Homogenization of the Ionic Transport Equations in Periodic Porous Media

Abstract

The transport of an N-component electrolyte in a dilute Newtonian solvent through a rigid, porous body subjected to a static (d.c.) electric field is studied. The microscopic description is given by the linearized ionic transport (electrokinetic) equations, including the effects of ion diffusion, electromigration and convection. Periodic homogenization is used to derive effective governing equations for the fluid velocity, ionic flux and current density that capture the macroscopic behavior. Explicit expressions for the transport coefficient tensors are given in terms of solutions to cell problems. Without any additional assumptions, we rigorously prove that these transport coefficient tensors obey certain fundamental thermodynamic requirements, namely, Onsager's reciprocal relations and the positive definiteness of the diagonal coefficient tensors.

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

“…Many of these works rely on a preliminary linearization of the problem, introduced by O'Brien et al [64]. Let us mention in particular the work of Looker and Carnie in [48], where the formal two-scale expansions were undertaken and the resulting Onsager relations written explicitly. We will present the rigorous justification of the homogenization result, following article [10].…”

“…The formal two-scale asymptotic expansion method from sections 1 and 2 can be applied to system (145)-(147), (149)-(151), as in [48] and [10]. Introducing the fast variable y = x/ε, it assumes that the solution of (145)- (147), (149)- (151) is given by…”

“…In the way analogous to the calculations in section 2, we identify the various powers of ε and obtain a cascade of equations from which we retain only the leading ones that constitute the two-scale homogenized problem. For details we refer to [48]. We will present a rigorous passing to the limit using the two-scale convergence from section 3.…”

“…Obviously, it requires to separate the fast and slow scale. It was undertaken by Looker and Carnie in [48] and further improved in [10] and [11]. The main idea is to recognize in the two-scale homogenized problem (159)-(165) that there are two different macroscopic fluxes, namely (∇ x p 0 (x) + f * (x)) and {∇ x Φ 0 j (x) + E * (x)} 1≤ j≤N .…”

Electrokinetic coupling in unsaturated porous media

“…The matrix L is symmetrical ( L = L T ), a property known as Onsager's reciprocity [19]. If the material is anisotropic, the tortuosity of the pore space and the permeability are second-order tensors with the same eigenvectors and each component of L is therefore a second-order tensor.…”