A spectral approach for homogenization of diffusion and heterogeneous reaction in porous media

Abstract

Macroscopic models for diffusion and heterogeneous reversible reaction of two species in porous media are developed by using coupled homogenization technique and spectral approach. Three representative cases related to the order of magnitude of the macroscopic Damköhler number DaL, namely predominating reaction, diffusion-reaction of the same order and dominating diffusion, are considered. The concentrations are developed as time series in an eigenfunctions basis associated with periodic spectral problems formulated in the unit-cell, thus forming a new microscopic problem to be homogenized. Such an approach represents a powerful tool to upscale diffusion-reaction microscopic problems, especially for high Damköhler number values where classical asymptotic development fails. It enables to capture the physics at very short times, when the characteristic time of reaction involved is much faster than the diffusion one. This work allows us to formulate the complex macroscopic laws describing the heterogeneous diffusion/reaction problem for two species in high Damköhler number regime.