Constitutive equations for coupled flows in clay materials

We first upscale the local transport (Stokes and Nernst-Planck) equations to the scale of a single capillary saturated by a binary 1:1 electrolyte. These equations are then upscaled to the scale of a network of tortuous capillaries embedded in a homogeneous and continuous mineral matrix, including the influence of the distribution of pore sizes but excluding the effect of connectivity between the pores. One of the features of our theory is to account for transport along the mineral surface in the so-called Stern layer because of recent evidence that this mechanism is effective in describing frequency-dependent electrical conductivity. Real clay materials are, however, not described by a set of capillaries, so we have to modify the model to include the effect of transversal dispersivity, for example. We found no evidence for transport in the Stern layer because of the discontinuity of the solid phase at the scale of a representative elementary volume in clay materials. The effect of the diffuse layer is accounted for through the use of a Donnan equilibrium approach to determine the effective concentrations of the ions in the pore space, which are different from the ionic concentrations of an ionic reservoir in local equilibrium with the porous material. We found that the diffuse layer controls various transport properties, including, for example, the DC electrical conductivity, the osmotic efficiency coefficient, the streaming potential coupling coefficient, and the macroscopic Hittorf numbers. Comparison to a large data set of experimental data, mainly on clay materials, confirms the validity of the derived relationships used to describe the material properties entering into the constitutive equations.