Numerical Method for a Filtration Model Involving a Nonlinear Partial Integro-Differential Equation

Dossan Baigereyev, Dinara Omariyeva, Nurlan Temirbekov, Yerlan Yergaliyev and Kulzhamila Boranbek
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Dossan Baigereyev: Department of Mathematics, High School of Information Technology and Natural Sciences, Amanzholov University, Ust-Kamenogorsk 070002, Kazakhstan
Dinara Omariyeva: Department of Engineering Mathematics, Faculty of Basic Engineering Training, East Kazakhstan Technical University, Ust-Kamenogorsk 070003, Kazakhstan
Nurlan Temirbekov: National Engineering Academy of the Republic of Kazakhstan, Almaty 050010, Kazakhstan
Yerlan Yergaliyev: Department of Mathematics, High School of Information Technology and Natural Sciences, Amanzholov University, Ust-Kamenogorsk 070002, Kazakhstan
Kulzhamila Boranbek: Department of Engineering Mathematics, Faculty of Basic Engineering Training, East Kazakhstan Technical University, Ust-Kamenogorsk 070003, Kazakhstan

Mathematics, 2022, vol. 10, issue 8, 1-24

Abstract: In this paper, we propose an efficient numerical method for solving an initial boundary value problem for a coupled system of equations consisting of a nonlinear parabolic partial integro-differential equation and an elliptic equation with a nonlinear term. This problem has an important applied significance in petroleum engineering and finds application in modeling two-phase nonequilibrium fluid flows in a porous medium with a generalized nonequilibrium law. The construction of the numerical method is based on employing the finite element method in the spatial direction and the finite difference approximation to the time derivative. Newton’s method and the second-order approximation formula are applied for the treatment of nonlinear terms. The stability and convergence of the discrete scheme as well as the convergence of the iterative process is rigorously proven. Numerical tests are conducted to confirm the theoretical analysis. The constructed method is applied to study the two-phase nonequilibrium flow of an incompressible fluid in a porous medium. In addition, we present two examples of models allowing for prediction of the behavior of a fluid flow in a porous medium that are reduced to solving the nonlinear integro-differential equations studied in the paper.

Keywords: nonequilibrium fluid flows in porous media; nonlinear integro-differential equation; finite element method (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2022
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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