EQUATIONS GASES

FOR IN

Yu.

POROUS

THE

MIGRATION

OF

NATURAL

MEDIA UDC 532.546

I. Stklyanin

Equations are given for the equilibrium and nonequilibrium migration of natural g a s e s in variable and invariable porous media. In n u m e r o u s works [1-4], m i g r a t i o n has been considered principally in g e o l o g i c a l - g e o c h e m i c a l t e r m s , the qualitative side of the phenomenon being mainly investigated; its p h y s i c o m a t h e m a t i c a l aspects have been inadequately studied [5, 61.

1~ Equations for Nonequilibrium Migration in Microvolumes of Invariable P o r o u s Media. In the e a r t h ' s crust, natural g a s e s a r e f o r m e d as a r e s u l t of chemical and r a d i o a c t i v e decomposition of substances d i s p e r s e d in the r o c k skeleton, and are distributed between its three phases: gas, liqui d , and solid. As they enter the p o r e space, they acquire a g r e a t mobility, migrating in the free state and with the formation o f liquids. T r a n s p o r t through m i c r o - and m a c r o f i s s u r e s o c c u r s with even g r e a t e r intensity. Beds with m a r k e d differences in p a r a m e t e r s and migration r a t e s are deposited in a large m a s s of r o c k s . The d i s tribution of gases between the volume elements of s e d i m e n t a r y r o c k s is t h e r e f o r e , as a rule, nonequilibrium. The equilibrium nature of the state of a gas in porous media is determined by a complex of c h a r a c t e r i s t i c p a r a m e t e r s : the value of the migration space element, which a v e r a g e s out the values under investigation; the integral of the duration of the p r o c e s s as a whole and its basic stages; and also variations in the r a t e of t r a n s p o r t of the g a s e s in different sections of the p o r o u s media. It is expedient to distinguish t h r e e types of nonequilibrium c h a r a c t e r in the state of the gas in porous media, depending on the size of the e l e m e n t a r y volume of the media. In considering the migration of g a s e s f r o m the skeleton of the r o c k into the f i s s u r e - p o r e space, it is n e c e s s a r y to consider the nonequilibrium nature of the state of the c o m ponents in a m i c r o v o l u m e , i.e. the interphase nonequilibrinm nature, which is connected with v e r y low diffusion coefficients of the substances in the r o c k skeleton. In porous media divided by a s y s t e m of m i c r o and m a c r o f i s s u r e s , migration will be nonequilibrium in the m a c r o v o l u m e . This is connected with the kinetics of the exchange of the components between different sections of the f i s s u r e d and porous media. In s t r a t a with different alternations of high- and l o w - p e r m e a b i l i t y r o c k s , it is possible to examine m a c r o nonequilibrium migration or to introduce different limiting values of the p a r a m e t e r s of t r a n s p o r t along and a c r o s s the stratification (kx,y = 0, Dx,y = 0; kx,y = ~o, Dx,y = ~). In the p r e s e n c e of p r e s s u r e , concentration, and t e m p e r a t u r e gradients (we will d i s r e g a r d the effect of the migration of substances on the t e m p e r a t u r e distribution in the stratum), migration of g a s e s in each of the phases takes place in the porous medium and interphase o v e r c u r r e n t s of the components o c c u r . The specific m a s s flow of the i-th component gti in the gas phase (index 1) will be [6-9] g~ ~ y ~ [ - - D l ~ g r a d l n C i - - D ~ r g r a d l n T - -

(- k-I

D~n AM~)gradpL]

(1ol)

Moscow. Translated f r o m Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 5, pp. 152158, S e p t e m b e r - O c t o b e r , 1971. Original a r t i c l e submitted October 28, 1970. 9 1974 Consultants Bureau, a division of Plenum Publishing Corporation, 227 g'est 17th Street, New York, N. Y. 100}i. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, microfilming, recording or otherwise, without written permission of the publisher. A copy of this article is available from the publisher for $15.00.

865

H e r e , C i is the v o l u m e c o n c e n t r a t i o n (mole fraction), T is the t e m p e r a t u r e , Pl is the p r e s s u r e in the gas p h a s e , k 1 is the p h a s e gas p e r m e a b i l i t y , #1 is the v i s c o s i t y of the g a s e o u s m i x t u r e , DIi is the effective diffusion coefficient, D~ is the effective c o e f f i c i e n t of t h e r m a l diffusion, Yi is the s p e c i f i c g r a v i t y of the c o m p o n e n t u n d e r r e s e r v o i r conditions yi

= y~~ / Po,

PI~= piCi

The quantity z(pl, T) t a k e s into a c c o u n t the deviation of the c o m p r e s s i b i l i t y of the r e a l m i x t u r e f r o m the ideal, pfi is the p a r t i a l p r e s s u r e , 7 i0 is the specific g r a v i t y of the c o m p o n e n t at n o r m a l t e m p e r a t u r e and a t m o s p h e r i c p r e s s u r e , AM i = (M i -- M)/M, M i is the m o l e c u l a r weight of the c o m p o n e n t , M is the a v e r a g e m o l e c u l a r weight of the m i x t u r e , M= EMiC i, and N is the n u m b e r of c o m p o n e n t s in the g a s p h a s e . It is obvious that

Z C~ =

t,

yiC~ = y,

t

(1.2)

i

The f o r m a t i o n liquid (index 2) will be r e g a r d e d as a m i x t u r e with w e i g h t c o n c e n t r a t i o n s of c o m p o n e n t s b i. If the i n t e r f e r e n c e of the c o m p o n e n t s d u r i n g t h e i r diffusion in the liquid and solid p h a s e s is d i s r e g a r d e d , the flows in the f o r m a t i o n liquid can be r e p r e s e n t e d by the equation g2, =--D2~F~,grad

p~' -- b,( D2,Tgradln T + k2 Po

gradp2)

V2

(i .3)

In o r d e r to allow f o r v a r i a t i o n of the p r o p e r t i e s of the f o r m a t i o n w a t e r s in d e p e n d e n c e on m i n e r a l i z a tion, t e m p e r a t u r e , p r e s s u r e , a n d t h e c o n c e n t r a t i o n s of the c o m p o n e n t s , the p a r t i a l p r e s s u r e of the d i s s o l v e d g a s e s P2i is i n t r o d u c e d : p~ bl = F~i

Po

L N,=I, t

p~i = / ' 2- - S C 2~S ~---N~

,

L C~S=t,

L

i

i

pzs

Fz~ Po

P2~=P/

(1.4)

(1.5)

i

Here, F2i is the solubility f a c t o r at the r e s e r v o i r t e m p e r a t u r e , pS is the s a t u r a t i o n p r e s s u r e of the m i x t u r e at the r e s e r v o i r t e m p e r a t u r e , cSi a r e the v o l u m e c o n c e n t r a t i o n s of the c o m p o n e n t s in the g a s p h a s e in e q u i l i b r i u m with the liquid at the s a t u r a t i o n p r e s s u r e , and N i is the m o l e f r a c t i o n of the c o m p o nent in the liquid p h a s e . The p r e s s u r e s in the g a s and liquid p h a s e s differ by the c a p i l l a r y p r e s s u r e d r o p Ap k, which is a function of the g a s s a t u r a t i o n a : p~ ~ p , .

Aph

(1.6)

In addition, salt m i g r a t i o n takes p l a c e in the liquid p h a s e gzj = -- Dzj grad b~-- Dff grad In T -- k~.. b~grad p2 ~.~

(1.7)

w h e r e bj is the weight c o n c e n t r a t i o n of salt ( g / c m 3 of liquid); j = N + 1 . . . . . n. The s a l t s d i s s o l v e d in the f o r m a t i o n w a t e r , while not having a significant effect on the diffusion of the o t h e r c o m p o n e n t s , do c h a n g e the solubility f a c t o r s and c o n s e q u e n t l y the p a r t i a l p r e s s u r e s and the o v e r a l l saturation pressure. A n a l o g o u s l y to (1.3), f o r the solid r o c k skeleton and the f o r m a t i o n liquid in the c o m b i n e d state (index 3),

866

g~ =--D~F~, grad p~.~- - D~d~ grad in T

(1.8)

w h e r e d i is the weight concentration of a component in the r o c k skeleton, F3i is a coefficient c h a r a c t e r i z i n g the absorptive capacity of the i-th component,

p~ = pod~

t

For the salts, g~j= --D3j grad d~ - - DaTdj grad la T

(1.9)

Accordingly, the continuity equations in each phase will be 0 my~

divg, =

Ot

]u" ~, f , ~ ( p . , p ~ , p 3 ~ ) - - qh~(y~)

div g~, = - - - ~ t [b!m ( l - - o) ] - - .12, -Jr- I=~(P., P=,, P,,) -- %,(b,)

div g~ :

--

O [d~(l--m)]--]~,q-fa~(p,,,p=,,p~,)--%,(d,) at 0

(1.10)

(1.11)

(i

.12)

div g=j -~- -- O - T [ b j m ( i - - ~)] - - ],j-}- f=j(b~, d ~ ) - %~(b~)

(1.13)

0 div g3j = -- - ~ [ d ~ ( l - - m) ] - - la~ -I- f3j(b~, di) - - %j(dj)

(1.14)

Here, cr is the gas saturation of the porous space, m is the porosity, and Jki is the intensity of g e n e r ation of each component in unit volume of r o c k in each phase. The function f k i c h a r a c t e r i z e s the kinetics of the exchange of the components between the phases, which, inthe simplest case of r e v e r s i b l e kinetics of exchange of the f i r s t kind between the phases and of H e n r y ' s law for the distribution of components in the equilibrium state, can be given in the f o r m 3

]k,:

(k = t, 2,3)

2a~..,(ph~--p~)

(I.15)

n=l

w h e r e C~kn,i a r e the exchange kInetic coefficients. However, in the genera/ case, the exchange kinetics a r e m o r e complex than (1.15) because the m e c h a n i s m s of the dissolution and evolution of the g a s e s are d i f f e r ent and the p r o c e s s cannot always be a s s u m e d to be r e v e r s i b l e . The kinetic coefficients a k n , i cannot be a s s u m e d to be the same for different exchange d i r e c t i o n s . The functions q0ki take into account the loss of a component in each of the phases owing to its d e c o m position, chemical r e a c t i o n , and microbiological oxidation. In the simplest case, they may be given in the form ~, =

where

Xki is a constant characterizing

(1.16)

~C~,

the rapidity of decomposition

or the rate of biochemical

reaction.

In addition, the equations of state of each phase must be added to (1.1)-(1.16) 'v, = ~,, (c . . . . . . c,~, p,, T)

(1.17)

867

t

u ~---

bl -+-

bj = ?2(b, . . . . . b~v, bN+, . . . . . b., pz, T)

(1.18)

The p a r a m e t e r s D, DT, k, X , a, F, J, and Apk a r e also known functions of p r e s s u r e , t e m p e r a t u r e , gas saturation, and composition of components in each phase. The lack of equilibrium between the gas-liquid mixture and the solid r o c k phase is of most i m p o r tance in (1.1)-(1.18), whereas the state of the g a s - l i q u i d s y s t e m will be mostly close to equilibrium. Equilibrium is established differently for different components. The g a s e s (helium, methane, h y d r o gen, etc.) a r e quite mobile and for them equilibrium is established rapidly. High-molecular p e t r o l e u m components and also g a s e s with high m o l e c u l a r weights and molecular d i a m e t e r s (argon, radon, etc.) have v e r y low diffusion coefficients and may r e m a i n under nonequiltbrium conditions for a long time. In (1.1)-(1.18), we have 2n + N - 2 , unknown concentrations, the p r e s s u r e p, and the gas saturation a , for the determination of which there a r e 2n + N equations. 2. Equations for Equilibrium Migration in Microvolumes of P o r o u s Media. If all f k i = 0, t h e r e is no exchange between the phases, and in a porous medium t h e r e a r e t h r e e independent flows in each phase. In the c a s e of high r a t e s of interphase o v e r c u r r e n t s , equilibrium is rapidly established between them. Under equilibrium conditions, the overall gas content of a r o c k a i can be given in the following f o r m :

(2.1) [~ = too-}- m(i

--

o)F21 + (t

--

m)r:,

(2.2)

w h e r e p is the p r e s s u r e , corresponding with the p r e s s u r e in the gas phase when ~ > 0 (when ~ = 0, p = Ps), and C i is the equilibrium concentration, which, when a > 0, is the volume (molar) concentration of the components in the gas phase. If a = 0, C i has the meaning of the concentration of components in the f r e e phase at the formation t e m p e r a t u r e and saturation p r e s s u r e . T r a n s p o r t of the components o c c u r s in each of the t h r e e phases, but by v i r t u e of the equilibrium nature of the state of the gas in a microvolume, the total m a s s flow gi can be r e p r e s e n t e d in the following manner:

g~ --~ yi[--D~ grad In Ci - - D~~ grad In T -- k? gradp -- k~" grad a] -- D~pgrad p~ Dir(a) ~-D,r

+ D 2 i r - [ - D ~ r,

D~(a) :

k,

Vl

(D~,r~+D~r~,)~,o/po

(2.3) (2.4)

A. '

In2

k~~ ~ - k~.~

(2.5)

P

k2 d h p k

(2.6)

~t~ da

The continuity equation will be 0 div g, = -- ~0---:(f3~y~)-- ]~ --}-q)(y,),

J,=L~-}-12,-~-Js~

(2.7)

Analogous equations can be written for the m i g r a t i o n of salts, but in the m a j o r i t y of c a s e s t h e i r state in porous media is nonequilibrium, and it is n e c e s s a r y to use (1.6), (1.13)-(1.15) to allow for the effect of mineralization on the migration of g a s e s . The equation of state of the mixture must be added to (2.1)-(2.7) with the aid of p - V - - T - - C r e l a t i o n ships. Thus, for N - 1 unknown concentrations, p r e s s u r e p, and gas saturation ~, t h e r e will be N equations (2.1)-(2.7) and the equation of state of the mixture.

868

The functions Di, DT, Dp' k p ' and k i depend on the gas s a t u r a t i o n ~. Analogously to the p h a s e p e r m e a b i l i t i e s , they m u s t be d e t e r m i n e d e x p e r i m e n t a l l y . 3. Equations for Nonequilibrium Migration of G a s e s in a M a c r o v o l u m e . If the c h a r a c t e r i s t i c m i g r a tion t i m e s a r e e x t r e m e l y long (millions of y e a r s ) , equilibrium is mostly e s t a b l i s h e d h~ the m i c r o v o l u m e of the p o r o u s m e d i u m . However, in l a r g e r o c k m a s s e s , c o n s i d e r a b l e fluctuations in flow a r e p o s s i b l e through f i s s u r e s o r highly p e r m e a b l e zones in c o m p a r i s o n with the main bulk of the porous m e d i a . The s t a t e of g a s e s in a m i c r o v o l u m e will be equilibrium, but in a m a c r o v o l u m e the s t a t e will be nonequilibrium. In this c a s e , the model of nonequilibrium m i g r a t i o n can also be used by inserting in each of the m e d i a its p a r a m e t e r s and mflcnown fucntions. The equations for the flows in each m e d i u m can be taken in the f o r m (2.3), and the continuity equation can be taken in the f o r m (2.7), g.~ ---~--~/.~[D~ grad In C.~ ~- k.~' grad p. -k D.~r grad In T ~- k.~~grad ~.] -- D.? grad p.~

(3,1)

0 (3.2) w h e r e n = 1, 2 designates the different media: the h i g h - p e r m e a b i l i t y m e d i u m has the index 1, and the lowp e r m e a b i l i t y m e d i u m has the index 2. The function f n i allows for the kinetics of the exchange of c o m p o nents between m e d i a . This function m a y be obtained in a c c o r d a n c e with (1.15). 4.__Equations for the Migration of G a s e s in V a r i a b l e P o r o u s Media with Mobile Boundaries. We will examine the unidimensional m i g r a t i o n of g a s e s , taking into account t h e i r g e n e r a t i o n and decomposition, in a v e r t i c a l s t r e a m tube of p o r o u s m e d i u m . Its b a s e lies on a s t a t i o n a r y i m p e r m e a b l e foundation, with which we will align the origin of the c o o r d i n a t e s . Rock p a r t i c l e s with a m a s s flow g3(t) a r e deposited on the upper section of the s t r e a m tube. Then, equations for equilibrium m i g r a t i o n of g a s - l i q u i d m i x t u r e s analogous to those examined above m a y be adopted for the m a s s flow of the components, the only difference being that the weight of the components and their t r a n s p o r t due to the consolidation of the r o c k s a r e taken into account:

p

OlnC~ OlnT _[_kz dAp~ 0~} -~Dl~ -t-D~r 0-----~ F~ F2~ -~o Ox nUvaa, --(D~,r~,+D~,r~) w~ Op, p--7 0--7 v~(x, t)~-~-Z3(x, t)/yz(x, t)

(4.1) (4.2)

It is obvious that

v, ~ ot

(4.3)

w h e r e Y3 is the specific g r a v i t y of the r o c k skeleton. In o r d e r to e s t a b l i s h the dependence y3(x, t), it is n e c e s s a r y to solve the p r o b l e m of the d e f o r m a t i o n of the r o c k s under the action of the r o c k p r e s s u r e p3(x, t), taking into account the p h y s i c o c h e m i c a l c o n v e r s i o n p r o c e s s e s of the r o c k s . C o n s t r u c t i v e simplifications of the p r o b l e m can be obtained in the following two w a y s . It can be a s s u m e d that changes in the r o c k take p l a c e only b e c a u s e of consolidation, and

W-~7~(P~),

p~(x,t)=p~(H,t)q-fysdx

(4.4)

w h e r e P3 is the r o c k p r e s s u r e , r (Y3) is the d e f o r m a t i o n law, and H is the coordinate of the upper mobile boundary of the s e d i m e n t a r y r o c k s . As

869

Ops Ot

= v~(H, t ) ' ~ ( H , t) = g~(H, t)

(4.5)

where vs(H, t) is the speed of travel of the upper section of the s t r e a m tube, then, assuming the p r o c e s s of consolidation to be quasistatic and taking (4.3) and (4.4) into account, the m a s s flow of the solid material of the r o c k skeleton is

g~(x, t) ~-- g~(H, t) ~ p ( ~ ) d x

(4.6)

0

w h e r e the integral takes into account the shrinking of the rock over a time dr. The second simplification can be the assumption that the functional dependence of a s e r i e s of p a r a m e t e r s (geostatic p r e s s u r e , formation p r e s s u r e , t e m p e r a t u r e , water minerlization, density and porosity of the rock) on depth is invariant. This w a s established by p r o c e s s i n g statistically a g r e a t deal of field material [10, 11, etc.]. This is important in the r e s p e c t that in examining migration in media with a variable deposit thickness we will r e g a r d these p a r a m e t e r s , in the f i r s t approximation, as being dependent explicitly on depth and implicitly (in t e r m s of depth) on time. For example, the dependence of porosity on depth h(t) = H(t) - x has been determined [10] by p r o c e s s i n g the p o r o s i t y m e a s u r e m e n t s of a l a r g e number of c o r e samples: m = m 0 - ~ l n h / h 0, where m 0 is the porosity at h 0. The value of ~ is roughly constant for individual t e r r i tories. We will w r i t e the continuity equation in the f o r m

Ox------- -

Ot

~- 1~ - - (p~

(4.7)

where ~i is a function allowing for the loss of m a t e r i a l due to biochemical reactions and its loss through the wall surface of the s t r e a m tube. The equations examined above contain a complete set of coefficients, which are the p a r a m e t e r s of migration and equilibrium kinetics in m i c r o - and m a c r o v o l u m e s of porous media. A considerable p r o p o r tion of them can be determined experimentally under l a b o r a t o r y conditions [6, 12-16], others can be estimated on the basis of indirect experiments, and, finally, a third type (the p a r a m e t e r s c h a r a c t e r i z i n g the section as a whole in the bed) can only be estimated by r e v e r s e methods on the basis of empirical m a t e r i a l . Thus, the effective diffusion coefficient of the r o c k s , which allows collectively for the t r a n s p o r t of gases under the action of concentration differences, for ethane, methane, helium, propane, and butane varies in the r a n g e f r o m 10 -2 to 10 -10 c m 2 / s e c . The coefficient of "concentration conduction" ~ = D / f i of methane for samples of r o c k s saturated with water and m i n e r a l i z e d water is of the o r d e r of 10 -7 cm2/~ec. The r a t e of deposit accumulation is of the o r d e r of 10-8-10 -l~ c m / s e c . The gas permeability of the samples is mostly 10 -5 darcy, but can be 10-7-10 -8 d a r c y . The p o r o s i t y of low-permeability rocks is a few p e r c e n t and r a r e l y exceeds 10-20%. Because of the low values of the t r a n s p o r t p a r a m e t e r s , the distribution of material in the e a r t h ' s c r u s t is for the most p a r t far f r o m equilibrium and is unsteady, i.e., as if the initial stage of the p r o c e s s were taking place. In this case, the p r o c e s s e s c h a r a c t e r i z i n g the kinetics of the p r o c e s s and the p a r a m e t e r s of concentration conduction play the main p a r t . Secondly, in local a r e a s with small " c h a r a c t e r i s t i c " linear dimensions (meters, centimeters), in a number of c a s e s the s y s t e m can be considered as being isolated over a sufficiently long time interval (thousands of y e a r s ) . Under these conditions, p r o c e s s e s in the m i c r o v o l u m e acquire a significance. For example, gas may be r e g a r d e d as stationary in a closed pore (10 -2 c m ) . If t h e r e a r e oil components in the surrounding rock, their vapors will enter the gas bubble. If p r e s s u r e diffusion is d i s r e g a r d e d , an equilibrium distribution of components is established between the gas and solid phases of the r o c k . However, owing to p r e s s u r e - d i f f u s i o n deposition, the h i g h - m o l e c u l a r c o m p o nents of the oil will be deposited on the "bottom" of the pore, forming droplets of free oil. This m e c h a n i s m can be considered as one of the possible p r o c e s s e s of concentration of the d i s p e r s e d m a t e r i a l inside the p o r e s . It is easy to check by calculations that oil can be f o r m e d in the p o r e s over a s h o r t time in geologi-

870

cal t e r m s (104-10 ~ y e a r s ) under the conditions of the initial g a s s y s t e m of the s t r a t u m . This m e c h a n i s m m a y be used to explain the accumulation of oil f r i n g e s in d e p r e s s e d p a r t s of the s t r a t u m in c o n c e n t r a t e d f o r m and not in the f o r m of l a y e r s s p r e a d out through the whole s t r a t u m . LITERATURE io 2.

CITED

V.A. Sokolov, Migration of Oil and Gas [in Russian], Izd. AN SSSR, Moscow (1956). V. Fo Linetskii, Migration of Off and the Formation of Deposits [in Russian], Naukova

Dumka,

Kiev

(1965). 3o 4. 5. 6.

7.

8.

9. i0. ii. 12. 13o 14. 15.

16.

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