Path Integral Renormalization of Flow through Random Porous Media

Abstract

The path integral for Darcy's law with a stochastic conductivity, which characterizes flow through random porous media, is used as a basis for Wilson renormalization-group (RG) calculations in momentum space. A coarse graining procedure is implemented by integrating over infinitesimal shells of large momenta corresponding to the elimination of the small scale modes of the theory. The resulting one-loop β-functions are solved exactly to obtain an effective conductivity in a coarse grained theory over successively larger length scales. We first carry out a calculation with uncorrelated Gaussian conductivity fluctuations to illustrate the RG procedure before considering the effect of a finite correlation length of conductivity fluctuations. We conclude by discussing applications and extensions of our calculations, including comparisons with the numerical evaluation of path integrals, non-Gaussian fluctuations, and multiphase flow, for which the path integral formulation should prove particularly useful.

Pressure and flow statistics of Darcy flow from simulated annealing

“…The usual procedure [46][47][48][49] is to use a functional Fourier transform of the δ function, which yields a complex action. This is useful for formal studies, such as various types of perturbation expansion [50,51], where the complex action yields real results, despite the complex nature of intermediate calculations. But the Markov chain Monte Carlo method relies on real variables from the outset, so we must choose a representation of the δ function in terms of a real exponential.…”