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Fractional Calculus and Applied Analysis

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Volume 22, Issue 6


A time-space Hausdorff derivative model for anomalous transport in porous media

Yingjie Liang
  • Institute of Soft Matter Mechanics, College of Mechanics and Materials Hohai University, No. 8 Focheng West Road, Nanjing, 211100, China
  • MOE Key Laboratory of Groundwater Circulation and Environmental Evolution, China University of Geosciences (Beijing), Beijing 100083, China
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  • De Gruyter OnlineGoogle Scholar
/ Ninghu Su
  • College of Science and Engineering, Tropical Water and Aquatic Ecosystem Research James Cook University, Cairns, Queensland 4870, Australia
  • College of Resources and Environmental Sciences, Ningxia University, Yinchuan, Ningxia 750021, China
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  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Wen Chen
  • Institute of Soft Matter Mechanics, College of Mechanics and Materials Hohai University, No. 8 Focheng West Road, Nanjing, 211100, China
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2019-12-31 | DOI: https://doi.org/10.1515/fca-2019-0079


This paper presents a time-space Hausdorff derivative model for depicting solute transport in aquifers or water flow in heterogeneous porous media. In this model, the time and space Hausdorff derivatives are defined on non-Euclidean fractal metrics with power law scaling transform which, respectively, connect the temporal and spatial complexity during transport. The Hausdorff derivative model can be transformed to an advection-dispersion equation with time- and space-dependent dispersion and convection coefficients. This model is a fractal partial differential equation (PDE) defined on a fractal space and differs from the fractional PDE which is derived for non-local transport of particles on a non-fractal Euclidean space. As an example of applications of this model, an explicit solution with a constant diffusion coefficient and flow velocity subject to an instantaneous source is derived and fitted to the breakthrough curves of tritium as a tracer in porous media. These results are compared with those of a scale-dependent dispersion model and a time-scale dependent dispersion model. Overall, it is found that the fractal PDE based on the Hausdorff derivatives better captures the early arrival and heavy tail in the scaled breakthrough curves for variable transport distances. The estimated parameters in the fractal Hausrdorff model represent clear mechanisms such as linear relationships between the orders of Hausdorff derivatives and the transport distance. The mathematical formulation is applicable to both solute transport and water flow in porous media.

MSC 2010: 26A24; 28A80; 35K57; 76S05

Key Words and Phrases: time-space dependent dispersion; partial differential equation; Hausdorff derivative; solute transport; porous media


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About the article

Received: 2019-03-11

Published Online: 2019-12-31

Published in Print: 2019-12-18

Citation Information: Fractional Calculus and Applied Analysis, Volume 22, Issue 6, Pages 1517–1536, ISSN (Online) 1314-2224, ISSN (Print) 1311-0454, DOI: https://doi.org/10.1515/fca-2019-0079.

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