Homogenization Approach to the Dispersion Theory for Reactive Transport through Porous Media

Abstract

We study the homogenization problem for a convection-diffusion equation in a periodic porous medium in the presence of chemical reaction on the pores surface. Mathematically this model is described in terms of a solution to a system of convection-diffusion equation in the medium and ordinary differential equation defined on the pores surface. These equations are coupled through the boundary condition for the convection-diffusion problem.

An Introduction to the Homogenization Modeling of Non-Newtonian and Electrokinetic Flows in Porous Media

“…The equivalence between the norms ∥D(w)∥ L q (Ω ε ) 9 and ∥∇w∥ L q (Ω ε ) 9 is due to Korn's inequality in porous media 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 00 11 11 00 [65] and references therein) For each w ∈ W 1,q 0 (Ω ε ) 3 , 1 < q < +∞, we have the inequality…”

“…v ε satisfies a priori estimate (9). Obtaining the a priori estimate for the pressure field p ε is more involved and we address the question in the next section.…”

“…Another example is the two-scale convergence with drift, designed to handle homogenization of reaction-diffusion equations with large Péclet and Damkohler's numbers. For details we refer to [54], [8] and [9].…”

Analysis and Upscaling of a Reactive Transport Model in Fractured Porous Media with Nonlinear Transmission Condition

“…Our approach is inspired from similar exercises carried out for precipitation-dissolution type models in [19,25], but uses the drift model discussed e.g. in [1,2,3], and assumes that the velocity is now very high, i.e. of order ε −1 .…”

Design of nanoporous materials with optimal sorption capacity

“…14 This can be accomplished with a variety of upscaling techniques, including volume averaging 15 and homogenization via multiple-scale expansions. [16][17][18] We use the latter approach to predict anisotropic diffusion coefficients and effective adsorption rates for a class of nanoporous materials, 19 whose synthesis allows one to control their pore structure. It is worthwhile emphasizing that the use of various homogenization techniques to derive effective (continuum-or Darcy-scale) models of reactive transport in porous media (primarily subsurface environments) is not new.…”