A Two-Scale Computational Model of pH-Sensitive Expansive Porous Media (original) (raw)
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Computers & Mathematics with Applications
The paper deals with the homogenization of deformable porous media saturated by twocomponent electrolytes. The model relevant to the microscopic scale describes steady states of the medium while reflecting essential physical phenomena, namely electrochemical interactions in a dilute Newtonian solvent under assumptions of a small external electrostatic field and slow flow. The homogenization is applied to a linearized micromodel, whereby the thermodynamic equilibrium represents the reference state. Due to the dimensional analysis, scaling of the viscosity and electric permitivity is introduced, so that the limit model retains the characteristic length associated with the pore size and the electric double layer thickness. The homogenized model consists of two weakly coupled parts: the flow of the electrolyte can be solved in terms of a global pressure and streaming potentials of the two ions, independently of then the solid phase deformations which is computed afterwards for the fluid stress acting on pore walls. The two-scale model has been implemented in the Sfepy finite element software. The numerical results show dependence of the homogenized coefficients on the microstructure porosity. By virtue of the corrector result of the homogenization, microscopic responses in a local representative cell can be reconstructed from the macroscopic solutions.
Computational Geosciences, 2013
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 Lebedev Physical Institute RAS, Leninski ave., 53, 119991 Moscow, Russia (andrey@sci.lebedev.ru) 1 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).
Physica D: Nonlinear Phenomena, 2014
This paper is devoted to the homogenization (or upscaling) of a system of partial differential equations describing the non-ideal transport of a N-component electrolyte in a dilute Newtonian solvent through a rigid porous medium. Realistic non-ideal effects are taken into account by an approach based on the mean spherical approximation (MSA) model which takes into account * This research was partially supported by the project DYMHOM (De la dynamique moléculaire, via l'homogénéisation, aux modèles macroscopiques de poroélasticité etélectrocinétique) from the program NEEDS (Projet fédérateur Milieux Poreux MIPOR), GdR MOMAS and GdR PARIS. G. A. is a member of the DEFI project at INRIA Saclay Ile-de-France. The authors would like to thank O. Bernard, V. Marry, P. Turq and B. Rotenberg from the laboratory Physicochimie des Electrolytes, Colloides et Sciences Analytiques (PECSA), UMR CNRS 7195, Universit P. et M. Curie, for helpful discussions.
Coupled Processes in Charged Porous Media: From Theory to Applications
Transport in Porous Media, 2019
Charged porous media are pervasive, and modeling such systems is mathematically and computationally challenging due to the highly coupled hydrodynamic and electrochemical interactions caused by the presence of charged solid surfaces, ions in the fluid, and chemical reactions between the ions in the fluid and the solid surface. In addition to the microscopic physics, applied external potentials, such as hydrodynamic, electrical, and chemical potential gradients, control the macroscopic dynamics of the system. This paper aims to give fresh overview of modeling pore-scale and Darcy-scale coupled processes for different applications. At the microscale, fundamental microscopic concepts and corresponding mass and momentum balance equations for charged porous media are presented. Given the highly coupled nonlinear physiochemical processes in charged porous media as well as the huge discrepancy in length scales of these physiochemical phenomena versus the application, numerical simulation of these processes at the Darcy scale is even more challenging than the direct pore-scale simulation of multiphase flow in porous media. Thus, upscaling the microscopic processes up to the Darcy scale is essential and highly required for large-scale applications. Hence, we provide and discuss Darcy-scale porous medium theories obtained using the hybrid mixture theory and homogenization along with their corresponding assumptions. Then, application of these theoretical developments in clays, batteries, enhanced oil recovery, and biological systems is discussed.
2001
A systematic development of the macroscopic field equations (conservation of mass, linear and angular momentum, energy, and Maxwell's equations) for a multiphase, multicomponent medium is presented. It is assumed that speeds involved are much slower than the speed of light and that the magnitude of the electric field significantly dominates over the magnetic field so that the electroquasistatic form of Maxwell's equations applies. A mixture formulation is presented for each phase and then averaged to obtain the macroscopic formulation. Species electric fields are considered, however it is assumed that it is the total electric field which contributes to the electrically induced forces and energy. The relationships between species and bulk phase variables and the macroscopic and microscopic variables are given explicitly. The resulting field equations are of relevance to many practical applications including, but not limited to, swelling clays (smectites), biopolymers, biological membranes, pulsed electrophoresis, and chromatography.
Computers and Geotechnics, 2010
In order to describe diffusive transport of solutes through a porous material, estimation of effective diffusion coefficients is required. It has been shown theoretically that in the case of uncharged porous materials the effective diffusion coefficient of solutes is a function of the pore morphology of the material and can be described by the tortuosity (tensor) (Bear, 1988 [1]). Given detailed information of the pore geometry at the micro-scale the tortuosity of different materials can be accurately estimated using homogenization procedures. However, many engineering materials (e.g., clays and shales) are characterized by electrical surface charges on particles of the porous material which strongly affect the (diffusive) transport properties of ions. For these type of materials, estimation of effective diffusion coefficients have been mostly based on phenomenological equations with no link to underlying micro-scale properties of these charged materials although a few recent studies have used alternative methods to obtain the diffusion parameters Revil and Linde, 2006 [2-4]). In this paper we employ a recently proposed up-scaled Poisson-Nernst-Planck type of equation (PNP) and its micro-scale counterpart to estimate effective ion diffusion coefficients in thin charged membranes. We investigate a variety of different pore geometries together with different surface charges on particles. Here, we show that independent of the charges on particles, a (generalized) tortuosity factor can be identified as function of the pore morphology only using the new PNP model. On the other hand, all electro-static interactions of ions and charges on particles can consistently be captured by the ratio of average concentration to effective intrinsic concentration in the macroscopic PNP equations. Using this formulation allows to consistently take into account electrochemical interactions of ions and charges on particles and so excludes any ambiguity generally encountered in phenomenological equations.
Journal of Materials Science, 2017
Analytical solutions of exact lithium-ion (Li-ion) concentration distributions and diffusion-induced stresses (DISs) within elliptical and spherical particles are obtained. A two-dimensional scanning electron microscopy image-dependent model of porous electrodes that accounts for the diffusion, DIS, and the size and shape polydispersity of electrode particles is developed. The effects of the size and shape polydispersity on concentration profiles, DIS evolution, and mechanical failure mechanisms are numerically discussed. Simulations show that small particles experienced less DIS than larger particles, primarily because of their reduced strain mismatch. In elliptical electrode particles, simple cracks appear at the endpoints of the major axis, while more complicated and severe cracks appear at the endpoints of the minor axis. Small particles with a spherical geometry are most favorable for alleviating DIS. Thus, the microscopic state of charge (SOC) values and mass fractions of differently sized and shaped particles should be considered simultaneously when determining the optimal macroscopic SOC of a porous electrode.
Incorporating ionic size in the transport equations for charged nanopores
Microfluidics and Nanofluidics, 2010
Nanopores with fixed charges show ionic selectivity because of the high surface potential and the small pore radius. In this limit, the size of the ions could no longer be ignored because they occupy a significant fraction of the pore and, in addition, they would reach unrealistic concentrations at the surface if treated as point charges. However, most models of selectivity assume point ions and ignore this fact. Although this approach shows the essential qualitative trends of the problem, it is not strictly valid for high surface potentials and low nanopore radii, which is just the case where a high ionic selectivity should be expected. We consider the effect of ion size on the electrical double layer within a charged cylindrical nanopore using an extended Poisson-Boltzmann equation, paying special attention to (non-equilibrium) transport properties such as the streaming potential, the counter-ion transport number, and the electrical conductance. The first two quantities are related to the nanopore selectivity while the third one characterizes the conductive properties. We discuss the nanopore characteristics in terms of the ratio between the electrolyte and fixed charge concentrations and the ratio between the ionic and nanopore radii showing the experimental range where the point ion model can still be useful. Even for relatively small inorganic ions at intermediate concentrations, ion size effects could be significant for a quantitative estimation of the nanopore selectivity in the case of high surface charge densities.
Electroosmosis law via homogenization of electrolyte flow equations in porous media
Journal of Mathematical Analysis and Applications, 2008
By the homogenization approach we justify a two-scale model of ion transport in porous media for one-dimensional horizontal steady flows driven by a pressure gradient and an external horizontal electrical field. By up-scaling, the electroosmotic flow equations in horizontal nanoslits separated by thin solid layers are approximated by a homogenized system of macroscale equations in the form of the Poisson equation for induced vertical electrical field and Onsager's reciprocity relations between global fluxes (hydrodynamic and electric) and forces (horizontal pressure gradient and external electrical field). In addition, the two-scale approach provides macroscopic mobility coefficients in the Onsager relations.
Influence of pH and Compression on Electrohydrodynamic Effects in Nanoporous Packed Beds
Physical Separation in Science and Engineering, 2009
Fluid flow and charge transport in fine structures can be driven both by pressure gradients and by electric fields if electrochemical double layers are present on the surfaces. The interrelated electrohydrodynamic effects may be used to drive liquids without moving parts, for example, in dewatering or in electroosmotic chromatography, or to generate small electric currents. While the electrohydrodynamic transport is well understood for simple geometries, models for porous structures are complex. Furthermore, the interconnected porous structure of a packed bed itself strongly depends on the electrochemical double layers. In this study, the electrohydrodynamic transport in packed beds consisting of boehmite particles with an average diameter of 38 nm is investigated. We describe a new approach to the electrokinetic effects by treating the packed beds as theoretical sets of cylindrical capillaries. The charge transport and the electrically driven fluid flow predicted with this model agree well with experimental results. Furthermore, the hydraulic permeability was found to be a nonlinear function of the porosity, independent of whether the porosity change is caused by changing the compression or the electrochemical double layer.