A locally conservative Eulerian-Lagrangian numerical method and its application to nonlinear transport in porous media

Computational Geosciences

Volumn 4, Issue 1, 2000, Pages 1-40

  Douglas Jr , Jim ^a   Pereira, Felipe ^b,c   Yeh, Li Ming ^d  

^a PURDUE UNIVERSITY   (United States)

^b LABORATÓRIO NACIONAL DE COMPUTAÇÃO CIENTÍFICA   (Brazil)

^c STATE UNIVERSITY OF RIO DE JANEIRO   (Brazil)

^d NATIONAL CHIAO TUNG UNIVERSITY   (Taiwan)

Author keywords

Miscible flow; Modified method of characteristics; Transport dominated diffusion processes; Two phase flow; Waterflooding

Indexed keywords

EULERIAN-LAGRANGIAN ALGORITHM; NON-LINEAR SYSTEM; POROUS MEDIUM; TWO-PHASE FLOW;

EID: 0034048046     PISSN: 14200597     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1011551614492     Document Type: Article

References (44)

1. C. Almeida, J. Douglas, Jr. and F. Pereira, From nonconservative to locally conservative Eulerian-Lagrangian methods for miscible displacement in heterogeneous formations, to appear.
2. S.N. Antontsev, On the solvability of boundary value problems for degenerate two-phase porous flow equations, Dinamika Splošnoǐ Sredy 10 (1972) 28-53 (in Russian).
3. T. Arbogast, A. Chilikapati and M.F. Wheeler, A characteristics-mixed finite element for contaminant transport and miscible displacement, in: Computational Methods in Water Resources IX, Vol. 1: Numerical Methods in Water Resources, eds. T.F. Russell et al. (Elsevier Applied Science, London, 1992), pp. 77-84.
4. T. Arbogast and M.F. Wheeler, A characteristics-mixed finite element method for advection-dominated transport problems, SIAM J. Numer. Anal. 32 (1995) 404-424.
5. D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modél. Math. Anal. Numér. 19 (1985) 7-32.
6. J.B. Bell, C.N. Dawson and G.B. Shubin, An unsplit high-order Godunov scheme for scalar conservation laws in two dimensions, J. Comput. Phys. 75 (1988) 1-24.
7. F. Brezzi, J. Douglas, Jr., R. Durán and M. Fortin, Mixed finite elements for second order elliptic problems in three variables, Numer. Math. 51 (1987) 237-250.
8. F. Brezzi, J. Douglas, Jr., M. Fortin and L.D. Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modél. Math. Anal. Numér. 21 (1987) 581-604.
9. F. Brezzi, J. Douglas, Jr. and L.D. Marini, Two families of mixed finite elements for second order elliptic problems, Numer. Math. 47 (1985) 217-235.
10. M.A. Celia, T.F. Russell, I. Herrera and R.E. Ewing, An Euler-Lagrangian localized adjoint method for the advection-diffusion equations, Adv. Water Resources 13 (1990) 187-206.
11. G. Chavent, A new formulation of diphasic incompressible flows in porous media, in: Applications of Methods of Functional Analysis to Problems in Mechanics, Lecture Notes in Mathematics, Vol. 503, eds. P. Germain and B. Nayroles (Springer, Berlin/New York, 1976) pp. 258-270.
12. G. Chavent and J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation (North-Holland, Amsterdam, 1986).
13. Z. Chen and J. Douglas, Jr., Prismatic mixed finite elements for second order elliptic problems, Calcolo 26 (1989) 135-148.
14. A. Chilakapati, A characteristic-conservative model for Darcian advection, Adv. Water Resources 22 (1999) 597-609.
15. 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 (1995) 223-250.
16. J. Douglas, Jr., Simulation of miscible displacement in porous media by a modified method of characteristics procedure, in: Numerical Analysis, Lecture Notes in Mathematics, Vol. 912 (Springer, Berlin, 1982).
17. J. Douglas, Jr., Finite difference methods for two-phase incompressible flow in porous media, SIAM J. Numer. Anal. 20 (1983) 681-696.
18. J. Douglas, Jr., Numerical methods for the flow of miscible fluids in porous media, in: Numerical Methods in Coupled Systems, eds. R.W. Lewis, P. Bettess and E. Hinton (Wiley, London, 1984) pp. 405-439.
19. J. Douglas, Jr., Superconvergence in the pressure in the simulation of miscible displacement, SIAM J. Numer. Anal. 22 (1985) 962-969.
20. J. Douglas, Jr., R.E. Ewing and M.F. Wheeler, The approximation of the pressure by a mixed method in the simulation of miscible displacement, RAIRO Anal. Numér. 17 (1983) 17-33.
21. J. Douglas, Jr., R.E. Ewing and M.F. Wheeler, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, RAIRO Anal. Numér. 17 (1983) 249-265.
22. J. Douglas, Jr., F. Furtado and F. Pereira, The statistical behavior of instabilities in immiscible displacement subject to fractal geology, in: Mathematical Modelling of Flow Through Porous Media, eds. A.P. Bourgeat, C. Carasso, S. Luckhaus and A. Mikelić (World Scientific, Singapore, 1995) pp. 115-137.
23. J. Douglas, Jr., F. Furtado and F. Pereira, On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs, Computational Geosciences 1 (1997) 155-190.
24. J. Douglas, Jr., C.S. Huang and F. Pereira, The modified method of characteristics with adjusted advection for an immiscible displacement problem, in: Advances in Computational Mathematics, Lecture Notes in Pure and Applied Mathematics, Vol. 202, eds. Z. Chen, Y. Li, C.A. Micchelli and Y. Xu (Marcel Dekker, New York, 1999) pp. 53-73.
25. J. Douglas, Jr., C.S. Huang and F. Pereira, The modified method of characteristics with adjusted advection, Numerische Mathematik 83 (1999) 353-369.
26. J. Douglas, Jr., I. Martínez Gamba and M.C. Jorge Squeff, Simulation of the transient behavior of a one-dimensional semiconductor device, Mat. Apl. Comput. 5 (1986) 103-122.
27. J. Douglas, Jr., F. Pereira and L.-M. Yeh, A parallelizable method for two-phase flows in naturally-fractured reservoirs, Computational Geosciences 1 (1997) 333-368.
28. J. Douglas, Jr. and T.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 (1982) 871-885.
29. J. Douglas, Jr. and Y. Yuan, Numerical simulation of immiscible flow in porous media based on combining the method of characteristics with mixed finite element procedures, in: Numerical Simulation in Oil Recovery, The IMA Volumes in Mathematics and its Applications, Vol. 11, ed. M.F. Wheeler (Springer, Berlin, 1988) pp. 119-131.
30. R.E. Ewing, T.F. Russell and M.F. Wheeler, Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics, Comput. Math. Appl. Mech. Engrg. 47 (1984) 73-92.
31. F. Furtado and F. Pereira, Scaling analysis for two-phase, immiscible flow in heterogeneous media, Comput. Appl. Math. 17 (1998) 233-262.
32. F. Furtado and F. Pereira, Mixing regimes and the scale up problem for multiphase flow, in: Com7 putational Methods in Water Resources XII, Vol. 2 (1998) pp. 83-90.
33. F. Furtado and F. Pereira, Fluid mixing regimes for two-phase flow in heterogeneous porous media, submitted.
34. J. Glimm, B. Lindquist, F. Pereira and R. Peierls, The fractal hypothesis and anomalous diffusion, Mat. Apl. Comput. 11 (1992) 189-207.
35. J. Glimm, B. Lindquist, F. Pereira and Q. Zhang, A theory of macrodispersion for the scale up problem, Transport in Porous Media 13 (1993) 97-122.
36. C.S. Huang, The modified method of characteristics with adjusted advection and an accelerated domain decomposition procedure, Ph.D. thesis (July 1998); currently available as Technical Report #318, Center for Applied Mathematics, Purdue University, and to appear in Computational Geosciences.
37. C. Johnson, Streamline diffusion methods for problems in fluid dynamics, in: Finite Elements in Fluids IV (Wiley, New York, 1993) pp. 97-122.
38. 3, Numer. Math. 35 (1980) 315-341.
39. D.W. Peaceman, Improved treatment of dispersion in numerical calculation of multidimensional miscible displacement, Soc. Petroleum Engrg. J. 6 (1966) 213-216.
40. P.-A. Raviart and J.M. Thomas, A mixed finite element method for second order elliptic problems, in: Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, Vol. 606, eds. I. Galligani and E. Magenes (Springer, Berlin, 1977) pp. 292-315.
41. T.F. Russell, An incompletely iterated characteristic finite element method for a miscible displacement problem, Ph.D. thesis, University of Chicago, Chicago (1980).
42. T.F. Russell, Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media, SIAM J. Numer. Anal. 22 (1985) 970-1013.
43. A.M. Spagnuolo, Approximation of nuclear contaminant transport through porous media, Ph.D. thesis (July 1998); currently available as Technical Report #319, Center for Applied Mathematics, Purdue University, and to appear in Computational Geosciences.
44. J.M. Thomas, Sur l'analyse numérique des methodes d'éléments finis hybrides et mixtes, Thèse, Université Pierre et Marie Curie, Paris (1977).