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 [9, 10] where a two-phase gas-liquid model without external forces was studied for the first time, and 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.