Numerical Studies of Three-dimensional Stochastic Darcy’s Equation and Stochastic Advection-Diffusion-Dispersion Equation

Abstract

Solute transport in randomly heterogeneous porous media is commonly described by stochastic flow and advection-dispersion equations with a random hydraulic conductivity field. The statistical distribution of conductivity of engineered and naturally occurring porous material can vary, depending on its origin. We describe solutions of a three-dimensional stochastic advection-dispersion equation using a probabilistic collocation method (PCM) on sparse grids for several distributions of hydraulic conductivity. Three random distributions of log hydraulic conductivity are considered: uniform, Gaussian, and truncated Gaussian (beta). Log hydraulic conductivity is represented by a Karhunen-Loeve (K-L) decomposition as a second-order random process with an exponential covariance function. The convergence of PCM has been demonstrated. It appears that the accuracy in both the mean and the standard deviation of PCM solutions can be improved by using the Jacobi-chaos representing the truncated Gaussian distribution rather than the Hermite-chaos for the Gaussian distribution. The effect of type of distribution and parameters such as the variance and correlation length of log hydraulic conductivity and dispersion coefficient on leading moments of the advection velocity and solute concentration was investigated.