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

Yong Zhang,Xingyuan Chen,Alexander Y Sun,Wei Wei,Hongguang Sun,Dongbao Zhou,Jonathan M Frame

doi:10.3390/math9070790

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.

Highlights

Transport of materials in Earth systems can exhibit scale-dependent dynamics in anomalous diffusion, since the functions affecting non-Fickian transport may vary, interact, and/or compete in a wide range of spatiotemporal scales and result in complex mobilization and/or retention of materials in natural geomedia [1]
The following fractional advection-dispersion equation (FADE) built upon fractional calculus was found to be applicable in capturing non-Fickian dispersion in complex media, including flow/transport in geomedia, which typically result in non-Fickian or anomalous behavior due to the multi-scale intrinsic physical/chemical heterogeneity of the media [11]: β∗
Fractional index in the FADEvalidated tends to apparently fluctuate (FigThisthe study proposed and partially the hierarchical fractional-derivative ure 6b), resulting in the variable-index for reach-scale anomalous transport. It iscorridor at all models in simulating anomalous scaling for pollutants transport in the river our important logic expectation that anomalous transport at the

Summary

Introduction

Transport of materials in Earth systems can exhibit scale-dependent dynamics in anomalous diffusion, since the functions affecting non-Fickian transport may vary, interact, and/or compete in a wide range of spatiotemporal scales and result in complex mobilization and/or retention of materials in natural geomedia (e.g., soil, slopes, rivers, and aquifers) [1]. Studies on stream transport (mass and chemical) in the last three decades have made significant contributions toward better understanding of mechanisms and hydrobiogeochemical implications of hyporheic flow and transport processes [3,4,5], there remains 4.0/). The following fractional advection-dispersion equation (FADE) built upon fractional calculus was found to be applicable in capturing non-Fickian dispersion in complex media, including flow/transport in geomedia, which typically result in non-Fickian or anomalous behavior due to the multi-scale intrinsic physical/chemical heterogeneity of the media [11]: β∗

Methods

Discussion

Conclusion