Fully coupled generalised hybrid-mixed finite element approximation of two-phase two-component flow in porous media. Part II: Numerical scheme and numerical results - Marchand, Estelle; Müller, Torsten; Knabner, Peter (Computational Geosciences, 2012)
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. © 2012 Springer Science+Business Media B.
Keywords: Mixed hybrid finite elements; Nonlinear systems of partial differential equations

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 - Marchand, Estelle; Müller, Torsten; Knabner, Peter (Computational Geosciences, 2013)
This paper is a prequel to that of Marchand et al. (Comput Geosci 16:691-708, 2012), 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 Lake (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. © 2013 Springer Science+Business Media Dordrech
Keywords: Complementarity problems; Compositional multiphase flow; Nonlinear PDEs

Results of the MoMaS benchmark for gas phase appearance and disappearance using generalized MHFE - Marchand, Estelle; Knabner, Peter (Advances in Water Resources, 2014)
This article addresses 8 testcases proposed in the MoMaS benchmark for phase (dis)appearance. It involves slightly compressible, compositional two-phase flow with the components hydrogen and water, where hydrogen can appear in gas form or dissolved in water. The simplest example illustrates phase transition. Other problem set-ups include one and two-dimensional heterogeneous porous media and discontinuous initial conditions. Generalized mixed hybrid finite elements (MHFE) are used, which offer flexibility in the choice of primary variables. In the most general case, a complementarity formulation for the local problems together with a semi-smooth Newton algorithm is used. The ability of the method to solve all of the compositional benchmark problems is shown. This set of simple examples indicates that in the view of risk assessment of underground nuclear waste repositories the phenomenon of phase transition cannot be ignored and can take place even after the sources of hydrogen are extinguished. Moreover it can be useful for a numerical code to enable several choices of primary variables in order to cover a large variety of situations. © 2014 Elsevier Lt
Keywords: Compositional modeling; Gas-liquid flow with phase transitions; Multiphase flow in porous medium domains; Nuclear and chemical waste storage