A Priori Estimates for the Solution of an Initial Boundary Value Problem of Fluid Flow through Fractured Porous Media
Alimbekova N. Berdyshev A. Baigereyev D.
August 2022MDPI
Axioms
2022#11Issue 8
The paper studies a model of fluid flow in a fractured porous medium in which fractures are distributed uniformly over the volume. This model includes a nonlinear equation containing several terms with fractional derivatives in the sense of Caputo of order belonging to the interval (Formula presented.). The relevance of studying this problem is determined by its practical significance in the oil industry, since most of the world’s oil reserves are in these types of reservoirs. The uniqueness of the solution to the problem in a differential form and its dependence on the initial data and the right-hand side of the equation is proved. A numerical method is proposed based on the use of the finite difference approximation for integer and fractional time derivatives and the finite element method in the spatial direction. A change of variables is introduced to reduce the order of the fractional derivatives. Furthermore, the fractional derivative is approximated by using the L1-method. The stability and convergence of the proposed numerical method are rigorously proved. The theoretical order of convergence is confirmed by the results of numerical tests for a problem of fluid flow in fractured porous media with a known exact solution.
Caputo fractional derivative , convergence , finite element method , fluid flow in fractured porous medium , stability
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Department of Mathematics, High School of Information Technology and Natural Sciences, 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
Department of Mathematics
Department of Mathematics and Mathematical Modeling
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