IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v12y2024i16p2519-d1456910.html
   My bibliography  Save this article

Finite Element Method for a Fractional-Order Filtration Equation with a Transient Filtration Law

Author

Listed:
  • Nurlana Alimbekova

    (Department of Mathematics, Higher School of IT and Natural Sciences, Sarsen Amanzholov East Kazakhstan University, 148 Shakarim Ave., Ust-Kamenogorsk 070002, Kazakhstan)

  • Abdumauvlen Berdyshev

    (Institute of Information and Computational Technologies, 28 Shevchenko Str., Almaty 050010, Kazakhstan)

  • Muratkan Madiyarov

    (Department of Mathematics, Higher School of IT and Natural Sciences, Sarsen Amanzholov East Kazakhstan University, 148 Shakarim Ave., Ust-Kamenogorsk 070002, Kazakhstan)

  • Yerlan Yergaliyev

    (Department of Mathematics, Higher School of IT and Natural Sciences, Sarsen Amanzholov East Kazakhstan University, 148 Shakarim Ave., Ust-Kamenogorsk 070002, Kazakhstan)

Abstract

In this article, a numerical method is proposed and investigated for an initial boundary value problem governed by a fractional differential generalization of the nonlinear transient filtration law which describes fluid motion in a porous medium. This type of equation is widely used to describe complex filtration processes such as fluid movement in horizontal wells in fractured geological formations. To construct the numerical method, a high-order approximation formula for the fractional derivative in the sense of Caputo is applied, and a combination of the finite difference method with the finite element method is used. The article proves the uniqueness and continuous dependence of the solution on the input data in differential form, as well as the stability and convergence of the proposed numerical scheme. The linearization of nonlinear terms is carried out by the Newton method, which allows for achieving high accuracy in solving complex problems. The research results are confirmed by a series of numerical tests that demonstrate the applicability of the developed method in real engineering problems. The practical significance of the presented approach lies in its ability to accurately and effectively model filtration processes in shale formations, which allows engineers and geologists to make more informed decisions when designing and operating oil fields.

Suggested Citation

  • Nurlana Alimbekova & Abdumauvlen Berdyshev & Muratkan Madiyarov & Yerlan Yergaliyev, 2024. "Finite Element Method for a Fractional-Order Filtration Equation with a Transient Filtration Law," Mathematics, MDPI, vol. 12(16), pages 1-20, August.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:16:p:2519-:d:1456910
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/12/16/2519/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/12/16/2519/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Kui Liu & Michal Fečkan & D. O’Regan & JinRong Wang, 2019. "Hyers–Ulam Stability and Existence of Solutions for Differential Equations with Caputo–Fabrizio Fractional Derivative," Mathematics, MDPI, vol. 7(4), pages 1-14, April.
    2. Abdumauvlen Berdyshev & Dossan Baigereyev & Kulzhamila Boranbek, 2023. "Numerical Method for Fractional-Order Generalization of the Stochastic Stokes–Darcy Model," Mathematics, MDPI, vol. 11(17), pages 1-27, September.
    3. Francesco Mainardi, 2018. "Fractional Calculus: Theory and Applications," Mathematics, MDPI, vol. 6(9), pages 1-4, August.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ana Laura Mendonça Almeida Magalhães & Pedro Paiva Brito & Geraldo Pedro da Silva Lamon & Pedro Américo Almeida Magalhães Júnior & Cristina Almeida Magalhães & Pedro Henrique Mendonça Almeida Magalhãe, 2024. "Numerical Resolution of Differential Equations Using the Finite Difference Method in the Real and Complex Domain," Mathematics, MDPI, vol. 12(12), pages 1-39, June.
    2. Abbaszadeh, Mostafa & Zaky, Mahmoud A. & Hendy, Ahmed S. & Dehghan, Mehdi, 2024. "Supervised learning and meshless methods for two-dimensional fractional PDEs on irregular domains," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 216(C), pages 77-103.
    3. Vasily E. Tarasov & Valentina V. Tarasova, 2019. "Dynamic Keynesian Model of Economic Growth with Memory and Lag," Mathematics, MDPI, vol. 7(2), pages 1-17, February.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:gam:jmathe:v:12:y:2024:i:16:p:2519-:d:1456910. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.