Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications
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 s...
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doaj-7128993635494b2d8e67182a6c059dca2021-04-06T23:03:56ZengMDPI AGMathematics2227-73902021-04-01979079010.3390/math9070790Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and ApplicationsYong Zhang0Dongbao Zhou1Wei Wei2Jonathan M. Frame3Hongguang Sun4Alexander Y. Sun5Xingyuan Chen6Department of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USAState Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing 210098, ChinaSchool of Environment, Nanjing Normal University, Nanjing 210023, ChinaDepartment of Geological Sciences, University of Alabama, Tuscaloosa, AL 35487, USAState Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Mechanics and Materials, Hohai University, Nanjing 210098, ChinaBureau of Economic Geology, Jackson School of Geosciences, University of Texas Austin, Austin, TX 78713, USAAtmospheric Sciences and Global Change, Pacific Northwest National Laboratory, Richland, WA 99352, USAFractional 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.https://www.mdpi.com/2227-7390/9/7/790fractional calculusanomalous diffusionmulti-scale modelpollutant transport |
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DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Yong Zhang Dongbao Zhou Wei Wei Jonathan M. Frame Hongguang Sun Alexander Y. Sun Xingyuan Chen |
spellingShingle |
Yong Zhang Dongbao Zhou Wei Wei Jonathan M. Frame Hongguang Sun Alexander Y. Sun Xingyuan Chen Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications Mathematics fractional calculus anomalous diffusion multi-scale model pollutant transport |
author_facet |
Yong Zhang Dongbao Zhou Wei Wei Jonathan M. Frame Hongguang Sun Alexander Y. Sun Xingyuan Chen |
author_sort |
Yong Zhang |
title |
Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications |
title_short |
Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications |
title_full |
Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications |
title_fullStr |
Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications |
title_full_unstemmed |
Hierarchical Fractional Advection-Dispersion Equation (FADE) to Quantify Anomalous Transport in River Corridor over a Broad Spectrum of Scales: Theory and Applications |
title_sort |
hierarchical fractional advection-dispersion equation (fade) to quantify anomalous transport in river corridor over a broad spectrum of scales: theory and applications |
publisher |
MDPI AG |
series |
Mathematics |
issn |
2227-7390 |
publishDate |
2021-04-01 |
description |
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. |
topic |
fractional calculus anomalous diffusion multi-scale model pollutant transport |
url |
https://www.mdpi.com/2227-7390/9/7/790 |
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