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

• Published in: Geophysical research letters Vol. 46; no. 23; pp. 13785 - 13795
• Main Authors: Hyman, Jeffrey D., Rajaram, Harihar, Srinivasan, Shriram, Makedonska, Nataliia, Karra, Satish, Viswanathan, Hari, Srinivasan, Gowri
• Format: Journal Article
• Language: English
• Published: Washington John Wiley & Sons, Inc 16.12.2019; American Geophysical Union (AGU)
• Subjects: Computer simulation; Decay; Decay rate; Diffusion; Diffusion rate; Distribution; Matter & antimatter; Particle decay; Particle tracking; Power law; Random walk; Scaling; Simulators; Travel; Travel time
• ISSN: 0094-8276; 1944-8007
• DOI: 10.1029/2019GL085454

We 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. Key Points A theoretical model is developed for power law tailing of breakthrough curves influenced by matrix diffusion and heterogeneous advection A threshold decay rate for the advective travel time distribution is identified,below which matrix diffusion produces tail decay rates >−3/2 Matrix diffusion is implemented in high‐fidelity three‐dimensioal discrete fracture network simulations to confirm theoretical predictions