STOCHASTIC FINITE-ELEMENT ANALYSIS FOR MULTIPHASE FLOW IN HETEROGENEOUS POROUS-MEDIA

This study is concerned with developing a two-dimensional multiphase m odel that simulates the movement of NAPL in heterogeneous aquifers. He terogeneity is dealt with in a probabilistic sense by modeling the int rinsic permeability of the porous medium as a stochastic process. The deterministic finite element method is used to spatially discretize th e multiphase flow equations. The intrinsic permeability is represented in the model via its Karhunen-Loeve expansion. This is a computationa lly expedient representation of stochastic processes by means of a dis crete set of random variables. Further the nodal unknowns, water phase saturations and water phase pressures, are represented by their stoch astic spectral expansions. This representation involves an orthogonal basis in the space of random variables. The basis consists of orthogon al polynomial chaoses of consecutive orders. The relative permeabiliti es of water and oil phases, and the capillary pressure are expanded in the same manner, as well. For these variables, the set of determinist ic coefficients multiplying the basis in their expansions is evaluated based on constitutive relationships expressing the relative permeabil ities and the capillary pressure as functions of the water phase satur ations. The implementation of the various expansions into the multipha se flow equations results in the formulation of discretized stochastic differential equations that can be solved for the deterministic coeff icients appearing in the expansions representing the unknowns. This me thod 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 comprehe nsive framework for problems involving random media. This paper provid es the application of this formalism to the problem of two-phase how i n a random porous medium.