Stochastic Finite Element Analysis for Multiphase Flow in Heterogeneous Porous Media

This study is concerned with developing a two-dimensional multiphase model that simulates the movement of NAPL in heterogeneous aquifers. Heterogeneity is dealt with in a probabilistic sense by modeling the intrinsic permeability of the porous medium as a stochastic process. The deterministic finite element method is used to spatially discretize the multiphase flow equations. The intrinsic permeability is represented in the model via its Karhunen–Loeve expansion. This is a computationally expedient representation of stochastic processes by means of a discrete set of random variables. Further, the nodal unknowns, water phase saturations and water phase pressures, are represented by their stochastic spectral expansions. This representation involves an orthogonal basis in the space of random variables. The basis consists of orthogonal polynomial chaoses of consecutive orders. The relative permeabilities of water and oil phases, and the capillary pressure are expanded in the same manner, as well. For these variables, the set of deterministic coefficients multiplying the basis in their expansions is evaluated based on constitutive relationships expressing the relative permeabilities and the capillary pressure as functions of the water phase saturations. The implementation of the various expansions into the multiphase flow equations results in the formulation of discretized stochastic differential equations that can be solved for the deterministic coefficients appearing in the expansions representing the unknowns. This method allows the computation of the probability distribution functions of the unknowns for any point in the spatial domain of the problem at any instant in time. The spectral formulation of the stochastic finite element method used herein has received wide acceptance as a comprehensive framework for problems involving random media. This paper provides the application of this formalism to the problem of two-phase flow in a random porous medium.