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In part I (Lima et al., Transp Porous Media, 2009), a three-scale model governing the movement of an aqueous saline solution containing four monovalent species (Na+, H+, Cl−, OH−) in kaolinite clays was derived. Unlike purely macroscopic approaches, the novelty of the formulation relied on the double averaging of the nanoscopic electro- chemistry of particle/electrolyte solution interface ruled by the electrical double layer coupled with protonation/deprotonation reactions. The passage from the nano to the micro (pore)-scale gave rise to ion-sorbed concentrations and slip velocity at the solid/fluid interface which are coupled with the microscopic Stokes problem and Nernst–Planck equations governing the hydrodynamics and ion transport in the micropores. Application of a formal homogenization procedure led to macroscopic governing equations with effective electro-chemical parameters, such as retardation coefficients, electro-osmotic permeability, and electric conductivity. In this study, we reconstruct the constitutive laws of the macroscopic coefficients by solving the nano and microscopic closure problems. New generalized isotherms for Na+ and H+ − OH− sorption are build-up based on a perturbation approach and the limitations of classical Freundlich isotherm for modeling ion sorption at the solid/fluid interface are discussed. The macroscopic governing equations are discretized by the finite volume method and numerical simulations of a transient electroosmosis experiment for desalination of a clay sample by electrokinetics are presented.

A new three-scale model to describe the coupling between pH-dependent flows and transient ion transport including sorption phenomena in kaolinite clays is proposed. The kaolinite is characterized by three separate nano-micro and macroscopic length scales. The (micro)-scale consists of micro-pores saturated by an aqueous solution containing four monovalent ionic species (Na+, H+, Cl−, OH−) and charged solid particles surrounded by thin electrical double layers. The movement of the ions is governed by the Nernst-Planck equations and the influence of the double layers upon the flow is dictated by the Helmholtz–Smoluchowski slip boundary condition in the tangential velocity. In addition, sorption interface conditions for ion transport are postulated in the sense of Auriault and Lewandowska (Eur. J. Mech. A 15:681–704, 1996) to capture the immobilization of the ions in the electrical double layer and on particle surface due to protonation/deprotonation reactions. The intensity of sorption relative to diffusion effects is quantified by the Damkohler number, whose order of magnitude is estimated by invoking the nanoscopic modeling of the thin EDL based on Poisson–Boltzmann problem for the local electric potential coupled with a non-linear surface charge density with constitutive law dictated by the protonation/deprotonation reactions. The two-scale nano/micro model including sorption and slip boundary condition is homogenized to the core scale leading to a derivation of macroscopic governing equations.

Boiling in a porous layer heated from below at constant heat flux and cooled at constant temperature from the upper permeable surface leads below the critical flux of appearance of a vapor zone at the bottom in the approximation of dominated reservoirs to two solutions: a liquid-dominated one and a vapor-dominated one. The stability of these two situations is analyzed. The approach allows to understand the respective roles played by the gravitational instability, natural convection in the liquid upper layer, and movement of the interface between the two regions. Taking into account the confinement allows the comparison of these theoretical results with experimental results obtained in the laboratory.

A new three-scale model to describe the coupling between electro-chemistry and hydrodynamics in non-swelling kaolinite clays in steady-state conditions is proposed. The medium is characterized by three separate nano-micro and macroscopic length scales. At the pore (micro)-scale the portrait of the clay consists of micro-pores saturated by an aqueous solution containing four monovalent ions (Na+, H+, Cl−, OH−) and charged solid particles surrounded by thin electrical double layers. The movement of the ions is governed by the Nernst–Planck equations and the influence of the double layers upon the hydrodynamics is modeled by a slip boundary condition in the tangential velocity governed by the Stokes problem. To capture the correct form of the interface condition we invoke the nanoscopic modeling of the thin electrical double layer based on Poisson–Boltzmann problem with varying surface charge density ruled by the protonation/deprotonation reactions which occur at the surface of the particles. The two-scale nano/micro model is homogenized to the macroscale leading to a precise derivation of effective governing equations. The macroscopic model is discretized by the finite volume method and applied to numerically simulate desalination of a clay sample induced by an external electrical field generated by the placement of electrodes. Numerical results indicate strong pH-dependence of the electrokinetics.