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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
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)
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)
Koval, E.J.: A method for predicting the performance of unstable miscible displacement in heterogeneous media. Soc. Pet. Eng. J. 3, 145–154 (1963)
Klieber, W., Rivière, B.: Adaptive simulations of two-phase flow by discontinuous Galerkin methods. Comput. Methods Appl. Mech. Eng. 196, 404–419 (2006)
Rivière, B., Wheeler, M.F.: Discontinuous Galerkin methods for flow and transport problems in porous media. Commun. Numer. Methods Eng. 79, 157–174 (2002)
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)
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)
Oden, J.T., Babuška, I., Baumann, C.E.: A discontinuous hp finite element method for diffusion problems. J. Chem. Phys. 146, 491–519 (1998)
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)
Wheeler, M.F.: An elliptic collocation-finite element method with interior penalties. SIAM J. Numer. Anal. 15, 152–161 (1978)
Arnold, D.N.: An interior penalty finite element method with discontinuous elements. SIAM J. Numer. Anal. 19, 742–760 (1982)
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)
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)
Dawson, C., Sun, S., Wheeler, M.: Compatible algorithms for coupled flow and transport. Comput. Methods Appl. Mech. Eng. 193, 2565–2680 (2004)
Ames, W.F.: Numerical Methods for Partial Differential Equations, 2nd edn. Academic, Boston (1977)
Eriksson, K., Estep, D., Hansbo, P., Johnson, C.: Computational Differential Equations. Cambridge University Press, Cambridge (1996)
Argyris, J.H., Scharpf, D.W.: Finite elements in space and time. Nucl. Eng. Des. 10, 456–464 (1969)
Fried, I.: Finite element analysis of time-dependent phenomena. AIAA J. 7, 1170–1173 (1969)
Oden, J.T.: A general theory of finite elements ii. applications. Int. J. Numer. Methods Eng. 1, 247–259 (1969)
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)
Hulbert, G.: Space-time finite element methods for second order hyperbolic equations. PhD thesis, Department of Mechanical Engineering, Stanford University, Stanford (1989)
Hulbert, G.M.: Time finite element methods for structural dynamics. Int. J. Numer. Methods Eng. 33, 307–331 (1992)
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)
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)
Baumann, C.E.: An hp-adaptive discontinuous finite element method for computational fluid dynamics. PhD thesis, The University of Texas at Austin (1997)
Hassanizadeh, S.M., Gray, W.G.: High velocity flow in porous media. Transp. Porous Med. 2, 521–531 (1987)
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)
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)
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)
Cockburn, B.: Discontinuous Galerkin methods. Z. Angew. Math. Mech. 11, 731–754 (2003)
Oden, J.T., Babuška, I., Baumann, C.E.: A discontinuous hp finite element method for diffusion problems. J. Comput. Phys. 146, 491–519 (1998)
Marsden, J.E., Hughes, T.J.R.: Mathematical Foundations of Elasticity. Dover, New York (1983)
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)
Bastian, P., Rivière, B.: Superconvergence and h(div)-projectin for discontinuous Galerkin methods. Int. J. Numer. Methods Fluids 42, 1043–1057 (2003)
Cockburn, B., Shu, C.W.: The Runge-Kutta discontinuous Galerkin method for conservation laws V. J. Chem. Phys. 141, 199–224 (1998)
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)
Ewing, R.E. (ed.): The Mathematics of Reservoir Simulation. SIAM, Philadelphia (1983)
Homsy, G.M.: Viscous fingering in porous media. Ann. Rev. Fluid Mech. 19, 271–311 (1987)
Hughes, T.J.R.: The Finite Element Method. Prentice-Hall, Englewood Cliffs (1987)
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)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author is grateful to the DFG (German Science Foundation—Deutsche Forschungsgemeinschaft) for the financial support under the grant number Di 430/4-2.
Rights and permissions
About this article
Cite this article
Chen, Z., Steeb, H. & Diebels, S. A space-time discontinuous Galerkin method applied to single-phase flow in porous media. Comput Geosci 12, 525–539 (2008). https://doi.org/10.1007/s10596-008-9092-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-008-9092-z