A stochastic analysis of steady state two‐phase flow in heterogeneous media

We present a novel approach to modeling stochastic multiphase flow problems, for example, nonaqueous phase liquid flow, in a heterogeneous subsurface medium with random soil properties, in particular, with randomly heterogeneous intrinsic permeability and soil pore size distribution. A stochastic numerical model for steady state water‐oil flow in a random soil property field is developed using the Karhunen‐Loeve moment equation (KLME) approach and is numerically implemented. An exponential model is adopted to define the constitutive relationship between phase relative permeability and capillary pressure. The log‐transformed intrinsic permeability Y ( x ) and soil pore size distribution β( x ) are assumed to be Gaussian random functions with a separable exponential covariance function. The perturbation part of these two log‐transformed soil properties is then decomposed into an infinite series based on a set of orthogonal normal random variables {ξ n }. The phase pressure, capillary pressure, and phase mobility are decomposed by polynomial expansions and the perturbation method. Combining these expansions of Y ( x ), β( x ) and dependent pressures, the steady state water‐oil flow equations and corresponding boundary conditions are reformulated as a series of differential equations up to second order. These differential equations are solved numerically, and the solutions are directly used to construct moments of phase pressure and capillary pressure. We demonstrate the validity of the proposed KLME model by favorably comparing first‐ and second‐order approximations to Monte Carlo simulations. The significant computational efficiency of the KLME approach over Monte Carlo simulation is also illustrated.