Non-local and localized analyses of non-reactive solute transport in bounded randomly heterogeneous porous media: Theoretical framework

Solute transport in randomly heterogeneous media is described by stochastic transport equations that are typically solved via Monte Carlo simulation. A promising alternative is to solve a corresponding system of statistical moment equations directly. We present exact (though not closed) implicit conditional first and second moment equations for advective–dispersive transport in finite domains. The velocity and concentration are generally non-stationary due to possible trends in heterogeneity, conditioning on data, temporal variations in velocity, fluid and/or solute sources, initial and boundary conditions. Our equations are integro-differential and include non-local parameters depending on more than one point in space-time. To allow solving these equations, we close them by perturbation and develop recursive moment equations in Laplace space for the special case of steady state flow, to second order in σY where σY2 is a measure of (natural) log hydraulic conductivity variance. We also propose a higher-order iterative closure. Our recursive equations and iterative closure are formally valid for mildly heterogeneous media, or well-conditioned strongly heterogeneous media in which the random component of heterogeneity is relatively small. The non-local moment equations suggest (and a companion paper [Morales Casique E, Neuman SP, Guadagnini A. Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: computational analysis. Adv Water Resour, submitted for publication] demonstrates numerically) that, in general, transport cannot be validly described by means of Fick’s law with a (constant or variable) macrodispersion coefficient. We show nevertheless 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 varies generally in space-time. In a companion paper [Morales Casique E, Neuman SP, Guadagnini A. Nonlocal and localized analyses of nonreactive solute transport in bounded randomly heterogeneous porous media: computational analysis. Adv Water Resour, submitted for publication] we present a high-accuracy computational algorithm for our iterative non-local and recursive localized moment equations, assessing their accuracy and computational efficiency in comparison to unconditional and conditional Monte Carlo simulations.