Numerical Simulation of Multiphase Multicomponent Flow in Porous Media: Efficiency Analysis of Newton-Based Method

Timur Imankulov,Beimbet Daribayev,Bazargul Matkerim,Danil Lebedev,Nurislam Kassymbek

doi:10.3390/fluids6100355

Timur Imankulov, Beimbet Daribayev + Show 3 more

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https://doi.org/10.3390/fluids6100355

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Abstract

Newton’s method has been widely used in simulation multiphase, multicomponent flow in porous media. In addition, to solve systems of linear equations in such problems, the generalized minimal residual method (GMRES) is often used. This paper analyzed the one-dimensional problem of multicomponent fluid flow in a porous medium and solved the system of the algebraic equation with the Newton-GMRES method. We calculated the linear equations with the GMRES, the GMRES with restarts after every m steps—GMRES (m) and preconditioned with Incomplete Lower-Upper factorization, where the factors L and U have the same sparsity pattern as the original matrix—the ILU(0)-GMRES algorithms, respectively, and compared the computation time and convergence. In the course of the research, the influence of the preconditioner and restarts of the GMRES (m) algorithm on the computation time was revealed; in particular, they were able to speed up the program.

Highlights

Computer Science Al-Farabi Kazakh National University, Almaty 050040, Kazakhstan; Abstract: Newton’s method has been widely used in simulation multiphase, multicomponent flow in porous media
Multicomponent flow in porous media, the phase properties can vary depending on the composition of phases, temperatures and pressure
We indicate the restart cycle number with a subscript: xi is the approximate solution after i cycles, or m × i total iterations, and ri is the corresponding residual. The shortcoming of this method is that it reduces the robustness of the original generalized minimal residual method (GMRES) algorithm and cannot guarantee the convergence of the algorithm

Summary

Introduction

The modeling of hydrodynamic processes that take place in oil reservoirs is one of the difficult problems of fluid mechanics. The commonly used restarted GMRES, denoted by GMRES(m) [19], performs m iterations of GMRES, and the resulting approximate solution is used as the initial guess to start another set of m iterations This process repeats until the residual norm is small enough. We indicate the restart cycle number with a subscript: xi is the approximate solution after i cycles, or m × i total iterations, and ri is the corresponding residual The shortcoming of this method is that it reduces the robustness of the original GMRES algorithm and cannot guarantee the convergence of the algorithm. This article considers the mathematical model describing the process of multiphase, multicomponent fluid flow linearized using Newton’s method in a simulation, and we solve the system of the algebraic equation with one of the Krylov subspace methods—the. We solve and analyze the problem with the pure GMRES method, restarted GMRES(m) and preconditioned ILU(0)-GMRES method

Mathematical Modeling

Numerical Method

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Results of and Discussion

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