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

Liang, Yingjie, Su, Ninghu, and Chen, Wen (2019) A time-space Hausdorff derivative model for anomalous transport in porous media. Journal of Fractional Calculus and Applied Analysis, 22 (6). pp. 1517-1536.

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Abstract

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.

Item ID: 62265
Item Type: Article (Research - C1)
ISSN: 1314-2224
Keywords: time-space dependent dispersion, partial differential equation, Hausdorff derivative, solute transport, porous media
Copyright Information: © 2019 Diogenes Co., Sofia
Sensitivity Note: This is a mathematical model applied to environmental problems, so it is general and can be made public from the time when embargo date expires.
Funders: Fundamental Research Funds for the Central Universities (FRFCU), National Natural Science Foundation of China (NNSFC), China Postdoctoral Science Foundation (CPSF), Opening Fund of MOE Key Labaoratory Groundwater Circulation and Environmental Evolution (MOE)
Projects and Grants: FRFCU No. 2019B16114, NNSFC No. 11702085, NNSFC No. 11772121, CPSF NO. 2018M630500, MOE No. 20185016112
Date Deposited: 12 Feb 2020 07:34
FoR Codes: 01 MATHEMATICAL SCIENCES > 0102 Applied Mathematics > 010203 Calculus of Variations, Systems Theory and Control Theory @ 40%
04 EARTH SCIENCES > 0404 Geophysics > 040403 Geophysical Fluid Dynamics @ 30%
04 EARTH SCIENCES > 0406 Physical Geography and Environmental Geoscience > 040603 Hydrogeology @ 30%
SEO Codes: 96 ENVIRONMENT > 9606 Environmental and Natural Resource Evaluation > 960608 Rural Water Evaluation (incl. Water Quality) @ 30%
96 ENVIRONMENT > 9606 Environmental and Natural Resource Evaluation > 960609 Sustainability Indicators @ 40%
96 ENVIRONMENT > 9609 Land and Water Management > 960904 Farmland, Arable Cropland and Permanent Cropland Land Management @ 30%
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