Theory of Diiusions Applied to Stochastic Ow in Porous Media

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 diierential equation is studied using results from the theory of diiusions 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 microdiiusion. This distribution is shown to also solve a deterministic transport equation containing an eeective diiusion constant, analogous to the hydrodynamic dispersion equation. This relates the stochastic and de-terministic 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.