compositional hydrocarbon reservoirs. continuous thermodynamics

7/29/2019 Compositional Hydrocarbon Reservoirs. Continuous Thermodynamics

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ELSEVIER

A b s t r a c t

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

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

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

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

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

( c o n t i n u o u s ) h e a v y f r a c t i o n s o f t h e o i l i s e s t a b l i s h e d a n a l y t i c a l l y u s i n g t h e method of moments . T h i s a l l o w s t h e

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

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

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

a v a i l a b l e

Keywords : T h e o r y , H y d r o c a r b o n r e s e r v o i r f l u i d s ; C o m p o s i t i o n a l g r a d i n g , C o n t i n u o u s m i x t u r e s , M e t h o d s o f c a l c u l a -

t i o n

F l u i d P h a s e E q u i l i b r i a 102 (1994) 143-158

Computation of compositional grading in hydrocarbon

r e s e r v o i r s . Application of continuous thermodynamics

C a r l o s L i r a - G a l e a n aa ,

*, Abbas Firoozabadi a, John M . P r a u s n i t zb

Reserv oir Engineering Research Institute, 845 Page Mill Road, Palo Alto, CA 94304 , US A

b Chemical Engineering Department, Univ ersity of California-Berkeley and L awrence Berkeley L aboratory,Berkeley, CA 94720, US A

R e c e i v e d O c t o b e r 2 6 , 1 9 9 3 , a c c e p t e d i n f i n a l f o r m J u l y 2 0 , 1 9 9 4

1 . I n t r o d u c t i o n

C o n s i d e r a b l e v a r i a t i o n i n c o m p o s i t i o n a n d PVT p r o p e r t i e s w i t h d e p t h h a s b e e n o b s e r v e d i nv a r i o u s o i l a n d g a s - c o n d e n s a t e f i e l d s a r o u n d t h e w o r l d ( M e t c a l f e e t a l . , 1988 ; Neveux and

Sathikumar, 1988 ; B a t h e t a l . , 1980 ; Espach and Fry, 1951) . F o r i n s t a n c e , b u b b l e p o i n t p r e s s u r emay change as much as 0 .07-0 .11 MPa m - i i n a n o i l r e s e r v o i r , w h i l e t h e A P I g r a v i t y m a y

* Corresponding author . P r e s e n t a d d r e s s : I n s t . M e x i c a n o d e l P e t r o l e o , S u b d i r e c c i o n d e T e c n o l o g i a d e E x p l o t a c i o n ,

Mexico 0 7 7 3 0 , Mexico .

S S D I 0 3 7 8 - 3 8 1 2 ( 9 4 ) 0 2 5 8 0 - 0


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3/16

C L i r a - G a l e a n a e t a l / F l u i d P h a s e E q u i l i b r i a 1 0 2 ( 1 9 9 4 ) 1 4 3 - 1 5 8 1 4 5

2 . C o m p o s i t i o n a l g r a d i n g w i t h a n E O S

H i r s c h b e r g ( 1 9 8 8) h a s g i v e n a n i l l u s t r a t i o n o f t h e r o l e o f t h e p l u s f r a c t i o n s i n c o m p o s i t i o n a l

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

c o l u m n c o n t a i n i n g a s p h a l t e n e s . T h i s s e c t i o n e x t e n d s H i r s c h b e r g ' s i d e a o f s e g r e g a t i o n t o s y s t e m s

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

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

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

s t a t e . The oil composition is characterized by a set of k discrete ( i . e i d e n t i f i a b l e ) l i g h t

components, and one (or more) heavy-hydrocarbon ensembles ( i . e . C 7 f r a c t i o n ) . Then Eq . ( 2 )

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

within the framework continuous thermodynamics . C o n t i n u o u s t h e r m o d y n a m i c s i s a n e x t e n s i o n

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

components (Cotterman et al ., 1 985 ; C o t t e r m a n a n d P r a u s n i t z , 1 9 8 5 ; D u and Mansoori, 1986) .

To account for the polydisperse nature of petroleum, continuous thermodynamics replaces

discrete values of composition by a continuous distribution function F ( 1 ) , where I i s a

c o n t i n u o u s c h a r a c t e r i z a t i o n v a r i a b l e . A suitable variable may be, for example, the normal

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

s u c h t h a t

JF(fldl=l ( 3)w h i c h s h o w s t h a t t h e i n t e g r a l o f F ( I ) d I r e p r e s e n t s t h e a v e r a g e m o l e f r a c t i o n f o r s p e c i e s w h o s e

m o l e c u l a r w e i g h t f a l l s i n t h e r a n g e b e t w e e n I and I + d L A n a l o g o u s t o c o n v e n t i o n a l p h a s e - b e -

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

tion function of one thermodynamic state (or phase) of the fluid to those of all other

t h e r m o d y n a m i c s t a t e s ( o r p h a s e s ) o f t h e s y s t e m .

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

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

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

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

( 1 9 8 5 ) ; o t h e r r e l a t e d e x a m p l e s h a v e b e e n p r o p o s e d i n t h e l i t e r a t u r e ( L i r a - G a l e a n a e t a l., 1 9 9 1 ;

Luks et al . , 1 9 9 0 ) . The second class for performing calculations takes advantage of the

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

equilibrium problem . One example of this second class is the Gaussian quadrature method

d e v e l o p e d b y C o t t e r m a n a n d P r a u s n i t z ( 1 9 8 5 ) . D e t a i l s o n a p p l i c a t i o n s o f c o n t i n u o u s t h e r m o d y -

namics can be found elsewhere (Chorn and Mansoori, 1989 ; Y i n g e t a l . , 1 9 8 9 ; L i r a - G a l e a n a e t

a l . , 1 9 9 2 ) .

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

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

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

p h a s e s e p a r a t i o n ( L u k s e t a l . , 1 9 9 0 ; S a n d i e r a n d L i b b y , 1 9 9 1 ) . T h e s e c r i t i c i s m s a r e n o t r e l e v a n t

h e r e . Since the method of moments is here applied to a single phase r e s e r v o i r- f l u i d c o l u m n ,

material balance problems are avoided . We select the two-parameter gamma distribution


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146 C L i r a - G a l e a n a e t a l / F l u i d P h a s e E q u i l i b i i a 1 0 2 ( 1 9 9 4 ) 1 4 3 - 1 5 8f u n c t i o n t o r e p r e s e n t F ( I ) . T h e c h o i c e o f t h i s d i s t r i b u ti o n i s b a s e d o n i t s a b i l i t y t o r e p r e s e n t t h e

f o r m o f t h e m o l a r d i s t r i b u t i o n ( i . e . m o l e f r a c t i o n / m o l e c u l a r w e i g h t r e l a t i o n s h i p ) o f t r u e r e s e r v o i r

f l u i d m i x t u r e s ( W h i t s o n , 1 9 8 3 ) . T h e g a m m a d i s t r i b u t i o n h a s t h e f o r m

F ( I ) - - ( I~ a 1 [ exp(- I t i/

( 4 )

w h e r e F i s t h e g a m m a f u n c t i o n . I n E q . ( 4 ) , a a n d / 3 d e t e r m i n e t h e s h a p e o f t h e d i s t r i b u t i o n . F o r

g a s - c o n d e n s a t e f l u i d s , w h e n I i s c h o s e n e q u a l t o M , a i s c l o s e t o 1 ; i n t h a t e v e n t , E q . (4 ) is a n

e x p o n e n t i a l - t y p e d i s t r i b u t i o n . P a r am e t e r y f i x e s t h e o r i g i n o f t h e d i s t r i b u t i o n . T h e f i r s t s t a t i s t i c a l

moment gives the mean, 0 . A f u n c t i o n o f t h e s e c o n d s t a t i s t i c a l m o m e n t g i v e s t h e v a r i a n c e , r -

0 = Mean =

J

I F ( I ) d I = a / 3 + y ( 5 )

T ' = Var = f( I - 0 ) ' ' F ( I ) d I = a / 3 ' ( 6)

In Eqs . ( 5 ) a n d ( 6 ) , a i s d i m e n s i o n l e s s w h i l e / 3 a n d y h a v e t h e s a m e u n i t s a s L

T o c a l c u l a t e t h e f u g a c i t i e s i n E q . ( 2 ) , a s u i t a b l e , y e t s i m p l e E O S o f t h e S R K f o r m d e v e l o p e d

by Cotterman and Prausnitz (1985) (P SRK-EOS) is employed in this work . A p p e n d i x A g i v e s

a d e t a i l e d d e s c r i p t i o n . Based on the linear dependence of the PS RK- EOS parameters a'

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

r u l e s f o r t h e s e t w o p a r a m e t e r s , t h e r a t i o o f f u g a c i t y c o e f f i c i e n t s f o r e a c h s p e c i e s w i t h i n t h e C 7

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

( D I (M)= exp(CG + C?M) ( 7 )

~ii(M)

w h e r e s u b s c r i p t s I a n d I I r e f e r t o t h e r m o d y n a m i c s t a t e s . I n E q . ( 7 ) , CG and CG are EOS-derivedp a r a m e t e r s t h a t a r e i n d e p e n d e n t o f t h e m o l e c u l a r w e i g h t . T h e e x p r e s s i o n s f o r CG and C 2 c a nbe found in Appendix B .

U s i n g t h e P S R K - E O S t h e d e f i n i t i o n o f f u g a c i t y f o r a s p e c i e s w i t h i n t h e C, + f r a c t i o n w h o s e

c o n t i n u o u s c h a r a c t e r i z a t i o n v a r i a b l e i s M s g i v e n b yf(M) = (D(M)gF(M)P ( 8 )

where 't(M) i s t h e f u g a c i t y c o e f f i c i e n t o f a s p e c i e s w i t h m o l e c u l a r w e i g h t Mn d q i s t h e ( o v e r a l l )m o l e f r a c t i o n o f t h e c o n t i n u o u s f r a c t i o n . I n E q . ( 8 ) , P i s t h e h y d r o s t a t i c p r e s s u r e , w h i l e t h eq u a n t i t y tjF(M), r e p r e s e n t s a n a v e r a g e m o l e f r a c t i o n f o r a s p e c i e s w i t h m o l e c u l a r w e i g h t e q u a l

t o MF o r t h e c o n t i n u o u s f r a c t i o n ( i . e . t h e C,+ f r a c ti o n ) , w e r e p l a c e f , and f i n E q . (2 ) b y f(M) andf(M) (from Eq ( 7 ) ) , r e s p e c t i v e l y . I n t h i s w a y , t h e i s o f u g a c i t y c o n d i t i o n f o r a s p e c i e s w i t h i n t h e

C7 - , f r a c ti o n i s g i v e n b yghFh(M)~Dh(M)Ph = rlhoFho(M)Oho M)P h o e x Mg ( h - h o ) ( 9 )p C -

T g ,

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


5/16

lhoR`' - 1 - /3ho(CG - b )

and

rlh = phexp[CG + y(CG - b)][ 1

J " h oho Ph 1 - Nho(CG _b)E qs . (11) and (12) are the continuous thermodynamic equations analogous to Eq . ( 4 ) i n t h e

H i r s c h b e r g p a p e r ( H i r s c h b e r g , 1 9 8 8 ) . H o w e v e r , u n l i k e p r e v i o u s f o r m u l a t i o n s , E q s . ( 1 1 ) a n d ( 1 2 )

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

means of an EOS .

3 . C o m p u t a t i o n a l a l g o r i t h m

The calculation method described here requires simultaneous solution of k i s o f u g a c i t ye q u a t i o n s d e s c r i b e d b y E q . ( 2 ) f o r t h e d i s c r e t e c o m p o n e n t s a n d E q ( 1 2 ) , w h i c h w a s d e r i v e d b y

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

n e c e s s a r y t o s o l v e a t o t a l n u m b e r o f k + 1 s i m u l t a n e o u s n o n - l i n e a r e q u a t i o n s w h o s e i n d e p e n d e n t

v a r i a b l e s a r e p r e s s u r e a n d k compositions. A multidimensional Newton-Raphson algorithm canb e u s e d t o s o l v e t h e s e e q u a t i o n s . E q . ( 1 1 ) i s a c o n s t r a i n t .

O n c e t h e t o t a l a l g o r i t h m h a s c o n v e r g e d , t h e m e a n m o l e c u l a r w e i g h t o f t h e C 7+ f r a c t i o n a t t h e

d e p t h o f i n t e r e s t ( i . e . 0 = a / 3 + y ) i s u s e d t o p s e u d o i z e t h e c o m p o s i t i o n o f t h e C 7 + f r a ct i o n . T h e

p s e u d o i z a t i o n p r o c e s s c a n b e p e r f o r m e d b y t h e q u a d r a t u r e m e t h o d ( C o t t e r m a n e t a l . , 1 9 8 6 ) o r

b y t h e g e n e r a l i z e d p r o p e r t i e s - o f - g r o u p s m e t h o d p r o p o s e d b y W h i t s o n ( 1 9 8 3 ) . T h e s e c o n d s t a g e

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

p s e u d o i z a t i o n p r o c e s s .

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

o f t h e E O S t o re p r e s e n t t h e v o l u m e t r i c b e h a v i o r o f t h e r e s e r v o i r - f l u i d c o l u m n . E q s . ( 1 1 ) a n d ( 1 2 )

provide the connection of volumetric behavior and the extent of segregation for a given

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

C L i r a - G a l e a n a e t a l . / F l u i d P h a s e E q u i l i b r i a 1 0 2 ( 1 9 9 4 ) 1 4 3 - 1 5 8 1 4 7we multiply Eq . ( 9 ) b y M . B e c a u s e c o n t i n u o u s t h e r m o d y n a m i c s i s f o r a s o l u t i o n c o n t a i n i n g a n

infinite number of components, Eq . ( 9 ) i s r e p r e s e n t a t i v e o f a m u l t i t u d e o f e q u a t i o n s , o n e f o r

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

d e p t h , i t i s n e c e s s a r y t o e x t e n d E q . ( 9 ) t o t h e e n t i r e Mdomain from y to c c u s i n g t h e m e t h o dof moments . The rth moment of F(M),

Mr,i s d e f i n e d b y Mr = f MrF(M) dM M u l t i p l y i n g

b o t h s i d e s o f E q . ( 9 ) b y M a n d r e a r r a n g i n g g i v e s

J XMF (M) dM = O,,

= C ghoPho / FMF o(M) exp(CG + CGM) exp(-Mb) dM ( 1 0 )

1 h P h

where b = (g/RT gj(h - h ) . T h e c h o i c e o f y i s s o m e w h a t a r b i t r a r y a n d d i r e c t e d b y c h e m i c a l

a n a l y s i s . T h e r e i s n o r e a s o n t o c h a n g e y w i t h d e p t h . T h e r e f o r e w e s e t 1'h = yho .T o a v o i d h i g h e r

o r d e r m o m e n t s , w e a l s o s e t a h = a h o .C a r r y i n g o u t t h e i n t e g r a t i o n , w e r e l a t e Fh (M) t o Fh o(M)

t h r o u g h t h e f o l l o w i n g e x p r e s s i o n s d e r i v e d f r o m E q . ( 1 0 ) :


6/16

1 4 8 C L i r a - G a l e a n a e t a l . / Fluid Phase Equilibria 102 (199 4) 143-1 58t h e d i s t r i b u t i o n f u n c t i o n p a r a m e t e r s o f t h e p l u s f r a c t i o n , a r e s t ro n g f u n c t i o n s o f t h e " v o l u m e t r i c "

parameters CG and CG . The PSRK-EOS used in this work was developed from vapor pressure

data of heavy hydrocarbons . W e o b s e r v e d t h a t i n c o r p o r a t i o n o f a s i n g l e b i n a r y i n t e r a c t i o n

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

o f t h e E O S u s e d i n t h i s w o r k . A p r a c t i c a l m e t h o d i s t o a d j u s t t h e b i n a r y i n t e r a c t i o n c o e f f i c i e n t

between methane and the C 7 c o n t i n u o u s f r a c t i o n t o h y d r o s t a t i c - p r e s s u r e a n d c o m p o s i t i o n d a t a

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

m a y b e u s e d t o i n c o r p o r a t e t h e v o l u m e t r a n s l a t i o n c o n c e p t ( P e n e l o u x e t a l . , 1 9 8 2 ) , i n a w a y s u i t e d

t o c o n t i n u o u s o r s e m i c o n t i n u o u s s y s t e m s .

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

f r a c t i o n , t h e p s e u d o i z a t i o n p r o c e s s i s p e r f o r m e d a n d t h e E O S i s u s e d t o c o m p u t e t h e v a r i a t i o n

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

a t h , c a l c u l a ti o n o f t h e s a t u r a t i o n p r e s s u r e b y t h e E O S r e q u i r e s m e t h a n e i n t e r a c t i o n c o e f f i c i e n t s

with the pseudocomponents determined from the pseudoization process . S u c h i n t e r a c t i o n

c o e f f i c i e n t s , a s e x p e c t e d , a r e s i m i l a r t o t h o s e o b t a i n e d f r o m V L E - b a s e d c o r r e l a t i o n s.

3 . 1 . Comparison with discrete multicomponent formalism

P r i o r t o a p p l i c a ti o n t o f i e l d d a t a , i t i s u s e f u l t o c o n s i d e r a c o m p a r i so n o f t h e p r o p o s e d m e t h o d

w i t h t h e d i s c r e t e m u l t i c o m p o n e n t f o r m a l i s m w h i c h u s e s o n l y E q . ( 2 ) . Consider a hydrocarbon

l i q u i d c o l u m n a t 3 9 4 . 4 K . T h e p r e s s u r e a t t h e r e f e r e n c e d e p t h i s 3 0 M P a . T h e l i q u i d i s c o m p o s e d

of an equimolar mixture of methane and 13 n-alkanes whose molecular weights fall between

o c t a n e a n d n - e i c o s a n e ( n - C 2 0 ) . T h e c o m p o s i t i o n s f o r t h e 1 3 n - a l k a n e s u s e d i n t h i s c o m p a r i s o n

a r e : ( a ) d i s c r e t e v a l u e s s h o w n i n F i g . 1 , a n d ( b ) g a m m a d i s t r i b u t i o n f u n c t i o n s h o w n i n F i g . 2 .

D e t a i l s o f t h e t w o e q u i v a l e n t c h a r a c t e ri z a t i o n s c a n b e f o u n d e l s e w h e r e ( L i r a - G a l e a n a e t a l ., 1991) .

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

distance of 500 m . F o r t h i s p a r t i c u l a r a p p l i c a t i o n , a l l b i n a r y i n t e r a c t i o n c o e f f i c i e n t s b e t w e e n

methane and each paraffin were computed as a function of the paraffin molecular weight,

according to Cotterman (1985) :

k c t i , = -0 . 1 0 0 + 0 .309 exp(-0 .006 88M)T a b l e 1

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

methane and 13 n-alkanes at 394 4 K at depths of 0 m and 500 m

C a l c u l a t e d p r o p e r t y Calculation method

Multicomponent Proposed method

0 m 500 m 0 m 500 m

H y d r o s t a t i c p r e s s u r e ( M P a ) 30 . 0 2 7 5 6 3 0 0 2 7 5 3

Bubblepomt pressure (MPa) 1 8 .3 6 1 9 .22 1 8 . 0 7 1928

Methane (mol%) 5 0 0 0 5 2 3 0 5 0 0 0 5 2 0 7


7/16

. 1 4

. 1 2

.10

0U. 0 8Vww00x . 0 6

.04

.02

. 0 0

1 0 0 1 2 0 1 4 0 1 6 0 1 8 0 2 0 0 2 2 0

Molecular Weight

Fig . 1 . D i s c re t e c o m p o s i t i o n r e p r e se n t a t i o n f o r a f ra c t i o n o f 1 3 n -a l k a n e s

The critical properties and acentric factors for all discrete components were taken from

Magoulas and Tassios (1990 ) . Table 1 shows the calculated hydrostatic and bubblepoint

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

changes for methane . F o r t h e p r o p o s e d m e t h o d , t h e m e a n m o l e c u l a r w e i g h t o f t h e c o n t i n u o u s

f r a c t i o n a t t h e re f e r e n c e l e v e l w a s u s e d t o e s t i m a t e t h e v a l u e o f kc 1 -M .

T a b l e 1 s h o w s t h a t t h e c a l c u l a t e d h y d r o s t a t i c p r e s s u r e s a r e v i r t u a l l y t h e s a m e f o r t h e t w o

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

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

4. C o m p a r i s o n w i t h f i e l d d a t a : E a s t P a i n t e r r e s e r v o i r

C r e e k a n d S c h r a d e r ( 1 9 8 5 ) h a v e r e p o r t e d s e g r e g a t i o n d a t a f o r t h e E a s t P a i n t e r f i e l d i n t h e

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

f r a c t i o n o f t h e C 7 f r a c t i o n i n c r e a s e s f r o m 5 . 7 t o 8 . 6 % o v e r a v e r t i c a l d i s t a n c e o f 2 3 0 m . T h e

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

C. L i r a - G a l e a n a e t a l . / F l u i d P h a s e E q u i l i b r i a 1 0 2 ( 1 9 9 4 ) 1 4 3 - 1 5 8

2 4 0 2 6 0 2 8 0

1 4 9


8/16

1 5 0 C Lira-Galeana et al . / Fluid Phase Equilibria 102 (1994) 143-158

100 150 200 250 35 0

Molecular Weight

F i g 2 C o n t i n u o u s c o m p o s i t i o n r e p r e s e n t a t i o n f o r a f ra c ti o n o f 13 n - al k a n e s

T a b l e 2

R e s e r v o i r f l u i d c o m p o s i t i o n f o r t h e E a s t P a i n t e r r e s e r v o i r a t t h e r e f e r e n c e d e p t h (h _ - 1297 m)

:11JiIII1111iiiii11 111

Component M o l e f r a c t i o n

Methane 0 . 6 5 70

Ethane 0 . 1 1 2 0

Propane 0 . 0 6 2 0

i s o - B u t a n e 0 0 1 6 2

n-Butane 00215

i s o - P e n t a n e 0 . 0 0 9 1

n-Pentane 0 . 0 0 8 3

Hexanes 0 . 0 1 0 8C7 + 0 . 0 8 5 9Carbon dioxide 00014

Nitrogen 00145

Dewpoint pressure = 30 7 MPaC7 + mean mol . w e i g h t = 1 5 8C7 + s p . g r a v it y = 0 7 9 6R e s e r v o i r t e m p e r a t u r e = 3 6 1 1 K

1 2

1 0

c n 8

*0a05 6r AhA1wIt0 4

2

0


9/16

C L i r a - G a l e a n a e t a l . / F l u i d P h a s e E q u i l i b r i a 1 0 2 ( 1 9 9 4 ) 1 4 3 -1 5 8 1 51T a b l e 2 s h o w s t h e m e a s u r e d c o m p o s i t i o n o f t h e r e s e r v o i r f l u i d ( w e l l 4 2 - 7 A ) a t t h e r e f e r -

ence depth . The measured characterization parameters for the C-, f r a c t i o n a r e m o l e c u l a r

w e i g h t , 1 5 8 a n d s p e c i f i c g r a v i t y , 0 . 7 9 6 . The mole percent of the C 7 f r a c t i o n a n d d e w p o i n t

p r e s s u r e o f t h e r e s e r v o i r f l u i d a t t h e r e f e r e n c e d e p t h ( - 1 2 9 7 m s u b s e a) , a r e 8 . 6 % a n d 3 0 .7 MPa,

r e s p e c t i v e l y .

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

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

t h e C7

f r ac t i o n . W h i t s o n ( 1 9 8 3 ) h a s s h o w n p r a c t i c a l p r o c e d u r e s f o r o b t a i n i n g t h e s e p a r a m e t e r s

f r o m a v e r ag e c h a r a c t e r i z a t i o n p r o p e r t i e s o f t h e h e av y f r a c t i o n . T h e e s t i m a t e d m e a n a n d v a r i a n c e

of the gamma distribution function are O = 158 and i 2 = 1 2 1 3 8 , r e s p e c t i v e l y . The molar

d i s t r i b u t i o n o f t h e p l u s f r a c t i o n i s s h o w n i n F i g . 3 . B a s e d o n d i s t i l l a t i o n d a t a a n d t h e r e l a t i v e l y

" l i g h t " n a t u r e o f t h e C7 f r a c t i o n , t h e s h a p e o f t h e d i s t r i b u t i o n i s e x p o n e n t i a l .

Creek and Schrader (1985) used a regressed Peng-Robinson-EOS (Peng and Robinson, 1 9 7 6 )

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

EOS, they used the measured PVT d a ta fo r t h e r e se r v oi r f l u i d . N o d e t a i l s w e r e p r o v i d e d o n t h er e g r e s s i o n p a r a m e t e r s a n d o n s p e c i f i c d a t a e m p l o y e d i n t h e r e g r e s s i o n . Using the tuned EOS ,

1 0

1 0 0 1 2 0 1 4 0 1 6 0

Molecular Weight

Fig 3 . M o l a r d i s t r i b u t i o n s o f t h e C 7 f r a c t i o n o f t h e E a s t P a i n t e r r e s e r v o i r f l u i d a t t w o d e p t h s

1 8 0


10/16

152 C Lira-Galeana et al . / Fluid Phase Equilibria 102 (1994 ) 143-158these authors matched the composition distribution of methane and C, + f r a c t i o n a n d t h e

s a t u r a t i o n p r e s s u r e v a r i a t i o n .

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

r e s e r v o i r f l u i d a l o n g a v e r t i c a l d i s t a n c e o f 2 3 0 m a b o v e t h e r e f e r e n c e p o i n t . P a r a f f i n i c c o r r e l a -

t i o n s ( C o t t e r m a n e t a l ., 1985) for determining the PSRK-EOS parameters for the continuous

f r a c t i o n w e r e e m p l o y e d . T a b l e 3 s h o w s t h e c o m p u t e d m e t h a n e a n d C , + c o m p o s i t i o n d i s t r i b u t i o n

obtained from the new procedure, compared with field measurements . To compute these

v a r i a t i o n s , w e u s e d a b i n a r y i n t e r a c t i o n p a r a m e t e r 0 .15 between methane and the continuous

C, + f r a c t i o n .

T a b l e 3 s h o w s th a t t h e c o m p u t e d s e g r e g a t i o n c h a r a c t e r i s t i c s o f t h e o i l c o l u m n a g r e e w i t h f i e l d

d a t a f o r b o t h m e t h a n e a n d t h e C , + f r a c t i o n . T h e p r e d i c t e d m o l e c u l a r w e i g h t v a r i a t i o n f o r t h e

C, + f r a c t i o n , c o m p u t e d w i t h E q s . ( 5 ) a n d (1 2 ) , a g r e e s v e r y w e l l w i t h t h e v a r i a t i o n r e p o r t e d a t

f i e l d s c a l e . Fi g . 4 s h o w s t h e v a r i a t i o n

O n c e t h e f i r s t ' p a r t o f t h e c a l c u l a t i o n h a s b e e n c o m p l e t e d , t h e s e c o n d p a r t o f t h e c a l c u l a t i o n

r e q u i r e s t h e p s e u d o i z a t i o n f o r t h e c a l c u l a t i o n o f t h e s a t u r a ti o n p r e s s u r e. F o r t h i s c a s e , w e u s e d

2 5 0

20 0

0

- 5 0

IIIIII 11 1 111III,,,IIIIIIIIIIIIfII11111!IIIIII111111IIIIIIIO

0

- - - Calculated0 Field Data

1 4 0 1 6 0 1 8 0 2 0 0 22 0

Molecular W eight

F i g 4 V a r i a t i o n o f m e a s u r e d a n d c a l c u l a t e d m o l e c u l a r w e i g h t s w i t h d e p t h f o r t h e C 7f r a c t i o n o f t h e E a s t P a i n t e r

r e s e r v o i r f l u i d

1 0 0 1 2 0


11/16

20 0

- 2 0 0

C Lira-Galeana et al / Fluid Phase Equilibria 102 (1 994) 143-1 58I I

. . . . . . . . . .

- - - - -

I I I 1 1 1 1 1A 1 I

Dewpoint DataO H y d r o s t a t i c p r e s s u r e D a t aC a l c u l a t e d s a t u r a t i o n p r e s s u r e

C a l c u l a t e d h y d r o s t a t i c p r e s s u r e

1 1 1 11 I 1 1 I

- I-28 29 30

1 5 3

3 1 32 3 3 3 4 3 5

Pressure, MPa

Fig 5 V ariation of calculated and measured saturation and hydrostatic pressures with depth for the E ast Painter

r e s e r v o i r f l u i d

Table 3

Measured (Creek and Schrader, 1 985) and calculated segregation characteristics for the East Painter reserv oir-fluid

column . C 7 and C, mole fraction, and C 7 + p ro p er ti e s

Property Reference depth = 0 m

0 m 77 m 154 m 230 mC, + Molecular weight 15 8 . 0 ( e x p ) 1 5 2 3 8 1 4 5 1 8 1 3 9 5 1

T, (K) 638 24 628 78 619 55 610 79Pc (MPa) 23351 2 4 0 5 1 2 . 4 7 3 2 5 3 8( j ) 06556 06291 06035 0 . 5 7 9 6

Mole fraction, data 00859 0 . 0 7 6 5 00691 00575

M o l e f r a c t i o n , c a l c 00859 00787 00712 006 34

C, Mole fraction, data 0 . 6 5 7 6 0 6750 0 . 6 8 0 2 0 6912

M o l e f r a c t i o n , c a l c 0 6 576 0 6 6 84 0 6782 0 689 6


12/16

1 5 4 C. L i r a - G a l e a n a e t a l . / F l u i d P h a s e E q u i l i b r i a 1 0 2 ( 1 9 9 4 ) 1 4 3 - 1 5 8a s i n g l e p s e u d o c o m p o n e n t r e p r e s e n t i n g t h e e n t i r e C 7 f r a c t i o n . T a b l e 3 p r e s e n t s t h e v a r i a t i o n o fC

7c h a r a c t e r i z a t i o n f r o m t h e f i r s t p a r t o f t h e a l g o r i t h m . T h i s t a b l e s h o w s t h a t t h e c r i t i c a l

p r o p e r t i e s a n d a c e n t r i c f a c t o r o f t h e C 7 p s e u d o c o m p o n e n t c h a n g e s u b s t a n t i a l l y , a s a r e s u l t o f

m o l e c u l a r w e i g h t v a r i a t i o n . W i t h t h e c h a r a c t e r i z a t i o n p a r a m e t e r s o f T a b l e 3 , t h e P S R K - E O S i s

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

computed . Fi g . 5 s h o w s t h e r e s u l t s . A s i n g l e i n t e r a c t i o n p a r a m e t e r b e t w e e n m e t h a n e a n d t h e C 7 +

f r a c t i o n ( -0 . 0 2 ) s u f f i c e s t o g i v e a g o o d r e p r e s e n t a t i o n o f t h e s a t u r a t i o n p r e s s u r e . F i g . 5 i m p l i e sn o g a s - o i l c o n t a c t ( G O C ) f o r t h e f l u i d c o l u m n .

5 . C o n c l u s i o n s

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

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

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

state . We have selected the gamma distribution function to represent the molecular weight

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

p a r a m e t e r s o f t h e d i s t r i b u t i o n f u n c t i o n ( / 3 ) i s a l l o w e d t o v a r y w i t h d e p t h T h e r e f o r e , o n l y

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

be no need to vary both a and /3 with depth . The method outlined above was also used to

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

between data and calculated results was obtained (the results are not shown) . The good

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

model .

Acknowledgments

T h i s w o r k w a s s u p p o r t e d b y t h e D i r e c t o r , O f f i c e o f E n e r g y R e s e a r c h , O f f i c e o f B a s i c E n e r g y

Science, Chemical Sciences Div ision of the US Department of Energy under Contract No

D E - A C 0 3 - 76 S F 0 0 0 9 8 . C . L .G . a c k n o w l e d g e s t h e N a t i o n a l C o u n c i l f o r S c i e n c e a n d T e c h n o l o g y

of Mexico (CONACyT) ; t h e C h e m i s t r y F a c u l t y o f t h e N a t i o n a l U n i v e r s i t y o f M e x i c o ( D E P F Q -

UNAM), and the Mexican Petroleum Institute (IMP), Mexico, for a fellowship . S u p p o r t f r o m

Norsk Hydro, S audi Aramco and Texaco Inc . i s g r e a t l y a p p r e c i a t e d . D r . B . D i n d o r u k o f t h e

Res. Engng . R e s . Ins t . a s s i s t e d w i t h s o m e o f t h e s a t u r a t i o n p r e s s u r e c a l c u l a t i o n s

L i s t o f s y m b o l s

a EOS (energy) parameter (1 MPa mol - 2 )

b gravity modulus

b b ( M ) EOS (covolume) parameter for discrete and continuous components (1 mol - ' )

C ' EOS-derived functions in Appendix B

f f u g a c i t y o f d i s c r e t e c o m p o n e n t i ( M P a )


13/16

f(M) fugacity of continuous component ( MPa)F(M) d i s t r i b u t i o n function, here taken as a gamma distributiong a c c e l e r a t i o n d u e t o g r a v i t y

g C proportionality constant

h , h depths o f i n t e r e st a n d r e f e r e n ce , r e sp e c ti v e l y (m)

i discrete component index

k t o t a l number of d i s c r e t e ( i . e . i d e n t i f i a b l e ) components

k c , , , m e t h a n e i n t e r a c t i o n co e f f i c i e n tM molecular weight (g mol - ` )N t o t a l number of componentsP p r e s s u r e (M P a )P c r i t i c a l p r e s s u r e ( M P a )R gas constantT temperature ( K)T c r i t i c a l t e m p e r a t u r e ( K )G r e e k l e t t e r s

x , f l , y y d i s t r i b u t i o n f u n c t i o n parameters

r gamma functionA f i n i t e - d i f f e r e n c e operatorr l o v e r a l l m o l e f r a c t i o n o f a c o n t i n u o u s f r a c t i o n

0 c o e f f i c i e n t for temperature dependence in appendix AO f i r s t s t a t i s ti c a l m o m e n t of F(M)(D(M) f u g a c i t y c o e f f i c i e n t of continuous s p e c i e s

w a c e n t r i c f a c t o r

C L i r a - G a l e a n a e t a l . / F l u i d P h a s e E q u i l i b r i a 1 0 2 ( 1 9 9 4 ) 1 4 3 - 1 5 8 1 5 5

Appendix A : the PSR K-EOS

The fugacity c o e f f i c i e n t o f a s p ec i es within a C71 f r a c t i o n u s i n g t h e P S R K - E O S i s given by

RT I n ~(M) = RT Inv +RTb(M) + ab(M)

C

I n

v + b b

v - b v - b b - v v +b/

2[Ex,a(M, j) +01F, (M+)a(M, M+) dM+1i r -n t)+ b -RT1nZ (A1)b vwhere v a n d Z a r e t h e m o l a r v o l u m e a n d c o m p r e s s i b i l i t y f a c t o r o f t h e m i x t u r e , r e s p e c t i v e l y , a n db(M), a(M, j) and a(M, M+) a r e f u n c t i o n s o f temperature and molecular weight according tob(M) =b +b,M

a(M, M)1"2=a(T) +a,(T )M

a(M,j) =al/2(M, M)a 1 1 2 ( j , j ) ( 1 - k , , , , )a(M, M+) = a'/ - (M, M)a'' 2 (M+, M+)(1 - k , , , , , , + )


14/16

1 5 6 C L i r a - G a l e a n a e t a l . / F l u i d P h a s e E q u a h b r i a 1 0 2 ( 1 9 9 4 ) 1 4 3 - 1 5 8T h e t e m p e r a t u r e d e p e n d e n c e f o r c o n s t a n t s a o (T) and a , ( T ) i s g i v e n b y

a o (T) =0, +0,T+0 3 T2a l (T) =0 4 +0 5 T+06 T 2where values of 0 , , 0 , , . . . , 0 6 , b o and b , parameters are given by Cotterman and Prausnitz

(1985) f o r p a r a f f i n i c , a r o m a t i c a n d n a p h t h e n i c h y d r o c a r b o n e n s e m b l e s

The following mixing rules are used for the PSRK-EOS parameters a and b

A k k l ra=~~x,x,(i,j)+2~~x,rl, F , ( M ) a ( i , M) dM

k

+ q, t), F , (M)F, (M+)a(M, M+) dM dM!

5r

b = L x,b, + Y , q , F,(M)b(M) dMI fW i t h t h e s e m i x i n g r u l e s , t h e P S R K - E O S i s f i n a l l y w r i t t e n f o r k discrete components and I

c o n t i n u o u s f r a c t i o n s a s

T aP(T, V, n,, n,A , r 1 i F,(M), ij,F,(M), . . . , r l r F , ( M ) ) = RV -b v(v +b)Appendix B : e x p r e s s i o n s f o r e v a l u a t i n g t h e p a r a m e t e r s i n E q . ( 7 )

C o t t e r m a n e t a l (1985) s t a t e d t h a t E q . ( A l ) c a n b e c a s t a s t h e f o l l o w i n g l i n e a r f u n c t i o n o f

m o l e c u l a r w e i g h t :

I n D(M) = C' + C2M

E x p l i c i t e x p r e s s i o n s t o c o m p u t e t h e p a r a m e t e r s C' and C 2 w e r e n o t r e p o r t e d b y C o t t e r m a n e t

a l (1985), b u t t h e y w e r e d e r i v e d i n t h i s w o r k . T h e e x p r e s s i o n s a r e

C'=boCb/-f fe )a o ( T ) - f dC b,(b)_fc ft-)a , (T )gwhere f b , f c , f d , and f g a r e g i v e n b y

f d = l n ( Z - B )f g = a(M, M) ` - a

k r

fe = 2 x,a(M,j) + 0, 5 F,(M+)a(M, M+) dM J


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AC

Z+B B 1fb=B I nZ+B +Z-Bf c = A I n

Z+BB ZaP

A = ( R T ) 2bPRT

Thus, parameters C' and CG are simply given by

C 1 = C' - C'G it n oand

CG=Cy, -Chu

B =

R e f e r e n c e s

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