Modeling diffusion processes in the presence of a diffuse layer at charged mineral surfaces: a benchmark exercise

Abstract

The electrostatic properties of clay mineral surfaces play a significant role in their diffusion properties. The negative electrostatic potential field at clay mineral surfaces results in the presence of a diffuse layer that balances the mineral surface charge. The diffusion properties of the porosity fraction that is affected by this phenomenon are different from the diffusion properties of electroneutral bulk water. These properties have attracted growing interest from diverse communities in the past years, especially in the field of study of radioactive waste disposal. The influence of the diffuse layer can be described at the continuum scale by a set of equations that are formulated in terms of the Nernst-Planck equation. The number of codes that can handle the coupling between transport properties in clay affected by the presence of a diffuse layer in the porosity and chemical reactions is very limited, and no benchmark exercises have been published yet that make it possible to validate the numerical implementation of these equations in reactive transport codes. The present study proposes a set of benchmark exercises of increasing complexity that highlight caveats related to the finite difference (volume) treatment of the Nernst-Planck equation in the presence of a diffuse layer in heterogeneous systems. Once these problems are identified and solved, the codes PHREEQC, CrunchClay, and a new Fortran routine written for this study gave results in very good agreement for most of the benchmark exercises. When present, the differences in results were directly traceable to the differences in averaging methods at grid cell boundaries, and to the consideration or the omission of the activity gradient term in the Nernst-Planck equation.

A model for discrete fracture-clay rock interaction incorporating electrostatic effects on transport

“…As discussed in [30], it is necessary to account for cross-fluxes between the two continua, the diffuse layer and bulk water pore space, in order to maintain mass balance. Therefore, two transport terms are not sufficient to describe the system under investigation.…”

“…However, between the fracture (100% bulk water porosity) and the matrix (consisting of both DL and bulk water porosity), the diffusive cross-flux is important. The full averaging rules for the cross-flux are presented in [27,30].…”

“…Most if not all analyses of fracture-matrix interaction have assumed that molecular diffusion can be described adequately with Fick's Laws. However, the limitations of Fick's Laws have become increasingly clear as researchers have demonstrated the need to describe diffusion with the Nernst-Planck equation [24][25][26][27][28][29][30][31][32][33][34][35]. This is because charged chemical species diffuse at different rates, thus creating an electric field or diffusion potential that creates a flux that is in addition to the pure Fickian diffusion due to a concentration gradient.…”

“…In this case, the Nernst-Planck equation needs to be supplemented with the Poisson equation to provide an accurate representation of charged chemical species fluxes. Thus, the Poisson-Nernst-Planck (or PNP) equation provides the most complete continuum representation of ion fluxes in the general case of charged porous media, including clay rock and shale [28,30,36]. The Poisson-Boltzmann can be used in the case of static or equilibrium system.…”

Engineered Barrier System Research Activities at LBNL (FY2020 Progress Report)

“…June 04, 2020 Modeling of diffusion experiments will be carried out using CrunchClay (Steefel et al, 2015;Tournassat and Steefel, 2019a) taking full advantage of the capabilities of this reactive transport code developed at LBL. CrunchClay is currently one of only two codes that can handle diffusion processes in the diffuse layer, in which the solution is not electroneutral, and for which coupled interdiffusion processes must be taken into account to model diffusion properties with a mechanistic approach (Tournassat and Steefel, 2019b).…”