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

Abstract

[1] 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 b(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 {x n }. The phase pressure, capillary pressure, and phase mobility are decomposed by polynomial expansions and the perturbation method. Combining these expansions of Y(x), b(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 firstand second-order approximations to Monte Carlo simulations. The significant computational efficiency of the KLME approach over Monte Carlo simulation is also illustrated.

Stochastic analysis of transient three-phase flow in heterogeneous porous media

“…Log intrinsic permeability, log pore size distributions, and log van Genuchten fitting parameter are treated as input stochastic properties, distributed normally with a separable exponential covariance function. The Karhunen-Loeve expansions of these stochastic input variables was presented in Chen et al (2005Chen et al ( , 2006 for two-phase flow, and the application to three-phase flow follows that expansion. Monte Carlo (MC) simulations are conducted to confirm the validity of the KLME analysis and its numerical implementation.…”

“…A novel stochastic approach based on the Karhunen-Loeve expansion and perturbation method (KLME) has been implemented in saturated flow and unsaturated one-phase flow by Zhang and Lu (2004) and Yang et al(2004). Chen et al (2005Chen et al ( , 2006 introduced KLME method to complex multiphase flow systems, and predicted mean and variance of fluid pressures, capillary pressure and saturations for the water-oil flow.…”

“…We use the same KLME approach presented in Chen et al (2005Chen et al ( , 2006 to approximate the solutions of these stochastic partial differential equations in terms of moments of pressure head. Based on Eqs.…”

Conditional simulations of water–oil flow in heterogeneous porous media

“…C a (x,y) is symmetrical and positive definite, whose eigenfunctions are mutually orthogonal and form a complete set spanning the function space to which a(x,h) belongs (Ghanem and Dham 1998). The mean-removed stochastic process a 0 (x,h) can be expanded as follows (Karhunen 1947;Loève 1948;Zhang and Lu 2004;Chen et al 2005):…”

“…Yang et al (2004) extended KLME to analysis of saturated-unsaturated flow described by Richard's equation. Chen et al (2005Chen et al ( , 2006 developed a stochastic multiphase flow model following the same approach. These studies demonstrated the accuracy and efficiency of KLME over traditional Monte Carlo simulation or other stochastic moment approaches.…”