Model Reduction and Multi-Scale Simulation for Solute Transport in Fractured Porous Media

Fractured-porous media (FPM) pose one of the largest unresolved challenges for simulating large hydrogeological systems. The high contrast in advective transport between fast conduits and low-permeability rock matrix, including complex mass transfer processes, leads to the typical complex characteristics of early bulk arrivals and long tailings. Adequate direct representation of FPM requires enormous numerical resolutions. For large scales, e.g. the catchment scale, and when allowing for uncertainty in the fracture network architecture or in matrix properties, computational costs quickly reach an intractable level. In such cases, upscaling techniques like the multi-rate mass transfer model (MRMT), originally introduced for heterogeneous porous media can be very useful. MRMT models represent small-scale information by a kinetic sorption-like process described by so-called memory functions. Memory functions, however, make the governing MRMT equations non-local in time, which introduces its own challenges for the efficiency of numerical solution schemes. In this work, we do a first-time application of MRMT to FPM. We choose a multi-scale simulation approach based on block upscaling, with individual blocks defined by and aligned with the local flow coordinates. Also, we use the concept of temporal moments and their generating equations to localize the MRMT equations, and save computational costs in the scale transition. With this tool, we can analyze the multi-scale arrival time statistics in FPM. We can investigate and rank the influence of fracture network properties, matrix properties and heterogeneity on arrival time statistics, and shed more light into the physical interplay between fracture and matrix flow, and mass transfer that leads to non-Fickian transport behavior. In summary, our framework is a first-time application of MRMT to FPM and makes upscaling of transport in FPM faster and more applicable for large-scale problems.