Probability Density Functions for Solute Transport in Random Field

Abstract

Effective description of flow and transport in irregular porous media, adequate understanding and reliable estimation of the uncertainty, all require stochastic approach. The primary problem is finding the relations between the non-random functionals of the unknown and the given random fields, that is moments, distribution functions, probability density distributions, etc. This paper considers the process of transport of non-reactive admixture in random media and attempts to develop a method for finding the probability density function of the concentration of the solute. We introduce the random functional p(x, t; c), the density distribution function (DDF) of the local random concentration c(x, t) in the one-dimensional phase space of its possible values c, where parameter x is multi-dimensional vector, and parameter t is time. By using the stochastic transport equation in the (x, t) space, one can write the so-called stochastic Liouville equation for the functional p(x, t; c) and this equation bears the form of the transport equation in the (x, t; c) space. The averaging of the new transport equation in the (x, t; c) space leads to equations for P(x, t; c) = 〈 p(x, t; c) 〉 – the probability density function (PDF) for c(x, t) and the corresponding power moments. We present the analysis examples of PDFs for the concentration c(x, t) in several different cases of flow velocity field and initial concentration distribution.