Simulation of Reactive Diffusion in Clays By a Continuous-Time Markovian Particle-Tracking Scheme

Abstract

Clay minerals and clay rocks are considered as efficient components of engineered and natural barriers in many high-level radioactive waste disposal programs worldwide because of their low permeabilities and high sorption capabilities. In this paper we present an approach for modeling solute diffusive transport in saturated clay minerals at the structure map scale based on an extension of the Kolmogorov−Dmitriev theory of stochastic branching processes. The proposed modeling framework allows a simple description of (i) the dual-porosity behavior typical of some compacted clay structures, where diffusion occurs both in the interlayers and in the pore space between clay particles, and (ii) the chemical interactions between the dissolved ions and the solid matrix surfaces (both at nonequilibrium and equilibrium conditions). Furthermore, the Markovian nature of the modeling approach lends itself to a continuous-time particle-tracking scheme of resolution of the model equations, which, in general, easily allows us to deal with the complex geometrical structures of the porous media. Finally, in order to account for the uncertainty in the geometrical properties of the clay samples within the proposed modeling framework, the structure map is represented as a stochastic network of interconnected, one-dimensional interlayers with different lengths and orientations. The combined methods are demonstrated on a two-dimensional literature case study of diffusion through a compacted clay sample, and the resulting average diffusion coefficients are in general agreement with those obtained by a different simulation-based technique found in the literature. Then, the methods are applied to model the diffusion of the weakly sorbing cation Na + , providing results in accordance with physical intuition.

Water retention and diffusion in unsaturated clays: Connecting atomistic and pore scale simulations

“…They served then as input parameters in random walk pore-scale simulations, leading to diffusion coefficients at the sample scale. The latter may alternatively be obtained by a homogenization approach (e.g., Tyagi et al, 2013), or by a continuous-time Markovian particle-tracking scheme as presented in Cadini & Zio (2013). Experimentally observed features, such as anisotropy of diffusion or differences in diffusion coefficients between anions and water tracers, could be reproduced in this way.…”

Further development of discrete computational techniques for calculation of restricted diffusion propagators in porous media

“…The use of Markov chains has been explored previously for the modelling of transport processes in heterogeneous systems [31][32][33]. A significant advantage of the MTM method is the computational complexity that scales with the number of time steps as log 2 N. This feature makes MTM ideally suited for very long diffusion times.…”