PNNL

Abstract

Perturbation-based moment equations for advection-dispersion equations with random advection have been shown to produce physically unrealistic multimodality. Despite similarities, macrodispersion theory applied to advection-dispersion equations produces moments that do not exhibit such multimodal behavior. This is because macrodispersion approximations, whether explicitly or implicitly, involve renormalized perturbations that remove secularity by including select higher-order terms. We consider basic differences between the two approaches using a low-order macrodispersion approximation to clarify why one produces physically meaningful behavior while the other does not. We demonstrate that using a conventional asymptotic expansion (in the order of velocity fluctuations) leads to equations that cannot produce physically meaningful (macro)dispersion, whether applied to moment equations or macrodispersion theory, proving that the resulting moment equations are in fact hyperbolic in one spatial dimension. We identify higher-order terms that must be added to the conventional expansion to recover second- and fourth-order macrodispersivity approximations. Finally, we propose a closed-form approximation to two-point covariance as a measure of uncertainty, in a manner consistent with the derivation of macrodispersivity. We demonstrate that this and all the macrodispersion-based approximations to moments are more accurate than the alternatives for an example of transport in stratified random media.