Global weak solutions for a compressible gas-liquid model with well-formation interaction
Journal article, Peer reviewed
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Original versionEvje, S. (2011) Global weak solutions for a compressible gas-liquid model with well-formation interaction. Journal of Differential Equations, 251(8), pp. 2352–2386 10.1016/j.jde.2011.07.013
Abstract. The objective of this work is to explore a compressible gas-liquid model designed for modeling of well °ow processes. We build into the model well-reservoir interaction by allowing °ow of gas between well and formation (surrounding reservoir). In°ow 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 °ow 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.