Weak solutions for a gas liquid model relevant for describing gas-kick in oil wells

Abstract

The purpose of this paper is to establish a local in time existence result for a compressible gas-liquid model. The model is a drift-flux model which is composed of two continuity equations and one mixture momentum equation supplemented with a slip relation in order to take into account the possibility of flows with unequal fluid velocities. The model is highly relevant for modeling of gas kick for oil wells, which in its worst case can lead to blowout scenarios. The mathematical study of such kinds of models is important for the development of simulation tools that can be employed for increased control of deep-water well operations. The liquid phase is assumed to be incompressible whereas the gas is described by a polytropic equation of state. The model is studied in a framework previously used for investigations of the single-phase compressible Navier–Stokes model. New challenges arise due to the appearance of a generalized pressure term that depends on fluid masses as well as gas velocity. The local existence result is obtained by introducing a suitable transformation along the line of the works [S. Evje and K. H. Karlsen, Commun. Pure Appl. Anal., 8 (2009), pp. 1867–1894, S. Evje, T. Flåtten, and H. A. Friis, Nonlinear Anal., 70 (2009), pp. 3864–3886] in a free boundary setting. This allows us to obtain sufficient pointwise control of the gas and liquid masses. The estimates are rather delicate as they must be fine enough to control a possible singular behavior associated with the pressure law as well as the slip relation. The existence result is obtained under the assumption of a sufficient small time interval combined with suitable assumptions on the regularity of the initial data, the parameters that control, respectively, the behavior of the initial masses at the boundaries of the flow domain and the decay properties of the viscosity term.