Transport of water and ions in partially water-saturated porous media. Part 2. Filtration effects
Introduction
The existence of cross-coupling effects in the transport equations of porous media has been observed in the 19th century (e.g., Reuss, 1909; Quincke, 1959). Since these first observations, a rich literature has emerged regarding the modeling of the cross-coupling effects in charged porous media. The fundamental aspects of some of the cross-coupling terms such as electro-osmosis was for instance developed by Gray and Mitchell (1967) driven by some applications in soil sciences and geo-mechanics and later on by microfluidics (e.g., Ghosal, 2010). Hanshaw and Coplen (1973) developed a theory of reverse osmosis based on the concept of excess of charge in the pore space of clayey materials. Revil and Leroy, 2004, Revil et al., 2005, and Leroy et al. (2006) developed a consistent theory for transport mechanism in saturated conditions in clayey materials accounting for the partition between the Stern and diffuse layer coating the surface of the clay minerals. This type of approach is complementary to molecular modeling that can be used to compute ionic transport in the interlayer of smectite (Birgersson and Karnland, 2009). Recently, Dominijanni and Manassero (2012) and Dominijanni et al. (2013) developed a complete theory in which the transport equations with coupling terms were developed in a hydromechanical framework including elastic and plastic deformations. The list of works on this subject is far from being exhaustive and the interested readers could also consult scientific papers dealing with osmotic effects in cells, bones, and the tendons of living beings (e.g., Masic et al., 2016).
In the previous paper of this series (Revil, 2016), I have developed a model describing the generalized constitutive equations of transport of water and ions in the case of a partially saturated charged porous medium and in a rigid skeleton. In this model, the water phase is considered to be the wetting phase (index w) for the solid grains and the non-wetting phase (air) is considered to be very compressible and electrically insulating. By charged porous materials, I mean that the interface between the solid phase and the pore water phase contains an electrical double layer coating the surface of the grains as for instance in soils (Sposito, 1991, Sposito et al., 1999, Sposito, 2008) and in consolidated rocks made of silicates and aluminosilicates (e.g., Loret et al., 2002). This electrical double layer includes a Gouy-Chapman diffuse layer of counterions and co-ions (with concentrations usually described by Poisson-Boltzmann distributions, see Gouy, 1910, Chapman, 1913) and a Stern layer of sorbed counterions (see Stern, 1924, Avena and De Pauli, 1998, Leroy and Revil, 2004). The air-water interface can also be the setting of such an electrical double layer (e.g., Leroy et al., 2012). By unsaturated conditions, I consider that the pressure of the air phase is constant and therefore the mechanical driving force for the flow of the pore water is the gradient of the water pressure alone.
My goal is to apply the model developed in Paper 1 to study the filtration properties of charged porous materials such as bentonite in unsaturated conditions. This model is easily testable since all the conductivity and coupling terms of the matrix of material properties L are written in terms of the same fundamental parameters (permeability and cementation exponent) that can be measured independently. This is in contrast with the phenomenological models found in the literature and based on non-equilibrium thermodynamics without a micro-macro description of the coupling properties (e.g., Mitchell, 1993, Sherwood and Craster, 2000 for the saturated case, Chen and Hicks, 2013, for the unsaturated one). In some mechanistic models, the Donnan approach (Donnan, 1911) is used to represent the concentrations in the pore space of the porous body (e.g., Katchalsky and Curran, 1965). However, there are criticisms in the use of this model in the case of thin electrical double layers (e.g., at high salinities). In addition, most of these models do not account for the partition between the Stern and diffuse layers. Other micro-macro models ignore for instance the current density and electrostatic effects (e.g., Loret et al., 2002, Chen and Hicks, 2013) or consider the pore water phase being neutral (Dormieux et al., 1995). These models cannot therefore pretend to represent accurately transport phenomena in clay-rich materials with an excess of charge present in the pore water phase.
The goal of the present paper is to test various aspects of the model developed in Paper 1 against a range of experimental data in the context of diffusion and filtration (reverse osmosis) of a simple 1:1 salt like NaCl or KCl. Thanks to its filtration properties, highly charged porous media like bentonite are indeed used as permeability barriers in the shallow subsurface for environmental purposes (e.g., Malusis et al., 2003, Kang and Shackelford, 2009, Kang and Shackelford, 2010, Kang and Shackelford, 2011, Bohnhoff and Shackelford, 2013, Zhang et al., 2014) and for the containment of radioactive wastes (e.g., Bonaparte et al., 2008, Gens, 2013). We use the theory developed in Paper 1 to obtain new expressions for the osmotic filtration coefficient and the diffusion coefficient of a 1:1 salt in a charged porous material.
Section snippets
Charge densities in the pore space
I clearly distinguish below what are the properties over the pore water phase and what are the properties in a fictitious reservoir locally in thermodynamic equilibrium with the pore water phase (Fig. 1). As mentioned above the pore water is affected by the presence of the electrical double layer both on the mineral surface and also on the air-water interface. At the opposite, the fictitious reservoir locally in equilibrium with the pore space is neutral (see for instance Leroy et al., 2007,
Theory
In order to understand the filtration efficiency of partially saturated charged porous media, we need to consider the mechanism of ion filtration by reverse osmosis as explained in Fig. 2. The charged porous material is behaving like a membrane for the salt and the salinity of the effluent is smaller than the salinity of the solution entering the porous material. For practical purpose, I consider the theory only to the case of a binary symmetric 1:1 electrolyte such as NaCl or KCl electrolytes
Filtration efficiency in saturated conditions
To test our model in saturated conditions, we first use the data from Dominijanni et al. (2013). The material is a bentonite with 98% smectite (ϕ = 0.81). The electrolyte is NaCl with concentrations from 5 mM to 100 mM. The measured CEC (methylene blue) is 1.05 meq g−1. The permeability at saturation is kS = 8.15 × 10−19 m2. A test of the model is shown in Fig. 4. Eqs. (68) and (23) provide an excellent fit to the data. We considered here the porosity known, we fixed the value of the partition coefficient
Conclusions
I have tested the model derived in Revil (2016) to understand the filtration and diffusion coefficients and their relationship. The present model explains the effect of the saturation, salinity, cation type, and porosity on the osmotic coupling coefficient. It also explains the dependence of the mutual diffusion coefficient of the salt with the salinity. Finally the model explains the dependency of the diffusion coefficient with the osmotic coefficient. The advantage of the present equations is
Acknowledgements
I thank the Referees and the Associate Editor for their time and careful reviews of the present work.
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