Macroscopic models for filtration and heterogeneous reactions in porous media

TL;DR

This paper develops a homogenisation-based method to derive macroscopic models for advection-diffusion with heterogeneous reactions in porous media, validated through numerical simulations of complex structures.

Contribution

It introduces an extended upscaling approach that handles strong coupling of scales and non-solenoidal velocities, applicable to complex porous structures.

Findings

Accurate macroscopic coefficients for various Péclet and Damköhler numbers.

Validation against detailed numerical simulations confirms method's effectiveness.

Applicable to complex 2D and 3D periodic structures.

Abstract

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 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 literature. 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. 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 generality of the upscaling approach. The effective velocity, dispersion and reaction coefficients are obtained for a wide range of P\'eclet and surface Damk{\"o}hler numbers, and for Coulomb-like forces to the grains.