Macroscopic models for filtration and heterogeneous reactions in porous media

Derivation of macroscopic models for advection-diffusion processes in the presence of dominant heterogeneous (e.g., surface) reactions using homogenisation theory or volume averaging is often deemed unfeasible (Valdes-Parada a al., 2011; Battiato and Tartakovsky, 2011a) due to the strong coupling between scales that characterise such systems. In this work, we show how the upscaling can be carried out by applying and extending the methods presented in Allaire and Raphael (2007), Mauri (1991). The approach relies on the decomposition of the microscale concentration into a reactive component, given by the eigenfunction of the advection-diffusion operator, the associated eigenvalue which represents the macroscopic effective reaction rate, and a non-reactive component. The latter can be then upscaled with a two-scale asymptotic expansion and the final macroscopic equation is obtained for the leading order. The same method can also be used to overcome another classical assumption, namely of non solenoidal velocity fields, such as the case of deposition of charged colloidal particles driven by electrostatic potential forces. The whole upscaling procedure, which consists in solving three cell problems, is implemented for arbitrarily complex two- and three-dimensional periodic structures using the open-source finite volume library OpenFOAM (R). We provide details on the implementation and test the methodology for two-dimensional periodic arrays of spheres, and we compare the results against fully resolved numerical simulations, demonstrating the accuracy and generally of the upscaling approach. The effective velocity, dispersion and reaction coefficients are obtained for a wide range of Peclet and surface Damkohler numbers, and for Coulomb-like forces to the grains. Noticeably, all the effective transport parameters are significantly different from the non-reactive (conserved scalar) case, as the heterogeneity introduced by the reaction strongly affects the micro-scale profiles.