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

In a companion paper we presented exact (though implicit and not closed) nonlocal conditional first and second moment equations for nonreactive advective–dispersive transport under both steady state and transient flow regimes in bounded, randomly heterogeneous porous domains. To allow solving our nonlocal equations we developed recursive moment equations in Laplace space for the special case of steady state flow to second order in the standard deviation of natural log hydraulic conductivity, Y, which is generally nonhomogeneous, and proposed a higher-order iterative closure scheme. We also showed that, under a limited set of conditions, the mean transport equation can be localized to yield a familiar-looking advection–dispersion equation with a conditional macrodispersion tensor that generally varies in space–time. The purpose of this paper is to explore the behavior and assess the accuracy and computational efficiency of our moment solutions in comparison to conditional and unconditional Monte Carlo simulations. To do so, we present a high-accuracy computational algorithm for our iterative nonlocal and recursive localized moment equations and corresponding computational results in two spatial dimensions conditional on measurements of Y. Our algorithm solves the moment equations by finite elements in Laplace-transformed space and inverts the solution numerically back into the time domain. Conditional results obtained with our iterative algorithm compare well with Monte Carlo simulations for log-conductivity variance of 0.3 and Peclet number Pe = 100 defined in terms of the integral scale of Y, and for Pe = 10 in the unconditional case. As log-conductivity variance, Pe and time increase the quality of our iterative moment solution deteriorates. We show that this is due to our disregarding velocity moments of order higher than two and propose that including such moments should render our iterative solution workable over a wider range of these parameters. Second-order recursive nonlocal and space-localized results are considerably less accurate than those obtained with our iterative nonlocal algorithm. Even though our moment solution does not require computing (space–time localized) macrodispersion coefficients, we nevertheless do so to examine the influence of boundaries and conditioning on their behavior. Our results support an earlier observation by the second author [Morales Casique E, Neuman SP, Guadagnini A. Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: theoretical framework. Adv Water Resour, in press., Neuman SP. Universal scaling of hydraulic conductivities and dispersivities in geologic media. Water Resour Res 1990;26(8):1749–58], based on world-wide tracer test results, that the rate at which apparent longitudinal dispersivity increases with scale diminishes with conditioning. In preliminary runs conducted on a relatively small grid without optimizing our algorithms and without parallelization, the moment solutions required considerably less computer time than did the Monte Carlo simulations.