Upscaling solute transport in rough single-fractured media with matrix diffusion using a time fractional advection-dispersion equation

Abstract

Complex structures in fractures and the rock matrix, which are difficult to map exhaustively at the local scale, can dominate contaminant transport in fractured aquifers. An upscaled model is therefore needed to effectively characterize solute transport in a heterogeneous fracture-matrix system on a large spatiotemporal scale. Most existing upscaling methods, however, do not specifically quantify the influence of matrix diffusion and fracture surface roughness on solute transport, and their upscaling parameters are usually difficult to obtain. To fill this knowledge gap, first, a time fractional advection–dispersion equation (t-FADE) model, which can accurately describe solute transport in straight fractured media, is proposed. Next, the influence of local roughness and tortuosity of fractures on transport is comprehensively considered by introducing trend lines. The t-FADE is used to characterize solute transport in each segment of rough fractures, and an upscaled model describing solute transport in rough fractures is established by averaging the governing equations of each segment. Finally, a time-dependent attenuation function is introduced into the convolutional function of the upscaled model to overcome the error caused by overlapping diffusion regions. Model comparison and analysis reveal that the rough wall of fractures strengthens matrix retention capacity, leading to a delayed peak of the breakthrough curve (BTC) with a lower peak concentration, which can be quantitatively characterized by the ratio of the actual fracture length to its longitudinal length; however, the fracture structure does not affect the solute BTC’s trailing concentration at the constant fracture mean flow velocity without considering mechanical dispersion, so the relatively simple straight fracture model can be used to describe the trailing stage of transport in rough-walled fracture media.