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

Dossan Baigereyev, Nurlana Alimbekova, Abdumauvlen Berdyshev and Muratkan Madiyarov
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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
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

Mathematics, 2021, vol. 9, issue 18, 1-25

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.

Keywords: fractured porous media; fluid flow; finite element method; convergence; Caputo fractional derivative (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (3)

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