Upscaling nonlinear adsorption in periodic porous media - Homogenization approach

TL;DR

This paper develops a homogenized model for reactive flows in porous media with nonlinear adsorption described by Langmuir isotherm, extending previous linear models to account for strong convection effects.

Contribution

It introduces a homogenization approach for a nonlinear adsorption model in porous media, using two-scale convergence with drift, and provides numerical validation.

Findings

Derivation of a nonlinear monotone diffusion equation as the homogenized model.

Extension of linear homogenization results to nonlinear Langmuir isotherm.

Numerical tests confirming theoretical predictions.

Abstract

We consider the homogenization of a model of reactive flows through periodic porous media involving a single solute which can be absorbed and desorbed on the pore boundaries. This is a system of two convection-diffusion equations, one in the bulk and one on the pore boundaries, coupled by an exchange reaction term. The novelty of our work is to consider a nonlinear reaction term, a so-called Langmuir isotherm, in an asymptotic regime of strong convection. We therefore generalize previous works on a similar linear model [G. Allaire et al, Chemical Engineering Science, 65 (2010), pp.2292-2300], [G. Allaire et al, SIAM J. Math. Anal., 42 (2010), pp.125-144], [Allaire et al, IMA J Appl Math., 77 (2012), pp.788-815]. Under a technical assumption of equal drift velocities in the bulk and on the pore boundaries, we obtain a nonlinear monotone diffusion equation as the homogenized model. Our main technical tool is the method of two-scale convergence with drift [Marusic-Paloka et al, Journal of London Math. Soc., 72 (2005), pp.391-409]. We provide some numerical test cases in two space dimensions to support our theoretical analysis.