A stochastic model for solute transport in porous media: mathematical basis and computational solution

Abstract

In this paper, we develop a computational model of solute dispersion in saturated porous media by considering fluid velocity as a fundamental stochastic quantity. The velocity can be considered as having a random part correlated in space and δ-correlated in time superimposed on the Darcian velocity. The spatial correlation depends on the geometric and other properties of porous media. The stochastic partial differential equation that describes the mass conservation of a solute in an infinitesimal volume is derived by assuming the variables are irregular, continuous random variables which require higher order terms in the Taylor series to model the spatial variation. This partial differential equation can be written in the form of a stochastic differential equation with a drift term and a diffusion term. The diffusion term can be expressed in terms of a Hibert space valued Wiener process which is a function of the spatial correlation of the random part of velocity. This spatial correlation is modelled in terms of a covariance function with an exponential kernel having a fixed correlation length, and Karhunnen-Loeve expansion based on the orthonormal basis functions for the exponential kernel is used in the solution. A numerical scheme was developed to solve the model based on the definition of Ito integral.