Constitutive equations for ionic transport in porous shales

[1] The constitutive coupled equations describing ionic transport in a porous shale are obtained at the scale of a representative elementary volume by volume averaging the local Nernst-Planck and Stokes equations. The final relationships check the Onsager reciprocity to the first order of perturbation of the state variables with respect to the thermostatic state. This state is characterized by a modified version of the Donnan equilibrium model, which accounts for the partition of the counterions between the Stern and diffuse Gouy-Chapman layers. After upscaling the local equations the material properties entering the macroscopic constitutive equations are explicitly related to the porosity of the shale, its cation exchange capacity, and some textural properties such as the electrical cementation exponent entering Archie's law. This new model is then applied to predict the salt filtering and electrodiffusion efficiencies of a shale layer.