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Fractional Calculus and Applied Analysis

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Volume 22, Issue 6

Issues

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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  • Other articles by this author:
  • 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:
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/ 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

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.

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

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

References

  • [1]

    A. Allwright, A. Atangana, Fractal advection-dispersion equation for groundwater transport in fractured aquifers with self-similarities. Eur. Phys. J. Plus 133 (2018), 1–20; .CrossrefWeb of ScienceGoogle Scholar

  • [2]

    A. Balankin, J. Bory-Reyes, M. Shapiro, Towards a physics on fractals: Differential vector calculus in three-dimensional continuum with fractal metric. Physica A 444 (2016), 345–359; .CrossrefWeb of ScienceGoogle Scholar

  • [3]

    A. Balankin, B. Elizarraraz, Hydrodynamics of fractal continuum flow. Phys. Rev. E 85 (2012), ID 025302; .CrossrefWeb of ScienceGoogle Scholar

  • [4]

    D. Baleanu, A. Fernandez, On fractional operators and their classifications. Math. 7 (2019), ID 830; .CrossrefGoogle Scholar

  • [5]

    D. Barry, G. Sposito, Analytical solution of a convection-dispersion model with time-dependent transport coefficients. Water Resour. Res. 25 (2010), 2407–2416; .CrossrefGoogle Scholar

  • [6]

    B. Bijeljic, P. Mostaghimi, M. Blunt, Signature of non-Fickian solute transport in complex heterogeneous porous media. Phys. Rev. Lett. 107 (2011), ID 204502; .CrossrefWeb of ScienceGoogle Scholar

  • [7]

    W. Cai, W. Chen, F. Wang, Three-dimensional Hausdorff derivative diffusion model for isotropic/anisotropic fractal porous media. Therm. Sci. 22 (2018), S1–S6; .CrossrefWeb of ScienceGoogle Scholar

  • [8]

    W. Cai, W. Chen, W. Xu, Characterizing the creep of viscoelastic materials by fractal derivative models. Int. J. Non-Linear Mech. 87 (2016), 58–63; .CrossrefWeb of ScienceGoogle Scholar

  • [9]

    C. Chang, H. Yeh, Investigation of flow and solute transport at the field scale through heterogeneous deformable porous media. J. Hydrol. 540 (2016), 142–147; .CrossrefWeb of ScienceGoogle Scholar

  • [10]

    W. Chen, Time-space fabric underlying anomalous diffusion. Chaos Soliton. Fract. 28 (2006), 923–929; .CrossrefGoogle Scholar

  • [11]

    Y. Liang, X. Wei, W. Chen, J. Weberszpil, From fractal to a generalized fractal: non-power-function structal metric. Fract. 27 (2019), ID 1950083; .CrossrefGoogle Scholar

  • [12]

    W. Chen, Y. Liang, X. Hei, Structural derivative based on inverse Mittag-Leffler function for modeling ultraslow diffusion. Fract. Calc. Appl. Anal. 19, No 5 (2016), 1250–1261; ; https://www.degruyter.com/view/j/fca.2016.19.issue-5/issue-files/fca.2016.19.issue-5.xml.CrossrefWeb of Science

  • [13]

    W. Chen, H. Sun, X. Zhang, D. Korosak, Anomalous diffusion modeling by fractal and fractional derivatives. Comput. Math. Appl. 59 (2010), 1754–1758; .CrossrefWeb of ScienceGoogle Scholar

  • [14]

    W. Chen, F. Wang, B. Zheng, W. Cai, Non-Euclidean distance fundamental solution of Hausdorff derivative partial differential equations. Eng. Anal. Bound. Elem. 84 (2017), 213–219; .CrossrefWeb of ScienceGoogle Scholar

  • [15]

    J. Crank, The Mathematics of Diffusion. 2nd Ed., Clarendon, Oxford (1975).Google Scholar

  • [16]

    A. Daus, E. Frind, E. Sudicky, Comparative error analysis in finite element formulations of the advection-dispersion equation. Adv. Water Resour. 8 (1985), 86–95; .CrossrefGoogle Scholar

  • [17]

    M. Dentz, A. Cortis, H. Scher, B. Berkowitz, Time behavior of solute transport in heterogeneous media: transition from anomalous to normal transport. Adv. Water Resour. 27 (2004), 155–173; .CrossrefGoogle Scholar

  • [18]

    M. Dentz, P. Kang, T. Borgne, Continuous time random walks for non-local radial solute transport. Adv. Water Resour. 82 (2015), 16–26; .CrossrefWeb of ScienceGoogle Scholar

  • [19]

    P. Estevez, M. Orchard, M. Cortes, A fractal time thermal model for predicting the surface temperature of air-cooled cylindrical Li-ion cells based on experimental measurements. J. Power Sources 306 (2016), 636–645; .CrossrefWeb of ScienceGoogle Scholar

  • [20]

    G. Gao, H. Zhan, S. Feng, B. Fu, Y. Ma, G. Huang, A new mobile-immobile model for reactive solute transport with scale-dependent dispersion. Water Resour. Res. 46 (2010), ID W08533; .CrossrefWeb of ScienceGoogle Scholar

  • [21]

    J. He, A tutorial review on fractal spacetime and fractional calculus. Int. J. Theor. Phys. 53 (2014), 3698–3718; .CrossrefWeb of ScienceGoogle Scholar

  • [22]

    R. Hilfer, Y. Luchko, Desiderata for fractional derivatives and integrals. Math. 7, No 2 (2019), ID 149; .CrossrefGoogle Scholar

  • [23]

    W. Jost, 1960. Diffusion in Solids. 3rd Ed., Academic Press, New York (1960).Google Scholar

  • [24]

    S. Lee, I. Yeo, K. Lee, W. Lee, The role of eddies in solute transport and recovery in rock fractures: Implication for groundwater remediation. Hydrol. Process. 31 (2017), 3580–3587; .CrossrefWeb of ScienceGoogle Scholar

  • [25]

    S. Lehnigk, The Generalized Feller Equation and Related Topics. Longman Sci. Tech., Harlow (1993).Google Scholar

  • [26]

    J. Li, M. Ostoja-Starzewski, Fractal solids, product measures and fractional wave equations. P. Math. Phys. Eng. Sci. 465 (2009), 2521–2536; .CrossrefGoogle Scholar

  • [27]

    Y. Liang, A. Ye, W. Chen, R. Gatto, L. Colon-Perez, T. Mareci, L. Magin, A fractal derivative model for the characterization of anomalous diffusion in magnetic resonance imaging. Commun. Nonlinear Sci. 39 (2016), 529–537; .CrossrefWeb of ScienceGoogle Scholar

  • [28]

    Y. Liang, W. Chen, A non-local structural derivative model for characterization of ultraslow diffusion in dense colloids. Commun. Nonlinear Sci. 56 (2018), 131–137; .CrossrefWeb of ScienceGoogle Scholar

  • [29]

    G. Lin, Analyzing signal attenuation in PFG anomalous diffusion via a modified Gaussian phase distribution approximation based on fractal derivative model. Physica A 467 (2017), 277–288; .CrossrefWeb of ScienceGoogle Scholar

  • [30]

    X. Liu, H. Sun, M. Lazarevic, Z. Fu, A variable-order fractal derivative model for anomalous diffusion. Therm. Sci. 21 (2017), 51–59; .CrossrefWeb of ScienceGoogle Scholar

  • [31]

    R. Metzler, J. Klafter, The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339 (2000), 1–77; .CrossrefGoogle Scholar

  • [32]

    S. Mishra, J. Parker, Analysis of solute transport with a hyperbolic scale-dependent dispersion model. Hydrol. Process. 4 (2010), 45–57; .CrossrefGoogle Scholar

  • [33]

    M. Moslehi, F. Barros, F. Ebrahimi, M. Sahimi, Upscaling of solute transport in disordered porous media by wavelet transformations. Adv. Water Resour. 96 (2016), 180–189; .CrossrefWeb of ScienceGoogle Scholar

  • [34]

    R. Muralidhar, D. Ramkrishna, Diffusion in pore fractals: A review of linear response models. Transport Porous Med. 13 1993, 79–95; .CrossrefGoogle Scholar

  • [35]

    S. Nie, H. Sun, X. Liu, Z. Wang, M. Xie, Fractal derivative model for the transport of the suspended sediment in unsteady flows. Therm. Sci. 22 (2018), S109–S115; .CrossrefWeb of ScienceGoogle Scholar

  • [36]

    M. Ortigueira, J. Machado, What is a fractional derivative? J. Comput. Phys. 293 (2015), 4–13; .CrossrefWeb of ScienceGoogle Scholar

  • [37]

    L. Pang, B. Hunt, Solutions and verification of a scale-dependent dispersion model. J. Contam. Hydrol. 53 (2001), 21–39; .CrossrefPubMedGoogle Scholar

  • [38]

    D. Pedretti, A. Molinari, C. Fallico, S. Guzzi, Implications of the change in confinement status of a heterogeneous aquifer for scale-dependent dispersion and mass-transfer processes. J. Contam. Hydrol. 193 (2016), 86–95; .CrossrefWeb of SciencePubMedGoogle Scholar

  • [39]

    J. Reyes-Marambio, F. Moser, F. Gana, B. Severino, W. Calderon-Munoz, R. Palma-Behnke, M. Sabahi, H. Montazeri, B. Sleep, Practical finite analytic methods for simulation of solute transport with scale-dependent dispersion under advection-dominated conditions. Int. J. Heat Mass Tran. 83 (2015), 799–808; .CrossrefWeb of ScienceGoogle Scholar

  • [40]

    M. Shlesinger, Fractal time in condensed matter. Annu. Rev. Phys. Chem. 39 (1988), 269–290; .CrossrefGoogle Scholar

  • [41]

    N. Su, G. Sander, F. Liu, V. Anh, D. Barry, Similarity solutions for solute transport in fractal porous media using a time- and scale-dependent dispersivity. Appl. Math. Model. 29 (2005), 852–870; .CrossrefGoogle Scholar

  • [42]

    N. Su, Mass-time and space-time fractional partial differential equations of water movement in soils: theoretical framework and application to infiltration. J. Hydrol. 519 (2014), 1792–1803; .CrossrefWeb of ScienceGoogle Scholar

  • [43]

    X. Su, W. Chen, W. Xu, Characterizing the rheological behaviors of non-Newtonian fluid via a viscoelastic component: Fractal dashpot. Adv. Mech. Eng. 9 (2017), 1–12; .CrossrefWeb of ScienceGoogle Scholar

  • [44]

    H. Sun, Z. Li, Y. Zhang, W. Chen, Fractional and fractal derivative models for transient anomalous diffusion: Model comparison. Chaos Soliton. Fract. 102 (2017), 346–353; .CrossrefWeb of ScienceGoogle Scholar

  • [45]

    H. Sun, M. Meerschaert, Y. Zhang, J. Zhu, W. Chen, A fractal Richard’s equation to capture the non-Boltzmann scaling of water transport in unsaturated media. Adv. Water Resour. 52 (2013), 292–295; .CrossrefGoogle Scholar

  • [46]

    H. Sun, Y. Zhang, W. Chen, D. Reeves, Use of a variable index fractional derivative model to capture transient dispersion in heterogeneous media. J. Contam. Hydrol. 157 (2014), 47–58; .CrossrefPubMedWeb of ScienceGoogle Scholar

  • [47]

    G. Uffink, A. Elfeki, M. Dekking, J. Bruining, C. Kraaikamp, Understanding the non-Gaussian nature of linear reactive solute transport in 1D and 2D. Transport Porous Med. 91 (2012), 547–571; .CrossrefWeb of ScienceGoogle Scholar

  • [48]

    S. Wheatcraft, S. Tyler, An explanation of scale-dependent dispersivity in heterogeneous aquifers using concepts of fractal geometry. Water Resour. Res. 24 (1988), 566–578; .CrossrefGoogle Scholar

  • [49]

    B. Yu, X. Jiang, H. Xu, A novel compact numerical method for solving the two-dimensional non-linear fractional reaction-subdiffusion equation. Numer. Algorithms 68 (2015), 923–950; .CrossrefWeb of ScienceGoogle Scholar

  • [50]

    H. Zhang, X. Jiang, X. Yang, A time-space spectral method for the time-space fractional Fokker-Planck equation and its inverse problem. Appl. Math. Comput. 320 (2018), 302–318; .CrossrefWeb of ScienceGoogle Scholar

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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