Theory of diffusions applied to stochastic flow in porous media

Abstract

Contaminant transport by liquid flow in a porous medium is modelled by the addition of a stochastic term to Darcy's flow equation. The resulting stochastic differential equation is studied using results from the theory of diffusions as embodied in the Dynkin formula. The resulting integral equation for the probability distribution of fluid elements is solved for the case of a spatially homogeneous medium without microdiffusion. This distribution is shown to also solve a deterministic transport equation containing an effective diffusion constant, analogous to the hydrodynamic dispersion equation. This relates the stochastic and deterministic approaches to the contaminant transport problem. The case of a non-homogeneous medium is discussed, leading to a tentative conclusion that the stochastic description will not reduce to a dispersion equation in general.