Multi-scale random walk models for reactive transport processes in fracture-matrix systems

Abstract

Random walk methods are particularly well suited for modeling (anomalous) transport processes in complex systems, from heterogeneous porous domains to fractured rocks. Taking full advantage of their multi-scale attributes, these methods can be integrated into multi-scale modeling strategies where they are used for (i) performing small-scale simulations with a high level of description of the heterogeneities, (ii) defining statistical functions that describe these heterogeneities at a larger scale, and (iii) providing upscaled descriptions of the processes while taking into account the impact of the small-scale heterogeneities. In this work, we define such a strategy considering (i) structural heterogeneities with heterogeneous porosity fields incorporated into fracture-matrix systems and (ii) the reactivity of the system with chemical reactions that are not restricted to linear first-order reactions as it is the case in existing upscaled formulations. To this end, we develop two new modeling methods: the Reactive-Time Domain Random Walk (R-TDRW) and Upscaled Reactive TDRW (UR-TDRW) approaches at small and large scale, respectively. The numerical methods and multi-scale strategy are presented with a general formulation and applied to single-species transport and reaction in fracture-matrix systems. We analyze the impact of different levels of structural heterogeneities, and the impact of the reactivity, on the breakthrough curves computed at both scales and on the statistical functions that are used in the multi-scale strategy. This strategy is validated by demonstrating the good agreement between the results obtained at different scales, showing promising applications for future work in large-scale fracture networks.