Jensen Et Al Math Geology 1995

8/8/2019 Jensen Et Al Math Geology 1995

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M at hemat i ca l Geo l ogy , Vo l . 28 , No . 4 , 1996

P e r m e a b i l i t y S e m i v a r i o g r a m s , G e o l o g i c a l S t r u c t u r e ,

a n d F l o w P e r f o r m a n c e 1

J . L . J e n s e n , z P . W . M . C o r b e t t , z G . E . P i c k u p , z a n d

P . S . R i n g r o s e z

Cl as t ic s ed i men t s m ay have a s t r ong de t e r mi n i s t i c com pon en t to the i r pe r m eab i l i t y var ia tion . Th i s

s t r uc tu r e m ay be s een i n t he exper i m en t a l s em i var i ogr am , bu t pub l i s hed geos t a t is t iea l s t ud i e s have

no t a l wa ys exp l o i ted t h i s f ea t u r e dur i ng da t a an a l y s i s and eovar i ance mod e l ing , h~ t h is pap er , we

des c r i be s ed i men t ar T o r gan i z a t ion , i ts i mpor t ance fi ) r f l o w m ode l i ng , and h ow t he s emi var i ogr a m

can be used f i~r identi fi ca tion o f s t ruc ture . Clas t i e sed im enta ry s t ruc ture o ccur s a t seve ral sca les

and i s l i nked m t he cond i ti ons o f depos i t ion . Lam i na t i on , bed , a nd beds e t s ca l e s s how r epe t i ti v e

and t r end f ea t u r e s t ha t s hou l d b e s am p l ed car e f u l l y to a s s e s s t he degr ee o f o r gan i z a t i on and l evel s

o f he te rogene iO, . I n t e r pre t a ti on o f s em i var i ogr am s i s under t aken be s t w i th an appr ec i a t i on v f t he s e

geo l og i ca l un i t s and ho w t he i r f ea t u r e s r e l a te t o t he s am p l i ng p r ogr am . Sam pl i ng a t inappr opr i a t e

i n te r va l s o r w i t h i n s t r umen t s hav i ng a l a r ge mea s ur em en t vo l ume , fi ~r exam pl e , m ay g i ve mi s l ead i ng

s emi var i ogr am s . F l ow si mu l a t i ons f o r mo de l s w h i ch i nc lude and i gnor e s t ruc t u r e s ho w tha t the

repe t i t ive f ea tur es in perm eab i l i t y can eha nge a ni so t ro py an d recov ery perf i~rmance s ign(f icant(v.

I f s y s t ema t i c varia tion i s p r e s en t , ca r e f i d de s i gn o f t he pe r m eab i l i O , f i e l d s t h e r t ~ r e i s impo r t an t

par t i cu lar ly to pre serv e the s t ruc ture e ff t, e ts .

K E Y W O R D S :semivariograms, sedimentary structure, permeability.

I N T R O D U C T I O N

Clastic sediments are rarely "uniform" in lithology, grain size, cementation,

and textural features and variations in these attributes can change the petro-

physical properties significantly. Permeability, which controls the flow of fluids

within the subsurface, may differ by several factors of ten. Indeed, as a result

of these factors, sandstone reservoirs with several orders of magnitude variation

in permeability (0.001-10000 mD) are widespread (Brendsdal and Halvorsen,

t Received 20 January 1995; accepted 15 August 1995.

-'Petroleum EngineeringDepartment, Heriot-Watt University, Riccarton, Edinburgh EHI4 4AS,

Scotland: jerry~jensen@pet,hw. ac. uk

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4 2 0 J e n s e n , C o r b e t t , P i c k u p , a n d R i n g r o s e

1 9 9 3; B a l l a n d o t h e r s , 1 9 9 4 ). F a c e d w i t h su c h l a r g e v a r i a b i l i t y a n d a p p a r e n t

d i s o r d e r, o b t a i n i n g r e a li s ti c m o d e l s o f p e r m e a b i l i t y s e em s d a u n t i n g .

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

s e d i m e n t s . A n i m p o r t a n t c o n t r o l o n p e r m e a b i l i t y i s t h e g r a i n s i z e ( B e a r d a n d

W e y l , 1 97 3 ; B r a y s h a w , D a v i e s , a n d C o r b e t t , 1 9 95 ). G r a i n - s i z e v a r i a t i o n in

s e d i m e n t s is a fu n c t i o n o f s u p p l y , s o r t i n g , a n d d e p o s i t i o n a l e n v i r o n m e n t . I n

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

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

c u r r e n ts , j u s t t o n a m e a f e w . T h e r e s u lt o f t h e s e q u a s i p e r i o d i c o r e p i s o d i c

d e p o s i t i o n a l e v e n t s i s a s e d i m e n t w h e r e t h e g r a i n s a r e d e p o s i t e d in e l e m e n t s

h a v i n g n e a r - r e g u l a r v a r i a t io n s a t s e v e r a l , w e l l - d e f in e d s c a l e s .T h e g e n e t i c c o n t r o l s o n th e a r r a n g e m e n t o f l a m i n a t i o n s a n d b e d s i n d i c a t e s

t ha t t h e r e s u lt in g l a m i n a - s e t s , b e d s , a n d b e d s e t s h a v e l i m i t e d a r c h i t e c t u r a l f o rm s

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

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

b e us e d to g i v e a n e n v i r o n m e n t a l i n t e r p r e ta t i o n . T h i s o r g a n i z a t i o n o f s e d i m e n -

t a ry v a r i a ti o n t r a n s l a te s t o s i m i l a r s t ru c t u r e s f o r p e r m e a b i l i t y v a r i a t io n . K n o w l -

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

v a r i a ti o n t b r f lo w s i m u l a t i o n s , g i v i n g b e t t e r f l o w - p r o p e r t y p r e d i c t i o n s ( R i n g r o s e

a n d o t h e r s , 1 9 9 3 b ; J o n e s a n d o t h e r s , 1 9 9 3 ).

I n t h i s p a p e r , w e g i v e a n o v e r v i e w o f s e d i m e n t a r y o r g a n i z a t i o n ( a r c h i t e c -

t u re ) a n d th e s c a l e s a t w h i c h it o c c u r s . W e d i s c u s s h o w t h e s e f e a t u r e s a p p e a r

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

r e l a t i o n t o t h e g e o l o g y , c a n a f f e c t t h e s e m i v a r i o g r a m f o r m . T h i s k n o w l e d g e

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

s e d i m e n t a r y o r g a n i z a t i o n , in a d d i t i o n to a m e t h o d o f d e t e r m i n i n g a c o v a r i a n c em o d e l f o r s i F n u l a t i o n . W h e n s t r u c t u r e i s p r e s e n t , a n a l t e r n a t i v e t o c o v a r i a n c e

m o d e l s i s t o m o d e l t he p e r m e a b i l i t y f ie ld d e t e r m i n i s t i c a l l y . W e c o n c l u d e w i th

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

S U R V E Y O F P E R M E A B I L I T Y M O D E L S I N T H E L I T E R A T U R E

S e v e r a l r e p o r t e d a t t e m p t s t o a n a l y z e a n d m o d e l p e r m e a b i l i t y f ie l d s o f s e d -

i m e n t a r y r o c k s u s i n g g e o s t a t i s ti c a l m e t h o d s h a v e n o t e x p l o i t e d t h e a r c h it e c t u r e .

T h e r e a s o n s f o r t h is i n c l u d e t he p o o r q u a l i ty o f c o n v e n t i o n a l ( e . g . , c o r e p l u g )

d a t a , t h e st a t is t ic a l v a r i a b i l i t y o f t h e s e m i v a r i o g r a m , a n d i n a d e q u a t e s a m p l i n g .

U n d o u b t e d l y , s e m i v a r i o g r a m a n a l y s i s h a s a l a rg e i n t e rp r e t a t i v e c o m p o n e n t : f e a -

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

s a m p l i n g v a r i a b i l it y , m e a s u r e m e n t e r r o r s , a n d p h y s i c a l ( g e o l o g i c a l ) c a u s e s .

U s u a l l y th e i n t e rp r e t a ti o n p a y s l it tl e r e g a r d t o t h e g e o l o g i c a l c a u s e s i n f a v o r o f

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



Sem ivar iograms, Struc ture , and F low Per forma nce 421

a d iagnos t i c too l and s imp ly u s e s i t a s a neces s a ry s t ep in f i t t ing a cova r iance

m ode l fo r s tochas t i c s imu la t ion . W e i l lu s t ra te the s e idea s wi th a few exam ple s

f rom the l i t e ra tu re .

Des ba ra t s (1994 ) recen t ly ha s u s ed a geos ta t i s t i c a l mode l to inves t iga te

r e s e r v o i r p e r f o r m a n c e in a n e s tu a r i n e ti d al c o m p l e x ( L o w e r C r e t a c e o u s , C r y s t a l

V i k i n g F i e l d , A l b e r t a ) . B a s e d o n c o r e p l u g p e r m e a b i l i t y d a t a , t h e s p a t i a l c o r -

r e la t io n s t ru c t u r e o f l o g - p e r m e a b i l i t y w a s c h a r a c t e r i z e d b y a n e x p o n e n t i a l m o d e l

w i t h a h o r iz o n t a l r a n g e s e t u n i f o r m l y a t 1 0 0 t i m e s t h e v e r t ic a l r a n g e . D e s b a r a t s

( 1 9 9 4 ) n o t e d t h a t tw o p r o b l e m s a r o s e d u r i n g t h e fi tt in g o f th e e x p o n e n t i a l m o d e l

t o t h e s a m p l e s e m i v a r i o g r a m :

( 1 ) t h e r e w a s a l a r g e d i s c r e p a n c y b e t w e e n t h e t h e o r e t i c a l s e m i v a r i o g r a m

s i l l and the s ample va r iance a t l ag 10 m, and

( 2 ) t h e r e w a s a l a r g e n u g g e t i n t h e s a m p l e s e m i v a r i o g r a m .

D es ba ra t s (19 94 ) a t t r ibu ted the fir st f a c to r to pos s ib le s ed im en ta ry pe r iod ic i ty

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

h i s f ir st obs e rv a t ion and s ug ges t s an a l t e rna t ive , geo log ic a l c aus e fo r the s eco nd

i tem.

A s t u d y o f t h e f ie ld d a ta ( R e i n s o n , C l a r k , a n d F o s c o l o s , 1 9 8 8) r e p o rt s th a t

8 - t o 1 2 - m - t h i c k c h a n n e l s a r e p r e s e n t w h i c h c o u l d e x p l a i n t h e h i g h v a r i a n c e

fea tu re . F luv ia l channe l s u s ua l ly have a cha rac te r i s t i c g ra in - s ize va r ia t ion and

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

n u g g e t c o u l d b e t h e r e s u lt o f t h e i n su f f ic i en t s a m p l i n g o f v a ri a b i li ty i n th e m o r e

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

d is cus s in m ore de ta i l l a te r , the s i z e o f the co re p lug s u s ed fo r the s tudy m ay

no t have b een s u f f ic ien t to f il t er ou t the s m a l l - s ca le va r iab i l i ty fo r the coa rs e rg r a i n e d s e c t i o n s . R e i n s o n , C l a r k , a n d F o s c o l o s ( 1 9 8 8 ) a l s o p o i n t o u t a N E - S W

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

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

T h e r e a r e o t h e r e x a m p l e s i n t h e l it e ra t u re , b o t h g e o l o g i c a l a n d e n g i n e e r i n g ,

o f sp h e r ic a l o r e x p o n e n t i a l m o d e l s b e i n g f i tt ed t o s a m p l e s e m i v a r i o g r a m s s h o w -

i n g d i s ti n c t s ig n s o f s e d i m e n t a r y p e r i o d i c i t y . G o g g i n a n d o t h e r s ( 1 9 8 8 , fig . 1 5)

s u g g e st s a e o li a n d u n e b e d s e t c y c l i ci t y o f a p p r o x i m a t e l y 4 0 m . W o o d b u r y a n d

S u d i c k y ( 1 9 9 1 , f i g . 3 ) i n d i c a t e a p e r i o d i c c o m p o n e n t a t 8 0 c m w a v e l e n g t h .

D a w s a n d P r o s s e r ( 1 9 9 2 , f ig . 1 0) s h o w a s tr o n g c y c l i c c o m p o n e n t at 3 0 t o 1 20

c m i n t id a l c h a n n e l s . T h e s e s e m i v a r i o g r a m s a r e n o t n e c e s sa r i ly s u f f e r in g s o

m u c h f r o m s a m p l i n g i n d u c e d v a r i a t i o n , r a t h e r t h e y a r e r ef l ec t in g th e s e d i m e n t a r y

induced s t ruc tu re .

P h i l li p s a n d W i l s o n ( 1 9 8 9 ) d i s c u s s t h e r o l e o f g e o l o g i c a l i n f o r m a t i o n i n

d e t e r m i n i n g t h e c o r r e l a t io n s c a le o f p e r m e a b i l i t y . T h e y a d v o c a t e r e f e re n c e t o a

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




i n f o rm a t i o n in ge o s ta ti st ic a l p a r a m e t e r s w o u l d b e a " s i g n i f i c a n t a d v a n c e . " T h e y

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

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

m e n t a r y s t r u c t u r e m a y b e r e c o g n i z e d e x p l i c i t l y a n d e x p l o i t e d i n r e s e r v o i r m o d -

e l ing .

I n t h e f o r w a r d to J o u r n e l a n d H u i j b r e g t ' s ( 1 9 7 8 , p . v ) b o o k , M a t h e r o n

c o m m e n t s t h a t g e o s t a t is t ic s h a s t h e a im o f r e c o g n i z i n g b o t h s t r u c tu r e a n d r a n -

d o m n e s s . T h e r e f o r e , o u r p r o p o s a l i s i n k e e p i n g w i th w h a t w a s i n t e n d e d w h e n

M a t h e r o n c o i n e d t h e t e r m " g e o s t a t i s t i c s " a b o u t 3 0 y e a r s a g o . S o m e i m p o r t a n t

ana ly s i s ha s been done a l ready (e .g . , Se r ra , 1967 , 1968 ; J ou rne l and Hu i jb reg t s ,

1 9 7 8 , c h a p . I V ; R e n d u a n d R e a d d y , 1 9 8 2 ) t o t i e g e o l o g i c a l s t r u c t u r e t o t h es e m i v a r i o g r a m i n n o n s e d i m e n t a r y e n v i r o n m e n t s , b u t s t r u c t u r e i s s o p e r v a s i v e i n

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

h a v i o r a n d m o d e l i n g i s n e e d e d .

S T R U C T U R E I N C L A S T I C S E D I M E N T S

W i n d a n d w a t e r , t h e t w o p r i n c ip a l c a r r i er s o f s e d i m e n t , h a v e s i g n if i c an t

p e ri o d ic c o m p o n e n t s to t h ei r e n e r g y s p e c tr a ( e . g . , L u m l e y a n d P a n o f s k y , 1 9 6 4,

p . 4 2 - 4 4 ; B r as a n d R o d r f g u e z - I t u rb e , 1 9 8 5, p . 2 0 2 - 2 0 5 ) . D u r i n g d e p o si ti o n ,

the s e repe t i t ive and s ys tema t ic fo rce s may con t r ibu te a s t rong ly de te rmin i s t i c

a s pec t to c la s t ic s ed im en ts a t s eve ra l s ca le s (Al len , 1985 , p . 243 -2 65 ) . Ev en in

s t e a d y , u n i d i r e c ti o n a l f l o w s , s e d i m e n t m o v e m e n t s a n d d e p o s it s m a y t a k e o n

p e r io d i c f o r m s ( Y a l i n , 1 9 7 7 , p . 2 0 9 - 2 3 6 ; A l l e n , 1 9 8 5 , p . 7 4 - 7 8 ) . T h e r e g u l a r

v a r i at i o n s ta k e t h e f o r m o f l a y e r s ( m m s t o c m s t h i c k ) o f d i f fe r i n g g r a in s i z e ,

g iv ing the s ed imen t a l amina ted appea rance (F ig . 1 , top ) . A t a l a rge r s ca le (cmst o m s ) a n d l o n g e r t i m e f r a m e , c y c l i c v a r i a t i o n s i n t h e s y s t e m g i v e r i s e t o b e d s ,

c o m p o s e d o f n u m e r o u s l a m i n a t io n s ( F i g . 1 , c e n t e r ). A t l a r g e r s c a l e s ( m s t o 1 0 0 s

o f m s ) , s e t s o f b e d s m a y d e v e l o p ( F i g . 1 , b o t t o m ) , s h o w i n g a s y s t e m a t i c v ar i -

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

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

S u c h s y s t e m a t i c a n d p e r i o d i c f e a tu r e s f o r m t h e b as i s f o r g e o l o g i c a l a n a l y s i s

( e . g ., S c h w a r z a c h e r , 1 9 75 , p . 2 2 3 - 3 2 0 ; A l le n , 1 9 85 , p . 6 3 - 6 6 ) . T h e s e " b u i l d -

i n g b l o c k s " ( st ra ta l e l e m e n t s ) h a v e b e e n d e f in e d f o r sh a l l o w m a r i n e s e q u e n c e s

(F ig . 2 ) , bu t a h ie ra rchy a l s o is r e cogn ized in the a rch i te c tu ra l e l em en ts in f luv ia l

s y s t e m s ( M i a l l , 1 9 8 8) a n d i n t h e b o u n d i n g s u r f a c e s i n e o l i a n s y s t e m s ( B r o o k -

f ie ld . 1977 ). M eand e r ing r ive rs m ay s ho w regu la r , s inuo us fo rm s (A l len , 1985 ,

p . 9 6 - 9 9 ) . E o l ia n d u n e s m o v e b e c a u s e o f t h e r e g u la r b u i l d -u p o f s e d im e n t w h i c h

t h e n a v a l a n c h e s d o w n t h e d u n e f a c e ( A l l e n , 1 9 8 5 , p . 6 8 - 7 0 ) . T i d a l p r o c e s s e s

a r e p a r t ic u l a rl y d e t e r m i n i s ti c n a tu r a l p h e n o m e n a . I n se d i m e n t s d e p o s i t e d b y t id a l

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

c ie s c an be de tec ted (M ar t ino and San de rs o n , 1993 ). S t ruc tu red pe rm eab i l i ty



Semivariograms, Structure, and Flow Performance 423

Figure I. Lamination, bed, and bedset scales of sedimentary

structure with schematic pemmability variations (after Rin-grose and others. 1993a).

f i eld s the re fo re a re cha rac te r i s t i c o f c l as t i c r ese rvo i r s and can be a f ea tu re in

c a r b o n a te r e s e r v o i r s ( G r a n t, G o g g i n , a n d H a m s , 1 99 4) .

Pos tdepos i t iona l p rocesses (d iagenes i s ) can mod i fy the p r imary s t ruc tu re ins e d i m e n t s . O n c e d e p o s i t e d , b i o l o g i c a l , c h e m i c a l , m i n e r a l o g i c a l , a n d m e c h a n i c a l

e f fec ts can chan ge the o r ig ina l sed im en t pa t t e rn s . The se chan ges can enhan ce

o r d imin i s h the s t ruc tu re . A lso , the chan ges may no t be cons tan t th rougho u t the

rese rv o i r ( e . g . , o i l v s . wa te r l eg s) . Thus , in som e res e rvo i r s , t he re may be a

r a n d o m ( u n p r e d i c ta b l e ) c o m p o n e n t t o t h e p e r m e a b i l i t y v a r i a t io n . E v e n i n s e d i-

Figure 2. Shallow marine stmtal elemenls (adapatedfrom van Wag,.mer and others , 1990).



424 Jensen, Corbett, Pick up , and Ringrose

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

p o r t an t c o n t r o l o n p e r m e a b i l i t y v a r i a t io n ( e . g . , H a r t k a m p - B a k k e r , 1 9 93 ).

S E M I V A R I O G R A M B E H A V I O R I N S T R U C T U R E D M E D I A

I f g e o l o g i c a l s t ru c t u re is p r e s e n t , t he s a m p l i n g p r o g r a m , i n t e r p r e ta t i o n , a n d

m o d e l i n g w i l l a l l b e af f ec t e d ( e . g . , W a r r i c k a n d M e y e r s , 1 9 87 : B a r n e s , 1 9 88 ).

T o b e d e t e c t e d , t h e s a m p l e s p a c i n g s h o u l d b e l e ss th a n o n e - h a l f o f t h e s t ru c t u r e

s iz e t o s a t is f y t he N y q u i s t c ri t e r io n ( L u m l e y a n d P a n o f s k y , 1 96 4, p . 5 4 - 5 5 ) .

O t h e r w i s e , a n y r e p e t i t i v e f e a tu r e w il l b e " a l i a s e d " i n to a l o w e r fr e q u e n c y

( g r e a t e r l e n g t h ) . R e p e t i t i v e f e a t u r e s t h a t a r e w e l l s a m p l e d w i l l p r o d u c e " h o l e s "in t h e s e m i v a r i o g r a m a t t h e a p p r o p r i a t e la g . U n d e r - s a m p l e d f e a t u re s m a y p r o -

d u c e a n o i sy s e m i v a r i o g r a m t ra c e a n d a la r g e n u g g e t ( e . g . , J o u r n e l a n d

H u i j b r e g t s , 1 9 7 8, p . 1 52 : C o r b e t t a n d J e n s e n , 1 9 9 2 : D a w s a n d P r o s s e r , 1 9 9 2) .

T h e e f fe c ts o f a p e r i o d i c c o m p o n e n t u p o n t h e c o r r e l a ti o n s t r u c tu r e o f a

s i gn a l a r e w e l l k n o w n ( e . g . , S c h w a r z a c h e r , 1 9 75 , p. 1 6 9 - 1 7 9 ) . W e w i ll r e v i e w

s o m e o f th e s a l ie n t p o in t s a s th e y a f fe c t t h e s a m p l e s e m i v a r i o g r a m . S u p p o s e

t h at r e s e r v o i r p e r n l e a b i l i t y k (z ) a l o n g a l in e h a s a p e r i o d i c c o m p o n e n t w i t h

w a v e l e n g t h X . T h e p e r m e a b i l i t y s e m i v a r i o g r a m , 3 ,( h) , w i ll e x h i b i t h o l e s , t h a t i s

r e d u c t i o n s i n t h e c o v a r i a n c e , a t l a g s h = iX , w h e r e i = 1, 2 . . . . ( F i g . 3 ) . T h e

d e p t h o f t he h o l e s w i ll d e p e n d u p o n t h e s i z e o f th e p e r i o d i c c o m p o n e n t c o m p a r e d

t o a n y r a n d o m c o m p o n e n t ( s ) p r e s e n t i n k ( z) . T h e d e p t h o f th e h o l e s a ls o m a y

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

s y s t e m a n d th e n a tu r e o f it s e n e r g y s o u r c e ( S c h w a r z a c h e r , 1 9 75 , p . 2 6 8 - 2 7 4 ) .

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

. =

2 .0

1.5-

1 . 0 -

0 . 5 -

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0 . 0 1 0 . 0 2 0 . 0 3 0 . 0

P e f f e c l l y c y c h c , k

. . .. . . . . . . . .. . . . . . . S l i g h t l y ; t ~ y c t i c , k '

4 0 . 0

Position

1 5

0 5

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0 .0 I .0 20 .0 30 .0 40 .0

Lag ( h )

Figure 3 . E ~ a r u pl e p em ~ e a b i l i ly p l o t s a n d n o r m a l i z e d s e r n i v a r i o g m m s c o n s t r u c te d

f r o m s e r i e s k ( z ,) = I + 0 . 4 s i n ( ~ z , / 6 ) + ( ) . 4 e , , w h e r e ~, - N ( 0 . 1 ) a n d .-., ( = 0 , I ,

. . . ) is p o s i t i o n , s h o w i n g e v i d e n c e o f s t r u c t u r e ( c y c l i c i t y w i t h ~ = 1 2 u n i t s ) .

A c y c l i c s e r i e s i s k '(= .,) = I + 0 . 4 s i n ( ~ z , / 6 + 6 ) + 0 . 4 ~ , , w h e r e 6 = 0 . 0 1 [ : , / 1 2 ]

a n d 1 : ,t l2 1 i s g r e a t e s t i n t e g e r l e s s t h a n o r e q u a l t o 7.,1 12 . L e f t f i g u r e s h o w s d e t e r -

m i n i s l i c c o m p o n e n t s o f k ( : ) a n d k ' ( : ) .



S e m i v a r i o g r a m s , S t r u c t u r e , a n d F l o w P e r f o r m a n c e 4 2 5

a n d , t h e re f o r e th e h o l e e f fe c t m a y b e w e a k e r t h a n o n e m a y e x p e c t . F o r e x a m p l e ,

a s li g h t l y a p e r i o d i c s i g n a l ( F i g . 3 , l e f t ) h a s a s m a l l e r f ir st h o l e ( F i g . 3 , r i g h t )

a n d s m a l l e r s u b s e q u e n t h o l e s th a n i ts p e r i o d i c e q u i v a l e n t . Y e t , t h e s t ru c t u r e

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

T h e s a m p l i n g p r o g r a m u s e d to m e a s u r e p e r m e a b i l i t y a t fi xe d l o c a ti o n s , z

= z j , j = 1, 2 . . . . . n , c a n h a v e a s t r o n g i n f l u e n c e u p o n h o w w e l l t h e s a m p l e

s e m i v a r i o g r a m , ~ ( h ) , r e p r o d u c es 7 ( h ) ( e .g . , O l e a , 1 9 8 4; W a r r i c k a n d M e y e r s ,

1 9 87 ). F o r m a n y m e a s u r e m e n t s c h e m e s , s a m p l e s a r e c o l le c te d o n a r e g u l a r g r id

o f c o n s t a n t m e s h s i z e A z = Izi - z , ~ ~1. B e c a u s e t h e r e g i o n o f i n t e r e s t a n d

s a m p l i n g b u d g e t a r e l i m i t e d , s a m p l e s a r e t a k e n o n l y f o r d i s t a n c e s u p to n A z ,

w h e r e n is th e m a x i m u m n u m b e r o f s a m p l e s t a k e n i n a g i v e n d i re c t io n . An o r m a l i z e d - ~ ( h ) f o r t h a t d i r e c t i o n c a n b e o b t a i n e d f r o m s o m e e s t i m a t o r s u c h a s

n - I

1 ~ [ k ( z i ) - k ( z i + j A Z ) I ~-~ ' ( J A z ) - 2 N ( j A z ) s ~ ~= I

w h e r e N ( j A z ) is t h e n u m b e r o f p a ir s a t l a g h = j A z a n d s ~ is t h e s a m p l e v a r i a n c e

o f k . T h e r e l a t i o n s h i p o f ~. t o A Z a n d n i n f l u e n c e s "~ (h ). I f th e w a v e l e n g t h X is

l a r g e w i t h r e s p e c t t o t h e s a m p l e s p a n n A z , t h e h o l e e ff e ct w i l l n o t a p p e a r . T h e r ea r e t w o r e a s o n s f o r t h i s: ( 1 ) s a m p l i n g v a r i a b i l i t y ( R u s s o a n d J u r y , 1 9 8 7 ) , o r ( 2 )

j b e i n g l e s s t h a n n / 2 . A t t h e o t h e r e x t r e m e , i f ~. < 2 A z , a h o l e w i l l n o t a p p e a r

a t la g h = X b u t a t t h e l o w e s t i n t e g e r m u l t i p l e o f X t h a t e x c e e d s 2 A z ( F i g . 4 ) .

T h e m e a s u r e m e n t m e t h o d f or o b t a i n i n g k ( z ~ ) a l s o ha s a n e f f e c t upon ~ , ( h ) .

I f t he m e a s u r e m e n t " w i n d o w " is a c o n s i d e r a b l e f r ac t i o n o f X, t he m e a s u r i n g

i n s t r u m e n t i s, i n e f fe c t , a c t i n g a s a l o w - p a s s f i l te r ( L u m l e y a n d P a n o t ~ k y , 1 9 6 4 ,

p . 4 6 - 5 1 ) . T h u s , n o m a t t e r h o w s m a l l zkz i s i n r e l a t i o n t o X , t h e i n s t r u m e n t is

f il te r i ng ( " a v e r a g i n g " ) o u t th e p e r io d i c c o m p o n e n t w i th w a v e l e n g t h X. N o h o l e

w i l l b e o b s e r v e d i n ~ ,(h ) a n d it t y p i c a l l y w i ll h a v e a s m a l l n u g g e t . F o r e x a m p l e ,

c o re p lu g s ( - 2 . 5 c m d i a m e t e r) m a y c o n t a i n s e v er a l l a m i n a t i o n s a n d co u l d n o t

b e u s ed t o d et e ct l a m i n a t i o n - s c a l e s t ru c t u r e . A n e x c e p t i o n t o t h is w o u l d b e i n

c o a r s e - g r a i n e d ( e . g . , c o n g l o m e r a t i c ) i n t e r v a l s , w h e r e t h e g r a i n si z e is s u f f i c i e n tl y

l a rg e t h a t c o r e p l u g s d o n o t a v e r a g e o u t t h e s m a l l - s c a l e v a r i a b i l i t y . I n t h i s

1 . 5 -

1 . 0 -

0 . 5 -

0 0 r

0 1 0

~ 2 4 . . . . O - - - - ~ 1 . 5 . . . . " l ll -- - -

I I

2 0 3 0 4 0

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Figure 4. Sem ivariogramsusing Figure 3 periodic series k(z,I

but sampled at oth er AZ. Sam pling at ,-~ = h/ I.5 is below

Ny quist criterion lbr cyclic corup onen t. For Az _< h/2.4. "rih)

loses evidence of structure because of increased aliasing ef-

fects of no ise comp onent, wh ose upper frequency limil ex-

ceeds cyclic comp onent.




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

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

v o l u m e i s s u e .

E v e n w i t h s u f fi c ie n t ly d e n s e m e a s u r e m e n t s a p p r o p r i a t e t o th e s c a l e s o f

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

s c ur e e v id e n c e o f s tr u c tu r e . C o n v e r s e l y , s a m p l i n g v a r i a b i l i t y c a n c a u s e s e m i -

v a r i o g r a m s to e x h i b i t h o le a n d n u g g e t e f f e c ts w h e r e n o n e a r e j u s t i f ie d . R o b u s t

m e t h o d s h a v e b e e n s u g g e s t e d ( e . g . , C r e s s i e , 1 9 91 , p . 7 4 - 8 3 ) , b u t f u r t h e r t e s t in g

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

f ie l d s. O u r e x p e r i e n c e h a s b e e n t h a t a j a c k k n i f e p r o c e d u r e , d i s c u s s e d in S h a f e r

a n d V a r l j e n (1 9 9 0 ) , is h e l p f u l to a s s e s s s e m i v a r i o g r a m v a r i a b i l i t y . T h i s a p -p r o a c h , h o w e v e r , r e q u i r e s t h a t a t l e a s t t o u r o r fi ve m e a s u r e m e n t s p e r s e d i m e n -

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

S e d i m e n t a r y b o d i e s m a y h a v e s e v e r a l , n e s t e d s t r u c tu r a l f e a tu r e s : o n e a t t h e

l a m i n a t i o n s c a l e (h j~ ,, - m m s ) ; o n e a t t h e b e d s c a l e ( h b ~ - m s ) : a n d a n o t h e r

a t t h e b e d s e t o r p a r a s e q u e n c e s c a l e s (X p~rj - 1 0s m ) . D e p e n d i n g u p o n t h e

m e a s u r e m e n t , t h e la m i n a t i o n c o m p o n e n t m a y b e fi lt e re d o u t an d u n d e r s a m p l i n g

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

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

s c a l e s . F i g u r e 5 ( to p ) s h o w s ~ ,(h ), w i t h e v e r - i n c r e a s i n g v a l u e s , r e f l e c ts t h e s y s -

t e m a t i c c o a r s e n i n g - u p s t r u c t u r e a t t h e p a r a s e q u e n c e s c a l e . W i t h t h e t y p i c a l i n -

d u s t ry p r a c ti c e o f s a m p l i n g w i t h o n e p l u g p e r 3 0 c m ( = A z ) , t h e s m a l l n u g g e t

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

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

L a r g e r s c a l e f e a t u r e s ( 2 0 m > 2~ > 1 m ) a re s a m p l e d a d e q u a t e l y , h o w e v e r ,

w i th f o u r o r m o r e s a m p l e s p e r fe a t u re . T h e h o l e s a t a p p r o x i m a t e l y 2 m a n d 16m s u g g e s t t w o f u r t h e r f e a t u r e s i n t h i s fo r n a a t io n : b e d - a n d b e d s e t - l e v e l s tr u c t u r e .

T h i s s t o m a - d o m i n a t e d s h o r e f a c e c o n s i s ts m o s t l y o f c ro s s - s t ra t i fi e d b e d s o f 1 - 2

m t h i c k n e s s , w h i c h a r e n o t a l w a y s s a m p l e d s u f f i c i e n t l y . T h e 1 6 m h o l e i n d i c a t e s

t ha t t h e re a r e a c t u a ll y tw o c o a r s e n i n g u p w a r d s c y c l e s i n th i s f o r m a t i o n , p e r h a p s

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

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

t h e p e r m e a b i l i t y p r o f i le ( F i g . 5 , t o p r i g h t ) .

M o r e d e t a i l e d e x a m i n a t i o n o f th e s h o r e f a c e d a t a ( F i g . 5 , c e n t e r ) s h o w s t h a t

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

p e r b e d c o m f o r t a b l y e x c e e d s t h e N y q u i s t c r i t e r i o n f o r t h i s i n t e r v a l . P r o b e p e r -

m e a m e t e r m e a s u r e m e n t s ( 4 m m o r i f i c e ) a t A z = 5 m m g i v e ~,(h ) w i t h a d i s t i n c t

h o l e a t hb~d = 1 .2 m , c o n f i r m i n g th e p l u g s e m i v a r i o g r a m h o l e a t t h a t d i s t a n c e .

W e c o u l d n ot d e t e rm i n e a n y g e o l o g i c a l si g n if i ca n c e fo r th e p l u g s e m i v a r i o g r a m

h o l e a t 0 . 6 m . T h e p r o b e " i'(h ) h a s a l a r g e n u g g e t : 3 0 % o f t o ta l v a r i a n c e . T h i s

" ' n o i s e " is t h e r e s ul t o f th e l a m i n a t i o n c y c l i c i t y s u p e r i m p o s e d o n t h e b e d s c a l e



S e m i v a r i o g r a m s , S t r u c t u r e , a n d F l o w P e r f o r m a n c e 4 2 7

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Core Plu~ [ ]000 1 , Co~e~,,~ ''10 . 0 0 0 . 5 0 1 . 0 0 1 . 5 0 1 1 0 I 0 0 1 0 0 0

L a g ( h ) m P e r m e a b i l i t y , m D

1 . 5 0 - 3 0 8 2 . 3 5 -

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Figure 5. Pemleability patterns and nomlalized sample semivariograms for shoreface

Ibnnation. Top: parasequence scale variation shows typical deterministic coarsening up-

ward effccts of grainsize on k and % Center: cyclic bedfonn (hummocky cross-stratif i-

cation) show s hole at approxim ately 1.2 m . B ottom: 2-cm thick laminations in a bed.

s t r u c tu r e ( F i g . 5 , b o t t o m ) . T h e o c c u r r e n c e o f a c o n s i d e r a b l e s i z e n u g g e t w h e n

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

l ey a n d P a n o f s k y , 1 9 6 4 , p . 4 4 ; J o u m e l a n d H u i j b r e g t s , 1 9 7 8 , p . 1 5 2 ) .

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

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

i n c l u d i n g t id a l ( D a w s a n d P r o s s e r , 1 9 9 2 ) , e o l i a n ( G o g g i n a n d o t h e r s , 1 9 8 8 ) ,

s h o r e f a c e ( C o r b e t t a n d J e n s e n , 1 9 9 2 ) , f lu v i a l ( H a r t k a m p - B a k k e r , 1 9 9 3 ) , a n d




submarine fan (Prosser and others, 1995). Tidal channel sandstones may exhibit

clear periodic semivariogram structures (Daws and Prosser, 1992) but insuffi-

cient sampling can make the smaller scale (lamination) periodicity difficult to

identify. In some of these examples, the permeability data and ~, helped the

sedimentologist to recognize the existence of structure, particularly if it occurred

over several lengths of core so that it was not visually obvious. Daws and Prosser

(1992) show that core plugs at l-foot spacing in channels are insufficient and

that "~s derived from plug data lack the detail seen by the probe permeameter.

Corbett and Jensen (1992) illustrate the semivariograms of probe and plug data

for comparable intervals and show that only if the sedimentary structure is

significantly larger than the sample spacing will the sample semivariogramsshow the structure which is there.

Careful sampling at a variety of scales and of representative elements clearly

is needed to reveal the structure of sedimentary rocks (Corbett and Jensen, 1992;

Hartkamp-Bakker, 1993; Brendsdal and Halvorsen, 1993). However, intensive

sampling may not be required throughout the reservoir. Once the lamination

level structure in a small number of prototype beds has been assessed, similar

beds can be assumed to have similar internal structure. A similar approach holds

at the larger levels. Once the structure has been assessed, further sampling needs

only to be done to ensure that the permeability variability and structure have

not departed significantly from the original assessments.

Strategic sampling in structured sediments has another advantage. Bed-scale

features may represent relatively rapid sediment deposition, which then are left

for a considerable period before the next deposit ional episode. During this hia-

tus, fine-grained material may be deposited on the top of the bedforna beiore

the next bed is deposited. These fine-grained, interbed deposits can be over-looked in sampling schemes which tend to sample for large intervals with nu-

merous beds. Once bed-scale structure is recognized, part of the strategic sam-

piing plan can test for these interbed features which, whereas being thin, can

reduce interbed flow considerably (e.g., Corbett, 1993).

INCORPOR ATING STRUCTURE IN GEOS TATI STIC AL MODELS

The sample semivariogram may be used to develop the covariance modelfor use in statistical simulation of permeabil ity fields. Where structure is present

in the form of holes, a limited selection of models is available (Journel and

Huijbregts, 1978, p. 168-171): "),(h) = 1 - sin(h)/h is valid for one-, two-, or

three-dimensional models, whereas --/(h) = 1 - cos(h) can be used for simu-

lating periodicity in one direction. Although such models can be fitted to ~,(h)

without any knowledge of the geological causes of the structure, we encourage

the statistical modeler to determine underlying reasons for the behavior. If a

geological cause can be established, the model will be more reliable and, in the



Sem ivar iograms, Struc ture , and F low Per formance 429

t e rm s o f C r e s s i e ( 1 9 9 1 , p . 1 1 2 - 1 1 4 ) , t h e g e o l o g i c a l i n f o r m a t i o n w i l l h e l p fix

t he " s c a l e s o f v a r i a t i o n " f o r p e r m e a b i l i t y .

I f t he g e o l o g i c a l c a u s e s f o r re p e t i t i v e fe a t u r e s c a n b e d e t e r m i n e d , a n a l t e r -

n a t iv e t o f it ti n g a c o v a r i a n c e m o d e l w i t h p e r i o d i c i t y ( p e r h a p s a t s e v e r a l s c a l e s )

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

o f st r u c t u r e r e v e a l e d i n -~ (h ) m a y p o i n t t o h i g h l y o r d e r e d s y s t e m s w h i c h a r e w e l l

h o n o r e d b y d e t e r m i n i s ti c p e r m e a b i l i t y m o d e l s . A n u m b e r o f s tu d i e s ( e . g . , R i n g -

r o s e a n d o t h e rs , 1 9 93 a a n d 1 9 9 3b ; J o n e s a n d o th e r s , 1 9 93 ; M o i s s i s a n d W h e e l e r ,

1 99 0) h a v e s h o w n t h a t s y s t e m a t i c p e r m e a b i l i t y v a r i a t i o n h a s a s i g n i f i c a n t l y l a r g e r

e f fe c t o n f lo w p r o p e r t i e s th a n r a n d o m , w e a k l y s tr u c t u r e d v a r i a t i o n . W h e r e n e s t ed

s t r u c t u r e s e x i s t , t h e s i m u l a t i o n s m a y n e e d t o b e s t a g e d , p r o d u c i n g p e r m e a b i l i t ym o d e l s f o r e a c h s c a le in s u c c e s s i o n ( e . g . , K o s s a c k , A a s e n , a n d O p d a l , 1 9 90 ).

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

s e d i m e n t (C o r b e t t a n d o th e r s , 1 9 92 ).

T h i s i s n o t to s u g g e s t th a t w e r e c o m m e n d c o m p l e t e a b a n d o n m e n t o f th e

s e m i v a r i o g r a m A s s h o w n in F i g u r e 5, i t c a n b e a u s e fu l i n d i c a t o r o f st r u ct u r e

w h i c h c a n m o t i v a t e fu r t h e r i n v e s t i g a t io n . A l t h o u g h o t h e r m e a s u r e s o f s tr u c tu r e

m a y b e m o r e u s e f u l i n s o m e c i r c u m s t a n c e s ( e . g . . J o u r n e l a n d D e u t s c h , 1 9 9 3 ;

H a r d y , 1 9 92 ), t h e s e m i v a r i o g r a m is a s t r a ig h t f o r w a r d a n d r e l a t i v e l y e a s y s t a t is t ic

t o c a l c u l a t e .

F L O W E F F E C T S F O R S T R U C T U R E D M E D I A

T h e s t r u c t u r e in r e s e r v o i r r o c k w i l l a f f e c t t h e f lo w p e r f o r m a n c e . A s a s i m p l e

e x a m p l e , b a s e d o n c o n c e p t s d e s c r ib e d a b o u t 5 0 y e a r s a g o b y M u s k a t ( 1 9 3 7 , p .

4 0 3 - 4 0 4 ) , c o n s i d e r h o w t h e s i n g l e - p h a s e e f fe c t iv e p e r m e a b i l i t y t~ )r l i n e a r fl o wc h a n g e s a s th e p e r m e a b i l i t y s t ru c t u r e o f a 2 - D d o m a i n is a l t e re d f r o m l a y e r e d

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

r e a l w i t h l o g - m e a n 4 . 6 a n d l o g - v a r i a n c e 3 [ i . e . , I n (k ) - N ( 4 . 6 , 3 ) ] , th e e f f e c t i v e

p e r m e a b i l i t y (k ,.) f o r l a y e r - p a r a l l e l f l o w i s 4 5 0 m D ( th e a r i t h m e t i c m e a n ) ; f o r

c r o s s - l a y e r f l o w , k , . = 2 2 m D ( th e h a r m o n i c m e a n ) ; a n d f o r t h e r a n d o m l y

o r g a n i z e d s i tu a t i o n , k,. = 1 00 m D ( th e g e o m e t r i c m e a n ) . F o r t h e l a y e r e d s i t u -

a t i o n , k , . is s e n s i t i v e t o t h e f l o w d i r e c t i o n r e l a t i v e to t h e l a y e r i n g a n d , a s i n t h i s

e x a m p l e , a n i s o t r o p y r a t io s o f 1 0 : 1 a r e e a s i l y a c h i e v e d . T h e r a n d o m l y d i s tr i b -u t ed p e r m e a b i l i t y s i t u a t io n , o n t h e o t h e r h a n d , i s i s o t r o p i c .

T h e i m p o r t a n c e o f s i n g l e - p h a s e a n i s o t r o p y c an h a v e c o n s i d e r a b l e e c o n o m i c

i m p a c t i n re s e r v o i r e x p l o i t a t i o n . A t th e l a m i n a t i o n an d b e d s c a l e s , h o r i z o n t a l

w e l l s a r e se n s i t iv e t o t h e lo c a l a n i s o t r o p y ( e . g . , O ' B y r n e , F l e m i n g , a n d P r e n t i c e,

1 99 1; S h e r r a r d , 1 9 9 5) . A t th e l a r g e r s c a l e s r e p r e s e n t e d i n f ie l d s i m u l a t i o n m o d e l s ,

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

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

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



430 Jensen, Corbett, Pickup, and Ringrose

reservoirs did not have bed- and lam inat io n-sca le s t ructure and on ly had ran-

domly d i s t ribu ted pe rm eabi l i ty va r ia t ions , the ve r t i ca l- to -hor izonta l pe rm eabi l i ty

ra t io would not be such an important parameter in many f low models and there-

fore would not require such deta i led s tudy (e .g . , Cor bet t , 1993; Sherrard, 1995).

There fore , the deve lopment of mode l s , e i the r based on the semivar iogram or

determinis t ica l ly , must be guided careful ly to ensure the appropria te scales and

levels of v ar iabi l i ty are incorporated.

The effec ts of smal l -scale laminated rock s t ructure are par t icular ly s igni f i -

cant for the f low of immiscible f luids , such as water displacing oi l , because the

capil lary forces at the f luid interfaces are sensi t ive to the local organizat ion of

pore throats , The effec ts of smal l -scale sediment archi tec ture on waterf loodbehavior have been desc r ibed ex tens ive ly e l sewhere (Kor tekaas , 1985; Corbe t t

and others , 1992: Ring rose and others , 1993a). Th e impo rtance of inc luding

s t ructure in models used in immiscible f low calcula t ions i s i l lus t ra ted here by a

s imple 2-D waterf lood in two permeabi l i ty f ie lds . The waterf lood perfomaance

depends s t rongly on the length-scale modeled, as capi l lary forces scale inverse ly

with the sys tem s ize . We therefore have se lec ted a length-scale which might

represent sedimentary bedding.

Figure 6 shows the s t ructured p erm eab i l i ty fie ld wi th a 0 .25 cm (h or izonta l )

by 1 cm (ver t ica l) gr id . T his f ield is based on a geom etr ica l ly s impl i f ied cross-

bedded system (i .e . , layers vert ical rather than incl ined) with a toe-set at the

bot tom. Other studies (e .g . , Ko rtekaas , 1985; Ringrose and others , 1993a) have

shown that th is geometr ica l approximat ion of cross-bed archi tec ture i s an ac-

ceptable s impl i f ica t ion for the purposes of f low model ing. The permeabi l i ty in

the cross-beds di f fers l inear ly f rom top to bot tom as usual ly i s observed, and

the permeabi l i ty contras t i s kept a t a mode s t 10:1 between the foreset layers .The f ie ld wi th low s t ructure was der ived f rom Figure 6 by exchanging perme-

abi l i t ies (us ing a random num ber genera tor) a long a l l hor izonta l l ines ( rows) .

This preserved the top- to-bot tom grading and the toe-se t , but caused the deg-

radat ion of the cross-bed layers . Us ing 10,000 cel l swaps , the st ructure of the

Figure 6. Cross-beddedpermeabilitymodel with bed-scale structure.




Figure 7. Oil recovery performances for the fieldof Figure 6 and a field without the cross bedding.

5 0 . 0 .

- ~ 4 0 . 0

30.012 0 . 0 -

to.o-

0 . 0

0 . 0

,...Y

f

- - C ro s s - b ~ d d e d f i e l d

. . . . . . . . . . . R a n d o m L z e d f i e l d

o 1 2 o i ~ o 1 6

P o r e v o l u m e s i n j e c t e d

0 . 8

smaller scale cross-bedding was thus erased while preserving the larger scale

structure and the permeability histogram for the two fields.

The flow direction tbr both fields was horizontal with no-flow boundaries

at the top and bottom. The frontal advance rate, 0.25 m per day, is a typical

field rate and the oil and water viscosities were 5 and 1 cp, respectively. Thecapillary pressure and relative permeability functions have been reported else-

where (Ringrose and others, 1993a). Figure 7 shows the oil recovery behavior

for the two models; the less structured model gives a considerably better recov-

ery performance. The water cut behavior was similar tbr both fields.

The "perfect" repetitive structure of the Figure 6 mode[ is seldom observed

in practice because adjacent beds may differ in thickness. We theretbre simulated

the recovery of a model with different layer thicknesses, with one layer of 5

cm, one of 3 cm, and one of 7 cm. The recovery performance was indistin-

guishable from the perfectly repetitive model.

This illustration of bed scale flow performance, where viscous and capillary

lbrces are competing, is just one of several flow dynamical effects that could

result from structure in geostatistical permeability fields. At the bed/bedset scales,

viscous-gravitational force competition would be present and could be affected

by the structure. The precise effects of the fluid-structure interaction depend

upon several factors, including the rock wettabil ity, geometry, level of hetero-geneity, and fluid properties, and need to be assessed on a case-by-case basis.

New methods of analysis and measurements (e.g., the probe permeameter

and high-resolution wireline resistivity and acoustic imaging tools) in elastic

reservoirs are leading to a better perception of the structured nature of sediments.

These structures, at several scales, can be incorporated into flow models to

interpret flow performance from core plug-sized volumes (e.g., Honarpour and

others, 1995), through the wellbore scale (Sherrard, 1995), right up to the full

field (Kossack, Aasen, and Opdal, 1990; Corbett e t a l . , 1993). If this structure




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

b e t t e r ( C o r b e t t a n d J e n s e n , 1 9 9 2: C o r b e t t a n d o t h e r s , 1 9 9 2 ).

C O N C L U S I O N S

W e h a v e o b s e r v e d t h at , a l t h o u g h c o n s i d e r a b l e a m o u n t s o f s t ru c t u re e x i s t

in e l a s ti c s e d i m e n t s , a n u m b e r o f m o d e l i n g s t u d i e s h a v e n e g l e c t e d it. P u b l i s h e d

p e r m e a b i l i t y s e m i v a r i o g r a m s s h o w c l e a r s ig n s o f t h is s tr u c t u re b u t th e c o v a r i a n c e

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

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

a n d d e p o s i t i o n m o d e s in t h e s e d i m e n t a r y e n v i r o n m e n t . S t r u c tu r e w i ll af fe c t t h e

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

m e n t a t i o n . A s i m p l e 2 -D e x a m p l e s h o w s b e d - s c a l e st ru c t u re e a s i l y c a n a l t e r

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

A w a r e n e s s o f st ru c t u r e i n s e d i m e n t s c a n l e a d t o m o r e e f f e c t iv e s a m p l i n g

p r o g r a m s , b e t t e r i n t e rp r e t a ti o n o f m e a s u r e m e n t s t a ti s ti c s , a n d b e t t e r fl ow m o d e l s .

T h e i m p r o v e m e n t is b a s e d o n t h e r e c o g n i t i o n t ha t th e g e o l o g i c a l c h a r a c t e r o f

t h e r e s e r v o i r is a n i m p o r t a n t e l e m e n t in th e d a t a c o l l e c t i o n , a n a l y s i s , a n d d e -v e l o p m e n t o f f lo w m o d e l s . E x p l o i t i n g g e o l o g i c a l i n f i~ r m a ti on l e a d s t o m o r e

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

A C K N O W L E D G M E N T S

W e t h a n k c o l l e a g u e s i n t h e E d i n b u r g h R e s e r v o i r D e s c r i p t i o n G r o u p f o r

t h e ir s u p p o r t a n d w e t h a n k t h e r e v i e w e r s f o r h e lp f u l c o m m e n t s . T h i s s t u d y isp a rt o f t he R e s e r v o i r H e t e r o g e n e i t y P r o j e c t t h n d e d b y A m e r a d a H e s s , B o w

V a l l e y , B r it is h G a s , C h e v r o n , C o n o c o , D e m i n e x , E l f, E x x o n , M o b i l , S h e ll ,

a n d t h e U . K . D e p a r t m e n t o f T r a d e a n d I n d u s t r y . I n t er a I n f o r m a t i o n T e c h n o l o -

g i e s i s t h a n k e d f o r u s e o f t he E c l i p s e f lo w s i m u l a t o r . P . W . M . C o r b e t t ' s p o s t

i s f u n d e d b y th e E l f G e o s c i c n c e R e s e a r c h C e n t e r .

R E F E R E NC E S

A l l e n , J . R . L . . 1 9 8 5. P r i n c i p l e s o f p h y s i c a l s e d i m e n t o l o g y : A l l e n & U n w i n , L o n d o n , 2 7 2 p .

B a l l , L . D . , C o r b e t t . P . W . M . , J e n s e n , J , L . . a n d L e w i s , J . M . , 1 9 9 4 , T h e r o l e o f g e o l o g y i n

t h e b e h a v i o r a n d c h o i c e o f p e r m e a b i l i ty p r e d ic t o rs : S P E p a p e r 2 8 4 7 7 i n T r a n s a c . 6 9 t h A n n ,

T e c h . C o n f . & E x h i b . S o c . P e t. E n g . ( N e w O r l e a n s ) p . 8 6 7 - 8 7 9 .

B a r n e s , R . J ., 1 9 8 8 , B o u n d i n g th e r e q u i r e d s a m p l e s i z e [ o r g e o l o g i c a l s i t e c h a r a c t e r i z a t i o n : M a t h

G e o l o g , , , v . 2 0 , n o . 5 . p , 4 7 7 - 4 9 0 .

B e a r d , D . C . . a n d W e y l . P K . , 1 9 7 3 , I n f l u e n c e o f t e x t u r e o n p o r o s i t y a n d p e m l e a b i l i t y o f u n c o n -

s o l i d a t e d s a n d s t o n e , A m . A s s c ,c . P e t r n l e u m G e o l o g i s t s B u l l . . v . 5 7 , n o . 2 , p . 3 4 9 - 3 6 9 .




B r a s , R . L . , a n d R o d r f g u e z - l t u r b e , I . , 1 9 8 5 , R a n d o m f u n c t i o n s a n d h y dr c~ l og y : A d d i s o n - W e s l e y ,

R e a d i n g . M a s s a c h u s e t t s , 5 5 7 p .

B r a y s h a w , A . C . , D a v i e s , G . W . , a n d C o r b e t t , P . W . M . . 1 9 9 5 . D e p o s it i o n al c o n t ro l s t~n p r i m a r y

p e n a l e a b i li t y a n d p o r o s i t y a t t h e b ed l't ~r m s c a l e i n f u v i a l r e s e r v o i r s a n d s t o n e s , i n D a w s o n . M . ,e d . , A d v a n c e , ,, i n fl u v i a l d y n a m i c s a n d s t r , t i g r a p h y : J ~ h n W i l e y & S ~ ms , C h i c h e s t e r , i n p re s~ ,.

B r e n d s d a l , A . , a n d H a l v o r s e n , C . . 1 9 9 2 . Q u a n t i f i c a ti o n o 1 p e m l e a b i l i t y v a r i a t i o n s a c r o s s th in l a m -

i n ae i n c r o s s b e d d e d s t a n d s t o n e , i n W o r t h i n g t o n , P . F . , a n d C h a r d a i r e - R i v 6 r e . c d s . . A d v an c c~ ,

i n c o r e e v a l u a t i o n 1 1 1: R e s e rv t ~ ir M a n a g e m e n t : G o r d c m a n d B r e a c h S c i e n c e P u b l . , P a n s . p .

2 5 - 4 2 .

B r o o k f ie l d , M . E . , 1 9 7 7 . T h e o r i g in o f b o u n d i n g s u r f a c e s i n a n c i e n t a c t d i a n s a n d s to n e s : S c d i m e n -

t o l o g y , v . 2 4 , n o . 3 . p . 3 0 3 - 3 3 2 .

C o r b e t t , P . W . M . , 1 9 9 3 . R e s e r v o i r c h a r a c t c r i s a l i ~ n ~1" a l a m i n a t e d ~ c d i m e n t : u n p u b l d o c t o r a l

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