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Fractional Vertical Infiltration

Author

Listed:
  • Carlos Fuentes

    (Mexican Institute of Water Technology, Paseo Cuauhnáhuac Núm. 8532, Jiutepec 62550, Morelos, Mexico)

  • Fernando Alcántara-López

    (Department of Mathematics, Faculty of Science, National Autonomous University of México, Av. Universidad 3000, Circuito Exterior S/N, Delegación Coyoacán 04510, Ciudad de México, Mexico)

  • Antonio Quevedo

    (Mexican Institute of Water Technology, Paseo Cuauhnáhuac Núm. 8532, Jiutepec 62550, Morelos, Mexico)

  • Carlos Chávez

    (Water Research Center, Department of Irrigation and Drainage Engineering, Autonomous University of Querétaro, Cerro de las Campanas SN, Col. Las Campanas 76010, Querétaro, Mexico)

Abstract

The infiltration phenomena has been studied by several authors for decades, and numerical and approximate results have been shown through the asymptotic solution in short and long times. In particular, it is worth highlighting the works of Philip and Parlange, who used time and volumetric content as independent variables and space as a dependent variable, and found the solution as a power series in t 1 / 2 that is valid for short times. However, several studies show that these models are not applicable to anomalous flows, in which case the application of fractional calculus is needed. In this work, a fractional time derivative of a Caputo type is applied to model anomalous infiltration phenomena. Fractional horizontal infiltration phenomena are studied, and the fractional Boltzmann transform is defined. To study fractional vertical infiltration phenomena, the asymptotic behavior is described for short and long times considering an arbitrary diffusivity and hydraulic conductivity. Finally, considering a constant flux-dependent relation and a relation between diffusivity and hydraulic conductivity, a fractional cumulative infiltration model applicable to various types of soil is built; its solution is expressed as a power series in t ν / 2 , where ν ∈ ( 0 , 2 ) is the order of the fractional derivative. The results show the effect of superdiffusive and subdiffusive flows in different types of soil.

Suggested Citation

  • Carlos Fuentes & Fernando Alcántara-López & Antonio Quevedo & Carlos Chávez, 2021. "Fractional Vertical Infiltration," Mathematics, MDPI, vol. 9(4), pages 1-14, February.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:4:p:383-:d:499472
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    References listed on IDEAS

    as
    1. Gorenflo, Rudolf & Mainardi, Francesco & Vivoli, Alessandro, 2007. "Continuous-time random walk and parametric subordination in fractional diffusion," Chaos, Solitons & Fractals, Elsevier, vol. 34(1), pages 87-103.
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