Abstract
The use of effective-medium treatments to estimate bulk properties pertaining to transport (of, for example, fluids, heat, particles or electricity) through random composite media (such as reservoir rocks), is widespread. This is because they are relatively simple, often reasonably accurate (on occasion, remarkably so) and in many cases yield closed-form expressions for the properties concerned. However, the single-bond effective-medium treatment (EMT) of random resistor networks that has been used to determine transport coefficients for various transport problems in pore networks is limited to some special isotropic networks with nearest-neighbour connections. We demonstrate here that transport through two different fracture system models, with stress-induced anisotropy, can be treated using an EMT originally applied to anisotropic resistor networks. The main purpose of the present contribution, however, is to present a new, more general effective medium formalism applicable to networks of arbitrary topology. This new generalised EMT is used to obtain a new criterion for percolation of an arbitrary conducting network under random dilution. A specific application to unsaturated flow through a pore network with nearest- and next-nearest-neighbour connections is also given.
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Baecher, G. B., Lanney, N. A., and Einstein, H. H., 1977, Statistical descriptions of rock properties and sampling,Proc. 18th U.S. Sympos. Rock Mechanics, Vol. 5C1, pp. 1–8.
Barton, C. C. and Larson, E., 1985, Fractal geometry of two-dimensional fracture networks at Yucca Mountain, southwestern Nevada,Proc. Internat. Sympos. Fundamentals of Rock Joints, Bjorkliden, 15–20 September 1985, pp.77–84.
Bear, J., Braester, C., and Menier, P. C., 1987, Effective and relative permeabilities of anisotropic porous media,Transport in Porous Media 2, 301–316.
Bernasconi, J., 1974, Conduction in anisotropic disordered systems: effective-medium theory,Phys. Rev. B9, 4575–4579.
Bruggeman, D. A. G., 1935, Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen,Ann. Phys. 24, 636–679.
Chatzis, I., Morrow, N. R., and Lim, H. T., 1983, Magnitude and detailed structure of residual oil saturation,Soc. Pet. Eng. J., April 1983, 311–326.
Domb, C. and Sykes, M. F., 1960, Cluster size in random mixtures and percolation processes,Phys. Rev. 122, 77–78.
Dullien, F. A. L., 1979,Porous Media: Fluid Transport and Pore Structure, Academic Press, New York.
Fatt, I., 1956, The network model of porous media, I, II and III,AIME Trans. 207, 144–181.
Gradshteyn, I. S. and Ryzhik, I. M., 1980,Table of Integrals, Series and Products, Corrected and enlarged edition, Academic Press, New York.
Hanai, T., 1968, Electrical properties of microemulsions, in P. Sherman (ed.),Emulsion Science, Academic Press, New York, Ch. 5, pp.354–478.
Kesten, H., 1982,Percolation Theory for Mathematicians, Birkhäuser, Boston, Ch. 3.
Kirkpatrick, S., 1973, Percolation and conduction,Rev. Mod. Phys. 45, 574–614.
Koplik, J., 1982, Creeping flow in two-dimensional networks,J. Fluid Mech. 119, 219–247.
Koplik, J. and Lasseter, T. J., 1984, One- and two-phase flow in network models of porous media,Chem. Eng. Commun. 26, 285–295.
La Pointe, P. R. and Hudson, J. A., 1985, Characterisation and interpretation of rock mass joint patterns,Geol. Soc. Amer. Special Paper 199.
Long, J. C. S., Remer, J. S., Wilson, C. R., and Witherspoon, P. A., 1982, Porous media equivalents for networks of discontinuous fractures,Water Resour. Res. 18, 645–658.
Long, J. C. S. and Witherspoon, P. A., 1985, The relationship of the degree of interconnection to permeability in fracture networks,J. Geophys. Res. 90, 3087–3098.
Mohanty, K. K. and Salter, S. J., 1982, Multiphase flow in porous media: II. Pore-level modelling, Society of Petroleum Engineers Paper 11018, presented at the 57th. Annual Fall Technical Conference of the Society of Petroleum Engineers of AIME, New Orleans, U.S.A., 26–29 Sept. 1982.
Mohanty, K. K. and Salter, S., 1983, Advances in pore-level modelling of flow through porous media, presented at AIChE Fall Meeting, 30 Oct.–4 Nov. 1983.
Robinson, P. C., 1984, Connectivity of fracture systems - a percolation theory approach, DPhil. thesis, Univ. of Oxford, UK.
Sen, P. N., Scala, C. and Cohen, M. H., 1981, A self-similar model for sedimentary rocks with application to the dielectric constant of fused glass beads,Geophys. 46, 781–795.
Singhal, A. K. and Somerton, W. H., 1977, Quantitative modelling of immiscible displacement in porous media, a network approach,Rev. Inst. Franc. Pet. 32, 897–920.
Veinberg, A. K., 1966, The magnetic permeability, electrical conductivity, dielectric constant and thermal conductivity of a medium containing spherical and ellipsoidal inclusions,Dokl. Akad. Nauk. SSSR 169, 543–546. English translation inSov. Phys. Dok. 11, 593 (1967).
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Harris, C.K. Application of generalised effective-medium theory to transport in porous media. Transp Porous Med 5, 517–542 (1990). https://doi.org/10.1007/BF01403480
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DOI: https://doi.org/10.1007/BF01403480