Advertisement

Computational Geosciences

, Volume 12, Issue 4, pp 525–539 | Cite as

A space-time discontinuous Galerkin method applied to single-phase flow in porous media

  • Zhiyun Chen
  • Holger Steeb
  • Stefan Diebels
Original paper

Abstract

A space-time discontinuous Galerkin finite element method is proposed and applied to a convection-dominant single-phase flow problem in porous media. The numerical scheme is based on a coupled space-time finite element discretization allowing for discontinuous approximations in space and in time. The continuities on the element interfaces are weakly enforced by the flux treatments, so that no extra penalty factor has to be determined. The resulting space-time formulation possesses the advantage of capturing the steep concentration front with sharp gradients efficiently. The stability and reliability of the proposed approach is demonstrated by numerical experiments.

Keywords

Discontinuous Galerkin methods Convection-dominant flow Porous media 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ewing, R.E., Wyoming, U., Russell, T.F., Wheeler, M.F.: Simulation of miscible displacement using mixed methods and a modified method of characteristics. Soc. Pet. Eng. J. 12241 (1983)Google Scholar
  2. 2.
    Douglas, J.: The numerical simulation of miscible displacement in porous media. In: Oden, J.T. (ed.) Computational Methods in Nonlinear Mechanics, pp. 225–237. North-Holland, Amsterdam (1980)Google Scholar
  3. 3.
    Koval, E.J.: A method for predicting the performance of unstable miscible displacement in heterogeneous media. Soc. Pet. Eng. J. 3, 145–154 (1963)Google Scholar
  4. 4.
    Klieber, W., Rivière, B.: Adaptive simulations of two-phase flow by discontinuous Galerkin methods. Comput. Methods Appl. Mech. Eng. 196, 404–419 (2006)zbMATHCrossRefGoogle Scholar
  5. 5.
    Rivière, B., Wheeler, M.F.: Discontinuous Galerkin methods for flow and transport problems in porous media. Commun. Numer. Methods Eng. 79, 157–174 (2002)Google Scholar
  6. 6.
    Rivière, B., Wheeler, M.F., Banaś, K.: Part II: Discontinuous Galerkin methods applied to a single phase flow in porous media. Comput. Geosci. 4, 337–349 (2000)zbMATHCrossRefGoogle Scholar
  7. 7.
    Nayagum, D., Schäfer, G., Mosé, R.: Modelling two-phase incompressible flow in porous media using mixed hybrid and discontinuous finite elements. Comput. Geotech. 8, 49–73 (2004)zbMATHGoogle Scholar
  8. 8.
    Oden, J.T., Babuška, I., Baumann, C.E.: A discontinuous hp finite element method for diffusion problems. J. Chem. Phys. 146, 491–519 (1998)zbMATHGoogle Scholar
  9. 9.
    Rivière, B., Wheeler, M.F., Girault, V.: Improved energy estimates for interior penalty, constrained and discontinuous Galerkin methods for elliptic problems. Part I. Comput. Geosci. 3, 337–360 (1999)zbMATHCrossRefGoogle Scholar
  10. 10.
    Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15, 152–161 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Dawson, C., Proft, J.: Coupled discontinuous and continuous Galerkin finite element methods for the depth-integrated shallow water equations. Comput. Methods Appl. Mech. Eng. 193, 289–318 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Sun, S., Wheeler, M.F.: Symmetric and nonsymmetric discontinuous Galerkin methods for reactive transport in porous media. SIAM J. Numer. Anal. 43(1), 195–219 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Dawson, C., Sun, S., Wheeler, M.: Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Eng. 193, 2565–2680 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ames, W.F.: Numerical Methods for Partial Differential Equations, 2nd edn. Academic, Boston (1977)zbMATHGoogle Scholar
  16. 16.
    Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Computational Differential Equations. Cambridge University Press, Cambridge (1996)zbMATHGoogle Scholar
  17. 17.
    Argyris, J.H., Scharpf, D.W.: Finite elements in space and time. Nucl. Eng. Des. 10, 456–464 (1969)CrossRefGoogle Scholar
  18. 18.
    Fried, I.: Finite element analysis of time-dependent phenomena. AIAA J. 7, 1170–1173 (1969)zbMATHCrossRefGoogle Scholar
  19. 19.
    Oden, J.T.: A general theory of finite elements ii. applications. Int. J. Numer. Methods Eng. 1, 247–259 (1969)zbMATHCrossRefGoogle Scholar
  20. 20.
    Hughes, T.J.R., Hulbert, G.M.: Space-time finite element methods for elastodynamics: Formulations and error estimates. Comput. Methods Appl. Mech. Eng. 66, 339–363 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Hulbert, G.: Space-time finite element methods for second order hyperbolic equations. PhD thesis, Department of Mechanical Engineering, Stanford University, Stanford (1989)Google Scholar
  22. 22.
    Hulbert, G.M.: Time finite element methods for structural dynamics. Int. J. Numer. Methods Eng. 33, 307–331 (1992)zbMATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hulbert, G.M., Hughes, T.J.R.: Space-time finite element methods for second-order hyperbolic equations. Comput. Methods Appl. Mech. Eng. 84, 327–348 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Chen, Z., Steeb, H., Diebels, S.: A time-discontinuous Galerkin method for the dynamical analysis of porous media. Int. J. Numer. Anal. Methods Geomech. 30, 1113–1134 (2006)CrossRefGoogle Scholar
  25. 25.
    Baumann, C.E.: An hp-adaptive discontinuous finite element method for computational fluid dynamics. PhD thesis, The University of Texas at Austin (1997)Google Scholar
  26. 26.
    Hassanizadeh, S.M., Gray, W.G.: High velocity flow in porous media. Transp. Porous Med. 2, 521–531 (1987)CrossRefGoogle Scholar
  27. 27.
    Diebels, S., Ehlers, W., Markert, B.: Neglect of the fluid-extra stresses in volumetrically coupled solid-fluid problems. Z. Angew. Math. Mech. 81, S521–S522 (2001)Google Scholar
  28. 28.
    Ehlers, W.: Foundations of multiphasic and porous materials. In: Ehlers, W., Bluhm, J. (eds.) Porous Media: Theory, Experiments and Numerical Applications, pp. 3–86. Springer, Berlin (2002)Google Scholar
  29. 29.
    Baumann, C.E., Oden, J.T.: A discontinuous hp finite element method for convection-diffusion problems. Comput Methods Appl. Mech. Eng. 175, 311–341 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Cockburn, B.: Discontinuous Galerkin methods. Z. Angew. Math. Mech. 11, 731–754 (2003)CrossRefMathSciNetGoogle Scholar
  31. 31.
    Oden, J.T., Babuška, I., Baumann, C.E.: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146, 491–519 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, New York (1983)zbMATHGoogle Scholar
  33. 33.
    Wriggers, P.: Konsistente Lineariserung in der Kontinuumsmechanik und ihre Anwendung auf die Finite-Element-Methode, Bericht Nr. F88/4 (1999) Institut für Baustatik und Numerische Mechanik, Univerität Honnover (1988)Google Scholar
  34. 34.
    Bastian, P., Rivière, B.: Superconvergence and h(div)-projectin for discontinuous Galerkin methods. Int. J. Numer. Methods Fluids 42, 1043–1057 (2003)zbMATHCrossRefGoogle Scholar
  35. 35.
    Cockburn, B., Shu, C.W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V. J. Chem. Phys. 141, 199–224 (1998)zbMATHMathSciNetGoogle Scholar
  36. 36.
    Hoteit, H., Ackerer, P., Mosé, R., Erhel, J., Philippe, B.: New two-dimensional slope limiters for discontinuous galerkin methods on arbitrary meshes. Int. J. Numer. Methods Eng. 61, 2566–2593 (2004)zbMATHCrossRefGoogle Scholar
  37. 37.
    Ewing, R.E. (ed.): The Mathematics of Reservoir Simulation. SIAM, Philadelphia (1983)zbMATHGoogle Scholar
  38. 38.
    Homsy, G.M.: Viscous fingering in porous media. Ann. Rev. Fluid Mech. 19, 271–311 (1987)CrossRefGoogle Scholar
  39. 39.
    Hughes, T.J.R.: The Finite Element Method. Prentice-Hall, Englewood Cliffs (1987)zbMATHGoogle Scholar
  40. 40.
    Baumann, C.E., Oden, J.T.: An adaptive-order discontinuous Galerkin method for the solution of the Euler equations of gas dynamics. Int. J. Numer. Methods Eng. 47(1–3), 61–73 (2000)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Chair of Applied MechanicsSaarland UniversitySaarbrückenGermany

Personalised recommendations