Coupled Processes in Charged Porous Media: From Theory to Applications (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.
Modelling of the transport in electrically charged porous media including ionic exchanges
Mechanics Research Communications, 2010
In this study, a multiscale model of interstitial fluid transport within bone tissues is proposed. Based on an asymptotic homogenization procedure, it takes into account the possible ionic exchanges at the pore level. Since these chemical reactions directly change the physico-chemical properties of the tissues, the macroscopically observed flow is also modified as shown by the calculated effective diffusion coefficients. Such an approach is interesting to study the bone remodelling signals expression. Due to the presence of charged surfaces in the bone porous matrix, the electrochemical phenomena occurring in the vicinity of mechano-sensitive bone cells, the osteocytes, are key elements in the cellular expression.
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.
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).
A Two-Scale Computational Model of pH-Sensitive Expansive Porous Media
Journal of Applied Mechanics, 2013
We propose a new two-scale model to compute the swelling pressure in colloidal systems with microstructure sensitive to pH changes from an outer bulk fluid in thermodynamic equilibrium with the electrolyte solution in the nanopores. The model is based on establishing the microscopic pore scale governing equations for a biphasic porous medium composed of surface charged macromolecules saturated by the aqueous electrolyte solution containing four monovalent ions (Na+,Cl-,H+,OH-). Ion exchange reactions occur at the surface of the particles leading to a pH-dependent surface charge density, giving rise to a nonlinear Neumann condition for the Poisson–Boltzmann problem for the electric double layer potential. The homogenization procedure, based on formal matched asymptotic expansions, is applied to up-scale the pore-scale model to the macroscale. Modified forms of Terzaghi's effective stress principle and mass balance of the solid phase, including a disjoining stress tensor and elect...
Mechano-chemical effects in weakly charged porous media
The paper is concerned with mechano-chemical effects, namely, osmosis and pressure‐driven separation of ions that can be observed when a charged porous medium is placed between two electrolyte solutions. The study is focused on porous systems with low equilibrium interfacial potentials (about 30 mV or lower). At such low po- tentials, osmosis and pressure‐driven separation of ions noticeably manifest themselves provided that the ions in the electrolyte solutions have different diffusion coefficients. The analysis is conducted by combining the irreversible thermodynamic approach and the linearized (in terms of the normalized equilibrium interfacial potential) version of the Standard Electrokinetic Model. Osmosis and the pressure‐driven separation of ions are considered for an arbitrary mixed electrolyte solution and various porous space geometries. It is shown that the effects under consideration are proportional to a geometrical factor which, for all the considered geometries of porous space, can be expressed as a function of porosity and the Λ- parameter of porous medium normalized by the Debye length. For all the studied geometries, this function turns out to be weakly dependent on both the porosity and the geometry type. The latter allows for a rough evaluation of the geometrical factor from experimental data on electric conductivity and hydraulic permeability without previous knowledge of the porous space geometry. The obtained results are used to illustrate how the composition of electrolyte solution affects the mechano- chemical effects. For various examples of electrolyte solution compositions, the obtained results are capable of describing positive, negative and anomalous osmosis, positive and negative rejection of binary electrolytes, and pressure‐driven separation of binary electrolyte mixtures.
Electrokinetic coupling in unsaturated porous media
Journal of Colloid and Interface Science, 2007
We consider a charged porous material that is saturated by two fluid phases that are immiscible and continuous on the scale of a representative elementary volume. The wetting phase for the grains is water and the nonwetting phase is assumed to be an electrically insulating viscous fluid. We use a volume-averaging approach to derive the linear constitutive equations for the electrical current density as well as the seepage velocities of the wetting and nonwetting phases on the scale of a representative elementary volume. These macroscopic constitutive equations are obtained by volume-averaging Ampere's law together with the Nernst Planck equation and the Stokes equations. The material properties entering the macroscopic constitutive equations are explicitly described as functions of the saturation of the water phase, the electrical formation factor, and parameters that describe the capillary pressure function, the relative permeability function, and the variation of electrical conductivity with saturation. New equations are derived for the streaming potential and electro-osmosis coupling coefficients. A primary drainage and imbibition experiment is simulated numerically to demonstrate that the relative streaming potential coupling coefficient depends not only on the water saturation, but also on the material properties of the sample, as well as the saturation history. We also compare the predicted streaming potential coupling coefficients with experimental data from four dolomite core samples. Measurements on these samples include electrical conductivity, capillary pressure, the streaming potential coupling coefficient at various level of saturation, and the permeability at saturation of the rock samples. We found very good agreement between these experimental data and the model predictions.
One- and Two-Equation Models to Simulate Ion Transport in Charged Porous Electrodes
Colloids and Interfaces
Energy storage in porous capacitor materials, capacitive deionization (CDI) for water desalination, capacitive energy generation, geophysical applications, and removal of heavy ions from wastewater streams are some examples of processes where understanding of ionic transport processes in charged porous media is very important. In this work, one-and two-equation models are derived to simulate ionic transport processes in heterogeneous porous media comprising two different pore sizes. It is based on a theory for capacitive charging by ideally polarizable porous electrodes without Faradaic reactions or specific adsorption of ions. A two-step volume averaging technique is used to derive the averaged transport equations for multi-ionic systems without any further assumptions, such as thin electrical double layers or Donnan equilibrium. A comparison between both models is presented. The effective transport parameters for isotropic porous media are calculated by solving the corresponding closure problems. An approximate analytical procedure is proposed to solve the closure problems. Numerical and theoretical calculations show that the approximate analytical procedure yields adequate solutions. A theoretical analysis shows that the value of interphase pseudo-transport coefficients determines which model to use.
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.
Preface on Physicochemical and Electromechanical Interactions in Porous Media
Transport in Porous Media, 2003
The focus of science and engineering shifts towards smaller length scales. Porous media mechanics has a vital role to play in the translation of microstructural data into macroscopic models of multicomponent systems. As the length scales shrink, more fundamental levels of understanding of natural laws, cause the boundaries between disciplines to blur. In particular, geosciences, polymer sciences and biosciences find a common ground of interest in high specific surface mixtures.