Two-level numerical and analytical homogenization of reactive transport in bi-disperse chemically heterogeneous porous media

Abstract:

Natural porous media are generally polidisperse and chemically heterogeneous, i.e. they exhibit a distribution of grain sizes and chemical reaction rates. The upscaling of systems exhibiting physical and chemical heterogeneity at the pore-scale poses challenges as classical upscaling approaches, e.g. homogenization, rely on idealized porous media with one single characteristic grain size: this treatment allows one to define a relatively small REV on which averaging is performed and a closure problem numerically solved. In presence of a distribution of grain size, one alternative is to increase the REV size and to perform a one-step upscaling. However, this comes to the detriment of numerical efficiency. To contain computational burden, we instead advocate a two-step sequential upscaling. In particular, we consider a porous medium where larger grains are embedded within a micro-porous matrix. By means of two-level homogenization, we determine the macroscale Advection-Reaction-Dispersion equation (ADRE) and identify the applicability regimes of the final upscaled model for reactive transport through  two-dimensional bi-disperse chemically heterogeneous porous media with reactive grains. The final applicability diagram depends on the P\'{e}clet number, two Damk\"{o}hler numbers and the ratio of spatial scales. Finally, we show the computational saving of the proposed approach relative to classical single-step homogenization and perform extensive numerical simulations to validate our theoretical results.