Clarifications about upscaling diffusion with heterogeneous reaction in porous media

Abstract

The upscaling process of coupled (single- and two-species) diffusion with heterogeneous chemical reaction in homogeneous porous media is revisited in this work with several important clarifications following the article from Bourbatache et al. (Acta Mech 234: 2293-2314, 2023. https://doi.org/10.1007/s00707-023-03501-w). It is shown that the upscaled model obtained from the volume averaging method (VAM) or, equivalently, following an adjoint and Green’s formulation technique provides a closed model without any a priori assumption on the form of the solution for the pore-scale concentration involved in the spectral approach used in the periodic homogenization method (PHM) reported in the above reference. Through comparison with direct pore-scale simulations, the VAM model is shown to outperform the predictions of the average concentration and average flux profiles for the simple two-dimensional configuration considered in Bourbatache et al. (Acta Mech 234: 2293-2314, 2023. https://doi.org/10.1007/s00707-023-03501-w) in comparison with the model obtained from PHM in this reference. Finally, identification of the apparent effective diffusion coefficient from these pore-scale simulations, which serve as in silico experiments, proves that the correct dependence upon the Damkhöler number is the one predicted by the model obtained with VAM, in contradiction with the conclusion put forth in Bourbatache et al. (Acta Mech 234: 2293-2314, 2023. https://doi.org/10.1007/s00707-023-03501-w). The physical explanation lies in the corrective contribution of the reactive part to the apparent effective diffusion coefficient, which is positive and adds up to the pure intrinsic diffusive part. The discrepancy between PHM and VAM approaches is proved to originate from the choice of changes of variables in the pore-scale concentration used in the spectral approach while employing PHM.