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Fractional Calculus and Applied Analysis

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Volume 22, Issue 6


Lagrangian solver for vector fractional diffusion in bounded anisotropic aquifers: Development and application

Yong Zhang
  • State Key Lab. of Hydrology-Water Resources and Hydraulic Engineering College of Mechanics and Materials, Hohai University, Nanjing, 210098, China
  • Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USA
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/ HongGuang Sun
  • State Key Lab. of Hydrology-Water Resources and Hydraulic Engineering College of Mechanics and Materials, Hohai University, Nanjing, 210098, China
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/ Chunmiao Zheng
  • Guangdong Provincial Key Lab. of Soil and Groundwater Pollution Control, School of Environmental Science & Engineering Southern University of Science and Technology, Shenzhen, Guangdong 518055, China
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Published Online: 2019-12-31 | DOI: https://doi.org/10.1515/fca-2019-0083


Fractional-derivative models (FDMs) are promising tools for characterizing non-Fickian transport in natural geological media. Hydrologic applications of FDMs, however, have been limited in the last two decades, due to the lack of feasible models and solvers to quantify multi-dimensional anomalous diffusion for pollutants in bounded aquifers. This study develops and applies FDM tools to capture vector fractional dispersion for both conservative and reactive pollutants in fractional Brownian motion (fBm) random fields with bounded domains. A d-dimensional anisotropic fBm field for hydraulic conductivity (K) is first generated numerically. A particle-tracking based, fully Lagrangian solver is then developed to approximate particle dynamics in the fBm K fields under various boundary conditions, where the governing equation is the vector FDM subordination to regional flow. Numerical experiments show that the Lagrangian solver can combine nonlocal anomalous transport and local aquifer properties to quantify pollutant transport in bounded aquifers. Application analyses further reveal that the K correlation can significantly enhance the spreading of conservative pollutant particles, and increase the reaction rate by enhancing the mobility and mixing of reactant particles undergoing bimolecular reactions.

Extension of the Lagrangian solver is also discussed, including modeling transient flow, generalizing boundary conditions, and capturing complex chemical reactions. This study therefore provides the hydrologic community an efficient Lagrangian solver to model reactive anomalous transport in bounded anisotropic aquifers with any dimension, size, and boundary conditions.

MSC 2010: Primary 26A33; Secondary 34A08; 34A34; 34K28; 35R11; 60G22; 65L10; 82C70; 86A05

Key Words and Phrases: vector fractional-derivative model; Lagrangian solver; fractional Brownian motion; bimolecular reaction; bounded domain

This paper is dedicated to the memory of the late Professor Wen Chen


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About the article

Received: 2019-07-12

Published Online: 2019-12-31

Published in Print: 2019-12-18

Citation Information: Fractional Calculus and Applied Analysis, Volume 22, Issue 6, Pages 1607–1640, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2019-0083.

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