Central schemes for porous media flows

Computational and Applied Mathematics

Volumn 28, Issue 1, 2009, Pages 87-110

  Abreu, E ^a   Pereira, F ^b   Ribeiro, S ^c  

^a IMPA   (Brazil)

^b UNIVERSITY OF WYOMING   (United States)

^c FLUMINENSE FEDERAL UNIVERSITY   (Brazil)

Author keywords

Central differencing; Hyperbolic conservation laws; Two phase flows

Indexed keywords

COMPUTATIONAL FLUID DYNAMICS; NUMERICAL METHODS; PETROLEUM RESERVOIR ENGINEERING; PETROLEUM RESERVOIRS; PHYSICAL PROPERTIES; POROUS MATERIALS; VELOCITY;

CENTRAL DIFFERENCING; CENTRAL DIFFERENCING SCHEMES; CENTRAL SCHEME; CONTROL VOLUMES; HETEROGENEOUS POROUS MEDIA; HYPERBOLIC CONSERVATION LAWS; NON-UNIFORM CONTROL; POROUS MEDIUM FLOW; SPEED OF PROPAGATION; TWO PHASES FLOW;

TWO PHASE FLOW;

EID: 69049103435     PISSN: 22383603     EISSN: 18070302     Source Type: Journal    
DOI: 10.1590/s1807-03022009000100005     Document Type: Article

References (36)

1. A.F.D. Loula, A.L.G.A. Coutinho, E.L.M. Garcia and J.L.D. Alves, Solution of miscible and immiscible flows employing element-by-element iterative strategies. 3rd SPE Latin American/ Caribean Engin. Conference, (1994), 431-444.
2. A. Kurganov and E. Tadmor, New high-resolution semi-discrete central scheme for Hamilton-Jacobi Equations. Journal of Computational Physics, 160 (2000), 720-742.
3. A. Kurganov and E. Tadmor, New high-resolution central schemes for nonlinear conservation laws and convection-diffusion equations. Journal of Computational Physics, 160 (2000), 241-282.
4. A.L.G.A. Coutinho and I.D Parsons, Finite element multigrid methods for two-phase immiscible flow in heterogeneous media. Communications in Numerical Methods in Engineering, 15 (1999), 1-7.
5. A.S. Francisco, F. Pereira, H.P. Amaral Souto and J. Aquino, A two-stage operator splitting algorithm for the numerical simulation of contaminant transport in porous media. International Journal of Computational Science, 2, 3 (2008) 422-436.
6. B. Lindquist, F. Pereira, J. Glimm and Q. Zhang, A theory of macrodispersion for the scale up problem. Transport in Porous Media, 13 (1993), 97-122.
7. B. Lindquist, J. Glimm, L. Padmanhaban and O. McBryan, A front tracking reservoir simulator: 5-spot validation studies and the water coning problem. Frontiers in Applied Mathematics, H.T. Banks, 1 (1983).
8. B. Van Leer, Towards the ultimate conservative difference scheme. v. a second order sequel to Godunov's method. Journal of Computational Physics, 32 (1979), 101-136.
9. C.C. Wu, E. Tadmor and J. Balbas, Non-oscillatory central schemes for one- and two-dimensional MHD Equations. Journal of Computational Physics, 201 (2004), 261-285.
10. C.N. Dawson, G.B. Shubin and J.B. Bell, An unsplit high-order Godunov scheme for scalar conservation laws in two dimensions. Journal of Computational Physics, 75 (1988), 1-24.
11. C.T. Lin and E. Tadmor, High-resolution non-oscillatory central scheme for Hamilton-Jacobi Equations. SIAM Journal on Scientific Computations, 21 (2000), 2163-2186.
12. C.W. Shu, Total variation diminishing time discretizations. SISSC, 6 (1988), 1073.
13. C.W. Shu and S. Osher, Efficient implementation of essentially non-oscillatory shock capturing schemes. Journal of Computational Physics, 77 (1988), 439.
14. E. Abreu, D. Marchesin, F. Furtado and F. Pereira, Transitional Waves in Three-Phase Flows in Heterogeneous Formations. Computational Methods for Water Resources, I.5 Unsaturated and Multiphase Flow, Edited by C.T. Miller, M.W. Farthing, W.G. Gray and G.F. Pinder, Series: Developments in Water Science, ELSEVIER, I (2004), 609-620.
15. D.W. Peaceman, Fundamentals of Numerical Reservoir Simulation. Elsevier (1977).
16. E. Abreu and F. Pereira, Desenvolvimento de Novas estratégias Para a Recuperação Avançada de Hidrocarbonetos em Reservatórios de Petróleo. Boletim Técnico da Produção de Petróleo (PETROBRAS-CENPES), 2 (2007), 83-106.
17. E. Abreu, F. Furtado and F. Pereira, Three-phase immiscible displacement in heterogeneous petroleum reservoirs. Mathematics and computers in simulation, 73 (2006), 2-20.
18. E. Abreu, J. Douglas, F. Furtado and F. Pereira, Operator splitting based on physics for flow in porous media. International Journal of Computational Science, 2 (2008), 315-335.
19. E. Tadmor and G.-S. Jiang, Non-oscillatory central schemes for multidimensional hyperbolic conservation laws. SIAM Journal on Scientific Computing, 19 (1998), 1892-1917.
20. th Triennal Issue on Images in Plasma Science, 33(2) (2005), 470-471.
21. F. Furtado and F. Pereira, Crossover from nonlinearity controlled to heterogeneity controlled mixing in two-phase porous media flows. Comput. Geosci., 7(2) (2003), 115-135.
22. F. Furtado, F. Pereira and J. Douglas Jr., On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs. Comput. Geosci., 1(2) (1997), 155-190.
23. F. Pereira, J. Douglas Jr. and Li-Ming Yeh, A locally conservative Eulerian-Lagrangian method for flow in a porous medium of a mixture of two components having different densities. In: "Numerical treatment of multiphase flows in porous media (Beijing, 1999)", volume 552 of "Lecture Notes in Phys.", pages 138-155. Springer, Berlin (2000).
24. G. Chavent and J. Jaffré, Mathematical Methods and Finite Elements for Reservoir Simulation., volume 17. Studies in Mathematics and its Applications, North-Holland, Amsterdam, studies in mathematics and its applications edition (1986).
25. H.K. Dahle, M.S. Espedal, O.Sævareid and R.E. Ewing, Characteristic adaptive subdomain methods for reservoir flow problems. Numer. Methods Partial Differential Equations, 6(4) (1990), 279-309.
26. H.K. Dahle, R.E. Ewing and T.F. Russell, Eulerian-Lagrangian localized adjoint methods for a nonlinear advection-diffusion equation. Comput. Methods Appl. Mech. Engrg., 122(3-4) (1995), 223-250.
27. H. Nessyahu and E. Tadmor, Nonoscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys., 87(2) (1990), 408-463.
28. H. Wang and R.E. Ewing, A summary of numerical methods for time-dependent advection-dominated partial differential equations. J. Comput. Appl. Math., 128(1-2) (2001), 423-445. Numerical analysis 2000, Vol. VII, Partial differential equations.
29. J. Douglas Jr. and Thomas F. Russell, Numerical methods for convection-dominated diffusion problems based on combining the method of characteristics with finite element or finite difference procedures. SIAM J. Numer. Anal., 19(5) (1982), 871-885.
30. nd order elliptic problems. In: "Mathematical aspects of finite element methods (Proc. Conf., Consiglio Naz. delle Ricerche (C.N.R.), Rome, 1975)", pages 292-315. Lecture Notes in Math., Vol. 606. Springer, Berlin (1977).
31. L.J. Durlofsky, A triangle based mixed finite element-finete volume technique for modeling two phase flow through porous media. Journal of Computational Physics, 105 (1993), 252-266.
32. M.J. Blunt, M.R. Thiele and R.P. Batycky, A 3d field scale streamline-based reservoir simulator. SPE Reservoir Engineering, November (1997), 246-254.
33. P.D. Lax, Weak solutions of non-linear hyperbolic equations and their numerical computation. Comm. Pure Appl. Math., 7 (1954), 159-193.
34. R.P. Batycky, A Three-Dimensional Two-Phase Field Scale Streamline Simulator. PhD thesis, Stanford University, Dept. of Petroleum Engineering, Stanford, CA (1997).
35. S.K. Godunov, A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics. Mat. Sb. (N.S.), 47(89) (1959), 271-306.
36. V.V. Rusanov, The calculation of the interaction of non-stationary shock waves with barriers. Ž. Vy̌cisl. Mat. i Mat. Fiz., 1 (1961), 267-279.