Abstract
An extended formulation of Darcy's two-phase law is developed on the basis of Stokes' equations. It leads, through results borrowed from the thermodynamics of irreversible processes, to a matrix of relative permeabilities. Nondiagonal coefficients of this matrix are due to the viscous coupling exerted between fluid phases, while diagonal coefficients represent the contribution of both fluid phases to the total flow, as if they were alone. The coefficients of this matrix, contrary to standard relative permeabilities, do not depend on the boundary conditions imposed on two-phase flow in porous media, such as the flow rate.
This formalism is validated by comparison with experimental results from tests of two-phase flow in a square cross-section capillary tube and in porous media. Coupling terms of the matrix are found to be nonnegligible compared to diagonal terms. Relationships between standard relative permeabilities and matrix coefficients are studied and lead to an experimental way to determine the new terms for two-phase flow in porous media.
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Abbreviations
- (B ji ):
-
inverse of the (C ji ) matrix
- B * :
-
flow constant
- C(z) :
-
fluid/fluid interface curvature
- C i :
-
curvature of the meniscus ∑ i ; C i = C(Z i ); i = 1, 2
- (C supjinfi ):
-
matrix of flows
- det(B supjinfi ):
-
determinant of the (B supjinfi ) matrix
- k :
-
intrinsic permeability of the medium
- k i :
-
mobility of phase i, i = 1, 2
- k ri :
-
relative permeability of phase i, i = 1, 2
- k supjinfri :
-
coefficient of relative permeability matrix i, j = 1, 2
- k supjinfi :
-
coefficient of mobility matrix i, j = 1, 2
- ℒ :
-
fluid/fluid interface in a cross-section
- P i :
-
pressure exerted by the system upon phase i, i = 1, 2
- P c :
-
capillary pressure (P c = P 1 − P 2)
- q i :
-
flow rate of phase i, i = 1, 2
- Q :
-
absolute value of q i
- r(z) :
-
half-side of a cross-section at position z
- s i :
-
saturation of phase i, i = 1, 2
- S i :
-
capillary cross-section at position Z i , i = 1, 2
- ℒ :
-
solid surface
- T :
-
temperature of the system
- \(\bar u_i\) :
-
velocity of phase i, i = 1, 2
- Z i :
-
position of meniscus ∑ i , i = 1, 2
- α :
-
function of \(\bar \mu\) and θ
- β :
-
shape factor
- γ :
-
interfacial tension of fluid/fluid interface
- λ i :
-
flow constant i = 1, 2
- μ i :
-
viscosity of fluid phase i, i = 1, 2
- \(\bar \mu\) :
-
visocity ratio: \(\bar \mu\) = μ1/μ2
- ϱ i :
-
mass per unit volume of fluid phase i, i = 1, 2
- θ :
-
contact angle
- ∑ i :
-
meniscus i of oil ganglion i = 1, 2
- φ :
-
porosity
- 1:
-
nonwetting fluid
- 2:
-
wetting fluid
References
Auriault J. L., 1987, Nonsaturated deformable porous media: Quasistatics, Transport in Porous Media 2, 45–64.
Auriault, J. L. and Sanchez-Palencia, E., 1986, Remarques sur la loi de Darcy pour les écoulements diphasiques en milieu poreaux, J. Theor. Appl. Mech. special issue, pp. 141–156.
Bourbiaux, B. and Kalaydjian, F., 1988, Experimental study of cocurrent and countercurrent flows in natural porous media, Soc. Petr. Eng. No. 18283.
Danis, M. and Jacquin, C., 1983, Influence du contraste des viscosités sur les perméabilités relatives lors du drainage. Expérimentation et modélisation, Rev. Inst. Franc. Petrol. 38, 723–733.
De Gennes, P.-G., 1983, Theory of slow biphasic flows in porous media, Phys. Chem. Hydr. 4, 175–185.
Glansdorff, P. and Prigogine, I., 1971, Structures, stabilité et fluctuations, Masson, Paris.
Kalaydjian, F., 1987, A macroscopic description of multiphase flow in porous media involving space-time evolution of fluid/fluid interface, Transport in Porous Media 2, 537–552.
Kalaydjian, F., 1988, Couplage entre phases fluides dans les écoulements diphasiques incompressibles en milieu poreux, Thesis Univ. Bordeaux I.
Kalaydjian, F. and Legait, B., 1987a, Ecoulement lent à contre-courant de deux fluides non miscibles dans un capillaire présentant un rétrécissement, C. R. Acad. Sc. Paris Ser. II 304, 869–872.
Kalaydjian, F. and Legait, B., 1987b, Perméabilités relatives couplées dans des écoulements en capillaire et en milieu poreux, C. R. Acad. Sc. Paris Ser. II 304, 1035–1038.
Kalaydjian, F. and Legait, B., 1988, Effets de la géométrie des pores et de la mouillabilité sur le déplacement diphasique à contre-courant en capillaire et en milieu poreux, Rev. Phys. Appl. 23, 1071–1081.
Legait, B., 1983a, Interprétation de certains types d'écoulements diphasiques en milieu poreux à partir des écoulements en capillaires, Thesis. Univ. Bordeaux I.
Legait, B., 1983b, Laminar flow of two phases through a capillary tube with variable square cross-section, J. Colloid Interface Sci. 96, 28–38.
Lelièvre, R.-F., 1966, Etude d'écoulements diphasiques permanents à contre-courants en milieu poreux, comparaison avec les écoulements de même sens. Thesis, Univ. Toulouse.
Mayer, R. P. and Stowe, R. A., 1965, Mercury porosimetry - breakthrough pressure for penetration between packed spheres, J. Colloid Interface Sci. 20, 893–911.
Ngan, C. G. and Dussan, V., E. B., 1982, On the nature of the dynamic contact angle: An experimental study, J. Fluid Mech. 118, 27–40.
Rose, W., 1972, Petroleum reservoir engineering at the crossroads (Ways of thinking and doing), The Iran. Pet. Inst. Bull. 46, 23–27.
Rose, W., 1974, Second thoughts on Darcy's law, The Iran. Pet. Inst. Bull. 48, 25–30.
Rose, W., 1988, Measuring transport coefficients necessary for the description of coupled two-phase flow of immiscible fluids in porous media, Transport in Porous Media 3, 163–171.
Singhai, A. K. and Somerton, W. H., 1970, Two-phase flow through a non-circular capillary at low Reynolds numbers, J. Can. Petrol. Technol. 197–205
Whitaker, S., 1986, Flow in porous media II: The governing equations for immiscible, two-phase flow, Transport in Porous Media 1, 105–125.
Wyckoff, R. D. and Botset, H. G., 1936, Flow of gas-liquid mixtures through unconsolidated sands, Physics 7, 325–345.
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Kalaydjian, F. Origin and quantification of coupling between relative permeabilities for two-phase flows in porous media. Transp Porous Med 5, 215–229 (1990). https://doi.org/10.1007/BF00140013
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DOI: https://doi.org/10.1007/BF00140013