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Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications

Author

Listed:
  • Yong Zhang

    (Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USA)

  • Dongbao Zhou

    (State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing 210098, China)

  • Wei Wei

    (School of Environment, Nanjing Normal University, Nanjing 210023, China)

  • Jonathan M. Frame

    (Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USA)

  • Hongguang Sun

    (State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing 210098, China)

  • Alexander Y. Sun

    (Bureau of Economic Geology, Jackson School of Geosciences, University of Texas Austin, Austin, TX 78713, USA)

  • Xingyuan Chen

    (Atmospheric Sciences and Global Change, Pacific Northwest National Laboratory, Richland, WA 99352, USA)

Abstract

Fractional calculus-based differential equations were found by previous studies to be promising tools in simulating local-scale anomalous diffusion for pollutants transport in natural geological media (geomedia), but efficient models are still needed for simulating anomalous transport over a broad spectrum of scales. This study proposed a hierarchical framework of fractional advection-dispersion equations (FADEs) for modeling pollutants moving in the river corridor at a full spectrum of scales. Applications showed that the fixed-index FADE could model bed sediment and manganese transport in streams at the geomorphologic unit scale, whereas the variable-index FADE well fitted bedload snapshots at the reach scale with spatially varying indices. Further analyses revealed that the selection of the FADEs depended on the scale, type of the geomedium (i.e., riverbed, aquifer, or soil), and the type of available observation dataset (i.e., the tracer snapshot or breakthrough curve (BTC)). When the pollutant BTC was used, a single-index FADE with scale-dependent parameters could fit the data by upscaling anomalous transport without mapping the sub-grid, intermediate multi-index anomalous diffusion. Pollutant transport in geomedia, therefore, may exhibit complex anomalous scaling in space (and/or time), and the identification of the FADE’s index for the reach-scale anomalous transport, which links the geomorphologic unit and watershed scales, is the core for reliable applications of fractional calculus in hydrology.

Suggested Citation

  • Yong Zhang & Dongbao Zhou & Wei Wei & Jonathan M. Frame & Hongguang Sun & Alexander Y. Sun & Xingyuan Chen, 2021. "Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications," Mathematics, MDPI, vol. 9(7), pages 1-15, April.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:7:p:790-:d:530937
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    References listed on IDEAS

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    1. Andersson, Martin & Yuan, Jinliang & Sundén, Bengt, 2010. "Review on modeling development for multiscale chemical reactions coupled transport phenomena in solid oxide fuel cells," Applied Energy, Elsevier, vol. 87(5), pages 1461-1476, May.
    2. Zhang, Yong & Sun, HongGuang & Stowell, Harold H. & Zayernouri, Mohsen & Hansen, Samantha E., 2017. "A review of applications of fractional calculus in Earth system dynamics," Chaos, Solitons & Fractals, Elsevier, vol. 102(C), pages 29-46.
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    Cited by:

    1. Alexander Sokolov & Irina Nikulina, 2022. "Choice of Regularization Methods in Experiment Processing: Solving Inverse Problems of Thermal Conductivity," Mathematics, MDPI, vol. 10(22), pages 1-16, November.

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