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Mathematics 2018, 6(1), 5; https://doi.org/10.3390/math6010005

Application of Tempered-Stable Time Fractional-Derivative Model to Upscale Subdiffusion for Pollutant Transport in Field-Scale Discrete Fracture Networks

1
Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USA
2
Department of Geosciences, Western Michigan University, Kalamazoo, MI 49008, USA
3
College of Mechanics and Materials, Hohai University, Nanjing 210098, China
4
School of Environmental Science & Engineering, Southern University of Science and Technology, Shenzhen 518055, Guangdong, China
*
Author to whom correspondence should be addressed.
Received: 10 December 2017 / Revised: 29 December 2017 / Accepted: 29 December 2017 / Published: 3 January 2018
(This article belongs to the Special Issue Fractional Calculus: Theory and Applications)
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Abstract

Fractional calculus provides efficient physical models to quantify non-Fickian dynamics broadly observed within the Earth system. The potential advantages of using fractional partial differential equations (fPDEs) for real-world problems are often limited by the current lack of understanding of how earth system properties influence observed non-Fickian dynamics. This study explores non-Fickian dynamics for pollutant transport in field-scale discrete fracture networks (DFNs), by investigating how fracture and rock matrix properties influence the leading and tailing edges of pollutant breakthrough curves (BTCs). Fractured reservoirs exhibit erratic internal structures and multi-scale heterogeneity, resulting in complex non-Fickian dynamics. A Monte Carlo approach is used to simulate pollutant transport through DFNs with a systematic variation of system properties, and the resultant non-Fickian transport is upscaled using a tempered-stable fractional in time advection–dispersion equation. Numerical results serve as a basis for determining both qualitative and quantitative relationships between BTC characteristics and model parameters, in addition to the impacts of fracture density, orientation, and rock matrix permeability on non-Fickian dynamics. The observed impacts of medium heterogeneity on tracer transport at late times tend to enhance the applicability of fPDEs that may be parameterized using measurable fracture–matrix characteristics. View Full-Text
Keywords: fractional partial differential equations (fPDEs); discrete fracture networks (DFNs); anomalous transport; fractional advection-dispersion equations fractional partial differential equations (fPDEs); discrete fracture networks (DFNs); anomalous transport; fractional advection-dispersion equations
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Lu, B.; Zhang, Y.; Reeves, D.M.; Sun, H.; Zheng, C. Application of Tempered-Stable Time Fractional-Derivative Model to Upscale Subdiffusion for Pollutant Transport in Field-Scale Discrete Fracture Networks. Mathematics 2018, 6, 5.

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