Percolation Models for Porous Media

Abstract

Recent progress in understanding the effective transport properties of percolation models for porous and conducting random media is reviewed. Both lattice and continuum models are studied. First, we consider the random flow network in ℤ^N, where the pipes of the network are open with probability p and closed with probability 1 — p. Near the percolation threshold p _c ,the effective permeability κ*(p) ~ (p — p _c )^e, p → p⁺_c, where e is the permeability critical exponent. In the limit of low Reynolds number flow, this model is equivalent to a corresponding random resistor network. Here we discuss recent results for the resistor network problem which yield the inequalities 1 ≤ e ≤ 2, N = 2, 3 and 2 ≤ e ≤ 3, N ≥ 4, assuming a hierarchical nodelink-blob (NLB) structure for the backbone near p _c . The upper bound t = 2 in N = 3 virtually coincides with a number of recent numerical estimates. Secondly, we consider problems of transport in porous and conducting media with broad distribution in the local properties, which are often encountered. Here we discuss a continuum percolation model for such media, which is exactly solvable for the effective transport properties in the high disorder limit. The model represents such systems as fluid flowing through consolidated granular media and fractured rocks, as well as electrical conduction in matrix-particle composites near critical volume fractions. Moreover, the results for the model rigorously establish the widely used Ambegaokar, Halperin, and Langer critical path analysis [AHL71].

Acknowledgements: K. Golden would like to specially thank S. Kozlov for many wonderful conversations on the transport properties of percolation models. He would also like to thank M. Aizenman, L. Berlyand, and P. Doyle for helpful conversations concerning the work reviewed here. L. Schwartz, M. Sahimi, VCH Publishers, and the Amercian Physical Society are kindly acknowledged for use of the figures that appear here. Finally, K. Golden gratefully acknowledges support from NSF Grant DMS-9307324 and ONR Grant N000149310141.