Here, 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. Yet, 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.
Hyman, Jeffrey D., et al. "Matrix Diffusion in Fractured Media: New Insights Into Power Law Scaling of Breakthrough Curves." Geophysical Research Letters, vol. 46, no. 23, Nov. 2019. https://doi.org/10.1029/2019GL085454
Hyman, Jeffrey D., Rajaram, Harihar, Srinivasan, Shriram, Makedonska, Nataliia, Karra, Satish, Viswanathan, Hari, & Srinivasan, Gowri (2019). Matrix Diffusion in Fractured Media: New Insights Into Power Law Scaling of Breakthrough Curves. Geophysical Research Letters, 46(23). https://doi.org/10.1029/2019GL085454
Hyman, Jeffrey D., Rajaram, Harihar, Srinivasan, Shriram, et al., "Matrix Diffusion in Fractured Media: New Insights Into Power Law Scaling of Breakthrough Curves," Geophysical Research Letters 46, no. 23 (2019), https://doi.org/10.1029/2019GL085454
@article{osti_1581275,
author = {Hyman, Jeffrey D. and Rajaram, Harihar and Srinivasan, Shriram and Makedonska, Nataliia and Karra, Satish and Viswanathan, Hari and Srinivasan, Gowri},
title = {Matrix Diffusion in Fractured Media: New Insights Into Power Law Scaling of Breakthrough Curves},
annote = {Here, 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 (α1, 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.},
doi = {10.1029/2019GL085454},
url = {https://www.osti.gov/biblio/1581275},
journal = {Geophysical Research Letters},
issn = {ISSN 0094-8276},
number = {23},
volume = {46},
place = {United States},
publisher = {American Geophysical Union},
year = {2019},
month = {11}}
Hyman, J. D.; Jiménez-Martínez, J.; Viswanathan, H. S.
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 374, Issue 2078https://doi.org/10.1098/rsta.2015.0426