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Universi^Micronlms

International300 N. Zeeb Road Ann Arbor, Ml 48106

8313780

Nakomthap, Kurujit

NUMERICAL SIMULATION OF MULTIPHASE FLUID FLOW IN NATURALLY FRACTURED RESERVOIRS

The University o f Oklahoma Ph.D. 1983

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Internstionsi 300 N. Zœb Road, Ann Arbor, MI 48106

Copyright 1983

by

Nakomthap, Kurujit

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UniversityMicrofilms

International

THE UNIVERSITY OF OKLAHOMA

GRADUATE COLLEGE

NUMERICAL SIMULATION OF MULTIPHASE FLUID FLOW

IN NATURALLY FRACTURED RESERVOIRS

A DISSERTATION

SUBMITTED TO THE GRADUATE FACULTY

i n p a r t i a l f u l f i l 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

DOCTOR OF PHILOSOPHY

BY

KURUJIT NAKORNTHAP

N o rm an , Oklahoma

1982

NUMERICAL SIMULATION OF MULTIPHASE FLUID FLOW

IN NATURALLY FRACTURED RESERVOIRS

A DISSERTATION

APPROVED FOR THE SCHOOL OF

PETROLEUM AND GEOLOGICAL ENGINEERING

BY

9 ^ - — ■

DISSERTATION COMMITTEE

ACKNOWLEDGMENTS

The a u t h o r i s d e e p l y g r a t e f u l t o D r . R. D. E v a n s ,

P r o f e s s o r o f P e t r o l e u m and G e o l o g i c a l E n g i n e e r i n g and

c h a i r m a n o f h i s d i s s e r t a t i o n c o m m i t t e e , f o r h i s g u i d a n c e ,

s u p p o r t , e n c o u r a g e m e n t and v a l u a b l e s u g g e s t i o n s . H i s p h i l o

s o p h y , h i s s t a n d a r d o f e x c e l l e n c e , and m o s t o f a l l , h i s

t r u s t i n t h e a u t h o r ' s a b i l i t y h a v e b e e n i n s p i r a t i o n a l a s

w e l l a s m o t i v a t i o n a l t h r o u g h o u t t h i s r e s e a r c h .

The a u t h o r a l s o w i s h e s t o e x p r e s s h i s s i n c e r e a p p r e

c i a t i o n t o P r o f e s s o r s D r . D. E. M e n z i e , D r . D. T i a b , D r .

E. F. B l i c k o f t h e S c h o o l o f P e t r o l e u m a n d G e o l o g i c a l E n g i

n e e r i n g , a n d D r . A. A. A ly o f t h e S c h o o l o f I n d u s t r i a l

E n g i n e e r i n g f o r s e r v i n g a s m em bers o f h i s d i s s e r t a t i o n

c o m m i t t e e .

F i n a n c i a l s u p p o r t t h r o u g h o u t t h i s a c a d e m i c e n d e a v o r

by t h e D e p a r t m e n t o f M i n e r a l R e s o u r c e s o f T h a i l a n d i s

g r a t e f u l l y a c k n o w l e d g e d . A p p r e c i a t i o n i s a l s o e x t e n d e d t o

t h e E n e r g y R e s o u r c e C e n t e r a n d t h e O klahom a M i n i n g and

M i n e r a l R e s o u r c e s R e s e a r c h I n s t i t u t e o f t h e U n i v e r s i t y o f

O klahom a f o r t h e i r f i n a n c i a l a s s i s t a n c e -

F i n a l l y , t h e a u t h o r w i s h e s t o d e d i c a t e t h i s d i s s e r

t a t i o n t o h i s p a r e n t s . D r . A r t h and M r s . T a p a n e e N a k o m t h a p

W i t h o u t t h e i r l o v e , s u p p o r t and u n d e r s t a n d i n g , t h i s d i s s e r

t a t i o n w o u ld n o t h a v e b e e n p o s s i b l e .

iii

ABSTRACT

A new m a t h e m a t i c a l f l o w m o d e l i s d e r i v e d f o r c h a r a c t e r

i z i n g t h e m u l t i p h a s e f l u i d f l o w p h enom enon i n n a t u r a l l y f r a c

t u r e d r e s e r v o i r s . T h i s new m o d e l t r e a t s t h e r e s e r v o i r a s a

d o u b l e p o r o s i t y medium c o n s i s t i n g o f a h e t e r o g e n e o u s , i s o t r o

p i c p r i m a r y r o c k m a t r i x and an a n i s o t r o p i c , h e t e r o g e n e o u s

f r a c t u r e . The s t a t i s t i c a l d i s t r i b u t i o n i n s p a c e and o r i e n

t a t i o n o f t h e f r a c t u r e i s m a n i f e s t i n t e r m s o f t h e f r a c t u r e

p e r m e a b i l i t y t e n s o r . The m o d e l p e r m i t s f l o w t h r o u g h t h e r o c k

m a t r i x and t h e f r a c t u r e t o b e d e s c r i b e d s i m u l t a n e o u s l y . The

f l u i d i n t e r a c t i o n t e r m s c o u p l i n g f l o w b e t w e e n t h e r o c k m a t r i x

and f r a c t u r e , t h e g r a v i t y e f f e c t s , t h e c a p i l l a r y f o r c e s and

t h e f r a c t u r e v e l o c i t y o f f l u i d s a r e a l l t a k e n i n t o c o n s i d e r

a t i o n .

A c c o r d i n g t o t h i s m o d e l , t h e n u m e r i c a l s i m u l a t i o n

p r o g r a m i s d e v e l o p e d i n FORTRAN IV c o m p u t e r c o d e u s i n g t h e

f i n i t e d i f f e r e n c e a p p r o x i m a t i o n t e c h n i q u e , t h e f u l l y i m p l i c i t

a p p r o a c h , a n d t h e P o i n t S u c c e s s i v e O v e r - R e l a x a t i o n m e t h o d .

E x t e n s i o n o f t h e s i m u l a t o r c a p a b i l i t y t o i n c o r p o r a t e t h e

M onte C a r l o t e c h n i q u e f o r r a n d o m l y g e n e r a t i n g f r a c t u r e l o

c a t i o n s i s a l s o d i s c u s s e d .

The s i m u l a t o r i s r u n and t e s t e d w i t h d a t a f r o m a f i e l d

e x a m p l e . The r e s u l t s show t h a t t h e p h a s e p r e s s u r e s r e s p o n d

t o e a c h o t h e r m ore s p o n t a n e o u s l y i n t h e f r a c t u r e t h a n t h e y do

i v

t o t h e p h a s e p r e s s u r e s i n t h e r o c k m a t r i x , and v i c e v e r s a .

F u r t h e r m o r e , t h e v e l o c i t y o f t h e f u i I d s i n t h e f r a c t u r e s

h a v e n e g l i g i b l e e f f e c t on t h e r e s u l t s . The r e s u l t s s u g g e s t

t h a t t h e c o n v e n t i o n a l way o f a s s u m i n g t h a t f l u i d s f i r s t f l o w

f r o m t h e r o c k m a t r i x i n t o t h e f r a c t u r e and t h e n f l o w t h r o u g h

t h e f r a c t u r e t o w a r d t h e w e l l b o r e may n o t b e an a d e q u a t e

d e s c r i p t i o n o f t h e f l o w p h e n o m en o n i n a n a t u r a l l y f r a c t u r e d

r e s e r v o i r .

TABLE OF CONTENTSPage

ACKNOWLEDGMENTS ................................................................................................... i i i

A B S T R A C T ................................................................................................................... i v

LIST OF T A B L E S .......................................................................................................... i x

LIST OF F I G U R E S ................................................................................................... x

CHAPTER

I . INTRODUCTION ........................................................................................... 1

I I . LITERATURE REVIEW ........................................................................... 3

I I I . MATHEMATICAL FORMULATION ........................................................... 14

3 . 1 B a c k g r o u n d and A s s u m p t i o n s .................................... 143 . 2 Summary o f G o v e r n i n g E q u a t i o n s ......................... 15

3 . 2 . 1 C o n t i n u i t y E q u a t i o n s .................................... 153 . 2 . 2 E q u a t i o n s o f M o t i o n s .................................... 163 . 2 . 3 F l u i d I n t e r a c t i o n Te rm s ......................... 163 . 2 . 4 P o r o s i t y E q u a t i o n s ......................................... 163 . 2 . 5 R e q u i r e d A u x i l i a r y E q u a t i o n s . . . . 17

3 . 3 B l a c k O i l M o d e l ................................................................... 173 . 4 T i l t e d R e s e r v o i r .............................................................. 193 . 5 D e r i v a t i o n o f t h e F lo w E q u a t i o n s .................... 21

3 . 5 . 1 Rock M a t r i x ........................................................ 213 . 5 . 2 F r a c t u r e ................................................................... 24

3 . 6 The Flow E q u a t i o n s i n T h r e e - D i m e n s i o n a lForms

3 . 6 . 1 Thet h e

3 . 6 . 2 The.the

3 . 6 . 3 Thet h e

3 . 6 . 4 Thet h e

29

The W a t e r D i f f u s i v i t y E q u a t i o n i nt h e M a t r i x R o c k .................................................... 29The O i l D i f f u s i v i t y E q u a t i o n i nt h e M a t r i x R o c k .................................................... 29The Gas D i f f u s i v i t y E q u a t i o n i nt h e M a t r i x .R o c k .................................................... 30The W a t e r D i f f u s i v i t y E q u a t i o n i nt h e f r a c t u r e s .......................................................... 30

VI

CHAPTER Page3 . 6 . 5 The O i l D i f f u s i v i t y E q u a t i o n i n

t h e F r a c t u r e s .......................................... ....3 . 6 . 6 The Gas D i f f u s i v i t y E q u a t i o n i n

t h e F r a c t u r e s .......................................... ....

31

32

IV . SIMULATOR DEVELOPMENT

F i n i t e D i f f e r e n c e A p p r o x i m a t i o n s . .E v a l u a t i o n o f C o e f f i c i e n t s .....................B o u n d a r y C o n d i t i o n s ..........................................T h r e e - P h a s e R e l a t i v e P e r m e a b i l i t y . . I n j e c t i o n a n d P r o d u c t i o n R a t e s . . . H y d r a u l i c R a d i u s o f a F r a c t u r e C o n d u i t .P o r o s i t y C a l c u l a t i o n .....................................V e l o c i t y C a l c u l a t i o n .....................................

4 .4 .4 .4 .4 .4 . 64 . 74 . 84 . 9 P o i n t S u c c e s s i v e O v e r - R e l a x a t i o n M ethod . .

35

3552555657585859 61

V. ALGORITHM 65

5 . 1 M a j o r P r o g r a m C o m p o n e n t s

5 . 1 . 1 M ain P r o g r a m . . .5 . 1 . 2 S u b r o u t i n e CODE5 . 1 . 3 S u b r o u t i n e COEFF .5 . 1 . 4 S u b r o u t i n e EXTRAP.5 . 1 . 5 S u b r o u t i n e FAVG. .5 . 1 . 6 S u b r o u t i n e INCON .5 . 1 . 7 S u b r o u t i n e LINTP .5 . 1 . 8 S u b r o u t i n e MATBAL5 . 1 . 9 S u b r o u t i n e POROSY5 . 1 . 1 0 S u b r o u t i n e PSOR5 . 1 . 1 1 S u b r o u t i n e PTPRES5 . 1 . 1 2 S u b r o u t i n e PTSAT .5 . 1 . 1 3 S u b r o u t i n e PTPOR .5 . 1 . 1 4 S u b r o u t i n e PTVELO5 . 1 . 1 5 S u b r o u t i n e RESDAT5 . 1 . 1 6 S u b r o u t i n e RESDIM5 . 1 . 1 7 S u b r o u t i n e TRAN5 . 1 . 1 8 S u b r o u t i n e VELOCY

65

656970 70 707071 71 71 71 71 71 73 737374 74 74

VI,5 . 2 F i e l d E x a m p le . .

DISCUSSION OF RESULTS

6 . 1 S i m u l a t o r C a p a b i l i t y . . . .6 . 2 R e s u l t s o f t h e F i e l d E x a m p le

6 . 2 . 1 D i f f e r e n c e i n t h e I n j e c t i o n R a t e s6 . 2 . 2 D i f f e r e n c e i n t h e C a p i l l a r y

P r e s s u r e D a t a ..........................................

74

82

8386

102

104

V I ]

CHAPTER Page6 . 2 . 3 D om in an ce o f t h e C a p i l l a r y -

P r e s s u r e R e l a t e d C o e f f i c i e n t s . . . . 104

6 . 3 E f f e c t o f L o w e r i n g I n j e c t i o n R a t e ................. 1106 . 4 E f f e c t o f F l u i d V e l o c i t y i n F r a c t u r e . . . . I l l6 . 5 R e c o m m e n d a t io n s f o r F u r t h e r I m p r o v e m e n t s . . 114

V I I . CONCLUSIONS.................................................................................. 116

NOMENCLATURE ......................................................................................................... 118

REFERENCES.............................................................................................................. 121

APPENDICES

A. C o m p a r i s o n o f U n i t s a n d C o n v e r s i o n F a c t o r s . . 124

B. C o m p a r i s o n o f E q u a t i o n s u n d e r SI U n i t s andO i l F i e l d U n i t s ...................................................................... 126

C. E x p r e s s i o n s f o r t h e F lo w C o e f f i c i e n t s .............. 130

D. D a t a P r e p a r a t i o n .................................................................. 153

E. A r r a n g e m e n t o f I n p u t D a t a ............................................. 165

F . C o m p u te r L i s t i n g o f t h e S i m u l a t i o n P r o g r a m . . 168

G. M o n te C a r l o P r o g r a m f o r G e n e r a t i n g F r a c t u r e dN o d e s ................................................................................................ 229

H. E s t i m a t i o n o f H y d r a u l i c R a d i u s f r o m Thomase t a l E x a m p l e ........................................................................... 230

I. Effect of Decreasing the Fracture HydraulicRadius...................................... 232

J. Effect of Smaller Grid S i z e s ............. 235

V l l l

LIST OF TABLES

TABLE Page1 . F i n i t e D i f f e r e n c e F o r m u l a e ........................................................... 38

2 . R e s e r v o i r an d G r i d D i m e n s i o n s an d RockP r o p e r t i e s ......................................................................................................... 77

3 . O i l PVT D a t a ...............................................................................................7 8

4 . W a t e r PVT D a t a ......................................................................................... 7 8

5 . Gas PVT D a t a ......................................................................................................79

6 . F l u i d D e n s i t i e s a t S t a n d a r d C o n d i t i o n s ............................... 79

7 . G a s - O i l R e l a t i v e P e r m e a b i l i t y and C a p i l l a r yP r e s s u r e D a t a ............................................................................................... 80

8 . W a t e r - O i l R e l a t i v e P e r m e a b i l i t y and C a p i l l a r yP r e s s u r e D a t a ........................................................................................ 81

9 . O u t p u t S a m p l e .................................................................................................87

I X

LIST OF FIGURES

FIGURE Page1 . A c t u a l N a t u r a l l y F r a c t u r e d R e s e r v o i r ........................... 4

2 . I d e a l i z a t i o n o f a N a t u r a l l y F r a c t u r e d P o r o u sMedium ( W a r r e n an d R o o t M ode l ) .......................................... 7

3 . I d e a l i z a t i o n o f a N a t u r a l l y F r a c t u r e d P o r o u sMedium (Kazemi M ode l ) .................................................................... 8

4 . C o o r d i n a t e C o n v e n t i o n ..................................................................... 20

5 . B l o c k - C e n t e r e d G r i d S y s t e m ..................................................... 36

6 . B l o c k M a t r i x Form o f t h e P r e s s u r e E q u a t i o n s( 1 9 - B a n d - D i a g o n a l m a t r i x ) .......................................................... 45

7 . F lo w C h a r t o f t h e S i m u l a t i o n P r o g r a m ........................... 66

8 . F low C h a r t o f PSOR S u b r o u t i n e ................................................ 7 2

9 . Random G e n e r a t i o n o f F r a c t u r e d N odes byM onte C a r l o M e t h o d ......................................................................... 83

10- F r a c t u r e 50% W a t e r S a t u r a t i o n P r o f i l e( C e n t e r C r o s s - S e c t i o n ) ............................................................... 99

1 1 . M a t r i x 50% W a t e r S a t u r a t i o n P r o f i l e( C e n t e r C r o s s - S e c t i o n ) ............................................................... 100

1 2 . W a t e r - O i l R a t i o v e r s u s t i m e ..................................................... 102

1 3 . F r a c t u r e 50%-an d M a t r i x 40%- W a t e r S a t u r a t i o nP r o f i l e s ( a f t e r Thoma'S e t a l e x a m p l e ) ...................................... 103

14 . W a t e r E q u a t i o n i n Rock M a t r i x and R e l a t i v eM a g n i t u d e s o f i t s C o e f f i c i e n t s .......................................... 106

1 5 . O i l E q u a t i o n i n Rock M a t r i x a n d R e l a t i v eM a g n i t u d e s o f i t s C o e f f i c i e n t s .......................................... 107

1 6 . W a t e r E q u a t i o n i n F r a c t u r e and R e l a t i v eM a g n i t u d e s o f i t s C o e f f i c i e n t s .......................................... 108

1 7 . O i l E q u a t i o n i n F r a c t u r e and R e l a t i v eM a g n i t u d e s o f i t s C o e f f i c i e n t s .......................................... 109

X

FIGURE Page1 8 . F r a c t u r e 50% W a t e r S a t u r a t i o n P r o f i l e

( C e n t e r C r o s s - S e c t i o n ; W a t e r I n j e c t i o nR a t e = 4000 S T B / d a y ) ...............................................................................112

1 9 . M a t r i x 50% W a t e r S a t u r a t i o n P r o f i l e ( C e n t e r C r o s s - S e c t i o n ; W a t e r I n j e c t i o nR a t e = 4000 S T B / d a y ...............................................................................113

1.1 Fracture 50% Water Saturation Profile(center cross-section) ........................... 233

1.2 Matrix 50% Water Saturation Profile(Center cross-section)............ 234

J.l Fracture 50% Water Saturation Profile(Center cross-section) ........................... 237

J.2 Matrix 50% Water Saturation Profile(Center cross-section) ........................... 238

X I

CHAPTER I

INTRODUCTION

N u m e r i c a l s i m u l a t i o n i s r e c o g n i z e d a s a v a l u a b l e t o o l

f o r r e s e r v o i r e n g i n e e r i n g s t u d i e s . S i m u l a t i o n e n a b l e s one t o

f o r e c a s t f i e l d p e r f o r m a n c e u n d e r a v a r i e t y o f p r o d u c t i o n

s c h e m e s . An o i l f i e l d c a n o n l y b e p r o d u c e d one t i m e ; and

o n c e t h e p r o d u c t i o n sch em e i s s e l e c t e d , i t i s o f t e n i r r e

v e r s i b l e . The a d v a n t a g e o f r e s e r v o i r s i m u l a t i o n l i e s i n t h e

f a c t t h a t t h e s i m u l a t i o n m o d e l c a n be r u n many t i m e s a t

r e l a t i v e l y low c o s t , a l l o w i n g e n g i n e e r s t o e x p e r i m e n t w i t h

s e v e r a l s e t s o f i n p u t d a t a . O b s e r v a t i o n o f m o d e l r e s u l t s

h e l p s e n g i n e e r s c h o o s e t h e b e s t s e t o f w e l l l o c a t i o n s and

p r o d u c t i o n schem es f o r t h e r e s e r v o i r .

U n d e r s t a n d i n g t h e f l u i d f l o w b e h a v i o r i s p a r t i c u l a r l y

e s s e n t i a l f o r t h e s u c c e s s f u l d e v e l o p m e n t o f a n a t u r a l l y f r a c

t u r e d r e s e r v o i r . I t i s e s t i m a t e d t h a t s u b s t a n t i a l am oun t o f

o i l r e s e r v e s l i e s u n d e r g r o u n d i n n a t u r a l l y f r a c t u r e d

r e s e r v o i r s . B e c a u s e o f t h e u s u a l l y low p r i m a r y p r o d u c t i o n

f r o m t h e s e r e s e r v o i r s a n d t h e i n c r e a s i n g s c a r c e n e s s o f new

o i l f i e l d s , t h e a p p l i c a t i o n o f E n h a n c e d O i l R e c o v e r y p r o c e s s e s

i n n a t u r a l l y f r a c t u r e d r e s e r v o i r s i s now r e c e i v i n g a g r e a t e r

a t t e n t i o n f r o m r e s e a r c h e r s . Y e t , t h e f l u i d f l o w phenomema

2i n s u c h r e s e r v o i r s a r e n o t a d e q u a t e l y u n d e r s t o o d ; a n d c o n

s i d e r a b l y more r e s e a r c h n e e d s t o be d o n e i n t h i s a r e a .

H e n c e , t h e r e i s a n e e d f o r d e v e l o p i n g new m a t h e m a t i c a l

s c h e m e s t h a t e n c o m p a s s many c o m p le x f e a t u r e s o f t h e s e r e s e r

v o i r s .

C o n s e q u e n t l y , t h e p u r p o s e o f t h i s s t u d y i s t o d e v e l o p

a t h r e e - d i m e n s i o n a l , f i n i t e - d i f f e r e n c e s i m u l a t i o n scheme

b a s e d on a new m a t h e m a t i c a l m o d e l f o r d e s c r i b i n g m u l t i p h a s e

f l u i d f l o w i n t h e n a t u r a l l y f r a c t u r e d p o r o u s m e d i a .

CHAPTER I I

LITERATURE REVIEW

A n a t u r a l l y f r a c t u r e d r e s e r v o i r c a n b e c o n s i d e r e d a s

a s y s t e m o f tw o p h y s i c a l d o m a i n s . The p r i m a r y r o c k m a t r i x

w h i c h c o n t a i n s l a r g e am o u n t o f f l u i d s h a s a r a t h e r low p e r

m e a b i l i t y ; a n d t h e f r a c t u r e w h i c h c o n s t i t u t e s a s m a l l vo lum e

b u t h a s t h e a b i l i t y t o t r a n s m i t a l a r g e p o r t i o n o f f l o w

t h r o u g h t h e p o r o u s m ed ium . As a r e s u l t , r e s e a r c h e r s t e n d t o

d e s c r i b e t h e f r a c t u r e d r e s e r v o i r a s a d o u b l e - p o r o s i t y medium.

The s t u d y o f t h e n a t u r a l l y f r a c t u r e d r e s e r v o i r h a s b e e n t h e

s u b j e c t o f n u m e r o u s p a p e r s o v e r t h e p a s t s e v e r a l y e a r s .

A c c o r d i n g t o W a r r e n and R o o t^ ^ t h e d o u b l e - p o r o s i t y

medium h a s tw o c l a s s e s o f p o r o s i t y . The p r i m a r y p o r o s i t y i s

i n t e r g r a n u l a r a n d c o n t r o l l e d by d e p o s i t i o n a n d l i t h i f i c a t i o n .

The v o i d s y s t e m s o f s a n d s , s a n d s t o n e s a n d o o l i t i c l i m e s t o n e s

a r e t y p i c a l o f t h i s c l a s s . The s e c o n d a r y p o r o s i t y , on t h e

o t h e r h a n d , i s f o r a m e n u l a r and i s c o n t r o l l e d by f r a c t u r i n g ,

j o i n t i n g a n d / o r s o l u t i o n i n c i r c u l a t i n g w a t e r . V u g s , j o i n t s ,

f i s s u r e s o r f r a c t u r e s w h i c h o c c u r i n f o r m a t i o n s s u c h as

s h a l e , s i l t s t o n e , s c h i s t , l i m e s t o n e o r d o l o m i t e a r e t y p i c a l

o f t h i s c l a s s o f p o r o s i t y .

The a c t u a l n a t u r a l l y f r a c t u r e d r e s e r v o i r may b e v i s u a l

i z e d a s shown i n F i g u r e 1 . Kazemi e t a l ^ ^ , D u g u id and Lee^,

F i g u r e 1 . A c t u a l n a t u r a l l y f r a c t u r e d r e s e r v o i r .

522 Q i n

Thomas e t a l , a n d E v a n s h a v e shown t h a t a n y m a t h e

m a t i c a l m o d e l f o r i t m u s t i n c l u d e t h e f l u i d i n t e r a c t i o n

t e r m s t o d e s c r i b e t h e m a ss f l u x o f f l u i d f r o m t h e p r i m a r y

r o c k m a t r i x i n t o t h e f r a c t u r e . F u r t h e r m o r e , s i n c e t h e f l o w

o f f l u i d i n t h e f r a c t u r e may be d e s c r i b e d a s c o n t i n u u m f l o w , i t

i s p o s s i b l e t h a t t u r b u l e n t f l o w may o c c u r when a b n o r m a l l y

l a r g e f r a c t u r e s a r e f o u n d i n t h e v i c i n i t y o f l a r g e p r e s s u r ey

g r a d i e n t s s u c h a s i n t h e e a r l y p h a s e s o f w e l l d raw d o w n . The

t e r m t u r b u l e n t f l o w i m p l i e s t h a t D a r c y ’ s l a w i s n o t a d e

q u a t e t o d e s c r i b e t h e f l o w p h e n o m e n a . When f l o w d e v i a t e s

f r o m t h e D a r c y ’ s l a w , t h e F o r c h h e i m e r e q u a t i o n i s n o r m a l l y

u s e d t o d e s c r i b e t h e f l o w p h e n o m e n a .

The d e v e l o p m e n t o f f r a c t u r e f l o w m o d e l s h a s p r o c e e d e d

a l o n g two d i f f e r e n t a p p r o a c h e s . T h e s e a r e t h e s t a t i s t i c a l9 10a p p r o a c h a n d t h e e n u m e r a t i v e a p p r o a c h . ’ The s t a t i s t i c a l

a p p r o a c h c h a r a c t e r i z e s a f r a c t u r e d r o c k m a ss a s a s t a t i s t i

c a l l y h o m o g e n eo u s medium c o n s i s t i n g o f a c o m b i n a t i o n o f

f r a c t u r e s and p o r o u s r o c k m a t r i x . The p r o b a b i l i t y o f f i n d i n g

a f r a c t u r e a t a n y p o i n t i n t h e s y s t e m i s t h e same a s f i n d i n g

one a t a n y o t h e r p o i n t . The e n u m e r a t i v e a p p r o a c h , h o w e v e r ,

a t t e m p t s t o m o d e l t h e a c t u a l g e o m e t r y o f b o t h t h e f r a c t u r e s

and t h e p o r o u s r o c k m a t r i x . The l o c a t i o n , o r i e n t a t i o n , and

a p e r t u r e w i d t h o f f r a c t u r e s become v e r y i m p o r t a n t i n t h i s

a p p r o a c h . An e x t e n s i v e d i s c u s s i o n o f b o t h a p p r o a c h e s a s9 10w e l l a s r e f e r e n c e s a r e r e p o r t e d by E v a n s . ’

S i n c e t h e t i t l e o f t h i s s t u d y d e a l s w i t h t h e n u m e r i c a l

6s i m u l a t i o n o f f r a c t u r e d r e s e r v o i r , a b r i e f r e v i e w o f m a t h e

m a t i c a l d e v e l o p m e n t on t h i s s u b j e c t seem s a p p r o p r i a t e and

s e r v e s t h e f u r t h e r p u r p o s e o f l a y i n g o u t t h e b a c k g r o u n d t h a t

l e d t o t h i s s t u d y .

A few m o d e l s h a v e e m e r g e d d u r i n g t h e p a s t two d e c a d e s

t o d e s c r i b e t h e b e h a v i o r o f n a t u r a l l y f r a c t u r e d r e s e r

v o i r s . M ost n o t a b l y a r e t h e m o d e l s p r o p o s e d b y W a r r e n and

R o o t ^ ^ i n 1963 a n d K a ze m i^ ^ i n 1 9 6 9 . B o th o f t h e s e m o d e l s

a l s o d e s c r i b e t h e f r a c t u r e d r e s e r v o i r a s a d o u b l e - p o r o s i t y

m edium .

I n t h e W a r r e n a n d R o o t m o d e l t h e m a t r i x r o c k c o n t a i n i n g t h e

p r i m a r y p o r o s i t y i s h o m o g e n e o u s a n d i s o t r o p i c and i s c o n

t a i n e d w i t h i n a s y s t e m a t i c a r r a y o f i d e n t i c a l , r e c t a n g u l a r

b l o c k s . S u p e r i m p o s e d on t h i s s y s t e m i s an o r t h o g o n a l s y s t e m

o f c o n t i n u o u s , u n i f o r m f r a c t u r e s w h i c h a r e o r i e n t e d s u c h

t h a t e a c h f r a c t u r e i s p a r a l l e l t o one o f t h e p r i n c i p a l a x e s

o f p e r m e a b i l i t y ( S e e F i g u r e 2 ) . F l u i d f l o w i n t h e r e s e r v o i r

o c c u r s t h r o u g h t h e f r a c t u r e s , w h i c h a r e a n i s o t r o p i c , w i t h

l o c a l e x c h a n g e o f f l u i d s b e t w e e n t h e f r a c t u r e s y s t e m and

m a t r i x b l o c k s , b u t f l o w c o m m u n i c a t i o n b e t w e e n m a t r i x b l o c k s

d o e s n o t o c c u r .

The Kazemi m o d e l d i f f e r s f r o m t h e W a r r e n and R o o t

m o d e l i n t h a t t h e r e s e r v o i r c o n s i s t s o f a s e t o f u n i f o r m l y

s p a c e d h o r i z o n t a l m a t r i x l a y e r s w i t h a s e t o f h o r i z o n t a l

f r a c t u r e s a s s p a c e r s (S e e F i g u r e 3 ) . The f r a c t u r e s a r e

a r r a n g e d h o r i z o n t a l l y w h e r e b y f l u i d s f l o w f r o m m a t r i x l a y e r s

%7TTyZ=Z%Z

y z

XZ==/Z

F i g u r e 2 . I d e a l i z a t i o n o f a n a t u r a l l y f r a c t u r e d p o r o u s medium (W ar ren and Root m o d e l ) .

F i g u r e 3 . I d e a l i z a t i o n o f a n a t u r a l l y f r a c t u r e d p o r o u s medium (Kazemi m o d e l ] .

9i n t o t h e f r a c t u r e s a n d t h e n t h e f l o w c o n v e r g e s r a d i a l l y

t o w a r d t h e w e l l b o r e . T h i s m o d e l i s u s e d m a i n l y i n p r e s s u r e

t r a n s i e n t a n a l y s i s .

N u m e r i c a l s i m u l a t i o n s t u d i e s o f f r a c t u r e d r e s e r v o i r s25h a v e b e e n r e p o r t e d by s e v e r a l r e s e a r c h e r s . Yamamoto e t a l

d e v e l o p e d a g a s - o i l c o m p o s i t i o n a l m o d e l o f a s i n g l e m a t r i x

b l o c k w i t h a f r a c t u r e system l o c a t e d a l o n g t h e m i d - d e p t h

o f t h e b l o c k . The f r a c t u r e r e p r e s e n t e d t h e b o u n d a r y c o n d i

t i o n s a r o u n d t h e b l o c k a n d f l o w a l o n g t h e f r a c t u r e was n o t

c o n s i d e r e d . R e c o v e r y m e c h a n i s m s f o r v a r i o u s s i z e b l o c k s w e re

s t u d i e d .

Kazemi e t a l ^ ^ u s e d t h e W a r r e n a n d R o o t m o d e l t o d e -

v e l o p a t h r e e - d i m e n s i o n a l s i m u l a t o r f o r a s i n g l e p h a s e o r

t w o - p h a s e f l o w o f w a t e r a n d o i l i n f r a c t u r e d r e s e r v o i r s .

T h e i r m o d e l s a c c o u n t e d f o r i m b i b i t i o n , g r a v i t y e f f e c t s ,

r e l a t i v e m o b i l i t y an d v a r i a t i o n s i n r e s e r v o i r p r o p e r t i e s .

The b l o c k G a u s s - S e i d e l m e t h o d was u s e d t o o b t a i n s o l u t i o n s .

R e s u l t s o f t h e s i m u l a t i o n r u n s w e r e g i v e n f o r a q u a d r a n t o f

a f i v e - s p o t p a t t e r n a n d a f i v e - w e l l r e s e r v o i r h a v i n g a d i p

a n d n a t u r a l w a t e r i n f l u x .

R o s s e n ^ ^ p r e s e n t e d t h e s i m u l a t i o n o f n a t u r a l l y f r a c

t u r e d r e s e r v o i r w i t h s e m i - i m p l i c i t s o u r c e t e r m s . The m o d e l

t r e a t e d o n l y t h e f l o w i n t h e f r a c t u r e s y s t e m w h i l e c o n s i d e r

i n g t h e r o c k m a t r i x b l o c k s a s s o u r c e t e r m s . The s o u r c e

t e r m s w e r e f u n c t i o n s o f t h e r o c k m a t r i x and t h e f l u i d p r o

p e r t i e s , w i t h f r a c t u r e s a t u r a t i o n s an d p r e s s u r e s d e f i n i n g

10t h e b o u n d a r y c o n d i t i o n s . M a t r i x b l o c k s s u r r o u n d e d by g a s

w e r e a s s u m e d n o t t o t r a n s f e r w a t e r f r o m t h e m a t r i x r o c k t o

t h e f r a c t u r e and m a t r i x b l o c k s s u r r o u n d e d by w a t e r w e r e

a s s u m e d n o t t o t r a n s f e r g a s f r o m t h e m a t r i x r o c k t o t h e f r a c

t u r e .O

D u g u id and Lee s t u d i e d t h e f l o w o f a s i n g l e - p h a s e

f l u i d t h r o u g h f r a c t u r e d p o r o u s m e d i a . T h e i r g o v e r n i n g

e q u a t i o n s a r e d e r i v e d f r o m t h e D a r c y ' s l a w f o r f l u i d f l o w

i n t h e p r i m a r y p o r e s , e q u a t i o n s o f m o t i o n f o r f l u i d f l o w i n

t h e f r a c t u r e s , an d tw o c o n t i n u i t y e q u a t i o n s . The r e s u l t s

w e r e two s e t s o f g o v e r n i n g e q u a t i o n s , one f o r e a c h t y p e o f

p o r o s i t y . T h e s e two s e t s o f e q u a t i o n s w e r e c o u p l e d b y t h e

f l o w i n t e r a c t i o n t e r m s b e t w e e n f l u i d i n t h e p r i m a r y p o r e s

a n d f l u i d i n t h e f r a c t u r e . The f i n i t e - e l e m e n t G a l e r k i n

m e t h o d was u s e d t o s o l v e t h i s c o u p l e d s y s t e m o f e q u a t i o n s

f o r t r a n s i e n t f l o w o f . w a t e r i n a c o n f i n e d l e a k y a q u i f e r .

D u g u id and L e e ' s w o rk was s i g n i f i c a n t i n t h a t i t a t t e m p t e d

t o m o d e l s i m u l t a n e o u s l y t h e f l o w i n b o t h t h e m a t r i x r o c k and

t h e f r a c t u r e , a n d i n t h e i r r e c o g n i t i o n t h a t a c c e l e r a t i o n

t e r m s m i g h t h a v e an e f f e c t on t h e e q u a t i o n o f m o t i o n i n t h e

f r a c t u r e .2 2Thomas e t a l d e s c r i b e d t h e d e v e l o p m e n t o f a t h r e e -

d i m e n s i o n a l , t h r e e - p h a s e m o d e l f o r s i m u l a t i n g w a t e r , o i l and

g a s f l o w i n a n a t u r a l l y f r a c t u r e d r e s e r v o i r . The r e s e r v o i r

was a s s u m e d t o b e c o m p r i s e d o f a c o n t i n u o u s f r a c t u r e s y s t e m

f i l l e d w i t h d i s c o n t i n u o u s m a t r i x b l o c k s . T h e r e f o r e , f l o w i n

11t h e r e s e r v o i r o c c u r r e d t h r o u g h f r a c t u r e s y s t e m w i t h l o c a l

t r a n s f e r o f f l u i d s b e t w e e n t h e f r a c t u r e s and t h e m a t r i x

b l o c k s ; an d t h e r e was no c o m m u n i c a t i o n b e t w e e n t h e m a t r i x

b l o c k s . As a r e s u l t , t h e g o v e r n i n g e q u a t i o n s w e r e d e r i v e d

f r o m t h r e e f r a c t u r e f l o w e q u a t i o n s , one f o r e a c h p h a s e ,

c o u p l e d w i t h t h r e e m a t r i x - f r a c t u r e f l o w t e r m s . The m a t h e

m a t i c a l f o r m u l a t i o n was i m p l i c i t i n p r e s s u r e , w a t e r s a t u r a

t i o n , an d g a s s a t u r a t i o n o r s a t u r a t i o n p r e s s u r e f o r b o t h t h e

f r a c t u r e f l o w a n d t h e m a t r i x - f r a c t u r e f l o w . A g e o m e t r i c a l

f a c t o r was u s e d i n t h e m a t r i x - f r a c t u r e f l o w t e r m s t o a c c o u n t

f o r t h e s u r f a c e a r e a o f t h e m a t r i x b l o c k s p e r u n i t v o lu m e

a n d a c h a r a c t e r i s t i c l e n g t h a s s o c i a t e d w i t h t h e t e r m s .

H y s t e r e s i s on r e l a t i v e p e r m e a b i l i t y and c a p i l l a r y p r e s s u r e s

a s w e l l a s t h e v a r i a t i o n o f t h e g a s - o i l i n t e r f a c i a l t e n s i o n

w e r e i n c o r p o r a t e d i n t o t h e m o d e l . S e v e r a l e x a m p l e s w e r e g i v e n

t o d e m o n s t r a t e t h e u t i l i t y o f t h e i r s i m u l a t i o n m o d e l .

B o s s i e - C o d r e a n u e t a l ^ d e s c r i b e d a new a p p r o a c h f o r

m o d e l in g f l u i d f l o w i n f u l l y o r p a r t i a l l y f r a c t u r e d r e s e r

v o i r . The a p p r o a c h a l s o a s s u m e d t h a t c o n t i n u o u s f l o w p a t h s

e x i s t e d o n l y i n t h e f r a c t u r e s y s t e m ; an d t h e f l o w i n t h e

m a t r i x r o c k was a c c o u n t e d f o r by t h e m a t r i x - f r a c t u r e f l o w

t e r m s . The m a t r i x - f r a c t u r e f l o w s w e r e d i r e c t l y c a l c u l a t e d

by t h e m o d e l , t a k i n g i n t o a c c o u n t c a p i l l a r i t y , g r a v i t y ,

p r e s s u r e d i f f e r e n c e b e t w e e n t h e m a t r i x b l o c k s and f r a c t u r e s

a n d p r e s s u r e g r a d i e n t s a c r o s s t h e m a t r i x b l o c k s . The b a s i c

f e a t u r e o f t h e i r m ode l was t h e c h e s s b o a r d a r r a n g e m e n t o f t h e

12f r a c t u r e an d m a t r i x c e l l s : e v e r y m a t r i x c e l l was s u r r o u n d e d

by t h e f r a c t u r e c e l l s . An exam ple was g i v e n f o r s i m u l a t i o n o f

a r e s e r v o i r c r o s s s e c t i o n w i t h b o t t o m w a t e r i n j e c t i o n .

I n su m m a ry , s i m u l a t i o n o f m u l t i p h a s e f l o w t h r o u g h

n a t u r a l l y f r a c t u r e d r e s e r v o i r s i s a c h a l l e n g i n g t a s k and

t h e s e e x i s t i n g m o d e l s r e p r e s e n t s i g n i f i c a n t a d v a n c e s t o w a r d th e

u n d e r s t a n d i n g o f t h i s p h e n o m e n o n . I t i s b e l i e v e d , h o w e v e r ,

t h a t c e r t a i n p h y s i c a l a s p e c t s m u s t a l s o b e t a k e n i n t o c o n

s i d e r a t i o n t o a d e q u a t e l y d e s c r i b e t h i s b e h a v i o r . F o r i n

s t a n c e , f l u i d s do f l o w i n b o t h t h e f r a c t u r e s and t h e m a t r i x

r o c k ; a n d n o t a l l t h e f r a c t u r e s a r e i n t e r c o n n e c t e d . I n a d d i

t i o n , b e c a u s e o f h i g h p e r m e a b i l i t y , f l o w c o n d u c t i v i t y i n t h e

f r a c t u r e s i s much h i g h e r t h a n t h a t i n t h e m a t r i x r o c k .

C o n s e q u e n t l y , f r a c t u r e f l o w s o m e t i m e s r e s e m b l e s f l o w i n a

c o n d u i t i n w h i c h t h e a c c e l e r a t i o n ( o r i n e r t i a l t e r m ) p l a y s

an i m p o r t a n t r o l e i n g o v e r n i n g t h e f l o w . T h u s , t h e u s e o f

D a r c y ' s l a w f o r f l o w i n f r a c t u r e s may b e i n a d e q u a t e .9 10R e c e n t l y , E v a n s ’ h a s i n t r o d u c e d a g e n e r a l m a th e m a

t i c a l m o d e l f o r m u l t i p h a s e f l o w t h r o u g h n a t u r a l l y f r a c t u r e d

r e s e r v o i r s w h i c h a d d r e s s e s t h e a b o v e s h o r t c o m i n g s . I t i s

b e l i e v e d t o be t h e m o s t g e n e r a l a n a l y t i c a l m ode l p u b l i s h e d

t o d a t e . The m o d e l t r e a t s f r a c t u r e s a s a g e n e r a l s t a t i s t i c a l

e n t i t y t h a t e x i s t s t h r o u g h o u t t h e r e s e r v o i r . T h i s p e r m i t s

f l o w t h r o u g h t h e p r i m a r y r o c k m a t r i x and t h e f r a c t u r e s t o b e

d e s c r i b e d s i m u l t a n e o u s l y w i t h a f l u i d i n t e r a c t i o n t e r m

c o u p l i n g t h e f l o w b e t w e e n t h e r o c k m a t r i x b l o c k s and t h e

13f r a c t u r e s . The e q u a t i o n s a r e g e n e r a l i n t h a t t h e y a l l o w

t i m e a n d s p a c e v a r i a t i o n s i n a l l f l u i d and r o c k p r o p e r t i e s .

The p r i m a r y o b j e c t i v e s o f t h i s d i s s e r t a t i o n a r e t o

d e r i v e f r o m t h e a n a l y t i c a l w o rk o f E v a n s a s e t o f d i f f u -

s i v i t y f l o w e q u a t i o n s and t h e n d e v e l o p a f i n i t e - d i f f e r e n c e ,

t h r e e - d i m e n s i o n a l m u l t i p h a s e s i m u l a t o r f o r f l o w o f f l u i d s

t h r o u g h n a t u r a l l y f r a c t u r e d r e s e r v o i r s .

CHAPTER I I I

MATHEMATICAL FORMULATION

3 . 1 B a c k g r o u n d a n d A s s u m p t i o n s

The new m a t h e m a t i c a l m o d e l a s p r o p o s e d by E v a n s ^ ’ ^^

f o r d e s c r i b i n g b e h a v i o r o f t h e n a t u r a l l y f r a c t u r e d r e s e r

v o i r s i s b a s e d u p o n t h e s t a t i s t i c a l a p p r o a c h . E v a n s i n t r o d u c e s

t h e f r a c t u r e m a t r i x f u n c t i o n f ( T , J 2 ) t o r e p r e s e n t t h e d i s t r i

b u t i o n o f f r a c t u r e s t h r o u g h o u t t h e r e s e r v o i r s y s t e m . F u r

t h e r m o r e , h e shows t h a t t h e i n t r i n s i c p e r m e a b i l i t y t e n s o r

f o r t h e f r a c t u r e i s r e l a t e d t o t h e f r a c t u r e m a t r i x

f u n c t i o n . H e n c e , t h e p r e s e n c e o r a b s e n c e o f f r a c t u r e a t an y

p o i n t i n t h e r e s e r v o i r s y s t e m , a s w e l l a s t h e o r i e n t a t i o n

and a n i s o t r o p y o f f r a c t u r e c a n b e a c c o u n t e d f o r by m a n i p u

l a t i n g e l e m e n t s i n t h e p e r m e a b i l i t y t e n s o r f o r t h a t p o i n t .

W i t h r e g a r d t o t h e m o d e l , t h e f o l l o w i n g a s s u m p t i o n s

a r e m a d e .

1 . The r e s e r v o i r i s t r e a t e d a s a d o u b l e p o r o s i t y

m ed ium . One p o r o s i t y i s a s s o c i a t e d w i t h t h e

p r i m a r y r o c k m a t r i x a n d t h e s e c o n d p o r o s i t y i s

a s s o c i a t e d w i t h t h e f r a c t u r e s .

2 . The r o c k i s an e l a s t i c i n c o m p r e s s i b l e s o l i d ; a n d

t h e c h a n g e s i n t h e r o c k m a t r i x and f r a c t u r e

14

15p o r i s i t i e s a r e due t o r e o r d e r i n g a n d / o r r e o r i e n

t a t i o n o f r o c k g r a i n s .

3 . The p r i m a r y r o c k m a t r i x i s i s o t r o p i c and h e t e r o

g e n e o u s , w h e r e a s t h e f r a c t u r e s a r e a n i s o t r o p i c and

h e t e r o g e n e o u s .

4 . F l u i d v e l o c i t i e s i n b o t h t h e p r i m a r y p o r e s a n d t h e

f r a c t u r e s a r e a s s u m e d t o b e s m a l l .

5 . P r i m a r y r o c k m a t r i x p o r e v o lu m e i s i n d e p e n d e n t o f

f r a c t u r e p r e s s u r e an d c o n v e r s e l y f o r t h e f r a c t u r e

p o r e v o l u m e .

O t h e r a s s u m p t i o n s a r e i n t r o d u c e d a s n e c e s s a r y .

3 . 2 Summary o f G o v e r n i n g E q u a t i o n s

The e q u a t i o n s n e c e s s a r y f o r d e r i v a t i o n s o f t h e m u l t i

p h a s e f l o w e q u a t i o n s a r e s u m m a r i z e d b e l o w f r o m E v a n s '

p a p e r . ^

3 . 2 . 1 C o n t i n u i t y E q u a t i o n s :

F r a c t u r e :

V - i J C i j f P j f V ^ j ] % - A^-Ay-Az

+ y E [ * f I C i j f P j f S j f ] = 0

R ock M a t r i x :^ i s ^ d s l

* " i j l ^ j l ^ l j ^ * ' Ax'AyAz

1 6

3 . 2 . 2 E q u a t i o n s o f M o t i o n

F r a c t u r e :

V' f j ■

Rock M a t r i x :

“ j f

t . ^ . V z l i^3 » j l

[VPI j

w h e re

”31 ^

K K KXX xy xz

K K Kyx y y yzK K Kzx zy zz

^ V d ]c

3 . 2 . 3 F l u i d I n t e r a c t i o n T e rm s :

g..

" j l[CP

- B - °® ■ B •: : - 2 l P r . e h

n = l

o r t h e s t e a d y s t a t e c o n d i t i o n :

6

w h e re

P j i

. V r l ü î 2

2 S, n j

3 . 2 . 4 P o r o s i t y E q u a t i o n s :

F r a c t u r e :

3fT t " " O f C l - O f j C f " â t

a P f 3P.

Rock M a t r i x :

1734>1T T

3Pi 3P.$ ^ ( 1 a t a t

3 . 2 . 5 R e q u i r e d A u x i l i a r y E q u a t i o n s

S a t u r a t i o n s :

®gf ■" ®wf + S q f = 1

®gl " S„ 1 + S o l = 1

C a p i l l a r y P r e s s u r e s :

^ c g o f ^ f g ^ f o

^ c o w f ~ ^ f o ^ fw

^ c g o l = P

P c o w l = P

C o n s e r v a t i o n o f m ass o f e a c h f l u i d p h a s e :

( r o c k m a t r i x )I C i g l '

I Ciol =

Ï C,i w l

P i

I = 1

g f

o f

I Cii w f

3 . 3 B l a c k O i l Mode

l o

o ■ ? l w

( f r a c t u r e )

( r o c k m a t r i x )

( f r a c t u r e )

( f r a c t u r e )





( f r a c t u r e )

( f r a c t u r e )

( f r a c t u r e )

The m u l t i p h a s e f l u i d f l o w i n p o r o u s medium r e f e r s t o

18t h e s i m u l t a n e o u s f l o w o f t h r e e f l u i d p h a s e s : g a s , o i l , and

w a t e r p h a s e s . I n a f u l l y g e n e r a l c o m p o s i t i o n a l m o d e l t h e r e

a r e N c h e m i c a l s p e c i e s o r c o m p o n e n t s , e a c h o f w h i c h may e x i s t

i n any o r a l l o f t h e t h r e e p h a s e s . H o w e v e r , t h i s v e r y g e n e r a l

c o m p o s i t i o n a l m o d e l i s e x t r e m e l y d i f f i c u l t t o s e t up and

s o l v e . A c c o r d i n g l y , a s i m p l i f i e d b l a c k o i l m o d e l ( o r tw o -

co m p o n e n t h y d r o c a r b o n s y s t e m ) i s u s e d i n t h i s s i m u l a t i o n

s t u d y . I t i s a s s u m e d t h a t n o m ass t r a n s f e r o c c u r s b e t w e e n t h e

w a t e r p h a s e , and t h e g a s and o i l p h a s e s . T h e r e a r e t h r e e com

p o n e n t s i n v o l v e d : w a t e r , o i l an d g a s c o m p o n e n t s ( i . e . , t h e

h y d r o c a r b o n s y s t e m h a s o n l y two c o m p o n e n t s : o i l an d g a s ) .

The o i l c o m p o n e n t i s t h e r e s i d u a l l i q u i d a t a t m o s p h e r i c p r e s

s u r e l e f t a f t e r a d i f f e r e n t i a l v a p o r i z a t i o n an d t h e g a s compo

n e n t i s t h e r e m a i n i n g f l u i d . To a v o i d c o n f u s i o n , t h e c a p i t a l

l e t t e r s u b s c r i p t s w i l l b e u s e d t o i d e n t i f y c o m p o n e n t s ( i ' s )

and t h e l o w e r - c a s e s u b s c r i p t s w i l l b e u s e d t o i d e n t i f y p h a s e s

( j ' s ) . F u r t h e r , t h e s u b s c r i p t S w i l l b e u s e d t o i n d i c a t e

s t a n d a r d c o n d i t i o n s .

L e t Pq 2 b e t h e d e n s i t y o f t h e o i l c o m p o n e n t , a n d Pgg

be t h e d e n s i t y o f t h e g a s c o m p o n e n t , b o t h m e a s u r e d a t s t a n d a r d

c o n d i t i o n s . I f t h e f o r m a t i o n v o lu m e f a c t o r f o r e a c h p h a s e i s

d e n o t e d by , a n d t h e d i s s o l v e d g a s - o i l r a t i o b y i t can

b e d e m o n s t r a t e d t h a t t h e r e s p e c t i v e m ass f r a c t i o n s i n t h e

g o v e r n i n g e q u a t i o n s f o r t h e b l a c k o i l m o d e l a r e :

F o r f r a c t u r e f l o w :

C|vgf = 0 ; Ô g f " ° ’ ^G gf " ^

19r = 0 • c - ÔSf . p _ * s o f P G s f

^Ww£ = 1 ; CQwf = 0 ; = 0

F o r r o c k m a t r i x f l o w :

^Wgl " ° ’ Ô gl " ° ' ^ G g l " ^

r = n . r = - ^ O S l . r - ^ s o l ^ G S l^Wr»1 ^ 9 ^ 0 ^ 1 T) A 9'Wol " • _ - O o l B o i O o i ' " G o l - B o i P o i

''Wwl ” ^ ' ‘'Owl ° ° ’ ‘'Gwl ' ®

F u r t h e r m o r e ,

and

P&Sl " GSf " ^GS

PQSl " PQSf PQS

PWSI " WSf " ^WS

_ PCSF _ C^sofPGSf^PoSF) . _ PwSfpgf - ’ Pof b;; ' ^

_ ^GSL . _ _ (*solPGSl+PoSl) . . _ Pwsip g i - B g , ' P o l - ' ' ^ 1 " " : ; z

3 . 4 T i l t e d R e s e r v o i r

The e f f e c t s o f g r a v i t y on f l u i d f l o w i n a t i l t e d r e s e r

v o i r a r e m a n i f e s t i n t e r m s o f VD i n t h e g o v e r n i n g e q u a t i o n s .

I n t h e p r e s e n t c o o r d i n a t e c o n v e n t i o n , a s shown i n F i g u r e 4 ,

t h e X, Y an d Z a r e p r i n c i p a l a x e s o f t h e r e s e r v o i r s t r a t u m ,

and D i s t h e v e r t i c a l e l e v a t i o n f r o m d a t u m p l a n e ( p o s i t i v e

d o w n w a r d ) . L e t ’ s d e f i n e ( s e e F i g u r e 4)

a = a n g l e o f t h e x a x i s w i t h t h e h o r i z o n t a l p l a n e

B = a n g l e o f t h e y a x i s w i t h t h e h o r i z o n t a l p l a n e

20

X

' »

F i g u r e 4 . C o o r d i n a t e c o n v e n t i o n .

21Y = angle of the z axis with the vertical plane

H e n c e ,

D = - x s i n a - y s i n g - z c o s y

And s i n c e

Thus ,

I f °

W ' ■

# = - C O S Y

F o r t h e s p e c i a l c a s e when t h e r e s e r v o i r i s h o r i z o n t a l :

o t = 0 , 6 = 0 , Y = 0 .

Thus ,

I E _ 0

3X

l E _ 0

37

^ =_1 32

3 . 5 D e r i v a t i o n o f t h e F low E q u a t i o n s

3 . 5 . 1 Rock M a t r i x

The c o n t i n u i t y e q u a t i o n f o r r o c k m a t r i x i n i t s g e n e r a l

f o r m i s w r i t t e n a s :

’ • t I + p j -

22f o r t h e c o m p o n e n t s : i = o i l , w a t e r , and g a s . T h i s e q u a t i o n

may b e w r i t t e n a s :

^ ' [ ^ ^ g l ^ g l ^ l g ^ ^ i o l * ^ o l ^ l o ^ ^ i w l ^ w l ^ l w ] ^ [^g ^ ô ^ ^wî

A x 'A y 'A z = q

( 2 )

Upon s u b s t i t u t i o n o f t h e a p p r o p r i a t e m ass f r a c t i o n s

( C ^ j ) f o r t h e b l a c k o i l m o d e l we h a v e :

F o r W a t e r Component

V. [ ^ ^ - - - ^ 3 + [T^ + r + r ^ ] - W j a s iJ I g o W'W A x * A y A z

f t = 0

F o r O i l Component

Pos „ PQSPOSIl o ] ^ g ^ o ^ w 0 A x * A y A z

. <j> 1 P S .

^ à : 1 = 0 C4)

F o r Gas Component

A - 0 C5)•’■ ^ g l ^ o l

A t t h i s p o i n t i t i s a s s u m e d t h a t w a t e r i s an i n c o m p r e s

s i b l e f l u i d ( i . e . , B^^ = B^^ = 1 and t h u s = p ^ f = P ^ ^ ) •

F u r t h e r , t h e f l u i d i n t e r a c t i o n t e r m s a r e a s s u m e d t o b e s t e a d y

s t a t e and f o l l o w t h e f o r m :

23

[ r j l i = - P f j ] m

A l t h o u g h t h e m ass f l o w r a t e f r o m t h e r o c k m a t r i x i n t o

t h e f r a c t u r e i s c o n s i d e r e d i n i t s o r i g i n a l fo r m t o b e u n s t e a d y ,

t h i s a s s u m p t i o n i s j u s t i f i e d b y t h e f a c t t h a t t h e i n t e r a c t i o n

e q u a t i o n show s t h e r a t e a p p r o a c h i n g s t e a d y s t a t e a s t i m e b e

comes l a r g e r . T h u s , u p o n s u b s t i t u t i o n o f a p p r o p r i a t e C ^ j ' s

f o r t h e b l a c k o i l m o d e l , i t f o l l o w s t h a t :

F o r W a t e r C om ponen t

c r g l „ - 0 C7)

t f o l w ' 0 C8)


[ T g ] o = 0 ( 1 0 )

t r „ I o = 0 ( 1 2 )

F o r Gas C om ponent

l i g l G ' ' " i * ■

( i . ' G = " ’i ° ‘

[ r „ ] g = 0 ( 1 5 )

T h e s e f l u i d i n t e r a c t i o n t e r m s t o g e t h e r w i t h t h e e q u a

t i o n s o f m o t i o n i n r o c k m a t r i x a r e s u b s t i t u t e d i n t o t h e

24r e s p e c t i v e c o n t i n u i t y e q u a t i o n s .

F o r W a t e r C om ponent

+ = — C16)Ax*Ay*Az ^1 9 1


" ' ' ’ ” 1 0 - ” 0 1 1 ’ ” ” ■ ‘ S t l ï l ' ” 1 0 - ” f o ' ’

* A x ^ y . A z ° *1 ^ ' ^ ° l ^ ” ° l ’ (17 )o ^

F o r Gas Component

^ l ^ r l g s'C

- ' ” i o - ” i o ’ >

" A x ^ T A z ' * 1 I t ' S g i / B g i * R s c l S o i / B o i ) d S )

3 . 5 . 2 F r a c t u r e

S i m i l a r t o t h e r o c k m a t r i x , t h e c o n t i n u i t y e q u a t i o n

f o r f r a c t u r e i n i t s g e n e r a l f o r m c a n b e w r i t t e n a s :

^ Âx*Ay*AZ I ^ i j f ^ j f ^ j f ^

= 0 (19)

25The s u b s t i t u t i o n o f t h e a p p r o p r i a t e m ass f r a c t i o n s

f o r t h e b l a c k o i l m ode l p r o d u c e :

F o r W a t e r C om ponent

[ B.wf

P WS%S1 A x * A y Az

I t [ * f P w s S w f / B w f ] - 0


V • [ - [r„ + + r, j

( 2 0 )

Pos ^ o s i

o f g W-'O Ax*Ay*Az

" I t [ * f P o s S o f / B o f ] ■ 0

F o r Gas C om ponen t

( 21 )

- [ r , . r „ . r j g -‘w G Ax « Ay ' AZ

[ * f ( p G s S g ^ / B g 2 + R s o f P c s S o f / B o f ) ] " ° ( 2 2 )

The e q u a t i o n o f m o t i o n i n t h e f r a c t u r e c o n t a i n s t h e t e n s o r

a s w e l l a s t h e v e c t o r t e r m s , a n d c a n b e w r i t t e n i n t h e fo r m

b e l o w ( f o r p h a s e s : j = o i l w a t e r a n d g a s ) •

Kxx V %xzS P f i

ax + P j f 1

§ ca V f I x

a t

k r j f« j f V S y * y z

•a P f .

ay + P j f1

Sc a t

• ' . y

aPf^.az + P j f

1

Sca V f j z

a t

26

* ^XZ ' t F ^ * 1 7 “ â f ^ ■ ‘’ j f ^ ü ”

? ! f i * ! ü ü f ü . p _s_ W )32 g^ 3 t g^ 32 ' '^\ z <3i Sj. 0 I- J i ë c

" " z .C - “ j f

Upon s u b s t i t u t i o n o£ t h e a b o v e e x p r e s s i o n s a n d t h e

f l u i d i n t e r a c t i o n t e r m s i n t o t h e r e s p e c t i v e c o n t i n u i t y e q u a

t i o n s f o r w a t e r , o i l and g a s c o m p o n e n t s t h e f r a c t u r e e q u a t i o n s

( 2 0 , (2 1 ) a n d (2 2 ) becom e

F o r W a t e r Com ponen t

Pwf ^ ^ fw y g_ 3D j ^ K (g^ 9 t Pwf ay '* xz az gç, a t ^wf g^ 3 :

27

* ' S m ^ I '” i«' '’*"■” ' / x”fy-T z ' *f & tS„,/B„,J (24)


— § F - P o f * %xz( ’ >

4- ) " y x ( - ) ^ * g j a ^ ' P o f ^ ) *

Kyy( ■ # " ^ S ^ - " o f ^ )

28

V r l o 4 > f _ . . . . S* ' r i ï u J - ( P l o - P f o ) ] " J x -AY-AZ = f t I t 1 (25 )

F o r Gas C om ponent

■ " g f ^ I f ’ * V ^ I f ” * ® ' " z x ( - #

* ^ ■ " g f ^ H ’ * " g f ^ )

* — ? r ' " o f ^ I f ’ * * x y ^ l 3 ^ * ^ - â f ^ - i ' o f ^ I f ’

* - P . t ^ I f ) } * f ( K y , ( % t M l ^ f o x’c

■ "of I f ’ * ' ^ y y ^ - 3 r ^ * ^ ~ é ^ ' " o f ^ ■ § ’ * ’' y z ' T T

" o f t I f ’ * ( " t x ( % e ^ ^ % ^ - " o f t i f ’

* ■'zyC % * % % - " o f t # ’ * ^ z z (

- P o f t I f ” ” * ' " V * ■ X H i i o i - B o i - " ’iP

■ P f o ’ ] * ISÎ^Psz ' * f Â ( S g f / B g f * P s o f W ® o f > (:«’

f o

293 . 6 The F low E q u a t i o n s i n T h r e e - D i m e n s i o n a l Forms

The f l o w e q u a t i o n s f o r w a t e r , o i l an d g a s c o m p o n e n t s

d e r i v e d i n S e c t i o n 3 . 3 c o n s t i t u t e t h e s i x g o v e r n i n g d i f f u s i -

v i t y e q u a t i o n s f o r m u l t i p h a s e , b l a c k - o i l , f l u i d f l o w i n a

f r a c t u r e d p o r o u s m e d i a . By u t i l i z i n g t h e a u x i l i a r y e q u a t i o n s

o f c a p i l l a r y p r e s s u r e s , s a t u r a t i o n s an d t i l t e d r e s e r v o i r ,

t h e s e s i x e q u a t i o n s c a n b e w r i t t e n i n t h r e e - d i m e n s i o n a l f o r m s

a s f o l l o w s .

I t s h o u l d b e p o i n t e d o u t t h a t t h e p a r a m e t e r s i n t h e s e

e q u a t i o n s h a v e t h e i r d i m e n s i o n s i n S I u n i t s . H o w e v e r , t h e

i n p u t d a t a a r e u s u a l l y o b t a i n e d i n p r a c t i c a l o i l f i e l d u n i t s .

A c c o r d i n g l y , some c o n v e r s i o n f a c t o r s h a v e t o b e i n c l u d e d . The

c o n v e r s i o n f a c t o r s b e t w e e n t h e s e two u n i t s y s t e m s a r e shown i n

A p p e n d i x A. The c o m p a r i s o n s o f t h e e q u a t i o n s u n d e r t h e SI

u n i t s and t h e o i l f i e l d u n i t s a r e g i v e n i n A p p e n d i x B.

3 . 6 . 1 The W a t e r D i f f u s i v i t y E q u a t i o n i n t h e M a t r i x Rock

* ' f e i * • Ï 5 1 •“ » f t " - " '

• f t " " ' '

3 . 6 . 2 The O i l D i f f u s i v i t y E q u a t i o n i n t h e M a t r i x Rock

S i n c e we a s s u m e t h a t w a t e r i s i n c o m p r e s s i b l e , t h i s

l a s t t e r m b e c o m e s z e r o ( — =®^lw

30

* * ' ( % * ^ o l ^ " » Y } ] - ( P i c - P f o î I * Â # ^

' h 'b ^ " S o l - â ^ 4 r l (28 )

3 . 6 . 3 The Gas D i f f u s i v i t y E q u a t i o n i n t h e M a t r i x Rock

^ è * P„ 1 ^ s i n e , ) .

- ' " i : - ' ÿ% A r - ' A &

( 2 9 )

3 . 6 . 4 The W a t e r D i f f u s i v i t y E q u a t i o n i n t h e F r a c t u r e s

31

^WS ^V fw z

®w£^c 9 t

^ws ^V fw xB „ £ 8 c 9 t

P ws ^V fw y

^ w £^c 9 t

P ws a V fw z

^w£®c 9 t

P WS BVfwx

^w£®c a t

P WS ^^£w y

B w fS c 9 t

P WS ^Vfwz

^ _o_ , ^ r w f ^ x z r®w£^w£ 3"

- * ' % 7 & ' ~ '

P WS

®w£ S c

WS _&_®w£ S c

p WS _&_\f S c

Pws

\f S c

Pws ,

B w f S c '

+ 9 r ^ r w f ^ z x , ^ ^ f w ^ _______

3 : B«f%wf B « f : c

+ r ^ l ^ r l w ^ £ , p p + ^WSf'■ Iw £wJJ A x ' A y A z

3 . 6 . 5 The O i l D i £ £ u s i v i t y E q u a t i o n i n t h e F r a c t u r e s

9 r^ r o £ ^ x >» ‘S J i ï V

+ r ^ r o f ^ x :

' W ' K f ' o i f

3P£o . P o l ^ ^ f o y n ' _g_By Sc 9 t ô l Sc

» f £ o , P o l ^ '^ foz ,0 ' _g_

9z Sc 9 t ô l Sc

3P£o ., P o l ®Vfox , n9x Sc 9 t Pol Sc

9P,

V r l o ^ f _ ,p _,p , _ 5 o S f _*■ ^h 1S.'o 2Vqj lo fo- AxAyAz

= , r 1 *Spf 3Pfo9t of 9P^q 9t (31)

3.6.6 The Gas Diffusivity Equation in the■Fractures

9x ^

_ 9 _ r ^ r g f ^ x y 9x BgfUgf

• È ' K f

. J . [ ^r8f*^y](ay BgfU gf

. a , ^TsfKyy 3y BgfWgf

, "Gs PCs §ay Bgfgc 9t Sr

, "Gs ^^fgz PGs gaz ®gf*c 9t ^g£ Sc

®*’fe , ' Gs pGs ga=t 9t Sc

®’’fE . "Gs '""fgy ?.Gs gay B^fgc 9t Sc

3 z

33 z

33x

33x

33x

33y

33y

_ 3ay

39z

33z

33z

^ r g f ^ y z r ^ ^ f g +

B g f ^ g f 9 :

^ r g f ^ z x { 3 P f g +

B

^ r g f ^ z y |-^ ^ f gB

g f : " g f

^ r g f ^ z z j- ^ f g +

B g f P g f 3z

33

P gs

Bgf®c 3 t

PGs ^ ^ f g xB g fS c 3 t

PGs ' " " f g y* g f S c 3 t

^Gs ^ ^ f g zB - f S cg]

^ s o f ^ r o f ^ x xo f

^ s o f ^ r o f ^ x z® o f ^ o f

^ s o f ^ r o f ^ z xB o f ^ o f

o f ' ' o f

^ s o f ^ r o f ^ z zB o f ^ o f

3P f o + ^ o l

3 t

3V

PGs

" g £ §c

PGs _g_" g £ Sc

PGs _g_®g£ Sc

PGs _H_®g£ Sc

c o s y ) ]

s i n g } ]

f o x

^ s o f ^ r o f ^ x y r

B o f % f

3x

3P f o

{

3 y

3P f o

B - s o f ^ r o f ^ y x ^

3 z

3P f o^ o f ^ o f

^ s o f ^ r o f ^ y y < B o f ^ o f

3x

3P f o

^ s o f ^ r o f ^ y z r B o f ^ o f

3y

3P f o

{

^ s o f ^ r o f ^ z y r B - ^

3 z

3x

3P f o3y

3P{■

f o3z

Sc 3 t-f-

^ o l Sc

P o l ^V foy + oSc 3 t P o l Sc

P o l ^ V fo z gSc 9 t P o l Sc

% 1 a V fox + n .g.Sc 3 t P o l Sc

% 1 ^V foy + n _g_Sc 3 t P o l Sc

% 1 aV fo z +Sc 3 t P o l Sc

% 1 a V fo x + _g_Sc 3 t P o l Sc

P. o l aV foy + n gSc 3 t P o l Sc

P o l SVfoz +Sc 3 t P o l Sc

34

^ J V r l g f f , p _ p 5 + V r l o i f ^ (p _ p . ‘’ G s f‘ r n l U g i B g i '■ I g fg- ' l o f o J i x - A y - A z

, r * s o f * S o f ^ ^ o f S * s o f 3 P f o . c p B P f oB o f 8 t a P f o ' 9 t S o f * s o f 3 t

(3 2 )

E q u a t i o n s (27 ) , ( 2 8 ) , ( 2 9 ) , (3 0 ) , (3 1 ) a n d (3 2 ) t h u s

f o r m t h e m a t h e m a t i c a l m o d e l t h a t s i m u l a t e s t h e f l u i d f l o w

b e h a v i o r i n n a t u r a l l y f r a c t u r e d r e s e r v o i r s . I n t h e n e x t

c h a p t e r , d e v e l o p m e n t o f t h e s i m u l a t i o n p r o g r a m b a s e d on t h i s

m o d e l w i l l b e d i s c u s s e d .

CHAPTER IV

SIMULATOR DEVELOPMENT

4 . 1 F i n i t e D i f f e r e n c e A p p r o x i m a t i o n s

The s i m u l a t i o n p r o g r a m i n t h i s s t u d y i s w r i t t e n u s i n g

t h e FORTRAN IV c o m p u t e r c o d e . The p r o g r a m i s c o m p i l e d and

r u n on t h e IBM s y s t e m 3081 (m o d e l D) m a c h i n e . I n o r d e r t o

d e v e l o p a c o m p u t e r p r o g r a m t h e d e r i v a t i v e s i n t h e g o v e r n i n g

f l o w e q u a t i o n s h a v e t o b e r e p l a c e d b y s u i t a b l e f i n i t e d i f f e r

e n c e a p p r o x i m a t i o n s . T h i s i n v o l v e s d i s c r e t i z a t i o n o f t h e

r e s e r v o i r s p a t i a l d o m a in i n t o a n u m b e r o f g r i d p o i n t s ( o r

n o d e s ) by e m p l o y i n g , i n t h i s c a s e , t h e b l o c k - c e n t e r e d r e c t a n

g u l a r g r i d s y s t e m (S ee F i g u r e 5 ) . The t i m e d o m a in i s a l s o

d i v i d e d i n t o a nu m b er o f t i m e s t e p s d u r i n g e a c h o f w h i c h t h e

s y s t e m o f l i n e a r e q u a t i o n s i s s o l v e d t o o b t a i n new v a l u e s o f

d e p e n d e n t v a r i a b l e s . The s o l u t i o n t o t h e e q u a t i o n s a t an y

p a r t i c u l a r t i m e s t e p t h u s c o n s i s t s o f a s e t o f n u m b e r s o b

t a i n e d f o r a l l g r i d p o i n t s w i t h i n t h e c o n t i n u o u s medium u n d e r c o n

s i d e r a t i o n .

The f i n i t e d i f f e r e n c e a p p r o x i m a t i o n s a r e e x p r e s

s i o n s d e r i v e d f r o m T a y l o r s e r i e s a n d t h e r e l a t i o n s h i p s

b e t w e e n t h e d i f f e r e n c e an d t h e d i f f e r e n t i a l o p e r a t o r s . I n

t h i s s t u d y , t h e d e r i v a t i v e s i n t h e s i x g o v e r n i n g f l o w e q u a

t i o n s c a n b e a p p r o x i m a t e d by t h e f i n i t e d i f f e r e n c e f o r m u l a e

35

36

# # # •

T‘iT

• • • •

# # # • • • • •TA Z

• • #

--------"-X

• • •

T o p V i e w S i d e V i e w

i-1, j . k / i . j .k■X

' - 1 1 + 1 , L k -

F i n i t e D i f f e r e n c e N o t a t i o n

F i g u r e 5 . B l o c k - c e n t e r e d g r i d s y s t e m

37g i v e n i n T a b l e 1.

The f u l l y i m p l i c i t a p p r o a c h i s u s e d t o s o l v e f o r t h e

new p r e s s u r e v a l u e s a t a l l g r i d p o i n t s s i m u l t a n e o u s l y . As

d e m o n s t r a t e d i n C h a p t e r I I I , t h e c a p i l l a r y p r e s s u r e e q u a t i o n s

t o g e t h e r w i t h t h e s a t u r a t i o n s e q u a t i o n s a r e e m p lo y e d t o e l i m

i n a t e a n d f r o m t h e s i x p a r t i a l

d i f f e r e n t i a l e q u a t i o n s r e s u l t i n g i n s i x e q u a t i o n s c o n t a i n i n g

t h e u nknow ns P ^ ^ , P £ ^ , P ^ ^ an d P ^ ^ . T h e s e s i x e q u a

t i o n s w r i t t e n f o r a l l n o d e s i n t h e g r i d s y s t e m a r e t o b e s o l v e d

by o n e o f t h e n u m e r i c a l m e t h o d s , p r e f e r a b l y an i t e r a t i v e m e t h o d23o v e r a d i r e c t m e th o d . The c a l c u l a t e d p r e s s u r e s a r e t h e n u s e d

i n t h e c a p i l l a r y p r e s s u r e a n d s a t u r a t i o n e q u a t i o n s t o c o m p u te

s a t u r a t i o n s . T h i s a p p r o a c h i s re c o m m en d e d b y F a r o u g A l i ^ ^ a s

b e i n g m ore s t a b l e a n d a c c u r a t e t h a n t h e c o n v e n t i o n a l I m p l i c i t -2 3P r e s s u r e - E x p l i c i t - S a t u r a t i o n (IMPES) a p p r o a c h . Thomas a l s o

s t a t e s t h a t t h i s a p p r o a c h p r o v i d e s a s m o o t h v a r i a t i o n o f u n

knowns i n s p i t e o f s a t u r a t i o n d i s c o n t i n u i t i e s . H o w e v e r , t h i s

a p p r o a c h r e q u i r e s n o n z e r o c a p i l l a r y p r e s s u r e d a t a . M o r e o v e r ,

t h e d e g r e e o f a c c u r a c y d e p e n d s p a r t i a l l y on t h e w ays t h e c o e f

f i c i e n t s o f t h e e q u a t i o n s a r e e v a l u a t e d . The s u b j e c t o f c o e f

f i c i e n t e v a l u a t i o n w i l l b e d i s c u s s e d i n t h e n e x t s e c t i o n .

The u s e o f f o r m u l a e s i m i l a r t o t h e o n e s i n T a b l e 1 t o

t r a n s f o r m t h e s i x g o v e r n i n g f l o w e q u a t i o n s i n t o s i x f i n i t e

d i f f e r e n c e e q u a t i o n s i s a v e r y cu m b er so m e a n d p a i n s t a k i n g

t a s k . F o r i n s t a n c e , t h e n u m b er o f c o e f f i c i e n t s i n v o l v e d i n

t h e s e e q u a t i o n s a m o u n t s t o 126 t e r m s , e a c h o f w h i c h i s c o m p r i s e d

o f a g r o u p o f p a r a m e t e r s . And s i n c e t h e g o v e r n i n g f l o w

38

T a b l e 1

F i n i t e D i f f e r e n c e F o r m u l a e

Ui+l , i ,k 'Ui ,1 ,k _ ,k-%i-l. i ,k9 r, 9u i, _ i + h , i Ax i - 3 s , 1 , k Ax

----------------

, Ui+l,1+l,k'Ui+l,1-l,k _ , Ui-l,i+l,k-Ui-l,i-l,k_9_ n ^ i+l,i ,k_________2Ax_____________ i - l . i ,k_________2Ax__________9x 9y -

• [ +1 •; V + 1 T + 1 V"î+l i -14AxAyi i + l , j , k ^ i + l , j + l , k i + l , j - l , k '

Su ^ ^ i , 1 , k ' ^ i , i , k9 t At

Su ^ U j + i j , k " U i - l , 1 ,k 9x 2 Ax

39e q u a t i o n s a r e n o n l i n e a r p a r t i a l d i f f e r e n t i a l e q u a t i o n s , e a c h o f

t h e s e c o e f f i c i e n t s h a s t o b e r e c o m p u t e d e v e r y t i m e s t e p . I n

t h i s c o n t e x t , d e v e l o p m e n t o f t h e f r a c t u r e d r e s e r v o i r s i m u l a t o r

i s d i f f i c u l t an d c h a l l e n g i n g .

The f i n i t e d i f f e r e n c e f o r m s o f t h e s i x g o v e r n i n g f l o w

e q u a t i o n s f r o m s e c t i o n 3 . 6 c o n s t i t u t e a s y s t e m o f s i m u l t a n e o u s

l i n e a r e q u a t i o n s a n d c a n b e w r i t t e n a s f o l l o w s . The e x p r e s

s i o n s f o r t h e c o e f f i c i e n t s o f t h e s e e q u a t i o n s a r e g i v e n i n

d e t a i l i n A p p e n d i x C.

i-l,j,k i+l,j,k

" * ( F " ) i . j . k p ? » ' " ' ™ ’ i . : . ki + l , j , k i , j + l , k i , j , k + l

" W i . j . k p ? ; ' * W i . j . k ' “ ' ' i . j . k (33 )i , j , k i , j , k

i . j . k p ; ; ' ' ( G O ) i . j . k p î o ' " ( p o ) i . j . k p % 'i , j , k i , j , k - l i , j - l , k

* t 'ô’ i . j . k p % ' " ( p o ) i . j . k p ; ; ' " ' “ ’ i . j . k p " : 'i - l , j , k i ,j-,k i t l , j . ,k

" (PO’ i . j . k ' (HO) i . j . k P ? r * '“ ’ i . j . k P?;'i , j + l , k i , j , k + l i , j , k

( * 0 ) i , j , k P f o ^ = DOi j % (3 4 ]

i , j ,k

40

C * * (E G O ) l , j , k F Î : '

i , j , k i , j , k - l i , j - l , k i - l , j , k

* * ( F G O ) . _ j _ ^ P j ; l

i , j , k i + l , j , k i , j + l , k

i , j , k + l i , j , k - l i , j - l , k

* i . j . k * ( : G ) i . j , k i . j . k " u 'i - l , j , k i , j , k i + l , j , k

* ( F G ) i . j . k " (« S ’ l . j . k ^ ( R G O ) i . j . ki , j + l , k i , j , k + l i , j , k

" f ’' " ’ i . j . k = ° = i . j , k (3 5 )i . j . k

( F R W ) i . j . k + ( F G W l ) i . j . k ♦ CF™2) i . j . k P îw "i , j , k i , j - l , k - l i - l , j , k - l

+ ( f G * 3 ) i . j . k " (FG"4) i . j . ki , j , k - l i + l , j , k - l

+ ( r G " 5 ) i . j . k P ? » " " (F C "1) i . j . k c ; » 'i , j + l , k - l i - l , j - l , k

" (FEW2 ) i _ j _ k P ? ; 3 + ( F E « 3 ) i . j . k P f « '

i , j - 1 , k i + 1 , j - 1 , k

41

i - l , j , k i , j , k i + l , j , k

* C F F W D i j . k f f w ' * ( F F " Z ) i . j . ki - l , j + l , k i , j + l , k

" ( F P W 3 ) i , j , k " (M W l) i . j . ki + l , j + l , k i , j - l , k + l

. (FHW2 ) j _ . _ k P ? ; ^ + ( P H W 3 ) l . j . k

i - l , j , k + l i , j , k + l

" CFHW4) j _ . _^ P ? ; 1 * ( P H " ' 5 3 i , j . k f f C '

i + l , j , k + l i , j + l , k + l

" ( F Ü W ) l . j . k = '’™ i , j , k

i J ,k

i . j . k p ; ; ' ' ( F v o j , p p ; i * ( F G o i ) i _ j _ ^ p ? ; i

i , j , k i , j , k i , j - l , k - l

* ( f G O Z ) i . j . k " ( f C 0 3 ) i . j . k *’? o "i - 1 , j , k - l i , j , k - l

+ ( F G 0 4 ) i . j . k " ( F G O S j i j t P ? ; l

i + l , j , k - l i , j + l , k - l

' ( ™ i 5 i . j . k i . j . ki - l , j - l , k i , j - l , k

* ( F E O S ) i . j . k P ? o " " ( F A O l ) i . j . k p ; : 'i + l , j - l , k i - l , j , k

42

" ( F B O ) i . j . k P f l ' " ( F C O l ) i . j . ki , j , k i + l , j , k i - l , j + l , k

* ( f r 0 2 ) i . i , k *’? o ^ * P j ; li , j + l , k i + l , j + l , k

* ( F H O l ) i . j . k " ( ™ 0 2 ) i , j , k

i , j - l , k + l i - l , j , k + l

+ c m 0 3 ) i , j , k " ( f H 0 4 ) i , j , ki , j , k + l i + l , j , k + l

+ (rH05)i,j,k * <™5i,j,k = DfOi,i.k (37)i , j + l , k + l i , j , k

( f % G o ) i . j . k i . j . k * ( f V G ) i , j , k p ; ; 'i ,j ,k i ,j ,k i ,3 ,k

" (FGRl)i.j.k " (FGK2)i.j.k PÎo"i ,j-1,k-l i-1jj,k-l

. ( F G R 3 ) i _ j _ k P j ; l + ( F G R 4 ) i _ j _ t Pg^^

i,j,k-l i+l,j,k-l

. ( F G R 5 ) i _ j _ k P ; ; l + C F E R l ) i _ . _ ] , P j ^ l

i,j+l,k-l i-l,j-l,k

* (FER2)l.j.k * (FER3)i.j.ki,j-l,k i + l,j-l,k

* (fAFl)i.j.k ’’îo" " (fSK)i.j.k ï’îo' * (FCM)i.j.ki-l.j.k i.j.k 1+1.3.R

43

+ * ( F F R 2 ) i . j . ki - l , j + l , k i , j + l , k

* ( ™ 3 ) i , j , k ^To"- "

i+l':+l'k i.j-l.k+1

* ( F H R 2 ) i . : , k P f o ^ " ( F " R 3 ) i . j . ki - l , j , k + l i , j , k + l

* CFHR4)i^._]^ P%;1 * ( m R 5 ) i , i . k P f l '

i + 1 , j , k + l i , j + 1 , k + l

* t ' ^ G ^ h . j . k P ? ; l

i , j - l , k - l i - l , j , k - l

* CFGG33i_-_^ - ( F G G 4 ) , _ j _ ^ P ? ; l

+ ( f G G S ) i _ j _ t P ? ; l + ( F C G l ) i . j . k P # ; '

i , j + l , k - l i - l , j - l , k

» ( F E G 2 ) i J . k ' ■ ? ; ' " P ? ; li , j - l , k i + l , j - l , k

* p g ; : ^ C B O i J . k

i - l , j , k i , j , k

' f p “ i ) i j , k " ( f f g i 3 j _ . _ , P , ^ * 1

i + l , j , k i - 1 , j + 1 , k

44

, n + l n+1* » f g

i + 1 , j + 1 , k

' i . j . ki , j - l , k + l i - 1 , j , k + l

, n + l n+1* i . j . k P f g " * ( 'H G ") i . j . k P f g

i , j , k + l i + l , j , k + l

= DFG..i , j ,ki , j + 1 , k + l

(38 )

T h e s e e q u a t i o n s , w r i t t e n f o r a l l N n o d e s i n t h e r e s e r

v o i r g r i d s y s t e m , p r o d u c e a b l o c k 1 9 - b a n d - d i a g o n a l c o e f f i c i e n t

m a t r i x f o r 6N s i m u l t a n e o u s l i n e a r e q u a t i o n s i n 6N unknow ns

w h i c h may be s o l v e d by d i r e c t o r i t e r a t i v e m e t h o d s . The s y s

tem o f e q u a t i o n s c a n b e w r i t t e n a s f o l l o w s i n b l o c k m a t r i x

fo rm [S ee F i g u r e 6 ) ,

w h e r e

= * i , j , k

^ ^ i , j , k ^ ^ i , j , k * W i , j , k

^ ° i , j , k ® ° i , j , k ^ ° i , j , k

^ ^ i , j , k ® ^ ° i , j , k ® ^ i , j , k

* ° i , j , k

* G O i , j , k * G i , j , k

FRO;i , j , k

™ ° i , j , k ™ i , j , l f ^ ^ i , j , k ™ i , j , k ™ i , j , k _

H 3m H5 X.

X .

X N

D 1

a

D N

45»on

F i g u r e 6 . B lo ck m a t r i x form o f t h e p r e s s u r e e q u a t i o n s ( 1 9 - b a n d d i a g o n a l m a t r i x )

46

C G O i , j , k C G i , j , k

V C i , j , k '

V \ j , k

A ° i , j , k

^ ^ ° i , j , k ^ ^ i , j , k

F A R l i ^ j ^ k ™ : l J , k

^ " i , j , k

l ° i , j , k

' l 2 i , j , k

' ^ ° i , 3 , k

% ,j ,k

47

0

“ r = “ i,j,k

0

0

48

G V G 3 i , j , k =

( ^ r = G 4 i , j , k =,k

FGR4, i , j ,kfGG4i^j,k

0

0FGW5.i ,5 ,k

FGR5i_.^^FGG5i_j_^

49

00

EZr ' E 2 i , j . k =,k

^ r = : " i j . k =0

™ i j . k

50

0

00

F V f 2 i , j . k =

™ : i , j . k

FFR2. . ,FFG2. .

"VRi.j.k=0

51

0

r » i i , 3 , i c

Ï H R l i , j , k * H , j , l c

0

0

Ï H 0 2 i , j , k

H W lJ .k

52

00

0

0

4 . 2 E v a l u a t i o n o f C o e f f i c i e n t s

The c o e f f i c i e n t s o f t h e d e r i v a t i v e s i n a n o n l i n e a r

p a r t i a l d i f f e r e n t i a l e q u a t i o n , i n a d d i t i o n t o b e i n g f u n c t i o n s o f

t h e i n d e p e n d e n t v a r i a b l e s , a r e a l s o f u n c t i o n s o f t h e d e p e n d e n t

v a r i a b l e s . F o r t h e p r e s e n t s y s t e m o f e q u a t i o n s , t h e c o e f f i

c i e n t s i n c l u d e p a r a m e t e r s s u c h a s f o r m a t i o n v o lu m e f a c t o r s ,

s o l u b i l i t y , r e l a t i v e p e r m e a b i l i t y , v i s c o s i t y , p o r o s i t y and

f r a c t u r e v e l o c i t y . T h e s e p a r a m e t e r s a r e e i t h e r p r e s s u r e - o r

s a t u r a t i o n d e p e n d e n t . C o n s e q u e n t l y , we a r e p r e s e n t e d w i t h

a d i l e m m a : how c a n we o b t a i n t h e s o l u t i o n t o t h e p r e s s u r e

53e q u a t i o n s when t h e s o l u t i o n d e p e n d s on t h e c o e f f i c i e n t s w h ic h

nt h e m s e l v e s d e p e n d on t h e s e p r e s s u r e s ? C r i c h l o w s u g g e s t s two

ways o u t o f t h i s s i t u a t i o n . F i r s t , t h e s e c o e f f i c i e n t s may be

e v a l u a t e d f r o m t h e p r e s s u r e s o r s a t u r a t i o n s o f t h e p r e v i o u s

t i m e s t e p . I m p l i e d i n t h i s a p p r o a c h i s t h e b e l i e f t h a t t h e s e

c o e f f i c i e n t s do n o t c h a n g e so r a p i d l y f r o m one t i m e s t e p t o

t h e n e x t . T h i s w o u ld b e t r u e i f p r e s s u r e s and s a t u r a t i o n s a r e

a l s o n o t c h a n g i n g s o r a p i d l y . S e c o n d , f o r an i t e r a t i v e m e t h o d ,

t h e s e c o e f f i c i e n t s a r e e v a l u a t e d f r o m t h e m o s t c u r r e n t p r e s s u r e s

o r s a t u r a t i o n s o f t h e l a s t i t e r a t i o n w i t h i n t h e same t i m e s t e p .

I n t h i s s t u d y , t h e f i r s t a p p r o a c h w i t h some m o d i f i c a

t i o n i s u s e d . P a r a m e t e r s s u c h a s [dS / d P ) and ( d S _ /w cow^ ga r e e v a l u a t e d f r o m a n d S ^ , r e s p e c t i v e l y . O t h e r p a r a - cgo^ w S

m e t e r s may b e e v a l u a t e d a c c o r d i n g t o t h e t e c h n i q u e shown by

F a r o u g A l i ^ ^ . T h a t i s t h e p r e s s u r e s a n d / o r s a t u r a t i o n s a r e

a v e r a g e d i n t i m e b e t w e e n t h e l a s t t i m e s t e p v a l u e s and t h e

" p r o j e c t e d " c u r r e n t v a l u e s . T h e n , t h e p a r a m e t e r s , B^,

d ( l / B Q ) d P o , d ( l / B g ) / d P g , d R g ^ / d P ) , u ^ , a n d a r e e v a l u a t e d

f r o m t h e P ^ * ^ v a l u e s , w h e r e a s t h e p a r a m e t e r s k ^ ^ ^

and k ^ „ a r e e v a l u a t e d f r o m t h e o r v a l u e s . Ther g w g" p r o j e c t e d " v a l u e s o f p r e s s u r e s o r s a t u r a t i o n s may b e c o m p u te d

18f r o m t h e f o l l o w i n g f o r m u l a e g i v e n b y S e t t a r i a n d A z i z

C39jA t*

W h e r e a s , t h e a v e r a g e v a l u e s i n t i m e may b e c o m p u te d f r o m

(40)

54S o m e t i m e s , when f l o w b e t w e e n tw o a d j a c e n t g r i d p o i n t s

a r e b e i n g c o n s i d e r e d , t h e p a r a m e t e r s h a v e t o be e v a l u a t e d

f ro m p r e s s u r e s / s a t u r a t i o n s a t m i d p o i n t i n s p a c e b e t w e e n t h o s e

two g r i d p o i n t s . T h e s e m i d p o i n t v a l u e s may b e c o m p u te d f ro m

(411^ i + % , j , k 2

W i th r e g a r d t o r e l a t i v e p e r m e a b i l i t y , t h e u p s t r e a m11 18w e i g h t i n g sch em e i s u s e d ’ . T h i s s c h e m e e v a l u a t e s t h e

r e l a t i v e p e r m e a b i l i t y f ro m s a t u r a t i o n s a t t h e u p s t r e a m n o d e ,

f o r e x a m p l e

^rw*^^w i , j , k ^ f l o w i s f r o m ( i , j , k ) t o

( i + l , j , k )

^rw^^w i + l , j , k ^ i f f l o w i s f r o m ( i + l , j , k ) t o

( i , j , k )

' rw i + h , j , k

The d i r e c t i o n o f f l o w c a n b e d e t e r m i n e d b y c o m p a r i n g t h e

f l o w p o t e n t i a l s o f t h e two g r i d p o i n t s .

T h e r e f o r e , a c o e f f i c i e n t s u c h a s (CO). ■ ■. f ro m A p p e n d i x

C may b e i n t e r p r e t e d

CO. , K l ^ r l oAx^ ^ o l ^ o l

^ r ^1 i + ? s , i , k ^ r l o ^ ^ w l > S g i ^ ) u p s t r e a m]

o l l o i + b , j , k - ' ^ o l l o i + % , j ,k ' '

w h e re

Kl i + J 5 , j , k 2^%1 i , j , k ^1 i + l , j , k ), n + 1 .n

Tjti+h _ l r l o i , i , k l o i , 1 , k . l o i + l , j , k l o i + l , i , k -il o i + h y j , k 2^ 2 2

55

V i oV es ^ ci*-'2 -Vrlo^^wl i,j,k ’ ^gl i,j,kJ

i f $*+% ,Tï+hl o i , j , k - ^ l o i + 1 , j ,k

rlo*- wl i + l , j , k ’ ^ g l i + l , j , k '.n+hi f $ l o i , j ,k < $ l o i , j ,k

F i n a l l y , i n e v a l u a t i n g f l u i d d e n s i t y i t w i l l be a s s u m e d

t h a t d e n s i t y a t t h e b o u n d a r y o f a g r i d b l o c k may be r e p r e s e n t e d

b y d e n s i t y e v a l u a t e d a t t h e c e n t e r o f t h a t g r i d b l o c k , i . e . .

^ i + î 2 , j , k P i - î 2 , j , k P i , i + % , k

P i , j , k - % P i , j , k

P i , i , k + %

4 . 3 B o u n d a r y C o n d i t i o n s

I n t h i s s t u d y , a c l o s e d o r n o - f l o w b o u n d a r y w i l l b e u s e d .

H e n c e , t h e Neumann c o n d i t i o n s a r e s p e c i f i e d on t h e r e s e r v o i r

b o u n d a r i e s .

on 3R (42 )

w h e r e 3R i s t h e r e s e r v o i r b o u n d a r y .

The Neumann c o n d i t i o n s c a n b e i n c o r p o r a t e d i n t o t h e11 23s i m u l a t i o n m o d e l i n tw o d i f f e r e n t w a y s : ’ b y m eans o f r e

f l e c t i o n n o d e s o r by s e t t i n g p e r m e a b i l i t y ( o r t r a n s m i s s i b i l i t y )

t o z e r o on t h e b o u n d a r y n o d e s . I n r e s e r v o i r s i m u l a t i o n , t h e

l a t t e r i s u s e d m ore f r e q u e n t l y . F o r an e x p r e s s i o n s u c h a s

56V ' [ K ' V $ ) , t h e r e q u i r e m e n t V$ = 0 on 3R c a n b e a s w e l l a c c o m

p l i s h e d by s e t t i n g c o e f f i c i e n t K = 0 on 9R.

4 . 4 T h r e e - P h a s e R e l a t i v e P e r m e a b i l i t y

N u m e r i c a l s i m u l a t i o n o f m u l t i p h a s e f l o w r e q u i r e s t h e

u s e o f t h r e e - p h a s e r e l a t i v e p e r m e a b i l i t y d a t a . E f f o r t i n

v o l v e d i n d e t e r m i n i n g t h e s e d a t a e x p e r i m e n t a l l y g e n e r a l l y

r u l e s o u t s u c h a d i r e c t a p p r o a c h . H o w e v e r , S t o n e ^ ^ ’ ^^ h a s

d e v e l o p e d a m e t h o d f o r e s t i m a t i n g t h r e e - p h a s e r e l a t i v e p e r m e a

b i l i t y f r o m tw o s e t s o f m ore e a s i l y o b t a i n e d t w o - p h a s e d a t a .

The r e q u i r e d d a t a c o n s i s t s o f a s e t o f w a t e r - o i l r e l a t i v e

p e r m e a b i l i t y d a t a a n d g a s - o i l r e l a t i v e p e r m e a b i l i t y d a t a . To

a l l o w f o r h y s t e r e s i s , t h e w a t e r and g a s s a t u r a t i o n s s h o u l d be

c h a n g i n g i n t h e same d i r e c t i o n i n t h e two s e t s o f t w o - p h a s e

d a t a a s d e s i r e d i n t h r e e - p h a s e s y s t e m . F o r e x a m p l e , i f b o t h

t h e w a t e r a n d g a s s a t u r a t i o n s a r e e x p e c t e d t o i n c r e a s e i n t h e

t h r e e - p h a s e s y s t e m , t h e n i m b i b i t i o n w a t e r - o i l d a t a a n d d r a i n

a g e g a s - o i l d a t a s h o u l d b e u s e d . H o w e v e r , i f o i l i s d i s p l a c i n g

g a s , f o l l o w e d b y w a t e r d i s p l a c i n g o i l , t h e i m b i b i t i o n d a t a

s h o u l d b e u s e d f o r b o t h t h e g a s - o i l a n d w a t e r - o i l s y s t e m s .

S t o n e ’ s m e th o d i s b a s e d on t h e c h a n n e l f l o w t h e o r y

c o m b in e d w i t h p r o b a b i l i t y c o n c e p t s . I t s t a t e s t h a t w a t e r

r e l a t i v e p e r m e a b i l i t y i s a f u n c t i o n o f w a t e r s a t u r a t i o n a l o n e

r e g a r d l e s s o f w h e t h e r t h e s y s t e m i s t w o - p h a s e o r t h r e e - p h a s e .

The same i s t r u e w i t h g a s r e l a t i v e p e r m e a b i l i t y a s a f u n c t i o n

o f g a s s a t u r a t i o n . T h u s , o n l y o i l r e l a t i v e p e r m e a b i l i t y

n e e d s t o b e r e d e t e r m i n e d i n t h e t h r e e - p h a s e s y s t e m .

S t o n e ' s e q u a t i o n f o r c a l c u l a t i n g o i l r e l a t i v e p e r m e a b i l i t y

57i s g i v e n a s

’" ro ' ( ’' r o w * k rw ) ( ' ' r o g * ’' rg> ‘ O '™ * ' ' r g ’ (43 )

N o te t h a t i f t h e a b o v e e q u a t i o n y i e l d s a n e g a t i v e

i t i m p l i e s c o m p l e t e b l o c k a g e o f o i l , o r k ^ ^ = 0 . T h i s e q u a

t i o n h a s a d e s i r a b l e p r o p e r t y i n t h a t i t y i e l d s t h e c o r r e c t

t w o - p h a s e d a t a when o n l y two p h a s e s a r e f l o w i n g , an d y e t p r o

v i d e s i n t e r p o l a t e d d a t a f o r t h r e e - p h a s e f l o w t h a t i s a c o n s i s

t e n t and c o n t i n u o u s f u n c t i o n o f t h e p h a s e s a t u r a t i o n s .

4 . 5 I n j e c t i o n and P r o d u c t i o n R a t e s

I n d i v i d u a l p h a s e r a t e s o f i n j e c t i o n a r e n o r m a l l y s p e c i

f i e d e x p l i c i t l y , f o r t h e u s u a l p r a c t i c e i s t o i n j e c t one

p h a s e , b e i n g w a t e r o r g a s , i n t o t h e r e s e r v o i r . The r a t e s a t

t h e p r o d u c i n g w e l l s , h o w e v e r , c a n n o t b e s p e c i f i e d s o e x p l i c i t

l y . T h i s i s t r u e b e c a u s e two o r m ore p h a s e s may b e p r o d u c e d

s i m u l t a n e o u s l y a n d t h e r e l a t i v e p r o d u c t i o n s o f t h o s e f l u i d p h a s e s

a r e n o t s u b j e c t t o c o n t r o l . A c c o r d i n g t o Peacem an^^ t h e y

a r e f u n c t i o n s o f s a t u r a t i o n s i n t h e v i c i n i t y o f t h e w e l l and

s h o u l d b e c o m p u te d i n p r o p o r t i o n t o t h e l o c a l m o b i l i t y o f

t h o s e p h a s e s . T h a t i s

% ' Ot i ; (44 )

% = ^ (45 )

Sg = S . 615 (4 6 )

w h e r e = t o t a l m o b i l i t y

58

4 •6 H y d r a u l i c R a d i u s o f a F r a c t u r e C o n d u i t

Ev an s e x t e n d e d t h e w ork o f D u g u id and Lee f o r a s i n g l e

p h a s e f l u i d and d e r i v e d t h e f l u i d i n t e r a c t i o n t e r m s f o r m u l t i

p h a s e f l o w a s shown i n s e c t i o n 3 . 2 . 3 . I n t h e d e r i v a t i o n , h e

i n t r o d u c t e d t h e p a r a m e t e r r ^ t o r e p r e s e n t t h e h y d r a u l i c r a d i u s

o f t h e f r a c t u r e c o n d u i t . C o m p a r i s o n b e t w e e n E v a n s ' and D u g u id

and L e e ' s d e r i v a t i o n s shows t h a t r ^ c a n b e r e l a t e d t o t h e

c h a r a c t e r i s t i c h a l f w i d t h o f t h e f r a c t u r e by

Th = ^ ( 4 7 )

w h e r e d = c h a r a c t e r i s t i c h a l f w i d t h o f t h e f r a c t u r e .

W i th r e g a r d t o t h e l e n g t h o f t h e f r a c t u r e , t h e c h a r a c t e r

i s t i c h a l f l e n g t h I o f t h e f r a c t u r e i s a s s u m e d t o e q u a l t h e

h a l f d i m e n s i o n o f t h e p r i m a r y g r i d b l o c k . T h u s ,

£ = ^ (4 8 )

4 . 7 P o r o s i t y C a l c u l a t i o n

I n t h i s s i m u l a t i o n p r o g r a m , p o r o s i t y w i l l b e e v a l u a t e d

f r o m o i l p r e s s u r e s o f t h e p r e v i o u s t i m e s t e p . The a s s u m p t i o n

h e r e i s t h a t p o r o s i t y v a r i e s v e r y s l o w l y a s a f u n c t i o n o f

p r e s s u r e s , a n d t h u s may b e c o n s i d e r e d c o n s t a n t d u r i n g t h e t i m e

s i n c e t h e p r e v i o u s t i m e s t e p t o t h e p r e s e n t . The o i l p r e s s u r e

i s a s s u m e d t o b e r e p r e s e n t a t i v e o f t h e f l u i d p r e s s u r e i n p o r e

s p a c e s . The j u s t i f i c a t i o n f o r t h i s a s s u m p t i o n i s b a s e d u p o n a

f i n d i n g by C o a t s , e t a l ^ t h a t e v a l u a t i o n o f c e r t a i n p r o p e r t i e s f rom

a s i n g l e p h a s e p r e s s u r e p r o d u c e d i n s i g n i f i c a n t e r r o r .

59T h e r e f o r e , t h e f i n i t e d i f f e r e n c e f o r m s o f t h e p o r o s i t y

e q u a t i o n s i n s e c t i o n 3 . 2 . 4 a r e g i v e n a s f o l l o w s .

F o r t h e r o c k m a t r i x :

, n ^ l ______ _ , n , n % ^^n*^l T,n'f’l i , j , k '*'1 i , j , k ( l " * l i , j , k ) C i ( P i o i , j , k ' ^ l o i , j , k )

" 1 i , j , k * f i , j , k "=f ^ ^ f o " i , j , k “P f o i , j , k )

" * 1 i , j , k C49)

F o r t h e f r a c t u r e s :

, n ^ l _________ , n ^ - 1 , n % r ^ n ^ l ^ n ^* f i , j , k " *^f i , j , k ( l " * f i , j , k ) C f ( P f o i , j , k " P f o i , j , k )

1 n ,n ' ^-..n+l ^ n *\■‘*’1 i , j , k i , j , k J C l ( P l o i , j , k ' ^ l o i , j , k J

,n , n + l >n

+(j)nf i , j , k

(5 0 )

4 . 8 V e l o c i t y C a l c u l a t i o n

The e q u a t i o n o f m o t i o n i n t h e f r a c t u r e f r o m s e c t i o n 3 . 2 . 2

f o r p h a s e j ( j = w a t e r , o i l a n d g a s ) c a n b e w r i t t e n i n x , y

a n d z c o o r d i n a t e s a s f o l l o w s :

^ £ j x y" r i f

^^xx^ 3x a t ^ f g( x y

+ ! if 3P ^ .a t

P j f C O S Y ) ]g

= - ^ U l I K‘i f

g

y . ^ - y x ' ^ Bx ■ g_ a t "P-j f g s im )+ K y y ( gy

(51)

^ ( " j f g ^ ? W ) + K y ^ C - a T +

(52)

60' ^ r i £ i P ; r 3 V r - 9 P r .

^ f j z " 3 x g ^ 9 t ^ j f g ^ s i n a ) +

+ ^ ^ - j f t - z z + Pj £ f ; C P S , ) 3(53 )

I n t h e s i m u l a t i o n p r o g r a m t h e s e v e l o c i t i e s a r e c a l

c u l a t e d e x p l i c i t l y f r o m t h e a b o v e e q u a t i o n s a f t e r p r e s s u r e s

a n d s a t u r a t i o n s h a v e b e e n d e t e r m i n e d . The a s s u m p t i o n i s t h a t t h e

v e l o c i t y c h a n g e s s l o w l y f r o m one t i m e s t e p t o t h e n e x t .

F o r e x a m p l e , t h e f i n i t e d i f f e r e n c e f o r m o f V^. may b er j X

w r i t t e n a s :

i j . k =

, i j . k , i s i n a )

P " * l . . - P Î t l

n+1 _ p n + lP f i i , i , k + l ^ f i i , i , k - l

^xz i , j ,k 2Az

' P?;\ J,k ^ PPPY) I (53)T h u s ,

f jx 1 ,3 ,k V n K Pif ,k... / r i f Pjf ^xx ,n+l

^ j f : c " Â r ) i , i , k

pn+1 _pn+l n+1TK ( i + l , i , k ^fj i - l , j ,_k Pjf i , i , k

XX i , j ,k 2Ax g^^t

61

^ f j x i , j , k " ^ s i * * ) " V i , j , k

V - p f L . V p ? : \ - . . 9 v . ./ f j i , j + l , k i , j - l , k ^ ^ j f i . j . k .n2Ay a t ) i , j , k

* t * " x z

. ^ C O S . ) , (5 4 )

The f i n i t e d i f f e r e n c e f o r m s o f Yn- and Y n . a r e s i m i l a rf j y f j zt o t h a t o f a n d , t h e r e f o r e , w i l l n o t b e shown h e r e .

I t s h o u l d b e p o i n t e d o u t t h a t i n i t s i n i t i a l s t a t e t h e

r e s e r v o i r i s s t a t i c o r u n d i s t u r b e d b y o u t s i d e f o r c e s . C o n s e

q u e n t l y , t h e r e i s n o f l u i d movement w i t h i n t h e r e s e r v o i r .

T h i s i m p l i e s t h a t t h e i n i t i a l f l u i d v e l o c i t y a n d a c c e l e r a t i o n

i n t h e f r a c t u r e s a r e z e r o .

4 . 9 P o i n t S u c c e s s i v e O v e r - R e l a x a t i o n M eth o d ÇPSOR)

PSOR i s one o f t h e v a r i o u s m e t h o d s a v a i l a b l e f o r s o l v i n g

a s y s t e m o f s i m u l t a n e o u s l i n e a r e q u a t i o n s . The d i s c u s s i o n o f

t h i s m e t h o d h a s b e e n w e l l d o c u m e n t e d i n many t e x t b o o k s a n d t h e

l i t e r a t u r e . ^ ^ T h e PSOR m e t h o d i s b a s e d on s o l v i n g

t h e e q u a t i o n s b y s u c c e s s i v e a p p r o x i m a t i o n s ( o r g u e s s e s ) u n t i l

t h e c o n v e r g e n c e o f r e s u l t s t o a p r e s e l e c t e d t o l e r a n c e l i m i t

i s a c h i e v e d o r an a c t i o n d u e t o f a i l u r e o f c o n v e r g e n c e i s

t a k e n .

S e v e r a l r e a s o n s w e r e t a k e n i n t o c o n s i d e r a t i o n i n

62c h o o s i n g t h i s m e th o d o v e r o t h e r o n e s f o r t h e s i m u l a t i o n p r o

g ram d e v e l o p e d i n t h i s d i s s e r t a t i o n .

1 . The PSOR i s a n i t e r a t i v e m e th o d w h i c h i s m ore s u i t

a b l e t h a n a d i r e c t m e t h o d f o r s o l v i n g a l a r g e s e t

o f e q u a t i o n s t h a t h a s a s p a r s e c o e f f i c i e n t m a t r i x

b e c a u s e i t t a k e s a d v a n t a g e o f t h i s s p a r s e n e s s by-

r e q u i r i n g c o m p u t i n g s t o r a g e o n l y f o r t h e n o n z e r o 12e l e m e n t s .

2 . The r o u n d o f f c h a r a c t e r i s t i c s o f an i t e r a t i v e m e th o d12s u c h a s t h e PSOR a r e b e t t e r . W i th a d i r e c t m e t h o d ,

r o u n d o f f e r r o r c a n b e i n c u r r e d w i t h e a c h m a t h e m a t i

c a l o p e r a t i o n an d s i m p l y a c c u m u l a t e s u n t i l t h e f i n a l

a n s w e r s a r e o b t a i n e d . W i th a n i t e r a t i v e m e t h o d ,

h o w e v e r , t h e p r e s e n c e o f r o u n d o f f e r r o r i n t h e

unknow ns a t t h e e n d o f an y i t e r a t i o n s i m p l y r e s u l t s

i n t h o s e unknow ns b e i n g som ew ha t p o o r e r e s t i m a t e s

f o r t h e n e x t i t e r a t i o n . T h u s , t h e r o u n d o f f e r r o r i n

t h e f i n a l c o n v e r g e d v a l u e s i s o n l y t h a t a c c u m u l a t e d

i n t h e f i n a l i t e r a t i o n .

3 . The s y s t e m o f e q u a t i o n s i n t h i s s t u d y h a s t h e fo r m

o f a 1 9 - b a n d d i a g o n a l c o e f f i c i e n t m a t r i x w h i c h i s

much l a r g e r and m o re c o m p l i c a t e d t h a n t h e c o n v e n

t i o n a l p e n t a - d i a g o n a l o r h e p t a - d i a g o n a l m a t r i c e s .

The PSOR i s p r e f e r r e d t o o t h e r i t e r a t i v e m e t h o d s

b e c a u s e i t s f o r m u l a i s s t r a i g h t f o r w a r d and c a n be

i n c o r p o r a t e d i n t o t h e s i m u l a t i o n p r o g r a m w i t h o u t

63a g r e a t d e a l o f d i f f i c u l t y .

The PSOR m e th o d e x p r e s s e s t h e n e x t a p p r o x i m a t i o n s o f

t h e s o l u t i o n a t one g r i d p o i n t e x p l i c i t l y i n t e r m s o f known

e s t i m a t e s o f o t h e r v a r i a b l e s i n an e q u a t i o n . T h u s , i n t h i s

s t u d y e q u a t i o n s ( 3 3 ) , ( 3 4 ) , ( 3 5 ) , ( 3 6 ) , ( 3 7 ) a n d (38 ) a r e

m a n i p u l a t e d i n s u c h a way t h a t t h e 6 unknow ns a t e a c h g r i d

noint pR+1 pn+1 pn+1 pn+1p o i n t ( P i ^ i , j , k ' ^ l o i , j , k ' ^ I g i , j , k ' ^ fw i , j , k ’ ^ f o i , j , k

and j a r e e x p r e s s e d i n t e r m s o f t h e same p h a s e

p r e s s u r e s a t o t h e r g r i d p o i n t s a n d / o r d i f f e r e n t p h a s e p r e s

s u r e s a t t h e same g r i d p o i n t . F o r e x a m p l e , t h e w a t e r f l o w

e q u a t i o n i n r o c k m a t r i x can b e w r i t t e n a s :

k+1 k+1

P l w ^ i , j , k " (BW)J-*■ J J > ^

k+1 k+1

( E W l i , j , k P ^ w ^ i , j - l , k ■ ( A W ^ i , j , k P l w ^ i - l , i , k

k k

k k

k

- CSS)

w h e r e s u p e r s c r i p t k i n d i c a t e s l e v e l o f i t e r a t i o n i n t i m e s t e p

( n + 1 ) .

A f t e r c o m p u t a t i o n a t e a c h g r i d p o i n t , t h e v a l u ek + 1

P ^ * ^ . . , i s r e l a x e d by t h e f o r m u l a :Iw 1 , 1 , k

64n i l k _ k + l k

’’ iw i , i , k ' *’lw i , i , k * “ t ^ l w ^ i , : , k " ( ^ 6 )

w h e r e u i s a r e l a x a t i o n p a r a m e t e r .

The i t e r a t i v e p r o c e s s c o n t i n u e s u n t i l t h e d i f f e r e n c e s

b e t w e e n j k ' ^ o f two s u c c e s s i v e i t e r a t i o n s f o r a l l g r i d

p o i n t s a r e l e s s t h a n a p r e d e f i n e d t o l e r a n c e l i m i t e .

k+1 kpn+1 _ pn+1

Iw i , j , k Iw i , j , k^ E (5 7 )

The same p r o c e d u r e d i s c u s s e d a b o v e a p p l i e s t o P ^ o ’ ^ I g ’

? £ ^ a n d ? £ g a s w e l l . E q u a t i o n s f o r t h e s e u n know ns a r e

s i m i l a r i n f o r m t o t h a t o f P^^., a n d f o r b r e v i t y w i l l n o t b e

shown h e r e .

W i t h r e g a r d t o t h e r e l a x a t i o n p a r a m e t e r w , i t s f u n c t i o n

i s t o s p e e d up t h e c o n v e r g e n c e r a t e . The s e l e c t i o n o f t h e

op t im u m v a l u e o f w u s u a l l y r e q u i r e s some t r i a l - a n d - e r r o r s i m u

l a t i o n r u n s w i t h v a r y i n g i d . The o p t im u m (d i s t h e one t h a t

p r o d u c e s t h e minimum n u m b er o f i t e r a t i o n p e r t i m e s t e p .

G e n e r a l l y , t h e m ore h o m o g e n eo u s t h e s y s t e m , t h e c l o s e r t h e

v a l u e o f CÜ i s t o 1 . 0 , and a s t h e d e g r e e o f a n i s o t r o p y i n

c r e a s e s , t h e m ore oj a p p r o a c h e s 2 . And i n p r a c t i c e , a n o v e r -7

e s t i m a t i o n o f w i s a l w a y s b e t t e r t h a n an u n d e r e s t i m a t i o n .

I n t h i s s t u d y a v a l u e o f w = 1 . 2 was u s e d a n d g a v e a r e a s o n

a b l e r a t e o f c o n v e r g e n c e .

CHAPTER V

ALGORITHM

I n t h i s c h a p t e r , a b r i e f d e s c r i p t i o n o f t h e m a j o r com po

n e n t s o f t h e s i m u l a t i o n p r o g r a m i s p r e s e n t e d . I n p u t d a t a f o r

a c a s e e x a m p l e i s a l s o g i v e n . The f l o w d i a g r a m o f t h e com pu

t a t i o n a l p r o c e d u r e i s i l l u s t r a t e d i n F i g u r e 7.

5 . 1 M a j o r P r o g r a m C om ponen ts

5 . 1 . 1 M a in P r o g r a m

The m a i n p r o g r a m s t a r t s w i t h d e f i n i n g a n d d i m e n s i o n i n g

v a r i a b l e s . P a r a m e t e r s s u c h a s f l o w c o e f f i c i e n t s , f r a c t u r e

v e l o c i t y a n d a c c e l e r a t i o n a r e i n i t i a l i z e d . T h e n , b a s i c d a t a

i s r e a d v i a s u b r o u t i n e s RESDIM, RESDAT, INCON a n d CODE.

C o n s t a n t t e r m s i n t h e t r a n s m i s s i b i l i t i e s o f t h e p r e s s u r e

e q u a t i o n s a r e e v a l u a t e d b y s u b r o u t i n e TRAN. N e x t , t h e p r o g r a m

p r o c e e d s t o c a l c u l a t e i n i t i a l f l u i d s i n p l a c e a n d r e s e r v o i r

p o r e v o l u m e . The s i m u l a t i o n p r o c e s s s t a r t s w i t h t h e r e a d i n g /

e s t a b l i s h i n g o f t i m e s t e p s i z e . S a t u r a t i o n c h a n g e s d u r i n g t h e

p r e v i o u s t i m e s t e p a r e c h e c k e d and a w a r n i n g p r i n t e d when t h e y

becom e s o e x c e s s i v e t h a t t i m e s t e p s i z e may h a v e t o be r e d u c e d

t o s m o o th t h e c h a n g e s . The i n j e c t i o n / p r o d u c t i o n r a t e s a r e

d e t e r m i n e d . The u p d a t i n g o f p r e s s u r e s a n d s a t u r a t i o n s i s d o n e

i n s u b r o u t i n e EXTRAP w h i c h a l s o p r o j e c t s t h e s e v a r i a b l e s t o

65

66

CS t a r t

Read R e s e r v o i r and G r i d D i m e n s i o n s

R ead F l u i d , P . V . T . and R e s e r v o i r D a t a

R ead I n i t i a l C o n d i t i o n s a n d P r o d u c t i o n D a t a

J

Read S i m u l a t i o n P a r a m e t e r s

S u b r o u t i n eRESDIM

S u b r o u t i n eRESDAT

S u b r o u t i n eINCON

S u b r o u t i n eCODE

S u b r o u t i n eTRAN

G e n e r a t e A P r i o r i T r a n s m i s s i b i l i t y

C a l c u l a t e R e s e r v o i r Volume a n d I n i t i a l F l u i d i n P l a c e

S i m u l a t i o n Loop B e g i n

ÔF i g u r e 7 . F low c h a r t o f t h e s i m u l a t i o n p r o g r a m ,

67

S u b r o u t i n ePSOR

S u b r o u t i n eEXTRA?

S u b r o u t i n ePOROSY

S u b r o u t i n eCOEFF

E s t a b l i s h Time S t e p S i z e f o r T h i s S t e p

C a l c u l a t e P o r o s i t y a s a F u n c t i o n o f P r e s s u r e s

S o l v e f o r New P r e s s u r e s b y PSOR I t e r a t i v e T e c h n i q u e

A s s i g n P r o p e r I n j e c t i o n / P r o d u c t i o n R a t e s f o r

t h i s S t e p

P r o d u c e W ar n in g When L a s t Time S t e p S i z e i s So L a r g e a n d S a t u r a t i o n C h an g es E x c e e d C o n t r o l L i m i t

F o r m u l a t e C o e f f i c i e n t M a t r i x a n d RHS V e c t o r f o r P r e s s u r e E q u a t i o n s

U p d a t e P r e s s u r e s and S a t u r a t i o n s ; and make

P r o j e c t i o n s t o H a l f Time S t e p

F i g . 7 ( c o n t ' d )

68

N > NMAX o r

T > TMAXNo

Yes

S t o p

S u b r o u t i n eMATBAL

S u b r o u t i n eVELOCY

O u t p u tO p t i o n s

M a t e r i a l B a l a n c e C a l c u l a t i o n s

S u b r o u t i n ePTPRESPTSATPTPORPTVELO

C a l c u l a t e F l u i d s ' V e l o c i t y a n d A c c e l e r a t i o n

D e t e r m i n e New S a t u r a t i o n s a s F u n c t i o n s o f C a p i l l a r y

P r e s s u r e s and C a l c u l a t e F l u i d i n P l a c e

F i g . 7 C c o n t ' d )

69(n + îs) t i m e s t e p t o p r o v i d e a b a s i s f ro m w h i c h t o e v a l u a t e

t r a n s m i s s i b i l i t i e s , and t o g i v e an i n i t i a l e s t i m a t e f o r t h e

p r e s s u r e . s o l u t i o n .

T r a n s m i s s i b i l i t y c o e f f i c i e n t s an d t h e r i g h t - h a n d - s i d e

v e c t o r a r e now c a l c u l a t e d i n s u b r o u t i n e COEFF. The . p r e s s u r e e q u a

t i o n s a r e s o l v e d s i m u l t a n e o u s l y i n s u b r o u t i n e PSOR. A f t e r

p r e s s u r e s o f t h i s t i m e s t e p a r e d e t e r m i n e d , s u b r o u t i n e POROSY

e v a l u a t e s p o r o s i t y a s a f u n c t i o n o f p r e s s u r e s . T h e n , new

s a t u r a t i o n s a r e d e t e r m i n e d f r o m new c a p i l l a r y p r e s s u r e s u s i n g

l i n e a r i n t e r p o l a t i o n f r o m t a b l e s o f l o o k - u p d a t a . S u b r o u t i n e

VELOCY c a l c u l a t e s f r a c t u r e v e l o c i t y b a s e d on new p r e s s u r e s

and s a t u r a t i o n s .

Tow ard t h e e n d o f t h e t i m e s t e p , m a t e r i a l b a l a n c e c a l c u

l a t i o n s a r e c a r r i e d o u t i n s u b r o u t i n e MATBAL. F i n a l l y , r e s u l t s

o f t h i s t i m e s t e p a r e p r i n t e d o u t v i a s u b r o u t i n e s PTPRES,

PTSAT, PTPOR and PTVELO. B e f o r e r e p e a t i n g t h e same p r o c e s s

a t a new t i m e s t e p , e l a p s e d t i m e a n d n u m b e r o f t i m e s t e p s a r e

c h e c k e d f o r t e r m i n a t i o n o f t h e s i m u l a t i o n .

5 . 1 . 2 S u b r o u t i n e Code

S i m u l a t i o n p a r a m e t e r s a r e r e a d i n t h i s s u b r o u t i n e .

T h e s e p a r a m e t e r s i n c l u d e maximum n u m b e r o f t i m e s t e p s , m a x i

mum s i m u l a t i o n t i m e , t o l e r a n c e l i m i t f o r PSOR, maximum n um ber

o f i t e r a t i o n s f o r PSOR, r e l a x a t i o n p a r a m e t e r , b u b b l e p o i n t

p r e s s u r e , a b a n d o n m e n t p r e s s u r e , maximum w a t e r - o i l r a t i o ,

maximum g a s - o i l r a t i o , and p r i n t c o d e s .

705 . 1 . 3 S u b r o u t i n e COEFF

T h i s s u b r o u t i n e f o r m u l a t e s t h e t r a n s m i s s i b i l i t y c o e f f i

c i e n t m a t r i x a n d t h e r i g h t - h a n d - s i d e v e c t o r f o r t h e p r e s s u r e

e q u a t i o n s . The s u b r o u t i n e i s q u i t e c o m p l i c a t e d and cum bersom e

b e c a u s e t h e r e a r e 126 c o e f f i c i e n t s an d 6 g o v e r n i n g e q u a t i o n s

t o b e e v a l u a t e d a t e a c h g r i d p o i n t . The f a c t t h a t e v e r y c o

e f f i c i e n t i s p r e s s u r e - and s a t u r a t i o n - d e p e n d e n t n e c e s s i t a t e s

t h e e x t e n s i v e u s e s o f a l i n e a r i n t e r p o l a t i o n s u b r o u t i n e (LINTP)

t o o b t a i n d a t a f r o m t a b l e s o f l o o k - u p . R e l a t i v e p e r m e a b i l i

t i e s a r e e v a l u a t e d u p s t r e a m and t h e r e f o r e f l o w p o t e n t i a l s and

d i r e c t i o n s o f f l o w h a v e t o b e d e t e r m i n e d f o r e v e r y f l u i d p h a s e

a t e v e r y g r i d p o i n t .

5 . 1 . 4 S u b r o u t i n e EXTRAP

T h i s s u b r o u t i n e u p d a t e s p r e s s u r e s an d s a t u r a t i o n s f r o m

t h e p r e v i o u s t i m e s t e p . I t a l s o e x t r a p o l a t e s t h e s e two

v a r i a b l e s t o t i m e s t e p ( n +%) t o p r o v i d e a b a s i s f r o m w h i c h

t o e v a l u a t e t h e t r a n s m i s s i b i l i t y c o e f f i c i e n t s i n s u b r o u t i n e

COEFF. The e x t r a p o l a t e d p r e s s u r e s a l s o s e r v e a s a n i n i t i a l

a p p r o x i m a t i o n o f t h e s o l u t i o n i n s u b r o u t i n e PSOR.

5 . 1 . 5 F u n c t i o n FAVG

T h i s f u n c t i o n s i m p l y c o m p u t e s t h e a v e r a g e v a l u e o f two

n u m b e r s .

5 . 1 . 6 S u b r o u t i n e INCON

I n i t i a l p h a s e p r e s s u r e s and s a t u r a t i o n s a r e r e a d i n t h i s

s u b r o u t i n e a l o n g w i t h g r i d l o c a t i o n s , r a t e s , t i m e - o n and t i m e -

71o f f o f t h e i n j e c t o r s a n d p r o d u c e r s .

5 . 1 . 7 S u b r o u t i n e LINTP

T h i s s u b r o u t i n e p e r f o r m s l i n e a r i n t e r p o l a t i o n f r o m

t a b u l a t e d d a t a .

5 . 1 . 8 S u b r o u t i n e MATBAL

M a t e r i a l b a l a n c e e r r o r s o f o i l , w a t e r and g a s d u r i n g a

t i m e s t e p a r e c a l c u l a t e d i n t h i s s u b r o u t i n e . F l u i d i n j e c t i o n s /

p r o d u c t i o n s a s w e l l a s t h e i r c u m u l a t i v e s a r e c o m p u t e d . W a t e r -

o i l r a t i o , g a s - o i l r a t i o a n d a v e r a g e r e s e r v o i r p r e s s u r e s a r e

a l s o c o m p u t e d .

5 . 1 . 9 S u b r o u t i n e POROSY

T h i s s u b r o u t i n e c a l c u l a t e s r o c k m a t r i x p o r o s i t y and

f r a c t u r e p o r o s i t y a s f u n c t i o n s o f p r e s s u r e s . (S e e s e c t i o n

4 . 7 . )

5 . 1 . 1 0 S u b r o u t i n e PSOR

; T h i s s u b r o u t i n e s o l v e s t h e s y s t e m o f p r e s s u r e e q u a t i o n s

by t h e P o i n t - S u c c e s s i v e - O v e r - R e l a x a t i o n m e t h o d . (S ee s e c t i o n

4 . 9 . ) A f l o w c h a r t o f t h i s s u b r o u t i n e i s shown i n F i g u r e 8.

5 . 1 . 1 1 S u b r o u t i n e PTPRES

T h i s s u b r o u t i n e p r i n t s p r e s s u r e d i s t r i b u t i o n s i n t h e

r o c k m a t r i x an d f r a c t u r e s . I t a l s o p r i n t s r e s u l t s f r o m s u b

r o u t i n e MATBAL.

5 . 1 . 1 2 S u b r o u t i n e PTSAT

S a t u r a t i o n d i s t r i b u t i o n s i n t h e r o c k m a t r i x and t h e

c

72S T A R T J

R e c e i v e N e c e s s a r y D a t a From Main P r o g r a m

Ik = 0

k = k + 1 rmax = 0

Do O v e r A l l G r i d P o i n t s

C a l c u l a t e .Iw 1 , j ,k

max - ER e l a x

k+1

Iw 1 , ] ,k

N N

ConvergenceRETURN Compute .

R e s i d u a l ^

k+1 kn + l _ pn+1Iw Iw

R e p e a t t h e P r e v i o u s T h r e e S t e p s

F o r P i Q , P i g , P f * , P£o a n d P^^

B e f o r e P r o c e e d i n g t o N e x t S t e p

maxY

Not C o n v e r g e n t RETURN

Residual >3 ^

N

r = Residualmax

F i g u r e 8. F low C h a r t o f PSOR s u b r o u t i n e .

73f r a c t u r e s a r e p r i n t e d i n t h i s s u b r o u t i n e .

5 . 1 . 1 3 S u b r o u t i n e PTPOR

P o r o s i t y d i s t r i b u t i o n s i n t h e r o c k m a t r i x a n d t h e

f r a c t u r e a r e p r i n t e d i n t h i s s u b r o u t i n e .

5 . 1 . 1 4 S u b r o u t i n e PTVELO

F l u i d v e l o c i t y d i s t r i b u t i o n s i n t h e f r a c t u r e s a r e p r i n t e d

i n t h i s s u b r o u t i n e .

5 . 1 . 1 5 S u b r o u t i n e RESDAT

T h i s s u b r o u t i n e r e a d s p o r o s i t y and p e r m e a b i l i t y d a t a

f o r t h e r o c k m a t r i x and t h e f r a c t u r e s . I t s h o u l d b e p o i n t e d

o u t t h a t b e c a u s e o f i t s t e n s o r f o r m , t h e r e a r e 9 t e n s o r

e l e m e n t s a s s o c i a t e d w i t h t h e f r a c t u r e p e r m e a b i l i t y . The s u b

r o u t i n e a l s o r e a d s f l u i d P . V . T . d a t a f o r o i l , w a t e r a n d g a s .

T h e s e i n c l u d e f o r m a t i o n v o lu m e f a c t o r s , v i s c o s i t i e s , s o l u t i o n

g a s - o i l r a t i o , d e r i v a t i v e s o f t h e i n v e r s e s o f f o r m a t i o n v o lu m e

f a c t o r s w i t h r e s p e c t t o p r e s s u r e s , an d d e r i v a t i v e o f s o l u t i o n

g a s - o i l r a t i o w i t h r e s p e c t t o p r e s s u r e . O t h e r d a t a r e a d a r e

o i l , w a t e r an d g a s d e n s i t i e s a t s t a n d a r d c o n d i t i o n s , f r a c t u r e

and r o c k m a t r i x c o m p r e s s i b i l i t i e s , and t i l t , a n g l e s ' o f t h e

f o r m a t i o n . G a s - o i l r e l a t i v e p e r m e a b i l i t y and c a p i l l a r y p r e s

s u r e d a t a an d d e r i v a t i v e s o f g a s s a t u r a t i o n w i t h r e s p e c t t o

g a s - o i l c a p i l l a r y p r e s s u r e a r e r e a d f o r t h e r o c k m a t r i x and

t h e f r a c t u r e . S i m i l a r l y , w a t e r - o i l r e l a t i v e p e r m e a b i l i t y and

c a p i l l a r y p r e s s u r e d a t a a n d d e r i v a t i v e s o f w a t e r s a t u r a t i o n

74w i t h r e s p e c t t o w a t e r - o i l c a p i l l a r y p r e s s u r e a r e r e a d f o r t h e

r o c k m a t r i x a n d t h e f r a c t u r e s . F o r l i n e a r i n t e r p o l a t i o n

p u r p o s e s , t h e w a t e r - o i l c a p i l l a r y p r e s s u r e v e r s u s w a t e r s a t u

r a t i o n d a t a a r e t o b e r e a r r a n g e d s u c h t h a t t h e w a t e r - o i l

c a p i l l a r y p r e s s u r e s a r e t a b u l a t e d i n i n c r e a s i n g o r d e r .

5 . 1 . 1 6 S u b r o u t i n e RESDIM

T h i s s u b r o u t i n e r e a d s n u m b e r s o f g r i d p o i n t s i n x - , y -

a n d z - d i r e c t i o n s a l o n g w i t h l e n g t h , w i d t h , and t h i c k n e s s o f

t h e r e s e r v o i r . T h i s d a t a i s u s e d t o c a l c u l a t e u n i f o r m g r i d

d i m e n s i o n s . I n a d d i t i o n , h a l f w i d t h o f t h e f r a c t u r e i s r e a d

and h y d r a u l i c r a d i u s c a l c u l a t e d .

5 . 1 . 1 7 S u b r o u t i n e TRAN

T h i s s u b r o u t i n e c o m p u t e s t h o s e p a r t s o f t h e t r a n s m i s s i b i l i t y

c o e f f i c i e n t s t h a t a r e c o n s t a n t a n d t h u s do n o t n e e d t o be

r e e v a l u a t e d e v e r y t i m e s t e p .

5 . 1 . 1 8 S u b r o u t i n e VELOCY

T h i s s u b r o u t i n e c a l c u l a t e s v e l o c i t y a n d a c c e l e r a t i o n o f

t h e f l u i d s i n t h e f r a c t u r e [ s e e s e c t i o n 4 . 8 ) .

5 . 2 F i e l d Exam ple22

D a t a f r o m one o f Thomas e t a l t h r e e - d i m e n s i o n a l e x a m p

l e s was u s e d t o d e m o n s t r a t e t h e u t i l i t y o f t h e s i m u l a t i o n

p r o g r a m d e v e l o p e d . I n t h e e x a m p l e , a r e s e r v o i r s e c t i o n b e t w e e n

a n i n j e c t i o n and p r o d u c t i o n w e l l i s m o d e l e d u s i n g a 10 x 2 x 5

g r i d s y s t e m .

75The r o c k m a t r i x p e r m e a b i l i t y and p o r o s i t y a r e s e t e q u a l

t o 1 md a n d 0 . 2 9 . The f r a c t u r e p e r m e a b i l i t y and p o r o s i t y a r e

10 md a n d 0 . 0 1 . B o th t h e r o c k m a t r i x a n d f r a c t u r e a r e hom o

g e n e o u s a n d i s o t r o p i c . H e n c e , t h e f o l l o w i n g v a l u e s w e r e u s e d ,

= 1 . 0 md

(j) 2 ~ 0 . 2 9

" 1 0 . 0 0 . 0 0 . 0 "

0 . 0 1 0 . 0 0 . 0 md

0 . 0 0 . 0 1 0 . 0

(j)£ = 0 . 0 1

I n t h e r o c k m a t r i x a s w e l l a s t h e f r a c t u r e , t h e i n i t i a l

r e s e r v o i r p r e s s u r e i s s e t a t 6 2 1 4 . 7 p s i a f o r t h e o i l p h a s e ,

6 1 6 4 . 7 p s i a f o r t h e w a t e r p h a s e , and 6 2 1 4 . 7 7 5 p s i a f o r t h e

g a s p h a s e . I n i t i a l s a t u r a t i o n s a r e 0 . 8 0 f o r o i l , 0 . 2 0 f o r

w a t e r a n d 0 . 0 f o r g a s .

B o th t h e i n j e c t i o n and p r o d u c t i o n w e l l s a r e p e r f o r a t e d

i n a l l f i v e l a y e r s . The p r o d u c t i v i t y i n d i c e s o f t h e f i v e

l a y e r s a r e a l l e q u a l . T h u s , i t i s a s s u m e d t h a t i n j e c t i o n /

p r o d u c t i o n r a t e s a r e d i v i d e d e q u a l l y among t h e s e f i v e l a y e r s .

The f o r m a t i o n i s h o r i z o n t a l , i . e . , a ~ 0 , 8 = 0 and y = 0 .

I n t h i s e x a m p l e , w a t e r i s i n j e c t e d i n t o t h e r e s e r v o i r

a t an i n i t i a l r a t e o f 7000 S T B / d a y . T o t a l f l u i d p r o d u c t i o n

r a t e i s s e t a t 4000 S T B / d a y .

I n Thomas e t a l p a p e r , t h e s i z e / g e o m e t r y o f f r a c t u r e

i s i m p l i c i t l y g i v e n i n t e r m s o f t h e m a t r i x s h a p e f a c t o r [ a )

w h i c h i s 0 . 2 5 f o r t h e e x a m p l e . B u t i n t h i s m o d e l , t h e

76f r a c t u r e h a l f w i d t h i s r e q u i r e d d a t a . S i n c e i t was n o t g i v e n ,

t h e f r a c t u r e h a l f w i d t h (d ) o f 0 . 0 0 0 5 f t w as a s s u m e d . T h i s

r o u g h e s t i m a t e i s o o t a i n e d f r o m c o m p a r i s o n o f Thomas e t a l

m a t r i x - f r a c t u r e f l o w e q u a t i o n t o E v a n s ' f l u i d i n t e r a c t i o n t e r m .

The c o m p a r i s o n i s e x p l a i n e d and i l l u s t r a t e d i n A p p e n d i x H.

I n f o r m a t i o n on r e s e r v o i r d i m e n s i o n s and r o c k / f r a c t u r e

p r o p e r t i e s i s g i v e n i n T a b l e 2 . PVT d a t a f o r o i l , w a t e r an d

g a s a r e shown i n T a b l e s 3 , 4 a n d 5 , r e s p e c t i v e l y . T a b l e 6

g i v e s f l u i d d e n s i t i e s a t s t a n d a r d c o n d i t i o n s .

G a s - o i l r e l a t i v e p e r m e a b i l i t y and c a p i l l a r y p r e s s u r e

d a t a i s g i v e n i n T a b l e 7 . W a t e r - o i l r e l a t i v e p e r m e a b i l i t y

an d c a p i l l a r y p r e s s u r e d a t a i s d i v e n i n T a b l e 8 . I n d e v i a t i n g

f r o m Thomas e t a l a p p r o a c h w h i c h a s s u m e s s t r a i g h t - l i n e r e l a

t i v e p e r m e a b i l i t i e s and z e r o c a p i l l a r y p r e s s u r e i n t h e f r a c

t u r e , d a t a i n T a b l e s 7 a n d 8 w e r e a p p l i e d i n b o t h t h e r o c k

m a t r i x and f r a c t u r e s . T h i s i s b e c a u s e t h e f u l l y i m p l i c i t

a p p r o a c h u s e d i n t h i s s i m u l a t i o n p r o g r a m r e q u i r e s n o n z e r o

c a p i l l a r y p r e s s u r e d a t a , e s p e c i a l l y t h e n o n z e r o s l o p e s o f

v e r s u s s a t u r a t i o n .

The b u b b l e p o i n t p r e s s u r e i s 5 5 5 9 . 7 p s i a . The a b a n

d o n m e n t p r e s s u r e o f 1500 p s i a was u s e d . A b a n d o n m en t w a t e r -

o i l r a t i o and g a s - o i l r a t i o a r e 100 STB/STB and 1 0 0 , 0 0 0 SPC/

STB, r e s p e c t i v e l y . A p r e s s u r e t o l e r a n c e l i m i t o f 0 . 2 p s i i s

u s e d i n t h e PSOR s u b r o u t i n e .

An e x a m p l e o f i n p u t d a t a a r r a n g e m e n t i s g i v e n i n

A p p e n d i x E.

77Table 2

R E S E R V O I R & G R I D D I M E N S I O N S AND ROC K P R O P E R T I E S * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *******

L E N G T H = 2 0 0 0 . 0 F E E T

WI D T H = 4 0 0 . 0 F E E T

T H I C K N E S S = 2 5 0 . 0 F E E T

NUMBER OF G R I D P O I N T S I N X - O I R E C T I U N = 1 0

NUMBER OF G R I D P O I N T S I N Y - D I R E C T I O N = 2

NUMBER OF G R I D P O I N T S I N Z - D I R E C T I Û N = 5

Û E L T A - X ( F T ) = 2 0 0 . 0 0

D E L T A - Y ( F T ) = 2 0 0 . 0 0

D E L T A - Z ( F T ) = 5 0 . 3 0

N A T U R A L F R A C T U R E H A L F - W I D T H = 0 . 0 0 0 5 0 0 F E E T

H Y D R A U L I C R A D I U S = 0 . 0 0 0 3 0 3 F E E T

F R A C T U R E C O M P R E S S I B I L I T Y = 0 . 3 5 0 0 0 - 0 5 P S I - 1

M A T R I X ROCK C O M P R E S S I B I L I T Y = 0 . 3 5 0 0 D - 0 5 P S I - 1

T I L T A N G L E S OF R E S E R V O I R F O R M A T I O N :

A L P H A = 0 . 0 D E G .

B E T A = 0 . 0 D c G .

GAMMA = 0 . 0 D E G .

78

T a b l e 3O I L P V T DATA * * * * * * * * * * * *

P O ( P S I )

BO( 8 B L / S T B )

D ( l / B O ) / D P ( S T B / B 3 L / P S 1 )

R S U ( S C F / S T B )

D R S O / O P ( S C F / S T B / P S I )

UU( C P )

1 4 , 7 1 . 0 0 0 0 - 0 , 1 3 7 9 0 D - 0 3 0 , 0 0 , 2 1 9 2 0 . 7 2 0 0

1 7 1 4 . 7 1 . 3 0 2 6 - 0 , 5 7 6 0 0 0 - 0 4 3 7 2 . 6 0 . 2 2 4 1 0 . 5 2 6 0

2 1 1 4 , 7 1 . 3 4 2 9 - 0 , 5 7 5 0 0 D - 0 4 4 6 3 , 2 0 . 2 3 4 5 0 . 4 8 0 0

2 6 1 4 . 7 1 . 3 9 6 9 - 0 , 5 7 9 0 0 D - 0 4 5 8 1 , 5 0 . 2 4 9 5 0 . 4 3 0 0

3 0 1 4 . 7 1 . 4 4 3 7 - 0 , 5 8 2 0 0 D - 0 4 6 8 1 , 5 0 . 2 8 0 2 0 . 3 9 6 0

3 6 1 4 . 7 1 . 5 2 0 4 - 0 , 5 9 4 0 0 D - 0 4 8 4 6 , 2 0 . 3 0 1 6 0 . 3 4 8 0

4 2 1 4 . 7 1 , 6 0 7 7 - 0 , 6 0 1 U O D - 0 4 1 0 2 9 , 7 0 . 3 2 9 5 0 . 3 0 3 0

4 6 1 4 . 7 1 , 6 7 2 5 - 0 , 6 0 8 0 0 0 - 0 4 1 1 6 3 , 3 0 . 3 6 3 2 0 . 2 7 4 0

5 0 1 4 , 7 1 . 7 4 4 2 - 0 , 6 4 0 0 0 D - 0 4 1 3 1 1 , 0 0 . 4 0 0 1 0 . 2 4 4 0

5 3 1 4 , 7 1 , 3 0 4 2 - 0 , 6 0 7 0 0 D - 0 4 1 4 3 1 , 2 0 . 4 0 3 5 0 . 2 2 6 0

5 6 1 4 . 7 1 , 8 6 4 9 - 0 , 5 8 0 0 0 0 - 0 4 1 5 5 7 , 0 0 . 5 0 1 2 0 . 2 0 6 0

7 0 1 4 . 7 2 , 1 9 7 8 - 0 , 5 7 9 0 0 D - 0 4 2 2 5 9 , 0 0 . 5 0 1 1 0 . 1 0 9 0

T a b l e 4

WATER P V T D A T A * * * * * * * * * * * * * *

PW ( P S I J

uw( C P )

1 4 . 7

7 0 1 4 , 7

0 , 3 5 0 0

0 , 3 5 0 0

79T a b l e 5

Ü A S P V T DATA * * * * * * * * * * * *

P G ( P S I )

BG( S 3 L / S C F )

D ( 1 / 0 G ) / D P ( S C F / B U L / P S I )

UG( C P )

1 4 . 7 0 . 1 7 8 0 9 0 + 0 0 0 . 2 9 8 3 5 0 . 0 1 2 0

1 7 1 4 . 7 0 . 1 9 4 8 5 D - 0 2 0 . 3 1 4 3 8 0 . 0 1 6 3

2 1 1 4 . 7 0 . 1 5 6 6 7 D - 0 2 0 . 3 0 4 5 3 0 . 0 1 7 3

2 6 1 4 . 7 0 . 1 2 6 7 1 D - Ü 2 0 . 2 3 5 6 3 0 . 0 1 8 6

3 0 1 4 . 7 0 . 1 1 0 7 2 0 - 0 2 0 . 2 4 0 7 4 0 . 0 1 9 7

3 6 1 4 . 7 0 . 9 4 9 3 0 0 - 0 3 0 . 2 2 7 7 1 0 . 0 2 1 4

4 2 1 4 . 7 0 . 8 4 1 8 0 Ü - 0 3 0 . 2 0 3 3 7 0 . 0 2 3 3

4 6 1 4 . 7 0 . 7 8 8 4 0 0 - 0 3 0 . 1 0 3 4 9 0 . 0 2 4 6

5 0 1 4 . 7 0 . 7 4 4 5 O D - 0 3 0 . 1 7 9 3 1 0 . 0 2 5 7

5 3 1 4 . 7 0 . 7 1 6 2 0 0 - 0 3 0 . 1 6 5 1 9 0 . 0 2 6 6

5 6 1 4 c ? 0 . 6 9 1 3 C D - 0 3 0 . 1 5 8 0 6 0 . 0 2 7 6

7 0 1 4 . 7 0 . 6 0 0 0 0 0 - 0 3 0 . 1 5 8 1 4 0 . 0 3 3 0

T a b l e 6

F L U I D D E N S I T I E S A T S T A N D A R D C O N D I T I O N S * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

S T A N D A R D P R E S S U R E = 1 4 . 7 0 0 0 P S I A

S T A N D A R D T E M P E R A T U R E = 6 0 . 0 D E G . F

O I L D E N S I T Y = 5 1 . 1 4 0 0 L 8 M / C U F T

WATER D E N S I T Y = 6 5 . 0 0 0 0 L B M / C U F T

G A S D E N S I T Y = 0 . 0 5 8 0 L B M / C U F T

80

T a b l e 7

G A S - O I L R E L A T I V E P E R M E A B I L I T Y & C A P I L L A R Y P R E S S U R E DATA ***** * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

S G KRG KROG P C G O O S G / D P C( P S I ) ( P S I - 1 )

0 . 0 0 . 0 1 . 0 0 0 0 . 0 7 5 1 0 . 0 0 0 0 0

0 . 1 0 0 . 0 1 5 0 . 7 0 0 0 . 0 8 5 1 0 . 0 0 0 0 0

0 . 2 0 0 . 0 5 0 0 . 4 5 0 0 . 0 9 5 1 0 . 0 0 0 0 0

0 . 3 0 0 . 1 0 3 0 . 2 5 0 0 . 1 1 5 5 . 0 0 0 0 0

0 . 4 0 0 . 1 9 0 0 . 1 1 0 0 . 1 4 5 3 . 3 3 0 0 0

0 . 5 0 0 . 3 1 0 0 . 0 2 8 0 . 2 5 5 0 . 9 1 0 0 0

0 . 5 5 0 . 4 2 0 0 . 0 0 . 3 8 6 0 . 3 8 0 0 0

1 . 0 0 1 . 0 0 0 0 . 0 1 . 5 6 5 0 . 3 8 0 0 0

81

T a b l e 8

W A T E R - O I L r e l a t i v e P E R M E A B I L I T Y & C A P I L L A R Y P R E S S U R E DATA * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *

sw KRW KROW pcow ( P S I )

D S W / D P C ( P S I - 1 >

0 . 0 0 . 0 1 . 0 0 0 2 1 4 . 0 0 0 - 0 . 0 0 1 2 2

0 . 2 0 0 . 0 1 . 0 0 0 5 0 . 0 0 0 - 0 . 0 0 1 2 2

0 . 2 5 0 . 0 0 5 0 . 8 6 0 9 . 0 0 0 - 0 . 0 0 1 2 2

0 . 3 0 0 . 0 1 0 0 . 7 2 3 2 . 0 0 0 - 0 . 0 0 7 1 4

0 . 3 5 0 . 0 2 0 0 . 6 0 0 0 . 5 0 0 - 0 . 0 3 3 3 3

0 . 4 0 0 . 0 3 0 0 . 4 9 2 0 . 0 - 0 . 1 0 0 0 0

0 . 4 5 0 . 0 4 5 0 . 3 9 2 - 0 . 4 0 0 - 0 . 1 2 5 0 0

0 . 5 0 0 . 0 6 0 0 . 3 0 4 - 1 . 2 0 0 - 0 . 0 6 2 5 0

0 . 5 5 0 . 0 3 2 0 . 2 2 5 - 2 . 6 0 0 - 0 . 0 3 5 7 0

0 . 6 0 0 . 1 1 0 0 . 1 5 4 - 4 . 0 0 0 - 0 . 0 3 5 7 0

0 . 6 5 0 . 1 4 3 0 . 0 9 3 - 7 . 0 0 0 - 0 . 0 1 6 7 0

0 . 7 0 0 . 1 8 0 0 . 0 4 2 - 1 0 . 0 0 0 - 0 . 0 1 6 7 0

0 . 7 5 0 , 2 3 0 0 . 0 - 4 0 . 0 0 0 - 0 . 0 0 1 6 7

1 . 0 0 1 . 0 0 0 0 . 0 - 1 9 0 . 0 0 0 - 0 . 0 0 1 6 7

CHAPTER VI

DISCUSSION OF RESULTS

I n t h i s c h a p t e r t h e r e s u l t s o f a n e x a m p l e c a s e s t u d y

a r e d i s c u s s e d a l o n g w i t h r e c o m m e n d a t i o n s f o r f u r t h e r i m p r o v e

m en t o f t h i s s i m u l a t i o n p r o g r a m .

So f a r , t h e d e v e l o p m e n t o f t h e f r a c t u r e d r e s e r v o i r

s i m u l a t o r may b e s u m m a r i z e d by t h e f o l l o w i n g s t e p s .

1 . The r e s e r v o i r i s d i s c r e t i z e d t h r e e - d i m e n s i o n a l l y

i n t o a s e r i e s o f g r i d p o i n t s , e a c h o f w h i c h c o n s i s t s

o f two p h y s i c a l d o m a i n s : t h e r o c k m a t r i x an d t h e

f r a c t u r e s .

2 . S i x g o v e r n i n g p a r t i a l d i f f e r e n t i a l e q u a t i o n s ( r e

f e r r e d t o i n t e r c h a n g e a b l y a s f l o w e q u a t i o n s ,

p r e s s u r e e q u a t i o n s o r d i f f u s i v i t y e q u a t i o n s ) a r e

f o r m u l a t e d t o d e s c r i b e t h e f l u i d m ovem ent a t e a c h

g r i d p o i n t . The b l a c k o i l m o d e l i s a s s u m e d t o

c h a r a c t e r i z e t h e c o m p o s i t i o n a l b e h a v i o r o f o i l ,

w a t e r a n d g a s . P h y s i c a l p r o p e r t i e s s u c h a s g r a v i t y

e f f e c t s and c a p i l l a r y f o r c e s a r e t a k e n i n t o c o n

s i d e r a t i o n .

3 . The f i n i t e d i f f e r e n c e schem e a n d f u l l y i m p l i c i t

a p p r o a c h a r e u s e d t o t r a n s f o r m t h e s i x g o v e r n i n g

p r e s s u r e e q u a t i o n s i n t o a s y s t e m o f a n a l o g o u s

82

83d i f f e r e n c e e q u a t i o n s . T h u s , f o r a t o t a l o f N g r i d

p o i n t s , t h e r e a r e 6N e q u a t i o n s i n 6N u n k n o w n s .

F l u i d and r o c k p r o p e r t i e s a t e a c h g r i d p o i n t a r e

e v a l u a t e d i n d i v i d u a l l y and t h e n u s e d i n c o m p u t i n g

t r a n s m i s s i b i l i t y c o e f f i c i e n t s . R e l a t i v e p e r m e a

b i l i t i e s m u s t b e e v a l u a t e d u p s t r e a m . A c o m p u t e r

c o d e i s w r i t t e n i n a c c o r d a n c e w i t h t h i s s c h e m e .

4 . The PSOR n u m e r i c a l m e th o d i s u s e d t o s o l v e t h e

r e s u l t i n g s y s t e m o f s i m u l t a n e o u s l i n e a r e q u a t i o n s

f o r t h e p r e s s u r e s a t a new t i m e s t e p .

5 . C a p i l l a r y p r e s s u r e s a r e c o m p u t e d , a n d s a t u r a t i o n s

a r e d e t e r m i n e d by i n t e r p o l a t i o n f r o m t h e t a b l e s o f

s a t u r a t i o n s v e r s u s c a p i l l a r y p r e s s u r e s . The

p r e s s u r e s a r e a l s o u s e d i n c a l c u l a t i n g p o r o s i t y

a n d f l u i d v e l o c i t y .

6 . 1 S i m u l a t o r C a p a b i l i t y

A l t h o u g h t h e r o c k m a t r i x and f r a c t u r e s a r e i s o t r o p i c a s

w e l l a s h o m o g e n e o u s i n t h e f i e l d e x a m p l e , t h e s i m u l a t i o n p r o

gram d e v e l o p e d i s c a p a b l e o f h a n d l i n g t h e i s o t r o p i c h e t e r o

g e n e o u s r o c k m a t r i x and t h e a n i s o t r o p i c h e t e r o g e n e o u s f r a c t u r e

H e n c e , a s t u d y o f a n i s o t r o p y i n f r a c t u r e c a n b e c o n s i d e r e d .

I n a d d i t i o n , t h e p r e s e n c e a n d / o r a b s e n c e o f f r a c t u r e

a t any g r i d p o i n t c a n b e a c c o u n t e d f o r b y m a n i p u l a t i n g e l e

m e n t s o f t h e p e r m e a b i l i t y t e n s o r . F o r e x a m p l e , a l l n i n e

e l e m e n t s o f t h e p e r m e a b i l i t y t e n s o r c o u l d h a v e b e e n z e r o i f

i t h a d b e e n d e t e r m i n e d t h a t t h e f r a c t u r e s a t a p a r t i c u l a r g r i d

84p o i n t e i t h e r d i d n o t e x i s t o r was n o t i n t e r c o n n e c t e d .

The f r a c t u r e o r i e n t a t i o n c a n a l s o be h a n d l e d i n a

s i m i l a r w ay . F o r i n s t a n c e , i f t h e p r e f e r e n t i a l d i r e c t i o n o f

t h e f l o w d e v i a t e s f r o m t h e p r i n c i p a l a x e s , t h e o f f - d i a g o n a l

e l e m e n t s o f t h e p e r m e a b i l i t y t e n s o r s may be a s s i g n e d some

r e a l n u m b e r s i n s t e a d o f z e r o .

The a b o v e c a p a b i l i t i e s l e n d t h e m s e l v e s r e a d i l y t o t h e

i n c l u s i o n o f a p r o b a b i l i s t i c a p p r o a c h s u c h a s t h e M onte C a r l o

m e th o d f o r g e n e r a t i n g t h e s t a t i s t i c a l d i s t r i b u t i o n o f f r a c t u r e

i n t h e r e s e r v o i r . T h i s a p p l i c a t i o n i s p a r t i c u l a r l y u s e f u l

f o r a n a t u r a l l y f r a c t u r e d r e s e r v o i r w h i c h h a s b e e n d e t e r m i n e d

( v i a w e l l t e s t i n g d a t a , f o r e x a m p l e ) t o h a v e n e t w o r k s o f

f r a c t u r e s , o r s a n d l e n s e s , o r s h a l e b a r r i e r s c o v e r i n g c e r t a i n

p e r c e n t a g e s o f t h e r e s e r v o i r a r e a . The M onte C a r l o m e th o d c an

t h e n b e u s e d t o g e n e r a t e r a n d o m l y t h e d i s t r i b u t i o n o f f r a c

t u r e d g r i d p o i n t s a c c o r d i n g t o t h o s e p e r c e n t a g e s . F o r e x a m p l e .

F i g u r e 9 i l l u s t r a t e s a r e s e r v o i r a r e a t h a t h a s b e e n d i s c r e

t i z e d i n t o a 1 0 x 7 g r i d s y s t e m . A 35 p e r c e n t c o n c e n t r a t i o n

o f f r a c t u r e i s a r b i t r a r i l y g i v e n f o r t h i s a r e a . T h i s t r a n s

l a t e s t o 25 o u t o f 70 g r i d p o i n t s c o n t a i n i n g f r a c t u r e s . And

c o n s e q u e n t l y , t h e r e m a i n i n g 45 g r i d p o i n t s w i l l h a v e t h e p e r

m e a b i l i t y t e n s o r s o f z e r o . I n F i g u r e 9 , t h e f r a c t u r e d g r i d

p o i n t s a r e d e n o t e d by " 1 " a n d t h e n o n f r a c t u r e d b y ” 0 . " The

i m p l i c a t i o n f r o m t h i s i s t h a t t h e Monte C a r l o m e t h o d may be

u s e d t o g e n e r a t e a p r i o r i t h e l o c a t i o n s o f g r i d p o i n t s c o n

t a i n i n g f r a c t u r e s . T h i s i n f o r m a t i o n i s u s e d a s i n p u t d a t a

85

1 0 1 ■J Ü 0 c 1 Û

0 1 0 1 1 0 A •j 0 J

0 1 1 u 1 0 1 Ü 0 Ü

1 1 0 J 1 0 1 0 J 0

1/ 0 1 Û 1 1 0 0 c

1 T; j 1 0 0 I 1 V 0

0 1 i ■J 1 1 0 1 Ü

F P A C T ü î ' r . " N' - J DEi - 1N U N - h RACTu ; - : Cù = 0

F i g u r e 9 . Random G e n e r a t i o n o f F r a c t u r e d Nodes by Monte C a r l o M e th o d .

86t o t h e s i m u l a t i o n p r o g r a m . An e x a m p l e o f t h e Monte C a r l o

p r o g r a m i s g i v e n i n A p p e n d i x G.

The s i m u l a t o r i s a l s o c a p a b l e o f n e g l e c t i n g t h e c a l c u

l a t i o n o f f r a c t u r e v e l o c i t y b y s i m p l y n o t c a l l i n g t h e VELOCY

s u b r o u t i n e a t t h e e n d o f e a c h t i m e s t e p . I n s u c h a c a s e ,

t h e e q u a t i o n s o f m o t i o n i n t h e f r a c t u r e s w o u ld b e i d e n t i c a l

t o t h e e q u a t i o n s o f m o t i o n i n t h e r o c k m a t r i x ( i . e . , D a r c y ' s

l a w ) . T h i s a l l o w s t h e s t u d y o f v e l o c i t y e f f e c t s on t h e

f l u i d movement t h r o u g h t h e n a t u r a l l y f r a c t u r e d r e s e r v o i r .

The p r e s e n t s i m u l a t i o n p r o g r a m i s d e s i g n e d f o r t h e 10

X 2 X 5 g r i d s y s t e m (100 g r i d p o i n t s ) . H o w e v e r , t h e COMMON/

DIMENSION s t a t e m e n t s a t t h e b e g i n n i n g o f e a c h p r o g r a m o r s u b

r o u t i n e c a n e a s i l y be m o d i f i e d t o f i t any o t h e r g r i d s y s t e m s .

6 . 2 R e s u l t s o f t h e F i e l d Exam ple

The d a t a g i v e n i n s e c t i o n 5 . 2 i s f e d i n t o t h e s i m u l a t i o n

p r o g r a m . F o r c o m p a r a t i v e s t u d y , i n j e c t i o n and p r o d u c t i o n

r a t e s a r e d e l e g a t e d t o t h e f l o w e q u a t i o n s f o r t h e f r a c t u r e s22b e c a u s e Thomas e t a l a s s u m e d t h a t f l o w t h r o u g h t h e r e s e r

v o i r o c c u r r e d p r i m a r i l y t h r o u g h t h e f r a c t u r e . H o w e v e r , i t

s h o u l d be n o t e d t h a t t h e s i m u l a t o r i s c a p a b l e o f d i s t r i b u t

i n g an y f l o w r a t e s t o b o t h t h e r o c k m a t r i x e q u a t i o n s a n d t h e

f r a c t u r e e q u a t i o n s . An o u t p u t s a m p l e i s g i v e n i n T a b l e 9-

The s a t u r a t i o n d i s t r i b u t i o n o u t p u t o v e r e x t e n d e d p e r i o d s o f

t i m e a r e u s e d i n p l o t t i n g t h e f r a c t u r e - and t h e m a t r i x - S O -

p e r c e n t w a t e r s a t u r a t i o n p r o f i l e s a s shown i n F i g u r e s 10

and 1 1 . N o te t h e g r a d u a l g r a v i t y e f f e c t shown by w a t e r

87

T a b l e 9

O u t p u t S am p le

S I M U L A T I O N R E S U L T S A F T E R T I M E S T E P

E L A P S E D T I M E = 1 4 . 8 2 D A Y S

T I M E S T E P S I Z E ( D A Y ) = 1 . 0 0

4 0

A V E R A G E R E S E R V O I R ( M A T R I X ) P R E S S U R E ( P S I A ) =

O I L M A T E R I A L B A L A N C E T H I S S T E P ( P E R C E N T ) =WATER M A T E R I A L B A L A N C E T H I S S T E P ( P E R C E N T ) =G A S m a t e r i a l B A L A N C E T H I S S T E P ( P E R C E N T ) =

C U M U L A T I V E O I L P R O D U C E D ( S T B ) =C U M U L A T I V E WATER P R O D U C E D ( S T B ) =C U M U L A T I V E WATER I N J E C T E D ( S T B ) =C U M U L A T I V E GAS P R O D U C E D ( S C F ) =C U M U L A T I V E GAS I N J E C T E D ( S C F ) =

W A T E R - O I L R A T I O =G A S - O I L R A T I O =

6 2 1 4 . 6 8

- 0 . 4 6 8 8 I . 0 8 0 5

- 0 . 4 6 7 2

5 9 2 9 9 . 1 .

1 0 3 7 7 5 . 1 1 0 1 3 2 3 9 5 ' .

0 .

0 . 03 7 1 4 . 6

O I L P R O D U C T I O N T H I S S T E P ( S T B ) WATER P R O D U C T I O N T H I S S T E P ( S T U ) WATER I N J E C T I O N T H I S S T E P ( S T B ) G A S P R O D U C T I O N T H I S S T E P ( S C F ) G A S I N J E C T I O N T H I S S T E P ( S C F )

- 3 9 9 9 . 9 2 - 0 . 0 8

7 0 0 0 . 0 0 • 7 4 2 5 4 9 0 . 3 5

0 . 0

Table 9 (cont'd)PHFJiSUHE DI STMIUUT IONS IN MATWIX HOCK

l a y e r n u m b e r ************

ROW NUMBER

PÜPWPC

6 2 1 3 . 56 1 6 3 . 66 2 1 3 . 6

6 2 1 3 . 66 1 6 3 . 6 6 2 1 J . 6

6 2 1 3 . 66 1 6 3 . 66 2 1 3 . 6

6 2 1 3 . 66 1 6 3 . 66 2 1 3 . 6

6 2 1 3 . 6 6 I 0 3 . 6 0 2 1 3 . 6

6 2 1 3 . 00 1 6 3 . 66 2 1 3 . 6

6 2 1 3 . 66 1 6 3 . 66 2 1 3 . 6

6 2 1 3 . 76 1 6 3 . 76 2 1 3 . 7

6 2 1 4 . 36 1 6 4 . 4 6 2 1 4 . 3

6 2 1 5 . 16 1 6 5 . 16 2 1 5 . 2

ROW NUMBER

POPWPC

6 2 1 0 . 8 6 1 6 1 . 0 6 2 1 0 . 7

6 2 1 3 . 5 0 1 o 3 . O6 2 1 3 . 6

6 2 1 3 . 06 1 6 3 . 0 6 2 1 3 . 6

6 2 1 3 . 66103.66 2 1 3 . 6

0 2 1 3 . 0 0 I 0 3 . 6 6 2 1 3 . 6

6 2 1 3 . 00163.06 2 1 3 . 0

6 2 1 3 . 66 1 6 3 . 7 6 2 1 3 . 6

6 2 1 4 . 06 1 6 4 . 16 2 1 4 . 1

6 2 1 5 . 06 1 6 5 . 06 2 1 5 . 1

6 2 1 5 . 26 1 6 5 . 26 2 1 5 . 3

LAYER NUMBER ************

ROW NUMBER

POPWPC

6 2 1 4 . 66 1 6 4 . 66 2 1 4 . 7

6 2 1 4 . 06104.0 6 2 1 4 , 7

6 2 1 4 . 66 1 6 4 . 60 2 1 4 . 7

6 2 1 4 . 06164.66 2 1 4 . 7

6 2 1 4 . 66164.6 6 2 1 4 . 7

0 2 1 4 . 0 b164,6 6 2 1 4 . 7

6 2 1 4 . 66 1 6 4 . 66 2 1 4 . 7

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 5 . 36 1 6 5 . 36 2 1 5 . 4

00006 2 1 5 . 3 6 1 6 5 . 26 2 1 5 . 4

ROW NUMBER

POPWPC

6 2 1 1 .a 6 1 6 2 . 0 6 2 1 1 . 0

6 2 1 4 . 66 1 6 4 . 66 2 1 4 . 7

6 2 1 4 . 06 1 6 4 . 66 2 1 4 . 7

6 2 1 4 . 6 6 1 6 4 . 6 6 2 1 4 . 7

0 2 1 4 . 66104.66 2 1 4 . 7

6 2 1 4 . 06164.0 6 2 1 4 . 7

6 2 1 4 . 66 1 6 4 . 66 2 1 4 . 7

6 2 1 5 . 06 1 6 5 . 06 2 1 5 . 1

6 2 1 5 . 2 6 1 6 5 . 16 2 1 5 . 3

6 2 1 5 . 36 1 6 5 . 36 2 1 5 . 4

l a y e r n u m b e r #*******#$*$

RUW n u m b e r

POPWPG

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

0 2 1 4 . 7 6 1 6 4 . 70 2 1 4 . 8

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 1 5 . 36 1 6 5 . 36 2 1 5 . 4

6 2 1 5 . 2 6 1 6 5 . I6 2 1 5 . 3

ROW NUMBER

POPWPG

6 2 1 1 . 9 6 102.I 6 2 1 1 . 9

6 2 1 4 . 76 1 0 4 . 76 2 1 4 . 8

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 4 . 76 1 6 4 . 7 62l4.a

6214.76164.76214.8

0 2 1 4 . 70 1 6 4 . 7 0 2 1 4 , 6

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 5 . 26 1 6 5 . 26 2 1 5 . 3

6 2 1 5 . 16 1 6 5 . 16 2 1 5 . 2

6 2 1 5 . 36 1 6 5 . 36 2 1 5 . 4

Table 9 (cont'd)LAYEk NUMOEU ************

HUM NUMüCK

POPWPG

6 2 1 4 . a 6 1 6 4 . 3 6 2 1 4 . 8

6 2 1 4 . 66 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 06 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 14 . 8 6 1 6 4 . 8 6 2 1 4 . 9

6 2 1 4 . 8 6164.66 2 1 4 . 9

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 96 1 6 4 . 9 6 2 1 5 . 0

6 2 1 5 . 4 6 1 6 5 . 36 2 1 5 . 5

6 2 1 5 . 16 1 6 5 . 16 2 1 5 . 2

ROW NUMOER

POPWPG

6 2 1 2 . 0 6 1 6 2 . 1 6 2 1 2 . 0

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 8

6 2 1 4 . 06 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 86164.86 2 1 4 . 9

6 2 1 4 . 66 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 1 5 . 16 1 6 5 . 16 2 1 5 . 2

6 2 1 5 . 1 6 1 6 5 . 06 2 1 5 . 2

6 2 1 5 . 36 1 6 5 . 36 2 1 5 . 4

LAYER NUMDER ************ 00«3

ROW NUMBER

POPWPG

6 2 1 5 . 8 6 1 6 5 . 06 2 1 5 . 9

6 2 1 5 . 86 1 6 5 . 8 6 2 1 6 . 0

6 2 1 5 . 86 1 6 5 . 8 6 2 1 6 . 0

6 2 1 5 . 8 6I65.8 6 2 1 6 . 0

6 2 1 5 . 8 6 1 6 5 . C 6 2 1 6 . 0

6 2 1 5 . 86 1 6 5 . 8 6216.0

6 2 1 5 . 86 1 6 5 . 8 6 2 1 6 . 0

6 2 1 5 . 9 6 1 6 5 . 8 6 2 1 6 . 0

6 2 1 5 . 6 6 1 6 5 . 56 2 1 5 .7

6 2 1 5 . 36 1 6 5 . 36 2 1 5 . 4

ROW NUMOER

POPWPG

6 2 1 3 . 0 6 1 6 3 . 26 2 1 3 . 1

6 2 1 5 . B6 1 6 5 . 86 2 1 5 . 9

6 2 1 3 . 86 1 0 5 . 8 6 2 1 6 . 0

6 2 1 5 . 8 6 1 6 5 . 8 6 2 1 6 . 0

6 2 1 5 . 8 6 1 6 5 . 3 6 2 1 6 . 0

6 2 1 5 . 86 1 6 5 . 8 6 2 1 6 . 0

6 2 1 5 . 9 6 1 6 5 . 8 6 2 1 6 . 0

6 2 1 5 . 5 6 1 6 5 . 46 2 1 5 . 6

6 2 1 5 . 26 1 6 5 . 26 2 1 5 . 3

6 2 1 5 . 4 6 1 6 5 . 36 2 1 5 . 5

Table 9 (cont’d)PRESSURE D I S T R I B U T I O N S IN FMACTUHE

LAYER NUMBER ************

RON NUMBER

PFOPFWPFG

6 2 1 3 . 26 1 6 3 . 3 6 2 1 2 . B

6 2 1 3 . 26 1 6 3 . 3 6 2 1 2 . 8

6 2 1 3 . 26 1 6 3 . 3 6 2 1 2 . 8

6 2 1 3 . 26 1 6 3 . 3 6 2 1 2 . 8

6 2 1 3 . 26 1 6 3 . 36 2 1 2 . 3

6 2 1 3 . 26 1 6 3 . 3 6 2 1 2 . 8

6 2 1 3 . 26 1 7 4 . 2 6 2 1 2 . 9

6 2 1 3 . S 6 2 2 0 . 0 6 2 1 3 . 2

6 214 .46381 .4 6214 .3

6 2 1 5 . 3 6 5 7 4 . 66 2 1 5 . 4

ROW NUMBER

PFOPFWPFG

6 2 0 9 . 36 1 5 9 . 3 6 2 0 7 . 9

6 2 1 3 . 26 1 6 3 . 3 6 2 1 2 . 8

6 2 1 3 . 26 1 6 3 . 3 6 2 1 2 . 8

6 2 1 3 . 26 1 6 3 . 3 6 2 1 2 . 8

6 2 1 3 . 26 1 6 3 . 3 6 2 1 2 . 8

6 2 1 3 . 26 1 6 3 . 46 2 1 2 . 8

6 2 1 3 . 26 1 9 0 . 3 6 2 1 2 . 9

6 2 1 4 . 26 2 4 6 . 3 6 2 1 4 . 0

6 2 1 5 . 16 4 7 4 . 76 2 1 5 . 3

6 2 1 5 . 46 9 0 0 . 26 2 1 5 . 7

LAYER NUMBER $****$»$****

OHOW NUMBER

PFO 6 2 1 4 . 6 6 2 1 4 . 6 6 2 1 4 . 6 6 2 14 . 6 6 2 1 4 . 6 6 2 1 4 . 6 6 2 1 4 . 6 6 2 1 4 . 9 6 2 1 5 . 4 6 2 1 5 . 4PFW 6 1 6 4 . 6 0 I 6 4 . 6 6 1 6 4 . 6 6 1 0 4 . 6 6 1 6 4 . 6 6 1 6 4 . 0 6 1 7 2 . 3 6 2 2 4 . 6 6 4 0 1 . 5 6 5 9 6 . 9PFG 6 2 1 4 . 6 6 2 1 4 . 6 6 2 1 4 . 6 0 2 1 4 . 6 6 2 1 4 . 6 6 2 1 4 . 0 0 2 1 4 . 6 6 2 1 5 . 0 6 2 1 5 . 6 6 2 1 5 . 7

ROW NUMBER 1

PFO 6 2 1 0 . 7 6 2 1 4 . 6 6 2 1 4 . 6 6 2 1 4 . 6 6 2 1 4 . 6 6 2 1 4 . 6 6 2 1 4 . 6 6 2 1 5 . 2 6 2 1 5 . 1 6 2 1 5 . 4PFW 6 1 6 1 . 1 6 1 6 4 . 6 6 1 6 4 . 0 6 1 6 4 . 6 6 1 6 4 . 6 6 1 6 4 . 7 6 1 9 2 . 2 6 2 6 4 . 9 6 4 9 6 . 9 6 9 2 2 . 6PFG 6 2 0 9 . 7 6 2 1 4 . 6 6 2 1 4 . 0 6 2 1 4 . 6 6 2 1 4 . 6 6 2 1 4 . 0 6 2 1 4 . 7 6 2 1 5 . 4 6 2 1 5 . 5 6 2 1 5 . 8

Table 9 (cont’d)LAYER NUMBER ************

ROW NUMBER

PFOPFWPFG

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 7

6 2 1 4 . 76 1 6 4 . 7 6 2 1 4 . ( 1

6 2 1 4 . 76 1 6 4 . 7 6 2 1 4 . U

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 4 . 7 6 1 7 6 . 36 2 1 4 . 8

6 2 1 5 . 1 6 2 2 7 . 56 2 1 5 . 2

6 2 1 5 . 4642 0 .76 2 1 5 .7

6 2 1 5 . 36 6 1 9 . 06 2 1 5 . 5

ROW NUMOER

PFOPFWPFG

6 2 1 0 . 9 6 1 6 1 . 26 2 0 9 . 9

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 7

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 4 . 76 1 6 4 . 76 2 1 4 . 8

6 2 1 4 . 76 1 6 4 . 86 2 1 4 . 8

6 2 1 4 . 7 6 1 9 8 . 26 2 1 4 . 8

6215.46 2 9 7 . 86 2 1 5 . 6

6215 .26 5 1 9 . 36 2 1 5 . 4

6 2 1 5 . 46 9 4 5 . 06 2 1 5 . 8

LAVER NUMOER

ROW NUMDER

PFOPFWPFG

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 80164.8 6 2 1 4 . 9

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 86 1 7 8 . 96 2 1 4 . 9

6 2 1 5 . 16 2 3 2 . 26 2 1 5 . 3

6 2 1 5 . 56 4 4 0 . 5 6 2 1 5 . 8

6 2 1 5 . 3 6 6 4 1 . 2 6 2 1 5 . 5 'O

ROW NUMBER

PFOPFWPFG

6 2 1 1 . 06 1 6 1 . 36 2 1 0 . 0

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 86164.86 2 1 4 . 9

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 86 1 6 4 . 86 2 1 4 . 9

6 2 1 4 . 8 6 1 6 5 . 06 2 1 4 . 9

6 2 1 4 . 96 2 1 8 . 06 2 1 5 . 0

6 2 1 5 . 2 6 3 2 1 . 9 6 2 1 5 . 4

6 2 1 5 . 26 5 4 1 . 76 2 1 5 . 4

6 2 1 5 . 46 9 6 7 . 4 6 2 1 5 . 8

LAYER NUMBER

ROW NUMBER

PFOPFWPFG

6 2 1 6 . 26 1 6 6 . 16 2 1 6 . 7

6 2 1 6 . 2 6 1 6 6 . 1 6 2 1 6 . 7

6 2 1 0 . 2 6 1 6 6 . 1 6 2 1 6 . 7

6 2 1 6 . 2 6160.1 6 2 1 6 . 7

6 2 1 6 . 2 6 1 6 6 . 1 0 2 1 6 . 7

6 2 1 6 . 26 1 6 6 . 16 2 1 6 . 7

6 2 1 6 . 2 6 1 7 9 . 8 6 2 1 6 . 7

6 2 1 6 . 36 2 3 5 . 66 2 1 6 . 8

6 2 1 5 . 76 4 6 1 . 36 2 1 6 . 1

6 2 1 5 . 46 6 6 3 . 66 2 1 5 . 7

ROW NUMBER

PFOPFWPFG

6 2 1 2 . 4 6 1 6 2 . 6 6 2 1 1 . 8

6 2 1 6 . 2 0160.1 6 2 1 6 . 7

6 2 1 0 . 2 6 I 6 6 . I 6 2 ( 0 . 7

0 2 1 6 . 2 6166.1 6 2 1 6 . 7

6 2 1 6 . 2 6166.1 6 2 1 6 . 7

6 2 1 6 . 26 1 6 8 . 46 2 1 6 . 7

6 2 1 6 . 36 2 1 6 . 96216.8

6 2 1 5 . 66 3 4 3 . 56 2 1 5 . 9

6 2 1 5 . 36 5 6 4 . 06 2 1 5 . 6

6 2 1 5 . 56 9 8 9 . 96 2 1 5 . 9

Table 9 (cont'd)SATURATION O I S T N IOUT I ÜN3 I N MATRIX UOCK

LAYER NUMBER 5****$*******

ROW NUMOER 2

SO 0 . 6 0 0 . 0 0 0 . 0 0 0 . 8 0 0 . 8 0 0 . 0 0 0 . 8 0 0 . 6 0 0 . 0 0 0 . 6 0SW 0 . 2 0 0 . 2 0 0 . 2 U 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0SG 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0

RUW NUMBER 1

SO 0 . 8 0 0 . 6 0 0 . 8 0 0 . 8 0 0 . 6 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0SW 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0SC 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0

LAYER NUMBER 4

RUW NUMBER 2

VOSC 0 . 8 0 0 . 6 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 t oSW 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0SG 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0

ROW NUMBER I

SO 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 6 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 6 0SW 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0SG 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0

LAYER NUMBER 3#»$*******#*

ROW NUMBER 2

SO 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0SW 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0SG 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0

ROW NUMOER I

SO 0 . 8 0 0 . 8 0 0 . 6 0 0 . 6 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0 0 . 8 0SW 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0 0 . 2 0

-rr n.n ,i.n ,i. n il.n n.il n.n 0.0 0.0 0. 0 0.0

Table 9 (cont'd)

LAYER NUMDER

RUW n u m b e r

SOSWSG

0.800 . 2 00 . 0

0.800 .200 .0

0 . BO 0 . 2 0 0 . 0

0.800 . 2 00.0

0.800 . 2 00 . 0

0.800 . 2 00 .0

0.800 .2 00 . 0

0.800.200 . 0

0.800.200 . 0

0.800 . 2 00.0

ROW NUMBER I

SOSWSG

0.800 . 2 00 . 0

0.800 . 2 00 .0

0. 80 0 .2 0 0 . 0

0.800 . 200. 0

0. 80 0 .2 0 0 . 0

0.800 . 200.0

0.800.200.0

0.80 0 . 2 0 0. 0

0.800.200.0

0.80 0.20 ,3 0 . 0 w

LAYER NUMBER

ROW NUMBER

SOSWSG

0.800 . 2 00.0

0.800.200 .0

0.80 0 . 2 0 0. 0

0. 80 0.20 0 . 0

0.800 . 2 00.0

0.800 . 2 00 . 0

0.800.200 .0

0.800.200.0

0.800.200.0

0.800.200 .0

ROW NUMBER I

SOSWSG

0.800.200 . 0

0.800 . 2 00 .0

0.80 0.20 0. 0

0. 80 0 . 2 0 0 . 0

0.800 . 2 00.0

O.BO0 . 2 00 . 0

0.800.200 . 0

0.800 . 2 00.0

0.800 .2 00.0

0.800.200 . 0

Table 9 (cont'd)SATURATION D I S T R I B U T I O N S I N FRACTURE

LAYER NUMBER

ROh NUMBER

SOF SWF SCF

ROW NUMBER

O.BO0 .2 00 . 0

I

O.BO 0 . 2 0 0 . 0

0 . uo 0 . 2 0 0 . 0

O.BO 0. 20 0 . 0

O.BO0 . 2 00 . 0

O.BO0 .2 00 .0

0 . 7 90 .210 . 0

0 . 3 6 0 . 6 4 0 . 0

0 . 0 40 . 9 60 . 0

0. 0 1.00 0.0

SOFSWFSCF

0 . 8 0 0 . 2 0 0 . 0

O.BO0.200 . 0

0 . BO 0 . 2 0 0 . 0

0 . UO 0. 20 0.0

O.BO 0 . 20 0 . 0

O.BO0 . 2 00. 0

0 . 7 70 . 2 30 . 0

0 . 2 6 0 . 7 4 0 . 0

0. 0 I . 0 0 0. 0

0.01 . 0 00.0

ROW NUMBER

SOFSWFSCF

0 . 0 00 . 2 00 . 0

O.BO0.200 . 0

0 . 80 0 . 2 0 0.0

0 . 00 0 . 2 0 0 . 0

0 . 0 0 0 . 20 0 . 0

O.BO0.200 . 0

0 . 7 90 . 2 10 . 0

0 . 3 10 . 6 90 . 0

0 . 0 10 . 9 90 . 0

lO

0.01 . 0 00 . 0

ROW NUMBER I

SOFSWFSCF

0.000.200 . 0

0 . 6 00 .2 00 .0

0 . 8 00 .2 00 . 0

O.BO 0.20 0 . 0

O.BO 0 . 20 0 . 0

O.BO0.200 . 0

0 . 7 70 . 2 30 . 0

0 . 2 30 . 7 70 . 0

0 . 0t . OO0 .0

0.01.000 .0

LAYER NUMBER * * # * * * » * $ $ * *

ROW NUMBER

SOFSWFSCF

O.BO0.200 . 0

0 . 0 00.200.0

0 . 8 00 .2 00 . 0

0 .0 00 . 2 00 . 0

0 . BO 0 . 2 0 0 . 0

0 . 0 00 . 2 00 . 0

0 . 7 90 .210.0

0 . 3 00 . 7 00 . 0

0. 0 1 .00 0 . 0

0 . 01 . 0 00 . 0

T&ble 9 (cont'd)ROW NUMBER

SOFSWFSCF

0 . 8 0 0 .2 0 0 .0

LAYER NUMBER * * * * * * * * * * * *

0 . 0 00 . 2 00.0

0 . 80 0.20 0 . 0

0 . 8 00.200.0

0 . 8 0 0 . 2 0 0 . 0

0 . 8 0 0.20 0 . 0

0 . 7 60 . 2 40.0

0 . 18 0 . 8 2 0 .0

0 . 01 .000 .0

0 . 01 .000 .0

ROW NUMBER

SOF SWF SOF

ROW n u m b e r

0 . 8 00 .2 00 .0

I

0 . 8 00 . 2 00.0

0 . 8 0 0 . 2 0 0 . 0

0 . 8 00.200 . 0

0 . 8 0 0 . 2 0 0 . 0

0 . 8 00.200 . 0

0 . 7 80 . 2 20.0

0 . 2 90 . 7 10 .0

0*0 1 . 0 0 0. 0

0 . 01 . 0 00 . 0

SOFSWFSCF

0 . 8 00.200 . 0

LAYER NUMBER * * * * * # * * ' * * » *

0 . 8 00 . 2 00 .0

0 . 8 0 0 .2 0 0.0

0 . 8 00 .200.0

0 . 8 00 . 2 00.0

0 . 8 00 .2 00 .0

0 . 4 30 . 5 70 . 0

0 . 14 0.86 0 .0

0.01 . 0 00.0

0.01 . 0 00 . 0

HOW NUMBER toi n

SOF SWF SCF

ROW NUMBER

0 . 8 0 0.20 0 . 0

I

O.BO 0 . 2 0 0 . 0

0 . 8 0 0 . 2 0 0 . 0

O.dO0.200 . 0

0 . 8 00 . 2 00 . 0

0 . 8 00 . 2 00 . 0

0 . 7 80 . 2 20 . 0

0 . 2 80 . 7 20 . 0

0 . 0t . OO0 . 0

0. 0 1 .0 0 0 . 0

SOFSWFSCF

0 . 8 00 . 2 00.0

0 . 8 00 . 2 00.0

0 . 8 0 0 . 2 0 0 . 0

0 . 8 00 . 2 00.0

0 . 8 00 . 2 00 . 0

0 . 8 00 .2 00 . 0

0 . 5 30 . 4 70.0

0 . 1 00 . 9 00.0

0 .0 1. 00 0.0

0.0 1 . 0 0 0 . 0

Table 9 (cont'd)FLUI U VELOCITY D I S T R I B U T I O N S IN FRACTURES * * » * * * * * * * * * * * * * * * * * * * * * * * * $ # * * * * * * * * * * * *

LAYER NUMBER $ # $ * * * » $ * * * »

ROW NUMBER

VOX 0 . 0 - 0 . 2 0 7 0 0 - 0 4 - O . I B 9 4 0 - O < iVOY 0 . 0 0 . 0 0 . 0WOZ - 0 . 9 3 5 4 0 - 0 1 - 0 . 9 3 5 4 0 - 0 1 - 0 . 9 3 0 4 0 - 0 1VWX 0 . 0 - 0 . 1 3 8 8 0 - 0 9 - 0 . 1 2 7 2 0 - 1 1VWY 0 . 0 0 . 0 0 . 0VWZ - 0 . 1 2 4 8 0 - 0 5 - 0 . 1 2 3 5 0 - 0 5 - 0 . 1 2 3 5 0 - 0 5VOX 0 . 0 0 . 0 0 . 0VGY 0 . 0 0 . 0 0 . 0VGZ 0 . 0 0 . 0 0 . 0

■ 0 . I 1 6 9 0 - 0 8 0 . 0 0 . 0 0 . 9 3 5 4 0 - 0 1 - 0 . 9 3 5 4 0 - 0 1

0 . 3 2 9 0 0 - 0 7 - 0 . 2 5 5 1 0 - 0 4 0 . 00 . 9 3 5 4 0 - 0 1

0 . 7 8 3 6 0 - 1 4 - 0 . 2 0 2 5 0 - 1 1 - 0 . 7 4 7 9 0 - 0 7 0 . 00 . 0

0.00 . 00.0

0.0

- 0 . 3 4 8 1 0 - 0 3 - 0 . 1 2 1 0 0 - 0 3 0*0 0.0

- 0 . 9 0 0 7 0 - 0 1 - 0 . 9 6 9 7 0 - 0 2 - 0 . 3 4 4 2 0 - 0 4 - 0 . 1 2 8 6 0 - 0 1 0.0 0.0

0 . 0 0 . 0 0 . 0 0.0 0 . 0

0 . 1 2 3 5 0 - 0 5 - 0 . 1 2 3 5 0 - 0 5 - 0 . 1 2 3 8 0 - 0 5 - 0 . 1 0 9 5 0 - 0 3 - 0 . 1 1 2 0 0 - 0 1 - 0 . 7 1 9 7 0 - 0 1 - 0 . 8 1 6 1 0 - 0 10.0 0.0 0.0 0.0 0.00 . 0 0 . 0 0 . 0 0 . 0 0 . 00.0 0.0 0.0 0.0 0.0

0.0 0.0 0 . 9

- 0 . 1 4 1 3 0 * 0 00.0

0.00 . 00 . 0

ROW NUMBER 1

VOXVOY

0.00.0

- 0 . 3 6 8 4 0 - 0 20 . 0

VOZ - 0 . 9 3 3 1 0 - 0 1 - 0 , 9 3 5 4 0 - 0 1 -0.0 0 . 0

VWZ - 0 . 4 4 1 8 0 - 0 5 VG>: 0 . 0

0 . 0 0.0

VWXVWY

VGYVGZ

- O . 2 4 2 9 0 - 0 7 0 . 0

- 0 . 1 2 4 8 0 - 0 5 0 . 0 0 . 0 0.0

0 . 2 0 6 7 0 - 0 40.00 . 9 3 5 4 0 - 0 1 0 . 1 3 8 2 0 - 0 9 0 . 00 . 1 2 3 5 0 - 0 5 0.0 0 . 0 0.0

• 0 . 9 5 6 6 0 - 0 7 - 0 . 4 2 9 8 0 - 0 6 - 0 . 7 3 4 2 0 - 0 4 0 . 0 0 . 0 0 . 0

- 0 . 9 3 5 4 0 - 0 1 - 0 . 9 3 5 4 0 - 0 1 - 0 . 9 3 5 1 0 - 0 1 - 0 . 6 0 2 5 0 - 1 2 - 0 . 6 0 7 0 0 - 0 9 - 0 . 3 2 1 8 0 - 0 6

0 . 0 0 . 0 0 . 0- 0 . 1 2 3 5 0 - 0 5 - 0 . 1 2 3 5 0 - 0 5 - 0 . 2 1 1 4 0 - 0 5 0 . 0 0 . 0 0 . 00 . 0 0 . 0 0 . 00.0 0.0 0.0

- 0 . 9 1 1 4 0 - 0 3 - 0 . 2 0 0 2 0 - 0 4 0.0 0.0

- 0 . 8 4 7 9 0 - 0 1 - 0 . 1 0 4 4 0 - 0 2

0 . 00 . 00.0

0 . 1 2 5 8 0 - 0 3 - 0 . 2 7 8 1 0 - 0 1 - 0 . 2 9 5 6 0 * 0 0

0 .0 0.0 0 . 0 0 . 0 0.0

- 0 . 2 7 4 0 0 - 0 3 - 0 . 1 7 6 9 0 - 0 1 - 0 . 8 1 6 1 0 - 0 1 - 0 . 8 1 6 1 0 - 0 10 . 0 0 . 0 0.0

0 . 0

0.0 0 . 0 0 . 0

0.00.00 . 00.0

0 . 0

0.00.00.0

lOo \

LAYER NUMBER

ROW NUMBER

VOXVOY

0.00.0

- 0 . 2 0 7 9 0 - 0 4 - 0 . 1 8 9 7 0 - 0 6 - 0 . 1 1 6 7 U - 0 80.00.0

VOZ - 0 . 8 7 7 5 0 - 0 1 - 0 . 8 7 7 5 0 - 0 1

O . 2 6 2 9 0 - 0 7 - 0 . 1 8 4 9 0 - 0 4 0 . 0 0 . 0

- 0 . 8 7 7 5 0 - 0 1 - 0 . 8 7 7 5 0 - 0 1 - 0 . 8 7 7 5 0 - 0 1 0 . 1 6 6 2 0 - 1 4 - 0 . 3 5 9 2 0 - 1 2 - 0 . 1 1 3 0 0 - 0 7 0 . 0 0 . 0 0 . 0

VWZ - 0 . 2 6 8 9 0 - 0 6 - 0 . 2 5 5 5 0 - 0 6 - 0 . 2 5 5 4 0 - 0 6 - 0 . 2 5 5 4 0 - 0 6 - 0 . 2 5 5 4 0 - 0 6 - 0 . 2 5 7 4 0 - 0 60 . 0 0 . 0 0 . 00 . 0 0 . 0 0 . 00 . 0 0 . 0 0 . 0

VWXVWY

0.00.0

- 0 . 2 9 5 0 0 - 1 00 . 0

VGX 0 . 0 VGY 0 . 0 VGZ 0 . 0

0 . 00 . 00.0

0 . 0 - 0 . 8 7 7 5 0 - 0 1 - 0 . 2 7 0 6 0 - 12 0.00 . 00 . 00 . 0

- 0 . 3 2 8 6 0 - 0 3 - 0 . 3 6 6 3 0 - 0 4 0 . 0 0 . 00.0 0.0 0.0 0.0

- 0 . 8 5 4 2 0 - 0 1 - 0 . 4 2 3 9 0 - 0 2 0 . 0 0 . 0- 0 . 2 5 5 7 0 - 0 4 - 0 . 1 8 1 7 0 - 0 1 - 0 . 1 6 4 9 0 * 0 0 0 . 00 . 0 0 .0 0 . 0 0 . 0

- 0 . 7 3 4 9 0 - 0 4 - 0 . 1 1 9 4 0 - 0 1 - 0 . 1 0 2 8 0 - 0 1 - 0 . 1 2 0 2 0 - 0 2 0 . 0 0 . 0 0 . 0 0 . 00.0 0.0 0.0 0.00 . 0 0 . 0 0 . 0 0 . 0

Table 9 (cont'd)- 0 . 3 6 B B 0 - 0 2 - 0.0 0.0

- 0 . 0 7 7 5 0 - 0 1 - 0 . 0 7 7SU-Ü1 - 0 . 5 3 5 6 0 - 0 0 - 0 . 2 9 3 O Ü - 1 0

0 . 0 0 .0VWZ —0 . 3 6 I 3 0 - 0 5 —0 . 2 6 6 9 0 - 0 6 - 0 . 2 5 5 4 0 - 0 6 VGX 0 . 0 0 . 0 0 . 0VGY 0 . 0 0 . 0 0 . 0VGZ 0 . 0 0 . 0 0 . 0

VOX 0 . 0 VOY 0 . 0 VOZ - 0 . 6 7 5 2 0 - 0 1 VWX 0 . 0 VWY 0 . 0

0 . 2 0 6 9 0 - 0 4 - 0 . 9 5 1 3 0 - 0 7 - 0 . 1 7 1 6 0 - 0 6 - 0 . 3 0 6 2 0 - 0 4 - 0 . 5 7 1 6 0 - 0 3 0 . 00.0 0.0 0.0 0.0 0.0 0 . 6 7 7 5 0 - 0 1 - 0 . 6 7 7 5 0 - 0 1 - 0 . 0 7 7 3 0 - 0 1 - 0 . 7 9 5 6 0 - 0 1 0 . 00 . 1 3 6 2 0 - 1 2 - 0 . 9 9 4 4 0 - 1 0 - 0 . 1 4 3 6 0 - 0 6 - 0 . 1 5 2 4 0 - 0 3 - 0 . 3 6 5 2 0 - 010 . 0 0 . 0 0 . 0 0 . 0 0 . 0

- 0 . 2 5 5 4 0 - 0 6 - 0 . 2 5 5 4 0 - 0 6 - 0 . 9 1 0 7 0 - 0 6 - 0 . 2 2 9 3 0 - 0 3 0.0 0.0 0.0 0.0 0.00 . 0 0 . 0 0 . 0 0 . 0 0 . 00 . 0 0 . 0 0 . 0 0 . 0 0 . 0

0 . 00.00.0

- 0 . 2 9 7 3 0 * 0 00 . 0

0 . 00.00.00.00.0

0 . 3 2 5 1 0 - 0 2 - 0 . 9 0 1 6 0 - 0 3 - 0 . 5 1 6 1 0 - 0 30 . 00 . 00.0

0 .00.00.0

LAYER NUMBER

ROW NUMBER

VOX 0 . 0 - 0 . 2 0 7 9 0 - 0 4 - 0 . 1 6 9 6 0 - 0 0 - 0 . 1 2 7 4 0 - 0 6 - 0 . 8 4 2 0 0 - 0 7 - 0 . 3 1 2 2 0 - 0 4 - 0 . 3 4 6 1 0 - 0 3 - 0 , 2 4 5 4 0 - 0 4 0 . 0VOY 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0VOZ - 0 . 9 2 7 0 0 - 0 1 - 0 . 9 2 7 0 0 - 0 1 - 0 . 9 2 7 0 0 - 0 1 - 0 . 9 2 7 0 0 - 0 1 - 0 . 9 2 7 0 0 - 0 1 - 0 . 9 2 6 9 0 - 0 1 - 0 . 6 9 0 4 0 - 0 1 - 0 . 3 5 6 5 0 - 0 2 0 . 0VWX 0 . 0 - 0 . 3 0 4 1 0 - 1 1 - 0 . 2 7 6 0 0 - 1 3 - 0 . 1 6 6 1 0 - 1 5 - 0 . 1 0 9 4 0 - 1 1 - 0 . 6 3 2 6 0 - 0 6 - 0 . 3 9 9 4 0 - 0 4 - 0 . 2 0 3 4 0 - 0 1 - 0 . 1 7VWY 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0VWZ - 0 . 4 1 0 7 0 - 0 7 - 0 . 2 7 0 7 0 - 0 7 - 0 . 2 6 9 6 0 - 0 7 - 0 . 2 6 9 5 0 - 0 7 - 0 . 2 6 9 5 0 - 0 7 - 0 . 9 6 3 0 0 - 0 7 - 0 . 9 8 0 3 0 - 0 4 VGX 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0VGY 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0VGZ 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0

0 .00 .00 . 00.00.0

- 0 . 1 2 4 7 0 - 0 1 - 0 . 1 1 0 1 0 - 0 1 - 0 . 1 3 9 5 0 - 0 20 .0 0 .0 0.0

ROW NUMBER I(O'-I

VOX 0 . 0VOY 0 . 0VOZ - 0 . 9 2 4 6 0 - 0 1 VWX 0 . 0VWY 0 . 0VWZ - 0 . 3 5 1 6 0 - 0 3 - 0 . 4 1 Ü 1 0 - U 7 VGX 0 . 0 0 . 0VGY 0 . 0 0 . 0VGZ 0 . 0 0 . 0

- 0 . 3 6 0 8 0 - 3 2 0.0

- 0 . 9 2 7 0 0 - 0 1- 0 . 7 9 4 4 0 - 0 90 . 0

- 0 . 2 0 7 U O - 0 40.0

- 0 . 9 2 7 0 0 - 0 1- 0 . 3 0 2 0 0 - 1 10.0

- 0 . 2 7 0 1 0 - 0 7 0.0 0.0 0.0

- 0 . 9 5 4 3 0 - 0 7 0 . 0

- 0 . 9 2 7 0 0 - 0 1 - 0 . 1 4 0 2 0 - 1 3 0.0

- 0 . 2 6 9 5 0 - 0 7 0.0 0.0 0.0

- 0 . 2 7 4 6 0 - 0 6 0.0

- 0 . 9 2 7 0 0 - 0 1 - 0 . 1 2 0 6 0 - 1 0 0.0

- 0 . 2 6 9 6 0 - 0 7 0.0 0 . 0 0.0

- 0 . 4 7 5 0 0 - 0 40 . 0

- 0 . 9 2 6 7 0 - 0 1- 0 . 1 5 3 1 0 - 0 60.0

-0 . 8 1 6 0 0 - 0 60.00 . 00 . 0

- 0 . 5 6 6 2 0 - 0 3 O. 0

- 0 . 6 1 9 1 0 - 0 1 - 0 . 2 4 5 5 0 - 0 3 0.0

- 0 . 1 4 3 0 0 - 0 3 0 . 0 0 . 0 0 . 0

0 . 0 0 . 0 0 . 00 . 0 0 . 0 0 . 00.0 0.0 0.0

- 0 . 6 4 9 9 0 - 0 1 - 0 . 2 9 2 5 0 * 0 0 0 . 0 0.0 0.0 0.00 . 9 5 9 1 0 - 0 20 . 00 . 00 . 0

- 0 . 6 4 6 9 0 - 0 3 - 0 . 5 7 5 0 0 - 0 30 . 0 0.0 0 .0

0 .00.00.0

LAYER NUMBER 2

ROW NUMBER

VOX 0 . 0VOYVOZVWXVWYVWZVGXVGYVGZ

0 .0 - 0 . 6 7 7 6 0 - 0 1 0 .0 0.0 0 . 0 0 . 0 0 . 0 0 . 0

- 0 . 2 0 8 0 0 - 0 4 - 0 . 1 6 9 9 0 - 0 6 - 0 . 1 2 9 6 0 - 0 8 - 0 . 6 4 4 4 0 - 0 7 - 0 . 2 0 8 9 0 - 0 4 - 0 . 2 7 9 5 0 - 0 3 - 0 . 1 9 6 5 0 - 0 4 0 . 00 . 0 0 . 0

- 0 . 6 7 7 6 0 - 0 1 - 0 . 6 7 7 6 0 - 0 10 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0

0.0 0. 0 0.0 0 . 0 0 . 0 0.0

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101

u n d e r r u n n i n g o i l . The w a t e r f r o n t s i n t h e r o c k m a t r i x move

s l o w e r t h a n t h e o n e s i n t h e f r a c t u r e s . W a t e r - o i l r a t i o

r e a c h e s 100 ( c o n d i t i o n f o r w a t e r b r e a k t h r o u g h ) i n a p p r o x i

m a t e l y 257 d a y s a s shown i n F i g u r e 1 2 .

Thomas e t a l o r i g i n a l r e s u l t s a r e i l l u s t r a t e d i n F i g u r e

13 . C o m p a r i s o n s b e t w e e n t h e two r e s u l t s i n d i c a t e t h a t w a t e r

moves much f a s t e r and t h u s b r e a k s t h r o u g h e a r l i e r i n t h i s

s t u d y t h a n w h a t h a d b e e n r e p o r t e d by Thomas e t a l .

The r e a s o n f o r t h e s e d i s c r e p a n c i e s may be e x p l a i n e d a s

f o l l o w s .

6 . 2 . 1 D i f f e r e n c e i n t h e I n j e c t i o n R a t e s

Thomas e t a l s e t i n i t i a l w a t e r i n j e c t i o n r a t e a t 7000

S T B /d ay w i t h a c o n s t r a i n t t h a t t h e r a t e w o u l d be a d j u s t e d and

r e d u c e d a s t i m e p r o g r e s s e d s o t h a t t h e f l o w i n g b o t t o m h o l e

p r e s s u r e w o u ld n o t e x c e e d 7400 p s i g . I n t h i s s t u d y , h o w e v e r ,

t h e c o n s t a n t w a t e r i n j e c t i o n r a t e o f 7000 S T B /d ay a n d p r o d u c

t i o n r a t e o f 4000 S T B /d ay a r e u s e d d u e t o t h e i n c o m p l e t e d a t a

g i v e n on p r o d u c t i v i t y i n d i c e s a n d r a t e s c h e d u l e s . T h i s p r o

d u c e s a n e t e f f e c t o f a d d i n g 3000 S T B /day w h i c h i n e v i t a b l y

l e a d s t o a b u i l d - u p o f t h e r e s e r v o i r p r e s s u r e . I n r e a l i t y ,

t h e r e s e r v o i r a s w e l l a s t h e w e l l b o r e e q u i p m e n t c a n n o t a c c o m

m o d a t e s u c h a b u i l d - u p f o r a l o n g p e r i o d o f t i m e w i t h o u t

a d j u s t m e n t o f t h e r a t e s .

T h i s d i f f e r e n c e i n i n j e c t i o n r a t e s c h e d u l e , t h e r e f o r e ,

c o n t r i b u t e s t o t h e e a r l i e r w a t e r b r e a k t h r o u g h r e p o r t e d i n

t h i s s t u d y .

102

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90

80

cc 7 0 O ^ 60

5 0

30

20

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0 2 0 4 0 6 0 8 0 100 120 140 160 180 2 0 0 2 2 0 2 4 0 2 6 0t i m e , d ays

F i g u r e 12 W a t e r - O i l R a t i o v e r s u s t i m e .

103

PRODUCTIONWELL WATER-OIL EXAMPLE

INJECTIONWELL

FRACTURE 50% SW CONTOUR

1 YEAR

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tWATER-OIL EXAMPLE,

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I

1 YEAR _

Figure 13. F r a c t u r e 50% W a t e r S a t u r a t i o n P r o f i l e and M a t r i x 40% W a t e r S a t u r a t i o n P r o f i l e ( a f t e r Thomas e t a l . e x a m p l e ) .

1046 . 2 . 2 D i f f e r e n c e i n t h e C a p i l l a r y P r e s s u r e D a t a

Thomas e t a l s e t t h e f r a c t u r e c a p i l l a r y p r e s s u r e t o

z e r o and u s e d s t r a i g h t - l i n e f r a c t u r e r e l a t i v e p e r m e a b i l i t i e s

i n t h e i r s t u d y . They w e r e a b l e t o n e g l e c t c a p i l l a r y p r e s

s u r e s i n t h e f r a c t u r e by v i r t u e o f t h e i r s o l u t i o n t e c h n i q u e .

T h e i r s ch e m e r e d u c e d t h e f l o w e q u a t i o n s i n t o t h r e e e q u a t i o n s

i n t h r e e f r a c t u r e d u n k n o w n s : p r e s s u r e , w a t e r s a t u r a t i o n and

g a s s a t u r a t i o n . I n t h i s s t u d y , t h e f o r m u l a t i o n schem e p r o

d u c e s s i x e q u a t i o n s i n s i x u n k n o w n s . The s i x unknow ns a r e

o i l - , w a t e r - , and g a s - p r e s s u r e s i n t h e r o c k m a t r i x and t h e

f r a c t u r e s . T h i s t y p e o f f o r m u l a t i o n r e q u i r e s n o n z e r o c a p i l

l a r y p r e s s u r e d a t a , p a r t i c u l a r l y t h e n o n z e r o s l o p e s o f

v e r s u s s a t u r a t i o n .

T h i s b a s i c d i f f e r e n c e i n p r o b l e m f o r m u l a t i o n t o g e t h e r

w i t h Thomas e t a l a s s u m p t i o n s o f z e r o f r a c t u r e c a p i l l a r y

p r e s s u r e s a r e b e l i e v e d t o be p a r t i a l l y r e s p o n s i b l e f o r t h e

d i f f e r e n c e s b e t w e e n t h e tw o r e s u l t s . I n t h e f o l l o w i n g s e c t i o n

i t w i l l b e shown t h a t t h e s o l u t i o n t o t h e p r e s s u r e e q u a t i o n s

i s , i n f a c t , i n f l u e n c e d l a r g e l y b y t h e c a p i l l a r y - p r e s s u r e -

r e l a t e d t e r m s .

6 . 2 . 3 D o m in an ce o f t h e C a p i l l a r y - p r e s s u r e - r e l a t e d

C o e f f i c i e n t s

One a d v a n t a g e o f u s i n g t h e PSOR m e t h o d t o s o l v e t h e

p r e s s u r e e q u a t i o n s i s t h e f a c t t h a t t h e m e th o d e x p r e s s e s an

unknown e x p l i c i t l y i n t e r m s o f known e s t i m a t e s o f o t h e r v a r

i a b l e s . T h e r e f o r e , i t i s e a s y t o o b s e r v e t h e r e l a t i o n s h i p s

105among v a r i a b l e s a n d how much i m p a c t a c e r t a i n v a r i a b l e h a s

on t h e f i n a l s o l u t i o n .

To i l l u s t r a t e t h i s p o i n t , a p o r t i o n o f t h e c o e f f i c i e n t

m a t r i x o f one g r i d p o i n t a t a c e r t a i n t i m e s t e p ( g r i d 5 , 1 , 1

a t t i m e s t e p 20) i s l i s t e d and shown i n F i g u r e s 1 4 , 1 5 , 16

a n d 17 f o r t h e w a t e r a n d o i l e q u a t i o n s i n t h e r o c k m a t r i x ,

and t h e w a t e r and o i l e q u a t i o n s i n t h e f r a c t u r e s , r e s p e c t i v e l y .

I n F i g u r e 1 4 , b e s i d e s c o e f f i c i e n t s (BW) a n d (DW), t h e

c o e f f i c i e n t (UW) h a s t h e l a r g e s t m a g n i t u d e among a l l o t h e r

c o e f f i c i e n t s . S i n c e t h e c o e f f i c i e n t (UW) b e l o n g s t o t h e

v a r i a b l e t h e s o l u t i o n f o r w i l l b e s o l e l y d o m i n a t e d

by t h e v a l u e o f P ^ q * shown i n A p p e n d i x C, t h i s (UW) i s

b a s i c a l l y a f u n c t i o n o f ( d S ^ ^ / d P ^ ^ ^ ^ ) . T h u s , i t i s a p p a r e n t

t h a t t h e v a l u e o f t h i s c a p i l l a r y - p r e s s u r e - r e l a t e d t e r m p l a y s

a m a j o r r o l e i n i n f l u e n c i n g t h e a n s w e r .

S i m i l a r l y , i n F i g u r e 1 5 , a s i d e f r o m (BO) and (DO), t h e

c o e f f i c i e n t (UO) h a s t h e l a r g e s t m a g n i t u d e . (RO), w h i c h i s

t h e n e x t l a r g e s t c o e f f i c i e n t , i s s e v e r a l o r d e r s o f m a g n i t u d e

l e s s t h a n (U O ). H e n c e , t h e s o l u t i o n f o r P ^ ^ w i l l b e d o m i

n a t e d m o s t l y b y t h e v a l u e s o f P ^ g w h i c h h a s t h e c o e f f i c i e n t

(UO). I n o t h e r w o r d s , t h e o i l p r e s s u r e i n t h e r o c k m a t r i x

i s r e l a t e d m o re c l o s e l y t o t h e o t h e r p h a s e p r e s s u r e s i n t h e

r o c k m a t r i x t h a n t o t h e o i l p r e s s u r e i n t h e f r a c t u r e s . T h i s

i s why f l u i d s i n t h e r o c k m a t r i x move a t a s l o w e r p a c e t h a n

t h e y do i n t h e f r a c t u r e s .

A f t e r i n s p e c t i n g F i g u r e s 16 and 1 7 , s i m i l a r f i n d i n g s

106

BU ** * * * * * * * * * ******************* -0 .5 6 0 1 d 3 1

DU * * * * * * * * * * * * * *************************** 0 . 2801d5

GU ** 0 .0

EU ** 0 .0

AU ** 0 .0

e u ** 0 .0

FU ** 0 .0

HU ** 0 .0

RU ** 0 .0

UU * ***************************** 0 .5 6 0 1 d 3

Iw i ,j ,k = ^ °^'i,3,k^lw\,0,k-1 - ^ 4 , j ,k ^ lC \ , j - 1 ,k - * \ , j , k ^ l w \ - 1 , j . k

■ ° ' ' i ,j ,k Iw i+1,j,k “ ^ i , j ,k ^ lw i,j+1,k" ®‘'i,j,k^lw i,j,k+1

BW _ Ijy \i.j.k^fw i , j , k "î,j,k^lo i , j , k ■’

F i g u r e 14 . W a t e r E q u a t i o n i n Rock M a t r i x and R e l a t i v e M a g n i t u d e s o f i t s C o e f f i c i e n t s .

107

BO ******************************************** 1 -0.2289d7 1DO *************************************** 0.2952d6VO *********************** 0.2790d3GO ** 0.0EO ** 0.0AO **** 0.1707d0CO **** 0.1707d0FO **** 0.1707d0HO ****************** 0.2731d2UO ******************************************** 0.2287d7RO *************************** 0.1739d4

^ - °i,j,k^lw i.j.k ~ °°i,j,kô i,j,k-1 ~ °i,j,k*lo i,j-1,ki.j.k

■ '^°i,j,kô\-l,3 ,k - ®°i,j,k^lo\+1,d,k ■ î.j.k^lî.j+l.k

\,a,k^lo i,j,k+1 “ i,j,k Is i,j,k i,j,k fo i.j.k

F i g u r e 15 . O i l E q u a t i o n i n Rock M a t r i x and R e l a t i v e M a g n i t u d e s o f i t s C o e f f i c i e n t s .

108FBU ********************** 1 -0.1931d2 1DFU ****************************** 0.96A9d3FRU ** 0.0FGUl ** 0.0FGU2 ** 0.0FGW3 ** 0.0FGU4 ** 0.0FGW5 ** 0.0FEWl ** 0.0FEU2 «* 0.0FEU3 ** 0.0FAUl ** 0.0FCUl ** 0.0FFUl ** 0.0FFW2 ** 0.0FFU3 ** 0.0FHWl ** 0.0FHU2 ** 0.0FHU3 ** 0.0FHW4 ** 0.0FHW5 ** 0.0FUU ********************** 0.1931d2

fw 1,j,k ■ ^l,j,k^lw i,j,k “ j.k^fw i,j-l,k-li,0 ,k

■ ^ ' ’' 2 i , j , k ^ w " ' i - 1 , j , k - 1 - ^ ' ' ' ^ i . 3 , k ^ f w ’' x , d , k - 1 - F ° ' ' \ . j , k P % \ + 1 , j , k - ^ = ” 5 i , 3 , k P ? w \ . 3 * 1 . k - 1

™ \ . j , k ^ : \ . i , j _ i , k - ^ • ' 2 i j ,Æ \ .3 - - i .k ■ ^ ^ i . j , k C \ + i , j - i . k - F A w \,j ,k ^ :\ . i , j ,k

-n+1

„n+1

- FHW1. . ,i , j , k f w i , j - 1 , k + 1 - FHW2. . . - FITW3.. PÎi* ,• - FHVvA. .i,j,k"fw i+l,j,k+l

Figure 16.- - ™ ' i ,d ,k ^ ; ’ i , i . k >W a t e r E q u a t i o n i n F r a c t u r e and R e l a t i v e M a g n i t u d e s o f i t s C o e f f i c i e n t s .

i u y

FBO ************************ ************** I -0.8063d5 IDFO ************************************* 0.1441d5FRO ********************************* 0.173?<MFVO ****************** 0.9620dlF601 ** 0.0FG02 ** 0.0FG03 ** 0.0FG04 ** 0.0FG05 ** 0.0FEOl ** 0.0FE02 ** 0.0FE03 ** 0.0FAOl ************** 0.1707«ilFCOl ************** 0.1707dlFFOl ** 0.0FF02 ************** 0.1707dlFF03 ** 0.0FHOl ** 0.0FH02 ** 0.0FH03 ********************** 0 .2 7 3 1 d 2

FH04 ** 0.0FH05 ** 0.0FUO *************************************** 0.7885d5

i . j . k

- FOOI^^j i , j - 1 , k - 1 ■ ^ ° ^ i , j , k ^ f o i - 1 , j , k - 1 " ^ ° ^ i , j , k ^ o i , j , k - 1

■ ^ ° ^ i , j , k ^ o i , j - l , k ■ ^ “' ° ^ i , ô , k ^ f o i + 1 , j - l , k " ^ ' ^ ° ' ' i , j , k ^ f o i - 1 , j . k

■ ^ ° \ , j , k ^ f r i + l , j , k ■ ^ ° \ , j , k ^ f o \ - 1 , j + l , k ■ ^ ° ^ i , j , k * * f o i , j + 1 , k

- FH03^ j i , j , k + 1 “ i + n . j . k + l " ^ “ ° ^ i , j , k ' ’f o i , j + 1 , k + 1

F i g u r e 17 . O i l E q u a t i o n i n F r a c t u r e and R e l a t i v e M a g n i t u d e s o f i t s C o e f f i c i e n t s .

110e m e r g e . The v a r i a b l e l a r g e l y r e s p o n s i b l e f o r t h e s o l u

t i o n i s w h i c h h a s t h e c o e f f i c i e n t (FUW) r e l a t e d t o

^ ^ c o w f ^ ' f o r t h e v a r i a b l e , i t s s o l u t i o n i s

l a r g e l y a f f e c t e d by t h e v a l u e o f P£g w h i c h h a s t h e c o e f f i

c i e n t (FUO) r e l a t e d t o ( d S ^ ^ / d P ^ ^ ^ ^ ) .

The i m p l i c a t i o n i s t h a t t h e p h a s e p r e s s u r e s i n t h e

f r a c t u r e s r e l a t e t o e a c h o t h e r m ore r a p i d l y a n d s p o n t a n e o u s l y

t h a n t h e y do t o t h e p h a s e p r e s s u r e i n t h e r o c k m a t r i x , and

v i c e v e r s a . S o , t h e f r a c t u r e i n j e c t i o n / p r o d u c t i o n r a t e s

w o u ld b e f e l t f a s t e r i n t h e f r a c t u r e s t h a n i n t h e r o c k m a t r i x ,

and c o n v e r s e l y f o r t h e r o c k m a t r i x .

C o n s e q u e n t l y , one c an make t h e f o l l o w i n g d e d u c t i o n :

t h e c o n v e n t i o n a l way o f a s s u m i n g t h a t f l u i d s f i r s t f l o w f ro m

t h e r o c k m a t r i x i n t o t h e f r a c t u r e s an d t h e n f l o w t h r o u g h t h e

f r a c t u r e s t o w a r d t h e w e l l b o r e may n o t b e a n a d e q u a t e d e s c r i p

t i o n o f t h e f l o w p h en o m en o n i n t h e n a t u r a l l y f r a c t u r e d r e s e r

v o i r . What i s l i k e l y t o o c c u r i s t h a t a p o r t i o n o f f l u i d s

w o u ld c o n v e r g e t o t h e w e l l b o r e t h r o u g h t h e r o c k m a t r i x w h i l e

a n o t h e r p o r t i o n i s c o n v e r g i n g t h r o u g h t h e f r a c t u r e s . H o w e v e r ,

t h e r a t i o o f t h e r o c k m a t r i x f l o w t o t h e f r a c t u r e f l o w i s

d i f f i c u l t , i f n o t i m p o s s i b l e , t o d e t e r m i n e .

6 . 3 E f f e c t o f L o w e r i n g I n j e c t i o n R a t e

As d i s c u s s e d e a r l i e r , t h e u s e o f c o n s t a n t 7000 S T B /day

w a t e r i n j e c t i o n r a t e i n e v i t a b l y l e a d s t o u n r e a l i s t i c p r e s s u r e

b u i l d - u p i n t h e r e s e r v o i r , and may r e s u l t i n a c o n v e r g e n c e

p r o b l e m . In o r d e r t o i n v e s t i g a t e t h i s r a t e e f f e c t , t h e w a t e r

I l l

i n j e c t i o n r a t e i s r e d u c e d t o 4000 S T B / d a y . The s i m u l a t i o n

p r o g r a m i s r u n w i t h t h i s new r a t e an d t h e r e s u l t s a r e shown

i n F i g u r e s 18 and 1 9 . I t i s n o t e d t h a t w a t e r b r e a k t h r o u g h

o c c u r s a t a p p r o x i m a t e l y 288 d a y s , an i n d i c a t i o n t h a t w a t e r

moves t h r o u g h t h e r e s e r v o i r a t a s l o w e r p a c e t h a n t h e c a s e

i n F i g u r e s 10 a n d 1 1 .

T h i s s u g g e s t s t h a t i n o r d e r t o k e e p t h e r e s e r v o i r

p r e s s u r e f r o m g r o w i n g o u t o f c o n t r o l , a r e a s o n a b l e r a t e

s c h e d u l e s h o u l d e m p l o y a s t a i r - s t e p p a t t e r n i n w h i c h one r a t e

w o u ld b e m a i n t a i n e d a t a c o n s t a n t f o r a c e r t a i n p e r i o d o f

t i m e f o l l o w e d by a new l o w e r e d r a t e w h i c h w o u ld be c o n s t a n t

f o r a n o t h e r p e r i o d o f t i m e , an d s o o n .

6 . 4 E f f e c t o f F l u i d V e l o c i t y i n t h e F r a c t u r e

The f r a c t u r e v e l o c i t y d i s t r i b u t i o n s s u c h a s t h e one

shown i n T a b l e 9 a r e o b s e r v e d o v e r a n e x t e n d e d p e r i o d o f t i m e ,

I t i s n o t e d t h a t t h e a b s o l u t e v a l u e s o f t h e s e v e l o c i t i e s a r e

l e s s t h a n 3 . 5 f t / d a y . W i t h s u c h a s m a l l v a l u e , one t e n d s t o

t h i n k t h a t f l u i d f l o w t h r o u g h t h e f r a c t u r e s w o u l d b e i n a

l a m i n a r r e g i o n , a n d t h e r e f o r e w o u ld n o t b e a f f e c t e d much by

t h e f r a c t u r e v e l o c i t y .

To c h e c k t h i s p o i n t , t h e s i m u l a t i o n p r o g r a m i s r e r u n

w i t h o u t c a l l i n g t h e VELOCY s u b r o u t i n e a t t h e e n d o f e a c h t i m e

s t e p . T h i s a m o u n t s t o k e e p i n g v e l o c i t y and a c c e l e r a t i o n a t

t h e i r i n i t i a l c o n d i t i o n s o f z e r o , an d i n e f f e c t , n e g l e c t i n g

t h e f l u i d v e l o c i t y i n t h e f r a c t u r e s . The r e s u l t s o f t h i s r u n

t u r n o u t t o b e i d e n t i c a l t o t h e f i r s t r u n ( F i g u r e s 10 and 11)

' r

250

200

(D(D

V)

^100Cu

0

■4- h - >2000500 1000

length (feet)1500

Figure 18. F r a c t u r e 50% Water S a t u r a t i o n P r o f i l e i n j e c t i o n r a t e = 4000 STB/day)

(Center c r o s s - s e c t i o n ; wa t er

A

250

200

d)0)

(A^100CXÜ!£

0

I

CaI

500 1000 length (feet)

1500 2 0 0 0

Fijîure 19. Mat r ix 50% Water S a t u r a t i o n P r o f i l e i n j e c t i o n r a t e = 4000 STB/day).

( c e n t e r c r o s s - s e c t i o n , water

114T h i s i n d i c a t e s t h a t f o r t h e p e r m e a b i l i t y d a t a and p r o d u c t i o n

d a t a u s e d , t h e e f f e c t o f f l u i d v e l o c i t y i n t h e f r a c t u r e s i s

n e g l i g i b l e .

I n c i d e n t a l l y , w i t h r e g a r d t o p o r o s i t y , i t s h o u l d b e m en

t i o n e d t h a t p o r o s i t y d i s t r i b u t i o n s i n b o t h t h e r o c k m a t r i x

and t h e f r a c t u r e r e m a i n s r e l a t i v e l y u n c h a n g e d t h r o u g h o u t t h e

s i m u l a t i o n p e r i o d .

6 . S R e c o m m e n d a t io n s f o r F u r t h e r I m p r o v e m e n t s

The s i m u l a t i o n p r o g r a m d e v e l o p e d i n t h i s s t u d y r e p r e s e n t s

a s i g n i f i c a n t s t e p t o w a r d a b e t t e r u n d e r s t a n d i n g o f t h e n a t u

r a l l y f r a c t u r e d r e s e r v o i r . H o w e v e r , b e c a u s e o f t h e c o m p l e x i t y

o f i t s m a t h e m a t i c a l m o d e l , t h e s i m u l a t o r may h a v e some f l a w s

i n a d v e r t e n t l y h i d d e n i n i t . And c e r t a i n l y , i t w i l l t a k e more

p r o g r a m m i n g t e s t s w i t h m ore f i e l d e x a m p l e s t o p e r f e c t t h e

s i m u l a t o r . I n t h i s s e c t i o n , s u g g e s t i o n s f o r i m p r o v e m e n t o f

t h e s i m u l a t i o n p r o g r a m a r e d i s c u s s e d .

The r a n g e an d m a g n i t u d e o f a c e r t a i n d a t a s e t i s v e r y

i m p o r t a n t i n i n s u r i n g t h e s t a b i l i t y o f t h e p r o b l e m . F o r

i n s t a n c e , t h e i n j e c t i o n / p r o d u c t i o n r a t e s s h o u l d n o t b e s o

u n r e a l i s t i c a l l y h i g h t h a t t h e f o r m a t i o n c a n n o t s u s t a i n i t .

An i m p r o v e m e n t c a n b e made i n t h i s r e g a r d by a d d i n g new

s t a t e m e n t s i n t o t h e p r o g r a m w h i c h s p e c i f y maximum/minimum

f l o w i n g b o t t o m h o l e p r e s s u r e s a n d r e l a t e th e m w i t h t h e b l o c k

p r e s s u r e an d p r o d u c t i v i t y i n d e x t o c o m p u te t h e f l o w r a t e s .

A n o t h e r i m p r o v e m e n t c a n b e a c h i e v e d by a d d i n g s t a t e m e n t s

115w h i c h a u t o m a t i c a l l y d e c r e a s e t h e t i m e s t e p s i z e i f t h e s a t u r

a t i o n c h a n g e s d u r i n g t h e p r e v i o u s t i m e s t e p a r e t o o l a r g e .

I t s h o u l d b e p o i n t e d o u t t h a t t h e l a r g e s t t i m e s t e p

s i z e u s e d i n t h i s s t u d y i s 1 d a y , f o r a c o n v e r g e n c e p r o b l e m

i s e x p e r i e n c e d when a l a r g e r s t e p s i z e i s u s e d .

A l o n g - t e r m i m p r o v e m e n t may b e t h e d e v e l o p m e n t o f a

s i m i l a r p r o g r a m f o r a l i m i t e d c o m p o s i t i o n a l h y d r o c a r b o n m o d e l .

M o r e o v e r , i n s t e a d o f a l i n e a r i n t e r p o l a t i o n r o u t i n e , t h e u s e

o f o t h e r t y p e s o f i n t e r p o l a t i o n t e c h n i q u e s h o u l d b e e x p l o r e d .

F i n a l l y , i t m u s t be e m p h a s i z e d t h a t t h e q u a l i t y o f t h e

s i m u l a t i o n r e s u l t s i s o n l y a s g o o d a s t h e q u a l i t y and a c c u r a c y

o f t h e i n p u t d a t a . T h e r e f o r e , s p e c i a l c a r e s h o u l d b e t a k e n

i n o b t a i n i n g o r m e a s u r i n g d a t a f o r t h e s i m u l a t o r .

CHAPTER V I I

CONCLUSIONS

T h i s s t u d y p r e s e n t s t h e d e v e l o p m e n t o f a n u m e r i c a l

s i m u l a t o r f o r n a t u r a l l y f r a c t u r e d r e s e r v o i r s . The p r i n c i p a l

c o n c l u s i o n s c a n b e s u m m a r i z e d a s f o l l o w s .

1 . The new m a t h e m a t i c a l f l o w m o d e l i s d e r i v e d f o r

c h a r a c t e r i z i n g t h e m u l t i p h a s e f l u i d f l o w t h r o u g h

n a t u r a l l y f r a c t u r e d r e s e r v o i r s . T h i s m o d e l p e r m i t s

f l o w t h r o u g h t h e r o c k m a t r i x a n d t h e f r a c t u r e t o be

d e s c r i b e d s i m u l t a n e o u s l y . The f l u i d i n t e r a c t i o n

t e r m s , t h e g r a v i t y e f f e c t s , t h e c a p i l l a r y f o r c e s

a n d t h e f r a c t u r e v e l o c i t y o f f l u i d s a r e a l l t a k e n

i n t o c o n s i d e r a t i o n .

2 . A f i n i t e - d i f f e r e n c e , t h r e e - d i m e n s i o n a l , b l a c k o i l

s i m u l a t o r h a s b e e n d e v e l o p e d a c c o r d i n g t o t h i s new

m o d e l .

3 . The s i m u l a t o r i s c a p a b l e o f h a n d l i n g t h e i s o t r o p i c ,

h e t e r o g e n e o u s r o c k m a t r i x a n d t h e a n i s o t r o p i c ,

h e t e r o g e n e o u s f r a c t u r e a s w e l l a s t h e f r a c t u r e

o r i e n t a t i o n .

4 . The p r e s e n c e / a b s e n c e o f f r a c t u r e s a t any p a r t i c u l a r

l o c a t i o n i n t h e r e s e r v o i r c a n b e s t a t i s t i c a l l y g e n

e r a t e d a p r i o r i by t h e Monte C a r l o m e t h o d ; t h i s

116

117r a n d o m l y g e n e r a t e d l o c a t i o n c a n t h e n b e i n p u t t e d

i n t h e s i m u l a t i o n p r o g r a m .

5 . F o r t h e d a t a u s e d i n t h i s s t u d y , t h e p h a s e p r e s

s u r e s r e s p o n d t o e a c h o t h e r m o re r a p i d l y an d s p o n

t a n e o u s l y i n t h e f r a c t u r e s t h a n t h e y do t o t h e

p h a s e p r e s s u r e s i n t h e r o c k m a t r i x , an d v i c e v e r s a .

6 . The c o n v e n t i o n a l way o f a s s u m i n g t h a t f l u i d s f i r s t

f l o w f r o m t h e r o c k m a t r i x i n t o t h e f r a c t u r e and

t h e n f l o w t h r o u g h t h e f r a c t u r e t o w a r d t h e w e l l b o r e

may n o t b e an a d e q u a t e d e s c r i p t i o n o f t h e f l o w p h e

nomenon i n t h e n a t u r a l l y f r a c t u r e d r e s e r v o i r s .

7. F o r t h e d a t a u s e d i n t h i s s t u d y , i t a p p e a r s t h a t

t h e f r a c t u r e v e l o c i t y o f f l u i d s h a s a n e g l i g i b l e

e f f e c t on t h e r e s u l t s .

8 . The s i m u l a t o r d e v e l o p e d r e p r e s e n t s s i g n i f i c a n t a d

v a n c e s t o w a r d a b e t t e r u n d e r s t a n d i n g o f t h e f l u i d

f l o w p h e n o m en o n i n t h e n a t u r a l l y f r a c t u r e d r e s e r

v o i r s .

NOMENCLATURE

B = N o n d i m e n s i o n a l c o n s t a n t ; o r f o r m a t i o n v o lu m e

f a c t o r ( b b l / S T B o r b b l / S C F )

= Mass f r a c t i o n o f c o m p o n e n t i i n p h a s e j• - 1c = C o m p r e s s i b i l i t y ( .ps i )

D = D e p t h m e a s u r e d p o s i t i v e downward ( f t )

d = F r a c t u r e h a l f w i d t h ( f t )

f ( r , f 2 ) = F r a c t u r e d i s t r i b u t i o n f u n c t i o n

g = A c c e l e r a t i o n o f g r a v i t y2

g^ = U n i t c o n v e r s i o n f a c t o r ( 3 2 . 1 7 4 I b m - f t / l b f s e c )

K = A b s o l u t e p e r m e a b i l i t y (md)

K£ = F r a c t u r e p e r m e a b i l i t y t e n s o r (md)

= R e l a t i v e p e r m e a b i l i t y o f p h a s e j

£ o r 1 = C h a r a c t e r i s t i c h a l f l e n g t h o f a f r a c t u r e ( f t )

N = T o t a l n u m b e r o f g r i d p o i n t s

P = F l u i d p r e s s u r e ( p s i )

q = F low r a t e , p o s i t i v e f o r i n j e c t i o n and n e g a t i v e f o r

P r o d u c t i o n (S T B /d ay o r S C F /d a y )

R = S o l u t i o n g a s - o i l r a t i o (SCF/STB)

r ^ = H y d r a u l i c r a d i u s f o r f r a c t u r e ( f t )

S = P h a s e s a t u r a t i o n

t = Time ( d a y )

Vg = B u lk v o lu m e ( f t ^ ) = Ax*Ay*Az

Vj = V e l o c i t y o f f l u i d p h a s e j ( f t / d a y )

118

119

x , y , z = R e c t a n g u l a r c o o r d i n a t e s

a .= A n g l e o f t h e x - a x i s t o t h e h o r i z o n t a l p l a n e

6 = A n g l e o f t h e y - a x i s t o t h e h o r i z o n t a l p l a n e ;

o r p a r a m e t e r i n f l u i d i n t e r a c t i o n t e r m

r = F l u i d i n t e r a c t i o n t e r m b e t w e e n t h e f r a c t u r e a n d

t h e r o c k m a t r i x

Y = A n g l e o f t h e z - a x i s t o t h e v e r t i c a l p l a n e

e = T o l e r a n c e l i m i t

X = F l u i d m o b i l i t y

y = F l u i d v i s c o s i t y ( c p )

p = F l u i d d e n s i t y ( I b m / f t ^ )

$ = F low p o t e n t i a l ( p s i )

4> = P o r o s i t y

(Ü = R e l a x a t i o n p a r a m e t e r

N o t e : U n i t s , i f g i v e n , a r e i n o i l f i e l d u n i t s .

S u b s c r i p t s

c = C a p i l l a r y p r e s s u r e

f = F r a c t u r e

G o r g = Gas

i = H y d r o c a r b o n c o m p o n e n t ; o r g r i d p o i n t i n x - d i r e c t i o n

j = H y d r o c a r b o n p h a s e ; o r g r i d p o i n t i n y - d i r e c t i o n

k = G r i d p o i n t i n z - d i r e c t i o n

1 = Rock m a t r i x

0 o r o = O i l

r = R e l a t i v e v a l u e

S = S t a n d a r d c o n d i t i o n s

120

W o r w = W a t e r

S u p e r s c r i p t s

k = I t e r a t i o n l e v e l

n = Time s t e p l e v e l

Sy m b o ls

= V e c t o r

= = T e n s o r

V = N a b l a d i f f e r e n t i a l o p e r a t o r

A = D e l t a o r i n c r e m e n t

i = U n i t v e c t o r i n x - d i r e c t i o n

j = U n i t v e c t o r i n y - d i r e c t i o n

k = U n i t v e c t o r i n z - d i r e c t i o n

REFERENCES

1 . A l - M a r h o u n , M. A . , " O p t i m a l N u m e r i c a l P r o c e d u r e t o S o l v eT w o - D i m e n s i o n a l T h r e e - P h a s e P e t r o l e u m R e s e r v o i r S i m u l a t o r , " P h . D . D i s s e r t a t i o n , S c h o o l o f P e t r o l e u m a n d G e o l o g i c a l E n g i n e e r i n g , U n i v e r s i t y o f Oklahom a ( 1 9 7 8 )

2 . A q u i l e r a , R . , " T h e U n c e r t a i n t y o f E v a l u a t i n g O r i g i n a l O i li n P l a c e i n N a t u r a l l y F r a c t u r e d R e s e r v o i r s , " SPWLA 1 9 t h A n n u a l L o g g i n g Sym pos ium , J u n e 1 3 - 1 6 , 1 9 7 8 .

3 . B i r d , R. B . , W. E. S t e w a r t and E. N. L i g h t f o o t , T r a n s p o r tP h e n o m e n a , W i l e y , New Y o r k , NY ( 1 9 6 0 ) .

4 . B o s s i e - C o d r e a n u , D . , P . R. B i a a n d J . S a b a t h i e r , "TheC h e c k e r M ode l on Im p r o v e m e n t i n M o d e l l i n g N a t u r a l l y F r a c t u r e d R e s e r v o i r s w i t h T r i d i m e n s i o n a l T r i p h a s i c B l a c k O i l N u m e r i c a l M o d e l , " SPE P a p e r No. 1 0 9 7 7 , SPE-AIME, D a l l a s , TX 1 9 8 2 .

5 . C o a t s , K. H . , W. D. G e o r g e , C. Chu and B. E . Marcum," T h r e e - D i m e n s i o n a l S i m u l a t i o n o f S t e a m f l o o d i n g , " SPE P a p e r No. 4 5 0 0 , SPE-AIME, D a l l a s , TX ( 1 9 7 3 ) .

6 . C o l l i n s , R. E . , F lo w o f F l u i d s T h r o u g h P o r o u s M a t e r i a l s ,The P e t r o l e u m P u b l i s h i n g C o . , T u l s a , OK ( 1 9 7 6 ) .

7 . C r i c h l o w , H. B . , M odern R e s e r v o i r E n g i n e e r i n g - AS i m u l a t i o n A p p r o a c h , P r e n t i c e - H a l l , NJ ( 1 9 7 7 ) .

8 . D u g u i d , J . 0 . a n d P . C. Y . L e e , " F l o w i n F r a c t u r e d P o r o u sM e d i a , " W a t e r R e s o u r c e s R e s e a r c h , V 1 3 , No. 3 , p 5 5 8 , J u n e 1 9 7 7 .

9 . E v a n s , R. D . , "A P r o p o s e d Model f o r M u l t i p h a s e F lowT h r o u g h N a t u r a l l y F r a c t u r e d R e s e r v o i r s , " SPE P a p e r No. 9 9 4 0 , SPE-AIME, D a l l a s , TX ( 1 9 8 1 ) .

1 0 . E v a n s , R. D . , " C h a r a c t e r i z a t i o n o f F l u i d F lo w i n N a t u r a l l yF r a c t u r e d R e s e r v o i r s , " R e s e a r c h R e p o r t No. DOE/BC/ 1 0 2 5 5 - 4 , U. S . D e p t , o f E n e r g y , B a r t l e s v i l l e E n e r g y T e c h n o l o g y C e n t e r , B a r t l e s v i l l e , OK ( 1 9 8 1 ) .

1 1 . F a r o u g , A l i , S . M . , N u m e r i c a l R e s e r v o i r S i m u l a t i o n ,P e n n s y l v a n i a S t a t e U n i v . , U n i v e r s i t y P a r k , PA ( 1 9 7 1 ) .

121

1221 2 . H o r n b e c k , R. W . , N u m e r i c a l M e t h o d s , Quantum P u b l i s h e r s ,

I n c . , New Y o r k , NY ( 1 9 7 5 ) .

1 3 . K a z e m i , H . , " P r e s s u r e T r a n s i e n t A n a l y s i s o f N a t u r a l l yF r a c t u r e d R e s e r v o i r s , " S o c i e t y o f P e t r o l e u m E n g i n e e r s J o u r n a l (D ec . 1 9 6 9 ) , p 4 5 1 .

14 . K a z e m i , H . , L. S . M e r r i l l , K. S. P o r t e r f i e l d a n d P . R.Zeman, " N u m e r i c a l S i m u l a t i o n o f W a t e r - O i l F lo w i n N a t u r a l l y F r a c t u r e d R e s e r v o i r s , " S o c i e t y o f P e t r . E n g r s . J o u r n a l (D ec . 1 9 7 6 ) , p 3 1 7 .

1 5 . P e a c e m a n , D. W . , F u n d a m e n t a l o f N u m e r i c a l R e s e r v o i rS i m u l a t i o n , E l s e v i e r S c i e n t i f i c P u b l . C o . , 1 9 7 7 .

1 6 . R o s s e n , R. H . , " S i m u l a t i o n o f N a t u r a l l y F r a c t u r e dR e s e r v o i r s w i t h S e m i - I m p l i c i t S o u r c e T e r m s , " SPE P a p e r No. 5 7 3 7 , SPE-AIME, D a l l a s , TX 1 9 7 6 .

1 7 . S a w y e r , W. K. an d J . C. M e r c e r , " A p p l i e d S i m u l a t i o nT e c h n i q u e s f o r E n e r g y R e c o v e r y , " R e s e a r c h R e p o r t No. M E T C / R I - 7 8 / 9 , U. S . D e p t , o f E n e r g y , M o rg an to w n E n e r g y T e c h n o l o g y C e n t e r ( A u g u s t 1 9 7 8 ) .

1 8 . S e t t a r i , A. a n d K. A z i z , " T r e a t m e n t o f N o n l i n e a r Termsi n t h e N u m e r i c a l S o l u t i o n o f P a r t i a l D i f f e r e n t i a l E q u a t i o n s f o r M u l t i p h a s e F low i n P o r o u s M e d i a , "I n t . J . M u l t i p h a s e F lo w , V o l . 1 , pp 8 1 7 - 8 4 4 , P e r g a m o n / E l s e v i e r , 19 7 5 .

1 9 . S m i t h , G. D . , N u m e r i c a l S o l u t i o n o f P a r t i a l D i f f e r e n t i a lE q u a t i o n s , O x f o r d U n i v . P r e s s , New Y o r k , NY ( 1 9 6 5 ) .

2 0 . S t o n e , H. L. , " E s t i m a t i o n o f T h r e e - P h a s e R e l a t i v eP e r m e a b i l i t y a n d R e s i d u a l O i l D a t a , " J . C a n a d i a n P e t r . T e c h . , ( O c t - D e c . 1 9 7 3 ) , V o l . 1 2 , No. 4 , p 5 3 .

2 1 . S t o n e , H. L . , " P r o b a b i l i t y Model f o r E s t i m a t i n g T h r e e -P h a s e R e l a t i v e P e r m e a b i l i t y , " J o u r n . o f P e t r . T e c h . ( F e b . 1970) , p 21 4 .

2 2 . T h o m as , L. K . , T . N. D ix o n a n d R. G. P i e r s o n , " F r a c t u r e dR e s e r v o i r S i m u l a t i o n , " SPE P a p e r No. 9 3 0 5 , SPE-AIME, D a l l a s , TX ( 1 9 8 0 ) .

2 3 . T h o m as , G. W . , P r i n c i p l e s o f H y d r o c a r b o n R e s e r v o i rS i m u l a t i o n , I n t e r n a t i o n a l Human R e s o u r c e s D e v e l o p m e n t C o r p . , B o s t o n , MA ( 1 9 8 2 ) .

2 4 . W a r r e n , J . E . a n d P . J . R o o t , "The B e h a v i o r o f N a t u r a l l yF r a c t u r e d R e s e r v o i r s , " S o c . o f P e t r . E n g r . J o u r n . ( S e p t . 1 9 6 5 ) , p 24 5 .

1232 5 . Yamamoto , R. H . , J . D. P a d g e t t , W. T . F o r d a n d A.

B o u b e g u i r a , " C o m p o s i t i o n a l R e s e r v o i r S i m u l a t o r f o r F i s s u r e S y s t e m s - The S i n g l e - B l o c k M o d e l , " S o c i e t y o f P e t r . E n g r s . J o u r n . ( J u n e 1 9 6 2 ) , p 1 7 7 .

APPENDIX A

C o m p a r i s o n o f U n i t s a n d C o n v e r s i o n F a c t o r s

Symbol D e f i n i t i o n SIU n i t s

D a r c y o f cfis u n i t s

O i l f i e l dU n i t s

**C o n v e r s i o nF a c t o r s

q F low R a t e m^/S cm^/S S T B /d ay 5 . 4 3 4 3 9 6 x 10^

Gas Flow R a t e m^/S cm^/S S F C /d a y 3 . 0 5 1 1 8 7 X 10^

k P e r m e -a b i l i t y m2 d a r c y md 1 . 0 1 3 2 5 X 10^^

y V i s c o s i t y P a . S cp cp 1 X 10^

p P r e s s u r e Pa a tm p s i 1 . 4 5 0 3 7 7 X 10 '^

x , y , z D i s t a n c e s m cm f t 3 . 2 8 0 8 4

t Time S 5 d a y 186400

c C o m p r e s - s i b i l i t y Pa‘ ^ a tm ^ p s i ‘ ^ 6 . 894757 x iO^

T T e m p e r a t u r e K K R 1 . 8

P D e n s i t y kg /m ^ gm/cm^ 6 . 2 4 2 7 9 7 x 10'2

g G r a v i t a t i o n a l A c c e l e r a t i o n m /s 2 c m /s2 f t / s 2 3 . 2 8 0 8 4

Sc C o n v e r s i o nC o n s t a n t 1 kg-m 2 gm cm

N . s 2 ^ d y n e s2o r

1Pa*S *m

3 2 . 1 7

I b ^ - S ^

3 2 . 1 7

h T h i c k n e s s m cm f t 3 . 2 8 0 8 4

V V e l o c i t y m / s cm /s f t / d a y 2 . 8 3 4 6 4 6 x 10^

**

The S I i s a c o h e r e n t u n i t , s o e q u a t i o n s w r i t t e n i n t h i s u n i t do n o t n e e d n u m e r i c a l c o n v e r s i o n f a c t o r .

To c o n v e r t f r o m t h e SI u n i t t o t h e o i l f i e l d u n i t , m u l t i p l y t h e SI u n i t by t h e c o r r e s p o n d i n g c o n v e r s i o n f a c t o r .

124

125Some U s e f u l C o n v e r s i o n F a c t o r s

A r e a :

P e r m e a b i l i t y :

V i s c o s i t y :

Volume :

P r e s s u r e :

1 a c r e =

1 d a r c y

43 5 6 0 f t '

1000 md

= 9 . 86923 X 10,-13 m'

9 . 86923 x 10’ ^ cm^

1 p o i s e = 1 gm

1 f f "

1 m3

1 a tm

cm* s

100 cp

0 . 1 Pa*S

= 6 . 7 1 9 6 8 9 X 10,-2 l b m

= 2 . 0 8 8 5 4 3 X 10r 3

f t *5 l b ^ * S

f t '

1 b b l = 5 . 6 1 4 5 8 3 f t '

42 g a l

1 g a l = 3 7 8 5 . 4 3 cm'

231

7 . 4 8 0 5

2 8 . 3 1 7

6 . 2 8 9 8

i n

g a l

l i t e r

b b l

= 760 mm Hg. (0®c)

= 3 3 . 8 9 9 5 f t w a t e r ( 4 ° c )

= 1 4 . 6 9 6 p s i

= 1 . 0 1 3 2 5 b a r

= 1 . 0 1 3 2 5 X 10^ Pa ( p a s c a l o r N/m^)

Mass : 1 I b ^ = 4 5 3 . 5 9 2 4 3 gm.

APPENDIX B

COMPARISON OF EQUATIONS UNDER SI UNITS AND OILFIELD UNITS

S I UNITS OILFIELD UNITS

D a r c y ’ s Law F o r F low i n Rock M a t r i x

K ,k , -Vt . = - 0 . 0 0 6 3 2 8 3 [VP

I j P j i IJ

Pz-iVD]144 ^ j lE q u a t i o n o f M o t i o n I n F r a c t u r e

' n " j f

- P j , ^V.D1

k - f K rV f . = - 0 . 0 0 6 3 2 8 3 - '[V P . .

±1 j f _ ^3V

+ 2 . 8 9 1 7 3 8 X 1 0 '^ ^ p . — ^] f 9 t

144 P; . VD]

W a t e r E q u a t i o n f o r F lo w i n Rock M a t r i x

v . [ V ï i ï ï (VP WS

® w l ^ l Iw B VD}]wl

- [^ l ^ r l w ^ f v s

1 3 t

K k0 . 0 0 1 1 2 7 V [ j } -y— {VP,

^ w l^ w lWS

144 B

-0.001127 [

wl

^l^rlw'J’fH ^^ l^ w l

VD}]

C^lw-^fw)]

5 7 ^ 5 hàXr a y -

1 2 6

127SI UNITS OILFIELD UNITS

Oil Equation for Flow in Rock MatrixK,k

i v D ) ]

, . V r l o * f t P lo - ^ f o L^H^ôlôl

A x - A 'y - ^ ' = "^1 & ( ^ o l / ^ o l ^

K k0 . 0 0 1 1 2 7 V - [ ^ L - ^ {VP -

® o l^ o

VD}]144 ^ o l

0 . 0 0 1 1 2 7 ' ^ £ o ^ ]

Ax*Ay*ST = sTgTs '^ l

Gas E q u a t i o n f o r F low i n Rock M a t r i x

K ,k

- - g # " " - ' " ' '

_ , V r l o * A o l^ H ^ ^ o l ^ o l

____________ 9 r ^ g l ^ ^ s o l ^ o lA x ' A y A z '*'1 9t ^ B+

’o l1

K k0 .0 0 1 1 2 7 V -

g l g l ®

m PglVD}] + 0 . 0 0 1 1 2 7 V'

- 0 . 0 0 1 1 2 7 [

- 0 . 0 0 1 1 2 7 [ ; l y Z l o l i _ s o l ( P i o " ^ f o ^ ]

V s cj).Ax-Ay-Az 5.615 1 9 t Bg l

^ so l^ o lBb l

128SI UNITS OILFIELD UNITSWater Equation for Flow in Fracture

V- •{7P^.+W w f ^ » w f 8 c

^ V D } ]wf S r

" 3 %

% S f = A rfwfi Ax'AyAz 9 t

0.001127A* [ = 2 ! ^ .{VP

+ 2 .891738x10<-14 * wsBwf

1 ^ws144 B VD}]

wf

. 0.001127

^wsfAx'AyAz 5.615 9 t ^wf

a [ ^ ]

O i l E q u a t i o n f o r F lo w i n F r a c t u r e

® o f ^ o f fo o l g^. 9 t

- C o l # - ™ ) ]

* ' T O f i

0 [ fo f ]Ax'AyAz f 9 t Bof

0.0011277.[ g _ ^ .{VP,*ofCof

+ 2.891738 X 10'- '"p9Vfo

o l 9 t

m C o l ™ } ]

0.001127 I ] }’lo -ô lô l

^ o s fAx'Ay.Az 5.615 ^ f 9 t of

129

S-I UNITS OILFIELD UNITS

Gas E q u a t i o n f o r F lo w i n F r a c t u r e

- ’

-, 3Vjf oo l 3 t - P o l ^ V D } ]

^G s£Ax*Ay*A z

^ s o f ^ o fBo f

k K0 . 0 0 1 1 2 7 V - [ ^ I S l ^ . { v p +

3V

g lf £ -2 . 8 91738 X 10’ ^^ p

Î & O g l ’ D> 1

+ 0 . 0 0 1 1 2 7 V * [ - 4 2 £ _ I £ i L ^ . { V P . _ rS t oo f ^ o f

+ 2 . 8 9 1 7 3 8 X 10’ ^^ p

P. 1 VD}]

9Vf oo l 3 1

144 ^ o l

+ 0.001127 [ f’’lo " ’’fo^]^ o l^ o l

+ "^Gsf _ 1 ,Ax*AyAz 5.615 *Pf 3t

r f g £ + ^SOfôf\ f ®of

APPENDIX C

EXPRESSIONS FOR THE COEFFICIENTS

W a t e r E q u a t i o n i n M a t r i x Rock = 1 when w a t e r i si n c o m p r e s s i b l e )

, ^ l ^ r l w

BW = - V r 2 < 1 rlw .n+% + 1_ f 1 rlM-,n-*-a1,3 B ^2( -k , j ,k-is ^^2 ’ 1, j ,kH-%

+ r . _1_ r ^ l^ r lw .Ay" ^wl^^wl I ^ i j + i , , k

. ( ^ l ^ r iw .n+h . r ^ l^ r lwAx" ^wl:"wl ^ i + i g j . k

+ V r l w p +%2 1V \ l ^ w l gn+3.

^n gn+3^. ^ i . 3 . k ' ' ^ i , j , k f _ l _ | ’nf% -*

it t Ji,j,k'

" i , j , k

* This last term is zero when water is assumedincompressible.

130

131

RW = . ^ l ^ r l w .n+%H w l^ w l

UW = V £ l 2 l i U ç t ® w l ' l , j , ki j , k

. , t V ? . i . k (^10- C . , )At- I T.+l< -‘•Jj»^^î,j,k " ^%slî,j,k At gii+a

X j . k

+ gn+^ f j ^ ' l ’n + î s p n *'" i,.j,k ®w] i,j,k ^î,j,k

- ^ H j . k ^ ) I : L . k - ( % ) I - L . k '

. ^ " " i L .k t ‘ ^ '

*This term is zero when water is assumed incompressible

132Oil Equation in Matrix Rock

GO- . = - ^ fh i } ^ o l ^ o l

Vt3 K . ,k_ ,_ Ti+hAn = B_ . 1 r l o ,Ax^

BO = -V r + _%_ r % i oi , k - [ ^^2 I J i j ,k-;, 2 i 4 j - ; , , kAy

A xZ ^B ol^ 'o l A x 2 ^ B o l : : o l

+ 1 . V r l o . 1 , n*hC-R ,, ) i iw-l. tr '"' ? ( R 1, ^4

Ay2 ®olôl Az2 ô lô l

n ,CS„-,)V 4+ ( ^ ^ i . i . k Ç ^ A l o ^ ^ * l ^ i , i

V ^01:^01 i ' i ' k^ r ' " g i ' i , i , k

X , j , k

133

CO = _Yë_ V r l o

FO. . = f ^ l ^ r l oi > j > k ^ y 2 ® o l ^ o l i , j + ^ , k

HO. . = Z g _ c ^ l ^ r l oAz^ ^ o l ^ o l i» j> k+ î2

UO. . , = J _ , k / ^ g l ^ i , j ,k1 >1 jk « rp nll +

% , j . k

H" " o l ^ o l

“ l i O . k

° 4 , j , k

_ ^ ^ g l ^ i j , k pn + pn i

< , j . k

- t

- V " o l ) i , j , k ^ - I 4 , j - > s , k '

. v ® o i ’ î j . k t “ S r ) w . k 4 - ( S r ^ W ' k - ^ >

134Gas Equation in Matrix Rock

v e , , .

GGO. = Z à . r ^ s o l ^ l ^ r l oi ' j ' k A ® o l ^ o l

( % X T ° j . k

1 , ^ s o l ^ l ^ r l o 1 ^ s o l ^ l ^ r l oA%2 * 0 ^ 0 1 6x2 ' Bolôl

1 , ^ s o A ^ f l o I 1 r ^ s o l ^ l ^ r l oAy2 Bolôl ^

+ ^ * f ' i . i . k . ^ A l o ^ s o l + ( * l ) j , 1 , k r ' ^ ^ g l ^ i , i , kV ®olôl i ' i ' k

2 ^ i , j , k

. ' V ' , ,^n+î2 %n+32

° ^ i , j , k ° H , j , k

fR’

' j 3 , ) »1 ,J jK

135

'B

Ax

VB

Ay'

^ s o l ^ l ^ r l o

G G i , j , k

A G i , j , k

* G i , j , k

'B

Az

VB

Az

VB

Ay

V

R s o l K l t r l o B o l ^ o l i ' j ' k + %

, K l ^ T l s

S g i V

, K l ^ r l c

B

Ax'

, C l a r i s

K , k

k

n+ h K , k . n+h- V r f 1 r l s ' + _ 2 ^ r 1 r i g 1 '

B ( Az2 ( B g l ^ g l AyZ ^ g l ^ g l

Ax

1

1 _ f K l^ T lE BglWgl

* 1 f V r l g h + h , j ,kAx B g lV g l

(K l ^ T l s n + 's

Ay) 4 4

1 r ^ l ^ r l gi , j + % , k Az^ B g i ^ g i i , j , k + %

n

+ S 'n+3s

2 l i , j , k

S ^ i , j , k B g i i , j , k

r s ’ 1 * s o l i , i , k l S g l J

.n + hB°î , j ,k

%'Ç'T TO,r MI >l‘ Ç‘ T IM %'C'T lOS +L U<1 uL »SJ 5,^yü

, ‘ C‘ x o i A T, ,U ^ ^ ‘ Ç‘ T lo s I '- 5j+uS‘ %+uS-T:J

5£+U*-

^ ' ^ luS +

’i‘ t ‘ î 3 i "‘‘ • ‘ y S a ,

^ S'Ç'TrlS . %'L'T lOS 5j+u I5j+U

n : ? % 'Ç 'T f T S 3 y i _ ^T f T ( i i g ) " '

U

V Ç‘ t . I ^ j

% .u a .11 (jj j A

S 'Ç 'T g O

% 'Ç 'T r t ° d t ° 3 . , T^J _ % 'P 'T o g y

= % 'Ç 'T ,r h * ç ‘ j . t s , f S R , i f !

’ Sa - — ' 3 J

>1‘ f ‘ ^ + î f _xv) % - = ■’■‘ f ' î s D

9£T

137

- a f . . « ( i j g p , ; % , y

. ÿ < " " > i J .» i . ! . « ( , ^ „1

W a t e r E q u a t i o n i n F r a c t u r e (B ^ £ = l when w a t e r i s i n c o m p r e s s i b l e )

' ( b j - i . k * ( ) w . k - i '

^ t e b - L . k * t b j . k - i î

C . k - ,

-Vt, K „ _ k „ . ^ n+î2 K, n+h

FGW5--Vg K . _ k „ . . ^ n + h K _ „ k „ . ^ n + i__________rr yz rwf . "zy'rwf ,

i j , k 4AyAz Bwf%wf i ' j + l ' k \ f ^ w £ i ' î ' k ' l

138

' f e f e f ' w - , . k

^ B « i . i , k 4 Â % [ ( C j . k t % f ) " j - l . k l

^ * 1 0 , 1 = ' -'^B ' i + % . : . k * I ? (

_ L _ ( K yy^rwf + _ i _ r * y y ^ rw fAyZ ( BwfVwf ) i , j + % , k AyZ ) i , j - % , k

, 1 ç ^zz^rw f 1 , ^zz^rw fAz2 BwfWwf ) i , j , k + % Az% ) i , i , k - %

^ ( * f ) i \ i . k ç h \ l w ^ V i . i . k % ^ ï , 1 ,kV \ l ^ w l At

4 V. n „n+% 1 ' ] '

+ l O f J j . i . k i , j , k ç 1 . ' n + % . *A t I 3 - ^ J i , j , k ]

- w i i , , . , = - ^ [ ( ) i : L . k c ) l l i . k !

FFW2„ , ,. = - ! ^ (l ' i ' k Ay2 Bwfl 'wf ^ i . 3 * > s . k

* This term is zero when water is incompressible.

139

™ « i j . k = ^ t c ) I j - l . k ^ ( % f ) w . k . l '

FHW2 - ~ r f '1^^^ + f ^ z x ^ w f ,

™ « i , ^ . k = J C . k * .

™ « i j . k = 4 ^ t ( % f ^ m j . k ^ t % f

= 4 ^ [ f

™ « i . j ,k = ( ë E C Ï .

° ™ i . j . k ' ■ ' • w s P i J . k . jH+isH j , k

> n f.n+h■ *fw i , i , k V £ f ^ L j J c ® ' w f ‘i , j , k , _2 _ -.'n+’-î pn

Bvif i 'k f ^ l , j , k

2Axg^ '■ B^jpy^ ' ' i j , k 3 t J i + l , j , k 3 t ' ' i - l , j , k n+h

This term is zero when water is assumed incompressible.

140

V r PwfVrvtf . ^^fwy n nJ i , j , k l l 9t J i + l , j , k - I at J i - l , j , k ]

\ ,k _g_ j. V rwf .n+% _ , V r w f ,

Pwf^xz^rwf ^ fwz n ^^fwz ,

2Axgc '’i , j , k ^ ' - 3t ' ' i + l , j , k 3t ^ i - l , j , k ^

Zb ^^w£î,j ,k _g_ r , ^xz^rw f .n+% _ . ^xz^rwfAx ^ % d*wf ) i+ % , j ,k C ) i - , ; , j , ^

^B , Pw £^>^nrf r ç ^^£wx . ^^fwx ,ZAygc I B ^ y ^ . ^ i , j , k I I 3t J i , j + I , k ■ 3t ' ' i , j - l , k J

s t ^ - n - , , . - ■ % w

B , Sv^rwfP-wf r( ZZfwy _ r fZfHZ )* ]f ë r B . y . ^ i , j , k i 3t ' ' i , j + l , k *■ 3t J i , j - l , k J

n+h

> t “ - « ï i g > " « . « - < f e

^B , Pw & z ^ r wf _ ^^fwz^wf^wf2Ay g r ( B . y . ) i , j , k [( 3t ) i , j + l , k - ( 3t ^ i,j-l,k ^

s t W ' ^ w

^B , Pwf^zx^rwf ■*■'2 _ (■ ^^fwx i

2Az g^ '• B ^ y ^ J i , j , k 3 t • ' i , j , k + l “ '■ 3t ' ' i , j , k - l

141n+h

. 1 c „ - , % C w

. r - ’ ” ' * c « - < % ■

f ^wf^zz^rwf . ^^fwz . ^^£wz ,I R 1, J i 4 ^ I I Ï + J i 4 V + 1 I -^+ J i 4 V- ' l - *2Az gg ®w£^tf£ i , ] , k 3t i , j , k + l 9t ^ i , j , k - l

- t - " % > " « - ' % > :!» >O i l E q u a t i o n i n F r a c t u r e

FGOl - r j- ^ y z ^ r o £ . ^ z y ^ r o f ,

FG02 = p , ^ x z ^ r o f + ç ^ z x ^ r o f •.i , j , k 4AxAz ® o f^ o £ ^ o f ^ o f i ' j ' ^ - l

h , i . k - h

FG05 - ~^B pp ^ y z ^ r o £ + r ^ z y ^ r o £ ii j , k 4AyAz ® o £ % £ ® o £ % £

F P m - ^B p p ^ x y ^ r o £ ^ p ^ y x ^ r o £ -.

i , j , k " 4 Ax Ay (- ®o£ô£ ^

142

= 4 ^ t( C i . j . k ' (

FBO. = -V r — f ^ y ^ o f V ■ + — ( _i ' j ' k B ^^2 ^QfVQf i + h . W ^ Ax^ ®o£ôf

+ ( V r o f + _ i_ r V r o fAy2 BofWof i ' i+ % 'k Ay^ ®ofôf

, 1 , K zz tro f ,*+% . 1 f K zz tro f .*+%i ; ? ( ) i , j , k + % Az2 ( B ofôf

f f O l i . j . k ' 4 Â 3 & ' f ’ i - l j , k * ( ) i . j + l , k '

FF03^

FHOl

. , V r o f . , V r c f ,i , j , k 4AxAy ® o £ ^ o f i + ô f 'ô f

_ -^B _ K yzkro£ . ^ z y ^ o £ ,i , j , k 4AyAz [ (- J i , j - l , k S o £ ^ o £

143

FH02 = r f x z ^ r o f + <- ^ z x ^ r o fi , j , k 4AxAz -’ i - l , j , k i , j , k + l ^

™ ' i . j . k - A ; t ( h l l : , , * C ) w . k . i '

FHOS = -T g — [ f * y z ^ r o f + f ^ z y ^ r o f ,

FUO = ^ B ( * f ) j . i . k ( ® g f ^ i , j , k

FVO = ' ’' ^ B ^ ^ f ^ l . i . k f ® w p ” . j , k

" C i . j . k

” F O i j , k = - ( % s f : ' i j , k -

ffYi c ^ + h _ qn+î w 1 -x’n + h[ { ( I ' S w f i , j , k ^ g f ! , ] , % ) ( B . J i , i , k

>T,+U

'o f

,ri r c i -i’.ïï+ îi C ( S v f ) i , i , k ■ ^ f o i , i , k

®of i , j , k

+B',n+h" i , j , k " (^^gf^ i j , k % i , j , k ^ ^

o f i , j , k

(. ^wl^xx^rof r f ^^fox _ r ^ ^ 5 5 . i2Â ^ c Bof:iof - 3 t ) i + i , i , k a t J i - i , ] , k

144

'"B - P p f V r c f n+% , f ! j ^ l ’' 12Ax I ô£ô£ i ' j ' k 9t ' ' i + l , j , l c '• 3 t ' ' i - l , j , k J

, % A A o f "*=« f i f o z - r ^ l " . 12Ax ®o£’ o£ i ' i ' k 9 t - ^ i+ l , j , k a t ' ' i - l , j , k

, ^ a ^ B ^ r o f ""== , f ! Ü o x " - r ^ l " 1

2Ay I ô£’ o£ i ' i ' k i , i + l , k 3t

< = » 6 è “ ’” ',1'■ % > L . « ■ ‘ % >".)

VB , PofK yzkrcf f^£gz v" _ , l^ £ o z' 2Ay I ^ i , j , k 3t '’i , j + l , k a t -’i , j

[ f g z ] a t J i , j - l , k

2 ^ ( ^ 4 ^ ) l lk '( % ,k*i - C % ) I j ,kal

145

V.

Âi"i X „ % < % C w

_ , P o f V r o f , , 5 s z : i " . r l " 12Az ®o£ô£ i ' j ' k ' ' i , j , k + l 3 t ■ ' i , j ,k - l^

■ s t “ " ' ^ < % ', , s v f o z ,

2Az ®o£ô£ i ' j ' k 3 t i , i , k + l 3 t - ' i , j , k - l ^

■ k t ‘ Ï S < %Gas E q u a t i o n i n f r a c t u r e

= 4 & " % C l . K * ' % ’ Î l l c - l l

FGG2 = , ^ x z ^ r g f , ^ z x ^ r g f ]f G G Z i , j , k 4 A X A Z I I B g ^ y g ^ J i - l j . k B g ^ P g ^ i , j , k - 1 ^

" = G 4 i j , k = 4 Â & : ( ’ Î j . k - l '

= % ; & ' ( % C j . , k * c % )T ik _ i'

" : " i j . k = 4 ^ [ ( ) I - W . k + f ) w - i . k '

146

P B G , , . , , . _v^ , 1 ( 7^ ( % üi x ‘ * g f% g f iX ^ “ gfW gf

_ L . r K y y ^ r g f * _ ± _ , V ^ r g f " * “=, y 2 f 5 i J * % , l c , „ 2 t ) l , j

. _ j _ f K z z k r g f .»+% , _ 1 _ f K ; z t r g f "*■*Az^ ' BgfWgf Az^ ' BgfWgf

A +" L -n+Lr o f _ R •'■1 4

- 1^+^ 1 es» ;)^ 1 + ( * f ) j , i , k f ^1 ^ r lg ,s o f i , j , k B g f ^ i J . k > r ^ l ^ i . j . k

f C G l l . j . k ' ^ ( T ^ iv

FFG3Vg K„..k___, n + h K r ^ K r r f ^+1___________ r r W r g f + r ^ x ^ r g f ,

i , j , k 4AxAy BgÛgf ' ' i + l , j , k ®gf l^gf %

147

= 4 Â ^ tt

B™ ^ i , j , k " 4AxAz '- Bg£Ug£ -*i + l , j , k '■ ^gf^^gf

= % & H % ( % ) w . k . i '

' G - ' i i . i . k = T w

^ « R ^ i j . k = f t * f ^ ^ % = ^ ' ) w , k - i '

- 4 & I( T § J ^ " C n , k * f % ^ ' ) I % k - i ]

™ i j . k = 4 ÿ t C ^ ^ ^ C Ï , 3 , k * t T ^ ' C - l , k ’

" ^ " ^ i j . k - ^ t ^ ^ § 5 ÿ < Î - % , k

148

p . „ , . _ , ^ s o f ^ x x ^ r o £ . n + %

FBR = -V r J L r x x ^ o f -s '*'' + _1_ r ^sof^xx^

. _J^ . *sofKyykrofxn+% + _X_ r ^sof^yy^rofAyZ I J i , i + % , k Ay2 I BofWof J i , j - % , ]

1 r ^ s o f ^ z z ^ r o f ^ n + ^ + 1 r ^ s o f ^ z z ^ r o f ^ n + ^7 I R r: V 4 4 V 4 -U 7 I T) ir ^ 4 4

A z ^ ®o£ô£ Az^ ^ o £ ^ o £ ^ » 3 » ^ ^

,n ^n j^+h

®g£ i , j , k ®o£ i , j , k

o £ i , j , k

, ( * £ ) i . i , k . ^ l ^ r l o ^ s o l ,r » ! ' S c l ^ o l "

r p n i _ ^B , ^so£^xx^rof> ,n+32i - j ’k ' 1 7 ' BofW of ' ’ l* = ^ .3 -k

™ i . j . k = 4 Â & 't

FFR^ = _ ^ s o £ V r o £ n+:3 + , ^ s o £ V r o £ .^ + ^ ,F F * 3 i , j , k 4AxAy ®o£ô£ ®o£ô£

F h r i = ~ r . + ( ^ s o f ^ z y \p £ ^^'^'^ ]

i , j , k 4AyAz [(- B^^Fg^ ^ i , j - l , k ^ i , l , k + l

P j jp . _ ’^B .Ç ^ so £ ^ x z \o £ .^ " '^ ^ r ^ s o £ S A p £ .^ '" '^ ]f H R 2 i , j , k - -4ÂEÂT [( Bof%of ) i - l , j , k ( \ f V o f ^ i ' j ' k + 1

149

FHR4 = ^ r ( ^ s o A z ^ r o f .^'^'^ + r ^ so f^ zx ^ ro fy"^^'i , j , k 4AXAZ I I ^ o f^ o f i + l ' j ' % ®o£ôf

FHR5 = ^ r r ^ so f^ y z^ ro f> '*'' r ^ so f^ zy ^ ro f. ” ' '’™ ^ i , j , k 4AyAz [ I B . y . '’i , j + l , k B . y . '’i j , k + l ^

-V fé R ^^^ rc ' - riFVG- - = B l * f J i , j , k % o £ i . i . k t \ f ^ i j , k

i , j , k Ato f i j ,k

' - c w -^ g f i , j , k

+ f _1_ \n+% _ \ o f i , i , k l ^ g f J i , j ,k , png f i , j , k l B r J i , j , k ^n+h fg i , j , k

% f i , j , k

fS’ R^"^^- ( ((s;,)" -

* g f i , j . k V i , j , l <

f R ' , , j- fl-S^ '"^ w s o f J i , i , k ,_ s o f i , j , k , 1 ) }

Swf i , j ,k V i J ,kJ gn+î2 B A , j , k 'o £ i , j , k

,n+h

^ o i , j , k + n°4 % ^ i , j , k ^fw i , j , k ô f i , j , k

- w , ( t ( ^ o .k- ( % I - i j ,ki

- 1 ' v ü . k ^ ^ ^ ) : L ,ki

150

1 t y ( % ) : LVg n+h 3V^„„ n 3V^„_ n

- ^ ) : L . k - ( % )::L - ,k=

- 1 ^ ( ) I 1 t c % )L n ..- ( % )L-I.k'

- 1 t ' ( % ’ r ? * ’- k - ( ^

- 7^ ( C , k ' ( ^ ) L . i , k - ( % : ) I j - i . k '

- 1 t - » " % ^ w ^ . . k - ( % ) w - . . k '

- ^ C . k " % C j . i , k - ( % C j - i . k '

- ^ ( ^ 4 ^ ) w . k '( % ) L . w - ( % ) i o , k - i i

- I t c % C . k _ , ,5gf«gf i , : , K » ^ V V

. u 'C g f g f

l A : ' ’ M , k t t % ) L . k . i - ( % ) L . k - % :

151

! bAz

1Ax

)I% k 1 ( % h . j ,k * r ( % C j . i c - i ’

I i c % ) w . k . , '

. P p A o f V r o f . S' fox " . r - É ^ , " ,2Ax ®ofô£ i ' j 'k 3t i+ l , i ,k 3t ^ i- l ,i ,k

^ i , j ,k _g_ i r r ^sof^xx^rof?" _ r ^sof^xx^ ,

^B . ‘ of^sof^xy^rof ^^foy _ f ^^foy T -,

2Axg^ ôfôf "ï'j'k 9t ''i+ l,j,k at ' ' i - l , j , r

^ (pof)i,j%k ^ ^ o g o f q

B , pQf^sof^xzôf , SVfoz ,2Axg^l ôf’ of i J at ''i+ l,j,k at ''i- l ,j ,k

^B . Ppf^sof^yx^rof r r ox ? _ / ^^fox ? ,2Ayg^'- Bof’ of i a t -'i,j+ l,k at - ' i j - l ,k

1Ay

^B . P o # s o fV r o f r r f oy T 1

2Ayg '- B^£%f -'ij.k '- at i j + 1 ,k at ''i,j-l,k

152

.n+h_ ^ < ^ P o £ ^ i , j ,k _g_ i ^ s o f V r o f _ Ç ^ s o f V r o f ^

. P o f R s o f K y z ^ r o f . ^ ^ £ o z ^ . ^ ^ £ o z ^ ,g r ^ Bmflkf t , j , k 3 t J i , j + l , k - (- 9 t J i , j - l , k J2Ay::c " o £ t £

_ Z â *-'ôPi,j,k_g_ cosyEC ^ s o ^ W r o f , ^so£S^z^ro£ Ây g ; C°SY[( 4 , j+ % ,k C B,fWof 4 , j - % ,k ]

^B f ô£^so£^zx^ro£ , ^^£ox •? ^^£ox •? ,2 A z B ^ f W o f J i , j , k I I 9 t ^ i , j , k + l " 3 t ^ i , j , k - l ^

^B , Po£*so£Kzykro£.P^% , , ^ ^ £ o y . ^ / ^ £ o y T ,m I R 1, Jt n V- I I Ï + J i 4 V j-1 I 3-h -<4 4 V _ l JZAzgî -%,j ,k I I 3 t ^ i , j , k + l 3 t J , k - l

^B P o £ ^ s o £ ^ z z W r r f Z f o z ^ ^ _ . ^^£oz ^ ,ZAzg^l 3 t " i , j , k + l '■ 3 t ' ' i , j , k - l ^

APPENDIX D

DATA PREPARATION

The i n s t r u c t i o n s f o r p r e p a r i n g i n p u t d a t a f o r t h i s s i m

u l a t i o n p r o g r a m a r e g i v e n b e l o w . The d a t a c a r d s m u s t be r e a d

i n t h e f o l l o w i n g o r d e r o f s u b r o u t i n e s .

S e t 1. RESDIM

S e t 2 . RESDAT

S e t 3 . INCON

S e t 4 . CODE

S e t 5 . MAIN PROGRAM

An e x a m p l e o f i n p u t d a t a a r r a n g e m e n t i s g i v e n i n A p p e n

d i x E.

S e t 1 . I n p u t d a t a f o r s u b r o u t i n e RESDIM

1 . T i t l e c a r d ___________________________ FORMAT (40A2)

2 . G r i d s i z e s a n d r e s e r v o i r d i m e n s i o n s _ _FORMAT( 3 I 5 , 3 F 1 0 * 0 )

NX c o l s 1 - 5 Number o f g r i d p o i n t s i n X - d i r e c t i o n

NY 6 - 1 0 Number o f g r i d p o i n t s i n Y - d i r e c t i o n

NZ 1 1 - 1 5 Number o f g r i d p o i n t s i n Z - d i r e c t i o n

XL 1 6 - 2 5 R e s e r v o i r l e n g t h ( f e e t )

YL 2 6 - 3 5 R e s e r v o i r w i d t h ( f e e t )

ZL 3 6 - 4 5 R e s e r v o i r t h i c k n e s s ( f e e t )

3 . F r a c t u r e h a l f w i d t h _________________ FORMAT (F IO .O )

WF c o l s 1 - 1 0 F r a c t u r e h a l f w i d t h ( f e e t )

153

154S e t 2 . I n p u t d a t a f o r s u b r o u t i n e RESDAT

1. T i t l e c a r d ___________________________ FORMAT (40A2)

2 . Code n u m b e r s f o r r o c k m a t r i x p o r o s i t y and p e r m e a

b i l i t y FORMAT ( 2 I 5 J

KPHIL c o l s 1 - 5 Code n u m b e r f o r r o c k m a t r i x

p o r o s i t y

KPERL 6 - 1 0 Code n u m b e r f o r r o c k m a t r i x

p e r m e a b i l i t y

I f KPHIL = 0 , p o r o s i t y i s u n i f o r m t h r o u g h o u t t h e

r e s e r v o i r .

I f KPERL = 0 , p e r m e a b i l i t y i s u n i f o r m t h r o u g h o u t

t h e r e s e r v o i r .

3 . T i t l e c a r d ( i f KPHIL = 0 ) _________ FORMAT (40A2)

4 . Rock m a t r i x p o r o s i t y ( i f K P H IL = 0 ) ___________ FORMAT (F IO .O )

PHIL c o l s 1 - 1 0 U n i f o r m r o c k m a t r i x p o r o s i t y

( f r a c t i o n )

5 . T i t l e c a r d ( i f KPHIL f 0 ) _________ FORMAT (40A2)

6 . P o r o s i t y d i s t r i b u t i o n i n r o c k m a t r i x ( i f KPHIL i- 0)

FORMAT ( 1 0 F 7 . 0 )

PHI ( I , J , K ) c o l s 1 - 7 P o r o s i t y d i s t r i b u t i o n f o r t h e

8 - 1 4 J ' t h ro w . NX v a l u e s m u s t be

1 5 - 2 1 r e a d - - c o n t i n u e on a d d i t i o n a l

c a r d s i f n e c e s s a r y . The row

o r d e r m u s t b e NY, NY-1 , NY-2 ,

1 . The l a y e r o r d e r m u s t

6 4 - 7 0 b e NZ, N Z - 1 , N Z - 2 , ------- 1 .

N o t e : New row o r l a y e r h a s t o b e s t a r t e d on a new c a r d .

1556 . T i t l e c a r d ( i f KPERL = 0 ) ____________ FORMAT(40A2)

7 . Rock m a t r i x p e r m e a b i l i t y ( i f KPERL = 0 ) ____ FORMAT

.(FIO.O)

PERL c o l s . 1 - 1 0 U n i f o r m r o c k m a t r i x p e r m e a b i l i t y

(md. )

8 . T i t l e c a r d ( i f KPERL f 0 ) ________________FORMAT(40A2)

9 . P e r m e a b i l i t y d i s t r i b u t i o n i n r o c k m a t r i x ( i f KPERL

f 0) ________________________________________ FORMAT(10F7.0)

K L ( I , J , K ) c o l s 1 - 7 P e r m e a b i l i t y d i s t r i b u t i o n f o r

8 - 1 4 t h e J ’ t h ro w . NX v a l u e s m u s t

1 5 - 2 1 b e r e a d - - c o n t i n u e on a d d i -

t i o n a l c a r d s i f n e c e s s a r y .

The row o r d e r m u s t b e NY, NY-1 ,

NY -2 , . . . . 1 . The l a y e r o r d e r

6 4 - 7 0 m u s t b e NZ, N Z -1 , N Z - 2 , ------- 1 .

N o t e : New row o r l a y e r h a s t o b e s t a r t e d on a new c a r d .

1 0 . T i t l e c a r d ---------------------------------------------------- FORMAT(40A2)

1 1 . Code n u m b e r s f o r f r a c t u r e p o r o s i t y and p e r m e a b i l i t y

___________ FORMAT(2I5)

KPHIF c o l s 1 - 5 Code n u m b e r f o r r o c k m a t r i x p o r o s i t y

NUMKF c o l s 6 - 1 0 Code n u m b e r f o r f r a c t u r e p e r m e a b i l i t y .

N o t e : F o r u n i f o r m d i s t r i b u t i o n s KPHIF = 0/NUMKF = 0 . As f o r

p e r m e a b i l i t y t e n s o r , i f n o t u n i f o r m (NUMKF 0 ) , one

s e t o f t e n s o r v a l u e s w i l l b e a s s i g n e d t o a l l n o d e s ;

t h e n NUMKF s e t s o f p e r m e a b i l i t y t e n s o r w i l l b e r e a d

f o r n o d e s t h a t a r e d i f f e r e n t .

1561 2 . T i t l e c a r d ( i f KPHIF = 0 ) _______________FORMAT(40A2)

1 3 . F r a c t u r e p o r o s i t y ( i f K P H I F = 0 ) _____ FORMAT(FIO.O)

PHIF c o l s 1 - 1 0 U n i f o r m f r a c t u r e p o r o s i t y

( f r a c t i o n )

1 4 . T i t l e c a r d ( i f KPHIF i 0 ) _______________FORMAT( 4 0A2)

1 5 . P o r o s i t y d i s t r i b u t i o n i n f r a c t u r e ( i f KPHIF f 0)

______________ FORMAT(10F7.0)

F P H I ( I , J , K ) c o l s 1 - 7 P o r o s i t y d i s t r i b u t i o n f o r

8 - 1 4 t h e J ' t h r o w . NX v a l u e s

— m u s t b e r e a d - - c o n t i n u e on

a d d i t i o n a l c a r d s i f n e c e s s a r y .

The row o r d e r m u s t b e NY, NY -1 ,

N Y -2 , . . . 1 . The l a y e r o r d e r

m u s t b e NZ, N Z -1 , N Z - 2 , . . . . l .

N o t e : New ro w o r l a y e r h a s t o b e s t a r t e d on a new c a r d .

1 6 . T i t l e c a r d ___________________________________ FORMAT(40A2)

1 7 . F r a c t u r e p e r m e a b i l i t y T e n s o r _________ _ FORMAT(9F8. 0 )

CFKXX c o l s 1 - 8 E l e m e n t s o f t h e p e r m e a b i l i t y

CFKXY 9 - 1 6

CFKXZ 1 7 - 2 4

CFKYX 2 5 - 3 2

CFKYY 3 3 - 4 0

CFKYZ 4 1 - 4 8

CFKZX 4 9 - 5 6

CFKZY 5 7 - 6 4

CFKZZ 6 5 - 7 2

1 8 .

1 9 .

157T i t l e c a r d ( i f NUMKF 0)

F r a c t u r e p e r m e a b i l i t y t e n s o r a n d g r i d p o i n t

l o c a t i o n s ( i f NUMKF # 0) ______ FORMAT( 3 1 5 , 9 F 6 . 0 )

I c o l s 1 - 5 '

J 6 - 1 0 G r i d p o i n t l o c a t i o n s

K 11-15-

F K X X ( I , J ,K ) 1 6 -2 1 E l e m e n t s o f t h e p e r m e a b i l i t y

F K X Y ( I , J ,K ) 2 2 - 2 7 t e n s o r (m d . )

F K X Z ( I , J , K ) 2 8 - 3 3

F K Y X ( I , J ,K ) 3 4 - 3 9

F K Y Y ( I , J ,K ) 4 0 - 4 5

F K Y Z ( I , J , K ) 4 6 - 5 1

F K Z X ( I , J , K ) 5 2 - 5 7

F K Z Y ( I , J , K ) 5 8 - 6 3

F K Z Z ( I , J , K ) 6 4 - 6 9

N o t e : NUMKF c a r d s a r e r e a d .

2 0 . T i t l e c a r d ________________________ FORMAT( 4 0A2)

2 1 . Maximum v a l u e s o f p r e s s u r e s an d s a t u r a t i o n s f o r

t a b u l a t e d d a t a _____ FORMAT ( 8 F 1 0 .0 )

PMAXO c o l s . 1 - 1 0 Maximum o i l p r e s s u r e i n t a b l e

( p s i a )

PMAXW 1 1 - 2 0 Maximum w a t e r p r e s s u r e i n t a b l e

( p s i a )

PMAXG 2 1 - 3 0 Maximum g a s p r e s s u r e i n t a b l e

( p s i a )

SGMAX 3 1 - 4 0 Maximum g a s s a t u r a t i o n i n t a b l e

( f r a c t i o n )

158SIVMAX cols. 41-50 Maximum water saturation in

table (fraction)22. Title card _______________________ FORMAT(40A2)23. Oil PVT tabulated data__________ FORMAT(6F12. 0)

POT(I) cols. 1-12 Oil pressure (psia)BOT(I) 13-24 Oil formation volume factor

(bbl/STB)DIBOT(I) 25-36 d(l/B^)/dP^ (STB/bbl/psi)RSOT(I) 37-48 Solution gas-oil ratio (SCF/STB)DRSOT(I) 49-60 dRg/dP^ (SCF/STB/psi)UOT(I) 61-72 Oil viscosity (cp)

Note: Repeat this card until all tabulated data are read.The POT(I) must be arranged in increasing order, with the last card having POT(I) = PMAXO to signal the end of data.24. Title card _______________________ FORMAT(40A2)25. Water PVT tabulated data ____ ___ FORMAT(6F12.0)

PWT(I) cols. 1-12 Water pressure (psia)IBVT(I) 13-24 Water viscosity (cp)

Note: Repeat this card until all tabulated data are read.The PWT(I) must be arranged in increasing order, with the last card having PWT(I) = PMAXW to signal the end of data.26. Title card _______________________ FORMAT(40A2)27. Gas PVT tabulated data___________FORMAT (6F12 .0)

PGT(I) cols 1-12 Gas pressure (psia)BGT(I) 13-24 Gas formation volume factor

(bbl/SCF)

159DIBGT(I) cols 25-36 d(l/Bg)/dPg (SCF/bbl/psi) UGTCI) 37-48 Gas viscosity (cp)

Note: Repeat this card until all tabulated data are read.The PGTCI) roust be arranged in increasing order, with the last card having PGT(I) = PMAXG to signal the end of data.

28. Title card________________________________FORMAT(40A2)29. Oil, water and gas densities at standard conditions ___

FORMAT(6F12.0)RHOSO cols 1-12 Oil density at standard conditions

(Ibm/cu.ft)RHOSW 13-24 Water density at standard conditions

(Ibro/cu.ft)RHOSG 25-36 Gas density at standard consitions

(Ibro/cu.ft)PSC 37-48 Standard pressure (psia)TSC 49-60 Standard teroperature (°F)

30. Title card ___ ________________ FORMAT(40A2)31. Fracture and rock matrix compressibility_ _FORMAT(2D12.4)

CF cols. 1-12 Fracture compressibility (psi )CL 13-24 Rock matrix compressibility (psi )

32. Title card________________________________FORMAT(40A2)33. Tilting angles of the formation _________ (6F12.0)

ALPHA cols. 1-12 Angle between X-axis and horizontalplane (DEGREE)

BETA 13-24 Angle between Y-axis and horizontalplane (DEGREE)

160GAMMA cols. 25-36 Angle between Z-axis and vertical

plane (DEGREE)34. Title card ____________________________ FORMAT(40A2)35. Gas-oil relative permeability data for rock matrix

MATRIX (6F12.0)SGT(I) cols. 1-12 Gas saturation (fraction)KRGT(I) 13-24 Gas relative permeability (fraction)KROGT(I) 25-36 Oil relative permeability (fraction)PCGOT(I) 37-48 Gas-oil capillary pressure (psi)DSGT(I) 49-60 dS^/dP^^^ (psi'b

Note: Repeat this card until all tabulated data are read.The SGT(I) must be arranged in increasing order, with the last card having SGT(I) = SGMAX to signal the end of data.

36. Title card ______ FORMAT(40A2)37. Water-oil relative permeability data for rock matrix

FORMAT(6F12.0)SWT(I) cols. 1-12 Water saturation (fraction)KRWT(I) 13-24 Water relative permeability

(fraction)KROWT(I) 25-36 Oil relative permeability (fraction)PCOWT(I) 37-48 Water-oil capillary pressure (psi)DSWT(I) 49-60 dS^/dPôw (psi” )

Note: Repeat this card until all tabulated data are read.The SWT(I) must be arranged in increasing order, with the last card having SWT(I) = SWMAX to signal the end of data-

16138. Title card ____________________________ FORMAT(40A2)39. Gas-oil relative permeability data for fracture _____

FORMAT(6F12.0)FSGT(I) cols. 1-12 Gas saturation (fraction)FKRGT(I) 13-24 Gas relative permeability (fraction)FKROGT(I) 25-36 Oil relative permeability (fraction)FPCGOT(I) 37-48 Gas-oil capillary pressure (psi)DSGT(I) 49-60 d S^/d (psi‘ )

Note: Repeat this card until all tabulated data are read.The FSGT(I) must be arranged in increasing order, with the last card having FSGT(I) = SGMAX to signal the end of data.

40. Title card _____ __ _ _ _ FORMAT (40A2)41. IVater-oil relative permeability for fracture_

FORMAT(6F12.0)FSWI(I) cols. 1-12 Water saturation (fraction)F K R U T (i) 13-24 Water relative permeability

(fraction)F K R O O T (i) 25-36 Oil relative permeability (fraction)FPCOWT(I) 37-48 Water-oil capillary pressure (psi)FDSWT(I) 49-60 d S ^ d Pô* (psi'^)

Note: Repeat this card until all tabulated data are read.The FSWT(I) must be arranged in increasing order, with the last card having GSWT(I) = SWMAX to signal the end of data.

162Set 3 Input data for subroutine INCON

1. Title card _________________________ FORMAT(40A2)2. Initial reservoir pressures _______ FORMAT(6F10.0)

PI cols. 1-10 Initial oil pressure in rock matrix (psia)PWI 11-20 Initial water pressure in rock matrix (psia)PGI 21-30 Initial gas. pressure in rock matrix (psia)PFI 31-40 Initial oil pressure in fracture (psia)PWFI 41-50 Initial water pressure in fracture (psia)PGFI 51-60 Initial gas pressure in fracture (psia)

3. Initial saturations ___ _____ _______ FORMAT(6F10.0)SOI cols. 1-10 Initial oil saturation in rock matrix

(fraction)SWI 11-20 Initial water saturation in rock matrix

(fraction)SGI 21-30 Initial gas saturation in rock matrix

(fraction)SOFI 31-40 Initial oil saturation in fracture

(fraction)SWFI 41-50 Initial water saturation in fracture

(fraction)SGFI 51-60 Initial gas saturation in fracture

(fraction)4. Title card _______ ______________ FORMAT (40A2)5. Number of variable rate nodes _________ FORMAT(15)

NVQN cols. 1-5 Number of variable rate nodes6.* Location of node and its active time FORMAT

(3I5,4F10.0)

163

QNICI) cols QN2CI)QN3CUTQl(I)TQ2(I)

Location of the gridpoint1-5 6 - 1 0

11-1516-25 Time the rate is on (day) 26-35 Time the rate is off (day)

Rates at this node FORMAT(8F9.0)

QVO(T) cols. 1-9 Oil rate in rock matrix (STB/day)QVW(I)QVG(I)QVT(I)QVO(I)QVWF(I)QVGF(I)QVTF(I)

Notes: (1)

( 2 )

(3)

(4)

10-18 Water rate in rock matrix (STB/day)19-27 Gas rate in rock matrix (SCF/day)28-36 Total fluid rate in rock matrix (STB/day) 37-45 Oil rate in fracture (STB/day)46-54 Water rate in fracture (STB/day)55-63 Gas rate in fracture (SCF/day)64-72 Total fluid rate in fracture (STB/day)

Cards 6 and 7 are read in sequence. NVQN sequences are read.Injection rate has positive sign, whereas production rate has negative sign.If the rates are to be determined based on phases' mobilities, QVT(I) and/or QVTF(I) must not be zero.If the rates of oil, water, and gas are to be explicitly defined, QVT(I) and/or QVTF(I) must be zero.

164Set 4 Input data for subroutine code

1. Title card _________________________ FORMAT(40A2)2. Simulation parameters _____________ F0RMAT(I5,5F10.0)

NMAX cols. 1-5 Maximum number of time stepsTMAX 6-15 Maximum simulation time (day)WORMAX 16-25 Maximum allowable water-oil ratioGORMAX 26-35 Maximum allowable gas-oil ratio (SCF/STB)PEP 36-45 Bubble point pressure (psia)PMIN 46-55 Abandonment pressure (psia)

3. Iterative parameters and print codes ___ FORMAT(5I5,2F10.0)MITER cols. 1-5 Maximum number of iterations for PSORLPT

LSI

LPOR

LVELO

OMEGATOL

6-10 Pressure distributions will be printed every LPT'th time step

11-15 Saturation distributions will be printed every LST'th time step

16-20 Porosity distributions will be printed every LPOR'th time step

21-25 Velocity distributions will be printed every LVELO’th time step.

26-35 Relaxation parameter 36-45 Tolerance limit for PSOR

Set 3 Input data for the MAIN program1. Time step size __________________ FORMAT(F10.0)

BELT cols. 1-10 Time step size (day)Note: At least one time step size must be read. If N cards are read,

all time steps after step N will have the same size as step N.

APPENDIX E ARRANGEMENT OF INPUT DATA

N U M d L R ÜF Ü H 1 Û 1 0 2

F k A C T U h i t HALF 0 . 0 5 0 0

C C O F N U M 5 2 R S P ü K 0 U

I N I T I A L 4 A T % i X F> 0 . 2 9

M A T R I X P t i R M f l A J I L l T Y 1 . 0

^ { N X , . j y . r 5 2 0 0 0 .■v I uTH ( F T )

: ) AND 4 0 0 .

R L à L k V w I i ;

2 5 0 .L E N G T H . A I D T H . T H I C K N E S S

4 A T 4 I X t - U K J S I T Y A! , U P E k i / E A r l I L 1 T Y

X ' U S I TY

( MD)

C E D E N UN I LF S t o y - F R A C T u ; <E P O R O S I T Y AND F»i_ R M E A B I L I TY0 0

I N I T I A L F K A C T U K E P U R O S I T Y0 . 0 0 0 5

F R A C TUNE Pr. . 7 W E A 3 I L 1 TY T E N S U M E L E M E N T S1 0 . 0 0 . 0 0 . 0 0 . 0 1 0 . 0 0 . 0 0 . 0 0 . 0 1 0 . 0MAX I / U 4 P R E S S U R E S An d S A T U R A T I O N S F U R P V T DA TA t a b l e s

7 0 1 4 . 7 7 0 1 * . 7 7 0 1 4 . 7 1 . 0 1 . 0O I L P V T T A S L w S

1 4 . 7 1 . 0 - 0 . 0 0 0 1 3 7 U 0 . 0 0 . 2 1 9 2 4 0 . 7 21 7 1 4 . 7 1 . 3 0 2 o - 0 . 0 0 0 0 5 7 6 3 7 2 . 8 C . 2 2 4 1 1 0 . 3 2 62 1 1 4 . 7 1 . 3 4 2 4 - 0 . 0 O O 0 5 7 5 4 6 3 . 2 0 . 2 3 4 4 7 0 . 4 82 t . 1 4 . 7 1 • 3 v o J - 0 . 0 0 0 0 5 7 9 S 3 1 . 5 0 . 2 4 9 4 7 0 . 4 3S u l 4 « 7 1 . 4 4 2 7 - 0 . 0 0 0 0 5 3 2 ô a 1 . 5 0 . 2 6 0 2 0 0 . 3 9 63 C 1 4 . 7 1 . 5 2 0 4 - 0 . 0 0 0 0 5 0 4 8 4 6 . 2 0 . 3 0 1 5 V 0 . 3 4 84 ^ 1 4 , 7 1 . 6 0 7 7 - 0 . 0 0 0 0 6 0 1 1 0 2 9 . 7 0 . 3 2 9 4 3 0 . 3 0 34 6 1 4 . 7 1 . 0 7 2 5 — 0 . OoOOôO- S 1 1 6 3 . 3 0 . 3 6 3 1 6 0 . 2 7 45 0 1 4 . 7 1 . 7 4 4 2 - 0 . 0 0 0 0 6 4 0 1 3 1 1 . 0 0 . 4 0 0 0 7 0 . 2 4 45 3 1 4 . 7 1 . 3 0 4 ^ - 0 . 0 0 0 0 o 0 7 1 4 3 1 . 2 0 . 4 0 3 4 O 0 . 2 2 05 6 1 4 « 7 1 . S t 4 J - 0 . 0 J 0 0 5 5 0 1 5 5 7 . 6 0 . 5 0 1 2 2 0 . 2 0 67 0 1 4 . 7 2 . 1 V 7 S - 0 . 0 0 0 0 5 7 V 2 2 5 9 . 0 0 . 5 0 1 1 1 0 . 1 0 9

* A T E F PVT T A r t C E1 4 . 7 0 . 3 57 U 1 4 . 7 0 . 3 5

G A S P V T 1 A E L ^ S1 4 . 7 0 . J 7 s O y 0 . 2 9 8 3 5 0 . 0 1 2 0 51 7 1 4 . 7 0 . 0 0 1 9 4 ^ 5 0 . 5 1 4 o 3 0 . 0 1 6 32 1 1 4 . 7 0 . J u l 5 ^ 6 7 0 . 3 0 4 5 3 0 . 0 1 7 32 0 1 4 . 7 0 . 0 0 1 6 6 7 1 O • 2 3 5 6 3 0 . 0 1 3 63 0 1 4 . 7 0 . 0 0 1 1 0 7 2 0 . 2 4 0 7 4 0 . 0 1 9 73 6 1 4 . 7 0 . 0 0 0 9 4 S 3 0 . 2 2 7 7 1 0 . 0 2 1 44 2 1 4 . 7 0 . 0 0 0 6 4 1 E 0 . 2 0 3 3 7 0 . 0 2 3 34 6 1 4 . 7 0 . 0 0 6 7 S 3 4 0 . 1 0 3 4 9 0 . 0 2 4 6S J 1 4 . 7 0 . 0 0 0 7 4 4 5 0 . 1 7 9 3 1 0 . 0 2 5 75 3 1 4 . 7 0 . 0 0 0 7 1 6 2 0 . 1 o 5 1 9 0 . 0 2 6 65 6 1 4 . 7 o.oooo 01 a C . 1 5 G 0 G 0 . 0 2 7 67 0 1 4 . 7 0 . OOOo C . 1 5 8 1 4 0 . 0 3 3 0

O I L WATER S A S P ' - l N o l T I E S AT s t a n d a r d p r e s s u r e a n d t e m p e r a t u r e

5 1 . 1 4 6 5 . '.) 0 . 0 5 8 1 4 . 7 6 0 . 0

165

166F K A C T U . - c ANÙ MAT,. 1 X NOCK C U M P K 5 S S 1 j 1 1_ I T Y

3 . 5 0 3 . 5 0 - 6T i i _ r I N o A(\ o L . L S : AL P HA « S E T A , GAMMA ( U L G R L E S )

0 . 0 0 . 0 0 . 0G A S U I L R ü L A l I VC P L RM 5 a j i l i t y i n r u c k m a t r i x

0 . 0 0 . 0 1 . 0 0 . 0 7 5 1 0 . 00 . 1 0 0 . 0 1 5 Û . 7 0 0 . 0 3 5 1 0 . 00 . 2 0 0 . 0 5 0 0 . 4 5 0 . 0 9 5 1 0 . 0O . S C O . l u J 0 . 2 5 0 . 1 1 3 5 . 0 0 0 00 . A G 0 . 1 0 0 0 . 1 1 0 . 1 4 5 3 . 3 3 0 0O . b O 0 . 3 1 0 0 . 0 2 8 0 . 2 5 3 0 . 9 1 0 00 . S 5 0 . 4 2 0 0 . 0 0 . 3 3 6 0 . 3 8 0 01 . 0 1 . 0 0 . 0 1 . 5 6 3 0 . J Ô O O

W A T E K - O I U K L L A T I v L P L k • 1 L A Ü I L I T Y I N RUCK M A T R I X0 . 0 0 . 0 1 . 0 2 1 4 . 0 - 0 . 0 0 1 2 20 . 1 0 0 . 0 1 . 0 5 0 . J - 0 . 0 0 1 2 20 . 1 5 0 . 0 0 5 0 . 8 0 9 . 0 - 0 . 0 0 1 2 20 . J O 0 . 0 1 0 . 7 2 3 2 . 0 - Û . 0 0 7 1 40 . 3 0 0 . 0 2 0 . 5 0 0 . 5 - 0 . 0 3 3 3 30 . 4 0 0 . 0 3 0 . 4 v 3 0 . 0 —0 . 10 . 4 5 0 . 0 4 5 0 . 3 * 2 - 0 . 4 - 0 . 1 2 50 . 5 0 0 . Ot. 0 . 3 0 4 - 1 . 2 - 0 . 0 6 2 50 . 5 5 0 . 0 3 2 0 . 2 2 3 — 2 . 6 - 0 . 0 3 5 70 . o 0 0 . 1 1 0 0 . 1 5 4 - 4 . 0 - 0 . 0 3 5 70 . o 5 0 . 1 4 3 0 . 0 R 3 - 7 . 0 - 0 . Ü 1 6 70 . 7 0 0 . 1 0 . 0 4 2 - 1 0 . 0 —0 . 0 1 6 70 . 7 5 0 . 2 3 0 0 . U - 4 0 . 0 - 0 . 0 0 1 6 71 . 0 1 . 0 0 . 0 - 1 9 0 . u - J . 0 0 1 6 7

G A S i ; i L Or: L A T I Vf. P L k MC a j i l I t y i n f p A L T U R L0 . 0 0 . j 1 . 0 0 . 0 7 5 1 0 . 00 . 1 0 0 . 0 1 5 0 . 7 0 0 . 0 6 5 1 0 . 00 . J 0 O . O u O 0 . 4 5 0 . 0 * 5 1 0 . 00 . J ' J 0 . 1 0 3 0 . 2 5 0 . 1 1 5 5 . 0 O O 0O . A J U . l v O 0 . 1 1 0 . 1 4 5 3 . 3 3 0 00 . 5 0 0 . 3 1 0 0 . 0 2 3 0 . 2 5 5 0 . 9 1 0 00 . 5 5 0 . 4 3 0 C . >) 0 . 3 2 5 0 . 3 8 0 01 . 0 1 . 0 0 . 0 1 . 5 6 5 0 . 3 8 0 0

* A T E k - ü l L R E L A T I V E P E R ^ L A B I L I T Y I N F R A C T U R E0 . 0 0 . 0 1 . 0 2 1 4 . 0 - 0 . 0 0 1 2 20 . 2 0 0 . 0 1 . 0 5 0 . 0 - 0 . 0 0 1 2 2C . 2 5 0 . 0 0 5 G . 3 6 J , 0 - 0 . 0 0 1 2 20 . 3 0 0 . 0 1 0 . 7 2 3 2 . 0 - 0 . O 0 7 1 40 . 3 5 0 . 0 ^ 0 . 6 0 0 . 5 - 0 . 0 3 3 3 3C . 4 0 0 . 0 3 0 . 4 * 2 0 . 0 —0 . 10 . 4 5 0 . C/4 5 0 . 3 9 2 — 0 . 4 - 0 . 1 2 50 . 5 0 0 . Oo 0 . 3 0 4 - 1 . 2 - 0 . 0 6 2 50 . 5 5 0 . 0 c 2 0 . 2 2 3 — 2 . 6 - 0 . 0 3 5 7O . o O 0 . 1 1 0 0 . 1 5 4 - 4 . 0 - 0 . 0 3 5 70 . 6 5 0 . 1 4 3 0 . 0 9 3 - 7 . J - 0 . 0 1 6 70 . 7 0 0 . 1 J O 0 . 0 4 2 - 1 0 . 0 —0 . 0 1 6 70 . 7 5 0 . 2 J 0 0 . 0 —4 0 . 0 - 0 . 0 0 1 6 71 . 0 1 . 0 0 . 0 — 1 9 0 . 0 - G . 0 0 1 6 7

I N I T I A L A j E f’ k J o S J R L S I t J P O C K M A T R I X ANO F R A C T U R d6 2 1 4 . 7 o l o 4 . 7 6 2 1 4 . 7 7 3 6 2 1 4 . 7 Ô 1 C 4 . 7 6 2 1 4 . 7 7 5O . Ü C 0 . 2 0 0 . 0 O . B O 0 . 2 0 0 . 0

167NUKr - t K C F V Ak I A.2LF: HiAT rl N O D E S

1C 01 1 1 0 . 0 5 0 0 . 0

0 . 0 0 . 0 0 . 0 u . 0 0 . 0 0 . 0 O . C - 8 0 0 . U1 1 2 0 . 0 5 0 0 . 0

0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 —6 0 0 . u1 1 3 0 . 0 S O u . O

0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 - 8 0 0 . 01 1 A 0 . 0 5 0 0 . 0

0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . o 0 . 0 - 8 0 0 . 01 1 5 0 . 0 5 0 0 . 0

0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 —S O U . V1C 1 1 0 . 0 5 0 0 . 0

0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 1 4 0 0 . 0 0 . 0 0 . 01 0 1 2 0 . 0 5 0 0 . 0

0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 14 0 0 . 0 0 . 0 C. 01 0 1 3 C . 0 5 0 0 . 0

0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 14 0 0 . 0 0 . 0 0 . c1 0 1 '* 0 . 0 5 O Ù . 0

0 . 0 0 . 0 o . c 0 . 0 C . 0 1 4 0 0 . 0 0 . 0 0 . 01 0 1 5 0 . 0 5 0 0 . 0

0 . 0 0 . 0 0 . 0 0 . 0 0 . 0 1 4 3 0 . 0 0 . 0 •- 0 . 0S I M U L A T I O N P A h Al ■'i T L.K5

■ 4 0 0 5 C 0 . 0 1 0 0 . 0 1 i O O O O . 0 5 5 5 0 . 7 1 5 0 0 . 01 0 0 2 0 2 0 3 0 4 0 1 . 2 0 . 2

0 . 0 50 . 0 30 . 0 50 . 0 30 . 0 "

0 . C 7 5

0 . 0 7 3 0 . 0 7 5 0 . 0 7 5 0 . 0 ? 5 Ü . l O

0 . 1 0

0 . 1 0

0 . 1 0 0 . 1 5 0 . 1 5 0 . 1 5 0 . 1 5 0 . 2 ^ 5

0 . 2 ^ 5 0 . 2 2 5 0 . 2 2 5 0 . 2 2 5 0 . 3 7 5 0 . 3 7 5 0 . 3 7 5 0 . 3 7 5 0 . 5 0 0 . 5 0 0 . 5 0 1 . 0

1 . 0

APPENDIX F

COMPUTER LISTING OF THE SIMULATION PROGRAM

168

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Z z z o 3 z 3 3 Z z G G z 3 z Z Z 2 G n r? n3 X M 3 3 3 Z C y > > r a

o o c o O O N o o o o X 3 O o 3 S X X N X X S N N X N N 3 N 3 N 3 Z n G C c Gn n n X y n y m > < X Z X y X G < X < -4

•0 X X X N X 2 X X X X X X X X Z y y z y y z z 3 3 n y m X X > X G fTÎ < c Z G Z XM y -4 -4 M m r y Q y y n? Z n c y G n m

c yt X y L- y y X y y y y X X X ■n X X X r c G y N y y G z r c T z >y y » z > U c n X X N < N N y r y a r rN s N. y y n N c N N y y « o y y G N 3 c c *

n n -a n -n o y G N < Ci O y 3 c G G H G X 3o r u't TT > y c c T c < < 3 G < c n C « % C G < G a z

r . r . a X c c c « % « X Z 3 y 3 3 Q S G G C X -n s C 3 G a C > > Z» a X u U Ci y u » c y % o X < < X 3 z 3 z C a .G z 1 >

X 3 X X G « G c 3 3 3 X G •4 Cc o X 3 C y 3 N < -4 X 3 X 3 a a ( • z z

c o a o c O c c o c 3 3 3 3 y X y y G O X X 3 X 3 X C c o >C X X y X X X -4 y X y X r z -4

% X X X X X X X X X X X X y z 3 y y X y X y r r r > c â OX y y y n y % y y y > C r z z fn 1

j î vT Cî yi y y y y y y y y y y 5 y c 3 3 X < y y y z z G > > rr Ny y G c y % G c r

c T? y y Ci 3 i r X X y X < X c C G z > G C r GT •*! D “ î r 3 c y 3 o % w> z 5 > z ■<C > c. y < % % y 3 c X y y 5 3 3 G G Z >

Cî C 3 c c c « X « X X c 3 o -n 3 m G yc . Cî c y X X 3 3 y 3 < z 3 X z 3 3 C < 2 a r.

3 y o y G c X 3 X m > > >e O O o o y X X X < < X y X X C z G n

o o 9 o o o o o X o 3 3 X 3 X G y X y 3 3 r y H GfO X X X X n y 2 y « y N M y y y c C r Gj X X X X X X X y X X X o y y C G y X y X y a > >ü: y y y y y < o 3 m n

( f o y im y y y y y y y 2 m 3 > 3 o o y y y G G n o r r3 y r 3 o % < < % « Ci >

3 T -n •n 3 c y X 3 X o z 3 Z r a c itr t T % 1» n T y m % X G X N 3 mc .T O 6 < c X 3 S 3 % n y X o y 3 X y 3 X O fR y r 3

O Ci X X X c C 3 % X X y 3 y X 3 3 Z n Xy X u X X y 3 X X y X e y X y a a <«

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r T A y T T! y 3 y 3 X Co c. r X m c % n y y X X 3 X X 3 3

r . f X X c c X « X X X X 3 X 3 y X z XX X M u X » u u y G y X y y X X c 3

y X % y X y r Nd c O o o o y y y >c C 3 o c o 3 3 y y 3N X X X X X O c V

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n w C 3 3 o etc r z C Z 9 N et N 9 9 N N V N V S

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IMPLICI T REAL40I A - I I . U - Z » INTEGER42 T I T L E ! 4 0 ) R E A O I S . I O O ) T I I L L

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D'OMAXIMUM SIMULATION TI ME (DAYS) = « , F 1 0 . 0 . / .D'OHAXIMUM WATER-OIL RATIO = ' . F I 0 . 0 . / .C'OMAXIMUM G AS -O I L RATIO = • . F 1 2 . 0 . / . • OUUnOLE POINT PRESSURE = ' . t r i o . I . • P S I A* . / . •OAIIANOONHENT PRESSURE = * . H 0 . 1 . * P S I A* I

PRINT CODES AND TOLERANCE L I MI T S

R E A 0 I 5 . 1 0 3 ) Ml T E H . L P T . L U T . LPOR.LVELO.OMEGA.TOL1 0 3 F O R M A T I 5 I 5 . 2 F I O . O I

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COMMON /TIIA/ TI OUI ( 1 0 , 2 , 0 1 ,Th (iOJ ( I U, 2 ,01 , TruU3 ( 10 , 2 , 0 1 , 1FUUAI I 0 , 2 , 05) , TF0U31 10,2,5» ,THt:ul I 1 0 , 2 , 5 ) ,T1FU2( I 0 , 2 , 5 ) , ITLO JI 1 0 , 2 , 5 ) ,TI AOI (0 I 0 , 2 , 5 ) , 11 ( III ( I 0 , 2 , 5 ) .1 FFUI ( 1 0 , 2 , 5 ) ,TF1 0 2 1 1 0 , 2 , 5 I , T H 0 3 1 l ' ) , 2 , 0 ) ,C TI'HOI ( 1 0 , 2 , 5 1 ,TI MO.'l 1 0 . 2 , 5 ) , 1111031 1 0 , 2 , 5 ) ,11 IIOAl 1 0 , 2 . 5 ) , 1111051 1 0 . t 2 , 5 ) , i r R 0 1 1 0 , 2 , 5 ) . i r U ( J l I O , 2 , 5 ) , 1 F V 0 1 1 0 , 2 , 5 )

COMMON /1l<5/ II or.) 1 I -3,2,5) ,1 F G U 2 11 0 , 2 , 5 ) ,1FUG31 10 , 2, 5 ) , IFtiU A 1 1 0 , 2 , t 5 ) , irOG511 0 , 2 , 5 ) , i n 01 I 1 0 , 2 , 5 ) ,1FLG21 1 0 , 2 , 5 ) ,1F1 0 3 1 1 0 , 2 , 5 ) , I F AGI I H O , 2 , 5 ) , II c o i l 1 0 , 2 ,5 1,11 Fo i l 1 0 ,2 , 5 ) , TFF021 1 0 ,2 .5 ) ,11 FÜ3I 1 0 , 2 , 5 ) , D i n i O l l 1 0 , 2 , 5 ) , II 11021 10 , 2 , 5 ) , IF 11031 10 , 2 , 5 ) .IF M G A l 10,2 ,51,11 11051 10, 12 , 5 ) , IFO R 1 I I 0 , 2 , 5 ) ,11-0I>21 10 ,2 ,5 ) , IFOF. i l I 0 , 2 , 5 ) , I I O R A tl 0 , 2 , 5 ) ,1. 1 F O H 5 1 1 0 , 2 , 5 ) , 1 1 I.III 1 1 0 , 2 , 5 ) , T F L H 2 1 1 0 , 2 , 5 ) , 1 I 1:1131 1 0 , 2 , 5 ) .11 AIM 10 1 0 . 2 . 5 ) , I F C H 11 1 0 , 2 , 5 ) , IP n i l 1 1 0 , 2 , 5 ) , 1 F F H 2 1 1 0 , 2 , 5 ) , 1 1 F U J I 1 0 , 2 , 5 ) , t 1FIII11 1 1 0 , 2 , 5 ) ,TFIII121 1 0 , 2 , 5 ) , T F M I ( J l l 0 , 2 , 5 ) , t l IIIIAl 1 0 , 2 , 5 ) ,11 MI1510 1 0 . 2 . 5 ) , 11-1101 1 0 , 2 , 5 ) , 11 n o u n 0 , 2 , 5 ) , 1 F V 0 1 1 0 , 2 , 5 )

CUHMUN / 5 L 1 2 2 VI) ,HFLX, UFLV , I>FL2, DtLT ,I1H CUM KIN / S L 1 J / C U N l ,C U N 2 , C U N 3 , C O N 4CUMMON / S L 1 1 / Ill lUOl), I1tin-3W,I1II03G, P S C , I S C , ALPHA, llli TA, OAMMA CUMMUN / L I Ml 1 / MPU1 , -11'WI ,MP01 ,MSG1 ,MS j l . M F S O I .MI S» I COMMUN / P A R M I / C L . C F , P H 1 1 1 0 , 2 , 5 ) . F P H I I 1 0 . 2 , 5 )CUMMUN / P A H H 2 / K L l 1 0 , 2 , 5 ) , F K X X l 1 0 , 2 , 5 ) .FKXYl 1 0 , 2 , 5 ) , FKXZ1 I 0 , 2 , 5 ) ,

t FKVXl 1 0 , 2 , 5 ) , 1 K Y Y l 1 0 , 2 , 5 ) , r KVZl 1 0 , 2 , 5 ) , F K Z X 1 1 0 , 2 , 5 ) , t F K Z Y l 1 0 , 2 , 5 ) , ) K/ ZI 1 0 , 2 , 5 )

CUMMON / P A P U J / P U 1 I 1 5 > , I I O P l 1 5 > , O t U U l l I S ) ,C USUI 1 1 5 ) , nil s u 11 1' , ) , UU1 1 1 5 ) ,PW11 I S ) ,uv*11 1 5 ) ,1*011 15 1 , 0 0 1 1 1 5 ) ,D 01 0 0 1 1 1 5 ) ,1101 1 ) 5 1 , 5 0 1 1 1 5 ) ,KI1GT1 1 5 ) , K R U 0 1 1 1 5 ) . PCGIII l 1 5 ) , U 5 0 1 1 1 5 ) ,D SW11 1 5 ) ,KP»i11 1 5 ) .KIIUa TI 1 5 ) .PCUV.T1 15 ) , U S W1 11 5 ) ,F SOI 1 1 5 ) ,FKI»011 1 5 ) , C FKIUIOt I I S I . I P C O I I I 1 1 5 ) ,1 U3G11 1 5 ) ,1 5*111 1 5 ) «FKI1W11 1 5 ) ,1 KI1UW1 1 1 5 ) ,D FPCnWTl 1 5 ) .FDSWTI 1 5 ) , I13WT 11 5 ) , IlFSWI 1 1 5 )

OlMb'NSIlIN W1 1 1 0 , . . > , 5 ) , W2I 1 0 , 2 , 5 ) , M 3 1 1 0 , 2 , 5 ) , H4 1 1 0 , 2 , 5 ) , M 5 ( l 0 , 2 , 5 )0 ,W611 0 , 2 , 5 ) ,W71)0 , 2 , 5 ) .Will 1 0 , 2 , 5 ) ,W H 1 0 , 2 , 5 ) , »lu l 1 0 , 2 , 5 ) ,M111 1 0 , 1.2 , 5 ) , M 121 1 0 , 2 , 5 ) ,W131 1 0 , 2 , 5 ) , H 1 4 l 1 0 , 2 , 5 ) , MI 51 1 0 , 2 , 5 ) ,Wlü 1 1 0 , 2 , 5 ) ,D Ml 7 1 1 0 , 2 , 5 ) ,W 111 1 1 0 , 2 . 5 ) ,W I V 1 1 0 , 2 , 5 ) ,W201 1 0 , 2 , 5 )

UU 5 0 K=1 ,112 DO 5 0 3 = 1 ,NY DU 5 0 1 = 1 , NX CALL L I N I P I M S ’I I . S W I 1 CALL L l N I P l M S M l . S W l I CALL I. 1N1P1MS01 , S' j 1 I CALL L I U I P I M S O I . S O I I CALL L IN1 P1 UI iW1,5WI CALL LINII ' IMI 3W1, .,WI CALL L1N1I-1MI S OI .S OI CALL L l N l l ' l ' J I S G I . - n i CALL L I N I P l r i P l ' I . P U l 1 CALL L1N1P1 I l 'WI.PWl 1 CALL L l N 1 P | M I ' i 1 , l * - . ; l l CALL L 1M1P1.11*111, PI 111 CALL L I N I P I IPHI ,1*1 Wl CALL L l N I P l i l l ' O I . P I ',1 CALL L l N I P l M P ' , 1 ,P!H 1 CALL H N 1 P (Il"-i1 , ” -j 1 1 CAI I Ll '111'1 I PU I . PI 111 CALL L IN ri'li lP-.t 1 , PI -, I CALL L INI P l i l l - ' i I , " U l I ( ALL LIM1l ' (MI"i 1 , PI III

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120 FORMAT 110)IF (NVOtI.I U.O) UU TO fib PRINT 1 2 1 , NVON

121 F O R M A T ! / / / / / / * NUMOLR OF VARIAI ILL HATL l'IOl)LS= * , I !i, / / 1

RLAU VARIAOLF. RAT F NUOL LOCATION AMI) TIML-ON ANU TIML-OI F FOR TllOSt; RATES Afi V.LLI AS I N JLC T I ON! ♦ ) /PRODUCT I UNI - ) RAILS

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DO 16 1 = 1 «NVONR E A D I S , 1 2 2 ) ONI I 1 ) , 0 0 2 1 1 1 . Q N 3 I 1 ) , T 01 I I I . TD2I 1 )

12.2 I 0 R M A T ! : U S , 4 I 1 0 , 0 )READIO, 1 2 3 ) D V D ! 1 ) , OVWI I I , O V U ! I ) ,OVTI 1 ) ,O VU F! 1 ) , O VW F I I ) , O V G F I 1 ) ,

D Q V T F I 1)1 2 3 FORMAT I , IF0 . 0 )

PRINT 1 2 4 . U M 1 I 1 ) , 0 0 : I ) ) , 0 N 3 I 1 ) , T0 1 I 1 ) , T 0 2 1 1 I ,OVOI 1 I ,OVWI 1 I , I V U I 1 ) , t t lVTI I ) ,OVUI I 1 ) ,OVWI I 1 I , OVUI I I ) , UVTF I I )

1 2 4 FORMAT I ' 0 R A 1 L NODE LOCAT ION= ' , 3 1 5 , / ' TIME ON IDAY) = ' , F 1 2 , . l ,t * TIME 01 I I DAY I = * , 1 1 2 . 2 , / * OVOISTU/D) = * , F 10 , 0 , * 0VW|5TÜ/D)=*& , I 1 0 . 0 , * OVUISCI / l , ) = * , F 1 0 . 0 , * DVTISTU/D) = * ,1 1 0 , 0 , / ,F* OVOI l :>TII /0 )= • ,1 " , 0 , * OVWI l 3 T n / D ) = * , l 1 0 , 0 , * OVUI I S c l / D » = * ,C F I O . O , ' OVTI ( S I O / D ) =* ,1-10 . 0 )

16 CONTINUE 5(> RETURN

END

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f. I’LWI 1 0 , 2 , 3 ) ,PI=; ; i 1 0 , 2 , 5 ) , P U I I ( 1 0 , 2 , 5 ) ,PWN( 1 0 , 2 , 5 ) , POM ( I 0 , 2 , 3 ) ,C PFONI 10 , 2 , 3 ) , PI IVMI I 0 , 2 , 5 ) . P F O N I 1 0 , 2 , 3 )

COM.4UN / P P O U / 0 0 ( 1 0 , 2 , 5 ) ,0V* ( 1 0 , 2 , 5 ) , U 0 ( I 0 , 2 , 5 ) ,OUI ( 1 0 , 2 , 3 ) ,0 Q W F ( 1 0 , 2 , 5 ) , 0 0 1 ( 1 0 , 2 , 5 )

COMMUN / L I M I T / Il’lIT , '1PWT ,MP«T , M3GT ,MSW1 , MFSÜ T , Ml 5WT COMMON / P A I I M 3 / PU f ( 15 ) , U O K I 5 ) , U 10 0 TI 15 ) ,

0 K 5 0 T ( 1 5 ) ,014501 ( 13 ) , UU T ( 15 ) , PW T ( I 5 ) , U W T ( 1 5 ) ,I>GT( I 5 I , DOT ( 1 5 ) ,C D i n O T C 1 5 ) , U G T ( 1 5 ) , 5 G T ( 1 5 ) ,KI?GT( 1 3 ) ,KMUGTI 1 3 ) , PCGUTI 1 5 ) , U 5 G T ( 1 5 ) , t 5 W T ( I 5 ) , K U W T ( 15 ) ,KI ! UWT( 1 5 ) , P C U W T ( I 5 ) , D S W T ( 1 5 ) ,T 5GT( 1 5 ) ,FKMGT( 1 3 ) , 0 F K U O G T I 1 5 ) . F P C G O r <15 I ,1 U 5 GT ( 1 5 ) , FSWT( 1 5 ) . F K P U T ( 1 3 ) , FKMUWf( 1 5 ) ,D FPCOWTI 1 3 ) ,1 U5W1 ( 13 ) , 0. ,Wt ( 1 5 ) . K F S W T ( 15 )

0 IMLN5 ION V P H I 117 , NY, I I ,V) ’F (NX,NY , 1 )BEAL+0 .MULU.MOLW ,MOI.GW A f 1 = 0 . 0W A T P = 0 , 0U I L P = 0 , 0G A S P = 0 . 0G A S I = 0 . 0P A V G = 0 . 0P A V G F = 0 . 00 0 15 K=I ,N. ' .1)0 15 3= I , N Y0 0 I S I = 1 ,N , <I F ( Q O ( I , J . K ) . L T . 0 . 0 ) U I L P = 0 I L P T 0 0 ( I , J , K )* OL L T I F ( o u r ( I , J . K ) . L I , 0 , 0 ) I ) 1 L P = 0 1 L P « Q U F ( I , J , K ) T U t L T I F ( O W ( I , J . K ) , L T . 0 . 0 ) WATP=WATPTUW( I , J . K ) * O L L T IF (OWI ( I , J , K ) . I T . 0 . 0 ) WATP=WATPFOWFII, J , K I* OL LT I F ( O W ( I , J . K I . G T . 0 . 0 ) WATI=WAT1F0WI1 , J , K ) ♦DELÎ lF(t)WI- ( I , J . K ) . o r . 0 . 0 ) WAT 1=WAT l + OWFC I , J . K )»UhLT 1 F ( Q G ( I . J . K I . L T . 0 . 0 ) 0 A 5 P = G A S P F U G ( I , J . K ) * 0 1 LT1 F ( OGI ( I . J . K ) . L T . 0 . 0 ) GASP--=GA5PFQGF( 1 , J , K ) * O L L T I F ( 0 G ( I , J , K ) . G T . O . O l G A 5 1 = G A S I t O C ( I , J , K ) * O L L T I F ( o u r ( I , J , K ) . G l . 0 . 0 ) G A 3 I = G A 3 1 F U G F I I , J , K ) * U L L T PAVG=PAVG+I’i.:( I , J , K ) 7 V P M | 1 , J . K )PAV3I =PAVGI *PI !J( 1 , J . K ) FVPI ( I , J . K )

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IM PL ICI T N E A L* 8 ( A - t l , Ü - Z )CtlMMUN / P H L S S / P U I 1 0 , 2 , ü l , P W ( 1 0 . 2 , 5 1 . PG< 1 0 . 2 , 5 I . P h U I 1 0 . 2 . 5 ) .

f. PFW( 1 0 . 2 , 5 ) . P r u ( 1 0 . 2 . 5 ) , PON ( 1 0 . 2 . 5 ) .PWNi 1 0 . 2 . 5 ) , PON ( 1 0 . 2 . 5 ) .C P F U N I 1 0 . 2 . 5 ) . PFWNI1 0 . 2 . 5 ) . P F G N I 1 0 . 2 . 5 )

COMMUN / C M T X I / G wl 1 0 . 2 . 5 ) . EWI 1 0 . 2 . 5 ) . AW I 1 0 . 2 . 5 ) . UV. I 1 0 . 2 . 5 ) .OCWI 1 0 . 2 . 5 ) . rWI 1 0 . 2 . 5 ) . MWI 1 0 . 2 . 5 ) .KWI 1 0 . 2 . 5 ) .UWMI 1 0 . 2 . 5 ) . DW1 1 0 . 2 . 5 )

COMMUN / C M T X 2 / VIII 1 0 . 2 , 5 ) .GUI 1 0 . 2 . 5 ) . EUI 1 0 . 2 . 5 ) . AU( 1 0 . 2 . 5 ) .C IJ U I 1 0 , 2 . 5 ) , CO I I 0 . 2 . 5 ) .FUI 1 0 . 2 . 5 ) , H U I 1 0 . 2 . 5 ) .UUMI 1 0 . 2 . 5 / .G KDI 1 0 . 2 . 5 ) . o n 1 1 0 . 2 . 5 )

c o m mo n / C M T X J / V G I 1 0 . 2 . 5 ) .GGUI 1 0 . 2 . 5 ) .EGOI 1 0 , 2 . 5 ) . A o u l 1 0 , 2 . 5 ) .G UGUI 1 0 . 2 . 5 ) .CUOI 1 0 . 2 , 5 ) .FGOI 1 0 . 2 . 5 ) .HGUI 1 0 . 2 . 5 ) . u O I 1 0 . 2 . 5 ) .(. FGI 10 . 2 . f ) . AGI 1 0 . 2 . 5 ) , n U I 1 0 . 2 . 5 ) . CGI 1 0 . 2 . 5 ) .1 Gl 1 0 . 2 . 5 ) .G IIGI 1 0 . 2 . 5 ) . IlGUI 1 0 . 2 , 5 ) .KOI 1 0 . 2 . 5 ) ,UGI 1 0 . 2 . 5 )

COMMUN / C M T X 4 / F H W l 1 0 . 2 . 5 ) . F G W I I 1 0 . 2 . 5 ) . F GW2 I 1 0 . 2 . 5 ) . F G W 5 I t 0 , 2 . 5 ) G . F G / * 4 l 1 0 . 2 . 5 ) . F G W 5 I 1 0 . 2 . 5 ) . F E W I I 1 0 . 2 . 5 ) . FEW21 1 0 . 2 . 5 ) . F E W 3 I 1 0 . 2 . 5 ) G . F AH I I 1 0 . 2 , 5 ) .1 tlWI 1/) , 2 ,5 ) . FC WI I 1 0 , 2 .5 ) , F F W 1 1 1 0 . 2 . 5 ) . FF W2110 ,2 ,5 ) G . F F W 3 I 1 0 . 2 , 5 ) . FHWl I 1 0 , 2 . 5 ) . FH W 2 I 10 , 2 . 5 ) . FHV.5I 1 0 . 2 . 5 ) . FH W 4 I 1 0 . 2 , 5 ) G . l HW5I 1 0 , 2 . 5 ) . FUWI 1 0 , 2 . 5 ) . UF W H O . 2 . 5 )

CUMMON / C M T X 5 / F Nu l I 0 . 2 . 5 ) . FVÜI 1 0 . 2 . 5 ) .F GUI I 1 0 . 2 . 5 ) . F G O 2 H 0 , 2 , 5 ) ,G FGU3I 1 0 . 2 . 5 ) . l u U 4 l 1 0 . 2 . 5 ) . F G 0 5 I 1 0 . 2 . 5 ) . F EUI I 1 0 . 2 . 5 ) . F E U 2 I 1 0 . 2 , 5 ) G.F EU5I 1 0 . 2 . 5 ) . F A U l 1 1 0 , 2 . 5 ) . F I I U I 1 0 . 2 . 5 ) . F C u l I 1 0 i 2 . 5 ) . F F U l I 1 0 . 2 , 5 ) .G FFU2 I 1 0 . 2 . 5 ) . F F 0 3 I 1 0 . 2 . 5 ) . F H U l I 1 0 . 2 . 5 ) . FHU2I 1 0 . 2 . 5 ) . F H 0 3 I 1 0 , 2 . 5 ) G . F H U 4 I l u . 2 . 5 ) . F H U 5 I 1 0 . 2 . 5 ) . F U O I 1 0 . 2 . 5 ) . J F U l 1 0 . 2 . 5 )

CUMMUN / C M T X 6 / FRGUI 1 0 . 2 . 5 ) . FRGI 1 0 , 2 . 5 ) . F VGI 1 0 . 2 . 5 ) . PGR I ( 1 0 , 2 , 5 ) ,G I G N 2 I 1 0 . 2 , 5 ) . F G N 3 I 1 0 , 2 . 5 ) . F GR 4 I 1 0 . 2 . 5 ) . FGR51 1 0 . 2 . 5 ) . F E R I I 1 0 , 2 , 5 ) G . F E N 2 I 1 0 . 2 , 5 ) . F E N 3 ( 1 0 , 2 , 5 ) , F A H 1 ( 1 0 , 2 , 5 ) . FURI 1 0 . 2 . 5 ) . F C N I I 1 0 , 2 , 5 ) ,G F F R I I 1 0 . 2 . 5 ) .FI N 2 I 1 0 . 2 , 5 ) , F F R 3 ( 1 0 . 2 . 5 ) . FHN1 I 1 0 . 2 . 5 ) . F H N 2 I 1 0 , 2 , 5 ) G . FH R 3I 1 0 . 2 . 5 ) . F HR 4 I 1 0 . 2 , 5 ) . FHR 5I 1 0 , 2 . 5 ) . F GG1 ( 1 0 . 2 . 5 ) . F G G 2 I 1 0 , 2 , 5 ) G. FG G3 I 1 0 . 2 , 5 ) . F G G 4 I 1 0 . 2 . 5 ) . FGG5I 1 0 . 2 . 5 ) , F E g 1 I 1 0 , 2 . 5 ) . F E G 2 I 1 0 . 2 . 5 ) G . F E G 3 I 1 0 . 2 . 5 ) . F A Gl I 1 0 . 2 . 5 ) . F U G H 0 . 2 . 5 ) . F C G II 1 0 . 2 . 5 ) . F F G I I 1 0 . 2 . 5 ) ,G FFG2I 1 0 . 2 . 5 ) . F F G J I 1 0 , 2 . 5 ) . F H G l I 10 . 2 . 5 ) . FHG2I 1 0 . 2 . 5 ) . F H G 3 I 1 0 , 2 . 5 ) G . F M G 4 I 1 0 , 2 . 5 ) . F H Ü 5 I 1 0 . 2 . 5 ) , U F G I 1 0 . 2 . 5 )

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G - E U I 1 . J . K ) 4 P 0 ( 1 . J - k . K ) - A U I l . J . K l ^ P U I 1 - l . J . K ) - C U < 1 . J . K ) ♦POI G l f 1 , J . K ) - I U ( 1 , J . K ) ^ P U I 1 , J H . K ) - H G I 1 . J . K ) F P U I 1 . J . K * 1 )G -UUMI 1 . J . K ) ♦PGI 1 . J . K ) -MUI 1 , J . K ) 4 P F U I 1 . J . K ) l / I I OI 1 . J ,K)

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101 FORMAT! ' OAVERACL RESERVOIR ( M A T RI X) PRESSURE I P S I A )t • UI L MATERIAL UALANCE THI S STEP (PERCENT)t « WATER MATERIAL BALANCE THIS STEP (PERCENT)t • GAS MATERIAL BALANCE THIS STEP (PERCENT)C • CUMULATIVE OIL PRODUCED ( S T B )C • CUMULATIVE WATER PRODUCED (STD)C • c u m u l a t i v e WATER INJECTED ( ST B)

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PRINT 1 0 410 4 FOR MA T C I PRESSURE D I ST R I B UT I UN 3 IN MATRIX R O C K ' . / .

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5 0 PRINT 1 0 6 . J1 0 6 FORMAT COROW NUMOL R' . 1 7 . / ' ---------------------- • / )

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PRINT 1 0 ) . I P , ( I . J . K ) . 1 = 1 . NX)1 0 9 FORMATC PG ' , 1 0 F I 2 . 1)

J = J - 1I F I J . C O . O ) l .U TO 12 GO TO 5 0

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t , «0 POROSI TY U I S T R I I I U T I U N 5 I N MATRIX R U C K * , / .C. t ******»******************************,//;0 0 12 L K = I , N Z K = N Z - L K f I P R I N T 1 0 3 , K J= NY

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12 CONTINUE P R I N T 1 8 2

1 5 2 F O R M A T ! / / / * ♦ ♦ ♦ POROSI TY D I S T R I l l U T I O N IN FRACTURE ♦ » * * , / ,C *

DO 18 L K = I , N Z K = N Z - L K f I P R I N T 1 0 3 , K 3=NY

3 1 P R I N T 1 0 4 , 3P R I N T 1 0 9 , ( F P I I I I I , 3 , K ) , 1 = 1 , NX)3 = 3 - 1I F ( 3 . E Q , 0 I GO TO 18 GO TO 31

10 CONTINUE1 0 3 FORMAT( «0LAYER NUMOER ' , I S , / ' ♦ ♦ * ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ * « / )1 0 4 FORHATCOROW NUMBER* , 1 5 , / * -----------------------* / )1 0 9 F O R M A T ! ! * 0 * , I 0 F I 2 . 4 ) I

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I M P L I C I T D C A L * » ( A - M . K . Ü - Z 1 INTLOER K . K P H I L . K P E I f L . KP HI FCOMMON / S L T l / 1111050, HH03W . HMOSG. P B C , T S C . ALPHA . ÜLTA .GAMMA COMMON / L I M I T / MPUT, MPWT, MPÜT, MSGT. MSWT, MFSGT. MF SWT COMMON / P A P M I / C L . C l . P H I ( 1 0 , 2 , 5 * . F P H I I 1 0 , 2 , 5 »COMMON / P A R M 2 / KL( 1 0 • 2 . 5 ) , F K X X l 1 0 , 2 . 5 ) » F K X V ( 1 0 • 2 . 5 ) . P K X Z ( 1 0 . 2 , S ) ,

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COMMON / PAMM3/ P01 ( 1 5 ) , UOT( 1 5 ) , D 1 DDT 1 I S ) ,C RS UTl 1 5 ) , ! )KSUT( I S ) . U O f l 1 5 ) ,P MT ( 1 S ) , U W T ( I S ) , P G T ( 1 5 ) , UGT I 1 5 ) ,C D I D G T l 1 5 ) . U G T ( I S ) , S G T ( 1 5 ) . KRGT( 1 5 > , K R 0 G T ( 1 5 ) , PCGUT( 1 5 ) , U S G T ( I S ) , e S W T ( 1 5 ) , Kl<WT(1 5 ) , K RO W T( 1 5 ) , PCOWT( 1 5 ) , DSWT( 1 5 ) , P SG T ( 1 5 ) , F K R G T ( 1 5 ) , C F K R U G T l I 5 I , F P C G 0 T ( 1 5 ) . F O S G T l 1 5 ) , F S H T ( 1 5 ) ,FKRWT( 1 5 ) , PKRUWT( 15 ) ,& F P C n W T I 1 5 ) , F D S W T ( 1 5 ) , R5WT( 1 5 ) , NFSWT( 1 5 )

I N T E ÜE R 42 T 1 T L L 1 4 0 )

READ CODE NUMliKR FOR ROCK-MATRIX POROSI TY AND P ER MEADI LI TY ( FOR UNIFORM D 1 S T R 1 HUT I O N : K P H 1 L = 0 , KP ER L= 0)

R E A D ( S , H 2 ) T I T L E R E A D ( 5 , 1 0 0 ) K P H1 L ,K P ER L I F I K P H I L . N L . O ) GO TO 2 0 R E A D I S . l 1 2 ) T I T L E R E A D ( 5 , 1 0 1 ) P H I L DO 10 K = 1 , N Z DO 10 3 = 1 , NY DO 1 0 1 = 1 . NX P H I ( 1 . 3 . K ) = P H 1 L

10 CONTINUE GO TO 21

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READ NUDES' P OR OS ITY STARTING FROM LAYER N Z , N Z - I , FOR EACH LAYER THE HOW ORDER I S : NY, N Y - 1 , N Y - 2 , , . , , , 3 , 2 , 1.

2 0 R E A D ( 5 , 1 1 2 ) T 1 TEE DO 11 L K = 1 , NZ K = N Z - L K f l3 = NY

2 2 R E A O ( 5 , 1 0 2 ) ( P H I ( 1 , 3 , K ) , 1 = 1 , NX)3 = 3 - 11 P ( 3 , E O , 0 ) GO TO 11 GO TO 2 2

11 CONTINUE21 PRINT 5 05 0 IORMAT( / / « *** "OI.OSITY DISTRIHUTIDN IN HOCK MATRIX ♦ ♦ ♦ • / / )

DU 12 LK = l , i l Z K = N Z- LKf 1 P RI NT 1 0 3 , K 3 = NY

2 3 PRINT 1 0 4 , I P H I I 1 , 3 , K | , 1 = 1 , N X )3 = 3 - 11 1 ( 3 , F I ) . 0 ) GO TU 13 GO 1 0 23

12 CONTINUE

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0 0 6 1 0 0 6 20 0 6 30 0 6 40 0 6 0 0 0 6 6 0 0 6 7 0 0 6 » 0 0 6 9

0 0 7 00 0 7 10 0 7 2 0 0 / 3 0 0 7 4 00 75 0 0 76 0 0 7 7 0 0 7 » 0 0 79 OOCO 0 0 » ! 0 0 0 2 00 « 3 00M4 OOMS 0 1(16

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I F ( K P E H L . N E . O ) GU TU 2 5 R E A D ( S f l l 2 ) T I T L E R L A » ( S . I 0 1 > PLRL DO 13 K = t . N /DU 13 J = 1 , M V DO 13 1 = 1 . MX K L l 1 . J f K ) = P E R L

13 CONTINUE GU TU 2 4

READ NUDES' PEUMEAll I L I TV STARTING FROM LAYER ...................FUR EACH l a y e r , THE ROW ORDER I S : N Y , N Y - 1 . N Y - 2 ............... 3 , 2 . 1 .

2 5 R E A D I S . l 1 2 ) T I T L E DU 14 L K = 1 , N Z K= NZ -L K+ 1J = NY

2 6 H E A I K S , 1 0 2 ) ( K L l I , J . K ) , 1 = 1 , NX)J = J - 1I F ( J , E O , 0 ) GO TO 14 GO TO 2 6

14 CONTINUE 2 4 P R I N T 151

151 FORMAT I / / ' I S O T R O P I C I 'ERMEAHIL ITY 1)1 STR I OUT I UN I N RUCK MATRIX& ♦ ♦ ♦ • / / )

DO I S L K = 1 , N Z K = N Z - L K + I P R I N T 1 0 3 . K J = NY

2 7 P R I N T 1 0 7 , ( K L ( I , J . K ) , 1 = 1 , NX)J = J - 1I F ( J . E O . O ) GU TO 15 GO r o 2 7

1 5 CONTINUE

READ CUDE NUMOERG FUR FRACTURE POROSI TY AND PKRMFAUILITY (FOR UNIFORM D I S TR I O U T I UN : KPIIIF = 0 , NUMKF=0)AS FUR PER MEAUI LI TY TENSOR, I F NUT UN IFORMI NUMKF . Nt; . O ) , UNE SET OF TENSOR VALUES WILL 01: ASSIGNED TO ALL NODES : THEN NUMKF S E T S OF PER MEADI LI TY TENSUI! WILL PL HEAD IN FOR NUDES THAT ARE DI FFERENT

R E A D I S , 1 1 2 ) T I T L E R E A U I S , 1 0 0 ) KPI I IF , NUMKF I F ( K P I I I F . N F . O ) GO TO 2 »R E A D I S , I 1 2 ) T I T L E R E A D I S , 1 0 1 ) PII IF DO 16 K = 1 , N 7 DU 16 J = 1 , N Y DU 16 1 = 1 , NX I P M I I 1 , J , K ) = PII1I

16 C U N T I NUL GO TU 2 9

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0 1 3 0 1 1 71 F O n M A T C O » . F l O . l . 2 K 1 0 . 4 »0 1 3 9 I 1 7 2 FORMAT( • 0 * .1 l O . I , 0 1 0 . 5 . F I O . S . F I O . 4 I0 1 4 0 I I 3 2 F O R M A T ! / / / ' O G A S PVT DATA T A O L C . / . « ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ' , / / . T 9 .

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0 1 4 2 1 1 9 1 F O R M A T ! / / ' T I L T I N G ANGLES OF R E SF RVOI R F O R M A T I O N : ' . / / .C O A L P H A = ' . 1 10 . 2 . ' D h G . ' / . 'OÜETA = ' . F I 0 . 2 . " D E C . ' . / .C'OGAMMA = ' . 1 1 0 . 2 . ' D U G . ' / / )

0 1 4 3 1 2 0 T 0 H M A T I 2 D I 2 . 4 )0 1 4 4 121 F O R M A T ! / / ' U F R A C T O R E C O M P R E S S I B I L I T Y = ' . 0 1 5 . 4 . ' P S I - 1 ' / .

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CDATA FOR MATRIX R O C K ' / . ' ♦ ♦ ♦ ♦ ♦ * * ♦ ♦ ♦ ♦ ♦ ♦ ♦ » ♦ ♦ ♦ * ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦, / / , T 7 . ' S G ' . T I 7 . ' K R G ' . T H 6 .

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0 1 4 7 1 1 3 5 F O R M A T ! / / / « 0 W A T E R - 0 1 L RELATI VE P ER MEABI LI TY C CAPILLARY PRESSURECDATA FOR MATRIX R O C K ' . / . ' ♦ ♦ » ♦ * ♦ ♦ ♦ ♦ * ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ * ♦ ♦ ♦ ♦ * * ♦ • N)C * * » * * * * * * * * * * * * * * * * * * * * * 4 * * * * * * * * * • . / / . T 7 . ' S W , T I 7 . ' K R W ' . T 2 6 .C ' K R O W * . T 3 0 . ' P C 0 W ' . T 4 5 . ' D S W / D P C , / . ^C T 3 7 . ' ! P S 1 ) ' . T 4 5 . ' I P S I - I ) ' . / .C ------------------------------------------------------------------------------------------------------------------------- ' / / I

0 1 4 B 1 1 3 6 FORMAT! / / / ' O G A S - O I L RELATIVE PER ME AB I LI T Y C CAPILLARY PRESSURECDATA FOR F R A C T U R E ' / . ' * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *C * * * * * * * * * * * * * * * * * * * * * * * * * * * * * . / / , T 7 . ' S G ' • T 1 7 . ' K R G ' . T 2 6 .C ' K R O G * . T 3 7 . ' P C G O ' . T 4 4 . ' D S G / O P C ' / .C T 3 6 . ' ! P S I ) ' . T 4 4 . ' I P 3 1 - 1 ) • . / .C -----------------------------------------------------------------------------------------------------------------------------' . / / )

0 1 4 9 1 1 3 7 F ' H M A T ! / / / ' 0 W A T E R - 0 I L R ELA TI VE P ERMEABI LI TY C CAPILLARY PRESSURECDATA FOR F R A C T U R E ' . / . ' * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * c * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * ' , / / , T 7 . ' S W ' . T 1 7 . ' K R W ' . T 2 6 .C ' K R O W ' . T 3 B . ' P C O W . T 4 5 . ' O S W / O P C . / ,C T 3 7 . ' I P S I ) ' . T 4 S . ' I P S I - I ) ' . / .C ------------------------------------------------------------------------------------------------------------------------- • / / »

CC P R I N T PER MEAUI LI TY TENSOR C

0 1 5 0 3 2 P R I N T 1 5 30 1 5 1 1 5 3 F O R M A T ! / / / ' * ♦ * AN I S O TR OP IC PERMEABI LI TY D I S T R l O U T I O N I N FRACTURE

C * * * ' / / )0 1 5 2 DO 01 L K = I . N 20 1 5 3 K=NZ-LKF10 1 5 4 P R I N T 1 0 3 . K0 1 5 5 J=NY0 1 5 9 3 3 PRINT 1 1 0 . 30 1 5 7 DO 0 2 1 1 = 1 . N X . 401 511 1END=I 1 * 30 1 5 9 I F ! l E N D . G T . N X ) IEND=NX0 1 GO PRINT I I I . II K X X ! I , 3 . K ) . F K X Y ! I . 3 .K ) . F K X Z ! I . J . K ) , 1 = I 1 . ILND)

I UPTKAM IV 01 H r L f ASE 2 . 0 HEiiOAT OATE = 0 2 J 3 J 2 0 / 3 7 / 0 7 PAOl: 0 0 0 0

0 1 6 1 P R I N T I I I . ( F K Y X I I , J . K I . H K Y Y I I . J . K ) . F K Y Z I I . J . K ) . 1 = I I . IENl>>0 1 6 2 P R I N T i n . (FKZXI I . J . K ) « I K Z Y I I . J . K ) . F K Z Z I I . J . K ) . 1 = 1 I t I C N U )0 I 6 3 P R I N T I J 50 1 6 4 I JO F O R M A T ! ' O ' )0 1 6 0 « 2 CONTINUE0 1 6 6 J = J - I0 1 6 7 I F ( J . E Q . O ) 0 0 TO (110 1 6 B GO TO 3 30 1 6 ' » 01 CONTINUE

CC READ MAXIMUM PREOSIJHES AND SATURATIONS FUR VALUES IN TAULES C

0 1 7 0 R E A D I S . 1 1 2 ) T I T L E0 ) 7 1 R E A D I S . 1 0 6 ) I 'MAXU.PMAXW.PMAXO.SOMAX.SWMAX0 1 7 2 P R I N T I IB.PHAXU. PMAXW. PMAXO. SGMAX. SHMAX

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CC READ WATER PVT DATA

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CC HEAD GAS PVT DATAC

0 1 0 ) 5 4 R E A D I S . I 1 2 ) T I T L E0 1 9 0 P R I N T 1 1 3 2O l ' J l DU SS 1 = 1 . 5 0 00 1 9 2 MPGT=I0 1 9 3 R E A U I S . 1 1 6 ) POT I I I . HOT I 1 ) . 0 1 UÜTI 1 ) . U O T 1 1 )0 1 9 4 P R I N T 1 1 7 2 . POT 1 1 ) . U O T I 1 I . D I D G T I 1 ) . U O T I 1 )0 1 9 5 IF I P G T I I ) . O L . P M A X ' , ) 0 0 TU 5 60 1 9 6 5 5 CONTINUE

CC READ O I L . WATER. GAS D E N S I T I E S I N t .UM/CU FT AT STANDARD CUN DITIU NSC

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READ G A S - U I L RELATIVE PER MEAUI LI TY C CAPILLARY PRESSURE FOR MATRIX RUCK

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5 9 CONTINUE

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1 0 1 r O R M A T ( 3 I S . 3 1 1 0 . 0 1 D L L X = X L / D r L U A T ( N X I D E L Y = Y L / D f L U A T I N Y )D E L Z = Z L / D F L Ü A T I M Z )P R I N T 1 0 2 , X L , Y L . Z L , N X , N Y , N Z , O E L X , D E L V , DELZ

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0 0 3 3 TAW( 1 J t K ) = T A W ( I , J . K ) » F A C T B0 0 3 » TCW( I J . K ) = i AVGI KLI I f t . J . K I . K U d . J . K ) )0 0 3 ! . TCWC 1 J , K » = T C W ( 1 , J . K l » F A C T 80 0 3 0 1FW( I J . K ) = I - A V G I K L I 1 . J F l .K I . K L d . J . K » )0 0 3 7 TFW( t J . K ) = T F W d . J . K ) * F A C T 900311 TMW( 1 J . K ) =1 AVGI KLI I . J . Fv F l ) , K L d . J «KI )0 0 3 0 THW( 1 J . K ) = T M W d . J . K ) * F A C T 1 0OOAÜ rt (W( 1 J . K I = K L ( r . J . K X ^ r ACT»0 0 » l TUW( 1 J , K I = r ACTJ

0 0 » 2 TVI K I J . K ) = I AC 130 0 » 1 TGO( I J . K ) = T G W d . J . K )0 0 » » T t U l l J . K ) = T i ; w d . J . K )00 » ! , T A I K I J . K ) = T A W d . J . K )0 0 » l . TCO( 1 J , K ) = T C W d . J . K )0 0 » 7 TFO< 1 J . K ) = T F W d . J . K )0 0 » H THO( I J , K ) = T I I Wd . J . K )0 0 » " TUÜ( 1 J . K ) = I A C T 30 0 5 0 Ï H O ( I J . K ) = T H W d , J . K )

OOUl TVG( I J , K ) - I A C T 300! , : ' T G G lK . J . K ) = T G W I 1 . J . K )0 0 5 3 TEGÜI , J , K ) = T C W ( 1 . J . K )0 0 5 » TAGO( . J . K ) = T A k { t . J . K )0 0 5 5 TCGUI , J . K ) = T C W ( I . J . K )0 0 5 b TFGUC . J . K ) = T I W ( 1 . J . K )0 0 5 7 THGU( , J . K ) = TIIW( I . J . K )0 0 5 0 TGGI 1 J . K ) = T G W d . J . K I005!» TLG( I J . K ) = T L W d . J . K )0 0 0 0 TAG! I J . K ) = T A W ( 1 , J . K )0 0 b I TCG( 1 J , K ) = T C W d . J . K )0 0 6 2 T F G ( 1 J , K ) = T r - W d . J . K )0 0 6 3 THG< I J . K ) = TIIW( I . J . K )0 0 6 » TRGIM . J . K ) — T l( V 1 . J . K )0 0 0 5 TRG( 1 J . K ) = TI (Wi : , J . K )

0 0 6 6 TFGWl 1 . J . K ) = F , V C T 50 0 ( , 7 TFGW2 1 . J . K ) =1 ACT' ,000(1 TFGW3 1 . J . K ) = r A V G I F K / / ( 1 . J . K ) . F K Z Z d . J . K00( , O TFGW3 I . J . K ) = TI G W J d . J . K ) * F A C T 1 00 0 7 0 TFGW» 1 . J . K ) = r A C T 60 0 7 1 TFGW5 1 . J . K ) -1 ACT50 0 72 TI e w i I . J . K ) = I ACT 70 0 7 3 TFEW2 1 . J . K ) =1 AVGIFKVY d . J . K I . F K Y Y I 1 . J - 10 0 7 » TFCW2 I . J . K ) =TFLW;>( I . J . K) ♦ l ACTV0 0 7 5 TI-nW3 1 . J . K ) = I ACT70 0 7 0 TFAWI 1 . J . K ) = r A V G ( l KXX ( I . J . K ) .1 K X X d - l . J00 77 TFAWI I . J . K ) = T I A K l d . J . K ) * F A C T O00711 TFCWl 1 . J . K ) =F A V G I F K X X d . J . K ) . F K X X d F l . J0 0 7 " TFCWl 1 . J . K ) - TI CWl ( I . J . K ) + F A c n i00(10 TFFWI I . J . K ) =1 AC I 700 fi t TF( W2 1 . J . K ) =1 A V G I F K Y Y ( I . J . K ) . F K Y Y < I . J F 1001»:' TFFW2 I . J . K ) - r I 1 W 2 d . J . K) f f a c t p

OOH t TFFW3 1 . J . K 1=1 AC r 700H» TFIIWI I . J . K ) = I ACT!,70'1' , TKI W2 1 . J . K ) =1 ACT' ,OOl'l- TF HW3 1 . J . K 1 =1 AV' . I I V ./ . / . ( I . J . K ) .1 K / / < I . J . K0007 TIMW3 I . J . K ) I I II ,.' .:( 1 . J . K ) F | ACTIO

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OOHIt TFMW4 1 1 . J , K ) = I AtTC.OOtP) T F H W S l 1 , J , K ) = I ACT50 0 0 0 T F R W l 1 . l , K ) = K L ( l . J , K ) 4 F A C 1 40 0 9 1 TFUWl 1 , J t K ) - - l AC.TJ

0 0 9 2 T F G D M 1 J . K ) = T F G W 1 (1 J . K )0 0 9 3 TFG112( I J . K ) =1 F G W 2 ( 1 J . K )0 0 9 A T F G O J l 1 J . K ) = T I G W J l 1 J . K )0 0 9 Ü T F G 0 4 ( t J . K ) = 1 I <5W4( 1 J . K )0 0 9 / . T F G D S l 1 J , K ) = T F Ü W Ü ( 1 J . K)0 0 9 7 T F E U K 1 J . K ) = TI E W I ( 1 J . K )0 0 9 0 T F E 0 2 1 1 J . K ) - T F L W2 11 J . K)0 0 9 9 T F E O J l I J . K ) - T F L W J l 1 J . K)0 1 0 0 T F A U l (1 J . K ) = r i A Wl 11 J . K )0 1 0 1 T F C M H 1 J . K ) - F F ( : W 1 ( 1 J . K )01 0 2 T F F O l (1 J . K l = T I - r W l < 1 J . K )01 0 3 T F F U 2 1 1 J . K ) - 1 I F W 2 ( 1 J . K)01 09 T F F U 3 1 I J . K ) = r r r w 3 ( 1 J . K)01 0 0 TFI I UI ( 1 J . K ) = 1 F I I W1 ( 1 J . K )01 Of. TFII<32( 1 J . K ) = T I MW2 . ( 1 J . K)0 1 0 7 T F M O J l 1 J . K ) = T I I I UJ l 1 J . K )01 0 0 T F H U 4 ( 1 J . K ) = T F I I W 4 ( 1 J . K )01 0 9 r r n o s i 1 j , K ) = T r Hwr>( 1 J .K)0 1 1 0 T F H U l I . J . K ) = T F M W ( 1 . J K)0 1 1 1 TFUCJl I . J , K ) = F A C T 30 1 1 2 T F V U l 1 , J , K ) = I A C T 3

01 13 T F G G H 1 J . K ) = T F G W 1 (1 J K)0 1 1 4 T F G Ü 2 1 1 J , K ) = T I G W 2 1 1 J K)O i l ! , T F G G 3 1 1 J . K ) = T F G W J ( 1 J K)01 If. T F G G 4 ( I J . K ) = T F ( i W4 ( 1 J K)01 17 T F G G 5 1 1 J . K ) = TFGW‘i ( 1 J K)011/ 1 T F E G K 1 J . K ) = T n : W l ( 1 J K )0 1 1 9 T F E G 2 1 1 J . K ) = T I L W 2 ( 1 J K)01 2 0 T F E G J t I J . K ) = r i I . HJ l 1 J K)0 1 2 1 T F A G I I 1 J , K ) = 1 I AWl ( 1 J K)01 2 2 T F C G K 1 J . K ) = T FLW1 < 1 J K)01 2 3 T F F G K I J . K ) = T IT K H 1 J K )0 1 2 4 T F F G 2 1 1 J , K ) = T F T W 2 ( 1 J K)0 1 2 0 T F F G 3 1 1 J . K l - T F I WJ( 1 J K)0121, TFI IGI ( 1 J . K ) =TITIW1 1 1 J K )01 2 7 TI I IG2( 1 J . K ) =TI H' . ' 2( 1 J K)0 1 2 0 T M I G J l 1 J . K ) = r n i W 3 ( 1 J K)0 1 2 9 TFHG4 ( 1 J . K) =TI TI W4 1 1 J K)0 1 3 0 T FHGSl 1 J . K ) = TITIWM 1 J K)0 1 3 1 TFGI U 1 1 J . K ) = T F ( i W l 1 1 J K )0 1 3 2 T F G M 2 1 1 J . K ) = TFGW2< 1 J K)0 1 3 3 T F G H J l 1 J . K ) = T r ü V I J < 1 J K I01 3 4 T F G M 4 I 1 J . K ) = T r . ' i W 4 ( 1 J K )0 1 3' j TFGI l Bl 1 J . K ) =TI 1 J K)0 1 J o Tf E M I ( I J . K ) = T I T . W 1 ( 1 J K )0 1 3 7 1 F E M 2 1 I J . K ) = r i LW2( 1 J K)01 3 0 T F E U J l 1 J . K | - 1 I L W 3 ( 1 J K)01 3 9 TI AMI ( I J . K ) = T I A Wl <1 J K )0 1 4 0 TF C M l (1 J , K ) = m . W l ( 1 J K)0 1 4 1 TI F n i I 1 J . K ) = r F F W l 11 J K)0 1 4 2 TI 1 M2( 1 j . t ; ) = Ti 1 W2( 1 J K)0 1 4 3 TI 1 M I( 1 J , K ) = 1 I 1 W.K 1 J K. I

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C 2.40ELY)IIIK.EO. 1 .lll'.K.LQ.MZ) DPU2 2=0.0IFIK.GT. I .AND. K.LT.N7 I l)Pl>22 = CUN4 ♦ (PI G(l . J.Kf 1 I-PI G( 1 . J . K-I ) I / (

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e I)PU2ZFCUN54ULNGF*1)VGZUT( I . J . K ) FCUN|.+DENGF*I)CUS( GAHH A ) )VGXII.J.K)=VGX(I.J.K)*FKUG/rACT VGYN=VGY( I . J .K )F.ACT=1 .-Fl Fl KYY( I.J.K)VGYII.J.K)=I KYYlI.J.K)*(UPU2Y-CUN5*0ENGF*VGYN/nKLTF

6 CONOFULNGFFDSINIMLTA))IFIFKYXII.J.K).NC.Ü.O) VGY(I.J.K)=VGY(I.J.K) F IKYX(I.J,K)♦(

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& l)PD2ZFCnN5FnnNGFFUVGZr)T < I . J , K ) FCMNO FOI.NGF F|)C(I5( GAMM 4 ) )VGY( I . J . K )= VG Y ( I . J . K ) FI-k UG/FACT VGZN = VG Z( I . J , K )FACT = I .-If +1 KZZ( I.J.K)VGZ( I. J.K)=FKZZI I.J.K) F(UPU2Z-C()N5F0FNCFFVGZN/UI:LI F

e CONGFOC.NGI «llCObl GAMMA))IF(FKZXII,J.K).ME.0.0) VÜZ(I.J.K)=VGZ(I.J.K) F IKZX(I.J.K)F(

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& 2 . F U h L X )I F ( J .F-Q. I . ( IP . J . l - a . N Y ) U P U 2 Y = 0 . 0II ( J . G T . I . A M U . J . L T . M Y ) UP02Y = COM4* ( ,’ F U ( I . J F ( . K ) - P I M( I . J - 1 . K ) ) / (

6 2 . FOLLY) .IF ( K . L Q . 1 . IIL.K . FU. r i Z ) P P U 2 Z = 0 . 0II (K.GT . 1 . A' lU. K . L T . l i Z ) 1>PI)2Z = CIJM4* IPFOI I . J . K F | ) - P l II ( I . J . K - I ) ) / (

r, 2 . FUCLZ )V O X N = V n X ( I . J . K )FKU )=-FKPU/UUIFF = FKUUFCGM5Fi,LIU)l / I l l .L Tr A C r = I . - F F F | KXX( I . J . K )VOX( I . J . K ) =1 KXXl I . J . K ) F I 0P'I2X-CMN5 FOLNGF F V'JXM/UI LT F

C UCMUFFCUMI.FUJ IM( ALPilA ) )I f (FKXY( I . J . K ) . N L . O . , ; ) VMX( I . J , K ) = VIIX( I . J . K ) F I KX> I I . J . K ) F(

C UP02YFCUM5 IU! M.ll MIVUYUI I I . J . K ) F CDMoFULNlIF F D j I Ml PL IA ) )I F ( F K X Z ( I , J . K ) . M L . 0 . 0 ) V U X ( I . J . K ) = VDX( I . J . K ) 1 I K X / ( I . J . K) F (

r, l ) l>027F( MM‘'< Jl MU) < UViJZUT ( I . J . K ) FCIIN',»UL,MUI FULMbI '.AMMA ) )

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VOX{ I I J . K ) =V l ) X ( I , J i K ) * f KUU/I-ACT VUYiy=VOY( 1 f.l ,K ) rACT=1 . - i r + I K Y Y < I i J . K )VO Y ( I f J . K ) =1 l',YY( I , J , K ) * ( Dl'UZ Y-CU1.5 *ULNOI *VUYN/Ut . l . 1 »

r, CUN'i*ULNi:l ♦ U o I M d i f . T A ) )IKITKYXII . J . K ) . M C . O .O ) VOY( I , J , K )=VUYII• J . K) + hKYX(I . J t K )* (

0 nPr)2XfCUM5*[)lf NU(:*l)Vl)X()T ( I , J t K)4COH6*ülNUF»D3 1M( ALPHA) ) I F I F K Y Z d . J . K ) .ML . 0 . 0 ) VUY ( I . J . K ) = VIJ Y(1 . J . K ) ^ ( KYZI I , J . K ) *(

r- l)PU2Z+C(INr,*l)l:MI)l *PVt)ZOT l l . J.K)4tON(,*üENOF*UtU3IGAIIMA) ) VOYI I , J . K ) = VUYH . J , K ) t l KUU/FACT VOZM-VOZII. J . K )F A C T = l . - r i *1 KZZI I . J . K )V U Z I I . J . K ) = I KZ / I I . J . K) ♦IUPUZZ-CUNr>*UfcN(JF*V(IZNZOLLrt

6 C0N64DI N()f *l)Llt3 IGAMIIA ) )I I ( FK ZX I I . J . K ) . I I I . 0 . 0 ) V U / I I . J . K ) = VUZI I . J . K ) 4 F K Z x l I . J . K ) * I

r. l )P02X4CUM0*l ) I lie F*I)VIIX1)II I . J . K | 4 C U N I , * 1 ) C N U F * 0 3 I I 4 | ALPHA) ) I F I F K Z Y I I . J . K ) . ' I F .0 . 0 ) V UZ I I . J . K ) = VUZI I . J .K ) 4 t KZYI I , J . K ) 4 |

r. DPD2Y4Cl)NS*DLIIUr 4UVUYD1 I I . J . K ) 4 CUN(,4>ULM0F«U'j 1 Ml IIL IA ) )VnZI I . J . K ) = V l ) Z I I . J . K ) ♦FKUU/FACT OVUXUTII . J . K ) - | V I I X | I . J . K) -VUXM)/ OF, Lr OVOYÜT I I . J . K ) = I VOY I I . J . K ) - VUYN ) / O L l . I DVOZDT I I . J . K ) = IVOZI I . J . K )-VIJZN ) /Ül .LT

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I F I I . K O . 1 . l . H . I . 1 : 9 . M X ) l ) P | ) 2 X = 0 . 0I F I I . G T . I . A M D . I . L I . N X ) l ) l > D 2 X = C U M A 4 11*FW I I 4 1 . J . K ) - f ’F W l I - I . J . K ) ) / |

D 2 . * D K L X )I F I J . K O . I . I I M . J . I f ) . M Y ) P P U 2 Y = 0 . 0I F I J . G T . I . A M I ) . J . L l . M Y ) i) l>IJ2Y = C U M A * | P F W I I . J 4 | . K ) - f I Y I I . J - I . K ) ) / l

C 2 . * 0 t L Y )I F I K . E . Q . I . U M . K . I . O . N Z ) l ) P J 2 Z = 0 . 0I F I K . G T . I . A M D . K . L Î . M Z ) D P U Z / r C O M A ♦ I P F W I I . J . K 4 I ) - P I v | I . J . K - 1 ) ) / l

e 2 . * D K L Z )V W X H = V W X I I . J , K )I K U W = - F K H V i / MI M I F = F K U W 4 C U I i r . « , | ) ( : i . W I V ! ) F L TI A C r = l . - F l *1 K X X l I . J . K )V WX I I . J . K ) = F K X X I I . J . K ) H U P n ? X - C l ) M 5 * D K M W F * V W X N / U l L T 4

C D C N W r * C I ) M i , 4 | ) ' J I N I A L P H A I )I I I F K X Y I I . J . K ) . : i f . 0 . J ) VVfXl I . J . K ) = V W X I I . J . K ) 4 ) K X Y I I . J , K ) *1

C D P D 2 Y 4 C l l M ! . , * D L M i - l * D V W Y D l I I . J . K ) 4 C U M i > 4 D E N V t F * D G I M | l > l T A ) )I I I I K X Z I I . J . K ) . M l : . 0 . 0 1 VWXI I . J . K ) = VWX| I . J . K ) 4 I KXZ. I I . J . K ) » l

t U P D 2 Z 4 ( : U M 9 * r i MW F * D V VIZ D T I I . J . K ) 4 C U M ' , * D L N W r * D L U ( , l GAMMA) )V W X I I . J . K ) = V W X I I . J . K ) ♦ F K U W / F A C T V W V N = V W Y l I . J . K )I A C T = 1 . - F l * F K Y Y l I . J . K )V W Y I I . J . K ) r | K Y Y l I . J . K ) 4 I D P D Z Y - C l l N Ü ♦ DLNWF ♦ V W V N / D L I . T 4

& CUMf , *L) LNWI 411 j I I I HL I A ) )I ) I F K Y X I I . J . K ) . M L . O . O ) VWYI I . J . K ) = VWYI I . J . K ) 4 I K Y X I I . J . K ) 4 |

6 D P D 2 X 4 C 1 1 M 3 4 D 1 M'.vl 4 D VWXUT I I . J . K ) 4 CUNi i ♦ Ü L N 4 I F 4 U 3 I Ml A l P I I A ) )I F I I K Y Z I I . J . K I . I I I . 0 . 0 ) VWYI I . J . K ) = VWY| I . J . K ) 4 ) ) , Y Z I I , J.K) 4 |

t D P D 2 Z 4 ( . ( I N S < D L M . V F * i i V y Z I ) I I I . J . K ) 4 C U N f i 4 0 L N W F 4 D C U S I GAM IA ) )V W Y I I . J . K ) - V W Y I I . J . K ) 41 K U W / F A C T V W Z N = VW / . I I . J . K )I A C T = I . - F l <1 K Z Z I I . J . K )V W Z I I . J . K ) M K Z Z I I , J . K ) 4 | D P D . - ' Z - C l ) N ; . * D t N W I ♦ V WZ N / I . ' I L I 4

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APPENDIX GMonte C a r l o P r o g r a m f o r G e n e r a t i n g F r a c t u r e d Nodes

Ù1 R E L L A S E 2 . 0 MA I N D A T E = 8 2 ^ 3 9

CC MCNTE C A R L O METHOD F O R G E N E R A T I N G F R A C T U R E C E L L SC T H I S o k u GHAM i s F O R 2 - D 1 M E N S i O N A L R E S E R V O I R AREAC G A U S S C E L L O R D E R I N G S C H E M E I S U S E D I N N U M O E R I N G THE N O D E Sc E . G . F O R A 1 0 * 7 G R I D S Y S T E M * T h L CR D C R I SCc 8 1 6 2 6 3 6 4 6 5 6 6 6 7 8Ô 6 9 7 0c 5 1 5 2 5 3 5 4 S 5 5 6 5 7 5 8 59 6 0c 4 1 4 2 4 3 4 4 4 5 4 6 4 7 4 8 4 9 50c 3 1 3 2 3 3 3 4 3 5 3 t . 3 7 3 8 3 9 4 0c 2 1 2 2 2 3 2 4 2 5 2 6 2 7 2 8 2 9 5 0c 1 1 12 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0c I 3 4 5 6 7 8 9 I 0

c t iX = NUMOER O F n u d e S I N X - D I R E C T I O Nc NY • = NUMOER OF N ODE S I N Y - D I R E C T I O Nc N P E R C T = P E R Ci _NTAGE OF F R A C T U R E D N O D E SC t n = I N I T I A L RANDOM NUMBERC

D I M E N S I O N N F C E L L ( 1 0 * 1 0 ) , N F ( 1 0 0 )D A T A N F C E L L / 1 0 0 * 0 /DATA NX*NY,NPCRCT, N l / 1 0 * 7 . 3 5 , 2 4 9 8 /N T A L = 0M T A L = 0N T U T A L = N X » N Y T O T A L = N T ü T AL M w= n P E KCT * NT OT AL / 1 0 0 M T = N R E R C T * N T U T A L - M M * 1 O u I F ( M T . G E . 5 0 ) MM=MM+1

1 N T A L = N T A L + 1C A L L R A N D U ( N 1 * N 2 , R N )R N F = 1 . + R N * ( T O T A L - 1 . )N F ( N T A L ) = R N FNROW..11 = ( N F ( N 7 A D - 1 ) / N XN R O W= nROW M1 + 1N C U L = N F ( NT A D - N X ^ N R O W M l1 F ( N F C E L L ( N C J L * N R O W) . £ 0 . 0 ) M T A L = N T A L + 1 N F C E L L ( N C U L * NRC W) = 1RF. i m 1 U 3 * N T A L * N 1 *RN*P.fvlF , N F ( N T A L ) , N R ü W , N C ü L

1 0 3 F OR MA T ( « O ' . I 5 * I I 0* 2 F I 0 . + * 3 I 1 0 )N 1 = N2I F ( M T A L . L T .M.M) GO TU 1 DO 1 2 0 = 1 * NY K J = N Y - J + 1P P I NT 1 0 1 , ( N F C E L L ( 1 * K J ) , 1 = 1 . N X )

1 0 1 F O R M A T ( • 0 * , 2 0 1 5 )12 C OtniNUE

P R I N T 9 09 0 F O R M A T ! / / / / / * F R A C T U R E D N O D E S = 1 * » / ,

8 • N u N - F R A C T U R E D N O D E S = 0 * / )S T u P E ND

229

APPENDIX H

E s t i m a t i o n o f H y d r a u l i c R a d i u s f r o m Thomas e t a l E xam ple

I n E v a n s ’ m o d e l , t h e f l u i d i n t e r a c t i o n t e r m s a r e

w r i t t e n i n t h e o i l f i e l d u n i t a s :

[ T . ] ^ = 0 . 0 0 1 1 2 7 ^ l ^ r l i ^ f ^ i j l P j l ^ ^ 1 1 ' ^ f j ^ (H-1)

F o r e x a m p l e , t h e w a t e r c o m p o n en t i n w a t e r p h a s e h a s :p

=Wwl “ » w l " f fW1H e n c e ,

[ F ] _ = 0 . 0 0 1 1 2 7 ^ l^r lw ^^f^W s *^^lw”^fw^

MN o t e : The d i m e n s i o n s o f [ T 1 a b o v e a r e .L t

2 2I n Thomas e t a l m o d e l , t h e m a t r i x - f r a c t u r e f l o w

e q u a t i o n s a r e g i v e n a s i l l u s t r a t e d b e l o w f o r t h e w a t e r p h a s e

( w r i t t e n i n t h i s s t u d y ’ s n o t a t i o n ) .

% 1 ^ w l ^ w l

The l e f t - h a n d - s i d e o f t h e a b o v e e q u a t i o n r e p r e s e n t s

t h e w a t e r f l o w r a t e f r o m t h e r o c k m a t r i x t o t h e f r a c t u r e s .

B e f o r e c o m p a r i s o n b e t w e e n t h e tw o m o d e l s c a n b e m a d e , t h e

e q u a t i o n s m u s t h a v e t h e same d i m e n s i o n s . T h e r e f o r e , t h e

d i m e n s i o n s o f Thomas e t a l e q u a t i o n a r e c o n v e r t e d f r o m C ~ )

230

231M P

t o ( — ) a s shown b e l o w . M u l t i p l y t h e e q u a t i o n by .L ^ t ' 'b

*** = - 0 . 0 0 1 1 2 7 a ^ ^ B ^ ^ l w ^fw^^w l ' 'b ^ w l^ w l

CH-4)

The l e f t - h a n d - s i d e now r e p r e s e n t s t h e m a ss f l o w r a t e

p e r u n i t v o lu m e a n d h a s d i m e n s i o n s o f C — ] . N o te t h a t "6L ^ t

s i g n i f i e s t i m e s t e p d i f f e r e n c e

Cl = - 6 ^ ) .

C o n s e q u e n t l y , c o m p a r i s o n b e t w e e n t h e tw o m o d e l s r e s u l t s

i n t h e f o l l o w i n g r e l a t i o n s h i p .

* fr ^ l (H -5)a =

4>-f’’h ' Î T f ” '® ’

I n t h i s s t u d y , 1 = ^ = 100 f t , <j>£ = 0 . 0 1 . From Thomas_ 2

e t a l e x a m p l e , t h e m a t r i x s h a p e f a c t o r (cr) e q u a l s 0 . 2 5 f t

T hus ,

0 - 2 5 X 100 ~ 0 *0004 f t .

From s e c t i o n 4 . 6 , t h e f r a c t u r e h a l f - w i d t h i s r e l a t e d t o t h e

h y d r a u l i c r a d i u s b y :

4 r „d = - Ü (H -7 )

TT

H e n c e , i n t h i s s t u d y :

d = ^ ^ 0 . 0 0 ,04 ^ 0 . 0 0 0 5 f t .

i^PENDIX I

Effect of Decreasing the Fracture Hydraulic RadiusThe effect of decreasing the fracture hydraulic

radius was investigated by reducing the value of the fracture half-width from 0.0005 to 0.0001 ft (i.e. decrease r„

f l

from 0.0004 to 0.00008 ft). Water breakthrough of this run occured at approximately the same time as the Original results (255 days). However^ it was observed that the water saturation profiles of this differed slightly from those of the original run in Figure 10 and 11 (indicated by dash lines in Figures I-l and 1-2). As the fracture half width decreases, the water fronts tend to move slower in the fractures while the water fronts in the rock matrix tend to move faster. This implies that as the fracture half-width approaches zero, the water fronts in the fractures and the rock matrix converge become identical when no fracture exists (i.e. d = 0).

232

t250

200

(g$100

I

I

CM

OOi

toww

500 1000le n g th ( f e e t )

1500— h->> 2000

Figure I-l Fracture 50% Water Saturation Profile (center cross-section)

250

200

(Dm

(/)*100cÜ

to

-T IH |

N)W

4-0 500 1000

le n g th ( f e e t )1500 2000

Figure 1-2 Matrix 50% Water Saturation Profile (Center cross-section)

APPENDIX J

Effect of Smaller Grid SizesFor comparative study the grid sizes used in this

22study were the same as those used in Thomas et al paper (i.e. AX = AY = 200 ft and AZ = 50 ft). To investigate the level of truncation errors caused by replacing derivatives in the partial differential equation with the finite difference formulae, the grid sizes AX and AY were decreased from 200 ft to 100 ft. Hence, the reservoir system consisted of 20x4x5 = 400 grid points (or 6x400 = 2400 simultaneous linear equations). As the grid sizes decreased, the fracture characteristics half width (£ = AX/2) decreased as well. In order to preserve the same valve of geometrical factor used by Thomas et al, the fracture half width (thus, the hydraulic radius) had to be proportionally increased according to equation H-5. Thus, the fracture half width used in this run was increased from 0.0005 to 0.001 ft.

The results of this run are shown in Figures J-1 and J-2 (original results of Figures 10 and 11 are shown by dash lines). It is noted that the water fronts from the two runs have similar profiles although the water fronts of this run move at slightly slower pace than those of the original run. Water breakthrough occurred at approximately 290 days (or 33 days later than the original results). The difference be

235

236

tween the two results, however, may be considered small and constitutes an acceptable level of truncation errors.

It should be pointed out that as the grid sizes decreases, the number of equations to be solved and the computing time increases considerably. Therefore, one should carefully weigh between the accuracy desired and the time and cost of computation involved.

0)ü)

(/)(/)0)Cü

250

200

00

100

row

!■" >20000 500 1 0 0 0

l e n g th ( f e e t )1500

Figure j.i Fracture 50% Water Saturation Profile (Center cross-section)

250

200

00 (NNW00

-4— ►0 500 1000

length (feet)1500 2000

Figuî'o J.2 Mc'jtrix 50% Water Saturation Profile [center cross-section)

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