University of Colorado at Denver and Health Sciences Center A Stochastic Diferential Equation Approach For Modeling of NAPL Flow in Heterogeneous Porous Media

This paper presents a stochastic differential equation approach to multiphase flow, a typical example of which is flow in the unsaturated domain. Specifically, a two phase problem is studied which consists of a wetting phase and a nonwetting phase. The approach given results in a nonlinear stochastic differential equation describing the position of the non-wetting phase fluid particle. The nonlinearity arises because both the drift and diffusion coefficients depend on the volumetric fraction of the phase which in turn depends on the position of the fluid particle in the experimental domain. Central to the development described in this report is the concept of a fluid particle. Expressions for both saturation and volumetric fraction are developed using the fluid particle concept. Darcy’s law and the continuity equation are then used to derive a FokkerPlanck equation using these expressions. The Itô calculus is then applied to derive a stochastic differential equation for the non-wetting phase. This equation has both drift and diffusion terms which depend on the volumetric fraction of the non-wetting phase. Computational aspects of the approach are discussed in some detail, and a sample problem is worked.