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On Stabilization of Fingers in A Slightly Cracked Heterogeneous Porous Medium
ON STABILIZATION OF FINGERS IN A SLIGHTLY CRACKED HETEROGENEOUS POROUS MEDIUM A. P. VERMA
ABSTRACT
The stabilization problem of fingers in a specific oil-water displacement process has been examined from a statistical viewpoint for a slightly cracked heterogeneous porous medium. A perturbation procedure has been employed for the analytical solution of the equation of motion, and experimental results of Evgenev, Mattax and Kyte, Bokserman et al. are taken into consideration. It is shown that the perturbation solution does produce “stable fingers” in at least one special case corresponding to the investigated problem.
1. INTRODUCTION
The growth of fingers in displacement processes for homogeneous porous media was statistically examined by Scheidegger and Johnson (S and J) [l], and they found that no stabilization of fingers is possible in the statistical theory. Bokserman, Zheltov and Kocheshkov (BZK) [ 2 ] have recently discussed the physics of oil-water motion in a cracked porous medium. Some special problems with additional physical phenomena have been investigated by the author [3, 4, 51. The present paper discusses the stabilization problem for a well developed finger flow in a slightly cracked heterogeneous porous medium from the S and J viewpoint, and using BZK description of a cracked medium. The finger flow is furnished by water displacing oil from an underground formation. It is shown, by using a perturbation technique, that the stabilization of fingers may occur in this particular case. Thus we have shown that, by changing the basic assumptions of S and J theory, it is possible to find one case in which “stable fingers” may be produced.
2.
STATEMENT OF PROBLEM
Water is injected with constant velocity Vinto an underground oil saturated seam which consists of a slightly cracked heterogeneous porous medium. The displacement of oil by water gives rise to a well developed finger flow. It is assumed that the entire oil on the initial boundary, x = 0 (x is measured in the direction of displacement) is displaced through a small distance due to 22 1
A . P. Verma
222
the impact of the injecting water. For definiteness, the laws of variation in the characteristics of the medium are taken in standard forms (Section 4) (as in [3, 51). The particular interest of the present investigation is to determine the stabilization of fingers under the special conditions of this problem
3.
STATISTICS OF FINGERS, AND CRACKED MEDIUM
In the statistical treatment of fingers [11, only the average cross-sectional area occupied by the fingers is taken into consideration, their individual size and shape are disregarded (see Fig. 1). With the introduction of the notion of fictitious relative permeability [13, this treatment of fingers becomes formally identical to the Buckley-Leverett description of immiscible fluids flow in porous media. The saturation of ith fluid ( S , ) is then defined as the average cross sectional area occupied by the ith fluid at the level x i.e., S , = S,(x, t ) . Thus the saturation of the displacing fluid in the porous medium represents the average cross sectional area occupied by fingers. The following relationship suggested by S and J has been considered. (3.1)
-
K,
=
S,,
-
KO = So = 1 - S , ,
denote the fictitious relative permeabilities of water and oil, where I?, and while S,, So are their saturations.
WATER
DISPLACEMENT
Fig. 1 . Schematic representation of fingers at level "x"
01L
Stubilizurion of Fingers in HererogeneouJ Porous Medium
223
Fig. 2. Diagram showing the impregnation of a cracked one dimensional porous seam with water. The numbers denote: 1 . Cracks 2. Completely impregnated blocks 3. Blocks being impregnated 4. Non-impregnated blocks
For flow in a cracked porous medium [3] (see Fig. 2 ) , the volume of water entering the blocks in. an elementary volume of seam (under the capillary action) is called the Impregnation function @ ( t ) , which is defined by Mattax and Kyte [6] (also, by Vezirov and Kocheshkov [7]) as below
(3.2)
D(E~)-*
=
@(t)
where D and E , depending on the nature of cracked medium, are constants (values are given in [3]); while t denotes the time. Ryzhik (as in [2]) has pointed out that the equations of continuity for flowing phases in cracked porous medium include an additional term, the capillary suction function @{T- 7 ( 0 ) which is defined as @{T- T(<)} (3.3)
=
T
D [ T - T({)]-*
= ~t
r(5)
=
5
=
fit2
(ci
is constant)
(3.3) X -
1
( I is the mean blocksize)
Since we are considering a slightly cracked porous medium thereforc, the capillary suction function is assumed small in the discussion.
A . P. Verma
224
4. LAWS
OF VARIATION IN THE CHARACTERISTICS OF POROUS MEDIUM
Folowins Oroveanu [S], we take the laws of variation in the porosity and permeability of the medium as 1
rn = m(x) = CI - bx’
(4.1) and,
K = K ( x ) = Ko(1
(4.2)
+u ~ x ) ,
where a, b, k, and a l are constants. Since m(x) cannot exceed unity, we assume further that a - bx 2 1 or, x 5 (a - l)/h
.
5.
FUNDAMENTAL EQUATIONS
The seepage velocity of water (u,) and oil (uo) may be written, from Darcy’s law, as
where K = K ( x ) is the variable permeability of the cracked medium, p is the common pressure of the phases (capillary pressure is neglected), and pw, p,, are the constant viscosities of water and oil respectively. Following Ryzhik (as in [2]), the equations of continuity for flowing phases (phase densities regarded constant) are written as
(5.3)
m- as, at
au, + +ax
@[T - T(<)]
=
0
(5.4) where m = m ( x ) is the variable porosity of the medium and remaining symbols are already defined in Section 3. Lastly, the definition of phase saturation [9] gives
(5.5)
&+So
6.
=
1
EQUATION OF MOTION FOR SATURATION
Putting the value of 0, and uo from equations (5.1) and (5.2) in (5.3) and (5.4) respectively, we get
Stabilization of Fingers in Heterogeneous Porous Medium
225
Combining equations (6.1) and (6.2), we get (6.3) Integrating equation (6.3) with respect to x , and evaluating the constant of integration by using the condition (6.4)
u,(O,t)
Eo(O,t) = 0
= V , and
(the later condition becomes obvious when we note that the entire oil from the initial boundary of the seam is displaced for all time by the injected water) we get
ap ax
V
+ -KKO
-K
*
PO
PO
From equations (6.5) and (6.1), we get
Using equation (3.1), we may write
R,
;z/LRr] -+Pw
=
P p=I”o. P-1+- 1 ’ P W SW
PI)
Differentiation gives
P y’(su) = [SJP-1)
+ 13’
Rewriting equation (6.6) with the help of equation (6.7)-(6.8), we have (6.9)
m -as, + V ( at [ S J P - 1) + 112
) as,+ ax Q { T - r ( t ) }
=
0.
This is the equation of motion for the saturation S ,
7. SOLUTION BY THE PERTURBATION METH3D Since the capillary suction function @{ T - T(C;)} is small in our problem (see remarks at the end of Section 3) therefore we use a perturbation procedure
226
A.
P. Verma
[lo] to solve equation (6.9). Thus neglecting @{T- r ( t ) } , in equation (6.9), we have (7.1) The characteristic equations of (7.1) are
P d-x -dt [ S , ( P - 1)
(7.2)
+ 112
V m(x)
(7.3) Substituting the value of m(x) from equation (4.1) in equation (7.2), integrating and applying the condition (7.4)
x = O , t = O
for all saturations (since all saturations start intially from the initial boundary), we have x = bQ [I - e x p (
(7.5)
-
+
[S,(P-bPVt 1) 112
I1
.
On transforming t into T , and substituting the value of @{T - z(t)} from equation (3.3), we may write equation (6.9) as (7.6)
as,
em - + V aT
P
as,+
([s,(p- 1) + 1 1 2 1 ax
D JT-
= 0 . RXZ
Recently, Evgenev [ll] has pointed out that the value of P , in most cases of practical interest, is large, and therefore we may consider 1/P to be a small quantity. Keeping these remarks in the mind, and substituting the value of T ( = e t ) from equation (7.5) in the last term of equation (7.6), we may write equation (7.6), after simplification, as (7.7)
as, + V
emaT
P
+D
The characteristic equations of this quasilinear equation are
This may be written in the equivalent form as
(7.9)
D (%)'x-*dx
1 = [S,(P - 1) + 13 dSm
=
0.
221
Stabilization of Fingers in Heterogeneous Porous Medium
(7.10)
x-i(a - b x ) d t =
D
1
(?ji
[S,(P-
1
dSlu 1) + 13
Integration of equation (7.9) gives 1
(7.1 1)
lOg[S,(P-
1)
+ 13 + E ,
where E is the constant of integration. Integrating equation (7.10) with the help of equation (7.11), we get Dug (;lit
(7.12)
[{S,(P-21)
=
a
2DJa(P - 1)’
+ 1)’ log{S,(P-
(3 1)
i-(2D,,/a(P JpI/E- 1)
6E ___
)’
3E’
+
+ I} - &{S,(P- 1) + 1}’
where F is the constant of integration. An arbitrary functional relation between the two integrals given by equations (7.11) and (7.12) gives a solution of equation (7.7). Since the saturation S, is defined as the average cross sectional area occupied by the fingers (Section 3), therefore S, = 0 may be regarded as a criteria for investigating their stabilization. It is observed from equations (7.1 1) and (7.12) that definite value of x and t correspond to the zero value of the saturation S-,, and this, in turn, implies that the stabilization of fingers is possible in the specific problem investigated. It may be mentioned here that the conclusion depends on a perturbation procedure. Notwithstanding the difficulty in using such a procedure for studying the long term behaviour of the solution (cf. [12]), we have employed it due to the special nature of the medium, and the particular interest of the present investigation viz., showing the occurrence of the “stable” fingers in at least one case of queer permeability-heterogeneoity and capillary suction term.
8.
PARTICULAR CASES
The following particular cases are discussed. (i) Heterogeneous m e d i u m without cracks For discussing this case, we put @ = 0 in equation (6.9) so that the equation of motion reduces to equation (7.1), and hence the solution may be written as
A . P. Verma
228
(4.1)
x =
_a [I - exp( b
+
[S,(P- bPVt 1) 13’
11
*
It follows from this equation that x tends to a definite limit a/b as time increases indefinitely. However, we cannot immediately infer the stabilization of fingers from the definiteness of alb , because our discussion is restricted (a - l)/b (Section 4). to x (ii) Homogeneous medium without cracks For discussing this case, we put b = 0 and @ = 0 in equation (6.9) so that the equation of motion becomes the same as that of S and J [l], and therefore no stabilization of fingers is possible. (iii) Slightly cracked homogeneous medium For discussing this case, we put b = 0 and @ # 0 in equation (7.7) so that the solution is given by equations (7.11) and (7.12) with b = 0 , and therefore the fingers may stabilize, in this case. 9. CONCLUSION We have shown that a perturbation solution does produce “stable” fingers in one special case corresponding to the conditions of the investigated problem. Three particular cases viz. heterogebeous medium without cracks, and homogoneous medium with or without cracks, are also discussed. REFERENCES
1 . SCHEIDEGGER, A. E. AND E. F. JOHNSON (1961), The statistical behaviour of instabilities in displacement processes in porous media. Canadian J. Physics, 39, 326. 2. BOKSERMAN, A. A., P Yu. ZHELTOV AND A. A. KOCHESHKOV, (1964), Motion of immiscible liquids in a cracked porous medium, Soviet Physics Doklady, 9, 4, 285. 3. VERMA,A. P. (1968), Motion of immiscible liquids in a cracked heterogeneous porous medium with capillary pressure, Rev. Roum. des Sci. techn. Mec. Appli., Rumania, 13, 2, 211. 4. VERMA, A. P. Motion of immiscible liquids in a cracked hetrogeneous porous medium with pressure dependcnt densities. Proc. Notional Institute of Sciences, India, Part A 35, 8, 458 (1969). 5. VERMA, A. P., Statistical behaviour of fingering in a displacement process in heterogeneous porous medium with capillary pressure, Canadian J. Phys., 47, 3, 319 (1969). 6. MATTAX,c. c. AND I. R. KYTE(1962), Pefroleum Engg. J., 2. 7. VEZIROV, D. SH. AND A. A. KOCHESHKOV, Izvestiya An SSR, Mekh. i mashinosf. 6. 8. OROVEANU, T (1963), Scurgerea Fluidelor prin medii poroase neomogene, Editura Academiei R.P.R., Rumania, pp. 92 and 328. 9. SCHEIDEGGER, A. E. (1960), The Physics of Flow Through Porous Media, University of Toronto Press, p. 216. 10. MORSE,P. M. AND FESHBACK, H. (1953), Methods of Theoretical Physics, McGraw Hill, p. 1001. 11. EVGENEV, A. E. (1965), Phase permeabilities in flltration of a two phase system through a porous medium, Soviet Physics Doklady, 10, 5 . 1 1 I . 12. SCHEIDEGGER, A. E. (1960, Growth of instabilities on displacement fronts in porous media, The Physics of Fluids, 3, 1.
DEPARTMENT OF MATHEMATICS, FACULTYOF TECHNOLCGY AND ENGINEERING, M.S. UNIVERSITY OF BARODA, BARODA,INDIA
ABSTRACT
The stabilization problem of fingers in a specific oil-water displacement process has been examined from a statistical viewpoint for a slightly cracked heterogeneous porous medium. A perturbation procedure has been employed for the analytical solution of the equation of motion, and experimental results of Evgenev, Mattax and Kyte, Bokserman et al. are taken into consideration. It is shown that the perturbation solution does produce “stable fingers” in at least one special case corresponding to the investigated problem.
1. INTRODUCTION
The growth of fingers in displacement processes for homogeneous porous media was statistically examined by Scheidegger and Johnson (S and J) [l], and they found that no stabilization of fingers is possible in the statistical theory. Bokserman, Zheltov and Kocheshkov (BZK) [ 2 ] have recently discussed the physics of oil-water motion in a cracked porous medium. Some special problems with additional physical phenomena have been investigated by the author [3, 4, 51. The present paper discusses the stabilization problem for a well developed finger flow in a slightly cracked heterogeneous porous medium from the S and J viewpoint, and using BZK description of a cracked medium. The finger flow is furnished by water displacing oil from an underground formation. It is shown, by using a perturbation technique, that the stabilization of fingers may occur in this particular case. Thus we have shown that, by changing the basic assumptions of S and J theory, it is possible to find one case in which “stable fingers” may be produced.
2.
STATEMENT OF PROBLEM
Water is injected with constant velocity Vinto an underground oil saturated seam which consists of a slightly cracked heterogeneous porous medium. The displacement of oil by water gives rise to a well developed finger flow. It is assumed that the entire oil on the initial boundary, x = 0 (x is measured in the direction of displacement) is displaced through a small distance due to 22 1
A . P. Verma
222
the impact of the injecting water. For definiteness, the laws of variation in the characteristics of the medium are taken in standard forms (Section 4) (as in [3, 51). The particular interest of the present investigation is to determine the stabilization of fingers under the special conditions of this problem
3.
STATISTICS OF FINGERS, AND CRACKED MEDIUM
In the statistical treatment of fingers [11, only the average cross-sectional area occupied by the fingers is taken into consideration, their individual size and shape are disregarded (see Fig. 1). With the introduction of the notion of fictitious relative permeability [13, this treatment of fingers becomes formally identical to the Buckley-Leverett description of immiscible fluids flow in porous media. The saturation of ith fluid ( S , ) is then defined as the average cross sectional area occupied by the ith fluid at the level x i.e., S , = S,(x, t ) . Thus the saturation of the displacing fluid in the porous medium represents the average cross sectional area occupied by fingers. The following relationship suggested by S and J has been considered. (3.1)
-
K,
=
S,,
-
KO = So = 1 - S , ,
denote the fictitious relative permeabilities of water and oil, where I?, and while S,, So are their saturations.
WATER
DISPLACEMENT
Fig. 1 . Schematic representation of fingers at level "x"
01L
Stubilizurion of Fingers in HererogeneouJ Porous Medium
223
Fig. 2. Diagram showing the impregnation of a cracked one dimensional porous seam with water. The numbers denote: 1 . Cracks 2. Completely impregnated blocks 3. Blocks being impregnated 4. Non-impregnated blocks
For flow in a cracked porous medium [3] (see Fig. 2 ) , the volume of water entering the blocks in. an elementary volume of seam (under the capillary action) is called the Impregnation function @ ( t ) , which is defined by Mattax and Kyte [6] (also, by Vezirov and Kocheshkov [7]) as below
(3.2)
D(E~)-*
=
@(t)
where D and E , depending on the nature of cracked medium, are constants (values are given in [3]); while t denotes the time. Ryzhik (as in [2]) has pointed out that the equations of continuity for flowing phases in cracked porous medium include an additional term, the capillary suction function @{T- 7 ( 0 ) which is defined as @{T- T(<)} (3.3)
=
T
D [ T - T({)]-*
= ~t
r(5)
=
5
=
fit2
(ci
is constant)
(3.3) X -
1
( I is the mean blocksize)
Since we are considering a slightly cracked porous medium thereforc, the capillary suction function is assumed small in the discussion.
A . P. Verma
224
4. LAWS
OF VARIATION IN THE CHARACTERISTICS OF POROUS MEDIUM
Folowins Oroveanu [S], we take the laws of variation in the porosity and permeability of the medium as 1
rn = m(x) = CI - bx’
(4.1) and,
K = K ( x ) = Ko(1
(4.2)
+u ~ x ) ,
where a, b, k, and a l are constants. Since m(x) cannot exceed unity, we assume further that a - bx 2 1 or, x 5 (a - l)/h
.
5.
FUNDAMENTAL EQUATIONS
The seepage velocity of water (u,) and oil (uo) may be written, from Darcy’s law, as
where K = K ( x ) is the variable permeability of the cracked medium, p is the common pressure of the phases (capillary pressure is neglected), and pw, p,, are the constant viscosities of water and oil respectively. Following Ryzhik (as in [2]), the equations of continuity for flowing phases (phase densities regarded constant) are written as
(5.3)
m- as, at
au, + +ax
@[T - T(<)]
=
0
(5.4) where m = m ( x ) is the variable porosity of the medium and remaining symbols are already defined in Section 3. Lastly, the definition of phase saturation [9] gives
(5.5)
&+So
6.
=
1
EQUATION OF MOTION FOR SATURATION
Putting the value of 0, and uo from equations (5.1) and (5.2) in (5.3) and (5.4) respectively, we get
Stabilization of Fingers in Heterogeneous Porous Medium
225
Combining equations (6.1) and (6.2), we get (6.3) Integrating equation (6.3) with respect to x , and evaluating the constant of integration by using the condition (6.4)
u,(O,t)
Eo(O,t) = 0
= V , and
(the later condition becomes obvious when we note that the entire oil from the initial boundary of the seam is displaced for all time by the injected water) we get
ap ax
V
+ -KKO
-K
*
PO
PO
From equations (6.5) and (6.1), we get
Using equation (3.1), we may write
R,
;z/LRr] -+Pw
=
P p=I”o. P-1+- 1 ’ P W SW
PI)
Differentiation gives
P y’(su) = [SJP-1)
+ 13’
Rewriting equation (6.6) with the help of equation (6.7)-(6.8), we have (6.9)
m -as, + V ( at [ S J P - 1) + 112
) as,+ ax Q { T - r ( t ) }
=
0.
This is the equation of motion for the saturation S ,
7. SOLUTION BY THE PERTURBATION METH3D Since the capillary suction function @{ T - T(C;)} is small in our problem (see remarks at the end of Section 3) therefore we use a perturbation procedure
226
A.
P. Verma
[lo] to solve equation (6.9). Thus neglecting @{T- r ( t ) } , in equation (6.9), we have (7.1) The characteristic equations of (7.1) are
P d-x -dt [ S , ( P - 1)
(7.2)
+ 112
V m(x)
(7.3) Substituting the value of m(x) from equation (4.1) in equation (7.2), integrating and applying the condition (7.4)
x = O , t = O
for all saturations (since all saturations start intially from the initial boundary), we have x = bQ [I - e x p (
(7.5)
-
+
[S,(P-bPVt 1) 112
I1
.
On transforming t into T , and substituting the value of @{T - z(t)} from equation (3.3), we may write equation (6.9) as (7.6)
as,
em - + V aT
P
as,+
([s,(p- 1) + 1 1 2 1 ax
D JT-
= 0 . RXZ
Recently, Evgenev [ll] has pointed out that the value of P , in most cases of practical interest, is large, and therefore we may consider 1/P to be a small quantity. Keeping these remarks in the mind, and substituting the value of T ( = e t ) from equation (7.5) in the last term of equation (7.6), we may write equation (7.6), after simplification, as (7.7)
as, + V
emaT
P
+D
The characteristic equations of this quasilinear equation are
This may be written in the equivalent form as
(7.9)
D (%)'x-*dx
1 = [S,(P - 1) + 13 dSm
=
0.
221
Stabilization of Fingers in Heterogeneous Porous Medium
(7.10)
x-i(a - b x ) d t =
D
1
(?ji
[S,(P-
1
dSlu 1) + 13
Integration of equation (7.9) gives 1
(7.1 1)
lOg[S,(P-
1)
+ 13 + E ,
where E is the constant of integration. Integrating equation (7.10) with the help of equation (7.11), we get Dug (;lit
(7.12)
[{S,(P-21)
=
a
2DJa(P - 1)’
+ 1)’ log{S,(P-
(3 1)
i-(2D,,/a(P JpI/E- 1)
6E ___
)’
3E’
+
+ I} - &{S,(P- 1) + 1}’
where F is the constant of integration. An arbitrary functional relation between the two integrals given by equations (7.11) and (7.12) gives a solution of equation (7.7). Since the saturation S, is defined as the average cross sectional area occupied by the fingers (Section 3), therefore S, = 0 may be regarded as a criteria for investigating their stabilization. It is observed from equations (7.1 1) and (7.12) that definite value of x and t correspond to the zero value of the saturation S-,, and this, in turn, implies that the stabilization of fingers is possible in the specific problem investigated. It may be mentioned here that the conclusion depends on a perturbation procedure. Notwithstanding the difficulty in using such a procedure for studying the long term behaviour of the solution (cf. [12]), we have employed it due to the special nature of the medium, and the particular interest of the present investigation viz., showing the occurrence of the “stable” fingers in at least one case of queer permeability-heterogeneoity and capillary suction term.
8.
PARTICULAR CASES
The following particular cases are discussed. (i) Heterogeneous m e d i u m without cracks For discussing this case, we put @ = 0 in equation (6.9) so that the equation of motion reduces to equation (7.1), and hence the solution may be written as
A . P. Verma
228
(4.1)
x =
_a [I - exp( b
+
[S,(P- bPVt 1) 13’
11
*
It follows from this equation that x tends to a definite limit a/b as time increases indefinitely. However, we cannot immediately infer the stabilization of fingers from the definiteness of alb , because our discussion is restricted (a - l)/b (Section 4). to x (ii) Homogeneous medium without cracks For discussing this case, we put b = 0 and @ = 0 in equation (6.9) so that the equation of motion becomes the same as that of S and J [l], and therefore no stabilization of fingers is possible. (iii) Slightly cracked homogeneous medium For discussing this case, we put b = 0 and @ # 0 in equation (7.7) so that the solution is given by equations (7.11) and (7.12) with b = 0 , and therefore the fingers may stabilize, in this case. 9. CONCLUSION We have shown that a perturbation solution does produce “stable” fingers in one special case corresponding to the conditions of the investigated problem. Three particular cases viz. heterogebeous medium without cracks, and homogoneous medium with or without cracks, are also discussed. REFERENCES
1 . SCHEIDEGGER, A. E. AND E. F. JOHNSON (1961), The statistical behaviour of instabilities in displacement processes in porous media. Canadian J. Physics, 39, 326. 2. BOKSERMAN, A. A., P Yu. ZHELTOV AND A. A. KOCHESHKOV, (1964), Motion of immiscible liquids in a cracked porous medium, Soviet Physics Doklady, 9, 4, 285. 3. VERMA,A. P. (1968), Motion of immiscible liquids in a cracked heterogeneous porous medium with capillary pressure, Rev. Roum. des Sci. techn. Mec. Appli., Rumania, 13, 2, 211. 4. VERMA, A. P. Motion of immiscible liquids in a cracked hetrogeneous porous medium with pressure dependcnt densities. Proc. Notional Institute of Sciences, India, Part A 35, 8, 458 (1969). 5. VERMA, A. P., Statistical behaviour of fingering in a displacement process in heterogeneous porous medium with capillary pressure, Canadian J. Phys., 47, 3, 319 (1969). 6. MATTAX,c. c. AND I. R. KYTE(1962), Pefroleum Engg. J., 2. 7. VEZIROV, D. SH. AND A. A. KOCHESHKOV, Izvestiya An SSR, Mekh. i mashinosf. 6. 8. OROVEANU, T (1963), Scurgerea Fluidelor prin medii poroase neomogene, Editura Academiei R.P.R., Rumania, pp. 92 and 328. 9. SCHEIDEGGER, A. E. (1960), The Physics of Flow Through Porous Media, University of Toronto Press, p. 216. 10. MORSE,P. M. AND FESHBACK, H. (1953), Methods of Theoretical Physics, McGraw Hill, p. 1001. 11. EVGENEV, A. E. (1965), Phase permeabilities in flltration of a two phase system through a porous medium, Soviet Physics Doklady, 10, 5 . 1 1 I . 12. SCHEIDEGGER, A. E. (1960, Growth of instabilities on displacement fronts in porous media, The Physics of Fluids, 3, 1.
DEPARTMENT OF MATHEMATICS, FACULTYOF TECHNOLCGY AND ENGINEERING, M.S. UNIVERSITY OF BARODA, BARODA,INDIA