Volume averaging for determining the effective dispersion tensor: Closure using periodic unit cells and comparison with ensemble averaging

Abstract

In this work, we use the method of volume averaging to determine the effective dispersion tensor for a heterogeneous porous medium; closure for the averaged equation is obtained by solution of a concentration deviation equation over a periodic unit cell. Our purpose is to show how the method of volume averaging with closure can be rectified with the results obtained by other upscaling methods under particular conditions. Although this rectification is something that is generally believed to be true, there has been very little research that explores this issue explicitly. We show that under certain limiting (but mild) assumptions, the closure problem provides a Fourier series solution for the effective dispersion tensor. When second‐order spatial stationarity is imposed on the velocity field, the method yields a simple Fourier series that converges to an integral form in the limit as the period of the unit cell approaches infinity. This limiting result is identical to the quasi‐Fickian forms that have been developed previously via ensemble averaging by Deng et al. [1993] and recently by Fiori and Dagan [2000] except in the definition of the averaging operation. As a second objective we have conducted a numerical study to evaluate the influence of the size of the averaging volume on the effective dispersion tensor and its volume averaged statistics. This second objective is complimentary in many ways to recent research reported by Rubin et al. [1999] (via ensemble averaging) and by Wang and Kitanidis [1999] (via volume averaging) on the block‐averaged effective dispersion tensor. The variability of the effective dispersion tensor from realization to realization is assessed by computing the volume‐averaged effective dispersion tensor for an ensemble of finite fields with the same (ensemble) statistics. Ensembles were generated using three different sizes of unit cells. All three unit cell sizes yield similar results for the value of the mean effective dispersion tensor. However, the coefficient of variation depends strongly upon the size of the unit cell, and our results are consistent with those developed by Fiori [1998] from the ensemble averaging perspective. This implies that in applications the actual value of the effective dispersion tensor may be significantly different than expected on the basis of unconditioned hydraulic conductivity statistics, and this variation should be considered when applying macrodispersion to real‐world systems.