Abstract
We consider the modeling and simulation of compositional two-phase flow in a porous medium, where one phase is allowed to vanish or appear. The modeling of Marchand et al. (in review) leads to a nonlinear system of two conservation equations. Each conservation equation contains several nonlinear diffusion terms, which in general cannot be written as a function of the gradients of the two principal unknowns. Also the diffusion coefficients are not necessarily explicit local functions of them. For the generalised mixed finite elements approximation, Lagrange multipliers associated to each principal unknown are introduced, the sum of the diffusive fluxes of each component is explicitly eliminated and the static condensation leads to a “global” nonlinear system of equations only in the Lagrange multipliers also including complementarity conditions to cope with vanishing or appearing phases. After time discretisation, this system can be solved at each time step using a semi-smooth Newton method. The static condensation involves “local” nonlinear systems of equations associated to each element, solved also by a semismooth Newton method. The algorithm is successfully applied to 1D and 2D examples of water–hydrogen flow involving gas phase appearance and disappearance.
Similar content being viewed by others
References
Bastian, P.: Numerical computation of multiphase flow in porous media. http://conan.iwr.uni-heidelberg.de/people/peter/pdf/Bastian_habilitationthesis.pdf (1999, Habilitationsschrift)
Bourgeat, A., et al.: Numerical test data base. http://momas.univ-lyon1.fr/cas_test.html (2009). Accessed 13 February 2012
Bourgeat, A., Jurak, M., Smaï, F.: Two-phase, partially miscible flow and transport modeling in porous media; application to gas migration in a nuclear waste repository. Comput. Geosci. 13(1), 29–42 (2009)
van Genuchten, M.T.: A closed-form for predicting the hydraulic conductivity of unsaturated soils. Soil Sci. Soc. Am. J. 44, 892–898 (1980)
Kröner, D.: Numerical Schemes for Conservation Laws. Wiley and Teubner, Chichester (1997)
Nedelec, J.: Mixed finite elements in ℝ3. Numer. Math. 35, 315–341 (1980)
Pop, I.S., Radu, F., Knabner, P.: Mixed finite elements for the Richards equation: linearization procedure. J. Comput. Appl. Math. 18, 365–373 (2004)
Raviart, P., Thomas, J.: A mixed finite element method for second order elliptic problems. In: Galligani, I., Magenes, E. (eds.) Mathematical Aspects of Finite Element Methods, no. 606 in Lecture Notes in Mathematics, pp. 292–315. Springer, Berlin (1977)
Roberts, J., Thomas, J.M.: Handbook of Numerical Analysis, vol. 2, Mixed and Hybrid Methods, chap. IV. North-Holland, Amsterdam (1991)
Schneid, E.: Hybrid-gemischte finite-elemente-diskretisierung der Richards-gleichung. http://fauams5.am.uni-erlangen.de/am1/de/theses.html (2000)
Smaï, F.: Développement dóutils mathématiques et numériques pour l’évaluation du concept de stockage géologique. Ph.D. thesis, Université de Lyon (2009)
Wieners, C.: Distributed point objects. A new concept for parallel finite elements. In: Kornhuber, R., Hoppe, R., Priaux, J., Pironneau, O., Widlund, O., Xu, J. (eds.) Domain Decomposition Methods in Science and Engineering. Lecture Notes in Computational Science and Engineering, vol. 40, pp. 175–183. Springer, Berlin (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Marchand, E., Müller, T. & Knabner, P. Fully coupled generalised hybrid-mixed finite element approximation of two-phase two-component flow in porous media. Part II: numerical scheme and numerical results. Comput Geosci 16, 691–708 (2012). https://doi.org/10.1007/s10596-012-9279-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10596-012-9279-1