Fully coupled generalized hybrid-mixed finite element approximation of two-phase two-component flow in porous media. I: Formulation and properties of the mathematical model. (English) Zbl 1382.76173
Summary: This paper is a prequel to that of E. Marchand et al. [“Fully coupled generalised hybrid-mixed finite element approximation of two-phase two-component flow in porous media. II: Numerical scheme and numerical results”, ibid. 16, No. 3, 691–708 (2012; doi:10.1007/s10596-012-9279-1)], where an efficient and accurate hybrid-mixed finite element approximation for a system of time-dependent nonlinear conservation equations has been formulated, implemented, and tested, which are general enough to represent most of the existing formulations for two-component liquid-gas flow in porous medium with phase exchange, also allowing for any (dis)appearance of one of the phases. Temperature variation is neglected, but capillary effects are included by extended Darcy’s law, and Fickian diffusion is taken into account. The efficiency and stability of the numerical method of L. W. Lake [Enhanced oil recovery. Englewood Cliffs, NJ: Prentice-Hall (1989)] relies on an equivalent reformulation of the otherwise commonly used model in terms of new principal variables and subsequent static (flash) equations allowing more generally for any (dis)appearance of one of the phases without the need of variable switching or unphysical quantities. In particular, the formulation in terms of complementarity conditions allows for an efficient and stable solution by the semismooth Newton’s method.
MSC:
| 76M10 | Finite element methods applied to problems in fluid mechanics |
| 76S05 | Flows in porous media; filtration; seepage |
| 76T10 | Liquid-gas two-phase flows, bubbly flows |
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