24 Four-Phase Flash Equilibrium Calculations for Multicomponent Systems Containing Water R. M . Enick, G. D. Holder, J . A. Grenko, and A. J . Brainard
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Chemical and Petroleum Engineering Department, University of Pittsburgh, Pittsburgh, PA 15261
A technique for predicting one- to four-phase flash equilibrium is presented for multicomponent systems containing water, such as CO /crude o i l / water mixtures which characterize the carbon dioxide miscible flooding of petroleum reservoirs. The Peng-Robinson equation of state is used to describe the aqueous and hydrocarbon phases. An accelerated and stabilized successive substitution method is used to obtain convergence, even in the near c r i t i c a l region. Additional hydrocarbon phases are introduced by using a fugacity based testing scheme. Water-free flash calculations are first performed, yielding one, two or three phase equilibrium. A comprehensive search strategy is then used to consider eleven general classifications of systems which may result, including water-rich liquid/hydrocarbon-rich liquid/CO -rich liquid/vapor equilibrium. Improved methods for obtaining i n i t i a l estimates of additional phases are presented and a reliable scheme of searching for additional hydrocarbon-rich phases is introduced which considers all three possible phase identities. 2
2
General Objectives. Multiple phase behavior is often encountered in the gas miscible flooding of petroleum reservoirs. In water-free carbon dioxide/crude o i l systems, for example, three phases often exist in equilibrium at low temperatures. (1) However, water is almost universally present, either interstitially or because i t is injected for mobility control;(2) hence its presence should also be considered in phase behavior studies. Because of the relatively low miscibility of water with both carbon dioxide and hydrocarbons at reservoir conditions, an aqueous phase will almost always exist, resulting in the possible presence 0097-6156/86/0300-0494$07.50/0 © 1986 American Chemical Society
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
24.
ENICK ET A L .
Four-Phase
Flash Equilibrium
Calculations
o f a s many a s f o u r p h a s e s . In a s p h a l t 1 c crude systems, a f i f t h p h a s e , a s o l i d p r e c i p i t a t e , may a l s o f o r m , ( 3 ) b u t p r e d i c t i n g i t s presence or c o m p o s i t i o n i s beyond t h e scope this study.
of
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G e n e r a l l y , w a t e r - f r e e f l a s h c a l c u l a t i o n s a r e made when t h e s e systems a r e modeled,(_4) a n d a n e q u a t i o n o f s t a t e i s u s e d t o c a l c u l a t e component f u g a c i t i e s . Temperature, pressure and o v e r a l l composition a r e t y p i c a l l y s p e c i f i e d i n the f l a s h c a l c u l a t i o n technique i n order to determine the amounts and c o m p o s i t i o n s of each p o s s i b l e p h a s e . The o b j e c t i v e o f t h i s s t u d y i s t o d e v e l o p a n e f f i c i e n t a l g o r i t h m for CC^/hydrocarbon/water systems i n which the n u m b e r o f p h a s e s , one t o f o u r , i s d e t e r m i n e d a n d i n w h i c h t h e c o m p o s i t i o n a n d amount o f e a c h p h a s e , i n c l u d i n g t h e a q u e o u s phase, i s a c c u r a t e l y described. Emphasis i s p l a c e d on f o u r phase e q u i l i b r i u m and the e f f e c t s of the presence o f water on the phase behavior of a CC^/hydrocarbon m i x t u r e . Review of the L i t e r a t u r e : Multiphase Flash Calculations. The m o d e l f o r t h e two p h a s e f l a s h p r o b l e m was p r e s e n t e d i n 1952 by R a c h f o r d a n d R i c e ( 5 _ ) a n d many i m p r o v e m e n t s w e r e i n t r o d u c e d i n subsequent s t u d i e s , ( 6 - 8 ) but multiphase f l a s h c a l c u l a t i o n s w e r e n o t a d d r e s s e d u n t i l 1969 when Deam a n d Maddox(^) presented the three-phase f l a s h e q u i l i b r i u m problem. S i n c e t h a t t i m e many i m p r o v e m e n t s i n t h e a l g o r i t h m s u s e d have l e d t o more r a p i d c o n v e r g e n c e r a t e s a n d t o t h e c o r r e c t i d e n t i f i c a t i o n o f t h e number a n d c o m p o s i t i o n o f t h e s t a b l e equilibrium phases. These improvements have a d d r e s s e d the two d i s t i n c t p r o b l e m s w h i c h c h a r a c t e r i z e f l a s h calculations. The f i r s t i s d e f i n i n g a t h e r m o d y n a m i c m o d e l , such as an equation of s t a t e , which g i v e s r e s u l t s i n agreement w i t h e x p e r i m e n t a l measurements. The s e c o n d p r o b l e m i s f i n d i n g a numerical s o l u t i o n to the f l a s h c a l c u l a t i o n w i t h the g i v e n thermodynamic model. The m a j o r c o n t r i b u t i o n s t o these improvements i n c l u d e e f f i c i e n t n u m e r i c a l t e c h n i q u e s such as the over-relaxâtion method, a c c e l e r a t e d successive s u b s t i t u t i o n , and the m u l t i - v a r i a n t Newton-Raphson t e c h n i q u e , the development of e q u a t i o n s of s t a t e , a l g o r i t h m s f o r o b t a i n i n g i n i t i a l estimates f o r the composition of each phase, and the a p p l i c a t i o n of Gibbs free energy m i n i m i z a t i o n p r i n c i p l e s for e v a l u a t i n g the s t a b i l i t y of any e q u i l i b r i u m phase.(10-25) T h e s o l u t i o n scheme p r e s e n t e d b y R i s n e s a n d D a l e n ( 2 5 ) serves as a foundation f o r t h i s work. Improved a l g o r i t h m s h a v e b e e n d e v e l o p e d f o r d e t e r m i n a t i o n o f t h e number o f p h a s e s , the i n i t i a l e s t i m a t i o n o f phase c o m p o s i t i o n s , and the s e a r c h s t r a t e g y f o r d e t e r m i n i n g i f a n a d d i t i o n a l phase s h o u l d e x i s t o r i f a n e x i s t i n g p h a s e s h o u l d be e l i m i n a t e d . Emphasis i s p l a c e d o n q u a n t i t a t i v e l y d e s c r i b i n g how t h e a d d i t i o n o f
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
EQUATIONS O F STATE: THEORIES A N D APPLICATIONS water to a hydrocarbon based m i x t u r e a f f e c t s compositions of e q u i l i b r i u m phases.
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Two
t h e number a n d
Phase And M u l t i p h a s e F l a s h C a l c u l a t i o n s
Governing Equations and Computational Algorithms. The e q u a t i o n s employed t o d e s c r i b e t h e p h a s e s , d e f i n e t h e r m o d y n a m i c e q u i l i b r i u m , f a c i l i t a t e t h e s o l u t i o n o f two phase a n d m u l t i p h a s e s o l u t i o n s , i n i t i a t e e s t i m a t e s o f a n y phase's composition, and e f f i c i e n t l y a d j u s t these i n i t i a l estimates a r e presented i n t h i s s e c t i o n . In a d d i t i o n t o the system temperature, pressure and o v e r a l l composition, the input data r e q u i r e d f o r the s o l u t i o n of these f l a s h e q u i l i b r i u m problems c o n s i s t of the c r i t i c a l temperature, c r i t i c a l p r e s s u r e a n d a c e n t r i c f a c t o r o f each component, a s w e l l a s a b i n a r y i n t e r a c t i o n parameter f o r each p a i r of components. A l l p h a s e s a r e d e s c r i b e d by a n e q u a t i o n o f s t a t e . The Peng-Robinson equation of s t a t e i s presented i n Table l a along with i t s corresponding expressions for compressibility f a c t o r , f u g a c i t y c o e f f i c i e n t and mixing r u l e s , i n Table l b , Equations 1-13. Table Ρ -
la.
Peng-Robinson
Equation of
RT _ a v-b v(v+b) + b(v-b)
m V
b = 0 . 0 7 7 8 0 RT / P c c a(T)
-
α ·
= 1 + in(l-Tr
0
5
m -
0.37464 +
Table
Z
3
0.45724
lb.
-
0 e 5
2
Ζ »
Β » k
-
)
1.54226ω -
(4)
0.26992ω
+ (A-3B -2B)Z 2
-
Pv/RT
A - aP/R T
Ψ
(3)
2
2
In
'
(5)
2
Compressibility Factor and Fugacity
(l-B)Z
where
A
(2)
R T 38
Σy = 1
β
(39)
i
thermodynamic e q u i l i b r i u m c r i t e r i a
f
(b)
system o f predicted
a
i
L
=
l
iL
f
=
2
f
iL
=
3
f
i V
< > 40
phases
must m i n i m i z e G i b b s
energy
The m u l t i p h a s e f l a s h c a l c u l a t i o n p r o c e d u r e i s a l s o q u i t e s i m i l a r t o t h a t o f t h e two phase s y s t e m . The e q u i l i b r i u m c o n s t a n t s a r e d e f i n e d i n r e f e r e n c e t o t h e vapor phase a s K
l i
- ?ΐ/ 11 χ
K
2 i- y i / 2 i x
K
The g a s p h a s e c o m p o s i t i o n ,
V
Z
i
/
[
1
+
l
L
"
1
}
+
L
3 i - *ΐ/*31
(
L Z(x -y ) 1
u
i
24 (
"
X
)
+
L
3 ^T (
+ L ^ x ^ - y ^ + L ^ x ^ - y . )
Defining t h e gj f u n c t i o n s , used t o determine distribution, g
j
( L
r
L
2
, L ) = « x 3
1
)
f o r Ν = 1, i s g i v e n b y
'
E l i m i n a t i n g V f r o m E q u a t i o n 38 a n d s u m m i n g o v e r components, g i v e s Ey.+
4
- y ^
(
4
2
)
a l l
= Σζ. = 1
(43)
t h e phase
(44)
Combining E q u a t i o n s 4 1 , 4 2 , 43 and 44
In Equations of State; Chao, K., et al.; ACS Symposium Series; American Chemical Society: Washington, DC, 1986.
24.
Four-Phase
ENICK ET A L .
(
1)ζ
-b"
s
g
Flash Equilibrium
Calculations
ι
i l
J
1
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