HYDRODYNAMIC
AND
CHARACTERISTICS FILTRATION
G.
OF
SPECIAL
UNSTEADY-STATE
PROCESSES
FISSURED-POROUS R.
THERMODYNAMIC
OF
NONEQUILIBRIUM
MULTIPHASE
MIXTURES
IN
MEDIA
Isaev
As is well known, m a n y d e p o s i t s in f i s s u r e d - p o r o u s r e s e r v o i r s a r e w o r k e d at s t r a t a l p r e s s u r e s below the s a t u r a t i o n p r e s s u r e ; as a r e s u l t , in s u c h s t r a t a t h e r e a r i s e s a flow of n o n h o m o g e n e o u s liquid in the p o r o u s b l o c k s and c r a c k s . In addition, the m i x t u r e being f i l t e r e d is i t s e l f not h o m o g e n e o u s in its c o m p o s i tion, but is a m i x t u r e of h y d r o c a r b o n s with d i f f e r e n t s t r u c t u r e s . If, in addition, it is taken into a c c o u n t that a f i s s u r e d - p o r o u s s t r a t u m is u s u a l l y m o d e l l e d as a r e s e r v o i r with double p o r o s i t y , i.e., that it c o n s i s t s of two p o r o u s m e d i a with p o r e s of d i f f e r e n t s c a l e s [1], the n o n e q u i l i b r i u m u n s t e a d y - s t a t e p r o c e s s of the f i l t r a tion of h y d r o c a r b o n m i x t u r e s in s u c h s t r a t a will be c h a r a c t e r i z e d by a whole s e r i e s of significant h y d r o d y n a m i c and t h e r m o d y n a m i c s p e c i a l c h a r a c t e r i s t i c s . We note that the m o t i o n of m u l t i e o m p o n e n t m i x t u r e s in c o l l e c t o r s with o r d i n a r y p o r o s i t y has b e e n d i s c u s s e d in the o r i g i n a l and i m p o r t a n t a r t i c l e s of R o z e n b e r g , Zheltov, K u r b a n o v , Shovkrinskii~ and o t h e r s [2-5]. The t h e r m o d y n a m i c s and h y d r o d y n a m i c s of the f i l t r a t i o n of h y d r o c a r b o n h o m o g e n e o u s ( s i n g l e - p h a s e ) liquids in f i s s u r e d - p o r o u s s t r a t a have b e e n i n v e s t i g a t e d in [6]. 1. Equations Components
of the Material Balance in the Phases of a Filter
of the Medium
Let us a s s u m e that a f i s s u r e d - p o r o u s s t r a t u m , the p o r o u s b l o c k s and c r a c k s of w h i c h a r e s a t u r a t e d with a n o n h o m o g e n e o u s (multiphase) m u l t i c o m p o n e n t m i x t u r e , can be r e g a r d e d as a t h e r m o d y n a m i c s y s t e m c o n s i s t i n g of the s o l i d m a t e r i a l of the s t r a t u m itself, of a m i x t u r e s a t u r a t i n g the p o r o u s b l o c k s , and of a m i x t u r e s a t u r a t i n g the s p a c e taken up by the c r a c k s . E a c h of the above m i x t u r e s is m u l t i p h a s e (the p h a s e s a r e d e n o t e d by ~, fl . . . . . ~), while the c o m p o n e n t s of the m i x t u r e will be d e n o t e d by the s y m b o l j (j v a r i e s f r o m 1 to n). We denote also the t h r e e m a i n c o m p o n e n t s by the s u b s c r i p t i: the solid m a t e r i a l of the s t r a t u m (i = 1), the m u l t i p h a s e m i x t u r e in the p o r o u s b l o c k s (i = 2), and the m u l t i p h a s e m i x t u r e in the c r a c k s (i = 3). We note a l s o that, in an n - c o m ponent rnultiphase m i x t u r e (in the b l o c k s o r in the c r a c k s ) , e a c h c o m p o n e n t m a y be c o n t a i n e d in all of the p h a s e s . A d s o r p t i o n and c a p i l l a r y e f f e c t s a r e n e g l e c t e d . Then, on the b a s i s of the above, the equations f o r the m a t e r i a l b a l a n c e of the j - t h c o m p o n e n t in the m i x t u r e (i = 2, i = 3) f o r the ~ p h a s e m a y be r e p r e s e n t e d in the f o r m
o 0--/~ ~ ~j p~ ) --
s L~'~ + div(L;' + m 0 ~ ~v ~', --
9r 1 6 2
] ~
=
~,,m~
H e r e 0 1j ~ is the s a t u r a t i o n of the p o r o u s b l o c k s and c r a c k s by the c o m p o n e n t j, l o c a t e d in the p h a s e ~; Pij is the d e n s i t y of the j c o m p o n e n t of the o~ p h a s e in the m i x t u r e filling the p o r o u s b l o c k s (i = 2) or the c r a c k s (i = 3); J ~ is the diffusion flow of the j c o m p o n e n t in the ~ p h a s e ; v ~ is the v e l o c i t y of the c e n t e r of m a s s of the s y s t e m (i = 2, i = 3) in the a p h a s e ; J ~ v is an i n t e r n a l s o u r c e of the j - t h c o m p o n e n t of the p h a s e due to p h a s e t r a n s i t i o n s in the m i x t u r e (i = 2, o r i = 3); J~,~i• 1,j is an i n t e r n a l s o u r c e of the j - t h c o m ponent of the a p h a s e due to r e t u r n flows f r o m the p o r o u s b l o c k s into the c r a c k s (from the p h a s e s of the
Grozny. Translated from Izvestiya Akademii 128-136, January-February, 1971.
Nauk SSSR, Mekhanika
Zhidkosti i Gaza, No. I, pp.
9 1973 Consultants Bureau, a division of Plenum Publishing Corporation, 227 IVest 17th Street, New York, N. Y. 10011. All rights reserved. This article cannot be reproduced for any purpose whatsoever without permission of the publisher. A copy of this article is available from the publisher for $15.00.
111
m i x t u r e i • 1; f o r J i , i + t,j t h i s i s t a k e n a t i = 2, a n d a t J i , i - t , j it i s t a k e n a t l i = 3). 2. Equations and Equations
of Motion of the Phases of the Energy Balance
of the
Mixture
U t i l i z i n g the w e l l - k n o w n e q u a t i o n s f r o m h y d r o m e c h a n i c s , t h e e q u a t i o n of m o t i o n f o r the w h o l e h y d r o d y n a m i c s y s t e m u n d e r c o n s i d e r a t i o n c a n b e w r i t t e n in the f o r m dvt
dvz
dv~
-- divF--)'
p~F~ = 0
l
H e r e F r s i s a c o m p o n e n t of t h e t e n s o r of the t o t a l s t r e s s e s in the f i s s u r e d - p o r o u s m e d i u m . T h e q u a n t i t y g r s is a c o m p o n e n t of t h e m e t r i c a l t e n s o r ; r i d e r s i s a c o m p o n e n t of the t e n s o r of t h e t r u e s t r e s s in t h e s o l i d s k e l e t o n . T h e l a s t t e r m on t h e l e f t - h a n d s i d e c h a r a c t e r i z e s
t h e e f f e c t of the e x t e r n a l m a s s f o r c e s .
F o r e a c h p h a s e of the m i x t u r e (i = 2, i = 3), t h e e q u a t i o n of m o t i o n c a n b e w r i t t e n in t h e f o l l o w i n g form: dv~ P~'O~ dt .
. gradP~ . .- - R~
~,
li. " ~~• + p ~ F , ~
(2.2)
m{
H e r e J9 i , i • ~~ v a r e t e r m s c h a r a c t e r i z i n g the t r a n s f e r o f m o m e n t u m due to r e t u r n f l o w s in t h e s y s t e m ; R i i s the v e c t o r o f t h e f o r c e of v i s c o u s f r i c t i o n (the f i c t i t i o u s f o r c e of f r i c t i o n in t h e s e n s e of Zhukov [7]). T h e e q u a t i o n of the b a l a n c e f o r t h e r a t e of c h a n g e o f the d e n s i t y of t h e k i n e t i c e n e r g y of t h e o~ c o m p o n e n t (i = 2 o r i = 3) i s w r i t t e n , on the b a s i s of w e l l - k n o w n g e n e r a l c o n s i d e r a t i o n s , in the f o r m fl
[T
'~ 0,
(2.3)
l = - div v=ct
T h e b a l a n c e e q u a t i o n of t h e i n t e r n a l e n e r g y f o r the ~ p h a s e of the m i x t u r e i = 2 o r i = 3, on the b a s i s o f t r i v i a l c o n s i d e r a t i o n s , c a n b e w r i t t e n in t h e f o r m ft
^ dat ~ 6Q~~ ~^= d t r n ~ t t ~ d t - - - dt -m~v'V~ - ~ " ~ +
R~ ( v ~ - - v ~ ) + p---~"
L~.F~
~--~L?,.,~) 7, ~.~
q~ p~
(2.4)
~=CL
~Q~ [ -9" \ ~ dt T h e t e r m s L i , i • t and ~ i , l • 1
- - div (Iq~mi0i) - external heat flux
in (2.3) and (2.4) t a k e a c c o u n t of t r a n s f e r in the s y s t e m due to r e t u r n
flows. We n o t e t h a t , w i t h a c o m p a r i s o n of E q s . (2.3) and (2.4), i t i s a s s u m e d t h a t it i s p o s s i b l e to n e g l e c t t h e f i c t i t i o u s f o r c e s of Zhukov at t h e i n t e r f a c e b e t w e e n t h e l i q u i d p h a s e s , in c o m p a r i s o n w i t h the a n a l o g o u s f o r c e s at t h e i n t e r f a c e w i t h t h e s o l i d b o d y . On t h e b a s i s of (2.3) and (2.4), t h e b a l a n c e e q u a t i o n f o r t h e t o t a l e n e r g y of the ~ p h a s e m a y be w r i t t e n as
~ t { m i p ~ 0 ~ [ _ _ _ ~ ( v ~ ) Z + a ~ ] } + d i v t m ~ p ~ 0 ~~r r i [ _ ~ ( v ~ 2 + u ~ ] v ~ } ) + div(Iq~m,~#) +
~
(2.5)
t~L ~,~"t~t l-1- r~"~) ~ , i• j~•T div(m,O~P~v,~) + 6A~t~ + q~ --- 0
w h e r e 6 A i l ~ i s t h e w o r k of the f o r c e s at the i n t e r f a c e b e t w e e n the s o l i d c o m p o n e n t and t h e ~ p h a s e of the mixture i.
112
An e x p r e s s i o n f o r 5Ai~ a c a n be found, if the equation f o r the d e n s i t y of the i n t e r n a l e n e r g y i8 a v e r aged o v e r the v o l u m e : d
m~p:'O~,~ ~
"~
+ m~P~O:div vi + div (Iq:miO~~) + q~ fl
I~.F:-[-6A~,~-~-p~vcgrad(m~O~ ~)
i2.6)
R ~ v~.~ s v=cz
A c o m p a r i s o n of (2.6) and (2.4) l e a d s to 6A ~~ = R ? . v , -4- P~ 3. Equation for the Balance Phases of the Mixture, and
0 (m,O~~) Ot
of the Entropy Phenomenological
(2.7)
of the Equations
A s s u m i n g the v a l i d i t y of the h y p o t h e s i s of local e q u i l i b r i u m , in o u r f u r t h e r d i s c u s s i o n s , we u s e the Gibbs t h e r m o d y n a m i c e q u a t i o n
ds da dv T - ~ = -~ + P d-~-- s
dc~ (p,-~
(3.1)
and the e q u a t i o n of the e n t r o p y b a l a n c e
0(ps)
0----~"-~ -- div I,,. + 71
(3.2)
w h e r e s is the e n t r o p y ; Is, a is the total flow of e n t r o p y ; V is the i n t e n s i t y of the s o u r c e of e n t r o p y due to i n t e r n a l i r r e v e r s i b l e p r o c e s s e s in the s y s t e m ; goi is a p a r t i a l s p e c i f i c Gibbs function; c i is the m a s s c o n c e n t r a t i o n of the c o m p o n e n t i. On the b a s i s of E q s . (3.1) and (3.2) we can obtain s u c h an equation f o r the ~ p h a s e of the c o m p o n e n t i (i = 2, i = 3)
Ot ~ .v. . . .
/=--div~
T
ip~-i i i -~
T~
T~
(3.3)
T~•
F o r the whole t h e r m o d y n a m i c s y s t e m , the e n t r o p y b a l a n c e e q u a t i o n will have the f o r m
O (m,m s d + ~ ( ~ , d-Y
m,p~"O~s , ~~\ ] = - - d i v (m, ms,vi
..~_~ ~Tt rn i pv'"$VV f u i i i ! "A miIq,--q)iI, Ti ~-~rnr
-- W
-- s
s
I,, grad T~ ---~I,
Iqt grad T~ -- L - - ~ v
~-[I:(T,
--i-
q~
) grad-~i -R : ( v : - - v,)
:;~
(I), ~
J~,i_.+_i T~
q)7•
T~•
~-
q,
(3.4)
I13
P
H e r e 4)i is the f i l t r a t i o n potential of the v p h a s e in the m i x t u r e (i = 2) o r (i = 3). To c l o s e the s y s t e m of e q u a t i o n s ~oalance) and the e n t r o p y b a l a n c e , it is n e c e s s a r y to s u p p l e m e n t (3.4) by a s e t of p h e n o m e n o l o g i c a l e q u a t i o n s , c o n n e c t i n g the flows and the f o r c e s . In a c o m p a r i s o n of the p h e n o m e n o l o g i c a l equations (in the l i n e a r region) it is n e c e s s a r y to b e a r in m i n d the C u r i e p r i n c i p l e with r e s p e c t to the identical n a t u r e of the t e n s o r d i m e n s i o n a l i t i e s of the flows and of the t h e r m o d y n a m i c f o r c e s . If all the p h e n o m e n o l o g i c a l equations a r e w r i t t e n out, then, as is known f r o m [8], in a c c o r d a n c e with the O n s a g e r p r i n c i p l e , the following r e l a t i o n s h i p s will e x i s t between the kinetic c o e f f i c i e n t s of t h e s e e q u a tions: L~ = L ~ ,
L~--~ L~,
L~ ~--L~ ....
(3.5)
The p h e n o m e n o l o g i c a l e q u a t i o n s will be built up in a c c o r d a n c e with the following type (we give only one f o r i l l u s t r a t i o n ; the r e m a i n i n g equations a r e obtained in an analogous fashion): m~Iq~ - - - - L . grad T~ Ti ~
(gradT~%+)T, _ ~
Ll~ grad T~ T~~
L~--T~grad T~~
~, L ~n (gradTiqh~+)T~
~, ~ L~ ( v J ~ v~)
v==-a i = 2
(3.6)
v=a i~2
In Eq. (3.6) we a s s u m e F~ --~ --grad W~, The e q u a t i o n f o r q = ql h a s the f o r m
q =
T,T~T---~
(s~ + e~ ) i 3 -~
T2T~ - -
(3.7)
~,=~
v----~
T~ /
Since the p e n u l t i m a t e t e r m s in the r i g h t - h a n d p a r t of (3.4) a l r e a d y take a c c o u n t of heat t r a n s f e r b e tween the c o m p o n e n t s and the p h a s e s , it c a n be a s s u m e d that Lt2 --~ Lzi ---- Lt3 . . . . .
0
(3.8)
We note also that, if we d i s c a r d the t e r m s Li] = Lji at i ~ 4, j > 4 (but i r j) f o r the s a m e r e a s o n as in [9], we then obtain the following s e t of p h e n o m e n o l o g i c a l e q u a t i o n s : m~I~l ~
~
grad T~,
Lz~ Tz~ grad T~,
m20~Iv2- -
isa m~0sIv~ ~ -- T~ grad T, L,, I, = -- T~ (grad(p'+)r"
I~" ~ R ~=
Is" = --
L55" T'2 (grad (p~'+)T"
L~J (grad ~3~+) ~, T3
(3.9)
-- - -
L77" (v~" - v , ) , Tz
- -
a~ ~
=
L88" ( v : - v , )
~
v
A s s u m i n g that L , / T I ~ -~. m ~ . i , L~7 v
9 T2
114
L2~/T2 ~ ~
m2~02,
m22~2 v
Lss v
k2~
T3
Lsa/Ts ~ ~
rns~,sOs
msu~ts v
k3*
(3.10)
p p w h e r e k i a r e the p h a s e p e r m e a b i l i t i e s (i = 2, 3);/t i a r e the v i s c o s i t i e s of the p h a s e s of the s y s t e m (i = 2), (i = 3); ~ i is the t h e r m a l c o n d u c t i v i t y c o e f f i c i e n t . It follows f r o m Eq. (3.9) that Iq, = -- 9~, grad T,, R~ ~-
I~z ------- )~ grad T~,
2 ~z ( v ~ _ v~), k---~
Iq~ = - - ~ grad T~
ms ~ta
R/--
(v~ ~ - - v~)
(3.11)
k~ ~
The g r a d i e n t s of the c h e m i c a l p o t e n t i a l s c a n be e x p r e s s e d in t e r m s of the c o n c e n t r a t i o n g r a d i e n t s n--i
V'~ ~ ~0~+grad c~~ (grad %~+) ~ -----~ff)~+ grad p ~ • , ~.~
(3.12)
where
Relationships (3.12) are usually simplified by assuming the absence of a pressure-diffusion effect (which is possible only in the first approximation); then, the equations for the diffusional flow of the components (i = 2), (i = 3) in the ~ phase have the form
I~ I~~
--(D2J grad c, + D~2~ grad c2~) - - ( D ~,~ gradc~+D~z ~ grad c2~)
( L ~ ~)~+~ \ D ~ ~(P /
(3.13)
w h e r e Dij is the t r a n s f e r c o e f f i c i e n t of the j - t h c o m p o n e n t . The t r a n s f e r of the j - t h c o m p o n e n t f r o m one p h a s e to a n o t h e r within the l i m i t s of the s a m e s y s t e m (i = 2) or (i = 3) is d e t e r m i n e d by the d i f f e r e n c e b e t w e e n the p a r t i a l s p e c i f i c Gibbs functions in t h e s e phases, i.e., J'~ ~- -- % where
~?ij is the transfer
(q~ -- (P'~)
(3.14)
coefficient of the j-th component.
4. Differential Equations for the Balance of Mass, Motion, and Total Energy of the Components of a Mixture, Taking Account of the Energy Balance Equation and of the Constituent Phenomenologieal Equations
a) M a t e r i a l B a l a n c e Equation. T o obtain an o v e r a l l m a t e r i a l b a l a n c e f o r the j c o m p o n e n t in the p h a s e of the m i x t u r e (i = 2) o r (i = 3), u s i n g the p h e n o m e n o l o g i c a l e q u a t i o n s , it is n e c e s s a r y to expand the v a l u e s of the t e r m s J~ij, Ji~, and Ji,i~v =~l,j i n E q . (1.1). A f t e r s i m p l e t r a n s f o r m a t i o n s , we have 0 r-, (raip~O,~ ~) .-q-~..~_~q~y((p~'~- - q~,/)+ div [m~p~j~Oi/'vi~ O--~
~--~,-
T,~,. = 0
(4.1)
If we r e g a r d the m o t i o n of the (~ p h a s e t o g e t h e r with the p h a s e v e l o c i t y , obeying the D a r c y law, then, u s i n g the p h e n o m e n o l o g i c a l e q u a t i o n s , the m a t e r i a l b a l a n c e f o r the ~ p h a s e c a n be p r e s e n t e d in the f o r m 0 (m~Pi~Oi~)+ Ot
~b, ~=L, 9
T~
T~•
+ div
v~ct
- - ~kg"~ radP,]--div[~ where
O?,
k" ~= ~i" grad P~ ] ==0
(4.2)
is the f i l t r a t i o n c a p a c i t y of the u p h a s e in the m i x t u r e (i = 2) o r (i = 3)
115
v __
E
m A ~
v
b) E q u a t i o n of Motion f o r the P h a s e s of the M i x t u r e {i = 2) o r (i = 3). T h e equation of m o t i o n f o r e a c h of the p h a s e s of the m i x t u r e (i = 2) o r (i = 3) is o b t a i n e d s i m p l y , f r o m (2.2), taking a c c o u n t of the v a l u e s of R ? and Ji,i -~l( ~ O-Y(m~p~O?v?) = - - div(m~p?O?v?.v~ -4- PO
~-~, (v? - - v,)
In the c a s e of f i l t r a t i o n obeying a n o n l i n e a r law (a t w o - t e r m d e p e n d e n c e ) , the p h e n o m e n o l o g i c a l e q u a tion f o r R ~ is c o n s t r u c t e d taking account of the e x t r e m a l p r i n c i p l e s of the t h e r m o d y n a m i c a l l y i r r e v ~ r s i b I e p r o c e s s e s ; the equation of m o t i o n of e a c h of the p h a s e s then a s s u m e s the f o r m
Z
Ot
(m,p,~'O,=vd') = -- div(m,p,~O#v, ~ .v, + PO - - ~
(v# - - vt)
k,
o
"4- b,: (v, r - - vt)" Iv, c' - - v, [ + o # F / ' - - / ,
(4.4)
,. ,• ( v # - - v,•
c) B a l a n c e Equation f o r the T o t a l and I n t e r n a l E n e r g y . The u s e of E q s . (2.5) and (2.4) f o r this p u r p o s e , as well as of the p h e n o m e n o l o g i c a l equations (3.11)-(3.13), l e a d s to the following d e p e n d e n c e , e x p r e s sing the d e n s i t y b a l a n c e of the total e n e r g y :
Ot
div (Iq,~m~O,~)
( m,p['O['e, ~') ~-- - - div ( m~o,~'O,~,e[,v,) -
g
-- div (mr
~) -- R~~ .v, --P~ O (m~0#)0_____~ q~ -- ~7 , -4.r~'(')~•
(4.5)
v ~
Ol
w h e r e e i is the total s p e c i f i c e n e r g y of the ~ p h a s e in the m i x t u r e i. U s i n g the t h e r m o d y n a m i c identity of the s e c o n d law of t h e r m o d y n a m i c s , it w a s found that the r e a l t i m e d e r i v a t i v e s of the e n t r o p y of the ~ p h a s e in the m i x t u r e (i = 2) o r (i = 3) have the f o r m ds, ~ m~pi~0t T~ --77-. dr
cp#rni dTi @ Aa~T,ra~O~ ~ dP, dt dt
(4.6)
H e r e a ~ is the c o e f f i c i e n t of v o l u m e t r i c e x p a n s i o n with h e a t i n g at a c o n s t a n t p r e s s u r e ; A is the t h e r m a l e q u i v a l e n t of w o r k ; Cpia is the h e a t c a p a c i t y of the a p h a s e at c o n s t a n t p r e s s u r e . On the o t h e r hand, the change in the e n t r o p y of the a p h a s e , on the b a s i s of (3.3), c a n be w r i t t e n as ds? ( m,O,~Iq,~ - - (p,~I,~ ) m,O, ~ m~p~0# d t = -- div Tt -- ~ Iq~" grad T~ Q.
--i-~-[ Ii~" (T, grad qg'~ -- F,~) ] q T~
T,
T~
T,
v
J,~+i
T,
T~• /
(4.7)
Now, c o m p a r i n g E q s . (4.6) and (4.7) and taking account of the p h e n o m e n o l o g i c a l equations (3.11)(3.13), we obtain the following h e a t - b a l a n c e e q u a t i o n (neglecting the s q u a r e s of the f i l t r a t i o n r a t e s ) : m~O~cp~
dTi dt
- - mA~. O~ A T~ - - F,.~
- - q),~D~#Ac, - - ~t~Di2~Ac#
116
~ i c~
- - ---~ u ( T ~ T 2
~ ~ .,-.~ ~ ~ .v c~~ --
s~T~Tt - - e3TIT2)
(4.8)
+ Aai T~
: - - ~ -c ~..~ " t• (
T~• )
T,~T, )
----"
"~=fz
Combining Eq. (4.8) with the analogous equations for the other phases and c a r r y i n g out a s u m m a t i o n over all the components of the s y s t e m (over all values of i; i = 1, 2, 3), in the final analysis, 'we obtain the following heat equation:
rnlcpi - - ~ +
v=g
~ Im,Oi'c~,~ dTictt
a
2
i=2
j=t
2
3
f~
j=l
i=2
v=~
L Aai TomiO~ T t -- Fl "
~ i~ . j
(4.9) Assuming in the approximation that the t e m p e r a t u r e s of the phases of the mixture are identical, from (4.9) we have the s i m p l e r equation (rniO~cv~v: 9 V Ti + Aa:ToraiO~*v:VPO
ot i~Z
2
B
v=~
i
~=i
V=~
i=i
2
j~l
i=2
v==
i~2 $
a
= -B
v=~
Q
(4.10)
m~),iaJ,_v iv
i=2
Thus, if we use the phenomenological
v~
equations of the given thermodynamic
system
for expansion of
the t e r m s entering into the m a t e r i a l balance of the components of the s y s t e m , into the equation of motion, and into the e n e r g y equations for the phases of the m i x t u r e s (i = 2), (i = 3), as well as the expansion of the analogous t e r m s in the c o r r e s p o n d i n g equations for the f i r s t component (i = 1), we can obtain a closed s y s tem of differential equations, in partial derivatives with r e s p e c t to the value of ~2(c - 1) of the m a s s c o n centrations, of the density of the mixture, and of the p r e s s u r e s and t e m p e r a t u r e s of the components of the m i x t u r e . It is u n d e r s t o o d that these equations must n e c e s s a r i l y be supplemented by a s y s t e m of constituent u dependences for m~ and m i as a function of the p r e s s u r e and for the saturation capacity0~ as a function of the p r e s s u r e , as well as by dependences of the phase p e r m e a b i l i t y as a function of the p r e s s u r e ~Ld the saturation capacity, by dependences of the phase v i s c o s i t i e s as functions of the concentrations and the p r e s s u r e s and, finally, by an equation for the potentials as a function of the r e t u r n flow. Some of these dependences a r e a s s u m e d as working hypotheses, while others are determined e x p e r i m e n t a l l y . F r o m the equations given above, there follows v e r y simply a s y s t e m of equations for the u n s t e a d y state filtration of a gasified liquid in plates with double porosity, as well as natural generalizations for the case of t h r e e - p h a s e filtration (petroleum, water, gas), and for f o u r - p h a s e filtration (petroleum, gas, water, heavy petroleum h y d r o c a r b o n s ) .
LITERATURE I~
2.
CITED
G. I. Barenblatt and Yu. P. Zheltov, "Basic equation of the filtration of homogeneous liquids in fissured rocks," Dokl. Akad. Nank SSSR, 132, No. 3 (1960). Yu. P. Zheltov, M. D. Rozenberg, and G. Yu. Shovkrinskii, "Investigation of the filtration of multicomponent mixtures," in: The Extraction of Petroleum, Theory and Practice, Yearly Publication [in Russian], Izd. Nedra, Moscow (1964).
117
3.
4.
5.
6o
7. 8. 9.
118
V. N. Nikolaevskii, l~. A. Bondarev, M. I. Mirkin, G. S. Stepanov, and V. P. Terzi, The Motion of Hydrocarbon Mixtures in a Porous Medium [in Russian], Izd. Nedra, Moscow (1968). A. K. Kurbanov, M. D. Rozenberg, Yu. P. Zheltov, and G. Yu. Shovkrinskii, "Investigation of the motion of multicomponent mixtures in a porous medium," in: The Extraction of Petroleum, Theory and P r a c t i c e , Yearly Publication [in Russian], Izd. Nedra, Moscow (1964). M. D, Rozenberg, S. A. Kundin, A. K. Kurbanov, N. I. Suvorov, and G. Yu. Shovkrinskii, The F i l t r a tion of a Gasified Liquid in Different Multieomponent Mixtures in Petroleum Strata [in Russian], Izd. Nedra, Moscow (1969). R. G. Isaev, "Thermodynamic investigation of filtration processes in fissured-porous deformed media," Rev. Roumaine Sci. Tech., Sec. Mec. Appl., 14, No. 3 (1969). L. S. Leibenzon, The Motion of Natural Liquids and Gases in a Porous Medium [in Russian], Izd. Gostekhizdat, Moscow-Leningrad (1947). S. De Groot and P. Mazur, Nonequilibrium Thermodynamics, Am. Elsevier (1962). R. G. Mokadam, "Thermodynamic investigation of the Darcy law," Prikl. Mekhan., No. 2 (1961).
