Global weak solutions for a gas liquid model with external forces and general pressure law

Abstract

In this work we show existence of global weak solutions for a two-phase gas-liquid model where the gas phase is represented by a general isothermal pressure law, whereas the liquid is assumed to be incompressible. To make the model relevant for pipe and well-flow applications we have included external forces in the momentum equation representing, respectively, wall friction forces and gravity forces. The analysis relies on a proper combination of the methods introduced in [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], where a two-phase gas-liquid model without external forces was studied for the first time, and on techniques that have been developed for the single-phase gas model. As a motivation for further research, some numerical examples are also included demonstrating the ability of the model to describe the ascent of a gas slug due to buoyancy forces in a vertical well. Characteristic features like expansion of the moving gas slug as well as counter-current flow mechanisms (i.e., liquid is moving downward due to gravity and gas is displaced upward) are highlighted. These examples are highly relevant for modeling of gas-kick flow scenarios, which represent a major concern in the context of oil and gas well control operations.