Fully coupled generalized hybrid-mixed finite element approximation of two-phase two-component flow in porous media. Part I: Formulation and properties of the mathematical model

Computational Geosciences

Volumn 17, Issue 2, 2013, Pages 431-442

  Marchand, E ^a   Muller T ^a   Knabner, P ^a  

^a UNIVERSITY OF ERLANGEN NUREMBERG   (Germany)

Author keywords

Complementarity problems; Compositional multiphase flow; Nonlinear PDEs

Indexed keywords

COMPLEMENTARITY; DARCY LAW; FINITE ELEMENT METHOD; GAS FLOW; MULTIPHASE FLOW; NUMERICAL MODEL; POROUS MEDIUM; TWO PHASE FLOW;

EID: 84874974217     PISSN: 14200597     EISSN: None     Source Type: Journal    
DOI: 10.1007/s10596-013-9341-7     Document Type: Article

References (41)

1. Abadpour, A., Panfilov, M.: Method of negative saturations for modeling two-phase compositional flow with oversaturated zones. Transp. Porous Media 79, 197-214 (2009).
2. Angelini, O., Chavant, C., Chénier, E., Eymard, R., Granet, S.: Finite volume approximation of a diffusion-dissolution model and application to nuclear waste storage. Math. Comput. Simul. 81(10), 2001-2017 (2010).
3. Battino, R., Clever, H. L.: The solubility of gases in liquids. Chem. Rev. 66(4), 395-463 (1966).
4. Bear, J., Verruijt, A.: Modeling Groundwater Flow and Pollution. Reidel, Asmterdam (1987).
5. Ben Gharbia, I., Gilbert, J.: Nonconvergence of the plain Newton-min algorithm for linear complementarity problems with a P-matrix. Math. Program. (2011). doi: 10. 1007/s10107-010-0439-6.
6. Ben Gharbia, I., Jaffré, J.: Gas phase appearance and disappearance as a problem with complementarity constraints. Math. Comput. Simul. (2013, in press). URL: http://arxiv. org/abs/1111. 3808.
7. Bird, W. E., Lightfoot, E. N.: Transport Phenomena. Second Edition. Wiley, New York (2002).
8. Bourgeat, A., et al.: Numerical test data base. URL: http://momas. univ-lyon1. fr/cas_test. html. Accessed 6 Feb 2013.
9. 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).
10. Bourgeat, A., Jurak, M., Smaï, F.: Modelling and numerical simulation of gas migration in a nuclear waste repository (2010). URL: http://arxiv. org/abs/1006. 2914.
11. Bourgeat, A. P., Granet, S., Smaï, F.: Compositional two-phase flow in saturated-unsaturated porous media: benchmarks for phase appearance/disappearance (2012, in press).
12. Buchholzer, H., Kanzow, C., Knabner, P., Kräutle, S.: The semismooth Newton method for the solution of reactive transport problems including mineral precipitation-dissolution reactions. Comput. Optim. Appl. 50, 193-221 (2011).
13. Chavent, G., Jaffré, J.: Mathematical Models and Finite Elements for Reservoir Simulation. North Holland, Amsterdam (1986).
14. Class, H., Helmig, R., Bastian, P.: Numerical simulation of nonisothermal multiphase multicomponent processes in porous media 1. An efficient solution technique. Adv. Water Resour. 25, 533-550 (2002).
15. Class, H., Dahle, H. K., Helmig, R. (guest eds.): Special issue: numerical models for carbon-dioxide storage in geological formations. Comput. Geosci. 13(4), 405-509 (2009).
16. Class, H., Ebigbo, A., Helmig, R., Dahle, H. K., Nordbotten, J. M., Celia, M. A., Audigane, P., Darcis, M., Ennis-King, J., Fan, Y., Flemisch, B., Gasda, S. E., Jin, M., Krug, S., Labregere, D., Beni, A. N., Pawar, R. J., Sbai, A., Thomas, S. G., Trenty, L., Wei, L.: A benchmark study on problems related to CO2 storage in geologic formations. Summary and discussion of the results. Comput. Geosci. 13, 409-434 (2009).
17. Darcy, H.: Détermination des lois d'écoulement de l'eau à travers le sable. In: Les Fontaines Publiques de la Ville de Dijon. V. Dalmont, Paris (1856). Appendice-Note D.
18. Deuflhard, P.: Newton methods for nonlinear problems. In: Springer Series in Computational Mathematics. Springer, Berlin (2004).
19. Ern, A., Mozolevski, I.: Discontinuous Galerkin method for two-component liquid-gas porous media flows. Comput. Geosci. 16, 677-690 (2012).
20. Gasmi, C. F., Voskov, D. V., Tchelepi, H. A.: A new method for thermodynamic equilibrium computation of systems with an arbitrary number or phases. In: SPE Reservoir Simulation Symposium, SPE 119060 (2009).
21. Ho, C. K., Webb, S. W. (eds.): Gas Transport in Porous Media. Springer, New York (2006).
22. Hou, M. Z., Xie, H., Yoon, J. (eds.): Underground Storage of CO2 and Energy. Balkema, Rotterdam (2010).
23. Ito, K., Kunish, K.: On a semi-smooth Newton method and its globalization. Math. Program. Ser. A 118, 347-370 (2009).
24. Jaffré, J., Sboui, A.: Henry's law and gas phase disappearance. Transp. Porous Media 82, 521-526 (2010).
25. Knabner, P., van Duijn, C. J., Hengst, S.: An analysis of crystal dissolutions fronts of flows through porous media. Part 1: Compatible boundary conditions. Adv. Water Resour. 18, 171-185 (1995).
26. Kräutle, S.: General multi-species reactive transport problems in porous media: efficient numerical approaches and existence of global solutions. Chapter 4: reactions with minerals: a complementarity problem solved by the semismooth Newton method (2008). URL: http://www1. am. uni-erlangen. de/~kraeutle/habil. pdf. Habilitationsschrift.
27. Kräutle, S.: The semismooth Newton method for multicomponent reactive transport with minerals. Adv. Water Resour. 34(1), 137-151 (2011).
28. Krishna, R., Wesselingh, J. A.: The Maxwell-Stefan approach to mass transfer. Chem. Eng. Sci. 52(6), 861-911 (1997).
29. Lake, L. W.: Enhanced Oil Recovery. Prentice-Hall, Englewood Cliffs (1989).
30. Landau, L. D., Lifshitz, E. M.: Statistical Physics Part 1. Pergamon, New York (1959).
31. Lauser, A., Hager, C., Helmig, R., Wohlmuth, B.: A new approach for phase transitions in miscible multi-phase flow in porous media. Adv. Water Resour. 34, 957-966 (2011).
32. Marchand, E., Müller, T., Knabner, P.: Fully coupled generalised hybrid-mixed finite element approximation of two-phases two-components flow in porous media. Part II: numerical scheme and numerical results. Comput. Geosci. 16, 691-708 (2012).
33. Michelsen, M. L.: The isothermal flash problem. Part II. Phase-split calculation. Fluid Phase Equilib. 9, 21-40 (1982).
34. Neumann, R., Bastian, P., Ippisch, O.: Modeling and simulation of two-phase two-component flow with disappearing nonwetting phase. Comput. Geosci. (2012). doi: 10. 1007/s10596-012-9321-3.
35. Orr, F. M. Jr., Jessen, K.: An analysis of the vanishing interfacial tension technique for determination of minimum miscibility pressure. Fluid Phase Equilib. 255, 99-109 (2007).
36. 1-contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Differ. Equ. 131, 20-38 (1996).
37. Panfilov, M., Rasoulzadeh, M.: Interfaces of phase transition and disappearance and method of negative saturation for compositional flow with diffusion and capillarity in porous media. Transp. Porous Media 83, 73-78 (2010).
38. Shapiro, A. A., Stenby, E. H.: Thermodynamics of the multicomponent vapor-liquid equilibrium under capillary pressure difference. Fluid Phase Equilib. 178, 17-32 (2001).
39. Smaï, F.: Développement d'outils mathématiques et numériques pour l'évaluation du concept de stockage géologique. Ph. D. thesis, Université de Lyon (2009).
40. o and up to 600 bar. Geochim. Cosmochim. Acta 67(16), 3015-3031 (2003).
41. Whitson, C. H., Michelsen, M. L.: The negative flash. Fluid Phase Equilib. 53, 51-71 (1989).