# On Grid Method for Entropy Solution of the Problem of Simultaneous Motion of Two-Phase Fluid in a Natural Reservoir

## Keywords:

Buckley-Leverettâ€™s problem, Non-convex state function, Entropy solution, Numerical solution in a class of discontinuous functions## Abstract

In this paper a method for obtaining an exact and numerical solution of the initial and initial-boundary value problems for a first order partial differential equation with a non-convex state function is suggested, which models the macroscopic motion of the two- phase fluids in a porous medium. For this aim, an auxiliary problem is introduced such a way that it has some advantages over the main problem, and it is equivalent to the main problem in a definite sense. By use of this auxiliary problem it is proved that the exact and numerical solutions of the investigated problems satisfy the entropy condition in the sense of Oleinik. To make use of this auxiliary problem, a method for fixing the location of shock which appears in the solution of the main problem and its evolution in time is offered. The proposed auxiliary problem permits us also to prove convergence in the meaning of a numerical solution to the exact solution of the main problem. Besides, the auxiliary problem permits us to write the higher sensitive and the higher order numerical scheme with respect to time variable whose solution expresses all the physical properties of the problem accurately. Using the suggested algorithms, two laboratory experiments were carried out.## References

Baikov, V.A., Ibragimov, N.H., Zheltova, I.S., Yakovlev, A.A., â€œConservation Laws for Two-Phase Filtration Modelsâ€, Commun. Nonlinear Sci. Numer. Simulat, 19 (2014), 383-389.

Buckley, S.E., Leverett, M.C., â€œMechanism of Fluid Displacement in Sandsâ€, Trans. AIME, 146 (1942), 107-116.

Collins, R.E., Fluids Flow in Porous Materials, Reinhold Publishing, New York, 1961.

Fritz, J., Partial Differential Equations, Springer-Verlag, New York, 1986.

Glimm, J., Singularities in Fluid Dynamics, Lecture Notes in Physics, 153 (1982), 86-97.

Godunov, S.K., â€œA Difference Scheme for Numerical Computation of Discontinuous Solutions of Equations of Fluid Dynamicsâ€, Mat. Sbornik, 47(89) (1959), 271-306.

Goritskii, A.A., Krujkov, S.N., Chechkin, G.A., A First Order Quasi-Linear Equations with Partial Differential Derivatives, Pub. Moscow University, Moscow, 1997.

Lax, P.D., â€œThe Formation and Decay of Shock Wavesâ€, Amer. Math. Monthly, 79 (1972), 227-241.

Lax, P.D., â€œWeak Solutions of Nonlinear Hyperbolic Equations and Their Numerical Computationsâ€, Comm. of Pure and App. Math., 7 (1954), 159-193.

LeVeque, R.J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, London, 2002.

Muskat, M., The Flow of Homogeneous Fluids through Porous Medium, McGraw-Hill, New York, 1946.

Noh, W.F., Protter, M.N., â€œDifference Methods and the Equations of Hydrodynamicsâ€, Journal of Mathematics and Mechanics, 12(2) (1963), 149-193.

Oleinik, O.A., â€œDiscontinuous Solutions of Nonlinear Differential Equationsâ€, Usp. Math. Nauk, 12 (1957), 3-73.

Rasulov, M.A., â€œOn a Method of Solving the Cauchy Problem for a First Order Nonlinear Equation of Hyperbolic Type with a Smooth Initial Conditionâ€, Soviet Math. Dokl., 43(1) (1991), 150-153.

Rasulov, M.A., Ragimova, T.A., â€œA Numerical Method of the Solution of Nonlinear Equation of a Hyperbolic Type of the First Orderâ€, Differential Equations, 28(7) (1992), 2056-2063.

Richmyer, R.D., Morton, K.W., Difference Methods for Initial Value Problems, Wiley, New York, 1967.

Samarskii, A.A., Theory of Difference Schemes, Nauka, Moscow, 1977.

Savioli, G.B., â€œFernandes-Berdaguer, E.M., The Estimation of Oil Water Displacement Functionsâ€, Latin American Applied Research, 37 (2007), 187-194.

Sinsoysal, B., Rasulov, M., â€œEfficient Numerical Method of the 1D Motion of the Two-Phase Fluid Through Porous Medium in a Class of Discontinuous Functionsâ€, Lecture Notes in Comp. Science, 5434 (2009) 532-539.

Tikhonov, A.N., Samarskii, A.A., Equations of Mathematical Physics, Nauka, Moscow, 1977.

Toro, E.F., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag, Berlin, 1999.

Whitham, G.B., Linear and Nonlinear Waves, Wiley, New York, 1974.

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*Asian Journal of Applied Sciences*,

*4*(2). Retrieved from https://ajouronline.com/index.php/AJAS/article/view/3694

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