A Theory for Mixing and Reactions in Porous Media

In this work, we report on the upscaling of transport in a porous medium containing two reacting chemical species. As an initial exploration of the role of mixing in such systems, we examine the linear case where the two reaction rate expressions take the form r_A = -k_A c_A c_B and r_B = -k_B c_B, where c_A and c_B are the two chemical species concentrations, and k_A and k_B are the reaction rate coefficients. We find that the upscaled form of the balance equations for each species is independent of the amount of microscale mixing that is present. However, a covariance term arises in the results whose value does depend upon the amount of microscale mixing. Essentially, this covariance term modifies the reaction rate that would be provided by using the average concentration values in such a way that it accounts for the amount of microscale mixing present. When the system is completely segregated, the covariance term is equal to the rate that would be computed by assuming that the average concentration values, but opposite in sign. The net result is to predict that the reaction rate is zero under these circumstances. Analogously, when the system is under conditions of maximum mixedness, the covariance term is identically zero. The role of mixing in the reaction process is, then, automatically accounted for in the upscaling process. We explore this situation by examining purely diffusive systems under a variety of initial configuration conditions. Using numerical simulations, we are able to compute the effective rate of reaction directly, and compare this with the results that are predicted by theory (via the formulation and solution of an ancillary closure problem for the system). We will present our numerical results, and make some comments on how the notion of configurational entropy relates to the effective rate of reaction for such a system.