Stochastic Fluid Travel Times in Heterogeneous Porous Media

Abstract

An analytic approach is developed for quantifying the distribution of fluid passage times in a heterogeneous porous medium. The basic methodology employed utilizes a diffusion theory description for the displacement of a purely advected fluid subject to a random field of hydraulic conductivity. Within this framework, a governing model is formed using the backward form of the statistical Kolmogorov equation, which yields the inverse Gaussian distribution as a solution to the fluid passage time problem. In proposing the methodology, a rationale is presented for quantifying the associated model parameters using a simple application of the mean and variance to Darcy's law, with subsequent comparisons being made to previous results obtained for perturbation solutions of the associated stochastic partial differential equations. In addition, the validity of the model is discussed within the bounds of a Markovian description for ground‐water flow under a continuum‐based modeling framework. Here, it is argued that acceptably accurate results may be achieved for a statistical Peclet number in excess of 70.