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Simulation of Single-Phase Multicomponent Flow Problems in Gas Reservoirs by Eulerian–Lagrangian Techniques

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Abstract

Over the past two decades most discussions of the simulation of miscible displacement in porous media were related to incompressible flow problems; recently, however, attention has shifted to compressible problems. The first goal of this paper is the derivation of the governing equations (mathematical models) for a hierarchy of miscible isothermal displacements in porous media, starting from a very general single-phase, multicomponent, compressible flow problem; these models are then compared with previously proposed models. Next, we formulate an extension of the modified method of characteristics with adjusted advection to treat the transport and dispersion of the components of the miscible fluid; the fluid displacement must be coupled in a two-stage operator-splitting procedure with a pressure equation to define the Darcy velocity field required for transport and dispersion, with the outer stage incorporating an implicit solution of the nonlinear parabolic pressure equation and an inner stage for transport and diffussion in which the mass fraction equations are solved sequentially by first applying a globally conservative Eulerian–Lagrangian scheme to solve for transport, followed by a standard implicit procedure for including the diffusive effects. The third objective is a careful investigation of the underlying physics in compressible displacements in porous media through several high resolution numerical experiments. We consider real binary gas mixtures, with realistic thermodynamic correlations, in homogeneous and heterogeneous formations.

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References

  • de Almeida, C. G.: 2000, Escoamentos miscíveis em formaç¸ Ões heterogêneas: novos métodos numéricos e modelagem estocástica, PhD Dissertation, Instituto de Matemática, Estatística e Computac¸ çao Científica, UNICAMP, Campinas, Brazil.

    Google Scholar 

  • Arbogast, T., Chilikapati, A. and Wheeler, M. F.: 1992, A characteristics-mixed finite element for contaminant transport and miscible displacement, in: T. F. Russel et al. (eds), Computational Methods in Water Resources IX, Vol. 1: Numerical Methods inWater Resources, Elsevier Applied Science, London, New York, pp. 77–84.

    Google Scholar 

  • Batycky, R. P., Blunt, M. J. and Thiele, M. R.: 1996a, A 3D field scale streamline simulator with gravity and changing well conditions, SPE 36726, Oct.

  • Bratvedt, F., Gimse, T. and Tegnander, C.: 1996b, Streamline computations for porous media flow including gravity, Transport in Porous Media 25, 63–78.

    Google Scholar 

  • Brezzi, F. and Fortin, M.: 1991, Mixed and Hybrid Finite Element Methods, Springer-Verlag, New York.

    Google Scholar 

  • Chavent, G. and Jaffré, J.: 1986, Mathematical Models and Finite Elements for Reservoir Simulation, North-Holland, Amsterdam.

  • Dafermos, C. M.: 1972, Polygonal approximations of solutions of the initial value problem for a conservation law, J. Math. Anal. Appl. 38, 33–41.

    Google Scholar 

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

    Google Scholar 

  • Douglas Jr., J.: 1983, Finite difference methods for two-phase incompressible ow in porous media, SIAM J. Numer. Anal. 20, 681–696.

    Google Scholar 

  • Douglas Jr., J.: 1984, Numerical methods for the flow of miscible fluids in porous media, in: R. W. Lewis, P. Bettess and E. Hinton (eds), Numerical Methods in Coupled Systems, Wiley, London, pp. 405–439.

    Google Scholar 

  • Douglas Jr., J., Hensley, J. L., Wei, Y., Yeh, L., Jaffré, J., Paes Leme, P. J., Ramakrishnam, T. S. and Wilkinson, D. J.: 1992, A derivation for Darcy's law for miscible fluids and a revised model for miscible displacement in porous media, in: T. F. Russell, R. E. Ewing, C. A. Brebbia, W. G. Gray and G. F. Pinder (eds), Mathematical Modeling in Water Resources, Vol. 2, Computational Mechanics Publications, Elsevier Applied Science, Southhampton, Boston, pp. 165–178.

    Google Scholar 

  • Douglas Jr., J., Ewing, R. E. and Wheeler, M. F.: 1983a, The approximation of the pressure by a mixed method in the simulation of miscible displacement, R.A.I.R.O. Anal. Numér. 17, 17–33.

    Google Scholar 

  • Douglas Jr., J., Ewing, R. E. and Wheeler, M. F.: 1983b, A time-discretization procedure for a mixed finite element approximation of miscible displacement in porous media, R.A.I.R.O. Anal. Numér. 17, 249–265.

    Google Scholar 

  • Douglas Jr., J., Furtado, F. and Pereira, F.: 1997, On the numerical simulation of waterflooding of heterogeneous petroleum reservoirs, Comput. Geosci. 1, 155–190.

    Google Scholar 

  • Douglas Jr., J., Huang, C.-S. and Pereira, F.: 1999a, The modified method of characteristics with adjusted advection, Numer. Math. 83, 353–369.

    Google Scholar 

  • Douglas Jr., J., Huang, C.-S. and Pereira, F.: 1999b, The modified method of characteristics with adjusted advection for an immiscible displacement problem, in: Z. Chen, Y. Li, C. A. Micchelli, Y. Xu (eds), Advances in Computational Mathematics, Lecture Notes in Pure and Applied Mathematics 202, Marcel Dekker, Inc., New York, Basel, Hong Kong, pp. 53–73.

    Google Scholar 

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

    Google Scholar 

  • Douglas Jr., J., Pereira, F. and Yeh, L. M.: 2000a, A locally conservative Eulerian-Lagrangian method for flow in a porous medium of a mixture of two components having different densities, in: Z. Chen, R. Ewing and Z.-C. Shi (eds), Numerical Treatment of Multiphase Flows in Porous Media, Lecture Notes in Physics 552, pp. 138–155.

  • Douglas Jr., J., Pereira, F. and Yeh, L. M.: 2000b, A locally conservative Eulerian-Lagrangian method and its application to nonlinear transport in porous media, Comp. Geosci. 4, 1–40.

    Google Scholar 

  • Douglas Jr., J. and Roberts, J. E.: 1983, Numerical methods for a model for compressible miscible displacement in porous media, Math. Comp. 41, 441–459.

    Google Scholar 

  • Douglas Jr., J. and Russell, T. F.: 1982, Numerical methods for convection-dominated diffussion problems based on combining the method of characteristics with finite element or finite difference procedures, SIAM J. Numer. Anal. 19, 871–885.

    Google Scholar 

  • Douglas Jr., J. and Spagnuolo, A. M.: (to appear), The transport of nuclear contamination in fractured porous media, Korean J. Math.

  • Douglas Jr., J. and Yuan, Y.: 1988, Numerical simulation of immiscible flow in porous media based on combining the method of characteristics with mixed finite element procedures, in: M. F. Wheeler (ed.),Numerical Simulation in Oil Recovery, The IMA Volumes in Mathematics and its Applications, Vol. 11, Springer-Verlag, Berlin, New York, pp. 119–131.

    Google Scholar 

  • Ewing, R. E. and Russell, T. F.: 1982, Efficient time-stepping methods for miscible displacement problems in porous media, SIAM J. Numer. Anal. 19, 1–67.

    Google Scholar 

  • Ewing, R. E., Russell, T. F. and Wheeler, M. F.: 1983, Simulation of miscible displacement using mixed methods and a modified method of characteristics, in: Proceedings, Seventh SPE Symposium on Reservoir Simulation, Paper SPE 12241, Society of Petroleum Engineers, Dallas, Texas, pp. 71–81.

  • Ewing, R. E., Russell, T. F. and Wheeler, M. F.: 1984, Convergence analysis of an approximation of miscible displacement in porous media by mixed finite elements and a modified method of characteristics, Comp. Meth. Appl. Mech. Eng. 47, 73–92.

    Google Scholar 

  • Ewing, R. E., Wang, H. and Weekes, S. L.: 1999, On the simulation of multicomponent gas flow in porous media, Appl. Num. Math. 31, 405–427.

    Google Scholar 

  • Ewing, R. E. and Wheeler, M. F.: 1980, Galerkin methods for miscible displacement problems in porous media, SIAM J. Numer. Anal. 17, 351–365.

    Google Scholar 

  • Glimm, J., Lindquist, B., Pereira, F. and Zhang, Q.: 1993, A theory of macro-dispersion for the scale up problem, Transport in Porous Media 13, 97–122.

    Google Scholar 

  • Holden, H., Holden, L. and Hϕegh-Krohn, R.: 1988, A numerical method for first order nonlinear conservation laws in one dimension, Comput. Math. Appl. 15, 595–602.

    Google Scholar 

  • Huang, C.-S.: (to appear), Convergence analysis of a mass-conserving approximation of immiscible displacement in porous media by mixed finite elements and a modified method of characteristics with adjusted advection, in Computational Geosciences; currently available as Technical Report #318, Center for Applied Mathematics, Purdue University.

  • Joseph, D. D. and Renardy, Y. Y.: 1993, Fundamentals of Two-Fluid Dynamics. Part II: Lubricated Transport, Drops and Miscible Liquids, Interdisciplinary AppliedMathematics, Vol. 4, Springer-Verlag, New York.

    Google Scholar 

  • Lohrenz, J., Bray, B. G. and Clark, C. R.: 1964, Calculating viscosities of reservoir fluids from their compositions, J. Pet. Tech., 1171–1176.

  • Martin, J. J.: 1979, Cubic equations of state-Which? Ind. Chem. Fundam. 18(2), 81–97.

    Google Scholar 

  • Peaceman, D. W.: 1966, Improved treatment of dispersion in numerical calculation of multidimensional miscible displacements, Soc. Pet. Eng. J. 6, 213–216.

    Google Scholar 

  • Peaceman, D. W.: 1997, Fundamentals of Numerical Reservoir Simulation, Elsevier, New York.

    Google Scholar 

  • Raviart, P.-A. and Thomas, J. M.: 1977, A mixed finite element method for second order elliptic problems, in: I. Galligani and E. Magenes (eds), Mathematical Aspects of the Finite Element Method, Lecture Notes in Mathematics, 606, Springer-Verlag, Berlin, New York, pp. 292–315.

    Google Scholar 

  • Reid, R. C., Prausnitz, J. M. and Poling, B. E.: 1987, The Properties of Gases and Liquids, McGraw-Hill, New York.

    Google Scholar 

  • Renard, G.: 1990, A 2D reservoir streamtube EOR model with periodical automatic regeneration of streamlines, In Situ 14.

  • Russell, T. F.: 1985, Time stepping along characteristics with incomplete iteration for a Galerkin approximation of miscible displacement in porous media, SIAM J. Numer. Anal. 22, 970–1013.

    Google Scholar 

  • Russell, T. F. and Wheeler, M. F.: 1983, Finite element and finite difference methods for continuous flows in porous media, in: The Mathematics of Reservoir Simulation, SIAM, Philadelphia.

    Google Scholar 

  • Smith, J. M. and Van Ness, H. C.: 1975, Introduction to Chemical Engineering Thermodynamics, McGraw-Hill, New York.

    Google Scholar 

  • Spagnuolo, A. M.: (to appear), The approximation of contaminant transport in porous media by mixed finite elements and a modified method of characteristics with adjusted advection.

  • Thiele, M. R.: 1994, Modeling multiphase flow in heterogeneous media using streamtubes, PhD Dissertation, Dept. of Petroleum Eng., Stanford Univ.

  • Thiele, M. R., Batycky, R. P., Blunt, M. J. and Orr Jr., F.M.: 1996, Simulating flow in heterogeneous sysytems using streamtubes and streamlines, SPE Resrvoir Engineering.

  • Wang, H., Liang, D., Ewing, R. E., Lyons, S. L. and Qin, G.: 2000a, An ELLAM-MFEM solution technique for compressible fluid flows in porous media with point sources and sinks, J. Comput. Phys. 159, 344–376.

    Google Scholar 

  • Wang, H., Liang, D., Ewing, R. E., Lyons, S. L. and Qin, G.: 2000b, An accurate simulator to compressible flow in porous media with wells, in: L. R. Bentley, J. F. Sykes, C. A. Brebbia, W. G. Gray and G. F. Pinder (eds),Computational Methods in Surface-Water Systems with Hydrology, Computational Methods in Water Resources, Balkema, Rotterdam, pp. 623–630.

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Authors and Affiliations

  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, U.S.A

    Jim Douglas Jr.

  2. Departamento de Mecânica Computacional, Laboratório Nacional de Computação Científica, Petrópolis, RJ, 25651-070, Brazil

    Diego Frías

  3. Instituto Politécnico, Universidade do Estado do Rio de Janeiro, Nova Friburgo, RJ, 28601-970, Brazil

    Nélio Henderson & Felipe Pereira

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  1. Jim Douglas Jr.
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Douglas, J., Frías, D., Henderson, N. et al. Simulation of Single-Phase Multicomponent Flow Problems in Gas Reservoirs by Eulerian–Lagrangian Techniques. Transport in Porous Media 50, 307–342 (2003). https://doi.org/10.1023/A:1021138131367

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