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Convergence Analysis of a Numerical Method for a Fractional Model of Fluid Flow in Fractured Porous Media

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  • Dossan Baigereyev

    (Department of Mathematics, Faculty of Natural Sciences and Technology, Amanzholov University, Ust-Kamenogorsk 070002, Kazakhstan)

  • Nurlana Alimbekova

    (Department of Mathematics, Faculty of Natural Sciences and Technology, Amanzholov University, Ust-Kamenogorsk 070002, Kazakhstan
    Department of Mathematics and Mathematical Modeling, Institute of Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical University, Almaty 050000, Kazakhstan)

  • Abdumauvlen Berdyshev

    (Department of Mathematics and Mathematical Modeling, Institute of Mathematics, Physics and Informatics, Abai Kazakh National Pedagogical University, Almaty 050000, Kazakhstan)

  • Muratkan Madiyarov

    (Department of Mathematics, Faculty of Natural Sciences and Technology, Amanzholov University, Ust-Kamenogorsk 070002, Kazakhstan)

Abstract

The present paper is devoted to the construction and study of numerical methods for solving an initial boundary value problem for a differential equation containing several terms with fractional time derivatives in the sense of Caputo. This equation is suitable for describing the process of fluid flow in fractured porous media under some physical assumptions, and has an important applied significance in petroleum engineering. Two different approaches to constructing numerical schemes depending on orders of the fractional derivatives are proposed. The semi-discrete and fully discrete numerical schemes for solving the problem are analyzed. The construction of a fully discrete scheme is based on applying the finite difference approximation to time derivatives and the finite element method in the spatial direction. The approximation of the fractional derivatives in the sense of Caputo is carried out using the L1-method. The convergence of both numerical schemes is rigorously proved. The results of numerical tests conducted for model problems are provided to confirm the theoretical analysis. In addition, the proposed computational method is applied to study the flow of oil in a fractured porous medium within the framework of the considered model. Based on the results of the numerical tests, it was concluded that the model reproduces the characteristic features of the fluid flow process in the medium under consideration.

Suggested Citation

  • Dossan Baigereyev & Nurlana Alimbekova & Abdumauvlen Berdyshev & Muratkan Madiyarov, 2021. "Convergence Analysis of a Numerical Method for a Fractional Model of Fluid Flow in Fractured Porous Media," Mathematics, MDPI, vol. 9(18), pages 1-25, September.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:18:p:2179-:d:630525
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    References listed on IDEAS

    as
    1. Wei, Qing & Zhou, Hongwei & Yang, Shuai, 2020. "Non-Darcy flow models in porous media via Atangana-Baleanu derivative," Chaos, Solitons & Fractals, Elsevier, vol. 141(C).
    2. Wenwen Zhong & Changpin Li & Jisheng Kou, 2013. "Numerical Fractional-Calculus Model for Two-Phase Flow in Fractured Media," Advances in Mathematical Physics, Hindawi, vol. 2013, pages 1-7, August.
    3. Hashan, Mahamudul & Jahan, Labiba Nusrat & Tareq-Uz-Zaman, & Imtiaz, Syed & Hossain, M. Enamul, 2020. "Modelling of fluid flow through porous media using memory approach: A review," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 177(C), pages 643-673.
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    Cited by:

    1. Shuai Yang & Qing Wei & Lu An, 2022. "Fractional Advection Diffusion Models for Radionuclide Migration in Multiple Barriers System of Deep Geological Repository," Mathematics, MDPI, vol. 10(14), pages 1-7, July.
    2. Dossan Baigereyev & Dinara Omariyeva & Nurlan Temirbekov & Yerlan Yergaliyev & Kulzhamila Boranbek, 2022. "Numerical Method for a Filtration Model Involving a Nonlinear Partial Integro-Differential Equation," Mathematics, MDPI, vol. 10(8), pages 1-24, April.
    3. Xiaoji Shang & Zhizhen Zhang & Zetian Zhang & J. G. Wang & Yuejin Zhou & Weihao Yang, 2022. "Fractal Analytical Solutions for Nonlinear Two-Phase Flow in Discontinuous Shale Gas Reservoir," Mathematics, MDPI, vol. 10(22), pages 1-14, November.

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