Abstract
The permeability of reservoir rocks is most commonly measured with an atmospheric gas. Permeability is greater for a gas than for a liquid. The Klinkenberg equation gives a semi-empirical relation between the liquid and gas permeabilities. In this paper, the wall-slip gas flow problem is homogenized. This problem is described by the steady state, low velocity Navier–Stokes equations for a compressible gas with a small Knudsen number. Darcy's law with a permeability tensor equal to that of liquid flow is shown to be valid to the lowest order. The lowest order wall-slip correction is a local tensorial form of the Klinkenberg equation. The Klinkenberg permeability is a positive tensor. It is in general not symmetric, but may under some conditions, which we specify, be symmetric. Our result reduces to the Klinkenberg equation for constant viscosity gas flow in isotropic media.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adzumi, H.: 1937a, Studies on the flow of gaseous mixtures through capillaries: I. The viscosity of binary gaseous mixtures, Bull. Chem. Soc. Japan 12, 199–226.
Adzumi, H.: 1937b, Studies on the flow of gaseous mixtures through capillaries: II. The molecular flow of gaseous mixtures, Bull. Chem. Soc. Japan 12, 285–291.
Auriault, J.-L.: 1991, Heterogeneous medium. Is an equivalent macroscopic description possible? Int. J. Engng Sci. 29(7), 785–795.
Auriault, J.-L., Strzelecki, T., Bauer, J. and He, S.: 1990, Porous deformable media saturated by a very compressible fluid: quasi-statics, Eur. J. Mech., A/Solids 9(4), 373–392.
Barenblatt, G. I., Entov, V. M. and Ryzhik, V. M.: 1989, Theory of Fluid Flows through Natural Rocks, Kluwer Academic Publishers.
Bensoussan, A., Lions, J.-L. and Papanicolaou, G.: 1978, Asymptotic Analysis for Periodic Structures, North Holland.
Cercignani, C.: 1988, The Boltzmann Equation and its Applications, Springer-Verlag.
Cercignani, C.: 1990, Mathematical Methods in Kinetic Theory, Plenum-Press.
Dagan, G.: 1989, Flow and Transport in Porous Formations, Springer-Verlag.
Ene, H. I. and Sanchez-Palencia, E.: 1975, Équations et phénomènes de surface pour l'écoulement dans un modeèle de milieu poreux, Journal de Mécanique 14(1), 73–108.
Firdaouss, M. and Guermond, J.: 1995, Sur l'homogénéisation des équations de Navier–Stokes à faible nombre de Reynolds, C.R. Acad. Sci. Paris, Série I 320, 245–251.
Hornung, U.: 1997, Homogenization and Porous Media, Springer.
Jones, S. C.: 1972, A rapid accurate unsteady-state Klinkenberg permeameter, Petrol. Engrs. J. 383–397.
Klinkenberg, L. J.: 1941, The permeability of porous media to liquids and gases, Drill. Prod. Prac., API, 200–213.
Ladyzhenskaya, O. A.: 1969, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Breach, New York.
Levy, T.: 1987, Fluids in porous media. Darcy's law, In: E. Sanchez-Palencia and A. Zaoui (eds), Homogenization Techniques for Composite Media, Lecture Notes in Physics 272, Springer.
Levy, T. and Sanchez-Palencia, E.: 1975, On boundary conditions for fluid flow in porous media, Int. J. Engng Sci. 13, 923–940.
Matheron, G.: 1967, Eléments pour une Théorie des Milieux Poreux, Masson & Cie.
Sanchez-Palencia, E.: 1980, Nonhomogeneous Media and Vibration Theory, Lecture notes in Physics 127, Springer.
Skjetne, E.: 1995, High-velocity flow in porous media; analytical, numerical and experimental studies, Doctoral Thesis, Department of Petroleum Engineering and Applied Geophysics, Faculty of Applied Earth Sciences and Metallurgy, Norwegian University of Science and Technology.
Skjetne, E. and Gudmundsson, J. S.: 1993, Model for wall-slip in the Darcy and Forchheimer gas flow regimes, In: Third Lerkendal Petroleum Engineering Workshop, January 20–21, Norwegian Institute of Technology, Tapir Publishers, Trondheim, pp. 111–122.
Skjetne, E. and Gudmundsson, J. S.: 1995, Coupling of wall-slip and high-velocity flow for determination of gas permeability, 1995 Int. Symp. of the Society of Core Analysts.
Strang, G.: 1988, Linear Algebra and its Applications, Harcourt Brace Jovanovich, San Diego.
Whitaker, S.: 1969, Advances in theory of fluid motion in porous media, Ind. Engng. Chem. 61(12), 15–28.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Skjetne, E., Auriault, JL. Homogenization of Wall-Slip Gas Flow Through Porous Media. Transport in Porous Media 36, 293–306 (1999). https://doi.org/10.1023/A:1006572324102
Issue Date:
DOI: https://doi.org/10.1023/A:1006572324102