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The path integral for Darcy's law with a stochastic conductivity, whichcharacterizes flow through random porous media, is used as a basis for Wilsonrenormalization-group (RG) calculations in momentum space. A coarse grainingprocedure is implemented by integrating over infinitesimal shells of largemomenta corresponding to the elimination of the small scale modes of thetheory. The resulting one-loop $\beta$-functions are solved exactly to obtainan effective conductivity in a coarse grained theory over successively largerlength scales. We first carry out a calculation with uncorrelated Gaussianconductivity fluctuations to illustrate the RG procedure before considering theeffect of a finite correlation length of conductivity fluctuations. We concludeby discussing applications and extensions of our calculations, includingcomparisons with the numerical evaluation of path integrals, non-Gaussianfluctuations, and multiphase flow, for which the path integral formulationshould prove particularly useful.

Path Integral Renormalization of Flow through Random Porous Media