Macroscopic models for filtration and heterogeneous reactions in porous media

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 [1,2] 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 [3], Mauri [4]. 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 generality of the upscaling approach. The effective velocity, dispersion and reaction coefficients are obtained for a wide range of Péclet and surface Damköhler 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.

Impact of water vapor diffusion and latent heat on the effective thermal conductivity of snow

“…At the same time, this model could be formulated to explicitly take into account macroscopic convection as this phenomenon has been observed in sub-arctic shallow snowpacks (Trabant and Benson, 1972;Sturm and Johnson, 1991). Its derivation could be achieved using standard homogenization methods, such as the two-scale asymptotic expansion (e.g., Municchi and Icardi, 2020) or volume averaging methods (e.g., Whitaker, 1977).…”

Heterogeneous Multi-Rate mass transfer models in OpenFOAM®

“…However, the MRMT model does not apply to the mobile region, where the fluid is flowing. Therefore, macroscopic quantities like the effective fluxes must be obtained using other methods (for example, classical volume averaging [67] or two-scale asymptotics [45]). But this does not play a role in the numerical implementation of the MRMT model, which governs the inter-region transfer process.…”

Upscaling of Solute Plumes in Periodic Porous Media Through a Trajectory‐Based Spatial Markov Model

“…This latter point is particularly challenging because the simulation of steady-state transport with Lagrangian methodologies typically entails very large computational costs, as well as the need to implement specific methods to impose boundary conditions that are consistent with those considered in classical Eulerian models based on partial differential equations (Koch & Nowak, 2014 To accomplish these goals, we consider a periodic medium showing a disordered periodic geometry. Periodicity is imposed in agreement with numerous theoretical upscaling approaches applied in porous media and are relevant to engineering systems across a wide range of applications (Gebäck & Heintz, 2019;Kim et al, 2013;Municchi & Icardi, 2020;Schmuck & Berg, 2014).…”

“…To accomplish these goals, we consider a periodic medium showing a disordered periodic geometry. Periodicity is imposed in agreement with numerous theoretical upscaling approaches applied in porous media and are relevant to engineering systems across a wide range of applications (Gebäck & Heintz, 2019; Kim et al, 2013; Municchi & Icardi, 2020; Schmuck & Berg, 2014).…”