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

Abstract

In this manuscript, we extend the stochastic analysis of transient two-phase flow (Chen et al., Water Resour Res 42:W03425, 2006) to three-phase flow, i.e., water, air, and NAPL. We use the van Genuchten model and the Parker and Lenhard three-phase model to describe the relationships between phase saturation, phase relative permeability, and capillary pressure. The log-transformations of intrinsic permeability Y(x) = ln k(x), soil pore size distribution parameter β ow (x) = ln α ow (x) between water and NAPL, and β ao (x) = ln α ao (x) between air and NAPL, and van Genuchten fitting parameter $$\bar{n}({\bf x}) = \ln {\left[{n{\left({\bf x} \right)} - 1} \right]},$$ are treated as stochastic variables that are normally distributed with a separable exponential covariance model. The Karhunen–Loeve expansion and perturbation method (KLME) is used to solve the resulting equations. We evaluate the stochastic model using two-dimensional examples of three-phase flow with NAPL leakage. We also conduct Monte Carlo (MC) simulations to verify the stochastic model. A comparison of results from MC and KLME indicates the validity of the proposed KLME application in three-phase flow. The computational efficiency of the KLME approach over MC methods is at least an order of magnitude for three-phase flow problems. This verified stochastic model is then used to investigate the sensitivity of fluid saturation variances to the input variances.