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

Nurlana Alimbekova (), Abdumauvlen Berdyshev, Muratkan Madiyarov and Yerlan Yergaliyev
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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

Mathematics, 2024, vol. 12, issue 16, 1-20

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.

Keywords: differential filtration equation with a transient filtration law; Caputo fractional derivative; finite element method; stability; convergence; Newton’s method; substep technique (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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