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Flow in random porous media: mathematical formulation, variational principles, and rigorous bounds

Published online by Cambridge University Press:  26 April 2006

Jacob Rubinstein  and
S. Torquato
Jacob Rubinstein
Affiliation:
Department of Mathematics, Stanford University, Stanford, CA 94305-2125, USA Present address: Department of Mathematics, Technion-I.I.T. Haifa 32000, Israel.
S. Torquato
Affiliation:
Department of Mechanical and Aerospace Engineering and Department of Chemical Engineering, North Carolina State University, Raleigh, NC 27695-7910, USA
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Abstract

The problem of the slow viscous flow of a fluid through a random porous medium is considered. The macroscopic Darcy's law, which defines the fluid permeability k, is first derived in an ensemble-average formulation using the method of homogenization. The fluid permeability is given explicitly in terms of a random boundary-value problem. General variational principles, different to ones suggested earlier, are then formulated in order to obtain rigorous upper and lower bounds on k. These variational principles are applied by evaluating them for four different types of admissible fields. Each bound is generally given in terms of various kinds of correlation functions which statistically characterize the microstructure of the medium. The upper and lower bounds are computed for flow interior and exterior to distributions of spheres.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 206 , September 1989 , pp. 25 - 46
Copyright
© 1989 Cambridge University Press

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References

Berryman, J. G. 1983 Random close packing of hard spheres and disks.. Phys. Rev. A 27, 1053.Google Scholar
Berryman, J. G. 1986 Variational bounds on Darcy's constant. In Homogenization and Effective Moduli of Materials. IMA Volumes in Mathematics and Its Application, vol. 1, Springer.
Berryman, J. G. & Milton, G. W. 1985 Normalization constraint for variational bounds on fluid permeability. J. Chem. Phys. 83, 754.Google Scholar
Brinkman, H. C. 1947 A calculation of the viscous force exerted by a flowing fluid on a dense swarm of particles. Appl. Sci. Res. A1, 27.Google Scholar
Caflisch, R. E. & Rubinstein, J. 1986 Lectures on the Mathematical Theory of Multiphase Flow. Courant Institute Notes Series, New York University.
Childress, S. 1972 Viscous flow past a random array of spheres. J. Chem. Phys. 56, 2527.Google Scholar
Doi, M. 1976 A new variational approach to the diffusion and flow problem in porous media. J. Phys. Soc. Japan 40, 567.Google Scholar
Dullien, F. A. L. 1979 Porous Media. Academic.
Elam, W. T., Kerstein, A. R. & Rehr, J. J. 1984 Critical properties of the void percolation problem for spheres. Phys. Rev. Lett. 52, 1516.Google Scholar
Haan, S. W. & Zwanzig, R. 1977 Series expansions in a continuum percolation problem.. J. Phys. A 10, 1547.Google Scholar
Happel, J. & Brenner, H. 1983 Low Reynolds Number Hydrodynamics. Nijhoff.
Hasimoto, H. 1959 On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres. J. Fluid Mech. 5, 317.CrossRefGoogle Scholar
Hinch, E. J. 1977 An averaged-equation approach to particle interactions in a fluid suspension. J. Fluid Mech. 83, 695.Google Scholar
Keller, J. B. 1980 Darcy's law for flow in porous media and the two-scale method. In Nonlinear P.D.E. in Engineering and Applied Sciences (ed. R. L. Sternberg, A. J. Kalinowski & J. S. Papadakis). Marcel Dekker.
Keller, J. B., Rubenfeld, L. & Molyneux, J. 1967 Extremum principles for slow viscous flows with applications to suspensions. J. Fluid Mech. 30, 97.Google Scholar
Lundgren, T. S. 1972 Slow flow through stationary random beds and suspension of spheres. J. Fluid Mech. 51, 273.Google Scholar
Neuman, S. P. 1977 Theoretical derivation of Darcy's law. Acta. Mech. 25, 153.Google Scholar
Prager, S. 1961 Viscous flow through porous media. Phys. Fluids 4, 1477.Google Scholar
Rubinstein, J. 1987 Hydrodynamic screening in random media. In Hydrodynamic Behavior and Interacting Particle Systems. IMA Volumes in Mathematics and Its Application. vol. 9 (ed. G. Papanicolao) Springer.
Rubinstein, J. & Keller, J. B. 1987 Lower bounds on permeability. Phys. Fluids 30, 2919.Google Scholar
Rubinstein, J. & Torquato, S. 1988 Diffusion-controlled reactions: mathematical formulation, variational principle, and rigorous bounds. J. Chem. Phys. 88, 6372.Google Scholar
Salacuse, J. J. & Stell, G. 1982 Polydisperse systems: statistical thermodynamics with applications to several models including hard and permeable spheres. J. Chem. Phys. 77, 3714.CrossRefGoogle Scholar
Sanchez-Palencia, E. 1980 Nonhomogeneous media and vibration theory. Lectures Notes in Physics, vol. 127. Springer.
Sangani, A. & Acrivos, A. 1982 Slow flow through a cubic array of spheres. Intl J. Multiphase Flow 8, 343.Google Scholar
Scheidegger, A. E. 1960 The Physics of Flow through Porous Media. Macmillan.
Tartar, L. 1980 Nonhomogeneous Media and Vibration Theory, Appendix 2. Lecture Notes in Physics, vol. 127. Springer.
Torquato, S. 1984 Bulk properties of two-phase disordered media: I. Cluster expansion for the effective dielectric constant of dispersions of penetrable spheres. J. Chem. Phys. 81, 5079.CrossRefGoogle Scholar
Torquato, S. 1986a Bulk properties of two-phase disordered media: III. New bounds on the effective conductivity of dispersions of penetrable spheres. J. Chem. Phys. 84, 6345.Google Scholar
Torquato, S. 1986b Microstructure characterization and bulk properties of disordered two-phase media. J. Statist. Phys. 45, 843.Google Scholar
Torquato, S. & Beasley, J. D. 1987 Bounds on the permeability of a random array of partially penetrable spheres. Phys. Fluids 30, 633.Google Scholar
Torquato, S. & Lado, F. 1986 Effective properties of two-phase disordered composite media: II. Evaluation of bounds on the conductivity and bulk modulus of dispersions of impenetrable spheres.. Phys. Rev. B 33, 6428.CrossRefGoogle Scholar
Torquato, S. & Rubinstein, J. 1988 Diffusion-controlled reactions. II. Further bounds on the rate constant. J. Chem. Phys. 90, 1644.Google Scholar
Torquato, S. & Stell, G. 1982 Microstructure of two-phase random media. I. The n-point probability functions. J. Chem. Phys. 77, 2071.Google Scholar
Weissberg, H. L. & Prager, S. 1970 Viscous flow through porous media. III. Upper bounds on the permeability for a simple random geometry. Phys. Fluids 13, 2958.Google Scholar
Whitaker, S. 1986 Flow in porous media: a theoretical derivation of Darcy's law. Trans. Porous Media 1, 3.Google Scholar
Widom, B. 1966 Random sequential addition of hard spheres to a volume. J. Chem. Phys. 44, 3888.Google Scholar
Zick, A. A. & Homsy, G. M. 1982 Stokes flow through periodic arrays of spheres. J. Fluid Mech. 115, 13.Google Scholar