Flux exchange between fracture and matrix dictates late-time tracer tailing

Understanding and quantifying the solute tailing process in fractured rocks is critical for various subsurface applications but remains challenging. Particularly, the effects of flux exchange between fracture and matrix on the tailing process are considered to be important but have not been explicitly investigated. To fill this knowledge gap, we estimate the power-law scaling of late-time tracer tailing, n, based on three dimensional direct numerical simulations that consider both solute diffusion and advection across fracture-matrix interface. The dependence of solute decay rate n on fracture-matrix interaction is elucidated by simulating cases over a wide range of matrix permeability (k_m). In the case with matrix diffusion but without advection between fracture and matrix, we find that matrix diffusion significantly enhances tracer tailing. Then, the consideration of advection does not show significant effect on n up to the matrix permeability k_m of 10⁻¹² m². When k is greater than 10⁻¹² m², n declines rapidly down to 1.0 at k = 10⁻⁹ m². We finally demonstrate that the upscaled model of continuous time random work can capture the late-time tailing behavior. This study provides a mechanistic explanation for the heavy tailing process commonly observed in fractured rocks.