Global weak solutions for a compressible gas-liquid model with well-formation interaction

The objective of this work is to explore a compressible gas-liquid model designed for modeling of well flow processes. We build into the model well-reservoir interaction by allowing flow of gas between well and formation (surrounding reservoir). Inflow of gas and subsequent expansion of gas as it ascends towards the top of the well (a so-called gas kick) represents a major concern for various well operations in the context of petroleum engineering. We obtain a global existence result under suitable assumptions on the regularity of initial data and the rate function that controls the flow of gas between well and formation. Uniqueness is also obtained by imposing more regularity on the initial data. The key estimates are to obtain appropriate lower and upper bounds on the gas and liquid masses. For that purpose we introduce a transformed version of the original model that is highly convenient for analysis of the original model. In particular, in the analysis of the transformed model additional terms, representing well-formation interaction, can be treated by natural extensions of arguments that previously have been employed for the single-phase Navier–Stokes model. The analysis ensures that transition to single-phase regions do not appear when the initial state is a true gas-liquid mixture.