SEM11 for Permeability Simulation

8/2/2019 SEM11 for Permeability Simulation

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M a t h e m a t i c a l G e o l o g y , V o l . 2 9 , No . 1 , 1 9 9 7

M a r k o v C h a i n M o n t e C a r l o M e t h o d s f o rC o n d i t i o n i n g a P e r m e a b i l i t y F i e l d t o P r e s s u r e D a t a

D e a n S . O l i v e r , z L u c i a n e B . C u n h a , 3'4 a n dA l b e r t C . R e y n o l d s 5

G e n e r a t i n g o n e r e a l i z a t i o n o f a r a n d o m p e r m e a b i l i t y f i e l d t h a t i s c o n s i s t e n t w i t h o b s e r v e d p r e s s u r ed a t a a n d a k n o wn v a r i o g r a m m o d e l i s n o t a di f fi c u lt p r o b l e m . I f , h o w e v e r , o n e wa n t s t o i n v e s t i g a t et h e u n c e r ta i n t y o f r e s e r v i o r b e h a v i o r , o n e m u s t g e n e r a t e a l a r g e n u m b e r o f r e a l i z a t i o n s a n d e n s u r et h a t t h e d is t r i b u ti o n o f r e a l i z a ti o n s p r o p e r l y r e f l e c t s t h e u n c e r t a i n ty i n r e s e r v o i r p r o p e r t i e s . T h em o s t ~ i d e l y u s e d m e t h o d f o r c o n d i t i o n i n g p e r m e a b i l i t y f i e l d s t o p r o d u c t i o n d a t a h a s b e e n t h e m e t h o do f s im u l a t e d a n n e a l i n g , i n wh i c h p r a c t i t i o n e r s a t t e m p t t o m i n i m i z e t h e d i f f e r e n c e b e t we e n t h e " t r u e "a n d s i m u l a t e d p r o d u c t i o n d a t a , a n d ' " t r u e " a n d s i m u l a t e d v a r i o g r a m s . U n f o r t u n a t e l y , t h e m e a n i n go f t h e r e s u lt i n g r e a l iz a t i o n i s n o t c l e a r a n d t h e m e t h o d c a n b e e x t r e m e l y s l o w . I n t h i s p a p e r , wep r e s e n ' a n a l t e r n a t i v e a p p r o a c h t o g e n e r a t i n g r e a l i z a t io n s t h a t a r e c o n d i t i o n a l t o p r e s s u r e d a t a ,f o c u s i n g o n t h e d i s t r ib u t i o n o f r e a l iz a t i o n s a n d o n t h e e f f i c ie n c y o f t h e m e t h o d . U n d e r c e r t a i nc o n d i t io n s t h a t c a n b e v e r i f ie d e a s il y , t h e M a r k o v c h a i n M o n t e Ca r l o m e t h o d i s k n o wn t o p r o d u c es t a t e s ~ v ho s e r e q u e n c i e s o f a p p e a r a n c e c o r r e s p o n d t o a g i v en p r o b a b i l i t y d i s t r i b u t i o n , s o we u s et h i s m e t h o d t o g e n e r a t e t h e r e a l i z a ti o n s . T o m a k e t h e m e t h o d m o r e e f f i ci e n t , we p e r t u r b t h e s t a t e si n s u c k a wa y t h a t t h e v a r i o g r a m i s s a t i s fi e d a u t o m a t i c a l l y a n d t h e p r e s s u r e d a t a a r e a p p r o x i m a t e l ym a t c h e d a t e v e r y s t e p . T h e s e p e r t u r b a t i o n s m a k e u s e o f s e n s i t iv i t y c o e ff i c i en t s c a l c u l a t e d f r o m t h er e s e r v o i r s i m u l a t o r .KEY WORDS: conditional simulation, Markov chain, Monte Carlo, sampling, pressure data,sensitivity, well t e s t .

I N T R O D U C T I O NBecause the process of predic t ing f low and t ransport in petroleum reservoirs isnonlinear , i t is general ly imposs ible to calcula te direct ly the probabil i ty dis t r i -bu t ion fo r fu tu re re s e rvo i r pe r fo rmance . Ins tead , we a re fo rced to e s t ima te theReceived 11 September 1995; revised 26 March 1996.

2Chevron Petrol. Tech. Co., P.O. Box 446, La Habra, California 90633-0446; e-mail:[email protected] University of Tulsa, Department of Petroleum Engineering, Tulsa, Oklahoma 74104.

4present address: Petr61eo Brasileiro Research Center, Ilha de Fundao, Q7, Rio de Janiero, Brasil21949--900; e-mail: lubo@cenpes, petrobras.gov.br

5The University of Tulsa, Department of Petroleum Engineering, Tulsa, Oklahoma 74104; e - m a i l :[email protected]

6 1

0882-8121/97/0100-0061512.50/I 9 1997 InternationalAssociationor Mathematical Geology


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62 Oliver, Cunha, and Reynolds

p r o b a b i l it y d i st r ib u t i on f r o m t h e o u t c o m e s o f f l o w p r e d i c t io n s f o r a l a r g e n u m b e ro f r e a l i z a tio n s o f th e r e s e rv o i r . F o r th i s me th o d to w o rk , i t is e s s e n t i a l t h a t th ep e rme a b i l i ty a n d p o ro s i ty r e a l i z a t io n s u s e d in th e f lo w s imu la t io n a d e q u a te lyre f l ec t t h e u n c e r t a in ty in r e s e rv o i r p ro p e r t i e s . T h u s , t h e r e a l i z a t io n s mu s t b ed r a w n f r o m t h e p r o b a b i l i t y d i st r i bu t i on f o r t h e r e s e r v o i r m o d e l s . I f w e g e n e r a t ea la rg e n u m b e r o f r e a l i z a t io n s o f th e r e s e rv o i r p e rme a b i l i ty f i e ld , w e s h o u lde x p e c t to g e n e r a t e a f e w m o d e l s f r o m r e g i o n s o f t h e m o d e l s p a c e t h a t h a v e l o wp r o b a b i l it y d e n si t y . M e t h o d s t h a t d o n o t g e n e ra t e m o d e l s f r o m t h e " t a i l s " o fth e d i s tr ib u t io n a re n o t s a m p l in g c o r re c t ly th e d i s t r ib u t io n o f p o s s ib le r e s e rv o i r s .

T h e re a re ma n y s to c h a s t i c me th o d s th a t wo rk we l l fo r mu l t in o rma l d i s t r i -b u t io n s , o r fo r in d e p e n d e n t s to c h a s t i c v a r i a b le s , b u t s to c h a s t i c s imu la t io n o fg e n e ra l mu l t iv a r i a t e d i s t r ib u t io n s c a n b e d i f f i c u l t , e s p e c ia l ly wh e n th e n u mb e ro f v a r i a b le s i s l a rg e . T h e ty p e o f p ro b le m th a t we e n v i s io n i s o n e in wh ic h th ep e r m e a b i l i t y i n a l a r g e n u m b e r o f s i m u l a t o r g r i d c e l l s m u s t b e d e t e r m i n e d f o rre s e rv o i r flo w a n d t r a n s p o r t p re d ic t io n . W e a s s u me th a t th e re a re a t l e a st s o mep r io r e x p e c ta t io n s r e g a rd in g th e d i s t r ib u t io n o f p e rm e a b i l i ty a n d th a t th e re a rea l s o s o me p ro d u c t io n d a ta a v a i l a b le . F o r o u r e x a mp le s , i t w i l l b e th e c o n d i -t io n in g to p ro d u c t io n d a ta th a t ma k e s th e g e n e ra t io n o f r e a l i z a t io n s d i f f i c u l t .

T h e p a p e r i s o r g a n i z e d a s f o l l o w s . W e b e g i n b y p r e s e n t i n g b a c k g r o u n dm a t e r ia l o n th e u s e o f M a r k o v c h a i n M o n t e C a r l o m e t h o d s f o r s a m p l i n g . T h i si s fo l lo we d b y a c r i t i q u e o f th e c u r re n t p ra c t i c e o f u s in g s imu la te d a n n e a l in g tomin imiz e o b je c t iv e fu n c t io n s a s we l l a s a d i s c u s s io n o f th e c o n d i t io n a l p ro b -a b i l i ty d i s t r ib u t io n fo r Ga u s s ia n ra n d o m f i e ld s . We th e n a p p ly th e s e id e a s to th es i m u l a ti o n o f G a u s s ia n r a n d o m f i el d s w i t h k n o w n m e a n a n d v a r i o g r a m . T h em e t h o d t h a t w e u s e h a s c l o s e c o n n e c t io n s t o t h e m e t h o d o f m o v i n g a v e r a g e s s owe d i s c u s s th a t r e l a t io n , a n d th e s h a p e o f th e s mo o th in g fu n c t io n s fo r o n e a n dtwo d ime n s io n s . A b r i e f d e s c r ip t io n o f th e a p p l i c a t io n to c o n d i t io n a l s im u la t io nwi th p o in t d a ta l e a d s u s to th e ma in to p ic wh ic h i s th e p ro b le m o f c o n d i t io n in gGa u s s ia n ra n d o m f i e ld s to n o n l in e a r d a ta s u c h a s t r a n s ie n t p re s s u re me a s u re -m e n t s .

G e n e r a l b a c k g r o u n d o n M a r k o v c h a i n M o n t e C a r l o ( M C M C ) m e t h o d s f o rc o n d i ti o n a l s i m u l a ti o n c a n b e o b t a i n e d f r o m H a m m e r s l e y a n d H a n d s c o m b( 1 9 6 4) , H a s t i n g s ( 1 9 7 0 ), G e m a n a n d G e m a n ( 1 9 8 4 ) , a n d N e a l ( 1 99 3 ) . U s e f u lr e f e r en c e s o n t h e a p p l i c at i o n o f M a r k o v c h a i n M o n t e C a r l o m e t h o d s t o t h e e ar t hs c ie n c e s in c lu d e p u b l i c a t io n s b y T je lm e la n d , O mr e , a n d He g s ta d (1 9 9 4 ) a n dHe g s ta d a n d o th e r s (1 9 9 3 ) .

M A R K O V C H A I N S F O R S A M P L I N GIn th e c o n te x t o f r e s e rv o i r c h a ra c te r i z a t io n , we c a n th in k o f a s e q u e n c e o f

p o s s ib le s t a t e s o r realizations o f th e r e s e rv o i r p e rm e a b i l i ty d i s t r ib u t io n a s aM a rk o v c h a in i f t h e p ro b a b i l i ty o f g e n e ra t in g s o m e p a r t i c u la r n e w re a l i z a t io n


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M a r k o v C h a i n M o n t e C a r l o M e t h o d s 6 3

d e p e n d s o n l y o n t h e p r e c e d i n g r e a l iz a t i o n i n th e s e q u e n c e . W e w i ll d e n o t e ap a r t ic u l a r s t o c h a s ti c r e a l i z a ti o n o r m a p o f r e s e r v o i r p r o p e r t ie s b y t h e s y m b o l ,m s , w h e r e t h e s u p e r s c r i p t i r e f e r s to t h e i t h p o s s i b l e r e a l i z a t i o n o f a d e n u m e r a b l es e t o f p o s s i b l e r e a l i z a t i o n s , n o t t o t h e i th r e a l i z a t i o n i n t h e M a r k o v c h a i n . E a c ho n e o f t h e s e p o s s i b l e r e a l i z a t i o n s , m i, w i l l h a v e a p r o b a b i l i t y , 7 rs , a s s o c i a t e dw i t h i t . I n o u r c o n t e x t 71 w i ll b e t h e p r o b a b i l i t y o f t h e i t h p o s s i b l e r e a l i z a t i o nb e i n g t h e c o rr e c t m a p o f re s e r v o i r p ro p e r t i e s . O f c o u r s e e a c h o f th e s e p r o b -a b i li ti e s i s e x t re m e l y s m a l l b u t s o m e r e a l i z a ti o n s w i lt b e f a r m o r e l i k el y t h a no t h e rs . I f w e w e r e t o g e n e r a t e m a n y r e a l iz a t i o n s f o r p r e d i c ti o n o f r e s e r v o i rp e r f o r m a n c e , w e w o u l d w a n t t he r e a l iz a t i o n s t o b e s a m p l e d w i t h th e c o r r e c tp r o b a b i l it y . U n f o r t u n a t e l y , i n m o s t r e a l a p p l ic a t i o n s w e c a n n o t c h a r a c t e r i z ee a s i l y t h e p r o b a b i l i t y o f th e i th r e a l i z a t io n s o w e m u s t i n s t e a d u s e a m e t h o d t h a tr e l i e s o n l y o n t h e r e l a t i v e p r o b a b i l i t i e s t o g e n e r a t e r e a l i z a t i o n s . T h i s c a n b e d o n eu s i n g M a r k o v c h a i n s i f w e a r e c a r e f u l i n t h e s p e c i f i c a t i o n o f t h e c o n d i t i o n a lp r o b a b i l i t y , P jk , o f t r a n s i t i o n t o r e a l i z a t i o n m k f r o m r e a l i z a t i o n m j .

I f w e c a n d e t e r m i n e t r a n s i t io n p r o b a b i l i t i e s P i9 s u c h t h a t i t i s p o s s i b l e t o g e tf r o m a n y o n e s t a t e t o a n o t h e r i n a f in i t e n u m b e r o f t r a n s i t i o n s a n d s u c h t h a t t h ep r o b a b i l i t y o f s t a te m j i s t h e s u m o f t h e p r o b a b i l i t i e s o f b e i n g i n a s t a t e m i t i m e st h e p r o b a b i l i t y o f t r a n s i t i o n f r o m m s t o m j , t h a t i s

~r = ~ ~rsPs9 (1 )st h e n t h e M a r k o v c h a i n w i l l b e s ta t i o n a ry a n d e r g o d i c ( i n d e p e n d e n t o f i n it ia lc o n d i t i o n s ) a n d ~ / w i l l b e th e p r o b a b i l i t y d i s t r i b u t i o n f o r t h e r e a l i z a t i o n s ( F e l l e r ,1 9 6 8 ; H a m m e r s l e y a n d H a n d s c o m b , 1 9 6 4 ) . In p r a c ti c e , i t m a y t a k e a l a r g en u m b e r o f t r a n s i t i o n s t o r e a c h th e s t a t i o n a r y d i s t r i b u t i o n .

G e n e r a t i o n o f a M a r k o v c h a i n w i t h t h e d e s i r e d d i s t r i b u t i o n w i l l r e q u i r ec a l c u l a t i o n o f t r a n s i t io n p r o b a b i l i t i e s P i9 t h a t s a t i s f y E q u a t i o n ( 1 ) . W e w i l l d i s -c u s s l a t e r th e c o m p o n e n t s o f ~ rj a n d w h y t h e y a r e s o d i f fi c u l t t o c a l c u l a t e . F o rn o w , w e w i l l s t a te s i m p l y t h a t a l t h o u g h i t m a y b e i m p o s s i b l e t o c a l c u l a t e 7 rj , i tm a y b e r e l a t i v e l y e a s y t o c a l c u l a t e t h e r a t i o , 7 rj /~ r,, o f th e p r o b a b i l i t y o f b e i n gi n s t a t e m j t o t h e p r o b a b i l i t y o f b e i n g i n s t a t e m i.

M e t r o p o l i s a n d o t h e r s ( 1 9 5 3 ) o b s e r v e d t h a t t h e p r o b l e m o f c a lc u l a t i n g ap e r m i s s i b l e t r a n s i t io n m a t r i x c o u l d b e s i m p l i f i e d i f t h e t r a n s i t io n m a t r i x s a t i s fi e da r e v e r s i b i l i t y c o n d i t i o n

~ r i P i j = 7 r j P j i (2 )T h e y a l s o p r o p o s e d s p l i t t i n g t h e m a t r i x i n t o t w o c o m p o n e n t s :

P i j = ~ (3 )w h e r e q i j i s t h e p r o b a b i l i t y o f p r o p o s i n g a t r a n s i t i o n f r o m s t a t e rn t o s t a t e m ja n d c~ i s th e p r o b a b i l i t y o f a c c e p t i n g t h e p r o p o s e d t r a n s i t i o n . T h e q o c a n b e


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64 Oliver, Cunha, and Reynoldsse lec ted somewhat a rbi t ra r i ly , subjec t to the res t r ic t ion tha t , because they arep robab i l i t i e s , t hey t ake va lue s be tween ze ro and one . I t s hou ld be c l ea r , how-eve r , t ha t s ome s e lec tions fo r q o wil l resul t in h igh probabi l i t ies of acceptanceof t rans i t ion and thus su bs tant ia l ly reduce the com puta t io n t ime. Th e c~ij a re notde te rmined comple te ly by Equa t ions (2 ) and (3 ) . Has t ings (1970) , t he re fo re ,proposed us ing

[ 7 1 " i q i j - IceU = sij 1 + (4)7 r j q j iwh ere s 0 is a sy m m etr ic ma trix sele cted to e nsu re th at 0 __< o#j _< I. Fro mEquat ion (4) and the cons tra ints on o#i we obta in

r j q o0 < s r < 1 + - -7rjqj ;Because [s~j] is symm etric , then fro m E quat ion (5) we a lso obta in

( 5 )

7 r j q j iO < s i j ~-~ 1 + - -~r q i9Based on these cons tra ints , Has t ings ' (1970) choice for si j is

s O = m i n I 1 + r i q i J , 1 + 7 F J q J irrj qji 7ri qij .)which g ive s the fo l low ing a c cep tance p robab i li ty fo r s t a t e j w i th re s pec t to s t a tei:

I 7 r j q j i (6 )cx~j = m in 1, z c ~q iJ )Note tha t i f t he p ropos ed t rans it ions a r e s ymm et r i c , t ha t i s i f qu = q j i , t hen

the ca lcu la t ion o f w he the r to a ccep t a t r ans i t ion i s ba s ed on ly on the ra t io o f thep robab i l ity o f be ing in the two s t at e s . I f t he p ropos ed t rans i tion i s r e j e c ted , t heo ld s t a te i s r e pea te d in the cha in . S wapp ing pe rm eab i l i ty va lue s in two random lys e le cted g r id b loc ks i s an exa m ple o f a s ymm et r ic t r ans i t ion tha t ha s been us edin re s ervo i r c ha rac te r i z a tion (Fa rmer , 1992 ; D eu t s ch and J ou rne l , 1992 ; Saga r ,Kelk ar , and Th om pson , 1993). W e wil l res t r ic t ourse lv es to the c~ij g iven byEqua t ion (6 ) be c a us e th i s s e l e c tion i s know n to be re l a t ive ly e f f i c ien t a t s ampl ingthe dis t r ibut ion compared to other poss ible se lec t ions (Ripley, 1987, p . 114) .

The focus o f ou r e f fo r ts w i l l be on improv ing the e f f i c i ency o f the ove ra l lme thod by inc rea s ing the accep tance p robab i l i ty fo r p ropos ed t rans i t ions q oB eca us e 7 f is a f ixed , bu t unkn ow n , func t ion o f the pe rm eab i l i ty d i s tr ibu t ion ,


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Mar k ov C h ai n Mon l e C ar l a Me t h od s 65

we can o n ly imp ro v e t h e e f f i c i en cy b y imp ro v in g t h e p ro cess fo r p ro p o s in gtransit ions.

" O B J E C T I V E F U N C T I O N S "M o s t p u b l i sh ed a t t emp t s t o co n d i t i o n p e rmeab i l i t y f i el d s t o p ro d u c t i o n d a t a

r e ly o n e i t h e r s imu la t ed an n ea l in g o r t h e g en e t i c a l g o r it h m to r ed u ce t h e e r ro rmi s f i t b e tween p ro d u c t i o n d a t a ca l cu l a t ed f ro m a t h eo re t i ca l mo d e l o f t h e r es -ervo i r and the actual p rodu ct ion data (Sen and o thers , 1992; Sa gar , K elkar , andThom pson , 1993; Deu tsch , 1992; Ou enes and o thers , 1993) . S tud ies that areb ased o n s imu la t i o n o f Gau ss i an p e rmeab i l i t y f i e l d s t y p i ca l l y a l so a t t emp t t omin imize t h e mi s f i t b e tween t h e ex p er imen ta l v a r i o g ram an d t h e mo d e l v a r i o -g ram (Deu t sch an d Jo u mel , 1 9 9 2 ) . Th e mo s t wid e ly u sed measu re o f mi s f i t i st h e su m o f t h e sq u ared d i f f e ren ces . T o f i t t h is i n to t h e o p t imiza t i o n f r am ewo rko n "o b j ec t i v e fu n c t i o n " i s fo rmed f ro m th e we ig h t ed su m o f t h e mi s f i t t e rms .Var /o u s au th or s h av e p ro p o sed d i f f e r en t min imiza t io n o b j ec t i v es ev en fo r p ro b -l ems t h a t a r e a s co n cep tu a l l y s imp le as t h e g e n era t i o n o f Gau ss i an r an d o m f i e ld s .P 6 rez (1 9 9 1 ) an d Deu t sch (1 9 9 2 ) p ro p o sed match in g t h e mo d e l v a r i o g ram,wh ereas Gu p ta (1 9 9 2 ) an d Ou en es (1 9 9 2 ) p ro p o sed match in g an ex p er imen ta lv a r i o g ram. R eg ard in g th e c lo sen ess o f ma t ch r eq u i red , De u t sch an d Jo u m el(1992 , p . 155) argue that the ob jec t ive funct ion should approach ze ro to ach ievea co n d i t i o n a l s imu la t i o n , a l t h o u g h o th e r s h av e su g g es t ed s t o p p in g wh en co n -v erg en ce i s u n accep t ab ly s l o w. Wi th f ew ex cep t i o n s (Heg s t ad an d o th e r s , 1 9 9 3;T j e lmelan d , Om re , an d H eg s t ad , 1 9 9 4) t h e c ri t e ri a u sed t o s t o p an n ea l i n g an dth e c r i te r i a u sed t o we ig h t t h e y a r i o u s mi s f i t te rms h av e b ee n em p i r i ca l l y d e t e r -min ed .

O N M A T C H I N G T H E V A R I O G R A MBecau se mo s t ad v o ca t es o f s imu la t ed an n ea l i n g fo r co n d i t i o n a l s imu la t i o n

p ro p o se t o min imize o b j ec t i v e fu n c t i o n s t h a t co n t a in measu res o f t h e mi s f i t o fthe var iograrn , i t i s reasonable to invest igate the degree of match requ i red inorder to generate a leg i t imate condi t ional s imulat ion .

To do th is we must d is t ingu ish , as Joumel and Hui jb reg ts (1978 , p . 192)d id , betwe en th ree d i f feren t var iogram s: the theoretical v ar io g ram, t h e experi-mental v ar io g ram, an d t h e local v ar io g ram. Th e ex p er imen ta l v a r i o g ram i s t h evar iogram that i s calcu lated f rom a l imi ted sampl ing of a random funct ion wi th ina l imi t ed d o main . Th e t h eo re t i ca l v a r i o g ram i s t h e v a r i o g ram th a t we wo u ldca l cu l a te b y av erag in g t h e v a r i o g rams ca l cu l a t ed fo r t h e en t i r e en semb le o fp o ss ibl e r e se rv o i rs o r f ro m ex h au s t i v e samp l in g o f a r e se rv o i r wh o se d im en s io n sa re man y t imes l a rg e r t h an t h e r an g e o f t h e co r r e l a ti o n .

In p rac t i ce , we can o n ly es t ima t e t h e t h eo re t i ca l v a r i o g ram, b ased o n a


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p r i o r i k n o w l e d g e o f w h a t i s r e a s o n a b le , o n c l o s e l y s p a c e d m e a s u r e m e n t s f r o man ou t c r op , on l i m i t ed s ampl e s ob t a i ned t h r ougho u t the f i e ld , o r on s om e com -b i na t i on o f a l l o f t he s e . T he i m por t an t po i n t i s t ha t t he va r i og r am mod e l u s edf o r k r ig i ng and s i mul a t i on i s an e s t i ma t e o f t he t heo r e t i ca l va r i og r am f o r theens embl e o f pos s i b l e r e s e r vo i r s o r , equ i va l en t l y , f o r a r e s e r vo i r o f i n fi n i te ex t en t .

T he l oca l va r i og r am, 3 'v (h ) , is t he va r i og r am t ha t one w ou l d ca l cu l a t e f r omcomp l e t e know l ed ge o f a s i ng l e r ea li za t ion o f t he r andom f i e ld w i t h i n a dom a i nv o f l i m i t ed ex t en t . T he e xpec t ed va l ue o f t he l oca l va r i og r am ( f o r lags t ha t c anbe ca l cu l a t ed ) is t he t heo r e t i ca l va r i og r am , bu t t he l oca l va r i og r am can f l uc t ua t es u b s ta n t ia l ly a r o u n d t h e e x p e c t e d v a l u e w h e n t h e d i m e n s i o n s o f t h e d o m a i n va r e on l y a f ew t i mes l a r ge r t han t he l ags o f i n t e r e s t f o r t he compar i s on o f l oca lva r i og r am. U nf o r t una t e l y , t he expec t ed f l uc t ua t i ons i n t he ca l cu l a t ed cova r i ancea r e s o m e w h a t d i ff ic u l t t o c o m p u t e b e c a u s e t h e y i n v o l v e t h e c a l c u la t io n o f m o -men t s o f o r de r 4 o f t he rando m va r i ab l e ( J ou r ne l and H u i j b r eg t s , 1978 , p . 193) .I t does s eem t ha t , f o r r andom va r i ab l e s w hos e s pa t i a l co r r e l a t i on is gove r nedby a l i nea r va r i og r am, t he r e l a t i ve f l uc t ua t i on va r i ance , t ha t i s , E { [ 3 ' ( h ) -" Y v ( h ) ] 2 } / [ ' y ( h ) ] 2 a p p r o a c h e s z e r o a s h / L f o r lags h t ha t a r e s ma l l comp ar ed t od i mens i ons L o f t he l oca l dom a i n ( J ou r ne l and H u i j b r eg t s , 1978 , p . 193) . W ew o u l d e x p e c t t h i s s a m e b e h a v i o r t o h o l d f o r o t h e r v a r i o g r a m m o d e l s , i n c l u d in gexponen t i a l and s phe r i ca l .

I n s t ead o f ca l cu l a t i ng t he f l uc t ua t i on va r i ance f o r la r ge l ags , w e ca n e s t i -ma t e t he va r i ab i l i t y o f t he d i s pe r s i on va r i ance C ( v , v ) o f a r a n d o m f u n c t i o n f o rt he doma i n v . T h e d i s pe r s i on va r i ance w i l l be a meas u r e o f the va r i ab i l i ty o ft he va r i ance o f po i n t va l ues w i t h i n a l im i t ed dom a i n . I n o t he r w or ds , i f w ec a l c u la t e th e v a r i a n c e o f s a m p l e v a l u e s a b o u t t h e s a m p l e m e a n f o r a d o m a i nw h o s e d i m e n s i o n s a r e l ar g e c o m p a r e d t o t h e r a n g e o f t h e v a r i o g r a m ( a s s u m i n gt ha t it ha s one ) , w e s h ou l d ex pec t t o ob t a i n t he s i ll va l ue f o r the va r i og r am . I f ,how eve r , ou r s ampl e doma i n i s s ma l l t hen a l l o f t he s ampl e s w i l l be co r r e l a t edh i gh l y and w e exp ec t to ca l cu l a t e a s ma l l s ampl e va r i ance . J ou r ne l andH ui j b r eg t s ( 1978 , p . 61 - 68) s how t ha t t he d i s pe r s i on va r i ance i n a doma i n v i sg i v e n b y

D z ( O / v ) = C ( O ) - C ( v , v ) ( 7 )w h e r e

- C ( v , v ) = - ~ d r ' d r C ( r - r ' )U t3

( 8 )U s i ng t h i s r e s u lt , w e ca l cu l a t e t ha t , f o r a one - d i m ens i ona l expo nen t i a l

v a r i o g ra m m o d e l , " r( h) = e x p ( - [ h i ~ L ) , t h e d i s p e r si o n v a r ia n c e r e a c h e s 8 0 % o fi ts m a x i m u m v a l u e w h e n t h e d o m a i n l e n g th i s 1 0 L a n d 9 0 % w h e n t h e d o m a i nl eng t h is 20L . B ec aus e t he p r ac t i ca l r ange o f t he expon en t i a l mode l i s 3L w es e e t h a t o n l y w h e n t h e d i m e n s i o n s o f t h e d o m a i n a r e o n t h e o r d e r o f I 0 t i m e s


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Markov Chain Monte Carlo Me t h o d s 6 7

t h e r an g e o f t h e v a r i o g ram can w e ex p ec t t ha t t h e v a r i an ce o f p o in t v a lu es wi th inthe dom ain wi l l be close to the var iance g iven b y the s i ll o f the var iogram .Un less o u r d o main i s l a rg e co m p ared t o t h e r an g e o f th e v a r i o g ram we sh o u ldnot expect to generate real izat ions whose var iance matches the s i l l o f the var-i o g ram, an d we sh o u ld n o t a t temp t t o ach i ev e co n d i t io n a l s imu la t i o n s b y match -in g t h e v a r i o g ram ex ac t l y .

Thus, a l though i t might be feasib le to es t imate the p robabi l i ty d is t r ibu t ionfo r t h e v a r i a ti o n o f t h e ex p er imen ta l v a r i o g ram ab o u t t h e t h eo re t i ca l v a r i o g ram,th e t e rms i n t h e o b j ec t i v e fu n c t i o n fo r a s imu la t ed an n ea l i n g ap p ro ach wo u ldh av e t o b e w e ig h t ed su ch t h at t h e r ea l i za t i o n s o f th e v a r i o g ram a re d rawn f ro mth is d is t r ibu t ion at the f inal temperature. Th is i s no t a new idea. In a d iscussiono f e rg o d i c i ty , Deu t sch an d Jo u mel (1 9 9 2 , p . 1 2 6) s t a te t h a t "ex ac t r ep ro d u c t i o nof the model s ta t i s tics by each s imulated real izat ion m ay no t be desi rab le , o rp o ss ib l e . " As u su a l l y p rac t i ced , h o wev er , t h e "o b j ec t i v e o f ( s imu la t i o n b ysimulated annea l ing) i s fo r the sem ivar iogram -~*(h) o f the s imu lated real izat ionto match th e p resp ec i fi ed semiv ar io g ram mo d e l 7 * (h ) " (Deu t sch an d Jo u rn e l ,1992, p. 154).

T H E C O N D I T IO N A L P R O B A B I L I T Y D I S T R IB U T I O NAlthough i t i s d i f f icu l t to es t imate the p robabi l i ty d is t r ibu t ion for the locat

var iogram when the theoret ical var iogram is known, i t i s s t raigh t forward toca l cu l a t e t h e p ro b ab i l i t y d i s t r i b u t i o n fo r t h e r an d o m v ar i ab l es u n d er t h e sameassum pt ion . Speci f ical ly , i f the theoret ical cova r iance , CM, and the m ean , m 0are b o th k n o wn a p r i o r i fo r a Gaussian random f ield then the p robabi l i ty d is-t r ibu t ion for the random var iab les i s mul t ivar iate normal :

J [ m - m o ] r C ~ t t [ m mo]} (9)I ~ e x p { - iTh e mean o f t h e r an d o m v ar i ab l e in Eq u a t i o n (9) can b e a fu n c t i o n o f p o s i t i o n ,as can the covar iance. Because th is i s a s imple d is t r ibu t ion , i t i s possib le to usean y n u m b er o f s t an d ard g eo s t a t i st i cal m e th o d s o f s imu la t i o n i n c lu d in g seq u en t i a lGaussian s imulat ion and the L U d eco mp o s i t i o n meth o d (S ch eu er an d S to l l e r,1 96 2 ). We wi l l d i scuss t h e ap p l i ca ti o n o f M ark o v ch a in M o n te C ar lo me th o d sto the sampl ing of th is d is t r ibu t ion in the fo l lowing sect ion .

Reca l l t h a t th e g o a l o f s imu la t i on i s n o t t o d e t e rmin e t h e m ax im u m o f t h eprobabi l i ty d is t r ibu t ion bu t to sample i t . Al though s imulated anneal ing i s anM C M C meth o d , wh en i t is u sed wi th o u t ca re fu l co n s id e ra ti o n o f t h e s to p p in g" t emp era tu re , " t h e r esu l t s w i l l n o t b e r an d o m rea l i za t i o n s f ro m th e co r r ec tp ro b ab i l it y d en s it y fu n c t i o n . I f , f o r ex amp le , s imu la t ed an n ea l i n g w ere u sed t od e t e rmin e t h e max imu m o f Eq u a t i o n (9 ) o r , eq u iv a l en t l y , t h e min imu m o f [m- m o ] r C M 1 [ m - - mo], one would ob tain m : m0 indep ende nt ly o f the var io -


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68 Oliver , Cunha, and Reynolds

g r a m. S o , i f we a r e n o t c a r e f u l , we o b ta in th e k r ig in g s o lu t io n in s te a d o f as to c h a s t i c s imu la t io n . I n c id e n ta l ly , s imu la te d a n n e a l in g h a s b e e n u s e d o c c a -s io n a l ly to s a m p le th e mo d e l s p a c e in su c h a wa y th a t th e d i s t r ib u t io n o f s a mp le sa p p r o x i m a t e s t h e a p o s t e r i o r i d i s t r ib u t io n in th e mo d e l s p a c e ( Ko r e n a n d o th e r s ,1 9 9 1 ; He g s ta d a n d o th e r s , 1 9 9 3 ) .

Co n d i t io n in g th e p e r me a b i l i ty f i e ld to p r o d u c t io n d a ta r e q u i r e s th a t we b ea b le to c h a r a c te r i z e th e lik e l ih o o d o f o b ta in in g th e me a s u r e d d a ta f r o m a s im-u l a t o r w h e n t h e t r u e v a l u e s o f th e r a n d o m p e r m e a b i l i t y f i el d a re k n o w n . T w os o u r c e s o f e r ro r e n te r th e p r o b le m : m e a s u r e m e n t o r o b s e r v a t io n e r r o r s r e s u lt int h e i n a c c u ra c y o f t h e m e a s u r i n g i n s t r um e n t o r s l o p p y t e c h n i q u e, a n d m o d e l i n ge r r o r s , wh ic h a r e th e r e s u l t o f o v e r s imp l i f i c a t io n o r in a c c u r a c y o f th e f lo w s im-la to r . I t g e n e r a l ly i s d i f f ic u l t to c h a r a c te r i z e th e d i s t r ib u t io n o f e r r o r s a c c u r a te ly ,b u t i f we a s s u me th a t th e e r r o r s o f b o th ty p e s a r e d i s t r ib u te d n o r m a l ly , th e n w ec a n c o mb in e th e m in to o n e d a ta e r r o r t e r m a n d wr i t e th e c o n d i t io n a l p r o b a b i l i tyd i s t r ib u t io n f o r th e p e r m e a b i l i ty f i e ld a s

H oc exp { - 8 9 [dobs - - g ( m ) ] r C D ' [ d o b s - - g ( m ) ]l [ m m o ] T C M I [ m m0]} (10)

w h e r e C o i s th e s u m o f th e d a ta e r r o r c o v a r ia n c e ma t r ix a n d th e mo d e l in g e r r o rc o v a r ia n c e ma t r ix ( T a r a n to ta a n d Va le t t e , 1 9 8 2 ) . T h r o u g h o u t , g ( m ) is the pre -d ic te d v a lu e o f th e p r o d u c t io n d a ta th a t o n e o b ta in s f r o m u s in g th e p e r me a b i l i tyf i e ld m in th e f lo w s imu la to r . A l th o u g h we h a v e ma d e th e a s s u mp t io n th a t th ee r r o r s in th e d a ta a r e d i s t r ib u te d n o r ma l ly , th e p r o b a b i l i ty d i s t r ib u t io n g iv e n b yE q u a t io n ( 1 0 ) wo u ld b e mu l t iv a r i a t e n o r ma l o n ly i f g ( . ) we r e a l in e a r f u n c tio no f th e r a n d o m v a r ia b le s m . T h e f u n c t io n g ( . ) i s n o t l in e a r f o r tr a n s ie n t p r e s s u r ed a ta a n d e v a lu a t io n o f E q u a t io n ( 1 0 ) r e q u i re s a f lo w s imu la t io n f o r e a c h s e le c t io no f th e r a n d o m v a r ia b le s . Be c a u s e , in th e s imp le s t s i tu a t io n , th e r e w i l l b e th o u -s a n d s to te n s o f th o u s a n d s o f g r id c e l l s , e a c h w i th a v a lu e o f p e r me a b i l i ty a s -s ig n e d to i t , a n d b e c a u s e a c o mp le te e v a lu a t io n o f th e p r o b a b i l i ty d i s t r ib u t io nw o u l d r e q ui r e a f l o w s i m u l a t i o n f o r e v e r y p o s s i b l e c o m b i n a t i o n o f p e r m e a b i l i -t i e s , c l e a r ly i t i s imp o s s ib le to s a mp le e x h a u s t iv e ly th e p a r a me te r s p a c e .

C O N D I T I O N I N G T O A V A R I O G R A M A N D A M E A N V A L U EAl th o u g h th e r e a r e ma n y s t a n d a r d me th o d s f o r g e n e r a t in g r e a l i z a t io n s th a t

a r e c o n d i t io n e d to a me a n a n d a v a r io g r a m ( o r c o v a r ia n c e ) , w e w i l l d e s c r ib e aM a r k o v c h a i n M o n t e C a r l o m e t h o d o f s a m p l i n g b e c a u s e i t w i l l p r o v i d e m u c ho f th e b a c k g r o u n d f o r t h e m o r e c o m p l i c a t e d e x a m p l e s t h a t f o l l o w . C o n s i d e r as t a t e i , wh ic h i s a n a r my o f lo g - p e r m e a b i l i ty v a lu e s , o n e f o r e a c h o f th e Mg r id b lo c k s o f a f lo w s im u la to r , a n d l e t


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M a r k o v C h a i n M o n t e C a r lo M e t h o d s 69

m i ___ ( m l , m 2 , m 3 , . . . , m M ) TI t i s w e l l k n o w n th a t t h e d i s t r ib u t io n o f s t a t e s i s m u l t iv a r i a t e n o rm a l , t h a t i s , m- N M ( m o , C M ) , i f an d o n ly i f

m i = m o + L Z i ( I 1 )w h e r e L i s a " m a t r i x s q u a r e r o o t " o f C M s a t i s fy in g

C M = L L r (12)a n d th e c o m p o n e n t s o f Z i a r e in d e p e n d e n t n o rm a l d e v ia t e s w i th m e a n v a lu e z e roa n d v a r i a n c e o n e (R a o , 1 9 6 5 , p . 4 4 0 ) . T h i s i s t h e b a s i s o f th e w h a t m a y b ete rm e d th e L U o r C h o l e s k y d e c o m p o s i t i o n m e t h o d o f c o n d i t i o n a l s i m u l a t i o n(Rip ley , 1981 ; Davis , 1987b; Alaber t , 1987) .B e c a u s e e v e ry a w a y Z i o f i n d e p e n d e n t n o rm a l d e v ia t e s i s r e l a t e d u n iq u e lyto a n a r r a y m o f lo g -p e rm e a b i l i t y v a lu e s b y E q u a t io n (1 1 ) , w e c a n d e f in e th et r a n s it i o n s a lo n g th e Ma rk o v c h a in in t e rm s o f p e r tu rb a t io n s o f Z in s te a d o f m .A par t icu la r va lue o f Z i , o r eq u iva l en t ly , m ~, is re fe rred to a s th e i th s ta te o fth e m o d e l o r Ma rk o v c h a in a n d i s o b ta in e d b y d e f in in g a c o m p o n e n t o f th era n d o m v e c to r Z fo r e a c h n o d e o f t h e r e s e rv o i r m o d e l ,

Z i = ( Z l , Z 2 , Z 3 . . . . . Z M ) TT o i m p l e m e n t t h e M a r k o v c h a i n M o n t e C a r l o m e t h o d , w e n e e d t o c a l c u la t e t hep ro b a b i l i t y o f d ra w in g a n e w s t a t e a n d th e p ro b a b i l i t y o f a c c e p t in g a t r a n s i t i o nto th e n e w s t a t e o n c e i t h a s b e e n d ra w n .

I N D E P E N D E N T R E A L I Z A T I O N S O F T H E P E R M E A B I L I T Y F I E L DC o n s id e r t h e t r a n s it i o n to a s t a t e j w h ic h i s o b ta in e d f r o m s t a te i b y d ra w in g

n e w v a lu e s o f Zk a t e v e ry g r id n o d e . I f w e d e n o te th e n e w v a lu e s w i th p r im e sto d i f f ere n t ia t e f ro m th e o ld v a lu e s , t h e n th e p ro p o s e d v a lu e s o f t h e n o rm a ldev ia tes a re

Z s = ( Z ' , , Z : 2, Z ~ . . . . . Z ~ ) r (1 3 )B e c a u s e th e s e a re d ra w n in d e p e n d e n t ly f ro m th e n o rm a l d i s t r ib u t io n N (0 , 1 ) ,w e c a n w r i t e t h e p ro b a b i l i t y o f d ra w in g th e s t a t e j w h e n th e c u r re n t s t a t e i s i a s

q u = c e x p ( - 8 9 ] . Z j )= c e x p [ _ 1 ( m j _ m o ) r C ; t l ( m j _ mo)] (14)

w h ic h i s i n d e p e n d e n t o f t h e c u r re n t s t a te .T h e p ro b a b i l i t y o f a c c e p t in g th e p ro p o s e d t r a n s i t i o n i s g iv e n b y E q u a t io n


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70 Oliver , Cunh a, and Reynolds

( 6 ) w h i ch co n t a i ns 7 r/ , r i , and q j,- i n add i t i on t o t he t e r m w e have j u s t c a l cu l a t ed .T he p r obab i l i t y , q j~ , o f p r opo s i ng t he r eve r s e t r ans it i on t o s t a t e i f r om t he s t a tej is

I z iq j i = c e x p ( - ~ 9 Z i)= c exp [ - 89 m i - - m o ) T C M l ( m i - - mo)]

and t he p r obab i l i ti e s o f be i ng i n s t a t e i o r j a r e(15 )

r i = c exp [ - 89 m i - m o ) T C M I ( m i - - m0) ] ( 16 )7rj = c ex p [ - 8 9 ( m j - m o ) r C ~ l ( m j - m0)] (17)

Fr om E qua t i ons ( 14 ) t h r ough ( 17 ) , i t i s e a s y t o s how t ha tr j q j i - 17ri q~i

s o t ha t e v e r y pr opos ed t r ans i t i on qu o f t he t ype g i ven by E qua t i on ( 13 ) w i l l beaccep t ed w i t h p r obab i l i t y a 0 = 1 .

I t m a y s e e m s t r a n g e t o t e r m t h i s a M a r k o v c h a i n M o n t e C a r l o m e t h o db e c a u s e t h e M a r k o v c h a i n t h a t is g e n e r a t e d i s s im p l y a s e q u e n c e o f i n d e p e n d e n tr ea l i za t i ons gene r a t ed us i ng t he L U d e c o m p o s i t i o n m e t h o d . T h e a d v a n t a g e i st ha t , un l i ke many o t he r me t hod s , t he r e i s no w as t ed e f f o r t i n p r opos i ng a ndr e j ec t ing l a r ge numb er s o f pe r t u r ba t ions o r t rans i t ions t ha t do no t me e t t heaccep t anc e c r it e r i a . A l s o , t he f ac t t ha t e ach e l eme n t o f t he cha i n i s an i nde -pend en t r ea l i za ti on makes t he ch a i n a mo r e e f f ic i en t s amp l e r o f the p r oba b i l i t yd i s t ri bu t i on t han a cha i n o f co r r e l a t ed r ea l i za t ions .

C O R R E L A T E D R E A L I Z A T I O N SM o s t p r o p o n e n t s o f s i m u l a t e d a n n e a l i n g f o r r e s e r v o i r c h a r a c te r i z a ti o n p r o -

pos e t r ans i t i ons t ha t a r e l oca l pe r t u r ba t i ons t o t he pe r meab i l i t y f i e l d . T w o pa r -t i c u l a d y w i d e s p r e a d m e t h o d s o f p e r t u rb i n g t h e f ie l d a r e to " s w a p " t h e v a l u e sa t t w o l oca t i ons , o r t o d r aw a new va l ue a t one l oca t i on f r om t he cumul a t i ved i s t r ibu t i on f unc t i on . M any o f t he t rans i t ions p r op os ed us i ng t he s w app i ng ap -p r oach a r e r e j ec t ed becau s e t hey v i o l a t e t he va r i og r am co ns t r a i n t . We w i l l ou t -l ine a t rans i t ion that i s local , tha t i s , i t on ly a l ters the pe rme abi l i t i es in a l imi tedr eg i on , bu t t ha t a l s o s a t i s f i e s t he va r i og r am.

S u p p o s e , f o r e x a m p l e , t h a t w e r a n d o m l y s e l e c t ( f r o m a u n i f o r m d i s tr i bu -t i on ) the k t h node , and d r aw a new va l ue o f Z k s o t ha t t he a r r ay Z j co r r e s po nd i ngto s ta te j i s g iven b y

z ~ = ( z ~ , z : . . . . . z ~ . . . . . z ~ ) ~


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Markov C h a i n M o n t e C a r l o M e t h o d s 71

B e c a u s e t h e Z k a r e i n d e p e n d e n t a n d o n l y o n e v a l u e i s b e i n g a l t e r e d , t h e p r o b -ab i l i ty o f se lec t ing s t a t e j wh en th e cu r re n t s t a t e i s i i s1 ( 1 )q i j M ~ e x p - ~ Z/~ 2T h e p r o b a b i l i t y o f p r o p o s i n g t h e r e v e r s e t r a n s i t io n i s

1q~i - M ~ ex p

T h e p r o b a b i l i ty o f b e i n g i n s t a te i i s] 7% = c ex p [ - ~ ( m ' - m o ) r C ~ ' ( m i mo ) ]

= c e x p [ - 89 i " Z i ]

= c e x p [ - 8 9 +Z~ + ' ' ' +Z,: + ' ' ' +Z2M)]a n d t h e p r o b a b i l i t y o f b e i n g i n s t at e j i s n e a r l y t h e s a me :

7% = c exp [ - 8 9 m j - m o ) r C ~ t l ( m j - mo)]

= c e x p [ - 8 9 j 9 Z j ]

= c e x p [ - 8 9 + Z ~ + . . - + Z ~ 2 + ' ' - + Z2M )]A g a i n , w e d e t e r m i n e t h a t

7r qji - - I7 r i q i js o t h a t a l l p r o p o s e d t r a n s i t i o n s a r e a c c e p t e d .

T h e d i f f e r e n c e b e t w e e n t h i s a p p r o a c h a n d t h e a p p r o a c h p r o p o s e d i n t h ep r e v i o u s s e c t i o n i s i n t h e s i z e o f t h e r e g i o n t h a t i s mo d i f i e d d u r i n g a t r a n s i t i o nf r o m o n e s t a te t o t h e n e x t . I n t h e p r e v i o u s s e c t i o n , w e p r o p o s e d t o d r a w n e wv a l u e s o f a l l t h e n o r m a l d e v i a t e s a t e a c h t r a n s i ti o n , w h e r e a s i n t h i s s e c t i o n , wep r o p o s e d d r a w i n g a n e w v a l u e o f Z k a t o n l y o n e l o c a t io n p e r t r a n s it i on . T h i sm a y s o u n d s i m i l a r to t h e s t a n d a rd p r a c t i c e o f d r a w i n g a n e w v a l u e o f p e r m e -a b i l it y f r o m t h e c u mu l a t i v e d i s t r i b u t i o n f u n c t i o n b u t o u r mo d i f i c a t i o n o f t h ep e r me a b i l i t y f i e l d c o me s t h r o u g h t h e r e l a t i o n s h i p b e t we e n Z i a n d m i. T h e c h a n g ei n m r e s u l t i n g f r o m t h e c h a n g e i n Z i s g i v e n i n E q u a t i o n ( 1 1 ) ,

6 m i j = L 6 Z ~j (18 )


12/31

72 Oliver, Cunha, and Reynoldsw h e r e

tsziJ = Z y _ Z i= ( 0 , 0 . . . . . z ; , - z ~ . . . . . 0 ) r= ( 0 , 0 . . . . . ~ z k . . . . . 0 ) r

so t h e p e r t u rb a t io n t o t h e p e rme ab i l i t y f ie l d t h a t r e su lt s f ro m a ch an g e , 5 Zk, inth e v a lu e o f t h e k th n o rmal d e v i a t e i s g iv en b y t h e p ro d u c t o f 6Zk wi th t h e k thc o l u m n o f L .

Al th o u g h o u r n o t a t io n an d o u r f r eq u en t r e f e r en ce t o t h e L U d e c o m p o s i t i o nmig h t seem to imp ly t h a t L i s a l o wer t r i an g u l a r ma t r i x , we o n ly r eq u i r e t h a t Lsa t is fy Eq u a t i o n (1 2 ) . Becau se t h e co lu m n s o f L d e t e rm in e t h e sh ap e o f t h ep er tu rb a t io n o f t h e p e rmeab i l i t y , we d ig res s f ro m th e p ro b l em o f g en era t i n g t h eM a r k o v c h a i n s t o e x a m i n e t h e c o l u m n s o f L f o r v a r i o u s d e c o m p o s i t i o n s .

S H A P E O F T H E P E R T U R B A T I O N SO n e - D i m e n s io n a l D o m a i n

I f CM i s t h e M M m at r i x o f co v ar i an ces o f t h e M ran d o m v ar i ab l es o nth e g r i d n o d es , t h en t h e d eco mp o s i t i o n o f CM in o t h e p ro d u c t o f a l o wer t r i an -g u l a r ma t r i x , L , an d i t s t r an sp o se L r , p ro v id es a se t o f we ig h t s t h a t can b eap p l i ed t o t h e a r r ay o f n o rmal d ev i a t es fo r t h e s imu la t i o n o f co r r e l a t ed r an d o mv ar i ab l es o n t h e n o d es . S imi l a r l y , we can co n s t ru c t a sy mmet r i c sq u are - ro o tmat r i x R su ch t h a t CM = RR ( see Ap p en d ix ) . Th e co lu mn s o f R can b e u sedl i k e th e co lu m n s o f L i n Eq u a t i o n (1 8 ) t o g en era t e a p e r t u rb a t i o n o f t h e p e rm e-ab i l it y f ie l d t h a t is co n s i s t en t w i th th e v a r i o g ram . F ig u res 1 an d 2 , r e sp ec t i v e ly ,sh o w se l ec t ed co lu mn s o f L an d R fo r an ex p o n en t i a l co v ar i an ce w i th a p r ac ti ca lr an g e o f 9 .

T w o - D i m e n s i o n a l P e r t u r b a t i o n sTh e sh ap es o f th e p e r t u rb a t io n s i n two an d t h r ee d imen s io n s a l so a r e d e -

t e rmin ed b y t h e co lu mn s o f t h e sq u are ro o t o f t h e co v ar i an ce mat r i x . T o v i su a l i zet h e " s h a p e , " w e a r r a n g e t h e e l e m e n t s o f o n e c o l u m n o f L o r R i n t he t w o -d imen s io n a l p o s i t i o n s co r r esp o n d in g t o t h e a r r an g emen t o f p e rmeab i l i t y v a lu es .In F ig u re 3 , we h av e p lo t t ed o n e p a r t i cu l a r co lu mn ( co r r esp o n d in g t o t h e p e r -t u rb a t i o n o f t h e 6 6 th co mp o n en t o f Z i) f o r a n L U d eco mp o s i t i o n o f a s t a t i o n aryex p o n en t i a l co v ar i an ce , an d fo r t h e sy m met r i c sq u are - ro o t d eco mp o s i t i o n . Th e" s y m m e t r i c " d e c o m p o s i t io n R i s s y m m e t r i c o n l y a p pr o x i m a t e l y b e c a u s e o fso me n u m er i ca l i n accu racy i n t h e ca l cu l a t io n o f t h e e ig en v ec to r s o f t h e co v ar i -an ce , wh ereas L i s l o ca l i zed b u t n o t sy mmet r i c . O l iv e r (1 9 9 5 ) h as d esc r i b ed a


13/31

73

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I 0 2 0A A --~A A.30

Markov Ch a in Monte Carlo Methods

4o

Fig ure 1 . 5 th , 15 th , 25 th , and 35 th co lumns o f lowe r t r i angu lar mat r ix L fo r expone n t i a l cova r i ancefunct ion wi th p rac t i ca l range o f 9 .

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15/31

Markov Chain Mo nte Carlo Methods 75m e th o d fo r c a l c u la tin g th e s h a p e s o f th e s y m m e t r i c p e r tu rb a t io n s , fo r s e v e ra lv a r io g ra m m o d e l s , w h ic h d o e s n o t r e q u i re c a l c u la t io n o f t h e s q u a re ro o t o f t h em a t r ix .

C O N D I T I O N I N G T O T R A N S I E N T P R E S S U R E D A T ASuppose tha t , in add i t ion to a p r i o r i k n o w le d g e o f t h e th e o re t r i c a l v a r io -

g ra m a n d m e a n , w e h a v e a c c e s s to m e a s u re d p ro d u c t io n d a ta th a t c a n b e u s e dfo r c o n d i t io n in g th e s im u la t io n s . W e k n o w th a t t h e r e a l i z a t io n s m u s t b e s a m p le df ro m E q u a t io n (1 0 ) . We c a n u s e th e s a m e p e r tu rb a t io n s in o u r Ma rk o v c h a inp ro c e d u re th a t w e re u s e d fo r c o n d it io n in g to th e v a r io g r a m , o r w e c a n im p ro v ec o n v e rg e n c e fu r th e r b y in c o rp o ra t in g th e " l in e a r p a r t " o f t h e p ro d u c t io n d a ta .We w i l l s t a r t w i th th e s im p le r m e th o d a n d th e n e x p la in th e im p ro v e m e n t s th a tc a n b e m a d e b y u s in g th e s e n s i t i v i ty fu n c t io n s in th e p e r tu rb a t io n s .

L o c a l P e r t u r b a ti o n s B a s e d o n t h e V a r i o g r a mC o n s id e r t h e t r a n s i t i o n f ro m o n e s t a t e t o th e n e x t , c o n s i s t in g o f a c h a n g e

in o n ly o n e o f th e c o m p o n e n t s o f th e v e c to r o f n o rm a l d e v ia t e s . T h e i t h p o s s ib l es t a te o f t h e m o d e l i s s p e c i f ie d b y th e v a lu e s o f t h e a r r a y o f n o rm a l d e v ia t e s :

Z i = ( Z 1 , ~ . . . . . Z k . . . . . ZM) rand the j th s ta te by

Z j = ( Z ~ , Z 2 . . . . , Z 'k . . . . . Z M ) r

B e c a u s e w e d ra w th e v a lu e s o f Zk f ro m th e n o rm a l d i s t r ib u t io n , t h e p ro b a b i l i t yof p ropos ing a t rans i t ion to s ta te j f ro m s ta te i i sq U - M ~ e xp - Z~2

a n d th e p ro b a b i l i t y o f p ro p o s in g th e r e v e r s e t r a n s i t i o n i s o b v io u s .W e c a l c u la t e th e p ro b a b i l i t y o f b e in g in th e s t a t e i f ro m E q u a t io n (1 0 ) , t h a t

is= i A d i T C D I A d i _ i7r c exp { - -~ 5 [m i - - m o ] T C M I [ r ni - - mo]}

- - I [ L z i ] T ( L T ) - I L - I [ L Z i ] }c e x p { - 8 9 A d i T C ~ l A d i

_ I Z ic exp {--89 A d i r C ~ l A d i ~ 9 Z i }w h e re w e h a v e in t ro d u c e d th e n o ta t io n , A d i - d o b s - - g ( m i ) fo r t h e m is f i t o fth e p ro d u c t io n d a ta .


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7 6 O l i v e r , C u n h a , a n d R e y n o l d s

Com bin ing the s e four re su l t s, and cance l ing the com mo n te rms , w e ob ta in~rj qo~r~qo

- ' Z j Z j ) e x p ( - 8 9exp (- 8 9 A d J r C D t A d j ~ 9- - i z i ic e x p ( - 89 d i r c o I A d i ~ " Z i ) exp ( - ~ Z~.2)

exp ( - 89 A d J r C D ' A d J ) exp [ - 89 Z ~ + ' ' 9 + Z ~ 2 + 9 9 9 + Z ~ t ) - g Z k l 'e x p ( - ~ ' A d i r C 1 ) l A d i) exp I - 89 (Z2 + " ' " Z~ + " ' ' + Z~ ) - ~'-~k'7'2'Je x p ( - 89A d J r C ~ ' A d j )e x p ( - ~ AdirCD A d i ) (19)

T hus , w hen a s qua re roo t ma t r ix , R o r L , i s u s ed to gene ra te the pe r tu rba t ionsto the pe rmeab i l i ty f i eld , the accep tance c r i t e r ion , E qua t ion (6 ) , depends on lyon the mis f i ts to the p roduc t ion da ta . I f the p ropos ed pe rmeab i l i ty f i e ld gene ra te sp roduc t ion da ta tha t f i t t he obs e rved da ta more c los e ly than the p rev ious pe rme-abi l i ty f ie ld , then w e accep t the perturbat ion a nd u pdate the f ie ld . I f the f i t isworse , we accept the t rans i t ion with probabi l i ty given by Equat ion (19). I t i sunneces s a ry to ca lcu la te the m is f it t o the va r iogram, and con ve rgence i s acce l -era ted because a l l proposed t rans i t ions are cons is tent with the probabi l i ty dis -t r ibu tion fo r the va r iogram .

P e r t u r b a t i o n s t h a t U s e S e n s i t i v i t y F u n c t i o n sW e can inc reas e fu r the r the p robab i l i ty o f gene ra t ing accep tab le t rans i t ions

by s e lec t ing t rans i t ions tha t a re approx ima te ly cons i s t en t w i th the p roduc t ionda ta . T o s ee how th i s migh t be done , l e t u s examine the l ike l ihood pa r t o f thetota l probabi l i ty dens i ty funct ion:

IIolM(dob~[m ) o~ exp {-- 89 [dob~ -- g(m)]Tc~)'[dob,~ -- g(m)]}where , in gene ra l , g ( m ) i s a non l inea r func t iona l o f m.

Se lec t a re fe rence s ta t e , s ay mr , and e xpand g ( m ) i n a T ay lo r s e r ie s a roundm r .

g ( m ) = g ( m D + G r ( m - m r ) + e ( m )where

e ( m ) -= g ( m ) - g ( m r ) - G r ( m - m r )an d G r m is a l i n e a r func t iona l o f m. Als o , de f ine a re fe rence da ta vec to r , d ,s uch tha t

d = d o b s - - g ( m r ) + G r m r


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M a r k o v C h a i n M o n t e C a r l o M e t h o d s 77S O

9 g o b s - - g(m) = g o b s - - [g (mr) + G,m - - G r m r "J r- 6 ( m ) ]= [dob~ - - g ( m r ) + G r m r ] - G r i n - e ( m )= d - G r m - e ( m )

T h u s , t h e l i k e l ih o o d o f o b t a i n i n g t h e o b s e r v e d d a t a , g i v e n a p a r t i c u l a r s e t o fr a n d o m v a r i a bl e s , m , c a n b e f a c t o r e d i n t o th e p r o d u c t o f G a u s s i a n a n d n o n -Ga u s s i a n d i s t r i b u t i o n s :

I IDlM(dob.~lm ) o: exp {-- 89 [d - G r i n - - e ( m ) l r C o ' [ g - - G r m - e ( m) l }

I [ d - G r m l r c [ ~ l [ d - G r m ]e x p { - g+ [ d - G r m l r c ~ l e ( m ) - 8 9 ( m ) r c ~ l e ( m ) }

T h e s a m e f a c t o r i z a t i o n t h e n c a n b e u s e d i n th e p r o b a b i l i t y d i s t r i b u t i o n f o r th er a n d o m v a r i a b l e s .

l [ m m o ] r C T u l [ m m o ] - l [ d G r m ] r c l [ d - G r m ] }I M ( m or e x p { - ~ - - ~ -

x e x p { [ d - G r m l T C D ' e ( m ) - - 8 9 ( m ) r C [ ~ ' e ( m ) }

e x p { - 8 9 [ m - I x l T c ~ I J [ m - - / x ] }

e x p { [ d - G r m l r C [ ~ J e ( m ) - 8 9 ( m ) r C [ ~ l e ( m ) } (20 )wh e r e t h e r e f e r e n c e me a n , / ~ , i s c a l c u l a t e d f r o m

T T= m o + C M G r [ G r C m G r + C o ] - l [ d - Gr mo ] ( 2 1 )a n d t h e r e f e r e n c e c o v a r i a n c e ma t r i x f o r t h e mo d e l i s

C ~ t ! = G r r C o ' G r + C M ~ (22 )I f t h e mo d e l i s l i n e a r , t h e p r o b a b i l i t y d i s t r ib u t i o n i n E q u a t i o n ( 2 0 ) i s Ga u s s i a n ,e (m) = 0 , a n d / ~ a n d C M , a r e t h e me a n a n d t h e c o v a r i a n c e o f t h e c o n d i t i o n a lp r o b a b i l i t y d i s t r ib u t i o n .

A g a i n w e c o n s i d e r tw o m e t h o d s o f g e n e r a ti n g a M a r k o v c h a i n f o r sa m p l i n gt h e p r o b a b i l i t y d i s t r ib u t i o n f o r t h e r a n d o m v a r i a b l e s . F i r s t , w e w i l l p r e s e n t am e t h o d f o r g e n e r a ti n g a c h a i n o f i n d e p e n d e n t r e a l i z a ti o n s o f t h e p e r m e a b i I i tyf ie l d b y r e s a mp l i n g a l l o f t h e n o r ma l d e v i a t e s s i mu l t a n e o u s l y . T h e n , w e p r e s e n ta n a lt e r n a t i v e i n wh i c h o n l y o n e o f t h e n o r m a l d e v i a t e s i s mo d i f i e d i n a n yt rans i t ion .


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78 Oliver, Cunh a, and ReynoldsC h a i n o f I n d e p e n d e n t R e a l i z a t i o n s

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

Z i = ( Z f , Z 2 , Z 3 . . . . . ZM) rt h a t a r e r e l a t e d t o t h e r a n d o m p e r m e a b i l i t y f i e l d a s

m i = ix + 1.7.. w h e r e C M , = L L TW e w i l l p r o p o s e t r a n s i ti o n s t o a n e w s t a t e , j , f o r w h i c h e v e r y e l e m e n t o f t h ea r r a y o f n o r m a l d e v i a t e s h a s b e e n r e s a m p l e d :

Z j = ( Z ' l, Z ~ , Z ~ . . . . . ZrM) TB e c a u s e t h e c o m p o n e n t s o f Z i a r e i n d e p e n d e n t , t h e p r o b a b i l i t y o f p r o p o s i n g t h et r a n s i t io n f r o m s t a t e i to s t a t e j i s

q ij = ( 2 7r ) - M /2 e x p ( - 8 9 Z j 9 Z j )F r o m E q u a t i o n s ( 2 0 ) a n d ( 2 2 ) , t h is i s e q u i v a l e n t t o

q i j o r e x p { - 8 9 [ m - l z ] r C ~ ! [ m j - # ] }

o c e x p { - 8 9 [ m - m o ] r C ~ t l [ m j - m 0 ][ d - G r m J ] r c ~ l [ d - G r m J ] } ( 2 3 )

T h e p r o b a b i l i t y o f p r o p o s i n g t h e r e v e r s e t r a n s it i o n , f r o m s t a te j t o i , is d e t e r-m i n e d b y r e p l a c i n g m j b y m i i n E q u a t i o n ( 2 3 ) .

R e c a l l t h a t w e c a n c a l c u l a t e t h e p r o b a b i l i t y o f b e i n g i n s ta t e i fr o m E q u a t i o n( 2 o ) :

r i o c e x p { - 8 9 [ m - p . ] T c ~ t l [ m i - / , I , ] }X e x p { [ d - G r m i ] T C D l e ( m i ) - - 8 9 e ( m i ) r c ~ l e ( m i ) }

C o m b i n i n g t h e s e f o u r re s u l ts , a n d c a n c e l i n g t h e c o m m o n t e r m s , w e o b t a i nr j q j i e x p { [ d - G r m i ] r c ~ l e ( m j ) - 89e ( m J ) r c ~ ' e ( m J ) }"giqij e x p { [ d - G r m i ] T C D l ~ _ ( m i ) - - 89e ( m i ) r C o l e ( m i ) }

e x p { [ d o b s - - g ( m J ) ] r c o ~ e ( m j ) + 89 e ( m J ) r C o l e ( m J ) }= l e ( m i ) T C D l e ( m i ) } ( 2 4 )xp { [ dobs - g ( m i ) ] r c o l e ( m i) +

w h i c h w e u s e i n H a s t i n g ' s c r i t e r i o n , E q u a t i o n ( 6 ), t o d e c i d e w h e t h e r to a c c e p tt h e t ra n s i ti o n t o s t a t e j . N o t e t h a t i f e ( m ) = 0 f o r a l l m t h e n t h e r a t i o i n E q u a t i o n


19/31

M a r k o v C h a i n M o n t e C a r l o M e t h o d s 7 9

(24) is equa l to one , and a l l p rop osed t ran s i t ions a re accep ted . I f e (m) is sm al l ,w e e x p e c t h ig h a c c e p ta n c e r a t io s .

In c id e n ta l ly , t h i s i s a p p ro x im a te ly th e a p p ro a c h u s e d p re v io u s ly b y th ea u th o r s (O l iv e r, 1 9 9 4 , 1 9 9 6 ; C h u , R e y n o ld s , a n d O l iv e r , 1 9 9 5 ) to g e n e ra t ere a l i za t io n s th at w e re c o n d i t io n a l t o w e l l t e s t d a ta . In th o s e s i tu a t io n s , h o w e v e r ,th e p ro p o s e d t r a n s i t i o n s w e re n o t c h e c k e d a g a in s t t h e a c c e p ta n c e c r i t e r io n b e -c a u s e th e n o n l in e a r i ty w a s p re s u m e d to b e s m a l l . We w i l l e x a m in e th e c o n s e -q u e n c e s o f r ig o ro u s ly a p p ly in g th e Ma r k o v c h a in Mo n te C a r lo m e th o d to th en u m e r i ca l p r o b l e m w h e n w e c o n s i d er c o m p u t a t i o n a l e x a m p l e s .

L o c a l P e r t u r b a t i o n s

O n e p ro b le m w i th m o d i fy in g a l l o f t h e r a n d o m v a r i a b le s a t e a c h t r a n s i t i o ni s t h a t th e n u m b e r o f a c c e p te d t r a n s it i o n s c a n b e lo w i f t h e n o n -G a u s s i a n p a r to f t h e d i s tr ib u t io n i s l arg e a n d th e o b s e rv e d p ro d u c t io n d a ta a re a c c u ra t e (b e c a u s ei f th e d a t a a re a c c u ra te , t h e n o n ly p e r tu rb a t io n s th a t c lo s e ly m a tc h th e o b s e rv e dd a ta a re a c c e p te d ) . T h e p ro b a b i l i t y o f p ro p o s in g a l a rg e t r a n s i t i o n th a t r e s u l t sin a g o o d m a tc h to a c c u ra t e p ro d u c t io n d a ta c a n b e s m a l l . A n a l t e rn a t iv e i s t om a k e a l a rg e n u m b e r o f s m a l l t r a n s i t i o n s a s d i s c u s s e d p re v io u s ly . In th i s a p -proach , the t rans i t ion f rom s ta te i ,

Z i = ( Z l , 7 --2 . . . . Z k , . . . . Z M ) rto s ta te j ,

z ~ = ( z l , ~ . . . . . Z ~ . . . . . Z M ) ri n v o lv e s c h a n g in g o n ly o n e c o m p o n e n t o f t h e a r r a y o f n o rm a l d e v ia t e s . T h ep ro b a b i l i t y o f p ro p o s in g th i s t r a n s i t io n i s

qo - M ~ exp - Z~2

w h e re a s th e p ro b a b i l i t y o f p ro p o s in g th e r e v e r s e t r a n s i t i o n i sq ii - M - - - ~ e x p - Z ~

T h e p ro b a b i l i t i e s o f b e in g in s t a t e s i o r j w e re r e p o r t e d in th e l a s t s e c t io nb u t t h e r e s u l ts a re m o re c o n v e n ie n t w h e n e x p re s s e d in t e rm s o f th e n o rm a ldev ia tes :

7 r o c e x p { - 8 9 2 + ' ' ' + Z , + ' ' ' + Z ~ ) }

exp { [d - Gr mi]TC D l e(mi) - 1 e(m i)TCD i c(mi }


20/31

8 0 O l i v e r , C u n h a , a n d R e y n o l d s

an d7r/ ~ e xp { - 8 9 (Z~ + ' ' ' + Z],= + ' ' " + Z~t)}

e x p { [d - G r m J ] r c [ ~ 1 e ( m j ) - 8 9 ( m J ) r c ~ l e ( m ) ) }C o m b i n i n g t h e s e f o u r re s u lt s , a n d c a n c e li n g t h e c o m m o n t e r m s , w e o b t a i n

7 r j q j i7 r i q ij

e x p { [ d - G r m J l r C ~ ' e ( m j ) - 8 9 ( m J ) r C D l e ( m J ) }e x p { [ d - G ~ m i ] r C o l e ( m i ) - 8 9 ( m i ) r c f i l e ( m i ) }

exp { [dobs - - g ( m J ) ] T C D l e ( m j ) + 8 9 ( m J ) T C D l e ( m J ) }exp { [dobS - - g ( m i ) ] r C ~ l ~ (m i ) + 8 9 ( m i ) r c ~ Ic(mi)}

(25)

w h ic h i s t h e s a m e a s E q u a t io n (2 4 ) . T h u s , t h e a c c e p ta n c e c r i t e r i a i s t h e s a m ew h e th e r w e d o lo c a l p e r tu rb a t io n s o r g lo b a l p e r tu rb a t io n s . T h e a c c e p ta n c e p r o b -a b i l i t y , h o w e v e r , w i l l b e d i f f e re n t fo r t h e tw o d i f f e re n t t y p e s o f t r a n s it i o n b e c a u s ein th e f ir s t a p p ro a c h m j i s " c l o s e " to m i w h e re a s in th e s e c o n d a p p ro a c h th etw o s t a t e s c a n b e d i f f e re n t. T h e d i s a d v a n ta g e i s t h a t m a n y o f t h e p ro p o s e dt ra n s i t i o n s w i l l t u rn o u t t o h a v e n o e f f e c t o n th e m is f i t o f t h e p ro d u c t io n d a tab e c a u s e , fo r e x a m p le , t h e p e r tu rb e d c e l l s a re lo c a t e d o u t s id e th e r e g io n o f i n -f lu e n c e o f t h e d a ta . I f t h i s w e re th e s i t u a t io n , w e w o u ld o b ta in g ( m i ) ' = g ( m i )a n d e (m ) = e (m ) s o th a t th e a c c e p ta n c e p ro b a b i l i t y , b y E q u a t io n (2 5 ) , w o u ldb e e q u a l t o o n e . C o n c e p tu a l ly , w e c a n d e te rm in e w h e th e r t h i s i s a g o o d a p -p ro x im a t io n w i th o u t ru n n in g th e f lo w s im u la to r b y c a l c u la t in g th e p ro d u c t o fth e s e n s i t i v i ty c o e f f i c i e n ts w i th th e p ro p o s e d t r a n s i t i o n . I f th e p ro d u c t a r ( m j -m ;) i s a p p ro x im a te ly z e ro , t h e n th e p ro d u c t io n d a ta a re in s e n s i t i v e to th e p ro -p o s e d c h a n g e , a n d i t i s u n n e c e s s a ry to ru n th e f lo w s im u la to r .

C O M P U T A T I O N A L E X A M P L E SH e r e , w e u s e M a r k o v c h a i n M o n t e C a r l o m e t h o d s t o g e n e r a t e r e a l i z a t i o n s

o f th e lo g -p e rm e a b i l i t y f i e ld th a t a re c o n s i s t e n t w i th t r a n s i e n t p re s s u re d a t a a n dc o r re c t ly s a m p le th e a p o s t e r i o r i p ro b a b i l i t y d e n s i ty fu n c t io n . F o r th e f i r s t e x -a m p l e t h e p e r m e a b i l i t y i s l o g - n o r m a l l y d i s tr i b u te d w i t h l o g v a r i a n c e o f 0 . 2 5 a n dlo g m e a n e q u a l t o 3 . 4 . In th e s e c o n d e x a m p le th e m e a n i s 3 . 4 b u t t h e v a r i a n c ei s i n c re a s e d t o 1 . 0. W h e n w e s o l v e t h e se e x a m p l e p r o b l e m s , w e a s s u m e t h a tth e p e rm e a b i l i t y o f e a c h g r id b lo c k i s a s c a l a r a n d th a t t h e 2 -D s p a t i a l c o n t in u i tyc a n b e d e s c r ib e d b y a n i s o t ro p i c s p h e r ic a l v a r i o g r a m m o d e l w i t h a r a n g e o f 6 0 0f t . In b o th s y n th e t i c e x a m p le s , t h e r e s e rv o i r i s s q u a re , 1 5 0 0 x 1 5 0 0 ft , w i th n of l o w b o u n da r i e s a l o n g t h e e d g e s. R e s e r v o i r p e r f o r m a n c e w a s s i m u l a t e d u s i n g au n i fo rm s p a t i a l g r id w i th 1 0 0 1 0 0 f l g r id b lo c k s , s o th e re a re 2 2 5 u n k n o w np e r m e a b i l i t y v a l u e s in e a c h m o d e l .


21/31

M a r k o v C h a i n M o n t e C a r l a M e t h o d s 8 1

Ten o b se rv a t i o n s o f p r es su re were o b t a in ed a t each o f f i v e we l l s wh o selo ca t io n s a r e sh o wn in F ig u re 4 . T h e cen t r a l , ac t i v e we l l p ro d u ced 7 5 0 b a r r e l sp e r d ay o f f l u id ; t h e o th e r fo u r a r e o b se rv a t i o n we l l s , u sed o n ly t o acq u i r ep res su re d a t a. M easu rem en t s o f t h e p res su re b eg an a t 0 .0 5 d ay s an d en d ed a t5 .7 d ay s . Th e e r ro r v a r ian ce fo r a ll p r es su re measu rem en t s i s a s su med t o b e0 .1 5 p s i 2 . A l l co mp u ta t i o n s were d o n e o n a P en t i u m-9 0 .

V A R I A N C E E Q U A L T O 0 .2 5Th e " t r u e " p e rmeab i l i t y d i s t ri b u t io n , sh o wn in F ig u re 4 , i s an u n co n d i -

t i o n a l s imu la t i o n g en era t ed u s in g t h e Ch o lesk y d eco mp o s i t i o n meth o d fo r asp h eri ca l v a r i o g ram. V a lu es o f In(k) r an g e f ro m 2 .0 t o 5 .0 , o r , i n t e rms o f k ,f ro m 7 t o 1 4 8 md . "T ru e " p res su re d a t a fo r t h e f iv e we l l s were g en era t ed u sin gthe t rue g r idb lock perm eabi l i t ies in a f in i te d i f ference f low s imulator . W e usedth ese d a ta i n th r ee o f th e M CM C meth o d s d esc r i b ed p rev io u s ly t o g en era t eM ark o v ch a in s o f p o ss ib le p e rm eab i l i ty f i e ld s . I n t h e f i r s t me th o d , t h e p ro p o sedp er tu rb a t i o ns a r e l o ca l an d a r e b ased o n t h e sq u are ro o t o f t h e p r i o r co v ar i an ce ,so t hey h o n o r o n ly t h e v a r i o g ram. In t h e seco n d an d t h i rd m eth o d s , t h e sen s i-t i vi t y co e f fi c i ent s f ro m th e r ese rv o i r s imu la to r a r e u sed t o m o d i fy t h e d i r ec t io nan d t h e mag n i tu d e o f t h e p ro p o sed p e r t u rb a t i o n s. Th e p e r t u rb a t io n s i n t h e seco n dmeth o d a r e l o ca l , i n t h e t h ird meth o d t h ey a r e g lo b a l , m ean in g t h a t t h e p e rm e-ab i l i ty in every gr idb lock i s modif ied in every t ransi t ion .

On e w ay t o co m p are t h e t h r ee meth o d s i s t o l o o k a t t h e r a t e o f g en era t i o no f accep t ab le i n d ep en d en t r ea l i za ti o n s fo r each . F ig u re 5 sh o w s t h e v a lu es o ft h e "o b j e c t i v e fu n c t i o n " [ i . e . , t h e a rg u men t o f t he ex p o n en t i a l i n Eq . (1 0 ) ] v s .t h e p e r t u rb a t i o n n u mb er . Th e p e r t u rb a t i o n n u mb er i n t h i s f i g u re r e f e r s t o t h en u mb e r o f t ran s i ti o n s p ro p o sed , b u t o n ly t h e v a lu es o f ac cep t ed t r an s i ti o n s a r ep lo t ted .

Accep t an ce o f a p e r t u rb a t i o n b ased o n Has t i n g ' s accep t an ce c r i t e r i o n ,Equat ion (6) , does no t necessar i ly ind icate that the resu l t ing s tate i s d rawn f romthe correct p robabi l i ty d is t ribu t ion . T here usua l ly i s a t ransien t per iod at theb eg in n in g o f a M ark o v ch a in t h a t mu s t b e d i sca rd ed o n t h e g ro u n d s t h a t eq u i -l i b ri u m h as n o t b een r each ed . F o r t h e M CM C meth o d s t h a t u se t h e a p o s t e r i o r icovar ian ce to genera te per tu rbat ions , the t ransien t per iod i s so shor t that i t i sd i ff i cul t t o see an y ev id en ce o f i t i n F ig u re 5 . C lo se ex am in a t i o n o f t h e d a t ashows that the t ransien t per iod ends af ter about 30 per tu rbat ions . I t migh t seemf ro m F ig u re 5 t h a t th e t r an s ien t p e r i o d fo r t h e f ir s t M CM C meth o d , wh ich u seslocal per tu rbat ion based on ly on the var iogram, also i s shor t . A carefu l inspec-t i o n , h o wev er , i n d i ca tes t h a t t h e v a lu es o f t h e o b j ec t i v e fu n c t i o n p ro b ab ly a r edecreasing af te r 50 ,000 per tu rbat ions so , fo r th is method , a l l o f the f irs t 50 ,000real izat ions are in the t ransien t per iod .

Wh en t h e s t a t i on ary d i s tr i b u ti o n o f t h e M a rk o v ch a in i s r each ed , t h e v a lu es


22/31

8~

~ ? -

z~

E ~

b~


23/31

Markov Chain Monte Carlo Me t h o d s

11=+3

8 3

o0C

i ,G).>_0

0

1 E + 2 -

1 E + 1

C M - L o c a l P e r t u r b a t i o n s( ~ ) C M , - L o c a l P e r t u r b a t i o n s< ~ C M , - G l o b a l P e r t u r b a t i o n s

I ' I ' I ' I '1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0

P e r t u rb a t io n N u m b e rFigur e 5. Values of objective function for three Markov chain Monte Carlo methods.

o f t h e o b j ec t i v e fu n c t i o n c lu s t e r a ro u n d a v a lu e g rea t e r t h an o r eq u a l t o M / 2wh ere M i s th e n u m b er o f mo d e l p a rame te r s . Bec au se t h e p res su re d a t a u sed i nth ese ex amp les t o ca l cu l a t e t h e max imu m a p o s t e r i o r i mo d el were ex ac t , an dthe pr io r model was close to the a p o s t e r i o r i mo d el , t h e av erag e v a lu e o f t h eo b j ec t i v e fu n c t i o n i n t h e s t a t i o n ary p h ase sh o u ld b e ap p ro x imate ly eq u a l t oM / 2 = 1 1 2 .5 . T h e o b se rv ed mean o f t h e o b j ec t i v e fu n c t i o n , co mp u ted fo r th ese t o f accep t ed r ea l i za t i o n s fo r t h e meth o d t h a t u ses l o ca l p e r t u rb a t i o n s b asedon the a p o s t e r i o r i co v ar i an ce i s 1 19 wi th a co r r esp o n d in g es t ima t e o f t h estandard dev iat ion g iven by 10 . The method that uses g lobal per tu rbat ions basedon the a p o s t e r i o r i co v ar i an ce r esu l t ed i n a mean v a lu e o f 1 1 1 an d a s t an d arddeviat ion of 10 .

Each o f t h e th r ee meth o d s r eq u i r ed ap p ro x im ate ly 1 3 h o u r s t o a t t emp t5 0 ,0 0 0 p e r t u rb a t i o n s . Th e M ark o v ch a in M o n te Car lo meth o d t h a t u sed l o ca lp e r t u rb a t i o ns b ased o n t h e v a r i o g ram d id n o t g en era t e an y l eg i t ima t e imag es o ft h e p e rmeab i l it y f ie l d d u r in g t hi s p e r i o d . Th e M CM C meth o d s t h a t u sed g lo b a lper tu rbat ions based on the a p o s t e r i o r i co v ar i an ce mat r i x g en era t ed mo re t h an200 leg i t imate, independent real izat ions . Using local per tu rbat ions based on thea p o s t e r i o r i co v ar i an ce mat r i x , ap p ro x imate ly 3 7 ,5 0 0 h ig h ly co r r e l a t ed imag eswere accep t ed . A s ex p ec t ed , u s in g l o ca l p e r t u rb a t i o n s g rea t l y in c reases t h e n u m-ber o f s ta tes acce p ted bu t success ive s tates , o r real izat ions , d i f fer on ly s l igh t ly .Recal l that the u l t imate goal usual ly i s no t to generate s imply possib ler ese rv o i r mo d e l s , b u t t o d raw in fe r en ces b ased o n p red i c t io n s f ro m th e r ea l -i za t i o n s . Th ese p red i c t i o n s may b e ex p en s iv e co mp u ta t i o n a l l y so i t u su a l l y i sb e t t e r t o h av e 1 0 -1 0 0 i n d ep en d en t r ea l i za ti o n s f ro m th e a p o s t e r i o r i dis t r ibu t ion


24/31

84

E

0L_>

120

Oliver, Cunha, and Reynolds

1 0 0

8 0

6 0

4 0

2 0

0 0 I I I I I I1 0 0 0 2 0 0 0 3 0 0 0L a g

F i g u r e 6 . E x p e r i m e n t a l v a r i o g r a m f o r o b j e c t i v e f u n c t i o n o f c h a i n o f n e wrea l iza t ions genera ted us ing loca l per tu rba t ions based on a p o s t e r i o r i c o v a r i -a n c e .

t han to have 10 , 000 dependen t rea l i za tions . Becaus e the dependence dec reas e swi th " l ag , " an approx ima te ly independen t s e t o f rea l i za t ions can be ob ta inedby re ta in ing eve ry r th s t a te f rom the cha in , w he re r i s the va lue o f the l ag fo rwhich the au tocor re la t ion i s approx ima te ly ze ro . In F igure s 6 and 7 we havep lo t t ed the " expe r imen ta l va r iogram" o f the cha in o f va lues o f the ob jec t ive

1 4 0

Ei-0

>

1 2 01 0 0

8 06 04 02 0

0 r I I I I I I0 5 1 0 1 5 2 0 2 5 3 0 3 5 4 0Lag

F i g u r e 7 . E x p e r i me n t a l v a r i o g r a m f o r o b j e c t iv e f u n c t i o n o f c h a i n o f n e w r e a l -i za t ions genera ted us ing g lobal per tu rbat ions based on a p o s t e r i o r i c o v a r i a n c e .


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M a r k o v C h a i n M o n t e C a r lo M e t h o d s 85fu n c t io n fo r t h e seco n d an d t h ird M C M C meth o d s . Th e ex p er imen ta l v a r i o g ramis calcu lated f rom

1 N s - ," y( s) = - ~ ( S ( m t ) - S (m t+ s) ) 2 (26)2Ns ,=o

w h e r e S(m,) i s th e v a lu e o f t h e a rg u men t o f t h e ex p o n en t i a l i n Eq u a t i o n (1 0 )for the tth accep te d t ransi t ion in the chain . I t i s c lear f rom Fig ure 7 that suc-cess iv e n ew s t a tes g en era t ed u s in g g lo b a l p e r t u rb a t i o n s a r e u n co r r e l a t ed wh ereasnew states generated f rom local per tu rbat ions are correlated s l igh t ly to lags o fap p ro x imate ly 1 2 0 0 . I f , f o l l o win g t h e su g g es ti o n o f T i em ey (1 9 9 4 ) , w e r e t a ino n ly ev ery 1 2 0 0 th n ew s t a te we wo u ld b e l e f t w i th 3 1 i n d ep en d en t r ea l i za t i o n sf ro m lo ca l p e r tu rb a t i o n s co mp ared t o 2 0 0 f ro m g lo b a l p e r t u rb a t io n s . F o r p ro b -ab i l i s t i c i n f e r en ce , each r ea l i za t i o n wo u ld b e we ig h t ed b y t h e f r eq u en cy o fo ccu r r en ce o f t h e s t a te w i th in t h e ch a in . Af t e r t h in n in g t o t h e se t o f i n d ep en d e n treal izat ions , the weigh t ing i s equal to the number o f a t t empted t ransi t ions be-tween t h e o c cu r r en ce o f t h e i n d ep en d en t s t a te s .

V A R I A N C E E Q U A L T O 1 .0In th is exam ple, the var ia nce of the log-permeab i l i ty f ie ld i s increased to

1 .0 , a l l o ther param eters o f the d is t r ibu t ion are the sam e as in the f i rst examp le.Th e " t ru e" l o g -p ermeab i l i t y f i e ld , g en era t ed u s in g t h e Ch o lesk y d eco m p o s i t i o nmeth o d wi th th e same r an d o m seed as th e p rev io u s ex am p le , i s sh o wn in F ig u re8 . Values o f ln (k) rang e f rom abo ut 1 .0 to about 7 .0 , co rrespo nding to a rangeo f pe rmeab i l i ti e s f ro m ab o u t 2 .7 rn d t o ab o u t 1 1 0 0 ro d. Th e " t ru e" p res su redata again are ob tained f rom a f in i te d i f ference s imulat ion of f low in the syn the t icreservo i r .

F ig u re 9 sh o ws v a lu es o f t h e "o b j e c t i v e fu n c t i o n " o f accep t ed s t a t e s v s .t h e p e r t u rb a ti o n n u mb er fo r t h e same t h ree M CM C meth o d s t h a t were co m p aredin th e p rev io u s ex am p le . W e see m o re c l ea r l y , in t h is ex am p le , t h a t t h e ch a ino f s t a te s g en era t ed f ro m lo ca l p e r t u rb a t io n s o f t h e v a r i o g ram d o es n o t r eacheq u i l i b r i u m in t h e f i r s t 5 0 ,0 0 0 p e r t u rb a t i o n s . Bo th M CM C meth o d s t h a t mak euse of the sensi t iv i ty in format ion , th rough the a pos ter ior i co v ar i an ce mat r i x ,have t ransien t per iods that are too shor t to be v is ib le in F igure 9 . Because oft h e g rea t e r n o n l i n ea r i ty o f t h e mo d e l , o n ly 2 5 ,0 0 0 p e r t u rb a ti o n s were accep t edwhen the per tu rbat ions were local . Using a g lobal per tu rbat ion , 14 s tates wereaccep t ed b u t s i x h ad t o b e d i sca rd ed b e cau se t h e ch a in h ad n o t r each ed eq u i l i b -r ium.

D I S C U S S I O NWe o u t l in ed sev era l d i f f e r en t p e r t u rb a t i o n meth o d s fo r u se i n M a rk o v ch a in

M o n te Car lo s imu la t i o n s an d g av e co r r esp o n d in g accep t an ce c r i t e r i a t h a t w i l l


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