Rock Flow Simulation by High-Order Quasi-Characteristics Scheme
https://doi.org/10.15514/ISPRAS-2018-30(5)-12
Abstract
A pure second-order scheme of quasi-characteristics based on a pyramidal stencil is applied to the numerical modelling of non-stationary two-phase flows through porous media with the essentially heterogeneous properties. In contrast to well-known other high-resolution schemes with monotone properties, this scheme preserves a second-order approximation in regions, where discontinuities of solutions arise, as well as monotone properties of numerical solutions in those regions despite of well-known Godunov theorem. It is possible because the scheme under consideration is defined on a non-fixed stencil and is a combination of two high-order approximation scheme solutions with different dispersion properties. A special criterion according to which, one or another admissible solution is chosen, plays a key role in this scheme. A simple criterion with local character suitable for parallel computations is proposed. Some numerical results showing the efficiency of present approach in computations of two-phase flows through porous media with strongly discontinuous penetration coefficients are presented.
References
1. B. Engquist, B. Sjogreen. High-Order Shock Capturing Methods. Computational Fluid Dynamics Review, John Willey and Sons, 1995, pp. 210-233.
2. E. Godlewsky, P.A. Raviart. Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, 1996, 524 p.
3. K.W. Morton. Numerical Solution of Convection-Diffusion Problems. Chapman and Hall, 1996, 384 p.
4. S.B. Hazra, P. Niyogi, S.K. Chakrabartty. Study in non-oscillatory schemes for shock computation using Euler equations. Computational Fluid Dynamics Journal, vol. 7, 1998, pp. 163-176.
5. Sh. Wo, B.M. Chen, J. Wang. A High-Order Godunov Method for One-Dimensional Convection-Diffusion-Reaction Problems. Numerical Methods for Partial Differential Equations, vol. 16, 2000, pp. 495-512.
6. M.P. Levin. A difference scheme of quasi-characteristics and its use to calculate supersonic gas flows. Computational Mathematics and Mathematical Physics, vol. 33, no. , 1993, pp. 113-121.
7. M.Yu. Zheltov, M.P. Levin. Application of the quasi-characteristics scheme for the two-phase flows through porous media. Computational Fluid Dynamics Journal, vol. 2, 1993, pp. 363-370.
8. M.P. Levin. Computation of 3-D supersonic flow with heat supply by explicit quasi-characteristics scheme. Computational Fluid Dynamics Journal, vol. 4, 1995, pp. 311-322.
9. M.P. Levin, L.V. Sidorov. Hybrid modification of the scheme of the method of quasi-characteristics on a pyramidal pattern. Computational Mathematics and Mathematical Physics, vol. 35, no. 2, 1995, pp. 253- 258.
10. M.P. Levin. Quasi-characteristics numerical schemes. In Hyperbolic Problems: Theory, Numerics, Application, Springer, 1999, pp. 619-628.
11. A.I. Ibragimov, M.P. Levin, L.V. Sidorov. Numerical investigation of two-phase fluid afflux to horizontal well by quasi-characteristics scheme. Computational Fluid Dynamics Journal, vol. 8, 2000, pp. 556-560.
12. V.M. Borisov, Yu.V. Kurilenko, I.E. Mikhailov, E.V. Nikolaevskaya. A method of characteristics for calculation of vortex spatial supersonic stationary flows. Computing Centre of USSR Academy of Sciences, Moscow, 1988 (in Russian).
13. E.V. Nikolaevskaya. One class of running finite difference schemes. Computing Centre of USSR Academy of Sciences, Moscow, 1987 (in Russian).
14. D.Y. Kwak, M.P. Levin. High-Resolution Monotone Schemes Based on Qasi-Characteristics Technique. Numerical Methods for Partial Differential Equations, vol. 17, 2001, 262-276
15. S.-H. Chou, Q. Li. Characteristics-Galerkin and mixed finite element approximation of contamination by compressible nuclear waste-disposal in porous media. Numerical Methods for Partial Differential Equations, vol. 12, 1996, pp. 315-332.
16. H. Wang, M. Al-Lawatia, A.S. Telyakovskiy. Runge-Kutta characteristic methods for first-order linear hyperbolic equations. Numerical Methods for Partial Differential Equations, vol. 13, 1997, pp. 617-661.
17. H. Wang, M. Al-Lawatia, R.C. Sharpley. A characteristic domain decomposition and space-time local refinement method for first-order linear hyperbolic equations with interfaces. Numerical Methods for Partial Differential Equations, vol. 15, 1999, pp. 1-28.
18. M. Marion, A. Mollard. A multilevel characteristics method for periodic convection-dominated diffusion problems. Numerical Methods for Partial Differential Equations, vol. 16, 2000, pp. 107-132.
19. C.N. Dawson, M.L. Martinez-Canales. A characteristic-Galerkin approximation to a system of shallow water equations. Numerische Mathematik, vol. 86, Issue 2, 2000, pp. 239-256.
20. Yu.P. Zheltov. Mechanics of Oil and Gas Bearing Formation. Nedra, Moscow, 1975 (in Russian).
Review
For citations:
Levin M.P. Rock Flow Simulation by High-Order Quasi-Characteristics Scheme. Proceedings of the Institute for System Programming of the RAS (Proceedings of ISP RAS). 2018;30(5):197-212. (In Russ.) https://doi.org/10.15514/ISPRAS-2018-30(5)-12
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