HR: 0830h
AN: H21E-0898    [PDF]
TI: A Stochastic Differential Equation Approach To Multiphase Flow In Porous Media
AU: * Dean, D
EM: ddean@carbon.cudenver.edu
AU: Russell, T
EM: trussell@carbon.cudenver.edu
AB: The motivation for using stochastic differential equations in multiphase flow systems stems from our work in developing an upscaling methodology for single phase flow. The long term goals of this project include: I. Extending this work to a nonlinear upscaling methodology II. Developing a macro-scale stochastic theory of multiphase flow and transport that accounts for micro-scale heterogeneities and interfaces. In this talk, we present 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 non-wetting phase. The approach given results in a nonlinear stochastic differential equation describing the position of the non-wetting phase fluid particle. Our fundamental assumption is that the flow of fluid particles is described by a stochastic process and that the positions of the fluid particles over time are governed by the law of the process. It is this law which we seek to determine. The nonlinearity in the stochastic differential equation 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 particles in the experimental domain. The concept of a fluid particle is central to the development of the model described in this talk. 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 Fokker-Planck equation using these expressions. The Ito 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. Standard stochastic theories based on the Ito calculus and the Wiener process and the equivalent Fokker-Planck PDE's are typically used to model dispersion processes. However, these models, in their usual form, cannot represent multiphase barrier effects. These effects occur at the interface between adjacent materials with different permeabilities. For example, in tracking a DNAPL plume, the behavior of the plume at an interface depends on the pressure-saturation curves involved in forming the interface. In the model, the control of the flow of DNAPL particles across an interface is accomplished using a jump term which derives from the Ito formula. This jump term is based on capillary diffusivity and the pressure-saturation curves of the sands forming the interface. Computational aspects of the approach are discussed in some detail along with results of applying the model.
DE: 1829 Groundwater hydrology
DE: 1832 Groundwater transport
SC: Hydrology [H]
MN: 2003 Fall Meeting