Finite volume approximation of degenerate two-phase flow model with unlimited air mobility

Boris Andreianov (1) , Robert Eymard (2) , Mustapha Ghilani (3) , Nouzha Marhraoui (3)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)
2 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
3 EMMACS

Models of two-phase flows in porous media, used in petroleum engineering, lead to a coupled system of two equations, one elliptic, the other degenerate parabolic, with two unknowns: the saturation and the pressure. In view of applications in hydrogeology, we are interested at the singular limit of this model, as the ratio $\mu$ of air/liquid mobility goes to infinity, and in a comparison with the one-phase Richards model. We construct a robust finite volume scheme that can apply for large values of the parameter $\mu$. This scheme is shown to satisfy a priori estimates (the saturation is shown to remain in a fixed interval, and a discrete $\ldehun$ estimate is proved for both the pressure and a function of the saturation) which are sufficient to derive the convergence of a subsequence to a weak solution of the continuous equations, as the size of the discretization tends to zero. At the limit as the mobility of the air phase tends to infinity, we obtain the two-phase flow model introduced in the work Henry, Hilhorst and Eymard \cite{MHenry-et-al} (see also \cite{Eymard-Ghilani-Marhraoui}) which we call the quasi-Richards equation.

Mots clés: Flow in porous media; two-phase flow model; infinite mobility limit; Richards model; Finite Volume method; discrete a priori estimates; convergence of approximate solutions

Domaines: Analyse numérique [math.NA]; Equations aux dérivées partielles [math.AP]

Boris Andreianov, Robert Eymard, Mustapha Ghilani, Nouzha Marhraoui. Finite volume approximation of degenerate two-phase flow model with unlimited air mobility. Numerical Methods for Partial Differential Equations, 2012, 29 (2), pp. 441-474. ⟨10.1002/num.21715⟩. ⟨hal-00606955⟩