Vertical matrix advection dominates transient anomalous diffusion within fracture-matrix systems using a modified diffusion model

Abstract

Pollutant transport in fractured media has been well analyzed using process-based diffusion models and concise, time-nonlocal models like the tempered fractional derivative (TFD) model. While traditional diffusion models are limited to specific transport processes, TFD model parameters lack hydrogeological meaning. In addressing these knowledge gaps, a modified diffusion model is proposed to augment the traditional diffusion model and offer hydrogeological interpretations for TFD parameters. This modified diffusion model incorporates transverse advection in low-permeability matrices, allowing for the quantification of transient anomalous diffusion in a single fracture-matrix system. This model offers a semi-analytical solution, systematically validated for its applicability. Memory function analysis establishes a quantitative connection between the modified diffusion model and the TFD model, uncovering a pivotal discovery in this study. Parameter sensitivity analysis further shows that the matrix characteristic rate defined as a function of (Vm)2/(4Dm), where Vm is the transverse matrix velocity and Dm is the transverse matrix diffusion coefficient, controls the temporal evolution of anomalous sub-diffusion in the fracture. Applications are performed to check the feasibility of the modified diffusion model. The developed modified diffusion model sheds light on quantifying complex transport in fracture-matrix systems and enhances the predictability of nonlocal transport models.