Impacts of local dispersion and first-order decay on solute transport in randomly heterogeneous porous media

Abstract

Stochastic subsurface transport theories either disregard local dispersion or take it to be constant. We offer an alternative Eulerian-Lagrangian formalism to account for both local dispersion and first-order mass removal (due to radioactive decay or biodegradation). It rests on a decomposition of the velocityv into a field-scale componentvω, which is defined on the scale of measurement supportω, and a zero mean sub-field-scale componentvs, which fluctuates randomly on scales smaller thanω. Without loss of generality, we work formally with unconditional statistics ofvs and conditional statistics ofvω. We then require that, within this (or other selected) working framework,vs andvω be mutually uncorrelated. This holds whenever the correlation scale ofvω is large in comparison to that ofvs. The formalism leads to an integro-differential equation for the conditional mean total concentration 〈c〉ω which includes two dispersion terms, one field-scale and one sub-field-scale. It also leads to explicit expressions for conditional second moments of concentration 〈c′c′〉ω. We solve the former, and evaluate the latter, for mildly fluctuatingvω by means of an analytical-numerical method developed earlier by Zhang and Neuman. We present results in two-dimensional flow fields of unconditional (prior) mean uniformvω. These show that the relative effect of local dispersion on first and second moments of concentration dies out locally as the corresponding dispersion tensor tends to zero. The effect also diminishes with time and source size. Our results thus do not support claims in the literature that local dispersion must always be accounted for, no matter how small it is. First-order decay reduces dispersion. This effect increases with time. However, these concentration moments 〈c〉ω and 〈c′c′〉ω of total concentrationc, which are associated with the scale belowω, cannot be used to estimate the field-scale concentrationcω directly. To do so, a spatial average over the field measurement scaleω is needed. Nevertheless, our numerical results show that differences between the ensemble moments ofcω and those ofc are negligible, especially for nonpoint sources, because the ensemble moments ofc are already smooth enough.