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Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation

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  • Abdon Atangana
  • P. D. Vermeulen

Abstract

The classical Darcy law is generalized by regarding the water flow as a function of a noninteger order derivative of the piezometric head. This generalized law and the law of conservation of mass are then used to derive a new equation for groundwater flow. Two methods including Frobenius and Adomian decomposition method are used to obtain an asymptotic analytical solution to the generalized groundwater flow equation. The solution obtained via Frobenius method is valid in the vicinity of the borehole. This solution is in perfect agreement with the data observed from the pumping test performed by the institute for groundwater study on one of their boreholes settled on the test site of the University of the Free State. The test consisted of the pumping of the borehole at the constant discharge rate and monitoring the piezometric head for 350 minutes. Numerical solutions obtained via Adomian method are compared with the Barker generalized radial flow model for which a fractal dimension for the flow is assumed. Proposition for uncertainties in groundwater studies was given.

Suggested Citation

  • Abdon Atangana & P. D. Vermeulen, 2014. "Analytical Solutions of a Space-Time Fractional Derivative of Groundwater Flow Equation," Abstract and Applied Analysis, Hindawi, vol. 2014, pages 1-11, January.
    • Handle: RePEc:hin:jnlaaa:381753
    DOI: 10.1155/2014/381753
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    Cited by:

    1. Sheikh, Nadeem Ahmad & Ali, Farhad & Khan, Ilyas & Gohar, Madeha, 2018. "A theoretical study on the performance of a solar collector using CeO2 and Al2O3 water based nanofluids with inclined plate: Atangana–Baleanu fractional model," Chaos, Solitons & Fractals, Elsevier, vol. 115(C), pages 135-142.
    2. Atangana, Abdon, 2018. "Non validity of index law in fractional calculus: A fractional differential operator with Markovian and non-Markovian properties," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 505(C), pages 688-706.
    3. Tassaddiq, Asifa & Khan, I. & Nisar, K.S., 2020. "Heat transfer analysis in sodium alginate based nanofluid using MoS2 nanoparticles: Atangana–Baleanu fractional model," Chaos, Solitons & Fractals, Elsevier, vol. 130(C).

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