Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media

Abstract

Solute transport in randomly heterogeneous media is described by stochastic transport equations that are typically solved by Monte Carlo simulation. A promising alternative is to solve a corresponding system of statistical moment equations directly. The moment equations are generally integro-differential and include nonlocal parameters depending on more than one point in space-time [Neuman, 1993; Zhang and Neuman, 1996; Guadagnini and Neuman, 2001]. We present recursive approximations, and a numerical algorithm, that allow computing lead ensemble moments of non-reactive solute transport in bounded, randomly heterogeneous media. Our recursive equations are formally valid for mildly heterogeneous aquifers with σ²ᵧ < 1, where σ²ᵧ is a measure of log-hydraulic conductivity variance, or well-conditioned highly heterogeneous aquifers. Our algorithm utilizes a finite element Laplace transform method (FELT) valid for steady state velocity fields. We solved the recursive moment equations up to second order in σᵧ. We also present an iterative improvement of the recursive equations which allows reaching a solution of order higher than two in σᵧ but does not reach third order accuracy because we do not include third order moments in the computations. Computational results in two spatial dimensions conditioned on synthetic measurements of K , hydraulic conductivity, compare well with Monte Carlo results for σ²ᵧ and Peclet number (in terms of the integral scale of K) as high as 0.3 and 100 respectively for the iterative approach. As these parameters increase, the quality of our iterative moment solution deteriorates. Without conditioning the quality of the solution deteriorates more rapidly as dimensionless time increases. The recursive solution without iteration is much less accurate and deteriorates more rapidly as σ²ᵧ , Peclet number, and/or dimensionless time increase. We infer that this loss in accuracy is due to higher order moments which become important as σ²ᵧ , dimensionless time, and/or Pe increase. We also evaluate a space-localized moment equation and show that the quality of its solution is of inferior accuracy than the iterative solution. In terms of computational efficiency, the recursive and iterative methods require less CPU time than Monte Carlo transport simulations using the same numerical solution method (FELT) and without parallelization.