Mass transfer process in a two-region medium

Two-region mass transfer models have been widely considered for describing flow and transport in highly heterogeneous porous media. Previous studies have indicated that in two-region systems the effective mass transfer coefficient is not a constant but depends upon a time convolution of the concentration history. Although such a representation does capture the mass transfer dynamics correctly, it can be difficult to apply because the convolution operation couples the effective mass transfer rate to the average concentration. In this work, we present the upscaling of a two-region system complete with a method for predicting the time-varying effective mass transfer coefficient that arises. We consider in detail the particular case that occurs when the convective flux in the low conductivity region can be neglected and the inclusions are spherical. This puts the problem in a classical context, and we develop explicit closed-form expressions for the time-varying effective mass transfer coefficient within the context of volume averaging. Our goals in this work are to (1) determine the general (convolution) form of the macroscale two-equation model for transport in such a medium using volume averaging, (2) assess the influence of several simplifications to the convolution form of the two-equation model, and (3) compare this result with existing classical results for the simplified geometry of spherical inclusions. A comparison of the results predicted by theory and experimental data from two macroscale, two-region systems is made. Additional discussion about how the method can be extended to other geometries and to cases where the convective flux is not negligible is also presented.