h e w e s t i m a t e d i n t h e v e r t i c a l m o d e l a n d i g n o r i n g t h e f l o w p e r p e n d i c u l a r t o t h e v e r t i c a l s e c t i o n , t h e r e s u l t s o f t h e . two d i m e n s i o n a l m o d e l i s s l i g h t l y c o n s e r v a t i v e . ( w i l l a m s , 1982) In b o t h h o r i z o n t a l a n d v e r t i c a l s e c t i o n m o d e l l i n g , t h e s t e a d y s t a t e f l o w e q u a t i o n was u s e d . T h i s means t h a t w a t e r c o n d i t i o n s a t no d r a i n a g e a n d a t t h e s t a g e when s t e a d y s t a t e f l o w a r e r e a c h e d a f t e r h o r i z o n t a l d r a i n i n s t a l l a t i o n c o m p l e t e d were m o d e l l e d . F l o w was a s s u m e d i n t h e s a t u r a t e d z o n e . H e t e r o g e n e i t i c a n d a n i s o t r o p i c p r o p e r t i e s o f s l o p e m a t e r i a l s c a n be m o d e l l e d . 12 The s t e a d y s t a t e f l o w i n a two d i m e n s i o n a l s a t u r a t e d medium i s g o v e r n e d by t h e f o l l o w i n g e q u a t i o n w h i c h was d e r i v e d f r o m D a r c y ' s Law a n d t h e c o n t i n u i t y o f w a t e r f l o w . ( F r e e z e & C h e r r y , 1979; CANMET, S u p p l e m e n t 4 - 1 , 1976) W h e r e : x a n d y a r e t h e c o o r d i n a t e s i n two d i r e c t i o n s p e r p e n d i c u l a r t o e a c h o t h e r , [ L ] Kx a n d Ky a r e t h e h y d r a u l i c c o n d u c t i v i t i e s i n x a n d y d i r e c t i o n s , [ L / T ] h i s h y d r a u l i c h e a d a s u n k n o w n , [ L ] Q ( x , y ) i s t h e f l o w i n f l u x i n t o o r o u t o f t h e r e g i o n , [ L / T ] T h e a n a l y t i c a l s o l u t i o n o f e q u a t i o n (1) w o u l d be t h e h y d r a u l i c h e a d a s a c o m p l i c a t e d f u n c t i o n o f x , y , K x , K y , a n d Q ( x , y ) . In most o f t h e c a s e s , an a n a l y t i c a l s o l u t i o n i s i m p o s s i b l e d u e t o t h e i r r e g u l a r s h a p e o f f l o w r e g i o n , h i g h h e t e r o g e n e i t y , a n d c o m p l i c a t e d b o u n d a r y c o n d i t i o n s . T h e n u m e r i c a l s o l u t i o n , on t h e o t h e r h a n d , w o u l d be a s e t o f h y d r a u l i c h e a d v a l u e s on a c e r t a i n number o f p o i n t s i n t h e r e g i o n . T o o b t a i n t h e s o l u t i o n o f e q u a t i o n (1) by a n y m e t h o d , b o u n d a r y c o n d i t i o n s a r e n e c e s s a r y t o be known. T h e s e c o n d i t i o n s a r e u s u a l l y d i f f e r e n t f r o m p r o b l e m t o p r o b l e m . T h e f l o w g o v e r n i n g e q u a t i o n (1) a n d b o u n d a r y c o n d i t i o n s d e s c r i b e t h e p r o b l e m u n i q u e l y , i . e . , t h e m a t h e m a t i c a l p r o b l e m i s p r e s e n t e d . T h e n e x t s t e p i s t o s o l v e t h e 13 m a t h e m a t i c a l p r o b l e m a n d o b t a i n t h e s o l u t i o n f o r h y d r a u l i c h e a d s i n t h e s t u d y r e g i o n . 2 . 3 F i n i t e E l e m e n t S o l u t i o n T h e r e a r e s e v e r a l a p p r o a c h e s b e i n g u s e d t o s o l v e b o u n d a r y v a l u e p r o b l e m s . T h e e x a c t s o l u t i o n c a n be o b t a i n e d by t h e a n a l y t i c a l m e t h o d i n some c a s e s . A n a l y t i c a l m e t h o d s a r e m a t h e m a t i c a l l y c o m p l i c a t e d a n d o n l y c a n be u s e d t o s o l v e p r o b l e m s h a v i n g s i m p l e b o u n d a r y c o n d i t i o n s i n a h o m o g e n e u o u s m e d i u m . N u m e r i c a l m e t h o d s , on t h e o t h e r h a n d , a r e a p p r o x i m a t e s o l u t i o n s w i t h o u t t h e a b o v e r e s t r i c t i o n s . W i t h t h e d e v e l o p m e n t o f t h e d i g i t a l c o m p u t e r , t h e y a r e b e c o m i n g a more p o w e r f u l t o o l i n t h i s a r e a . The f i n i t e e l e m e n t m e t h o d a n d t h e f i n i t e d i f f e r e n c e m e t h o d a r e two n u m e r i c a l m e t h o d s commonly u s e d i n g r o u n d w a t e r m o d e l i n g . C o m p a r e d w i t h t h e f i n i t e d i f f e r e n c e m e t h o d , t h e f i n i t e e l e m e n t m e t h o d h a s more a d v a n t a g e s . I t c a n h a n d l e f l o w r e g i o n s h a v i n g c o m p l i c a t e d s h a p e s a n d d e a l w i t h h e t e r o g e n e o u s m a t e r i a l s w i t h e a s e . In a d d i t i o n , t h e f i n i t e e l e m e n t m e t h o d c a n s o l v e t h e p r o b l e m w i t h a m o v i n g w a t e r t a b l e w i t h o u t g r e a t d i f f i c u l t y b e c a u s e o f t h e f l e x i b i l i t y i n v a r y i n g e l e m e n t s h a p e d u r i n g t h e p r o c e s s i n g o f t h e s o l u t i o n . B a s i c T h e o r y T h e v a r i a t i o n p r i n c i p l e i n m a t h e m a t i c s s t a t e s t h a t t h e s o l u t i o n o f a 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 w i t h c e r t a i n t y p e s o f b o u n d a r y c o n d i t i o n s i s i d e n t i c a l t o m i n i m i z i n g t h e 1 4 f u n c t i o n a l w h i c h i s f o u n d b y a c e r t a i n m a t h e m a t i c a l p r o c e d u r e . F o r t h e 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 (1) a n d t h e b o u n d a r y c o n d i t i o n s a s s o c i a t e d w h i c h a r e d i s c u s s e d l a t e r i n t h i s s e c t i o n , t h e f u n c t i o n i s f o u n d a s I . ( R e m s o n , 1 9 7 1 ) I = / X [ ^ K x ( | | ) 2 + l K y ( | | ) 2 + Q h ] d x d y ( 2 ) W h e r e : R i s t h e r e g i o n u n d e r s t u d y . A c c o r d i n g t o t h e v a r i a t i o n p r i n c i p l e , t o s o l v e e q u a t i o n (1) f o r h y d r a u l i c h e a d i s e q u i v a l e n t t o f i n d t h e h y d r a u l i c h e a d f u n c t i o n h ( x , y ) w h i c h c o u l d m i n i m i z e t h e f u n c t i o n a l I , i . e . h ( x , y ) s h o u l d b e s u c h t h a t i s s a t i s f i e d o n t h e r e g i o n R . T h e f i n i t e e l e m e n t m e t h o d i s a n a p p r o x i m a t e m e t h o d u s e d t o f i n d t h e h w h i c h s a t i s f i e s e q u a t i o n ( 3 ) . I n t h i s p r o c e d u r e , t h e c o n t i n o u s r e g i o n i s d i v i d e d i n t o a n u m b e r o f d i s c r e t e a r e a s o r e l e m e n t s . T h e s h a p e s o f t h e e l e m e n t s c a n b e t r i a n g u l a r o r r e c t a n g u l a r a s s o c i a t e d w i t h t h r e e o r f o u r n o d e s . T h e h y d r a u l i c h e a d s a t t h e s e n o d e s a r e u n k n o w n s . I f t h e r e g i o n i s d i v i d e d i n t o N e l e m e n t s w i t h M u n k n o w n h e a d n o d e s , 3hi 15 o v e r t h e w h o l e r e g i o n r e q u i r e s i = 1 ,M on e v e r y unknown h e a d n o d e . T h e r e w o u l d be one e q u a t i o n f o r e a c h n o d e , t h e r e f o r e , M e q u a t i o n s c o u l d be f o r m e d by c e r t a i n p r o c e d u r e s . C o n s e q u e n t l y , t h e unknowns c o u l d be o b t a i n e d by s o l v i n g t h e e q u a t i o n s a n d c o r p o r a t i n g i n t o t h e s p e c i f i e d b o u n d a r y c o n d i t i o n s i n t h e s o l u t i o n e q u a t i o n p r o c e s s . F i n i t e E l e m e n t P r o c e d u r e T h e f i r s t s t e p i n t h e f i n i t e e l e m e n t m e t h o d i s t o d i v i d e t h e r e g i o n i n t o s m a l l s u b r e g i o n s . T h e s m a l l e r t h e s u b r e g i o n s , t h e more a c c u r a t e t h e r e s u l t s . T h e n , t o f o r m u l a t e t h e f o l l o w i n g e q u a t i o n s f o r e a c h n o d e . A s i l l u s t r a t e d i n F i g u r e 4 , a n o d e i s a s s o c i a t e d w i t h a number o f e l e m e n t s , e a c h o f t h e s e e l e m e n t s a l l c o n t r i b u t e t o 31 t h e v a l u e o f •g^r* S i m i l a r i l y , an e l e m e n t i s s u r r o u n d e d by a number o f n o d e s ( u s u a l l y 3 o r 4 ) , t h e e l e m e n t h a s c o n t r i b u t i o n t o a l l o f i t s s u r r o u n d i n g n o d e s . F o r e a c h e l e m e n t i n t h e r e g i o n , t h e f o l l o w i n g e q u a t i o n s c o u l d be o b t a i n e d . T h e d e r i v a t i o n p r o c e d u r e i s d e t a i l e d i n A p p e n d i x 31 - 0 i = 1 , M on R 3 h , 16 Figure 4. —Associated Nodes and Elements 17 ( 1 ) . f o r t r i a n g u l a r e l e m e n t s , e K1 1 K12 K1 a rai ^ 31 ' 3 h , 31 ^3h,J K 2 1 K 2 2 K 2 3 K 3 , K 3 2 ^ 3 3 <hi c 1 C 1 ( 4 ) ( 2 ) . f o r r e c t a n g u l a r e l e m e n t s , f ah, 31 3 h , 31 3 h , 31 L 9 h U K 1 i K 1 2 K 1 3 K, „ K 2 1 K 2 2 K 2 3 K 2 o Kg i K 31 ft32 ^ 3 3 ft3« K: K„ , K « 2 R«-3 r -\ h , C t h , C 1 < \ h , C 1 v- J (5) W h e r e : s u p e r s c r i p t e a n d f r e p r e s e n t t h e e l e m e n t n u m b e r , [ k ] i s 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 , w h i c h i s a f u n c t i o n o f m a t e r i a l p r o p e r t i t i e s a n d e l e m e n t g e o m e t r y {h} i s t h e h y d r a u l i c h e a d v e c t o r , unknown {c} i s t h e f o r c e v e c t o r , w h i c h i s a f u n c t i o n o f b o u n d a r y c o n d i t i o n s a n d e l e m e n t g e o m e t r y As (4) a n d (5) were f o r m e d f o r e a c h e l e m e n t , t h e n e x t s t e p i s t o a s s e m b l e a g l o b a l s t i f f n e s s m a t r i x [K] a n d t h e v e c t o r {C} . T h i s was d o n e s t r i c t l y i n t e r m s o f g e o m e t r y , n o d e l a b e l i n g a n d e l e m e n t l a b e l i n g o f t h e e l e m e n t m e s h . (Wang & A n d e r s o n , 1 9 8 2 ) . T h e p r o c e s s o f a s s e m b l y i s a l s o • i l l u s t r a t e d i n A p p e n d i x A . 18 The r e s u l t o f t h e a s s e m b l y i s a s e t o f g l o b a l e q u a t i o n s r e s u l t i n g f r o m t h e number o f s e t s o f e q u a t i o n s . [K]{H}+{C}=0 (6) W h e r e : [K] i s t h e g l o b a l s t i f f n e s s m a t r i x {h} i s t h e h y d r a u l i c h e a d v e c t o r on t h e w h o l e r e g i o n , unknown {C} i s t h e g l o b a l f o r c e v e c t o r The p r o b l e m r e m a i n i n g i s t o s o l v e e q u a t i o n (6) t o o b t a i n h v a l u e s . B o u n d a r y C o n d i t i o n s To o b t a i n t h e s o l u t i o n o f e q u a t i o n (6) a n d t o s a t i s f y t h e g i v e n b o u n d a r y c o n d i t i o n s , a l l b o u n d a r y c o n d i t i o n s h a v e t o be i n c o r p o r a t e d i n s o l v i n g t h e e q u a t i o n s . F i g u r e 5 shows t h e b o u n d a r y c o n d i t i o n s a p p l i e d i n t h e s i m u l a t i o n . T h e r e a r e t h r e e d i f f e r e n t t y p e s o f b o u n d a r y c o n d i t i o n s i n v o l v e d . 1. Known H e a d B o u n d a r y : In F i g u r e 5 , B G , H I , M L , a r e a t d r a i n p o s i t i o n s a t w h i c h t h e p r e s s u r e h e a d i s z e r o . T h i s a l s o a p p l i e s f o r HM, t h e t o e o f t h e s l o p e . A t t h e s e p o s i t i o n s , h y d r a u l i c h e a d s a r e e q u a l t o t h e i r e l e v a t i o n h e a d s w h i c h a r e d e t e r m i n e d a c c o r d i n g t o a s e l e c t e d d a t u m . D E , J K a r e c o n s i d e r e d f a r f r o m t h e t o e o f t h e s l o p e , t h e r e f o r e , t h e s p e c i f i e d h y d r a u l i c h e a d v a l u e s a r e a s s u m a b l y d e t e r m i n e d by t h e r e g i o n a l g r o u n d w a t e r c o n d i t i o n s a n d k e e p c o n s t a n t . 1 9 Free Surface after Drainage (a). Vertical section (b). Horizontal section Figure 5.—Boundary Conditions Applied 20 2 . Known F l o w R a t e B o u n d a r y : A F , I J , a n d LK a r e i m a g i n a r y i m p e r m e a b l e b o u n d a r i e s h a v i n g z e r o f l o w r a t e b e c a u s e o f t h e symmetry o f t h e f l o w p a t t e r n . No w a t e r f l o w s c r o s s t h e s e b o u n d a r i e s . 3 . t h e C o m b i n a t i o n o f 1 a n d 2 , f r e e s u r f a c e b o u n d a r y : C u r v e BD i s t h e t h i r d t y p e o f b o u n d a r y . No f l o w c r o s s e s t h e f r e e s u r f a c e a n d t h e p r e s s u r e h e a d s on t h e s u r f a c e a r e z e r o . C o n d i t i o n 1 c a n be i n c o r p o r a t e d i n s o l v i n g e q u a t i o n s w i t h e a s e . C o n d i t i o n 2 was h a n d l e d by a d d i n g t h e i n f l o w c o n t r i b u t i o n o f t h e e l e m e n t t o t h e v e c t o r {c} u s i n g f o r m u l a ( A - 1 0 ) i n A p p e n d i x A . The t h i r d t y p e o f c o n d i t i o n i s more d i f f i c u l t t o d e a l w i t h . An i t e r a t i v e a p p r o a c h was u s e d i n w h i c h a f r e e s u r f a c e was e s t i m a t e d , t h e n a no f l o w b o u n d a r y c o n d i t i o n was a p p l i e d t o s o l v e t h e g l o b a l e q u a t i o n s . A f t e r o b t a i n i n g t h e s o l u t i o n , t h e h y d r a u l i c h e a d s a l o n g t h e f r e e s u r f a c e were i n v e s t i g a t e d t o s e e i f t h e y e q u a l t h e i r e l e v a t i o n h e a d s b e i n g s a t i s f i e d w i t h i n a s p e c i f i e d t o l e r a n c e . I f n o t , a new p o s i t i o n o f t h e f r e e s u r f a c e was e s t i m a t e d f r o m t h e h y d r a u l i c h e a d c a l c u l a t e d p r e v i o u s l y a n d t h e s o l u t i o n p r o c e s s was r e p e a t e d u n t i l s a t i s f i e d r e s u l t s a r e o b t a i n e d ( D e s a i , 1 9 7 2 ) . 2 . 4 C o m p u t e r P r o g r a m m i n g a n d P r o g r a m V e r i f i c a t i o n T h e e q u a t i o n s were s o l v e d by t h e G a u s s i a n E l i m i n a t i o n M e t h o d . The c o m p u t e r p r o g r a m , F i n i t e E l e m e n t S i m u l a t i o n o f H o r i z o n t a l D r a i n D r a i n a g e i n M i n i n g S l o p e s , FESHDMS, was 21 w r i t t e n b y t h e a u t h o r i n F O R T R A N I V r u n o n M T S ( M i c h i g a n T e r m i n a l S y s t e m ) a t T h e C o m p u t i n g C e n t e r i n T h e U n v e r s i t y o f B r i t i s h C o l u m b i a . T h e p r o g r a m c o u l d d o a l l t h e c o m p u t a t i o n s d e s c r i b e d i n t h i s c h a p t e r f o r e i t h e r t h e v e r t i c a l o r h o r i z o n t a l m o d e l . T h e d e t a i l s o f t h e c o m p u t e r p r o g r a m o r g a n i z a t i o n a n d a n e x a m p l e o f i n p u t d a t a a n d o u t p u t a r e p r e s e n t e d i n A p p e n d i x B . O n e i m p o r t a n t p r o c e d u r e i n a n y n u m e r i c a l m o d e l i n g i s t o v e r i f y t h e v a l i d i t y o f t h e r e s u l t o f t h e c o m p u t e r s i m u l a t i o n . I n m a n y c a s e s , o u r u n d e r s t a n d i n g o f t h e p h y s i c s o f g r o u n d w a t e r i s s u f f i c i e n t t o e v a l u a t e w h e t h e r a s o l u t i o n i s r e a s o n a b l e . N o r m a l l y , t h e r e a r e s e v e r a l m e t h o d s t o v e r i f y a m o d e l . T h e n u m e r i c a l s o l u t i o n c o u l d b e c h e c k e d b y 1 . A n a l y t i c a l S o l u t i o n A n a l y t i c a l s o l u t i o n s a r e n o t e a s i l y a v a i l a b l e f o r p r o b l e m s o f p r a c t i c a l i n t e r e s t . I t s a p p l i c a b i l i t y i s v e r y l i m i t e d . 2 . P h y s i c a l L a b o r a t o r y M o d e l P h y s i c a l m o d e l s a r e d i f f i c u l t t o b u i l d t o s i m u l a t e r o c k s l o p e s w h i c h u s u a l l y a r e c h a r a t e r i z e d b y t h e i r h e t e r o g e n e i t y . A l s o m u c h e q u i p m e n t m a y b e n e e d e d t o c o n s t r u c t a l a b o r a t o r y m o d e l . 3 . F i e l d M e a s u r e d D a t a T h e m o d e l c a n b e t e s t e d a g a i n s t t h e f i e l d m e a s u r e d d a t a i f t h e d a t a a r e a v a i l a b l e a n d r e l i a b l e . T h i s i s t h e m e t h o d e m p l o y e d i n t h i s s t u d y . 2 2 T h e f i e l d d a t a u s e d i n t h e m o d e l v e r i f i c a t i o n w a s o b t a i n e d f r o m t h e L O R N E X M i n e i n t h e H i g h l a n d V a l l y , B r i t i s h C o l u m b i a . I n t h i s m i n e s o m e i n t e n s i v e h y d r a u l i c a l i n v e s t i g a t i o n s w e r e d o n e b y L O R N E X s t a f f w i t h G o l d e r A s s o c i a t e s a n d C . 0 . B r a w n e r p r o v i d i n g s p e c i a l i s t a d v i c e . M o s t o f t h e g r o u n d w a t e r i n f o r m a t i o n w a s s u m m a r i z e d i n G o l d e r R e p o r t # 3 5 . L O R N E X M i n e i s a l a r g e o p e n p i t c o p p e r m o l y b d e n u m p r o d u c e r l o c a t e d i n t h e c e n t r a l o f B r i t i s h C o l u m b i a p r o v i n c e . T h e m a j o r i n s t a b l i t y i n t h e m i n e h a s b e e n r e c o r d e d o n t h e w e s t w a l l o f t h e p i t . G r o u n d w a t e r p r e s s u r e w a s c o n s i d e r e d a s a m a j o r c a u s e o f t h e i n s t a b i l i t y . A v e r t i c a l s e c t i o n w a s s e l e c t e d o n t h e w e s t w a l l a l o n g t h e A B l i n e i n F i g u r e 6 w h i c h s h o w s t h e p l a n v i e w o f t h e o p e n p i t . T h e r o c k s l o p e i s c o m p r i s e d o f c o m p e t e n t , w e a t h e r e d a n d m o d e r a t e l y a l t e r e d B e t h s a i d a G r a n o d i o r i t e w i t h t w o m a j o r f a u l t s . F i g u r e 7 s h o w s t h e s l o p e s e c t i o n a n d t h e c o n d u c t i v i t y d i s t r i b u t i o n s . A t a n u m b e r o f p o i n t s a s m a r k e d " * " i n t h e s e c t i o n , t h e h y d r a u l i c h e a d s w e r e m e a s u r e d a n d t h e d a t a w e r e a v a i l a b l e f r o m G o l d e r R e p o r t # 3 5 . T h e s e d a t a w e r e u s e d t o c h e c k t h e r e s u l t s f r o m t h e c o m p u t e r s i m u l a t i o n . F i g u r e 8 s h o w s t h e c o m p a r i s o n o f m e a s u r e d d a t a a n d c o m p u t e r o u t p u t . I t c a n b e s e e n f r o m F i g u r e 8 t h a t t h e r e l a t i v e l y g o o d c o n s i s t e n c y h a s b e e n o b t a i n e d . T h e s l i g h t l y d i f f e r e n c e i s e a s i l y u n d e r s t a n d a b l e d u e t o s o m e a s s u m p t i o n s m a d e i n t h e m o d e l i n g . 23 In a d d i t i o n , t h e c o m p u t e r m o d e l was a l s o t e s t e d by an e x a m p l e t a k e n f r o m INTRODUCTION TO GROUNDWATER MODELING (Wang & A n d e r s o n , 1 9 8 2 ) . T h e d a t a i n t h e e x a m p l e i n t h i s b o o k , C h a p t e r 6 was p u t i n t o t h e c o m p u t e r m o d e l , t h e c o m p u t e r o u t p u t i s c o m p a r e d w i t h t h e r e s u l t o f t h e e x a m p l e a s shown i n T a b l e I. I t c a n be c o n c l u d e d t h a t t h e c o m p u t e r m o d e l h a s b e e n v e r i f i e d by t h e f i e l d d a t a w i t h s a t i s f a c t o r y r e s u l t s . Golder Associates F i g u r e 6 . T h e P l a n V i e w o f L O R N E X M i n e K - C o n d u c t i v i t y [ m / s ] 1- - K = 3 E - 8 6 - -- K = 3 E - 8 2 - - K = 2 E - 8 7 - - K = 2 E - 8 3 - - K = 9 E - 9 8 - - K = 2 E - 8 4 - - K = 2 E - 8 9 - - K = 2 E - 6 5 - - K = 7 E - 6 1 0 -- - K = 7 E - 6 Figure 7.-The Model Section on the West Wall of Lornex Mine and the Conductivity Distribution "1 I I I I I I I I • I 1 1 0 50 100 150 200 250 300 350 400 450 500 550 Depth from 4752ft Bench(ft) Legend O Measured Values X Computed Values Figure 8.—The Compar ison of Computed and Measured Hydraulic Head Values in LORNEX Mine 27 T A B L E I. T h e C o m p a r i s o n o f Two S e t s o f C o m p u t e r O u t p u t s Node Number H y d r a i FESHDMS i l i c H e a d V a l u e s ( m ) Wang & A n d e r s o n 1 4 . 0 0 0 4 . 0 0 2 4 . 0 0 0 4 . 0 0 3 4 . 0 0 0 4 . 0 0 4 4 . 0 0 0 4 . 0 0 5 3 . 7 2 0 3 . 7 2 6 3 . 6 9 5 3 . 6 9 7 3 . 6 8 8 3 . 6 8 8 3 . 6 8 6 3 . 6 8 9 3 . 3 9 2 3 . 4 0 10 3 . 3 6 6 3 . 3 7 1 1 3 . 3 5 7 3 . 3 6 12 3 . 3 5 4 3 . 3 5 13 3 . 0 0 0 3 . 0 8 14 3 . 0 0 0 3 . 0 0 15 3 . 0 0 0 3 . 0 0 16 3 . 0 0 0 3 . 0 0 28 C h a p t e r 3 COMPUTER SIMULATIONS A f t e r t h e c o m p u t e r m o d e l h a s b e e n e s t a b l i s h e d a n d v e r i f i e d , i t i s u s e d t o s o l v e t h e p r o b l e m s p e c i f i e d i n C h a p t e r 2 . T h e c o m p u t e r s i m u l a t i o n s were p e r f o r m e d on b o t h h o r i z o n t a l a n d v e r t i c a l s e c t i o n s . T h e h o r i z o n t a l s e c t i o n m o d e l was u s e d t o a n a l y z e t h e d r a i n a g e e f f e c t s o f h o r i z o n t a l d r a i n s w i t h d i f f e r e n t s p a c i n g s a n d l e n g t h s . The v e r t i c a l s e c t i o n was u s e d t o e s t i m a t e t h e s u i t a b l e v e r t i c a l d i s t a n c e s b e t w e e n d r a i n r o w s . A s d i s c u s s e d i n s e c t i o n 2 . 2 , i t s h o u l d be n o t e d t h a t i n v e r t i c a l s e c t i o n m o d e l i n g , t h e h o r i z o n t a l s p a c i n g a n d l e n g t h o f d r a i n s h a v e b e e n t a k e n i n t o a c c o u n t . T h e s i m u l a t i o n s a l s o i n c l u d e d t h e m o d e l i n g on t h e h o r i z o n t a l s e c t i o n s f o r t h e d r a i n a g e p e r f o r m a n c e i n t h e s l o p e w i t h a n i s o t r o p i c m a t e r i a l s a n d f o r d i f f e r e n t d r a i n p a t t e r n l a y o u t s . 3.1 S i m u l a t i o n s on t h e H o r i z o n t a l S e c t i o n T h e p u r p o s e o f t h e s i m u l a t i o n on h o r i z o n t a l s e c t i o n s was t o a n a l y z e t h e d r a i n a g e e f f e c t s o f t h e h o r i z o n t a l d r a i n s y s t e m w i t h d i f f e r e n t s p a c i n g s a n d l e n g t h s . T h e h y d r a u l i c h e a d d i s t r i b u t i o n s on t h e l e v e l where d r a i n s a r e i n s t a l l e d was s t u d i e d . T h e m o d e l s e c t i o n g e o m e t r y a n d t h e b o u n d a r y c o n d i t i o n s a p p l i e d h a v e b e e n p r e s e n t e d i n t h e p r e v i o u s c h a p t e r , i n F i g u r e 2 . (b) a n d F i g u r e 5 ( b ) . The f i n i t e e l e m e n t mesh f o r t h i s m o d e l i s shown i n F i g u r e 9 . Drain 50 40 30 20 10 I 1 1 1 1 I 1 1 1 — I — 1 — 1 — 1 — I — ' — ' — ' — I — • — 1 — 1 — I — 1 — • — 1 — I — 1 — 1 — 1 — I — 1 — • — ' — I — ' — ' — ' I • ' ' I ' ' ' I 0 40 80 120 160 200 24 0 280 320 360 400 440 X(m) Figure 9.-Finite Element Mesh for Horizontal Model 3 0 In F i g u r e 9 , i t c a n be s e e n t h a t t h e d r a i n s p a c i n g i s r e p r e s e n t e d by v a r i a b l e y . In t h e s i m u l a t i o n by v a r y i n g t h e c o o r d i n a t e s i n t h e y d i r e c t i o n , t h e d i f f e r e n t s p a c i n g s c a n be m o d e l l e d . T h e s p a c i n g s r a n g e f r o m 80m t o 10m. The d i f f e r e n t d r a i n l e n g t h s were m o d e l l e d by v a r y i n g t h e l e n g t h o f z e r o p r e s s u r e h e a d on t h e u p p e r and l o w e r b o u n d a r i e s . The l e n g t h s r a n g e f r o m 50m t o 100m. A t o t a l o f 120 c o m p u t e r r u n s were made f o r a l l t h e c o m b i n a t i o n s o f d r a i n s p a c i n g s a n d l e n g t h s u n d e r t h r e e d i f f e r e n t s l o p e h e i g h t s , 150m, 175m, a n d 200m. The c o m p u t e r s i m u l a t i o n o u t p u t s a r e t h e h y d r a u l i c h e a d v a l u e s a t a number o f s e p a r a t e d n o d e s on t h e h o r i z o n t a l s e c t i o n . T h e h y d r a u l i c h e a d s on t h e l i n e midway b e t w e e n two p a r a l l e l d r a i n s a r e most i m p o r t a n t f o r s t a b i l i t y a n a l y s i s . T h e y were i n t e r p r e t e d i n t o a s e r i e s o f d i m e n s i o n l e s s p l o t s . F i g u r e 10 t o F i g u r e 14 show 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 f o r a s l o p e h e i g h t o f 150m, F i g u r e 15 t o F i g u r e 19 f o r t h e s l o p e h e i g h t o f 175m, a n d F i g u r e 20 t o F i g u r e 24 f o r t h e s l o p e h e i g h t o f 200m. I t i s o b v i o u s l y i m p r a c t i c a l t o do t h e s i m u l a t i o n s f o r a l l p o s s i b l e s l o p e h e i g h t s . In m i n i n g p r a c t i c e , a g r e a t number o f s l o p e s w h i c h n e e d t o be d r a i n e d f a l l i n t o t h e r a n g e o f 150m t o 200m. F o r a n y p a r t i c u l a r p r o b l e m , t h e s i m u l a t i o n c a n be d o n e s p e c i f i c a l l y . 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 d t h a t t h e r e i s a s i g n i f i c a n t r e d u c t i o n o f h y d r a u l i c h e a d i n a s l o p e f r o m 31 no d r a i n s t o e v e n v e r y s p a r s e l y s p a c e d d r a i n s . T h e v a r i a t i o n s i n d r a i n s p a c i n g i n d u c e d t h e d i f f e r e n t d r a i n a g e r e s u l t s . W i t h i n a d i s t a n c e o f 1 . 0 - 1 . 5 t i m e s t h e d r a i n l e n g t h f r o m t h e s l o p e t o e , t h e d i f f e r e n c e s i n h y d r a u l i c h e a d i s s i g n i f i c a n t . B e y o n d t h a t d i s t a n c e , d r a i n s p a c i n g s h a v e much l e s s i n f l u e n c e on t h e h y d r a u l i c h e a d d i s t r i b u t i o n . 3 2 X/L h-Woter Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=90m ot depth 150m F i g u r e 1 0 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 150m, d r a i n L e n g t h 90m h-Water Table Height at Distance x H-Slope Height L—Distance from Toe to Headwater Drain Length l=80m at depth 150m F i g u r e D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 150m, d r a i n L e n g t h 80m 3 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X/L h-Water Table Height at Distance x H—Slope Height L—Distance from Toe to Headwater Drain Length l=70m qt depth 150m F i g u r e 12 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 150m, d r a i n L e n g t h 70m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X/L h—Water Table Height at Distance x H-Slope Height L—Distance from Toe to Headwater Drain Length l=60m at depth 150m F i g u r e 1 3 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 150m, d r a i n L e n g t h 60m 3 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X/L h-Water Table Height at Distance x H-Slppe Height L-Distance from Toe to Headwater Drain Length l=50m at depth 150m F i g u r e 14 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 150m, d r a i n L e n g t h 50m 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X/L h—Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=100m at depth 175m F i g u r e 1 5 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 175m, d r a i n L e n g t h 100m h—Water Table Height at Distance x H-Slope Height L—Distance from Toe to Headwater Drain Length l=90m at depth 175m F i g u r e 16 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 175m, d r a i n L e n g t h 90m \ 3 9 X/L h-Water Table Height at Distance x H-Slope Height L—Distance from Toe to Headwater Drain Length 1=8Om at depth 175m F i g u r e 1 7 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 175m, d r a i n L e n g t h 80m 4 0 X / L h—Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=70m at depth 175m F i g u r e 18 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 175m, d r a i n L e n g t h 70m 41 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X / L h-Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=60m at depth 175m F i g u r e 1 9 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 175m, d r a i n L e n g t h 60m h-Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length 1=10Om at depth 200m F i g u r e 2 0 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 2 0 0 m , d r a i n L e n g t h 1 0 0 m 4 3 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X/L h-Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=90m at depth 200m F i g u r e 2 1 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 200m, d r a i n L e n g t h 90m 4 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X/L h-Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=80m at depth 200m F i g u r e 2 2 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 200m, d r a i n L e n g t h 80m 4 5 h-Water Table Height at Distance x H—Slope Height L-Distance from Toe to Headwater Drain Length l=70m at depth 200m F i g u r e 2 3 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 200m, d r a i n L e n g t h 70m 4 6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X/L h-Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=60m at depth 200m F i g u r e 2 4 . D i m e n s i o n l e s s H y d r a u l i c H e a d D i s t r i b u t i o n f o r t h e S l o p e H e i g h t o f 200m, d r a i n L e n g t h 60m 4 7 3 . 2 S i m u l a t i o n s on t h e V e r t i c a l S e c t i o n T h e p u r p o s e o f t h e c o m p u t e r s i m u l a t i o n on t h e v e r t i c a l m o d e l was t o e s t i m a t e t h e s u i t a b l e v e r t i c a l d i s t a n c e b e t w e e n d r a i n l e v e l s . The m o d e l s e c t i o n a n d t h e b o u n d a r y c o n d i t i o n s a p p l i e d h a v e been shown i n F i g u r e 3 (b) a n d F i g u r e 5 . T h e f i n i t e e l e m e n t mesh f o r t h i s s e c t i o n i s i l l u s t r a t e d i n F i g u r e 2 5 . The f i r s t row o f d r a i n s was a s s u m e d t o be i n s t a l l e d a t t h e l e v e l w i t h t h e s l o p e h e i g h t o f 100m. A s t h e p i t g o e s d e e p e r , t h e s l o p e h e i g h t b e c o m e s h i g h e r a n d h i g h e r , t h e s e c o n d row o f d r a i n s were i n s t a l l e d . T h e v e r t i c a l d i s t a n c e s be tween two d r a i n l e v e l s were m o d e l l e d r a n g i n g f r o m 20m t o 100m, i . e . , t h e s e c o n d row o f d r a i n s were i n s t a l l e d a t t h e s l o p e h e i g h t o f 120m t o 200m. The m o d e l s e c t i o n i s a t t h e m i d d l e o f two s e c t i o n s w h i c h a r e w i t h d r a i n s . T h e d i s t a n c e f r o m t h e t o e o f t h e s l o p e t o t h e i n t e r s e c t i o n o f t h e w a t e r t a b l e s u r f a c e a t t h e d r a i n l e v e l , w, was u s e d i n s t e a d o f t h e r e a l d r a i n l e n g t h 1 i n t h e v e r t i c a l s e c t i o n where d r a i n s a r e i n s t a l l e d . w i s s m a l l e r t h a n 1 a n d i t was e s t i m a t e d f r o m h o r i z o n t a l s i m u l a t i o n s a n d a b o u t 0 . 6 t i m e s t h e d r a i n l e n g t h . S i m i l a r t o t h e D u p u i t a s s u m p t i o n i n t h e h o r i z o n t a l m o d e l i n g , t h e a s s u m p t i o n h a v i n g been made f o r t h e v e r t i c a l s e c t i o n m o d e l i n g was no f l o w a c r o s s t h e s e c t i o n . In t h i s c a s e , t h e s e c t i o n i s a s y m m e t r i c s e c t i o n b e t w e e n two s e c t i o n s w h i c h a r e w i t h d r a i n s . The w a t e r must f l o w t o w a r d t h e d r a i n s o r s l o p e f a c e . T h e r e f o r e , t h e r e i s , a c t u a l l y , 300-1 2 5 0 -2 0 0 -v§, 150-100-A / / 0 - i i ' — r h — ' '—i—' — i — 1 1 — i — 1 0 50 100 150 200 250 300 350 400 450 500 550 600 X ( m ) Figure 25.-Finite Element Mesh for Vertical Model 4 9 some w a t e r f l o w i n g away f r o m t h e s e c t i o n m o d e l l e d . I t i s u n d e r s t a n d a b l 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 i n t h e f i e l d w o u l d be l o w e r t h a n t h a t c a l c u l a t e d f r o m t h e c o m p u t e r s i m u l a t i o n s i n w h i c h o u t g o i n g w a t e r f l o w i s i g n o r e d . T h e r e s u l t w o u l d be s l i g h t l y c o n s e r v a t i v e . A w=40m was u s e d , w h i c h r e p r e s e n t e d t h e d r a i n a g e s y s t e m s h a v i n g a | = 0 . 8 - 0 . 8 5 W h e r e : S i s t h e h o r i z o n t a l s p a c i n g b e t w e e n two p a r a l l e l d r a i n s , [ L ] 1 i s t h e d r a i n l e n g t h , [ L ] T h e h y d r a u l i c h e a d d i s t r i b u t i o n s on t h e v e r t i c a l s e c t i o n were o b t a i n e d a s t h e r e s u l t s o f c o m p u t e r c a l c u l a t i o n s . The w a t e r t a b l e p o s i t i o n s i n t h e s l o p e f o r e a c h s l o p e h e i g h t were p l o t t e d . F i g u r e 26 t o F i g u r e 31 show t h e w a t e r t a b l e b e f o r e a n d a f t e r d r a i n s were i n s t a l l e d . T h e r e s u l t s s u g g e s t e d t h a t 40m t o 60m w o u l d be a s u i t a b l e r a n g e o f t h e v e r t i c a l d i s t a n c e f o r s u c h a s l o p e c o n d i t i o n . L e s s t h a n t h a t r a n g e , t h e s a t u r a t e d z o n e u n d e r t h e f i r s t row o f d r a i n s a n d a b o v e t h e p i t f l o o r i s n o t r e l a t i v e l y h i g h a n d c a n m a i n t a i n a s t a b l e c o n d i t i o n . G r e a t e r t h a n t h a t v e r t i c a l d i s t a n c e r a n g e , t h e s a t u r a t e d z o n e u n d e r t h e f i r s t row b e c o m e s h i g h e r a n d h i g h e r a s t h e p i t e x t e n d s d e e p e r , t h e s e e p a g e f o r c e b e c o m i n g l a r g e r a n d l a r g e r . T h e s e a r e v e r y u n f a v o u r a b l e f a c t o r s f o r s l o p e s t a b i l i t y . F r o m F i g u r e 2 9 , Figure 26.—Water Table Position in the Slope 51 Figure 27.-Water Table Position in the Slope 5 2 Figure 28.-Water Table Position in the Slope 53 Figure 29.—Water Table Position in the Slope •9 Figure 30.-Water Table Position in the Slope Figure 31.-Water Table Position in the Slope 56 3 0 , a n d 3 1 , i t c a n be s e e n t h a t t h e w a t e r t a b l e i s v e r y n e a r t h e s l o p e f a c e . The d r a i n s a t t h e s e c o n d l e v e l a r e n o t s u f f i c i e n t t o d r a i n s u c h a h e i g h t o f s a t u r a t e d z o n e . T h e r e f o r e , t h e y n e e d t o be i n s t a l l e d a t an e a r l i e r s t a g e . O t h e r w i s e , i n c r e a s i n g o r i g i n a l d r a i n l e n g t h o r r e d u c i n g d r a i n s p a c i n g a r e n e c e s s a r y t o i n s u r e g o o d d r a i n a g e r e s u l t s . S l o p e s t a b i l i t y r e q u i r e s a c e r t a i n w i d t h o f d r a i n e d z o n e b e h i n d t h e s l o p e f a c e , u s u a l l y i t i s a b o u t an h a l f o f t h e s l o p e h e i g h t o r 100 m e t e r s maximum. 3 . 3 A n i s o t r o p i c M a t e r i a l S l o p e A n a l y s i s B e c a u s e o f t h e d i s c o n t i n u i t i e s i n r o c k s l o p e s , t h e a n i s o t r o p y i s a s p e c i a l f e a t u r e o f s u c h s l o p e s . T h e c o m p u t e r s i m u l a t i o n s were made t o s t u d y t h e i n f l u e n c e o f a n i s o t r o p i c c h a r a c t e r i s t i c s o f r o c k m a t e r i a l s on t h e d r a i n a g e . A w i d e r a n g e o f t h e r a t i o s o f c o n d u c t i v i t i e s i n two d i r e c t i o n s were a s s u m e d : R O C = 0 . 0 0 1 , 0 . 0 1 , 0 . 1 , 1, 10 , 100 , 1000 W h e r e : ROC i s t h e R a t i o Of C o n d u c t i v i t i e s , Kx a n d Ky A l t h o u g h t h e s i m u l a t i o n was made on one s l o p e m o d e l w i t h a h o r i z o n t a l d r a i n d r a i n a g e s y s t e m h a v i n g t h e l e n g t h o f 70m, s p a c e d a t 60m a n d a h e i g h t o f 150m, t h e r e s u l t s c a n q u a n t i t a t i v e l y i n d i c a t e t h e d r a i n a g e n a t u r e o f t h e a n i s o t r o p i c s l o p e . T h e c o m p u t e r s i m u l a t i o n r e s u l t i s p l o t t e d i n F i g u r e 3 2 . I t c a n be s e e n t h a t t h e a n i s o t r o p y o f t h e r o c k s l o p e m a t e r i a l s h a s a s i g n i f i c a n t i n f l u e n c e on 5 7 h-Water Table Height at Distance x H-Slope Height L-Distance from Toe to Headwater Drain Length l=70m at depth 150m Drain Spacing S=60m  Figure 32.—The Drainage Effect of Anisotropic Materials 58 d r a i n a g e e f f e c t s . T h e l a r g e r c o n d u c t i v i t y i n t h e d i r e c t i o n p a r a l l e l t o t h e s l o p e f a c e , i . e . , p e r p e n d i c u l a r t o t h e d r a i n s i s more f a v o r a b l e t o h o r i z o n t a l d r a i n a g e . When t h e ROC r e a c h e s a v a l u e w i t h an o r d e r o f 2 o r g r e a t e r , w i t h f u r t h e r c h a n g e i n R O C , t h e c h a n g e i n h y d r a u l i c h e a d d i s t r i b u t i o n become much s m a l l e r . 3 . 4 The C o m p a r i s o n o f D i f f e r e n t D r a i n P a t t e r n L a y o u t s In s l o p e d r a i n a g e p r a c t i c e , t h e d r a i n s v e r y o f t e n a r e i n s t a l l e d i n a f a n n e d p a t t e r n i n o r d e r t o s a v e e q u i p m e n t m o v i n g t i m e , make w a t e r c o l l e c t i o n e a s i e r , p a r t i c u l a r l y u n d e r f r e e z i n g c o n d i t i o n s , a n d c r o s s more d i s c o n t i n u i t i e s o r i e n t e d i n d i f f e r e n t d i r e c t i o n s . T o e s t i m a t e t h e d r a i n a g e e f f e c t s o f d i f f e r e n t d r a i n p a t t e r n s , a c o m p u t e r s i m u l a t i o n was made t o c o m p a r e t h e d r a i n a g e p e r f o r m a n c e o f a f a n n e d d r a i n l a y o u t a n d p a r a l l e l d r a i n l a y o u t . A s shown i n F i g u r e 3 3 , two s e c t i o n s u s e d i n t h e s i m u l a t i o n h a v e t h e same number o f d r a i n s o v e r t h e same l e n g t h o f s l o p e . T h e same b o u n d a r y c o n d i t i o n s s p e c i f i e d i n F i g u r e 5 , i n s e c t i o n 2 . 3 , a r e a p p l i e d t o two s e c t i o n s . The f i n i t e e l e m e n t mesh f o r t h e f a n n e d d r a i n p a t t e r n i s shown i n F i g u r e 3 4 . F o r t h e p a r a l l e l d r a i n l a y o u t , t h e f i n i t e e l e m e n t mesh i s shown i n F i g u r e 9 , C h a p t e r 2 . A s s u m i n g t h a t t h e d r a i n l e v e l s were a t 1. 150m w i t h a d r a i n l e n g t h 60m a n d s p a c i n g 60m, 2 . 175m w i t h a d r a i n l e n g t h 80m and s p a c i n g 80m, 3 . 200m w i t h a d r a i n l e n g t h 100m a n d s p a c i n g 100m. (a). Parallel Drain Pattern (b). Fanned Drain Pattern Figure 33.-Parallel and Fanned Drain Patterns £ > / > / > ' V 0 8 Drain 0 12 0 16 0 2( i , . )0 240 280 3: X(m) 20 3( 50 4( )0 4* Figure 34.-Finite Element Mesh for Fanned Drain Pattern 61 T h e - c o m p u t e r s i m u l a t i o n s were made on f a n n e d a n d p a r a l l e l d r a i n m o d e l s u n d e r t h e t h r e e a s s u m e d c o n d i t i o n s . Homogeneous m a t e r i a l s l o p e was o n l y c o n s i d e r e d . T h e most i m p o r t a n t h y d r a u l i c h e a d on t h e c e n t r a l s y m m e t r i c l i n e f o r b o t h s e c t i o n s were p l o t t e d i n F i g u r e 3 5 , 3 6 , a n d 3 7 . By a n a l y z i n g t h e s e r e s u l t s , i t i s known t h a t w i t h i n t h e d i s t a n c e o f d r a i n l e n g t h f r o m t h e t o e o f t h e s l o p e , t h e h i g h e s t h y d r a u l i c h e a d v a l u e i n a f a n n e d p a t t e r n i s a b o u t a s t w i c e a s t h e h i g h e s t v a l u e i n a p a r a l l e l d r a i n p a t t e r n . F u r t h e r b a c k i n t o t h e s l o p e , t h e d i f f e r e n c e becomes s m a l l e r . A t a d i s t a n c e t w i c e t h a t o f t h e d r a i n l e n g t h f r o m t h e t o e , t h e h y d r a u l i c h e a d s i n two m o d e l s a r e n e a r l y t h e s a m e . T h e s e r e s u l t s i n d i c a t e t h a t o v e r a c e r t a i n l e n g t h o f a s l o p e , i f t h e number o f d r a i n s h a s b e e n d e c i d e d , t h e p a r a l l e l and f a n n e d d r a i n p a t t e r n l a y o u t s i n d u c e a d i f f e r e n c e h y d r a u l i c h e a d r e d u c t i o n , e s p e c i a l l y , w i t h i n t h e d i s t a n c e o f d r a i n l e n g t h f r o m t h e s l o p e t o e . I f no f a v o u r a b l e l o c a l g e o l o g i c a l c o n d i t i o n s e x i s t f o r a f a n n e d d r a i n p a t t e r n , s u c h a s d i f f e r e n t o r i e n t e d d i s c o n t i n u i t i e s , i t i s recommended t h a t more d r a i n s , i . e . , s m a l l e r s p a c i n g b e t w e e n t h e e n d s o f f a n n e d d r a i n s s h o u l d be u s e d i n o r d e r t o a c h i e v e t h e d r a i n a g e r e s u l t s e q u i v a l e n t t o t h a t a c h i e v e d by p a r a l l e l d r a i n s . H o w e v e r , i n many c i r c u m s t a n c e s where t h e l o c a l g e o l o g y i s g o o d f o r a f a n n e d d r a i n p a t t e r n , t h e f a n n e d l a y o u t c a n be u s e d t o y i e l d b e t t e r d r a i n a g e p e r f o r m a n c e and make t h e d r a i n a g e s y s t e m i n s t a l l a t i o n more e f f i c i e n t a n d more c o s t e f f e c t i v e . 6 2 1-O.h 1 O.Oh 7 : O.OOI-Legend Fanned _ parallel 0 0.1 0.2 0.3 0. I 1— 4 0.5 0.6 0.7 0.8 0.9 X/L h-Water Table Height at Distance x H—Slope Height L-Distance from Toe to Headwater Drain Length l=60m, Spacing s=60m, at depth 150m F i g u r e 3 5 . C o m p a r i s o n o f F a n n e d a n d P a r a l l e l D r a i n P a t t e r n s F o r t h e S l o p e H e i g h t o f 150m 63 0.1 X t / t —> / T f Tanned parallel ,/ J 0 .1 0 2 0 3 0. 4 0 5 0 6 0 i i 7 0.8 0.9 1 X/L h—Water Table Height at Distance x H—Slope Height L-Distance from Toe to Headwater Drain Length l=T0m, Spacing s=60m, at depth 175m F i g u r e 3 6 . C o m p a r i s o n o f F a n n e d a n d P a r a l l e l D r a i n P a t t e r n s F o r t h e S l o p e H e i g h t o f 175m 0.1 > 0.01 * } J $ / t 1 t / * i —' — 1 -t _ 1 ranned / Parallel 3 0 .1 0 2 0 3 0 4 0 5 0 6 0 l i 7 0.8 0.9 1 X / L h—Water Table Height at Distance x H—Slope Height L-Distance from Toe to Headwater Drain Length 1=80rri. Spacing s=60m, at depth 200m F i g u r e 3 7 . C o m p a r i s o n o f F a n n e d a n d P a r a l l e l D r a i n P a t t e r n s F o r t h e S l o p e H e i g h t o f 2 0 0 m 6 5 C h a p t e r 4 A P P L I C A T I O N S The p r a c t i c a l p u r p o s e o f t h i s s t u d y i s t o p r o v i d e an a i d f o r h o r i z o n t a l d r a i n s y s t e m d e s i g n i n m i n i n g s l o p e s . In t h i s c h a p t e r , a s a m p l e c a s e a n d t h e a p p l i c a t i o n t o t h e LORNEX M i n e i s p r e s e n t e d t o i l l u s t r a t e t h e p r o c e d u r e s f o r e s t i m a t i n g t h e l e n g t h a n d t h e s p a c i n g f o r a h o r i z o n t a l d r a i n d r a i n a g e s y s t e m . 4.1 A H y p o t h e t i c E x a m p l e F o r t h i s e x a m p l e , t h e p r o c e d u r e s f o r u s i n g t h e r e s u l t s o f t h i s s t u d y i n h o r i z o n t a l d r a i n s y s t e m d e s i g n f o r a s l o p e h a v i n g r e l a t i v e l y i d e a l c o n d i t i o n s a r e o u t l i n e d . T h e c r o s s s e c t i o n g e o m e t r y o f t h e s l o p e i s shown i n F i g u r e 3 8 . Two a s s u m p t i o n s were made . F i r s t , t h e i n i t i a l w a t e r t a b l e p o s i t i o n i s d e t e r m i n e d by r e g i o n a l g r o u n d w a t e r c o n d i t i o n s a n d l o c a l c l i m a t i c e n v i r o n m e n t . The h y d r a u l i c h e a d on t h e r i g h t b o u n d a r y a r e k e p t c o n s t a n t a n d e q u a l t o t h e s l o p e h e i g h t . S e c o n d , t h e s l o p e m a t e r i a l i s c o n s i d e r e d t o be h o m o g e n e o u s . I f t h e s l o p e m a t e r i a l i n a a c t u a l m i n e i s h i g h l y h e t e r o g e n e o u s , i t i s s u g g e s t e d t o u s e t h e c o m p u t e r p r o g r a m FESHDMS s i m u l a t i o n t o o b t a i n t h e d e t a i l a n d more r e a l i s t i c h y d r a u l i c h e a d d i s t r i b u t i o n i n s t e a d o f g o i n g t h r o u g h f o l l o w i n g t h e p r o c e d u r e s . I H -x 5 0 0 m Figure 3 8 . - T h e S lope Sec t ion of the Examp le 67 (1) . To Assume a D r a i n L e n g t h A s s u m i n g l=70m, d e p e n d i n g on t h e d r a i n i n s t a l l a t i o n e q u i p m e n t , s p e c i f i e d l o c a l g e o l o g y a n d t h e s l o p e h e i g h t . (2) . T o O b t a i n t h e h / H V a l u e s F o r t h e l e n g t h s e l e c t e d l=70m a n d t h e s l o p e h e i g h t h = l 7 5 m , u s e F i g u r e 18 t o o b t a i n h / H v a l u e s a l o n g t h e i n t e r s e c t i o n l i n e o f t h e d r a i n l e v e l a n d t h e v e r t i c a l s e c t i o n b e t w e e n two d r a i n s f o r e a c h d r a i n s p a c i n g r a n g i n g f r o m 10m t o 80m. (3) . T a b l e C a l c u l a t i o n I n s e r t t h e d a t a o b t a i n e d f r o m s t e p (2) i n t o t h e C a l c u l a t i o n T a b l e I I . T a b l e I I I i s T a b l e II a f t e r c o m p l e t i n g d a t a i n s e r t e d f o r t h i s e x a m p l e . T h e d a t a u n d e r l i n e d a r e o b t a i n e d i n T a b l e I I I f r o m s t e p ( 2 ) , o t h e r s were c a l c u l a t e d by t h e v e r y s i m p l e e q u a t i o n h = ( h / H ) * H W h e r e : h / H a n d h a r e a l l known f r o m p r e v i o u s s t e p s . (4) . T o P l o t W a t e r T a b l e P o s i t i o n s P l o t t h e e s t i m a t e d w a t e r t a b l e i n t h e s l o p e by u s i n g t h e d a t a p a i r s , ( x , h ) i n T a b l e I I I . a s shown i n F i g u r e 39 i n w h i c h t h e w a t e r t a b l e c o r r e s p o n d s t o s=80m (5) . T o S e l e c t a S p a c i n g a n d C h e c k S t a b i l i t y R e q u i r e m e n t S e l e c t i n g a s p a c i n g w h i c h s a t i s f i e s t h e s l o p e s t a b i l i t y r e q u i r e m e n t s . B e c a u s e t h e s t a b i l i t y a n a l y s i s i s b e y o n d t h e e x t e n t o f t h i s s t u d y , i t i s n o t d i s c u s s e d h e r e . TABLE I. Calculation Table s (m) X/L=0 x = h/H 05 h x/L=0. x= h/H 075 h x/L=0 x = h/H 1 h x/L=0. x= h/H 15 h x/L=0. x= h/H 2 h x/L=0. x= h/H 3 h 10 20 30 40 50 60 70 80 TABLE I I . Example C a l c u l a t i o n Table X/L=0. 05 x/L=0. 075 x/L=0. 1 x/L=0. 15 x/L=0.2 x/L=0.3 s x=25m x=37.5m x=50m x=75m x=100m x=150m (m) h/H h h/H h h/H h h/H h h/H h h/H h 10 0.00 0.00 0.00 0.00 0.024 4.20 0.087 15.23 0.15 26.25 0.28 49.00 20 0.00 0.00 0.00 0.00 0.03 5.25 0.09 15.75 0.15 26.50 0.283 49.53 30 0.00 0.00 0.005 0.78 0.037 6.48 0.10 17.50 0. 155 27. 13 0.286 50.05 40 0.002 0.26 0.01 1 .66 0.045 7.88 0.11 19.25 0.16 28.0 0.289 50.58 50 0.004 0.68 0.016 2.80 0.05 8.75 0.115 20. 13 0. 165 28.88 0.292 51.10 60 0.007 1 .23 0.022 3.85 0.058 10.15 0.12 21 .0 0.17 29.75 0.295 51 .63 70 0.01 1 .75 0.028 4.90 0.065 1 1 .38 0. 125 21 .88 0. 175 30.63 0.298 52. 15 80 0.015 2.63 0.035 6. 13 0.071 12.43 0.13 22.75 0.18 31.5 0.30 52.50 L=500m, H=175m 70 Figure 39.—Estimated Wctre Table Position in the Exampls? 71 In t h i s c a s e , i f t h e w a t e r t a b l e i s a t t h e p o s i t i o n c o r r e s p o n d i n g t o s=80m, t h e s l o p e s t a b i l i t y a n a l y s i s shows a s a t i s f a c t o r y s a f e t y f a c t o r , t h e n t h e p r i m a r i l y d e s i g n f o r t h e d r a i n a g e s y s t e m w o u l d be a d r a i n l e n g t h e q u a l t o 70m a n d s p a c i n g 80m. I f t h e w a t e r t a b l e i s a t l=70m a n d s=80m c a n n o t s a t i s f y t h e s t a b i l i t y r e q u i r e m e n t , o t h e r o p t i o n s a r e e i t h e r k e e p i n g l=70m, r e d u c i n g t h e s p a c i n g , a n d go b a c k t o t h e s t a b i l i t y a n a l y s i s o r t o i n c r e a s e t h e d r a i n l e n g t h a n d r e p e a t s t e p (2) t o ( 5 ) . . O t h e r C o n s i d e r a t i o n s O t h e r c o n s i d e r a t i o n s s u c h a s l o c a l h e t e r o g e n e i t y , a n i s o t r o p y p r o p e r t i e s , e t c . a r e n e e d e d t o be t a k e n i n t o a c c o u n t t o make t h e f i n a l d e c i s i o n . 72 4 . 2 The A p p l i c a t i o n t o LORNEX Open P i t M i n e T h e r e a l s i t u a t i o n i n m i n e s u s u a l l y a r e f a r f r o m i d e a l . I t i s u n l i k e l y f o r d e s i g n g u i d a n c e t h a t t h e s e c h a r t s w i l l be a p p l i a b l e w i t h o u t some m o d i f i c a t i o n . In t h i s s e c t i o n , a p r o b l e m i n t h e LORNEX Open P i t M i n e i s d i s c u s s e d . The s l o p e c o n d i t i o n i n t h i s m i n e i s n o t c o n s i s t e n t w i t h i d e a l c o n d i t i o n s u n d e r w h i c h t h e g r a p h s were p r o d u c e d . T h e g e n e r a l i n f o r m a t i o n f o r t h i s m i n e h a s been d e s c r i b e d i n C h a p t e r 2 . The mass h y d r a u l i c c o n d u c t i v i t y o f t h e w e s t w a l l o f t h e p i t i s i n t h e r a n g e o f I 0 " 7 m / s t o I 0 " 8 m / s , w i t h an e x t r e m e r a n g e o f 1 0 " 6 m / s t o I 0 ~ 9 m / s . In t h i s r a n g e , i t i s l i k e l y t h a t t h e w a l l c o u l d be e f f e c t i v e l y d e p r e s s u r i z e d by h o r i z o n t a l d r a i n s o r by a d r a i n a g e a d i t s y s t e m . The h o r i z o n t a l d r a i n s w o u l d d e p r e s s u r i z e t o s h a l l o w d e p t h s i n t h e s l o p e , 100m t o 150m. W h e r e a s t h e a d i t w o u l d be e f f e c t i v e f o r d e e p d r a i n g e . ( G o l d e r A s s o c i a t e s , 1983) T h e p i t c r e s t on t h e w e s t w a l l i s a t t h e 5 2 0 0 f t l e v e l . T h e h o r i z o n t a l d r a i n s y s t e m i s i n s t a l l e d f r o m t h e 4 7 5 2 f t b e n c h t o r e d u c e t h e l i k l i h o o d o f t h e u p p e r s l o p e m o v i n g . The s l o p e h e i g h t a t t h e 4 7 5 2 f t l e v e l i s 4 4 8 f t ( 1 3 5 m ) . The d i s t a n c e f r o m s l o p e t o e t o t h e o t h e r v e r t i c a l b o u n d a r y where t h e h y d r a u l i c h e a d s a r e k e p t c o n s t a n t i s e s t i m a t e d t o be 1 3 0 0 f t ( 4 0 0 m ) w h i c h c o u l d be c o n s i d e r e d t o be f a r e n o u g h f r o m t h e d r a i n s t o e n s u r e t h e d r a i n a g e w o u l d n o t r e a c h s u c h a f a r r e g i o n . 73 T h e same p r o c e d u r e s a s o u t l i n e d i n s e c t i o n 4.1 were f o l l o w e d . The s p e c i f i c c o n d i t i o n s o f t h i s m i n e a r e c o n s i d e r e d i n e a c h s t e p . (1) . T o Assume a D r a i n L e n g t h A s s u m i n g l=90m ( 3 0 0 f t ) w h i c h i s f e a s i b l e f o r d r a i n i n s t a l l a t i o n e q u i p m e n t i n t h i s m i n e . ( G o l d e r R e p o r t #35) (2) . T o O b t a i n t h e h / H V a l u e s F o r t h e l e n g t h s e l e c t e d l=90m a n d t h e s l o p e h e i g h t h=135m, u s i n g F i g u r e 10 i n w h i c h t h e s l o p e c o n d i t i o n s a r e c l o s e s t t o t h e r e a l m i n e s i t u a t i o n s t o o b t a i n h / H v a l u e s a l o n g t h e i n t e r s e c t i o n l i n e o f t h e d r a i n l e v e l a n d t h e v e r t i c a l s e c t i o n b e t w e e n two d r a i n s f o r e a c h d r a i n s p a c i n g r a n g i n g f r o m 10m t o 80m. (3) . T a b l e C a l c u l a t i o n I n s e r t t h e d a t a o b t a i n e d f r o m s t e p (2) i n t o t h e C a l c u l a t i o n T a b l e I I . R e s u l t s a r e shown i n T a b l e V I . (4) . T o P l o t W a t e r T a b l e P l o t t i n g t h e e s t i m a t e d w a t e r t a b l e i n t h e s l o p e by u s i n g t h e d a t a p a i r s , ( x , h ) i n T a b l e VI a s shown i n F i g u r e 40 i n w h i c h t h e w a t e r t a b l e c o r r e s p o n d s t o s=20m (5) . T o S e l e c t a S p a c i n g a n d C h e c k S t a b i l i t y R e q u i r e m e n t S e l e c t i n g a s p a c i n g w h i c h s a t i s f i e s t h e s l o p e s t a b i l i t y r e q u i r e m e n t s . In t h i s c a s e , f o r t h e w a t e r t a b l e a t t h e p o s i t i o n c o r r e s p o n d i n g t o s=80m, t h e s l o p e s t a b i l i t y a n a l y s i s shows an u n s a t i s f a c t o r y s a f e t y f a c t o r . F u r t h e r r e d u c i n g t h e s p a c i n g t o 60m, 40m, a n d 20m, ' the p r i m a r y d e s i g n f o r t h e d r a i n a g e s y s t e m s u g g e s t s t h a t t h e d r a i n s T A B L E I I I . C a l c u l a t i o n T a b l e f o r LORNEX M i n e X / L = 0 05 x / L = 0 . 075 x / L = 0 . 1 x / L = 0 . 15 x / L = 0 . 2 x / L = 0 . 3 s x = 20m x=30m x = 40m x=60m x = 80m x=120m (m) h / H h h / H h h / H h h / H h h / H h h / H h 10 0 . 0 0 0 0 . 00 0 .00 0 .00 0 .00 0 .00 0 . 0 0 0 .00 0 . 0 0 2 0 .27 0 . 1 3 1 7 . 5 5 20 0 . 0 0 0 . 00 0 .00 0 .00 0 .00 0 .00 0 . 0 0 0 . 0 0 0.01 1 . 3 5 0 . 1 3 1 7 . 5 5 30 0 . 0 0 0 . 00 0 .00 0 .00 0 .00 0 .00 0 . 0 0 2 0 . 2 7 0 . 0 1 8 2 . 43 0 . 1 4 1 8 . 9 40 0 . 0 0 0 . 00 0 .00 0 .00 0 .000 0 .000 0 . 0 0 5 0 . 6 7 5 0 . 0 2 5 3 . 3 7 5 0 . 1 4 1 8 . 9 50 0 . 0 0 0 . 00 0.001 0 . 135 0 .002 0 .27 0 . 0 0 9 1 .215 0 . 0 3 4 . 0 5 0 . 1 5 2 0 . 2 5 60 0 . 0 0 0 . 00 0.001 0 . 135 0 .002 0 .27 0 . 0 1 4 1 . 8 9 0 . 0 4 5 .40 0 . 1 5 2 0 . 2 5 70 0 . 0 0 2 0 . 27 0 .004 0 .54 0 .007 0 .945 0 . 0 2 2 . 7 0 0 . 0 4 8 6 .48 0 . 1 6 21 .60 80 0 .004 0 . 54 0 .007 0 . 9 4 5 0.01 1 . 3 5 0 . 0 2 5 3 . 3 7 5 0 . 0 5 2 7 .02 0 . 1 6 21 . 6 0 L=400m, H=135m Figure 40 .—Es t imated Watre Table Pos i t ion in LORNEX Mine 76 h a v e l=90m, s=20m. ( 6 ) . O t h e r C o n s i d e r a t i o n s The g e o l o g i c i n f o r m a t i o n r e p o r t e d i n G o l d e r R e p o r t #35 shows t h a t t h e 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 d i r e c t i o n p a r a l l e l t o t h e s l o p e f a c e i s g r e a t e r w h i c h i s f a v o u r a b l e f a c t o r f o r s u c h a d r a i n a g e s y s t e m . T h e r e f o r e , t h e s p a c i n g c o u l d be i n c r e a s e d i n some a r e a s . The v e r t i c a l s p a c i n g o f d r a i n l e v e l s c o u l d be i n t h e r a n g e o f 40m-60m a c c o r d i n g t o t h e s t u d y r e s u l t s i n s e c t i o n 3 . 2 . In c o n c l u s i o n , t h e recommended h o r i z o n t a l d r a i n s y s t e m f o r LORNEX M i n e i n t h e u p p e r w e s t w a l l i s : d r a i n l e n g t h = 9 0 m , h o r i z o n t a l s p a c i n g 20m-30m, v e r t i c a l s p a c i n g 40m-60m. 77 C h a p t e r 5 CONCLUSIONS AND SUMMARY 5.1 C o n c l u s i o n s F r o m t h e c o m p u t e r s i m u l a t i o n s d o n e i n t h i s r e s e a r c h p r o g r a m , t h e f o l l o w i n g c o n c l u s i o n s a r e m a d e . (1) . T h e r e i s a s i g n i f i c a n t h y d r a u l i c h e a d r e d u c t i o n a t t h e d r a i n l e v e l i n a s l o p e w i t h no d r a i n s c o m p a r e d t o h a v i n g a s p a r s e l y s p a c e d d r a i n s y s t e m . (2) . D i f f e r e n t s p a c i n g s h a v e d i f f e r e n t d r a i n a g e e f f e c t s w i t h i n t h e d i s t a n c e o f 1 .5 t o 2 . 0 t i m e s o f t h e d r a i n l e n g t h . B e y o n d t h a t d i s t a n c e , d r a i n s p a c i n g h a s l i t t l e i n f l u e n c e on r e d u c t i o n i n h y d r a u l i c h e a d . (3) . A s u i t a b l e v e r t i c a l d i s t a n c e t o i n s t a l l t h e s e c o n d row o f d r a i n s i s s u g g e s t e d i n t h e r a n g e o f 40m t o 60m. (4) . T h e a n i s o t r o p y o f r o c k s l o p e s p l a y s an i m p o r t a n t r o l e i n d r a i n a g e p e r f o r m a n c e . H i g h e r c o n d u c t i v i t y v a l u e s i n a d i r e c t i o n p a r a l l e l t o s l o p e f a c e i s more f a v o u r a b l e f o r s u c h a d r a i n a g e s y s t e m . I f t h e r a t i o o f Kx t o Ky h a s a v a l u e i n t h e o r d e r o f 2 o r g r e a t e r , t h e c h a n g e i n h y d r a u l i c h e a d d i s t r i b u t i o n b e c o m e s s m a l l e r w i t h f u r t h e r c h a n g e i n t h e r a t i o . (5) . F a n n e d d r a i n s a n d p a r a l l e l d r a i n s i n s l o p e d r a i n a g e p r o d u c e d i f f e r e n t r e s u l t s i n t h e r e g i o n n e a r t h e s l o p e 78 face The hydraulic head after the i n s t a l l a t i o n of fanned drain drainage i s higher than for p a r a l l e l drain drainage within the distance about twice of the drain length from the slope toe. However the p r a c t i c a l benefits and cost benefits of fanned drains are usually substantial. 5.2 Summary This study comprises a series of computer simulations to model horizontal drain drainage in an open p i t slope. It includes the analyses for horizontal and v e r t i c a l spacing of drains, drainage c h a r a c t e r i s t i c s in anisotropic rock slopes and the drainage e f f e c t s of di f f e r e n t drain pattern layouts. The two dimensional f i n i t e element program was developed and used in the simulation. The computer program was v e r i f i e d by the f i e l d data from the LORNEX Mine. The procedures for using the res u l t s of the study are outlined in examples in Chapter 4. Several assumptions were made in the study. (1) . Darcy's law i s v a l i d , (2) . Flows are in two dimensional saturated f i e l d s . (3) . The graphic results are based on simulations in homogeneous materials. i 79 In the a r e a of h o r i z o n t a l d r a i n system d e s i g n f o r m i n i n g s l o p e s , t h e r e are a few a s p e c t s t h a t s h o u l d be s t u d i e d i n the f u t u r e as an e x t e n s i o n of t h i s r e s e a r c h program. 1. To t a k e the d r a i n a g e time i n t o account by modeling on t h e t r a n s i e n t f l o w models, 2. To model the d i r e c t i o n a l d i s c o n t i n u i t i e s of rock s l o p e i n more d e t a i l . 80 B IBLIOGRAPHY A b r a m o w i t z , M. a n d S t e g u m , I . A . , Handbook M a t h e m a t i c a l F u n c t i o n s w i t h F o r m u l a s G r a p h s , a n d M a t h T a b l e s , A p p l i e d M a t h . S e r i e s 5 5 , N a t i o n a l B u r e a u o f S t a n d a r d , W a s h i n g t o n , D. C , 1964 B i s e , C . J . a n d S c y o c , R . L . , C o m p u t e r A i d e d A n a l y s i s o f M i n e D r a i n a g e S y s t e m s , M i n i n g E n g i n e e r i n g , SME o f A I M E , S e p t e m b e r , 1984 B r a w n e r , C O . , G e n e r a l R e p o r t on M i n e D r a i n a g e , M i n e P r o c e e d i n g s o f t h e F i r s t I n t e r n a t i o n a l M i n e D r a i n a g e S y m p o s i u m , D e n v e r , C o l o r a d o , May 1979 B r a w n e r , C O . , S t a b i l i t y o f Rock S l o p e s , S t a b i l i t y i n S u r f a c e M i n i n g , V o l u m e 3 , 1982 B y r n e , P . M . a n d W a l t e r J a n z e n , S O I L S T R E S S - A C o m p u t e r P r o g r a m f o r N o n l i n e a r A n a l y s i s o f S t r e s s e s a n d D e f o r m a t i o n s i n S o i l , S o i l M e c h a n i c s S e r i e s N o . 5 2 , D e p a r t m e n t o f C i v i l E n g i n e e r i n g , t h e 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 , V a n c o u v e r , C a n a d a , D e c e m b e r 1981 C r a i g , R . F . S o i l M e c h a n i c s , 2nd E d i t i o n , 1978 , V a n N o s t r a n d R e i n h o l d ( l 5 ) C o . L t d . D e s a i , C . S . a n d C h r i s t i a n , J . T . , N u m e r i c a l M e t h o d s i n G e o t e c h n i c a l E n g i n e e r i n g , M c G r a w - H i l l , 1979 F r e e z e , R . A . a n d C h e r r y , J . A . , G r o u n d w a t e r , P r e n t i c e - H a l l , I n c . E n g l e w o o d C l i f f s , N . J . , 1979 F r e e z e , A . R . , M a t h e m a t i c a l M o d e l s o f H i l l s l o p e H y d r o l o g y , i n H i l l s l o p e H y d r o l o g y ( M . J . K i r k b y , E d ) J o h n W i l e y , a n d S o n s , P 1 7 7 - 1 2 5 , 1978 81 G e , S h e m i n , L e c t u r e N o t e s o f GEOL 5 6 2 , 1983 ; GEOL 3 4 2 , 1 9 8 3 ; C I V L 5 7 3 , 1984 and GEOL 5 6 4 , 1984 , t h e 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 , V a n c o u v e r , C a n a d a G o l d e r A s s o c i a t e s , D r a i n a b i l i t y o f t h e West W a l l , (LORNEX M i n e ) , R e p o r t #35, A u g u s t 1983 H o l l a n d , I. a n d B e l l , K o l b e i n , F i n i t e E l e m e n t M e t h o d , T a p i r , T h e T e c h n i c a l U n i v e r s i t y o f N o r w a y , T r o n d h e i m - N o r w a y , 1969 H u e b n e r , K . H . , T h e F i n i t e E l e m e n t M e t h o d f o r E n g i n e e r s , J o h n W i l e y a n d S o n , I n c . , 1 9 4 5 K e n n e y , T . C . , P a z i n , M. a n d C h o i , W . S . , D e s i g n o f H o r i z o n t a l D r a i n s f o r S o i l S l o p e s , J o u r , o f t h e G e o t e c h . E n g . D i v . , A S C E . November 1977 M u r r a y , W . A . a n d M o n k m e y e r , P . L . , V a l i d i t y o f D u p u i t - F o r c h h e i m e r E q u a t i o n , J o u r . H y d r a u l . D i v . , Am. S o c . C i r . E n g . , V 9 1 , S e p t . 1973 M a r l o n - L a m b e r t , J . , P i t S l o p e M a n u a l , C o m p u t e r A n a l y s i s o f G r o u n d w a t e r S e e p a g e , S u p p l e m e n t 4 - 1 , CANMET, CANMET R e p o r t 7 7 - 3 0 , D e c e m b e r 1977 R e m s o n , I., H o r n b e r g e r , G . M . a n d M o l z , F . J . , N u m e r i c a l M e t h o d s i n S u b s u r f a c e H y d r o l o g y , J o h n W i l e y a n d S o n s . I n c . , 1971 R u l o n , J . , t h e D e v e l o p m e n t o f M u l t i p l e S e e p a g e F a c e s a l o n g H e t e r o g e n e o u s H i l l s l o p e s , P h . D T h e s i s , t h e U n i v r e s i t y o f B r i t i s h C o l u m b i a , J u n e 1984 S e e g m i l l e r , B . L . , H o r i z o n t a l D r a i n s - T h e i r U s e i n Open P i t M i n e D e w a t e r i n g , M i n e D r a i n a g e , P r o c e e d i n g s o f t h e 82 F i r s t I n t e r n a t i o n a l Mine D r a i n g a e Symposium , Denver, C o l o r a d o , May 1979, P268 Sharp, J . C , Ley, G.M.M. and Sage, R., P i t S l o p e Manual Chapter 4 Groundwater, CANMET, CANMET Report 77-13, November 1977, P240 Sm i t h , I.M. Programming the F i n i t e Element Method With A p p l i c a t i o n t o Geomechanics, John W i l e y and Sons. I n c . , 1982 S t e r r e t t , R .J. and E d i l , T.B., Ground-Water Flow System and S t a b i l i t y of a S l o p e , Gorund Water, 20, 1982, P5-11 T e s a r i k , D.R. and K e a l y , C D . , E s t i m a t i n g H o r i z o n t a l D r a i n D e s i g n by the F i n i t e Element and F i n i t e D i f f e r e n c e  Methods, Bureau of Mines Report of I n v e s t i g a t i o n s : 8 8 7 5 , 1984, U. S. Department of the I n t e r i o r W i l l i a m s , R., Bloomsburg, G. and W i n t e r , G., I n f l o w of H o r i z o n t a l D r a i n s i n T a i l i n g s Embankments, A M i n i n g R e s e a r c h C o n t r a c t R e o p o r t , August 1982, Bureau of Min e s , U. S. Department of I n t e r i o r Wang, H.F. and Anderson, M.P., I n t r o d u c t i o n t o Groundwater M o d e l i n g - F i n i t e D i f f e r e n c e and F i n i t e Element  Methods, W. H. Freeman and Co. San F r a n c i s c o , 1982, P288 Z i e n k i e w i c z , O.C, The F i n i t e Element Method, 3 r d E d i t i o n , M c G r a w - H i l l , New York, 1977, P787 83 APPENDIX A F I N I T E ELEMENT FORMULATION T h e p r i n c i p l e o f t h e f i n i t e e l e m e n t m e t h o d was b r i e f l y d i s c u s s e d i n C h a p t e r 2 . T h e c o n t e n t i n t h i s a p p e n d i x i s t h e f o r m u l a t i o n p r o c e s s o f e q u a t i o n (4) a n d (5) i n C h a p t e r 2 . We know t h a t t h e w a t e r f l o w i n a two d i m e n s i o n a l s a t u r a t e d r e g i o n i s g o v e r n e d by e q u a t i o n ( 1 ) : ( F r e e z e & C h e r r y , 1979) T h e f u n c t i o n a l o f ( A - 1 ) i s I: I=/;t5Kx(||)24Ky(||)2+Qh]dxdy ( A - 2 ) To m i n i m i z e I, s o a s t o o b t a i n t h e s o l u t i o n f o r ( A - 1 ) , t h e f o l l o w i n g c o n d i t i o n m u s t be s a t i s f i e d : | £ = 0 on R ( A - 3 ) o r i f t h e r e g i o n R i s c o m p r i s e d o f N s u b r e g i o n s w i t h M unknown h e a d n o d e s , ( A - 3 ) i s e q u i v a l e n t t o : 9hf u i . e . ^ T = ; ; [ K x ( | | ) | K i ( | | ) + K y ( | | ) | K ( | | ) + Q f 7 ] a x a y = 0 (A-4) i = 1 ,M 84 T h e r e f o r e , M equations can be obtained, the s o l u t i o n of M equations are M h y d r a u l i c head v a l u e s . The f o r m u l a t i o n of (A-4) depends on the shape of the subregions. Two types of element shape were used, t r i a n g u l a r and r e c t a n g u l a r . Formulation f o r T r i a n g u l a r Elements Y (I. 3 ( x 3 , y 3 ) 2 ( x 2 ,y 2) w i t h i n each element, the h y d r a u l i c heads are assumed to be a f u n c t i o n of t h e i r p o s i t i o n , i . e . h=h(x,y), which i s c a l l e d the i n t e r p o l a t i o n f u n c t i o n . The g e n e r a l i z e d form of i n t e r p o l a t i o n f u n c t i o n f o r the t r i a n g l e i s : h=a,+a 2x+a 3y (A-5) =(1 x y ){a} where: {a} i s g e n e r a l i z e d c o o r d i n a t e s x,y are p o s i t i o n c o o r d i n a t e s The h y d r a u l i c heads at three nodes of the t r i a n g l e should 85 s a t i s f y ( A - 5 ) , i . e . 1 x , y , a. 1 x 2 y 2 < h 3 1 x 3 y 3 ( A - 6 ) T h e m a t r i x f o r m i s : {h}=[B]{a} ( A - 7 ) {a} i s f o u n d t o b e : {a} = [ B ] - 4 h } =[A]{h} ( A - 8 ) S u b s t i t u t i n g ( A - 8 ) i n t o ( A - 5 ) : h-(1 x y ) [ A ] { h } =[N]{h} ( A - 9 ) In ( A - 9 ) , [N] i s a f u n c t i o n o f x , y , x 1 f y 1 f x 2 , y 2 , x 3 , y 3 9h _ _ . • „ \ 9h 9h ' 9 / 9 h \ 9 / 9 h \ In e q u a t i o n ( A - 4 ) , ^ , ^ , ^ ( ^ ) , ^ ^ ) , a h . . i=1 , 3 a r e n e e d e d t o be known. A l l o f t h e t e r m s c o u l d be d e r i v e d f r o m ( A - 9 ) : 9x 9x 9x 9x 9 1 1 ^ 9 ^ 9 ^ 9 ^ 9y 9y 9y 9y 9_,9hx 9N_ 9 h f 9 x ' " 9 x ( A - 1 0 ) 86 9hT" N l S u b s t i t u t i n g ( A - 1 0 ) i n t o ( A - 4 ) , t h e c o n t r i b u t i o n o f e l e m e n t e t o node i i s o b t a i n e d f r o m t h e f o l l o w i n g f o r m a t i o n : ^ Li *r Tr]dr+K?ldT ay 97^ ]"oT^ ] { h } Q N l } d x d y W h e r e : Ae i s t h e a r e a o f e l e m e n t e . T h e s i m p l i f i e d f o r m b e c o m e s : ( | ^ T ) = [ k , i k 2 ; k 3 i ] { h } + { C j } ( A - 1 1 ) k H = K x ( f i ) ( | ^ ) A e + K y ( f i ) ( M i ) A e C i = / / Q i N i d x d y Ae i = 1 , 3 a n d j = 1 , 3 k f j a n d C i c o u l d be c o m p u t e d by k n o w i n g t h e c o o r d i n a t e s o f t h r e e c o r n e r p o i n t s . F o r m u l a t i o n f o r R e c t a n g u l a r E l e m e n t s T h e i n t e r p o l a t i o n f u n c t i o n f o r r e c t a n g u l a r e l e m e n t s i s : h=a , + a 2 x + a . 3 y + a ( | x y ( A - 1 2 ) F o l l o w i n g t h e same p r o c e d u r e s d e s c r i b e d a b o v e f o r t r i a n g u l a r e l e m e n t s , e q u a t i o n ( A - 1 3 ) i s o b t a i n e d : ( |^-) = [ k , i k 2 1 - k 3 ; k , i ] { h } + {Ci} ( A - 1 3 ) 87 C i = ; / Q f N j d x d y Ae i = 1 , 4 a n d j = 1 , 4 A s s e m b l y o f t h e G l o b a l M a t r i x A f t e r t h e f o r m u l a t i o n f o r i n d i v i d u a l e l e m e n t s i s f i n i s h e d , t h e n e x t s t e p i s t o a s s e m b l e them i n t o a s e t o f g l o b a l e q u a t i o n f o r t h e w h o l e s y s t e m . The g l o b a l e q u a t i o n s h a s t h e f o r m o f : [K ] {h}+ [C]=0 ( A - 1 4 ) T h e way t o f o r m t h e g l o b a l e q u a t i o n ( A - 1 4 ) f r o m t h e e q u a t i o n o f e a c h e l e m e n t i s d o n e s t r i c t l y i n t e r m s o f t h e g e o m e t r y , node n u m b e r i n g a n d e l e m e n t n u m b e r i n g . F i g u r e A-1 shows t h e p r o c e s s . ( W a n g & A n d e r s o n , 1982) Element contributions are computed, dispersed, and summed to form the global matrix X X X X X X X X X e = 5 e = 6 Global matrix Element No. e Nodes i j m 1 1 i 3 2 2 5 3 3 T 4 5 4 4 6 5 5 5 6 7 6 6 8 7 F i g u r e A . 1 . - T h e G l o b a l M a t r i x A s s e m b l y f r o m I n d i v i d u a l E l e m e n t s 89 APPENDIX B COMPUTER PROGRAM B.1 I n t r o d u c t i o n The computer program, FESHDMS ( F i n i t e Element S i m u l a t i o n f o r H o r i z o n t a l D r a i n Drainage i n Mining S l o p e s ) , d e s c r i b e d i n t h i s appendix was developed to determine the h y d r a u l i c head d i s t r i b u t i o n , water pressure as w e l l as flow r a t e s and flow d i r e c t i o n of steady s t a t e flow i n a two dimensional and s a t u r a t e d f i e l d . The c a p a b i l i t y of the program permits the a n a l y s i s of groundwater flow f o r the f o l l o w i n g problems: 1. the l o c a t i o n of the f r e e water s u r f a c e f o r water flow i n open p i t s l o p e s ; 2. h y d r a u l i c head d i s t r i b u t i o n on a v e r t i c a l or h o r i z o n t a l s e c t i o n ; 3. flow r a t e s and d i r e c t i o n s i n the flow r e g i o n . The computer program performances are sub j e c t e d t o the f o l l o w i n g c o n d i t i o n s : 1. Darcy's law i s v a l i d f o r flows; 2. m a t e r i a l c o n d u c t i v i t i e s can be heterogeneous and a n o i s o t r o p i c ; 3. flow region i s c o n f i n e d or unconfined. The f i n i t e element method by the v a r i a t i o n a l p r i n c i p l e approach was used i n the s i m u l a t i o n . The region of a n a l y s i s 90 i s d i v i d e d i n t o a number o f d i s c r e t e e l e m e n t s . The r e s u l t s o f t h e t h e f i n i t e e l e m e n t f o r m u l a t i o n i s a s e t o f l i n e a r e q u a t i o n s w i t h t h e h y d r a u l i c h e a d s a s u n k n o w n s . T h e m a t r i x f o r m o f t h e e q u a t i o n s i s g i v e n b y : [K]{h}+{C}=0 ( B - 1 ) w h e r e : [K] i s t h e g l o b a l s t i f f n e s s m a t r i x {h} i s t h e unknown h y d r a u l i c h e a d v e c t o r {C} i s t h e c o l u m n v e c t o r a s s o c i a t e d w i t h t h e f l o w f l u x T h e d e r i v a t i o n o f e q u a t i o n ( B - 1 ) h a s b e e n shown i n A p p e n d i x A . B . 2 C o m p u t e r P r o g r a m O r g a n i z a t i o n  F l o w c h a r t A f l o w c h a r t f o r t h e c o m p u t e r p r o g r a m i s p r e s e n t e d i n F i g u r e B . 1 . A c e r t a i n amount o f comment c o n t a i n e d i n t h e p r o g r a m l i s t i n g c o u l d make t h e p r o g r a m d e t a i l s r e l a t i v e l y s t r a i g h t f o r w a r d . S u b r o u t i n e s MAIN t h e m a i n r o u t i n e i n t i t i a t e s s t o r a g e s p a c e , r e a d s t h e g l o b a l g e o m e t r y a n d g l o b a l c o n t r o l v a r i a b l e s , a l l o c a t e s s t o r a g e a n d a c t s a s a g e n e r a l f l o w r o u t i n g r o u t i n e . 91 S t a r t — r R e a d h e a d i n g a n d g e o m e t r y S e t d i m e n s i o n A l l o c a t e s t o r a g e i 1 C a l l NODEIN C a l l ELEMTIN R e a d more d a t a P A , W T , t i m e p e r i o d C a l l CONDUC 1 C a l l T R I A N *1 C a l l QUADS C a l l ECHO C A L L IN IT I t = 0 C a l l BUILD * C a l l ASSEMB C a l l SOLVER i C a l l COMPA IF H=y ? NO y e s | C a l l PHOUT C a l l PLOUT S t o p y e s C a l l BUILD C a l l ASSEMB C a l l SOLVER C a l l COMPA IF H= • y ? C a l l PHOUT * C a l l PLOUT T C a l l TR IAN u C a l l QUADS No F i g u r e B.1 F l o w C h a r t f o r t h e C o m p u t e r P r o g r a m FESHDMS 92 NODEIN T h i s r o u t i n e i s co m p r i s e d of two p o r t i o n s . The f i r s t p a r t reads i n d i v i d u a l node d a t a from u n i t 5, i n t e r p o l a t e s m i s s i n g nodes and checks e r r o r s i n node d a t a i n p u t . The second p a r t c a l c u l a t e s t he number of unknowns. ELEMTIN T h i s s u b r o u t i n e r e a d s i n d i v i d u a l element d a t a from u n i t 5, checks f o r e r r o r s and c a l c u l a t e s t he maximum bandwidth. CONDUC S u b r o u t i n e t o r e a d c o n d u c t i v i t i e s of m a t e r i a l s and re a d s p e c i f i e d known h y d r a u l i c heads from u n i t 5. ECHO S u b r o u t i n e t o p r i n t t o u n i t 6 a l l node, element and m a t e r i a l d a t a f o r v e r i f i c a t i o n . I NITI For t r a n s i e n t p r o b l e m s , t h i s s u b r o u t i n e does i n i t i a l c o n d i t i o n c o m p u t a t i o n . BUILD The main s u b r o u t i n e t o form element m a t r i x f o r each element and s t o r e them f o r a s s e m b l i n g the g l o b a l m a t r i x . TRIAN and QUADS C a l l e d by BUILD, they c a l c u l a t e the element m a t r i x f o r t r i a n g u l a r and r e c t a n g u l a r e l e m e n t s . ASSEMB Assembly i n d i v i d u a l element m a t r i x i n t o t he g l o b a l 93 equation. COMPA For f r e e s u r f a c e problems, t h i s s u b r o u t i n e compares the computed h y d r a u l i c heads on f r e e s u r f a c e with t h e i r e l e v a t i o n heads. PHOUT Subroutine to p r i n t out the h y d r a u l i c head and pre s s u r e head at the end of computation. PLOUT Subroutine to p l o t the slope s e c t i o n , f i n i t e element mesh and water t a b l e p o s i t i o n . An example of the computer input and output i s presented at the end of t h i s Appendix. 94 V a r i a b l e Name L i s t i n g ( a c c o r d i n g t o t h e i r a p p e a r i n g s e q u e n c e i n t h e p r o g r a m ) NNODE t o t a l Number o f NODEs NU t o a t a l Number o f Unknow h e a d s n o d e s NELT t o t a l Number o f E L e m e n T s NMAT t o t a l Number o f d i f f e r e n t M A T e r i a l t y p e s NTRNGS t o t a l Number o f T R i a N G u l a r e l e m e n t s NQUADS t o t a l Number o f Q U A D i l a t e r a l e l e m e n t s NB B a n d w i d t h NHK Number o f Known H e a d n o d e s PA A t o m s p h e r i c P r e s s u r e WT u n i t W e i g h T o f w a t e r P1 t h e f i r s t P e r i o d o f d r a i n a g e P2 t h e s e c o n d P e r i o d o f d r a i n a g e P3 t h e t h i r d P e r i o d o f d r a i n a g e DELT1 t h e t i m e i n t e r v a l i n f i r s t d r a i n a g e p e r i o d DELT2 t h e t i m e i n t e r v a l i n s e c o n d d r a i n a g e p e r i o d D E L T 3 t h e t i m e i n t e r v a l i n t h i r d d r a i n a g e p e r i o d IT IME I n d e x o f T IME s t e p MAXIT MAXimum I T e r a t i o n t i m e s X E L T ( 1 , I ) X c o o r d i n a t e o f I t h node i n a ELemenT X E L T ( 2 , I ) Y c o o r d i n a t e o f I t h node i n a ELemenT E S M ( 4 , 4 ) E l e m e n t S t i f f n e s s M a t r i x E V E ( 4 ) E l e m e n t V E c t o r B M ( 2 , 4 ) i n t e r v a r i a b l e s 95 CONX C O N d u c t i v i t y i n X d i r e c t i o n CONY C O N d u c t i v i t y i n Y d i r e c t i o n STOR STORage c o e f f i c i e n t HIN I N i t i a l H e a d H H y d r a u l i c H e a d MB Maximum B o u n d a r y c o r n e r p o i n t s BX(MB) X c o o r d i n a t e o f c o r n e r p o i n t s on b o u n d a r y BY(MB) Y c o o r d i n a t e o f c o r n e r p o i n t s on b o u n d a r y XMAX MAXimum X c o o r d i n a t e YMAX MAXimum Y c o o r d i n a t e TOL s p e c i f i e d T O L e r a n c e KC Known h e a d C o n d i t i o n o f e a c h n o d e , 0 - u n k n o w n ; 4 - k n o w n ; 5 - f r e e s u r f a c e MC M o v a b l e C o n d i t i o n o f e a c h n o d e , 0 - f r e e t o m o v e ; 1 - f i x e d MESH 0 - n o mesh p l o t r e q u i r e d ; 1-mesh p l o t KCE Known f l u x C o n d i t i o n o f e a c h E l e m e n t , 0 - e l e m e n t w i t h no known f l o w b o u n d a r y ; 1 - e l e m e n t w i t h known f l o w b o u n d a r y N S E C T O - v e r t i c a l m o d e l ; 1 - h o r i z o n t a l m o d e l NSTADY O - t r a n s i e n t ; 1 - s t e a d y s t a t e C o m p u t e r S t o r a g e A l l o c a t i o n D i a g r a m V a r i a b l e X C O N X C O N Y S T O R H I N H G V E A r r a y A ( 2 0 0 0 0 ) S t o r a g e P o s i t i o n i n A r r a y S i z e N N O D E N N O D E N M A T N M A T N M A T N N O D E N N O D E N N O D E N N O D E G S M ( N N O D E ) N1 = 1 N 2 = N 1 + N N O D E N 3 = N 2 + N N O D E N 4 = N 3 + N M A T N 5 = N 4 + N M A T N 6 = N + N M A T N 7 = N 6 + N N O D E N 8 = N 7 + N N O D E N 9 = N 8 + N N O D E N 1 0 = N 9 + N N O D E N 1 1 = N 1 0 + ( N N O D E ) H K N H K V a r a b l e K C MC I T Y P E I M A T N C K C E A r r a y I A ( 2 0 0 0 ) S t o r a g e P o s i t i o n i n A r r a y S i z e N N O D E N N O D E N E L T N E L T J E L T 4 ( N E L T ) N E L T N E L T I N 1 I N 2 = I N 1 + N N O D E I N 3 = I N 2 + N N O D E I N 4 = I N 3 + N E L T I N 5 = I N 4 + N E L T I N 6 = I N 5 + 4 ( N E L T ) I N 7 = I N 6 + N E L T Free node • Fixed nixie F i g u r e B . 2 . An E x a m p l e t o Show t h e C o m p u t e r I n p u t 1St1ng of WAND at 21:47:31 on JAN 30. 1985 for CC1d=BRAW Page 1 HEAD,EXAM, 2 16,18.1,8,0,0, 1 , 3 4.60..60.. 4 0.,0., 5 6.,0.. e 6.,3.. 7 0. .4. , 8 0, g 1,0.,4..4,1, 10 2,0.,2..4,1, 11 3,0.,1.,4, 1, 12 13 5,2..3.7,5,0, 14 6,2.,2.,0,1, 15 16 8,2.,0.,0.1, 17 9.4. ,3.36,5,0, 18 10,4.,2.,0,1, 19 11.4.,1.,0,1, 20 21 13,6..3..4,1, 22 14,6.,2.,4,1, 23 15,6.,1.,4,1, 24 16,6.,0.,4,1, 25 1.1.1.0,1.6,5, 26 2,1,1,0,2,6,1, 27 3,1,1.0,2,7,6, 28 4,1,1,0.3,7,2, 29 5,1,1,0,3.8,7, 30 6,1,1.0.4,8,3, 31 7. 1,1,0,5,10.9, 32 8,1,1.0,6,10,5, 33 9,1,1.0,6.11,10, 34 10,1,1.0.7.11,6, 35 11,1.1,0,7,12.11. 36 12,1,1.0,8.12,7. 37 13,1,1.0,9,14,13, 38 14,1,1,0,10,14,9, 39 15,1,1,0,10,15,14, 40 16, 1 , 1 ,0, 1 1 , 15, 10, 41 17,1,1.0.11,16,15. 42 18,1,1.0,12,16.11. 43 ROCK, 44 9.8.0.,0.. 45 5.0.5, 45.5 1 , 45.7 1,0..0.0001,1., 46 1,4., 47 2,4. , 48 3,4. , 49 4,4. , 50 5,3. , 51 6,3. . 52 7,3. , 53 8,3. , 100 L i s t i n g 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 of WAND, at 12:02:29 on JAN 30, 1985 for CC1d=BRAW Page 1 ^Execution begins 6 MAXIT= 5 ********** NODAL INFORMATION ********** Total number of nodes= 16 Total number of unknowns3 8 Node Coordinates Known condition Movable condition X Y 1 0.0 4.000 4 1 2 0.0 2.000 4 1 3 0.0 1.000 4 1 4 0.0 0.0 4 1 5 2.000 3.700 5 0 6 2.000 2.000 0 1 7 2.000 1.000 0 1 8 2.000 0.0 0 1 9 4.000 3.360 5 0 10 4.000 2.000 0 1 11 4.000 1.000 0 1 12 4.000 0.0 0 1 13 6.000 3.000 4 1 14 6.000 2.OOO 4 1 15 6.000 1.000 4 1 16 6.000 0.0 4 1 ********** ELEMENT INFORMATION ********** Total number of element 3 18 Maximum bandw1dth= 6 Number of quadrllaterals= 0 Number of tnangles= 18 Element types: 1-Triangle 2-QuadMlateral Element Material Element Nodes Assoiated with Type Type Element Code • 1 1 1 1 6 5 2 1 1 2 6 1 3 1 1 2 7 6 4 1 1 3 7 2 5 1 1 3 8 7 6 1 1 4 8 3 .7 1 1 5 10 9 8 1 1 6 10 5 9 1 1 6 1 1 10 10 1 1 7 1 1 6 1 1 1 1 7 12 1 1 12 1 1 8 12 7 13 1 1 9 14 13 14 1 1 10 14 9 15 1 1 10 15 14 16 1 1 1 1 15 10 101 it Ing of WANO. at 12: 02:29 on JAN 30, 1985 for CC id=BRAW Page 61 17 1 1 1 1 16 15 62 18 1 1 12 16 11 63 64 65 66 **«*«*****Mater1al propert1es********** 67 68 69 Unit weight of water************ 9.80 70 Material type Conductivity Conductivity 71 Number in X d i r e c t i o n in Y d i r e c t i o n 72 1 0.00010 0.00010 73 IT= 1 74 75 76 ************** + * * * * * 0 u t p U t ^nformation ********* 77 RATIO OF Ky/Kx IS 1.00000 78 79 DRAIN LENGTH3 0.0 (M) 80 81 NODE NUMBER HYDRAULIC HEAD(m) WATER PRESSUI 82 1 4 .000 0 .0 83 2 4 .000 19 .600 84 3 4 .000 29 .400 85 4 4 .000 39 .200 86 5 3 .720 0 . 193 87 6 3 .695 16 .613 88 7 3 .688 26 .343 89 8 3 .686 36 .121 90 9 3 .392 0 .312 91 10 3 .366 13 .383 92 1 1 3 .357 23 .095 93 12 3 . 354 32 .868 94 13 3 .000 0 .0 95 14 3 .000 9 .800 96 15 3 .000 19 .600 97 16 3 .000 29 .400 98 ELEMENT Vx Vy THIT 99 1 0.000013801424 -0.000001439324 -5. 95373 100 2 0.000015240745 -0.0 0. 0 101 3 0.OOOO15240745 -O.OOO00O711495 -2. 67283 102 4 0.OOOO15596495 -0.0 0. 0 103 5 0.OOOO15596495 -0.000000222816 -0. 81848 104 6 O.000015707890 -O.O 0. 0 105 7 0.000016062913 -0.000001928770 -6. 84705 106 8 0.000016478938 -0.000001439324 -4 . 99172 107 9 0.000016478923 -0.000000901329 -3 . 13071 108 10 0.000016573846 • -0.000000711495 -2. 45812 109 1 1 0.000016573846 -0.000000273438 -0. 94519 1 10 12 0.000016599151 -0.000000222816 -0. 76905 1 1 1 13 0.OOOO19591855 -O.O O. O 112 14 0.000018280305 -0.000001928770 -6. 02302 113 15 0.000018280291 -0.0 0. 0 1 14 16 0.000017829632 -0.000000901329 -2. 89396 1 15 17 0.000017829632 -0.0 0. 0 1 16 18 0.000017692902 -0.000000273438 -0. 88541 117 118 *** UBC Plot Subroutines - End of P l o t t i n g * * * Storage 0.0 102 L i s t i n g of WAND, a t 12:02:29 on JAN 30, 1985 f o r CC1d=BRAW Page 3 119 120 Number of p l o t frames g e n e r a t e d = 1 121 I f t h i s p l o t 1s queued f o r p l o t t i n g 1t w i l l take a p p r o x i m a t e l y 2 minutes to 122 p l o t a t an approximate c o s t of 0.11 d o l l a r s ( U n i v e r s i t y r a t e s ) and use 123 26 i n c h e s of paper. A p p r o x i m a t e l y 36% of the time w i l l be spent p l o t t i n g 124 w i t h the pen r a i s e d . 125 ********** CALCULATION END ********** 126 /^Execution t e r m i n a t e d 

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