Implications of Alternative Macroscopic Descriptions Illustrated by General Balance and Continuity Equations

Abstract

This article demonstrates the outstanding challenges of the application of well-established averaging procedures and rules, and shows that the results are not universal and alternative forms differ by inherent realization, difficulties, and conveniences for implementation in practical analyses of processes in porous media. Two alternative forms of the general macroscopic balance equation are derived by means of the representative elementary volume and mass-weighted-volume averaging rules and illustrated for the equation of continuity. Differences in the resulting mathematical expressions and their implications for practical problems are delineated. The macroscopic equation of continuity derived by combining the volume and mass-weighted-volume averaging rules may be convenient for conforming to the mathematical form of the microscopic equation of continuity. Using only the volume averaging rules yields an additional term compared to the conventional microscopic equation of continuity, which is referred to as transport by hydraulic dispersion due to convective mixing and spreading in porous media and contains a hydraulic dispersion parameter. An equation for empirical determination of the hydraulic dispersion parameter is also derived. Both averaging approaches are shown to be equally applicable when used with proper designations of various quantities as being the volume or mass-weighted-volume averaged. The macroscopic equation of continuity using only the volume-averaged quantities and conforming to the mathematical form of the microscopic equation of continuity is proven to involve an inherent error.