www.elsevier.com/locate/mcm Theory of Difiusions Applied to Stochastic Flow in Porous Media

Abstract|Contaminant transport by liquid °ow in a porous medium is modeled by the addition of a stochastic term to Darcy’s °ow equation. The resulting stochastic difierential equation is studied using results from the theory of difiusions as embodied in the Dynkin formula. The resulting integral equation for the probability distribution of °uid elements is solved for the case of a spatially homogeneous medium without microdifiusion. This distribution is shown to also solve a deterministic transport equation containing an efiective difiusion constant, analogous to the hydrodynamic dispersion equation. This relates the stochastic and deterministic approaches to the contaminant transport problem. The case of a nonhomogeneous medium is discussed, leading to a tentative conclusion that the stochastic description will not reduce to a dispersion equation in general