Well-posedness of a Compressible Gas-liquid Model with a Friction Term Important for Well Control Operations

In this work we continue our investigations of a compressible gas-liquid model with special focus on inclusion of external frictional forces in the momentum balance. The model is often used for multiphase well flow modeling important for different well control operations. The frictional forces have a major impact on the pressure gradient, which determines the pressure distribution along the wellbore. Compression and decompression of gas in turn strongly depend on the pressure level along the wellbore. A precise understanding of these mechanisms is important since gas-kick scenarios and blow-out behavior is strongly linked to decompression effects. This work is a continuation of the recent work [”Global weak solutions for a gas-liquid model with external forces and general pressure law”, SIAM J. Appl. Math. 71 (2), pp. 409–442, 2011]. The novelty of the present work lies in the fact that: (i) we consider a full momentum equation whereas a simplified one was used in the first work; (ii) the gas and liquid masses vanish at the boundaries making the analysis more involved; (iii) special care must be given to the frictional term to make sure that it is balanced with other terms such that a well-defined model is obtained. The analysis ensures that global existence of weak solutions is obtained under suitable assumptions on initial data (e.g. decay rate at the boundaries for gas and liquid mass) and parameters that determine growth rate of mass terms associated with, respectively, the wall friction term and viscous term. Subject classification. 76T10, 76N10, 65M12, 35L60