C a e l q p c e u u a r r i l m v a e P R e n h - - PDF document

SLIDE 1 Advancesi W a ? e R e s

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2 5 3 1 ~ 1 9 E l s e S c i L A l r i g h r e s e r P r i n i G r B r P I I : S 3 9

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SLIDE 2 2 5 4 P h . R e n a r d G d M a r s i s u p p

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

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d i t T g l

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SLIDE 3 C a l c u l a t i n g e q u i v a l e n p e r m e a b i l i 2 ( i i ) S u f f i c i e n t c

• n

d i t i

• n

: L e t z = l / u ~ U e ( x ) d x b e t h e d i s s i p a t e d e n e r g y i n t h e m a c r

• m

e s h w a n d Z= – g r a d H . K b g r a d H t h e e n e r g y d e f i n e d f r

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t h e m a c r

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c

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i c m a g n i t u d e s ; t h e n , t h e c

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d i t i

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s : E ( z ) = E ( e ) , C ; ( r ) = C , ( r ) ( 5 i m p l y t h e n e c e s s a r y c

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d i t i

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. C r e p r e s e n t s t h e c

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a r i a n c e . N

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e t h a t i f t h e b l

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k

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e r w h i c h t h e u p

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c a l i n g i d

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e i s l a r g e e n

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g h , t h e u p

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

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a r d t h e e f f e c t i v e p e r m e a b i l i t y i f i t e x i s t s . l i l l i l ~ K b = K , f ( 6 I t i s m

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t i m p

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k p e r m e a b i l i t y i s n

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u n i q u e . I t d e p e n d s

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t h e b

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n d a r y c

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d i t i

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s . F

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e x a m p l e , f

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t h e s t r a t i f i e d m e d i u m i F i g . 1 , u s i n g t h e c r i t e r i

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9.05 K ~ e r m e a m e t e r

• 9

“ 5 T h u s , b l

• c

k p e r m e a b i l i t y i s n

• t

a n i n t r i n s i c c h a r a c t e r

• i

s t i c

• f

t h e m e d i u m a s

• p

p

• s

e d t

• e

f f e c t i v e p e r m e a b i l i t y . M

• r

e

• v

e r , s u b j e c t t

• c

e r t a i n b

• u

n d a r y c

• n

d i t i

• n

s f

• c

e r t a i n m e d i a , i t h a s b e e n s h

• w

n t h a t t h e b l

• c

k p e r m e a b i l i t y t e n s

• r

i s n

• n
• s

y m m e t r i c . 4 ’ 8 5 ’ 1 1 * ’ 1 1 5 W e h a v e d i v i d e d t h e m e t h

• d

s f

• r

c a l c u l a t i n g t h e e f f e c t i v e

• r

b l

• c

k p e r m e a b i l i t y i n t

• t

h r e e g r

• u

p s : d e t e r m i n i s t i c , s t

• c

h a s t i c a n d h e u r i s t i c . T h e f i r s t

• n

e s F i g . 1 . R e s u l t s

• f

a s i m u l a t i

• n
• n

a m e d i u m s t r a t i f i e d a t 4 5 ° i r e l a t i

• n

t

• t

h e x

• a

x i s w i t h u n i f

• r

m b

• u

n d a r y c

• n

d i t i

• n

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

• n

g t h e s i d e s ) . T h e d a r k b a n d s h a v e a p e r m e a b i l i t y

• f

1m D a n d t h e l i g h t

• n

e s a p e r m e a b i l i t y

• f

I O O m D . T h e l i n e s r e p r e s e n t t h e h y d r a u l i c h e a d c

• n

t

• u

r s a n d t h e a r r

• w

s t h e f i l t r a t i

• n

v e l

• c

i t y . a s s u m t h a t h g e

• l
• g

i m

• d

i p e r f k n a

• p

p

• s

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• c

h a s t e c h n t h

• n

a s a a p p r

• x

i m a k n

• w

l e

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h m

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a b i l i s t v i e T h h e u r i m e t p r

• r

f

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a l c u l a t i p l a u s i e q u i v a p e r m e a F

• e

a c g r

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• d

i f f c a l c t e c h n i q u e c a b u s e a n a l y t

• n

u m e e x

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p p r

• x

i m a t e A f i n c r i t e

• c

l a s s i i t w h i c d i s t i n g u i s b e t w l

• c

a n

• n
• m

e I t h n

• n
• l
• c

m e t h

• t

h p e r m e a

• a b

d e p e n d b

• t
• t

h i n t e r a t e x t p e a b i l i t v a l u e I t h l

• c

m e t h i o n d e

• t

h e l e m e n t a p e r m e a b i v a l i n s t b l M

• t

h m e t h

• p

r e s e n h e a r l

• n

T h a r t i c i l a i

• u

a f

• l

l S e c 2 i d e t t h b

• u

n d

• t

h e q u i v a p e r m e a w h a t h e u s e i S e c t i 3 i d e a l w i h e u r m e t S e c t i

• n

4 a n 5 d e a r e s p e c t w i d e t e r a s t

• c

h a s t i t e c h n i q u S e c t 6 c

• m

p t d i f t e c h n i q u e a n

• f

f e s

• r

e c

• m

m e n 2 I N E Q U A L I T I F O T H E Q U I V P E R M E A B I L I 2 . W i e n e b

• u

n T h i i n e q u a l i i a l c a l l t h f u n d a i n e b e c a u s i i a l w a v a l i I h a b e d e m

• n

b a g r e a n u m b e

• a

u t h

• e

. W i e n e C a r a P a r s

• n

s 1 M a t h e r

• n

a n D a g * a m

• t

p h < K e < P ( w i t P = h a r m

• n

m e a n P = a r i t m e 2 . H a s h i a n S h t r i k b

• u

T h e a r u s e f

• i

s

• t

r

• b

i n m e d ( w h e r ~ a n ~ a r t h f r a c t

• t

p e r m p h a s e k a n k , kl is higher than k. and ~ a = ~

• k
• +

~ l k . T h d e m

• n

s t r a t

• H

a s a S h t r i b a s e

• a m
• d
• t

h m e d b u

• c
• m

s p h e r e ( s e F i g 3 ) E a c

• m

p

• s

p h i d e b a i s

• t

r

• p

s p h e w i a c

• n

s p e r m e K a a i s

• t

r

• p

c

• n

c e n t s h e w i a K O p e r m B u s i n t h m e t h

• s

e l f

• c
• n

m e ( S e c t i

• 4

. 3 i i p

• s

s i t c a l c u a a p p r v a l u f

• t

h p e r m e a b i

• t

h m e d T m a

SLIDE 4 2 5 6 P h . R e n a r d G d M a r s i , m x I m x , , . 0.1 0.2 LM a4 ~: 0.6 0.7 0.8 0.9 , ( a ) ( b F i g . 2 . B

• u

n d s f

• r

t h e e q u i v a l e n t p e r m e a b i l i t y i n a b i n a r y m e d i u ( k = 1 k = 1 ( a T h a r i t h m g e

• m

a h a m e a n s a r e p l

• t

t e d v e r s u s t h e p r

• p
• r

t i

• n
• f

p h a s e z e r

• i

n t h m e d i u m ( b t h H a s h a n S h t r i b

• u

a p l

• v

e t p r

• p
• r

t i

• n
• p

h a s p e r m e a b i l i t y i s

• b

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

• w
• p

e r m e a b i l i t y m e d i u m a n d t h e s h e l l s a r e t h h i g h

• p

e r m e a b i l i t y

• n

e ( K i n = k

• ,

I & = k l ) . T h e m i n i m u m p e r m e a b i l i t y i s

• b

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

• n

d u c t i v e s p h e r e s a n d r e s i s t a n t s h e l l ( K i n = k l , K O U , = k O ) . A m

• r

e r i g

• r
• u

s d e m

• n

s t r a t i

• n

, b a s e d

• n

a v a r i a t i

• n

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

• m

p t s t h e a u t h

• r

s t

• c
• n

c l u d e t h a t t h e s b

• u

n d s a r e t h e b e s t p

• s

s i b l e i n t e r m s

• f

v

• l

u m e t r i c f r a c t i

• n

s . s 4 B u t , F i g . 2 s h

• w

s t h a t t h e r e i s n

• l

a r g e d i f f e r e n c e s b e t w e e n W i e n e r ( F i g . 2 a ) a n d H a s h i n a n d S h t r i k m a n ( F i g . 2 b ) b

• u

n d s . 2 . 3 C a r d w e l l a n d P a r s

• n

s b

• u

n d s C a r d w e l l a n d P a r s

• n

s ’ G u s e d a n e l e c t r i c a n a l

• g

y a n d L L

• c

’ h 7 4 u s e d a v a r i a t i

• n

a l m e t h

• d

t

• s

h

• w

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

• n

i s b

• u

n d e d b ( F i g . 4 ) : t h e a r i t h m e t i c m e a n

• f

t h e h a r m

• n

i c m e a n s

• t

h p

• i

n t p e r m e a b i l i t i e s , c a l c u l a t e d

• n

e a c h c e l l l i n p a r a l l e l t

• t

h e g i v e n d i r e c t i

• n

( l

• w

e r b

• u

n d ) ; t h e h a r m

• n

i c m e a n

• f

t h e a r i t h m e t i c m e a n s

• t

h p

• i

n t p e r m e a b i l i t i e s c a l c u l a t e d

• v

e r e a c h s l i c e

• a

c e l l p e r p e n d i c u l a r t

• t

h e g i v e n d i r e c t i

• n

( u p p e r b

• u

n d ) . F i g . 3 . A s s e m b l a g e

• f

c

• m

p

• s

i t e s p h e r e s ( f r

• m

H a s h i n s 4 ) . . . z e r i t h m e d i u K = p ; ( p j ( p < K : 2 . M a t h e r b

• u

n I t h c a s

• a

i s

• t

r t w

• d

i m m

• s

a i w i t t w p h a s M a t h

• b

t a ~ > 0 . 5 K e ~ K , ~ < 5 K , < K , . f = + K e = m w i t [ K , = ; ( f – f

• )

– / t + ( . – . f

• )

– k

• + 4

k

1

( r a ( N

• t

t h a t h i r e l a t r e m a v a i t m

• i

i n v a r i a b a 9 r

• t

a t F u r t h e M a t d e m

• n

s t r a t t h a – k

• k

,

. A + . + ( ) – k

( , f

• k

+ , m J k { ) – k w i t m = , f k + , f

• (

P , = . f k + . f

• k

c r = , f ; (

• k

( F i g u r 5 i l l u s t r t h t d i f f b e t m a x i m u a n t h m i n i b

• u
• M

a t i m u c s m a l l t h t h d i f f e b e t W i

• H

a s h i a n S h t r i k m a x i a m i n

SLIDE 5 C a l c u l a t i n g e q u i v a l e n p e r m e a b i l i m l & “ “ ” ” ” : ; ? p ; ’ ’ m r d w e ’ i \ x + ‘m7 Jo C a r d M F L O W D I R E C T I O N P ( P : K ? P ( P ( G ) F i g . 4 . C a l c u l a t i

• n
• f

C a r d w e l l a n d P a r s

• n

s ’ b

• u

n d f

• f

l

• i t

h x

• d

i r e c 2 . 5 E n e b

• u

n d s E n e 3 5 u s e s t h e m e t h

• d
• f

h

• m
• g

e n i z a t i

• n

( s e e S e c t i

• n

4 . 7 ) a n d

• b

t a i n s b

• u

n d s t h a t g e n e r a l i z e t h

• s

e

• f

R u b i n

• s

t e i n a n d T

• r

q u a t

• 9

9 i n t h e a n i s

• t

r

• p

i c c a s e . T h e s e b

• u

n d s a r e w r i t t e n a s a s e t

• f

d i f f e r e n t i a l e q u a t i

• n

s t h a t r e q u i r e a n u m e r i c a l s

• l

u t i

• n

f

• r

t h e g e n e r a l c a s e . 3 H E U R I S T I C M E T H O D S 3 . 1 S a m p l i n g T h e f i r s t t e c h n i q u e i s s i m p l y n

• t

t

• c

h a n g e s c a l e s . A b l

• c

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

• m

m

• n

l y u s e d i n t h e p e t r

• l

e u m i n d u s t r y a n d c

• n

s i s t s i n p a s s i n g f r

• m

a m e a s u r e m e n t a t h e 1 c m s c a l e t

• a b

l

• c

k

• n

t h e m e t e r s c a l e . 3 . 2 A v e r a g i n g m e a n s T h e g e n e r a l i d e a i s t

• t

a k e a v a l u e b e t w e e n t w

• t

h e

• r

e t i c a l b

• u

n d s .

S

• “

0.1 0 . 2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 f

• F

i g . 5 . M a t h e r

• n

b

• u

n d s f

• r

t h e e q u i v a l e n t p e r m e a b i l i t y i n a b i n a r y m e d i u m . ( k O = 1 , k , = 1 ) a s a f u n c t i

• n
• f

t h e p r

• p
• r

t i

• n
• f

p h a s e z e r

• i

n t h e m e d i u m K m a n d K , c a r e t h e b

• u

n d s d e f i n e d i n e q n s ( 1 ) a n d ( 1 1 ) . 2 3 . 2 . F u n d a m e n t b

• u

n M a t h e r

• n

7 s u g g e s a f

• r

m w h t e f f p e r m e a b i l i i a w e i g h a v e r

• W

i e b

• K

. = p ~ ~ ~

• w

i a E [ 1 ( I t h m e d i u i s t a t i s t i h

• m
• g

a i t r

• p

i c t h e x p

• n

e a i g i v b t e q u a D – Q = D ( T h i f

• r

m u i r i g

• r
• e

x a i t c

• a

f a c t

• r

i z e p e r m e a b i l i t i .

• t

f

• K

( x = k ( x ) k ~ ( x . . .k ~ ( I t h c a s

• a

a n i s

• t

r b s t a t i s h

• g

e n e

• u

m e d i u A b a b t a k @ = ( V : i ) ( f l ; ~ = ( – l h / l i ) ( w h e r / i t h c

• r

r e l a t l e n i t r e l d i r a n l i t h h a r m

• n

m e

• t

h c

• r

r e l e i t h p r i n c i p d i r e c t i

• a

n i s

• t

3 . 2 . C a r d w e a n P a r s b

• u

A c c

• r

d i n t K r u e l

• R
• m

e G u e r i l p r

• t

w t a k t h g e

• m

e t m e

• t

t C a r a P a r s

• n

b

• u

n d a n L e m

• u

g e n e r t h i b p r

• p
• s

i n g f

• t

h r e e

• d

i m e n s m e d

SLIDE 6 2 5 8 P h . R e n a r d G d M a r s i i n f l u e n c e

• f

a n i s

• t

r

• p

y . I n t h r e e d i m e n s i

• n

s , h e w r i t e s ( @ , , @ : 3 + d , 2 d , , 3 ) K ( 1

• d

, 3

• @

, 3 ) K ( l

• @

y 2 ) ~ , 3 K ( l

• @

, , )

• ,

> , ( 1 ( $ K ; x = K 1 2 3 4 w i t h

$ “ 2 = a

r c t a n ~ 1 9 Z 2 = a r c t a n @ j r r / 2 7 r / 2 O J > 3 = O Y 2 ( 1 – o z ~ ) 9 , 3 = 6 , 2 ( 1 – q 2 ) 1 – o @ z 2 1 – o y 2 e z 2 a j a n d a . r e p r e s e n t t h e a n i s

• t

r

• p

y f a c t

• r

s t h a t a r e d u t t h e f l a t t e n i n g

• f

t h e g r i d a n d t

• t

h e a n i s

• t

r

• p

y

• t

h l

• c

a l p e r m e a b i l i t i e s ( w h i c h i s a s s u m e d c

• n

s t a n t

• v

e r a l t h e m e s h e s ) . 3 . 3 P

• w

e r a v e r a g e J

• u

r n e l e t a l . 6 p r

• p
• s

e t h a t K e f b e e q u a l t

• a p
• w

e a v e r a g e (

• r

a v e r a g e

• f
• r

d e r p ) w i t h a n e x p

• n

e n t p i t h i n t e r v a l b e t w e e n – 1 a n d + 1 , d e p e n d i n g

• n

t h e s p a t i a d i s t r i b u t i

• n
• f

t h e p e r m e a b i l i t i e s . f l f ’ = ’ k p ) ’ ’ ’ = ( w p d v ) ” p ( 1 7 N

• t

e t h a t p = – 1 c

• r

r e s p

• n

d s t

• t

h e h a r m

• n

i c m e a n l i m p + O ~ P t

• t

h e g e

• m

e t r i c m e a n a n d p = 1 t

• t

h a r i t h m e t i c m e a n . F

• r

s t a t i s t i c a l l y h

• m
• g

e n e

• u

s a n i s

• t

r

• p

i c m e d i a ,

• n

e

• b

t a i n s : g g p =

1 –

( 1 8 T h i s v a l u e f

• r

t h e e x p

• n

e n t p c

• r

r e s p

• n

d s t

• M

a t h e r

• n

’ s c

• n

j e c t u r e a n d i s d i s c u s s e d i n m

• r

e d e t a i l s f u r t h e r ( s e S e c t i

• n

5 . 2 ) . I n t h e c a s e

• f

a l

• g
• n
• r

m a l m e d i u m , A b a b

• u

e a Z .

• b

s e r v e t h a t : F

• r

a b i n a r y m e d i u m , K e ~ = [ , f i k ; +f l k ~ ] l ’ p e q n ( 1 7 ) b e c

• m

e s : ( 1 9 D e u t s c h 2 7 r e c

• m

m e n d s t h e u s e

• f

t h i s t y p e

• e

m p i r i c a l f

• r

m u l a f

• r

b l

• c

k p e r m e a b i l i t y ; t h e e x p

• n

e n t p i s

• b

t a i n e d b y f i t t i n g . 3 . 4 F l e x i b l e g r i d O n e d i f f i c u l t y c

• m

m

• n

t

• a

l l m e t h

• d

s i n v

• l

v i n g b l

• c

p e r m e a b i l i t y i s t h a t t h e m

• r

e t h e p e r m e a b i l i t y v a r i a b i l i t y i n c r e a s e s i n t h e v

• l

u m e u s e d f

• r

t h e a v e r a g i n g , t h g r e a t e r i s t h e u n c e r t a i n t y

• n

t h e u p

• s

c a l e d p e r m e a b i l i t y . O n e c a n m i n i m i z e t h e e r r

• r

b y a u t

• m

a t i c a l l y a d j u s t i n g t h l

• c

a s i z

• t

h l a r g e

• g

r a a f u n

• f

e x a m p l t h p e r m e a b i v a r i

• t

s m a g r i d T h i a p p r

• w

a f i r u s b G a e a a s e e m v e r p

• p

u l i n f

• e

x a m t p e t i n d u s t r R e c e n t D u r l

• e

a l T a J

• u

r n e 1 1 a n Y a m a 1 h a s h e x a w t h i t e c h n i q w a a p p l t m u l t i f l T a r s e v e r t y p

• f

l e x g r i g r w h t g e

• m

e t d e p e n

• n
• t

p e r m e f a t h

• s

t h a t a k i n a c c

• t

f l c

• n

d S a r d e f

• r

m r e g u g r i

• t

h a m

• l

n e m e s h e s . 4 D E T E R M I N I S M E T I t h d e t e r m i n i m e t h t p e r m f K ( x , y z a n t h b

• u

n d c

• n

d i a a s t b k n

• w

F

• a s

u f f i c i s i m p e r m f ( e . g a s t r a t i f m e d i u e x a a n a l s

• l

c b f

• u

n F

• m
• g

e n e c a s t h a t h ( p e r c

• l

a t i

• e

f f e c t m e d i s t r e a r e n

• i
• n

t h a c a b u s t m a a p p r

• x

c a l c w i t v a r y i p r e c i s A a p p r w h i i p r i n c i p l m

• r

g e n e c

• n

s i s

• l

n u m t h d i f f u s i e q u a t i F i n a t h

• m
• g

t h e

• r

y t h t h e

• a

v e r a w i c l

• a

t m e t h

• m
• m

e n r e p l t d i s

• b

e q u i v a l e p r

• b

l e t h a l r e q n u m s

• l

T h l a s t h r t h e

• r

a r c l a s s a h

• m
• e

q u a t i m e t h

• d

4 . A n a l y t i c s

• l

u t

• t

h d i f f e q u T h d i f f u s i e q u a t ( d i v [ K g = O c b s

• a

n a l y t i c a l f

• u

n i f f l i s t r a m e I t b e s k n

• w

r e s u l a r t h t e q u i p e r m f

• a f

l

• p

a r a l t t h s t r i t a r i t m

• t

h p

• i

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

• m

e y , T h r e c b u s e w h e t h p

• i

p e r m e a b a s c W t h l

• c

a p e r m e a b i l i a r t e n s

• Q

u i a W h i t a k e r s h

• t

h a f

• a s

t r a t m e c

• t

w p h a s w i p e r m e a b K a K t e f f e c t i p e r m e a b i t e n i e x p r b K e f = f O K + . f I w h e r e f = V

• V

t i t h f r a c

• t

m e w K p e r m e a b i l i a n A = V 1 / V i t f r a

• t

m e d i u w i t K p e r m e a b i T h r e i o e x

• t

h u s

• t

h t h e

• l

a s c a v e r N t h a t h s a m r e s u w a

• b

t a b K a a L a

SLIDE 7 C a l c u l a t i n g e q u i v a l e n p e r m e a b i l i 2 T a b l e 1 . P e r c

• l

a t i

• n

t h e

• r

y , v a l u e

• f

t h e c

• e

f f i c i e n t s i e q ( 2 1 T y p e

• f

m e d i u m A n P R e f e 2 D i s

• t

r

• p

i c . 1 1 2 D D e p e n d s

• n

m e d i u m D e p e n d

• m

e d i 1 . 2 3 D D e p e n d s

• n

m e d i u m D e p e n d

• m

e d i 1 . 2 3 D D e p e n d s

• n

m e d i u m D e p e n d

• m

e d i 1 4 w i t h a g r

• u

p i n g

• f
• n

l y t w

• b

l

• c

k s w i t h k n

• w

n p e r m e a b i l i t i e s . K a s a p a n d L a k e 6 3 s u g g e s t t h e u s e

• t

h i s r e s u l t a n d

• f

t h e r

• t

a t i

• n

f

• r

m u l a s ( c h a n g e

• r

e f e r e n c e a x i s ) f

• r

a n a l y t i c a l c a l c u l a t i

• n

s

• f

b l

• c

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

• n

s . 4 . 2 P e r c

• l

a t i

• n

t h e

• r

y T h e p e r c

• l

a t i

• n

t h e

• r

y d e a l s s t a t i s t i c a l l y w i t h t h e p r

• b

l e m

• f

c

• m

m u n i c a t i

• n

a c r

• s

s c

• m

p l e x s y s t e m s c

• n

s t i t u t e d b y

• b

j e c t s t h a t m a y

• r

m a y n

• t

b c

• n

n e c t e d . A c l a s s i c a l e x a m p l e z l i s t h a t

• f

a s e a d

• t

t e d w i t h i s l a n d s . I f t h e s e a l e v e l r e c e d e s , t h e i s l a n d s b e c

• m

e l a r g e r a n d g r a d u a l l y , s

• m

e

• f

t h e m a r e s t r u n g t

• g

e t h e r b c

• a

s t a l b a r r i e r s . I f t h e s e a c

• n

t i n u e s t

• s

h r i n k , a c

• n

t i n e n t w i l l e m e r g e a t a c r i t i c a l p

• i

n t i n t h e s e a l e v e l . T h e r e i s t h e n a c

• n

t i n u

• u

s p a t h w a y t h r

• u

g h t h e n e t w

• r

k

• f

c

• a

s t a l b a r r i e r s b y w h i c h a l l t h e i s l a n d s a r e l i n k e d . T h i s t r a n s i t i

• n

i s c a l l e d p e r c

• l

a t i

• n

t r a n s i t i

• n

a n d t h e c r i t i c a l p

• i

n t i s c a l l e d p e r c

• l

a t i

• n

t h r e s h

• l

d . T h i s t y p e

• f

t r a n s i t i

• n

a p p e a r s i n a g r e a t v a r i e t y

• p

r

• b

l e m s : v i s c

• s

i t y

• f

p

• l

y m e r m a t e r i a l s , c

• n

d u c t i

• n

i h e t e r

• g

e n e

• u

s m e d i a , d i f f u s i

• n
• f

i n f

• r

m a t i

• n
• r
• a v

i r u s i n a p

• p

u l a t i

• n

, e t c . M

• r

e s p e c i f i c a l l y , G u y

• n

e t a l . , 4 g a n d m

• r

e r e c e n t l y , B e r k

• w

i t z a n d B a l b e r g , 1 2 h a v e s t u d i e d t h e a p p l i c a t i

• n

s

• f

t h i s t h e

• r

y t

• t

h e p h y s i c s

• f

p

• r
• u

s m e d i a . I n t h e f i e l d

• f

e q u i v a l e n t p e r m e a b i l i t y , t h e p e r c

• l

a t i

• n

t h e

• r

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

• m

a t e r i a l s w i t h t w

• p

h a s e s ,

• n

e

• f

w h i c h i s n

• n

p e r m e a b l e . L e t n b e t h e p r

• p
• r

t i

• n
• p

e r m e a b l e m e d i u m . C l

• s

e t

• t

h e p e r c

• l

a t i

• n

t h r e s h

• l

d : n < n c ~ K e f = O n > n c

• i

K e f = A x ( r f – n c ) ” ( 2 1 ) + =

• J

. a

2

‘ W ; ‘ > . / / w h e r p i t a k e t d e p e

• n
• t

s p d i m a n A a n n d e p e

• t

h g e

• m
• t

n e ( s e T a b l 1 ) 4 . T h e

• r
• e

l % e t m e d A p a r f r

• e

f f e c t m e d i t h e ( E M t h a

• t

h e n a m e f

• t

h t h e

• s

e l f

• c
• n

a p p u s e b M a t h e r

• e

m b e d m a t m e t u b D a g a n T h h e t e r

• g

e n m e d c

• n

s b h

• m
• g

e n e

• b

l

• c

p l a c s i b s i i r e p b a s i n g l i n c l u s i

• K

p e r m e a b e m b i a h

• m
• g

e n e

• m

a t r w i a u n k p e r m K ( F i ( j T h b

• u

n d c

• n

d i a f e n a w a f r

• t

h i n c l u s f

• t

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

• n

s t a n a r

• u

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

• r

t h e i a a n a l y s

• l

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

• u

t s i T h y p a r a f

• l

l

• w

t h p e r t u r b a

• t

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

• i

n t e r w i t p e r t u c a u s e b a n

• t

h i n c l u s a n K = K e D a g a n ’ m a d t h c a l c u l a f s p h e i n s i

• n

a n f

• u

n t h f

• l

l

• w

e x p r e s K . = ~ [ m f (K)dK ‘1 D O K(D – 1) + K.f1 ( w h e r f ( K ) i t h p r

• b

a b i d e n f u n

• t

p e r m e a b i l i t D i t h s p a d i m e n E q u ( c a b i n t e g r a t n u m e r i c b i n c r e K u n t i e q u a l i i

• b

t a i n K i n p r

• p

a s i f

• r

m u l a D a g a n a n P

• l

e g i a m

• c
• f
• r

m u l f

• e

l l i p s

• i

i n c l u s b w h t h

• b

a a n i s

• t

r

• p

e f f e c t p e r m e a b t e n — . * K K —

‘ w +

( a ) ( b ( c F i g . 6 . S c h e m a t i c r e p r e s e n t a t i

• n
• f

t h e s e l f

• c
• n

s i s t e n t m

• d

e l f r

• D

a g a n . 1 ( a H e t e r

• g

e n e m e d c

• n

s t b p i w c

• n

s t a n t p e r m e a b i l i t y p u t s i d e b y s i d e , ( b ) i n c l u s i

• n

e m b e d d e d i a u n i f

• r

m a t r w i t c

• n

s t p e r m e a K ( i n c r e p r e s e n t e d b y a s p h e r

• i

d e m b e d d e i a i n f i n i m e d i u

SLIDE 8 2 6 P h . R e n a r d G d M a r s i

W

( a ) ( b ) F i g . 7 . A p p r

• x

i m a t e m e t h

• d

f

• r

c a l c u l a t i n g t h e l e n g t h

• s

t r e a m t u b e : ( a ) r e a l s t r e a m l i n e ; ( b ) a p p r

• x

i m a t i

• n

b y s t r a i g h t

• l

i n e s e g m e n t s ( f r

• m

B e g g a n d K i n g ’ ” ) . I n t h e c a s e

• f

b i n a r y m e d i a , e q n ( 2 2 ) b e c

• m

e s : K e ~ = ~ [ h + h

1

– 1 D k . + ( D – l ) K e f k [ + ( D – l ) K e f ( 2 3 A c c

• r

d i n g t

• D

e s b a r a t s , 2 5 t h i s f

• r

m u l a i s s a t i s f a c t

• r

y f

• r

~ < 0 “ 6 w i t h p h a s e O r e p r e s e n t i n g t h e l

• w

e s p e r m e a b i l i t y . 4 . 4 S t r e a m l i n e T h e s t r e a m l i n e m e t h

• d

i s u s e d t

• c

a l c u l a t e t h e v e r t i c a p e r m e a b i l i t y

• f

a b i n a r y s a n d – c l a y s y s t e m . T h e c l a f

• r

m a t i

• n

s a r e d e s c r i b e d b y r e c t a n g l e s ( i n 2 D )

• f

l a p a r a l l e l e p i p e d s ( 3 D ) w i t h z e r

• p

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

• h

a v e k ~ j a n d k V j a n i s

• t

r

• p

i c p e r m e a b i l i t i e s . T h e u p

• s

c a l e d p e r m e a b i l i t y i s

• b

t a i n e d b y c a l c u l a t i n g t h e h e a d l

• s

s e s a l

• n

g a t

• r

t u

• u

s t u b e c i r c u l a t i n g i n s i d t h e s a n d m a t r i x ( F i g . 7 ) . 5 2 ’ 7 8 A n i m p r

• v

e m e n t

• n

t h i m e t h

• d

i s

• b

t a i n e d b y i n c

• r

p

• r

a t i n g s t a t i s t i c a l p a r a m e t e r s

• n

t h e s i z e a n d n u m b e r

• f

c l a y i n c l u s i

• n

s a n g e n e r a l i z i n g i s t

• s

t r a t i f i e d m e d i a . l ” T h e f

• l

l

• w

i n g f

• r

m u l a i s t a k e n f r

• m

t h e r e v i e w b F a y e r s a n d H e w e t t : 3 8 ~ = ( 1

• F

J H 2 N S 1 “ ‘ s ~ S i s , i ( 2 4 w h e r e F Y i s t h e f r a c t i

• n
• f

c l a y i n c l u s i

• n

s , N ~ t h e n u m b e r

• f

s e l e c t e d s t r e a m l i n e s , H t h e f

• r

m a t i

• n

t h i c k n e s s , S t h l e n g t

• t

h i t s t r e a m a S t l e w e b t h p e r m e a b i l w h e r n i t h n u m b

• i

m p e r i n c l f i l a y e r T h h

• r

i z

• n

l e n g

• t

k s i t i s t r e a m l i

• t

h j t l a y i g i b d i i = r l q k l

• r

2 i j k w w h e r r l t i a n r 2 a r u n i f r a n n u a l i a n W q a r t h l e n g a n t h i c k

• t

i n c 4 . R e n

• r

m a l i z a t H i s t

• r

i c a l l r e n

• r

m a l i z h i

• r

i s t a p h y s i c 6 2 , 1 I t a p p l i c a t t p r

• c

a l i n t h e q u i v a l p e r m e a b i g e n e a s s w i t K i n g . H

• w

e v a m e t

• t

s t w p r

• p
• s

b L L

• c

’ ( p 1 1 R e n

• r

m a l i z a t i a r e c u r a l g

• T

e q l e n p e r m e a b i l

• a f

i n e

• g

r i d e t e b a s e r i e

• s

u c c e s s a g g r e g a t F e x a i a s

• d

i m e n s i D o n p r

• c

e f r a g r

• 2

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

• n

w i 2 1 ” m e T p r

• c

e d u i t h e r e p e a u n a g r

• t

d e s i f

• u

n d u l t i m a t

• n

s i n m e ( F 8 T h b a s

• p

e r a t i t e q u i p e r m c a l c u l a t i

• a c

e w i 2 m e s S i t h i n e x a c f

• r

m u i t w a n t h d i m e t c a l c u l a t i h a t b a p p r

• x

K i u t a n a l

• g
• a

e l e c t n e t w t r e p a p

• m

e d i u m T h p e r m e a b i

• b

t a b s u c s t a r – t r i a n g t r a n s f

• r

m a T h e x p r i e q l e n t a s

• l

u t

• t

h f l p r

• b

a f d i f f e r e n m e t h w i a c e n t f

• r

m u S e v e r v a r i a t i h a b e p r

• p

a d e b e l

• w

T h e u s u a l c

• n

c t t y

• f

i d i f f

• r

m u l a t i a n t h b

• u

n d c

• n

d i F e x a d i r e c f

• r

m u l a t s e e m

• e

x ( K r u e t h p e r m e a b i l i a s s i g t e a l i b e t

“

* n D m e ~ h e ~ 2 ( n

• l

) D ~ = h e

• n

s i n v a F i g . 8 . G e n e r a l p r i n c i p l

• r

e n

• r

m a l i z a t i

SLIDE 9 C a l c u l a t i n g e q u i v a l e n p e r m e a b i l i 2 F i g . 9 . C e n t r e d a n a l

• g

u e E l e c t r i c a n a l

• g

u e s u s e d t

• c

a l c u l a t e t h e u p

• s

c a l e d p e r m e a b i l i n

• d

e s , a n d n

• t

t

• e

a c h b l

• c

k s u r r

• u

n d i n g a n

• d

e ) . G a u t i e r a n d N c e t i n g e r 4 2 c a l c u l a t e a c

• m

p l e t e t e n s

• r

b u s i n g p e r i

• d

i c b

• u

n d a r y c

• n

d i t i

• n

s . F u r t h e r m

• r

e , t h e y u s e t h i s m e t h

• d

t

• c

a l c u l a t e t h e v e l

• c

i t y f i e l d i n t h m e d i u m . F i n a l l y , L e L

• c

’ h 7 4 p r

• p
• s

e s a n

• r

i g i n a l m e t h

• d

w h e r e t h e m e s h e s a r e

• n

l y g r

• u

p e d t w

• b

y t w

• .

4 . 5 . 1 D i r e c t i

• n

a l p e r m e a b i l i t y I n t w

• d

i m e n s i

• n

s , t h e c a l c u l a t i

• n

c a n b e c a r r i e d

• u

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

• r

m a t i

• n

s

• b

y w r i t i n g a l i n e a r s y s t e m

• f

t h e f i n i t e d i f f e r e n c e t y p e . T h e t w

• p

r

• c

e d u r e s a r e e q u i v a l e n t a n d p r

• d

u c e t h f

• r

m u l a : 6 5 ~ : x = 4 ( r k , + k q ) ( k z + k t ) [ k z k i ( k l + k ~ ) + k l k q ( k z + k ~ ) ] [ k , k 4 ( k , + k , ) + k , k , ( k 2 + k i ) ] [ k , + k z + k q + k i ] + 3 ( k , + k j ) ( k ~ + k ~ ) ( k l + k j ) ( k 2 + k ~ ) ( 2 5 I n t h e i r a r t i c l e , H i n r i c h s e n e t a l . 5 6 g i v e t h e F

• r

t r a n c

• d

e f

• r

c a l c u l a t i n g t h i s e x p r e s s i

• n

, i n t w

• a

n d t h r e e d i m e n s i

• n

s . 4 . 5 . 2 T e n s

• r

G a u t i e r a n d N c e t i n g e r 4 2 p r

• p
• s

e a m e t h

• d

f

• r

c a l c u l a t

• i

n g t h e c

• m

p l e t e t e n s

• r

. T h e b

• u

n d a r y c

• n

d i t i

• n

s a r p e r i

• d

i c a n d t h e f i n i t e d i f f e r e n c e f

• r

m u l a t i

• n

d i r e c t . T a k e a s q u a r e n e t w

• r

k

• f

f

• u

r m e s h e s . T h e m e s h e s a r e n u m b e r e d f r

• m

1 t

• 4 (

s e e F i g . 9 ) . T

• s

i m p l i f y t h n

• t

a t i

• n

,

• n

e a s s u m e s t h a t t h e s i z e

• f

t h e m e s h e s i s 1 x 1 a n d t h e e x p

• n

e n t f

• r

t h e d i a g

• n

a l t e r m s

• f

t h p e r m e a b i l i t y t e n s

• r

s ( k ” = k ’ ) i s n

• t

r e p e a t e d . T h e b a l a n c e i s w r i t t e n a t e a c h

• f

t h e f

• u

r n

• d

e s . T h e n , t h c

• e

f f i c i e n t s

• n

t h e h e a d s a r e f a c t

• r

i z e d . T h i s r e s u l t s i n a s y s t e m

• f

f

• u

r e q u a t i

• n

s w h e r e t h e f

• u

r u n k n

• w

n s a r t h e h e a d s . N

• t

e t h a t t h i s s y s t e m i s l i n k e d a n d t h a t i i t h e r e f

• r

e p

• s

s i b l e t

• f

i x a h e a d v a l u e ( e . g . h 4 = O ) a n d s

• l

v e

• n

l y a 3 x 3 s y s t e m .

()(

k ; k ; h l A h2 = k ; y – k ; – k ; y k ] h ~ ~ k : y – k : Y – k ; )

“( )

A H X A H Y D i r e c a n a l

• g
• a

e l e m e n t c e i t d i m e n w i t T h e n t h g l

• b

f l r a t

• v

t d

• a

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• e

a c m e s h [ U = – k ; h + ( k + k ; Y – k : + ( k + k j ) A + k ~ y U = – k ~ h – k j + ( k + k – k : + k ; Y A + ( + k ; ) U s i n t h e s e q u a t i

• U a

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SLIDE 10 2 6 2 P h . R e n a r d G d M a r s i e t a t F i g . 1 . E x a m p l e i n t w

• d

i m e n s i

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s

• f

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h s u c c e s s i g r

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a l c u l a t p e r m e a l x T s y m b

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, u ~ r e p r e s e n t s t h e h a r m

• n

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t w

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e s h e g r

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p e i t h x

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i r e c t K i t h a r i t h m i y h a r m

• n

i c m e a n . F

• r

e x a m p l e , f

• r

a f l

• w

i n x t h m a x i m u m a n d m i n i m u m v a l u e s a r e : A t t h e e n d

• f

t h e r u n , w e t a k e t h e g e

• m

e t r i c m e a n

• t

h e s e t w

• v

a l u e s . ( 2 7 4 . 6 N u m e r i c a l s

• l

u t i

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t h e d i f f u s i

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e q u a t i

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T h e r e a r e s e v e r a l t e c h n i q u e s f

• r

s

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v i n g p a r t i a l d i f f e r e n t i a l e q u a t i

• n

s n u m e r i c a l l y : f i n i t e d i f f e r e n c e s , f i n i t e l e m e n t s , m i x e d f i n i t e e l e m e n t s , s p e c t r a l e l e m e n t s , e t c I t h e c a s e

• f

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p e r m e a b i l i t y . A d i r e c t i

• n

a l f l

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

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k p e r m e a b i l i t y i s g i v e n b y : K, - Q L S A H ( 2 8 w h e r e L i s t h e d i s t a n c e b e t w e e n t h e t w

• s

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t u s t h i t e c h n i q W i t t h e s i m u l a t i i i p

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y d r a u l h e a h ( y i a v e l U y i a a small scale for the i = 1, . ,n d i f f b

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SLIDE 11 C a l c u l a t i n g e q u i v a l e n p e r m e a b i l i a s s i g n e d t

• t

h e c e n t e r

• f

t h e l a r g e m e s h e s . / ~/ z ( x , y , i ) S ( x , y ) d V H(x, y, i) = ~

j v s ( w ) d

W h i t e a n d H

• r

n e l l l u s e f i n i t e d i f f e r e n c e s f

• r

s u c h c a l c u l a t i

• n

s . T h e l a r g e

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c a l e v e l

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

• f

t h e v e l

• c

i t i e s

• b

t a i n e d

• t

h e s m a l l s c a l e

• v

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

• c

k s , 1 / U ( x , y , i ) = ~ ~ U ( X , y , i ) d A H a v i n g t h u s

• b

t a i n e d t h e l a r g e

• s

c a l e v e l

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i t i e s a n d h e a d s ,

• n

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T TXY U ( x , y , i ) = T b g r a d [ K ( x , y , i ) ] , T~ = T“ T yx yy T h e u n k n

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• n

t h e f l

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s . T h e l e a s t s q u a r e s c r i t e r i

• n

i s w r i t t e n : <= ~ [ U ( i ) – T b Z ( i ) ] 2 = m i n i m u m i = l w h e r e Zi s t h e h e a d g r a d i e n t b e t w e e n t w

• n

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s : H ( x + 1 , y , i ) – ~ ( x , y , i ) I X ( X + 1 / 2 , y , i ) = A x I j , ( x + 1 / 2 , y , i ) [ – 1 I

• I

( x + l , y + 1 , i ) – I I ( x + l , y – 1 , i ) — T A y + H ( x , y + 1 , i ) – H ( x , y – 1 , i ) A y

1

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• n
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• r

e x a m p l e , T X X a n d T X . a r e s

• l

u t i

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s

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: ( 3 ) w i t h a = ~ I X ( i ) 2 b = ~ Z Y ( i ) Z J , ( i ) d = ~ Z Y ( i ) 2 i n l i = l i c l a n d A s i m i l a r s y s t e m m a k e s i t p

• s

s i b l e t

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e t e r m i n e T Y X a n d T Y Y . T , X i s r e p l a c e d b y T Y X a n d T X ~ , b y T Y Y , a n d S 1 a n d S

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h r i g h t

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q ( 3 a r e p S = ~ Z X ( i ) U Y S = ~ Z Y U Y i c i 2 b U s

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• t

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t

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1

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m h y p

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p p e r i

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d

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a i a a w h

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b a c s u b j e c t e t p e r i

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d i ( F 1 T h i t

• p

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• d

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• d

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b e t p e r i

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r b

SLIDE 12 2 6 4 P h . R e n a r d G d M a r s i F i g . 1 1 . ( b t h a n t h

• s

e

• f

a p e r m e a m e t e r

• r

u n i f

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m t y p e . T h e l a t t e a r e p a r t

• f
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h e r , e q u a l l y c

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s t r a i n i n g , a s s u m p t i

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s

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• r

e x a m p l e , i n t h e c a s

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c

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s

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t h e p e r m e a m e t e r t y p e , t h e n

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l

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b

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n d a r y c

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r e s p

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d s t

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n a x i s

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s y m m e t r y . M

• r

e

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• n

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e f

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a c a s e w h e r e a l a r g e

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

• t

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a k e t h e n e c e s s a r y a p p r

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c a l e e q u a t i

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• s

t e p s F i r s t ,

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l e m r e g a r d i n g t h a n c i l l a r y v a r i a b l e s b l , b 2 , . . . , h ~ . K i t a m d n “ “ 6 6 w r i t e s : V . ( K g r a d b i ) = V . K i i = 1 , . . . ,~ ( 3 1 s u b j e c t t

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v e d n u m e r i c a l l y . F

• r

e x a m p l e , D y k a a r a n d K i t a n i d i s 3 3 ’ 3 4 u s e a s p e c t r a l m e t h

• d

a n d N j i f e n j

• u

8 7 u s e s m i x e d h y b r i d f i n i t e e l e m e n t s . T h e t e r m s

• t

h e q u i v a l e n t p e r m e a b i l i t y t e n s

• r

a r e t h e n c a l c u l a t e d w i t t h e f

• r

m u l a : f u n c t i

• t

h e q u i v a p e r m e a i e x p i t e r m

• a s

t

• c

h a d i f f e r e q u i a a d i f f e r e n t i e q u a t l i n k s e v r a v a r A c

• m

p l e s

• l

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• p

r

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i s t r i b u t i f

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• n

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

• (

m e a n v a r i a n c

• v

a r i e t

• a

l e t a p p r

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i m a v a l u A g r e n u m

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e — w h i c c a n n b a d d r e h e w i t d i g f t h m a i f

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h a r t i — h a b e u s s p m e t h

• p

e r t u r b a m e t h f i t h M

• C

a r l m e t h

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t W w i r e s t t d i s t t h m a i r e s u l M a j r e f e r i t a a M a t h e r

• 7 D

a g 1 a n G e l h a c r i r e

• t

h d i f f e r e m e t h

• c

a b f

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K i t a 5 . R u l

• g

e

• m

e t a v e r a O n

• t

h f e e x a r e s u i t r u

• g

e a v e r a g i n I w a d e m

• n

s t b M a t 7 T v e r e l e g a d e m

• n

s t r a i b a

• t

f t i a t w

• d

i m e n s i

• s

p a c a 9 r

• t

t r a a g r a d i e v e c t i n a c

• n

s e r v e a v v e r s a S u b j e t t h c

• n

d i t t h t p e r m k a n i t i n v e r k = 1 / a r r a n f u n a d t h s a m p r

• b

a b i d i s t r i b a t t f u n c t i

• a

r i n v a r i b a 9 r

• t

a t e f p e r m e a b i l i i e x p r e s i t f

• l

l f

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, = p = e x p [ E ( k ) ( T h p r e c e d h y p

• t

h a s a t i i t s p c a s

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i s

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r

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e d a a c h b

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r b i n a d e s i H

• w

e t h

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h

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(

• p

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

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a d i f l

• M

a t h e i n d t i r a d i a f l

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v e ( f l a d i s s h

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5 . A p p r

• x

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• n

j

+

J

V v

K Q d V i , j = 1 , . . . ,D ( 3 2 5 . 2 . A 4 a t h e r

• c
• n

j e c L a n d a a n L i f s h i 7 f

• t

h e l e c t r

• d

‘ A n a n a l y t i c a l s

• l

u t i

• n

w a s c a l c u l a t e d b y Q u i n t a r d a n 7 f

• u

n i f l i i s

• a

t i

• n

s a n M a t h e r W h i t a k e r 9 4 f

• r

a s t r a t i f i e d b i n a r y m e d i u m ( e q n ( 2 ) ) s t a t i

• n

a p

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• m

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e x t e r e s u l i t w d i m e n s i t D d i m e n s p w a f

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m u w h i i t h f i r s t

• a

p p r

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5 S T O C H A S T I C M E T H O D e f f e c t i p e r m e a b i l I n

• r

d e r t

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e a l w i t h t h e u n c e r t a i n t y a r i s i n g f r

• m

a K , = p : D

• l

) ’ D p ( p a r t i a l k n

• w

l e d g e

• f

t h e r e s e r v

• i

r p r

• p

e r t i e s , t h s t

• c

h a s t i c m e t h

• d

c

• n

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

• t

c a

• a l
• g

r a n d

• m

f u n c t i

• n

s i n s p a c e . T h e d e f i n i t i

• n
• t

h p e r m e a b i l i d i s t r i b u t h f

• r

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

• n

t h e n

• t

i

• n
• m

a t h e m a t i c a l e x p e c t a t i

• n

( e q n ( 3 ) ) . [ ( :

• 3

K e t _ = P g e x f f l n ( T h e d e t e r m i n a t i

• n
• f

t h e p r

• b

a b i l i t y d i s t r i b u t i

• n

SLIDE 13 C a l c u l a t i n g e q u i v a l e n p e r m e a b i l i 2 w h e r e C & k r e p r e s e n t s t h e v a r i a n c e

• f

t h e p e r m e a b i l i t y l

• g

a r i t h m . A v a s t n u m b e r

• f

a u t h

• r

s h a v e e x a m i n e d t h v a l i d i t y

• f

t h i s f

• r

m u l a . I n p a r t i c u l a r , G u t j a h r e t a l . , 4 s u s i n g t h e m e t h

• d
• f

p e r t u r b a t i

• n

s a n d a s s u m i n g t h a t

• &

~ ) i s s m a l l , f

• u

n d : ( 3 6 ) E q u a t i

• n

( 3 6 ) p r

• v

i d e s f i r s t

• r

d e r a p p r

• x

i m a t i

• n

s

• f

t h c l a s s i c a l e x a c t r e s u l t s i n

• n

e a n d t w

• d

i m e n s i

• n

s ( h a r m

• n

i c m e a n a n d g e

• m

e t r i c m e a n , r e s p e c t i v e l y ) . D a g a n 2 c a r r i e s t h e c a l c u l a t i

• n

s t

• t

h e f

• u

r t h

• r

d e r a n d

• b

t a i n s : ( 3 7 ) E q u a t i

• n

s ( 3 6 ) a n d ( 3 7 ) c a n t h e n b e s e e n a s , r e s p e c t i v e l y , t h e f i r s t

• a

n d s e c

• n

d

• r

d e r t e r m s

• f

T a y l

• r

’ s s e r i e s e x p a n s i

• n
• f

t h e e x p

• n

e n t i a l i n e q n ( 3 5 ) . T h e y h a v e t h e r e f

• r

e b e e n u s e d t

• j

u s t i f y c

• n

j e c t u r e ( 3 5 ) . T h e a d v a n t a g e

• f

e q n ( 3 5 )

• v

e r e q n s ( 3 6 ) a n d ( 3 7 ) i s t h a t i i s n

• t

l i m i t e d t

• s

m a l l v a r i a n c e s . K i n g 6 4 a n d l a t e r , N c e t i n g e r , 8 8 c

• m

b i n e d t h e f i e l d t h e

• r

y a n d G r e e n ’ s f u n c t i

• n

s t

• e

x p a n d t h e s e r e s u l t s . N c e t i n g e r 8 8 s h

• w

s t h a t a s u b

• s

e r i e s

• f

t h e p e r t u r b a t i

• n

e x p a n s i

• n

c a n b s u m m e d a g a i n a t a l l

• r

d e r s . H e d e m

• n

s t r a t e s t h a t e q n ( 3 5 ) h

• l

d s f

• r

a n u n c

• r

r e l a t e d i s

• t

r

• p

i c l

• g
• n
• r

m a l m e d i u m b u t t h a t t h e f

• r

m u l a r e m a i n s a n a p p r

• x

i m a

• t

i

• n

. F u r t h e r m

• r

e , h e s u g g e s t s t h a t e q n ( 3 5 ) c a n b e x p r e s s e d i n t h e f

• l

l

• w

i n g f

• r

m : & = (

k l

• (

2 / D ) ) [ l

• (

2 / D ) l

• ’

( 3 8 ) H

• w

e v e r , t h e d i s c u s s i

• n
• f

t h i s t

• p

i c i s s t i l l c

• n

t i n u i n g . A b r a m

• v

i c a n I n d e l m c a r r t e x p t t h s i x t h

• r

d a n D W i t u s p e r t e x p a n s i

• n

s h

• t

h f

• c
• r

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• t

m e t h c

• e

f f i c i e i a ~ n ( d e p e

• t

f

• t

c

• r

r e l a t i f u n c t i T h i c

• n

t t M a t c

• n

j e c t u r I i s t i l t r u h

• w

e v t h f

• r

( 3 h b s u c c e s s f u l t e s t b m a a u t h e v f m e w c

• r

r e l a t i l e n g t g r e a t h t s i

• t

m e ( t h c

• r

r e l a t i l e n g i d e f i a t d i s a w t h c

• v

a r i a n h a b e r e d u b a f a e f i x f e x a m s t t v a l u a l a z e r

• D

e s b a r p

• w

e m e a a n f

• u

e x p e r i m e t h i t d i m e n s i

• n

t h e x p

• n

i 1 / f a l

• g
• n
• m

e d i u m T h i c

• r

r e s p

• t

t h f

• r

p r

• b

N m t i n g e ( e q ( 3 8 ) D y k a K i t a n f

• a

d e v i a t i

• n

l 4 Y b e t w c a l c u l m w t h s p e c t r n u m e r i m e t a M a t h c j e c t u r ( s e F i g 1 2 T h e r e s u a c

• n

b N e w m a a n O r r u t t h v a l

• a

= 7 S a n c h e z

• V

i e a l . l c

• m

p t r e s

• b

t w t h r e e a p r i

• r

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

• (

e q ( 2 8 ) t h m e t

• R

u a G

• H

e r n a n d e z , e q ( 4 a n M a t h e c

• n

j T s h

• n

u m e r i c a l t h t h e t h r t e c h p r c

• m

p a r a b r e s u l f

• a l
• g
• n

m e w & ~ < 1 F i n a l l c

• m

p a r i w i p e r m e m s u r e m e n t m a d a s e v e s c a

• s

a

• s

a n d s t

• n

a n l i m e s t

• a

r a l c

• n

s w M a t h e r

• n

c

• n

j e c t u r 5 . 2 . T a k i n a n i s

• t

r i n a c c

• I

t h c a s

• a

a n i s

• t

m e d w a e x p

• n

e n t i c

• v

a r i a n

• n

c a c a l c t e f f p e r m e a b i l i t i i t h m a d i r e c

• a

n i s w 2 8 2 6

1

. 2 . 4 — s e u ~ . — 2 N 2 p m 9 ? h s t 1 4 m P 2 2 , 8 ’

• w

m c l w ’ l c O w j u t w / 1 2 1 . 8 1 . 8 1 . 4 1 2 1 1 L . .

• ,

1 . 2 . s

• 4

. 5 . 6 . V w i m c O t L

• g

( K F i g . 1 2 . C

• m

p a r i s

• n

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

• b

t a i n e d b t h e f f e c t i v e

• m

e d i t h e

• (

l a b e s e l f

• c
• i t

f i t h e p e r t u r b a t i

• n

m e t h

• d

( l a b e l e d s m a l l p e r t u r b a t i

• n

) , t h e m

• m

e n m e t h

• a

n M a t h e r

• c
• n

j e c f a i s

• t

l

• g
• 3

D m e d i u m . N

• t

e t h e g

• d

a g r e e m e n t b e t w e e n t h e n u m e r i c a l r e s u l t ( m

• m

e n t a n M a t h e r c

• n

j e ( f D y a K i t a n i d i s 3 4

SLIDE 14 2 6 6 P h . R e n a r d G d M a r s i t h e f

• r

m u l a 4 3 ) U e x p a n d e d i n t h e f i r s t

• r

d e r w i t h r e g a r t

• a

: ” ( k ) :

“ ’ =

w i t h ( 3 9

,

, = 1 ~ ~ g i i = 1 , , w h e r e l i i s t h e c

• r

r e l a t i

• n

l e n g t h i n t h e i d i r e c t i

• n

a n k i s t h e l

• c

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

• n

. T h i s f

• r

m u l a i s

• n

l y v a l i d f

• r

s m a l l

• &

k ) . A s b e f

• r

e , i f t h e p r e c e d i n g e q u a t i

• n

i s c

• n

s i d e r e d a s a T a y l

• r

s e r i e s e x p a n s i

• n
• t

h e e x p

• n

e n t i a l f u n c t i

• n

t r u n c a t e d t

• t

h e f i r s t

• r

d e r i ‘ & [ ’ ) , a p l a u s i b l e g e n e r a l i z a t i

• n
• f

e q n ( 3 9 ) f

• r

h i g v a l u e s

• f

a & k ) i s : ( 4 A l

• n

g t h e s a m e l i n e s , A b a b

• u

l t a k e s a n i s

• t

r

• p

y i n t a c c

• u

n t w i t h a s i m p l i f i e d f

• r

m u l a : w h e r e I i i s t h e c

• r

r e l a t i

• n

l e n g t h i n t h e c

• n

s i d e r e d d i r e c t i

• n

a n d 1 ~ t h e h a r m

• n

i c m e a n

• f

t h e c

• r

r e l a t i

• n

l e n g t h s i n t h e p r i n c i p a l d i r e c t i

• n

s

• f

a n i s

• t

r

• p

y . N e u m a n 8 4 p r

• p
• s

e s t h e s a m e t y p e

• f

f

• r

m u l a :

‘ e f = ~ g e x

w i t h O< P < D . M

• r

e p r e c i s e l y , ~ = 1 i n t h e i n f i n i t e i s

• t

r

• p

i c c a s e , O< ~ < 1 i n t h e f i n i t e i s

• t

r

• p

i c c a s e a n 1 < ~ < D i n t h e i n f i n i t e a n i s

• t

r

• p

i c c a s e . H e d

• e

s n

• t

h

• w

e v e r , e x p l a i n h

• w

t

• c

a l c u l a t e @a p r i

• r

i . T h e d e v e l

• p

m e n t a t

• r

d e r t w

• f
• &

’ ) 5 8 l e a d s t a f

• r

m u l a w i t h a n

• r

d e r

• t

w

• c
• e

f f i c i e n t t h a t d e p e n d s

• t

h e v a r i a n c e a n d

• n

t h e s p a c e d i m e n s i

• n

b u t i a l s d e p e n d e n t

• n

t h e f

• r

m

• f

t h e c

• v

a r i a n c e f u n c t i

• n

C ( Y ) = a & k ) p ( r ) a n d n

• t
• n

l y

• n

t h e a n i s

• t

r

• p

y r a t i

• s

p ( r ) i s t h e a u t

• c
• r

r e l a t i

• n

f u n c t i

• n

. ( 4 1

“ ’ = ~ g { l +

+ [ ( ~

• ~

i ) ’ + i ] a f n k } / 4 % ) i = l D % = ~2n) k , . . . > 1 I s d q d p . q ’ + p ’ ( ) : + % 2 ~ P ( 9 ) M P ) 9 “ P 7’= I (27r) (q+p)2 9 P w h e r ~ r e p r e s e t h F

• u

t r a n

• t

a u t

• c
• r

r e l a t f u n c t i j = f d r e x p N t h a g i = ~ i t h c a w h t c

• v

i e x p

• n

e n t i a T h u t h c

• n

j e

• G

e a A x n e s s e q ( 3 9 i n

• s

a t i s a o r t 5 . 2 . C a u t i T h r e s u l g i v a b

• m

a l e

• t

b e l t t a r s i m p r u l f

• c

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

• x

s e T i l l t h i s M a t h e r

• n

u s S c h w y d m e — p e a b i l i t i w r i t t

• t

h f

• k

( = E ( [ + ~ ( w E ( 7 ) t h m a t h e m a t e x p e c t

• ~ e

q t z — t c a r r

• u

c a l c u l a t i t f

• a d

e v t r u n c a t a

• r

d t h r f

• t

h m

• d

i m e d i a T h f i r

• n

i a r a n c r

• s

p u T s e c

• n

i a m e d i d i v i i n r a c

• p
• l

y g

• b

P

• i

s s l i n T l a

• i

a m e w i t a i n f i n i t d i v i s l a T r e s f t t m

• d

e l a r e [ 1) Ke~= m 1 – E ( ~ ~ + O “ 2 7 9

1

[ 2) K.~ = m 1 – E ( ~ ~ + O “ 3 7 1 [ 3) K.f = m 1 – E ( ~ ~ + 0 . 2 5

1

N

• t

t h a i a l t h r c a s t c

• e

f f

• t

t h

• r

d e t e r i d i f f e r T h m e t h f M a t h e r i n s i m p l u n i f

• v

a r u f a v e t p e r m e a b i l i t i n

• e

v i t h c a

• i

s

• r

a m e d i a T h i f

• l

l

• t

h s a l i

• r

e a s a t

• I

n d e l m a n A b r a m

• v

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

• A

2 f l

• g

m

• v

i c a n I n d e l m a n D W i s

• t

r

• p

m e d i 5 . B l

• c

p e r m e a b i R u b i a n G

• m

e z

• H

e r n a n w e t f i a u t s t u d t h p r

• b

l

• b

l

• p

e r m e f a s t

• c

h a s t s t a n d p

• i

T h m a p u r

• t

h w w a t f

• r

m u l t h e x p e c t E ( a t c

• v

a r i a n

• b

l

• p

e r m e a b C a t c r v a r i a n C & , l * b e t w b l p e r m a l

• c

a p e r m e a b i l b a s

• t

d i s t r f u

• t

h p e r m e a b i l T h e e x p r e w

• b

a n a l y t i c a l f

• b

l

• p

e r m e a i t c w t h l

• c

a p e r m e a b i i i s

• t

r a i l

• g

h a a m u l t i

• g

a u s d i s t r i b 9 T h y p a r e i n f i n i m e d i u n i f f l a s v a r i a n

• t

h l

• g

a r i

• k

F t p e r m

SLIDE 15 C a l c u l a t i n g e q u i v a l e n p e r m e a b i l i 2 G 6 m e z

• H

e r n a n d e z 4 5 w r i t e s : [ a ? n ( k ) E ( K ~ ) = p g 1 + ~ 1 ( + ~ a ? n ( k ) , ~ x – C l n ( k ) , . l . a ) ] c ~ , ( r ) = p

~ c h ( k ) (

‘ + r ) c ~ b , l n ( k ) ( ~ ) r ) = # g C l n ( k ) ( V > ( 4 2 ) V r e p r e s e n t s t h e s i z e

• f

t h e b l

• c

k , r i s a t r a n s l a t i

• n

v e c t

• r

, J . i s t h e m

• d

u l u s

• f

t h e a r i t h m e t i c m e a n

• f

t h e h y d r a u l i c g r a d i e n t v e c t

• r

J , J X i s t h e x c

• m

p

• n

e n t

• f

J c & ~ ) , ~ x i s t h e s t a t i

• n

a r y c r

• s

s

• v

a r i a n c e b e t w e e n t h e p e r m e a b i l i t y l

• g

a r i t h m a n d t h e h y d r a u l i c g r a d i e n t J X T h e m e a n c

• v

a r i a n c e s ~ a r e g i v e n b y : 1 [ S C M k ) , . J , = @ “ ~ ~ , n ( k ) , ~ z ( X – x ’ ) d ~ ‘ x ’ 1 J / ~ l n ( k ) ( ~ , ‘ + r ) = ~ ~ V + , C [ n ( k ) ( x , X ’ ) ~ X ~ X ’ ~ l n [ k ] ( ~ , r ) = j C l n ( k ) ( X ’ , X + ‘ ) ~ x ~ x ’ T h e c r

• s

s

• v

a r i a n c e b e t w e e n t h e p e r m e a b i l i t y l

• g

a

• r

i t h m a n d t h e h y d r a u l i c g r a d i e n t c l n ( k ) , ~ r c a n b e f

• u

n d i D a g a n ’ 9 ( s e c . 3 . 7 ) . T h e s e a n a l y t i c a l r e s u l t s a r e c

• n
• f

i r m e d b y M

• n

t e

• C

a r l

• s

i m u l a t i

• n

s . N

• t

e t h a t q u a l i t a t i v e l y , i n t w

• d

i m e n s i

• n

s : } n l

• E

( K b ) = # a l i m l l ( K b ) = p g V * W T h e v a r i a n c e

• f

K b d e c r e a s e s w h e n V i n c r e a s e s b u t i t r e m a i n s n

• n
• n

e g l i g i b l e f

• r

l a r g e v

• l

u m e s . T h e c

• v

a r

• i

a n c e

• f

K b i s d i f f e r e n t f r

• m

t h a t

• f

t h e l

• c

a l p e r m e a b i l i t i e s . T h e m a i n a t t r a c t i

• n
• f

t h i s a p p r

• a

c h i t h a t i t m a k e s i t p

• s

s i b l e t

• c
• n

d i t i

• n

t h e b l

• c

k p e r m e a b i l i t i e s b y l

• c

a l m e a s u r e m e n t s

• f

h a n d k R u b i n a n d G

• m

e z

• H

e r n a n d e z 9 8

• b

s e r v e t h a t t h e e f f e c t

• f

t h e c

• n

d i t i

• n

i n g b y h i s w e a k e r t h a n t h a t

• t

h e c

• n

d i t i

• n

i n g b y i n ( k ) . I n b

• t

h c a s e s , t h e v a r i a n c e

• f

K b d e c r e a s e s b u t i t d e c r e a s e s m

• r

e s t r

• n

g l y f

• r

a c

• n

d i t i

• n

i n g b y i n ( k ) . M

• r

e r e c e n t l y , F e n t

• n

a n d G r i f f i t h s 3 9 s t u d i e d t h e s a m e p r

• b

l e m i n t w

• d

i m e n s i

• n

s u s i n g M

• n

t e

• C

a r l

• s

i m u l a t i

• n

s . T h e y v a r i e d t h e m e s h s i z e w h i c h g a v e t h e m t h e

• p

p

• r

t u n i t y t

• i

n v e s t i g a t e t h e e f f e c t

• f

a n i s

• t

r

• p

y

• n

b l

• c

k p e r m e a b i l i t y . T h e y c

• n

c l u d e t h a t f

• r

b l

• c

k s t h a t h a v e n

• t

b e e n f l a t t e n e d ( i . e . w h e r e t h e s i z e

• f

t h e m e s h i s i d e n t i c a l i n t h e h

• r

i z

• n

t a l a n d v e r t i c a l d i r e c t i

• n

s ) , t h e b l

• c

k p e r m e a b i l i t i e s a r e p r

• b

a b l y l

• g
• n
• r

m a l l y d i s t r i b u t e d . F

• r

s u c h a c a s e , t h e y p r

• p
• s

e e x p r e s s i

• n

s

• f

t h e e x p e c t a t i

• n

a n d v a r i a n c e

• f

b l

• c

k p e r m e a b i l i t y : E [ l n ( K b ) ] = E [ l n ( k ) ] ~ h ( & = W

k )

( 7 = ‘ Y ( R R i t h v a r i a f u n c t R i t a n t r

• p

r a t i

• t

h m e s h R = ~ T v a r f u n c t i

• i d

e f i n b y : * 4 ~ ( R 1 R , = ( R , R , R, Rz x H ( R – r 1 ) r z ) r 2 ) F e n t

• a

n G r i f f i t h a l n

• t

h t f l a f a c t

• h

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

• r

r e l a t l e n g I t h c a s

• a m

e d i w h t p e r m i n

• l
• g
• n
• r

m a d i s t r i b a m b a n t r

• p

i c G

• m

e z

• H

e r n a n a n J

• u

r a G

• H

e r n a n d e z 4 s u g g e t h u s

• a n

u m e m e t T

• b

j e c t i v i t m a k s

• f

i n e

• s

i m u l

• t

p e r m e a b i l i f i e l F

• a

l s i m u l a t

• c

a l c t i n t e r

• b

l

• c

p e r m e a b i l t e n b s

• l

t d i f e q u a t i

• n

u m e r i c a l T h m e t i t

• d

e s a b

• v

i S e c t i 4 . 6 u s

• t

h l

• n

e i g h T h e s b l

• c

k

• s

c a p e r m e a b i l a u s t

• b

t e x p e c t a t i

• a

n t h c

• v

a r i

• T

a t c r c

• v

a r i a n c c ~ ~ , l ” ( I i t h p

• s

s t s i m d i r e c t l t h i n t e r

• b

l p e r m e a b i T h m e i a l s u s e b T r a 1 6 D I S C U S S I 6 . E f f e c t i

• b

l

• p

e r m e a b i T h f i r s p

• s

s i b c h

• i

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

• b

a s e

• v

a l u ( b l

• p

e r m e a b D u r l c

• m

p a r e t h r e t w

• d

i m e n s m e t f m e w i t c

• r

r e l a t l

• g
• n
• r

p e r m e a d i s t r T h e s m e t h

• a

r e ( 1 a t e c h n k n a g l w h e r t h h e t e r

• g

e n e m e d i r e p l b a u n e f f e c t i v p e r m e a b i l i ( 2 t h t e c h n

• s

a m ( a l

• c

a t e c h n i q w h i c

• n

s i i c a l c u t b l p e r m e a b i l i b t a k i t h g e

• m

e m e

• t

v a i t h b l

• c

k T h p r e c i s i

• a

l t h r t e c h n i m e a b c

• m

p a r i n t h s

• l

u t i a f t t h s c c h a t t s

• l

u t i

• b

e f

• t

h c h a n T h u D u r l

• c

a l a t e m e a q u a d r a e r r

• (

. L f t h e a t f l

• r

a t b e t w e t h r e s u

• t

n u m e s i t i

• n

( f i n i d i f f e r e n

• t

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

• s

c a H t h s t t h e v

• l

u t i

• L

a a f u n c

• t

t

• b
• u

n d a r c

• n

d i t i

• t

h c

• r

r e l l e n t s

• t

h b l

• c

k a n t h v a r i a

• t

p e r m l

• g

a r i t h m H

• b

s e r t h t h q u a d e r c

SLIDE 16 2 6 8 P h . R e n a r d G d M a r s i

S a m p l i n \ L 2 d G l

• b

L

• c

F i g . 1 3 . E v

• l

u t i

• n
• f

t h e q u a d r a t i c e r r

• r

( L z ) w i t h t h e s i z

• t

h e b l

• c

k s ( d ) f

• r

t h r e e u p

• s

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

• r

t h e s a m p l i n g t e c h n i q u e . I f d > d c , t h e g l

• b

a l m e t h

• i

s m

• r

e a c c u r a t e t h a n s a m p l i n g ( f r

• m

D u r l

• f

s k y 3 ) . b e a p p r

• x

i m a t e d b y L z = a d b w h e r e d i s t h e s i z e

• t

h b l

• c

k s f

• r

t e c h n i q u e s 2 a n d 3 . I n t h e c a s e

• f

t h e g l

• b

a m e t h

• d

, L 2 i s i n d e p e n d e n t

• f

d . F

• r

e a c h c

• n

f i g u r a

• t

i

• n

, h e d e t e r m i n e s a c r i t i c a l v a l u e d C , b e y

• n

d w h i c t h e t e c h n i q u e

• f

s a m p l i n g

• r

t h e l

• c

a l m e t h

• d

b e c

• m

e l e s s a c c u r a t e t h a n t h e g l

• b

a l

• n

e ( s e e F i g . 1 3 ) H c

• n

c l u d e s : — F

• r

f l

• w

r e s u l t i n g f r

• m

p e r m e a m e t e r

• t

y p e b

• u

n d a r y c

• n

d i t i

• n

s , t h e m e t h

• d
• f

s a m p l i n g i s m

• r

a c c u r a t e t h a n t h e g l

• b

a l

• n

e w h e n t h e s i z e

• t

h b l

• c

k s i s s m a l l e r t h a n t h e c

• r

r e l a t i

• n

l e n g t ( d C = A ) . T h i s r e s u l t i s i n d e p e n d e n t

• f

t h e v a r i a n c e

• f

t h e p e r m e a b i l i t y l

• g

a r i t h m ( & k ) ) . F

• r

f l

• w

d r i v e n b y a u n i f

• r

m s

• u

r c e t e r m , h i g h e r e s

• l

u t i

• n

i s n e c e s s a r y t

• m

a k e t h e s a m p l i n g m

• r

a c c u r a t e t h a n t h e g l

• b

a l m e t h

• d

. M

• r

e

• v

e r , d d e c r e a s e s w i t h a & ~ ) . — I n a l l c a s e s , t h e m e t h

• d
• f

l

• c

a l a v e r a g i n g i m

• r

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

• r

p r e c i s e t h a n t h e g l

• b

a l m e t h

• d

. I

• t

h e w

• r

d i t h b l

• p

e r m e i c a l w i t a r e a s

• n

a r e l i a m e t i g i a h a c c u r a c H

• w

e v w i a m u c r m e ( s a m p l i n g i t h s i

• t

h b l

• i

l a t t c

• r

r e l a t i l e n g t i i b e t t u a u n i e f p e r m e a b i l i 6 . N u m e r i c

• a

n a l y t m e t B a n a l y t i c m e t h w m e s i m a l g a v a

• p

p

• s

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

• l

u t

• p

a r t d i f f e r e q u A l t h a s s u m p t i

• t

h t y

• m

e a f p e r t a i n i t t h t e c h n i d e s c h e a l i i T a b l 2 ~ h i c l e a s h

• t

h a n a l t e c h a v a m u c m

• l

i m i r a

• a

p p l t n u m e r i c m e t h

• F
• e

x a m t h

• g

e r e s u l f

• s

t r a t i f

• i

s

• t

r m e w l

• g

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

• c

a s e s t h e r i a r i i n v

• i

u s t h N u m t e c h n i q u e

• t

h c

• n

t r a m

• g

e n t c b u s e

• a

n p e r m e a b f i e T h m d r i t h a t h e a r e x p e n s i t e

• m

e s p a c

• m

p u t t i m T h c h

• i

b e t w a n a l y a n u m i t h e r e f

• s

i m p l i o n h a a c a f w h a n a m e t h

• d

h a v b e d e v e l

• s

i a p t t h e

• r

e t i c f

• r

m u O t h e r

• m

r e t n u m e r i c m e t h

• I

i s

• c
• u

p

• s

t t s i z

• t

h p r

• b

l m a k i i m p

• s

t u n u t e c h n i q u e O n c a t h t u t h e u r

• a

p m a t m e t h

• d

A m

• t

h l a t t m

• g

e

• i

r e n

• r

m a l i z a t i 6 . W h a a b

• r

e n

• r

m a l i z T h r e s u l

• b

t a i b r e n

• r

m a l a s i t t h

• s
• b

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

• v

i t h t f l l i a n t 1 3 . w . w 1 3 . * m

• m

a s * @ .

• 4

. 3 . O . m

,

.@.@l 0.1 02 . 3 . 4

n

0.s 0.6 0.7 0.0 0.9 F i g . 1 4 . D e v i a t i

• n

b e t w e e n t h e p e r m e a b i l i t y

• b

t a i n e d b y s

• l

v i n t h d i f f u s i e q u a t i a n u s i r e n

• r

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

• f

a l

• w
• p

e r m e a b i l i t y m e d i u m f

• t

h r e a n i s

• t

r

• p

r a t i

• t

h m e s h ( f r M a l a H e w

SLIDE 17 T a b l e 2 . C

• m

p a r i s

• n
• f

u p

• s

c a f i n g t e c h n i q u e s . T h e c

• l

u m n S p e e d g i v e s t h e r a p i d i t y w i t h w h i c h t h e t e c h n i q u e c a n b e u s e d . T h e c

• l

u m n T e n s

• r

i n d i c a t e s w h e t h e r

• n
• t

h t e c h n i q u i c a p a h

• c

a l c u l a a p e r m e t e n s

• r

a n d i f S O , w h e t b e r i t i s d i a g

• n

a l ( D i a g . ) , s y m m e t r i c ( S y m . )

• r

n

• n
• s

y m m e t r i c ( N . S . ) A s s u m p t i

• n

s M e t h

• d

M e d i u m F l

• w

S p e e d T e n s

• r

R e m a r I n e q u a l i t i e s f

• r

t h e e q u i v a l e n t p e r m e a b i l i t y W i e n e r b

• u

n d s C a r d w e l l a n d P a r s

• n

b

• u

n d s H a s h i n a n d S h t r i k m m r b

• u

n d s M a t h e r

• n

b

• u

n d E n e b

• u

n d s H e u r i s t i c m e t h

• d

S a m p l i n g P

• w

e r a v e r a g e s A v e r a g i n g b

• u

n d s A n a l y t i c a l s

• l

u t i

• n
• f

t h e d i f f u s i

• n

e q u a t i

• n

A r i t h m e t i c m e a n H a r m

• n

i c m e a n Q u i n t a r d a n d W h i t a k e r ’ s f

• r

m u l a S t

• c

h a s t i c m e t h

• d

G e

• m

e t r i c m e a n P

• w

e r m e a n w i t h e x p

• n

e n t 1 / 3 P e r t u r b a t i

• n

E f f e c t i v e m e d i u m t h e

• r

y S t r e a m t u b e P e r c

• l

a t i

• n

t h e

• r

y R e n

• r

m a l i z a t i

• n

S i m p l i f i e d S t a n d a r d T e n s

• r

i a l N u m e r i c a l s

• l

u t i

• n
• f

t h e d i f f u s i

• n

e q u a t i

• n

P e r r n e a m e t e r b

• u

n d a r y c

• n

d i t i

• n

s P e r i

• d

i c b

• u

n d a r y c

• n

d i t i

• n

s U n i f

• r

m b

• u

n d a r y c

• n

d i t i

• n

s P e r t u r b e d b

• u

n d a r y c

• n

d i t i

• n

s G e n e r a l t e n s

• r

s c a l i n g ( G T S ) U s e

• f

l

• c

a l n e i g h b

• r

h

• d

s H

• m
• g

e n e

• u

s

• e

q u a t i

• n

m e t h

• d

H

• m
• g

e n i z a t i

• n

A v e r a g i n g w i t h c l

• s

u r e M

• m

e n t m e t h

• d

N

• n

e N

• n

e I s

• t

r

• p

i c b i n a r y M

• s

a i c , r a n d

• m

i s

• t

r

• p

i c 2 D P e r i

• d

i c N

• n

e N

• n

e N

• n

e S t r a t i t i e d S t r a t i f i e d B i n a r y s t r a t i f i e d 2 D , K a n d i t s i n v e r s e h a v e t h e s a m e d i s t r i b u t i

• n

a n d a r e i n v a r i a n t b y 9 ° r

• t

a t i

• n

3 D , l

• g
• n
• r

m a l i s

• t

r

• p

i c S m a l l v a r i a n c e H e t e r

• g

e n e i t i e s d

• n
• t

i n t e r a c t B i n a r y ,

• n

e i m p e r v i

• u

s p h a s e C l

• s

e t

• t

h e p e r c

• l

a t i

• n

t h r e s h

• l

d L

• c

a l l y i s

• t

r

• p

i c L

• c

a l l y i s

• t

r

• p

i c N

• n

e N

• n

e P e r i

• d

i c N

• n

e N

• n

e N

• n

e N

• n

e P e r i

• d

i c P e r i

• d

i c P e r i

• d

i c U n i f

• r

m U n i f

• r

m U n i f

• r

m U n i f

• r

m P e r i

• d

i c N

• n

e N

• n

e U n i f

• r

m P a r a l l e l t

• t

h e s t r a t a P e r p e n d i c u l a r t

• t

h e s t r a t a U n i f

• r

m U n i f

• r

m U n i f

• r

m U n i f

• r

m U n i f

• r

m V e r t i c a l U n i f

• r

m N

• n

e N

• n

e N

• n

e U n i f

• r

m U n i f

• r

m N

• n

e U n i f

• r

m P e r i

• d

i c P e r i

• d

i c P e r i

• d

i c + + + + — + + + + + + + + + + + + + + + + — — N

• D

i a g . N

• N
• S

y m . N

• D

i a g . D i a g , N

• N
• S

y m . N

• N
• D

i a g . D i a g . N

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