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A parallelizable method for two‐phase flows in naturally‐fractured reservoirs

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Abstract

A parallelizable, semi‐implicit numerical method is proposed for the study of naturally‐fractured reservoir systems. It has proved to be computationally efficient in producing accurate numerical solutions for the dual‐porosity model for immiscible, two‐phase flow in such reservoirs. The method combines hybridized mixed finite elements, a new version of the modified method of characteristics, a sophisticated operator‐splitting procedure for separating the pressure calculation in the fractures from that of the saturation, another operator splitting to handle the interaction of the matrix blocks and the fractures, and domain decomposition iterative procedures for both the pressure and the saturation. It permits moderately long time steps for the pressure and the saturation in the fractures and matrix blocks by using short, inexpensive microsteps to treat the transport portion of the saturation equation in the fractures. This paper is devoted to the formulation of the method and a discussion of numerical results for five‐spot and vertical cross‐section examples.

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References

  1. S.N. Antontsev, On the solvability of boundary value problems for degenerate two-phase porous flow equations, Dinamika Sploshn. Sredy 10 (1972) 28–53. In Russian.

    Google Scholar 

  2. T. Arbogast, A simplified dual-porosity model for two-phase flow, Computational Methods in Water Resources 9 (1992) 419–426.

    Google Scholar 

  3. D.N. Arnold and F. Brezzi, Mixed and nonconforming finite element methods: implementation, postprocessing and error estimates, RAIRO Modèl. Math. Anal. Numèr. 19 (1985) 7–32.

    Google Scholar 

  4. 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.

    Google Scholar 

  5. F. Brezzi, J. Douglas, Jr., M. Fortin and L.D. Marini, Efficient rectangular mixed finite elements in two and three space variables, RAIRO Modèl. Math. Anal. Numèr. 21 (1987) 581–604.

    Google Scholar 

  6. 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.

    Google Scholar 

  7. 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 503, eds. P. Germain and B. Nayroles (Springer, Berlin, 1976) pp. 258–270.

    Google Scholar 

  8. G. Chavent and J. Jaffré, Mathematical Models and Finite Elements for Reservoir Simulation(North-Holland, Amsterdam, 1986).

  9. Z. Chen and J. Douglas, Jr., Prismatic mixed finite elements for second order elliptic problems, Calcolo 26 (1989) 135–148.

    Google Scholar 

  10. B. Després, Méthodes de décomposition de domaines pour les problèmes de propagation dóndes en régime harmonique, Ph.D. thesis, Université Paris IX Dauphine, UER Mathématiques de la Décision (1991).

  11. B. Després, Domain decomposition method and the Helmholz problem, in: Mathematical and Numerical Aspects of Wave Propagation Phenomena, eds. G. Cohen, L. Halpern and P. Joly (SIAM, Philadelphia, PA, 1991) pp. 44–52.

    Google Scholar 

  12. J. Douglas, Jr., Simulation of miscible displacement in porous media by a modified method of characteristics procedure, in: Numerical Analysis, Lecture Notes in Mathematics 912 (Springer, Berlin, 1982).

    Google Scholar 

  13. 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.

    Google Scholar 

  14. J. Douglas, Jr., Finite difference methods for two-phase incompressible flow in porous media, SIAM J. Numer. Anal. 20 (1983) 681–696.

    Google Scholar 

  15. J. Douglas, Jr. and T. Arbogast, Dual porosity models for flow in naturally fractured reservoirs, in: Dynamics of Fluids in Hierarchical Porous Formations, ed. J.H. Cushman (Academic Press, London, 1990) pp. 177–221.

    Google Scholar 

  16. J. Douglas, Jr., T. Arbogast, P.J. Paes Leme, J.L. Hensley and N.P. Nunes, Immiscible displacement in vertically fractured reservoirs, Transport in Porous Media 12 (1993) 73–106.

    Google Scholar 

  17. 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.

    Google Scholar 

  18. 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.

    Google Scholar 

  19. J. Douglas, Jr., F. Furtado and F. Pereira, On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs, Computational Geosciences 1 (1997) 155–190.

    Google Scholar 

  20. J. Douglas, Jr., J.L. Hensley and T. Arbogast, A dual-porosity model for waterflooding in naturally fractured reservoirs, Comput. Methods Appl. Mech. Engrg. 87 (1991) 157–174.

    Google Scholar 

  21. J. Douglas, Jr., J.L. Hensley and P.J. Paes Leme, A study of the effect of inhomogeneities on immiscible flow in naturally fractured reservoirs, in: Flow in Porous Media, International Series of Numerical Mathematics 114, eds. J. Douglas, Jr. and U. Hornung (Birkhäuser, Basel, 1993) pp. 59–74.

    Google Scholar 

  22. 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.

    Google Scholar 

  23. J. Douglas, Jr., P.J. Paes Leme, F. Pereira and L.M. Yeh, A massively parallel iterative numerical algorithm for immiscible flow in naturally fractured reservoirs, in: Flow in Porous Media, International Series of Numerical Mathematics 114, eds. J. Douglas, Jr. and U. Hornung (Birkhäuser, Basel, 1993) pp. 75–94.

    Google Scholar 

  24. J. Douglas, Jr., P.J. Paes Leme, J.E. Roberts and J. Wang, A parallel iterative procedure applicable to the approximate solution of second order partial differential equations by mixed finite element methods, Numer. Math. 65 (1993) 95–108.

    Google Scholar 

  25. J. Douglas, Jr., F. Pereira, L.M. Yeh and P.J. Paes Leme, Domain decomposition for immiscible displacement in single porosity systems, in: Finite Element Methods: Fifty Years of the Courant Element, Lecture Notes in Pure and Applied Mathematics 164, eds. M. Křížek, P. Neittaanmäki and R. Stenberg (Marcel Dekker, New York, 1994) pp. 191–199.

    Google Scholar 

  26. J. Douglas, Jr., F. Pereira and L.M. Yeh, A parallelizable characteristic scheme for two phase immiscible flow in porous media I: Single porosity models, Comput. Appl. Math. 14 (1995) 73–96.

    Google Scholar 

  27. 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.

    Google Scholar 

  28. 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 11, ed. M.F. Wheeler (Springer, Berlin, 1988) pp. 119–131.

    Google Scholar 

  29. 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. Methods Appl. Mech. Engrg. 47 (1984) 73–92.

    Google Scholar 

  30. B.X. Fraeijs de Veubeke, Stress function approach, in: International Congress on the Finite Element Method in Structural Mechanics, Bournemouth (1975).

  31. J. Glimm, B. Lindquist, F. Pereira and R. Peierls, The fractal hypothesis and anomalous diffusion, Mat. Appl. Comput. 11 (1992) 189–207.

    Google Scholar 

  32. 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.

    Google Scholar 

  33. P.L. Lions, On the Schwarz alternating method III: a variant for nonoverlapping subdomains, in: Domain Decomposition Methods for Partial Differential Equations, eds. T.F. Chan, R. Glowinski, J. Periaux and O.B. Widlund (SIAM, Philadelphia, PA, 1990) pp. 202–223.

    Google Scholar 

  34. M. Murad and J. Cushman, A multiscale theory of swelling porous media, II: Dual porosity models for consolidation of clays incorporating physiochemical effects, Transport in Porous Media 28 (1997) 69–108.

    Google Scholar 

  35. J.C. Nedelec, Mixed finite elements in R3, Numer. Math. 35 (1980) 315–341.

    Google Scholar 

  36. 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 606, eds. I. Galligani and E. Magenes (Springer, Berlin, 1977) pp. 292–315.

    Google Scholar 

  37. T.F. Russell, An incompletely iterated characteristic finite element method for a miscible displacement problem, Ph.D. thesis, University of Chicago, Chicago (1980).

    Google Scholar 

  38. 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.

    Google Scholar 

  39. J.M. Thomas, Sur l'analyse numérique des méthodes d'éléments finis hybrides et mixtes, Ph.D. thesis, Universite Pierre et Marie Curie, Paris (1977).

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Jim Douglas Jr.

  2. Laboratório Nacional de Computação Científica, Rio de Janeiro, Brazil

    Felipe Pereira

  3. Department of Applied Mathematics, National Chiao‐Tung University, Hsinchu, Taiwan, R.O.C

    Li‐Ming Yeh

Authors
  1. Jim Douglas Jr.
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  2. Felipe Pereira
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  3. Li‐Ming Yeh
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Douglas, J., Pereira, F. & Yeh, L. A parallelizable method for two‐phase flows in naturally‐fractured reservoirs. Computational Geosciences 1, 333–368 (1997). https://doi.org/10.1023/A:1011581631813

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