Path-integral variational methods for flow through porous media

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Abstract
We characterize a porous medium as a statistically homogeneous continuum with local fluctuations in physical parameters. We consider (1) the steady-state flow of a single incompressible fluid through the medium, and (2) the dispersion of a passive tracer in such a flow. For each problem we average a path-integral expression for the Green’s function over parameter fluctuations, and obtain large-distance, long-time effective parameters via Feynman’s variational method. For the permeability problem, and the tracer problem at small Peclet number P, the variational results are consistent with results obtained by first-order perturbation theory. For the tracer problem at large P, the variational method predicts the expected linear dependence of the effective dispersion tensor on P, which perturbation theory does not. This indicates that, for the problems considered here and others like them, a first-order perturbation expansion can be of limited utility.

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