Effective dispersion coefficients for the upscaling of pore-scale mixing and reaction

Abstract

We upscale reactive mixing using effective dispersion coefficients to capture the combined effect of pore-scale heterogeneity and molecular diffusion on the evolution of the mixing interface between two initially segregated dissolved species. Effective dispersion coefficients are defined in terms of the average spatial variance of the solute distribution evolving from a pointlike injection, that is, the transport Green function. We numerically investigate the temporal behavior of the longitudinal effective dispersion coefficients for two porous media of different pore-scale heterogeneity as measured by the statistics of the flow speed, and different Péclet numbers. We find that the effective dispersion coefficients evolve with time, or equivalently travel distance. As the solute samples the pore-scale flow heterogeneity due to advection and transverse diffusion, the effective dispersion coefficients evolve from the value of molecular diffusion to the corresponding hydrodynamic dispersion coefficients. Thus, at times smaller than the diffusion time over a characteristic pore length, the effective dispersion coefficients can be significantly smaller than the hydrodynamic dispersion coefficients. This difference can explain frequently observed mismatches between pore-scale reactive mixing data, and predictions using Darcy scale transport descriptions based on hydrodynamic dispersion coefficients that are constant in time. This suggests that the notion of incomplete mixing on the support scale can be quantified in terms of effective pore-scale dispersion coefficients. We use effective dispersion in order to approximate the transport Green function in terms of a Gaussian-shaped distribution that is characterized by the effective variance. This is approximation is termed dispersive lamella. Based on this representation, we study reactive mixing between two initially segregated solutes. The dispersive lamella approach accurately predicts the evolution of the product mass of an instantaneous bimolecular reaction obtained from direct numerical simulations. This demonstrates that effective dispersion is an accurate measure for width of the mixing interface between the two reacting species. These results shed some new light on pore-scale mixing, the notion of incomplete mixing, and its prediction and upscaling in terms of an effective mixing model.