Matrix Diffusion in Fractured Media: New Insights Into Power Law Scaling of Breakthrough Curves

Abstract

AbstractWe develop a theoretical model for power law tailing behavior of transport in fractured rock based on the relative dominance of the decay rate of the advective travel time distribution, modeled using a Pareto distribution (with tail decaying as ∼ time−(1+α)), versus matrix diffusion, modeled using a Lévy distribution. The theory predicts that when the advective travel time distribution decays sufficiently slowly (α<1), the late‐time decay rate of the breakthrough curve is −(1+α/2) rather than the classical −3/2. However, if α>1, the −3/2 decay rate is recovered. For weak matrix diffusion or short advective first breakthrough times, we identify an early‐time regime where the breakthrough curve follows the Pareto distribution, before transitioning to the late‐time decay rate. The theoretical predictions are validated against particle tracking simulations in the three‐dimensional discrete fracture network simulator dfnWorks, where matrix diffusion is incorporated using a time domain random walk.