Ion transport in porous media: Derivation of the macroscopic equations using upscaling and properties of the effective coefficients

In this work, we undertake a numerical study of the effective coefficients arising in the upscaling of a system of partial differential equations describing transport of a dilute N-component electrolyte in a Newtonian solvent through a rigid porous medium. The motion is governed by a small static electric field and a small hydrodynamic force, around a nonlinear Poisson-Boltzmann equilibrium with given surface charges of arbitrary size. This approach allows us to calculate the linear response regime in a way initially proposed by O'Brien. The O'Brien linearization requires a fast and accurate solution of the underlying Poisson-Boltzmann equation. We present an analysis of it, with the discussion of the boundary layer appearing as the Debye-Hückel parameter becomes large. Next, we briefly discuss the corresponding two-scale asymptotic expansion and reduce the obtained two-scale equations to a coarse scale model. Our previous rigorous study proves that the homogenized coefficients satisfy Onsager properties, namely they are symmetric positive definite tensors. We illustrate with numerical simulations several characteristic situations and discuss the behavior of the effective coefficients when the Debye-Hückel parameter is large. Simulated qualitative behavior differs significantly from the situation when the surface potential is given (instead of the surface charges). In particular, we observe the Donnan effect (exclusion of co-ions for small pores).