Numerical Method for a Filtration Model Involving a Nonlinear Partial Integro-Differential Equation
Baigereyev D. Omariyeva D. Temirbekov N. Yergaliyev Y. Boranbek K.
April-2 2022MDPI
Mathematics
2022#10Issue 8
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 integrodifferential 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 secondorder 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 mediumIn 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.
finite element method , nonequilibrium fluid flows in porous media , nonlinear integro-differential equation
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Department of Mathematics, High School of Information Technology and Natural Sciences, Amanzholov University, Ust-Kamenogorsk, 070002, Kazakhstan
Department of Engineering Mathematics, Faculty of Basic Engineering Training, East Kazakhstan Technical University, Ust-Kamenogorsk, 070003, Kazakhstan
National Engineering Academy of the Republic of Kazakhstan, Almaty, 050010, Kazakhstan
Department of Mathematics
Department of Engineering Mathematics
National Engineering Academy of the Republic of Kazakhstan
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