The hyperbolic nature of a system of equations describing three-phase flows in wellbores

K. Sinkov, P. Spesivtsev, A. A. Osiptsov

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review

1 Citation (Scopus)

Abstract

The problem of the formulation of compressible, isothermal, multi-phase (3-phase) flow in wellbores is considered. One such approach is that provided by the drift-flux model. According to this model, in the three-phase case (typically, oil, water, and gas) the governing system of equations consists of three continuity equations, one for each phase, and a single equation for the conservation of momentum for the mixture. The system is closed by equations-of-state and algebraic relations for determining the individual phase velocities. The detailed characteristic analysis of both two- and three-phase problems is carried out to determine the domains where the system of equations is hyperbolic and, therefore, whether they are suitably well-posed, stable and robust for application in the numerical solution of this type of hyperbolic problem. The transient, steady-state, and non-inertial forms of the momentum conservation equation encountered in the literature are considered. The influence of mass exchange terms, responsible for the solubility of the gas-phase in liquid on the hyperbolicity is also studied. The analysis demonstrated that the system is found to be hyperbolic in the two-phase case and conditionally hyperbolic in the three-phase case, with eigenvalues being functions of the problem variables. It is also shown that the mixture momentum equation can be transformed to the so-called "advection" equation for pressure which possesses a real eigenvalue. The analysis presented suggests recommendations on the domains to which hyperbolicity is valid to a system of equations and on the specification of boundary conditions for the drift-flux model.

Original languageEnglish
Title of host publication14th European Conference on the Mathematics of Oil Recovery 2014, ECMOR 2014
PublisherEuropean Association of Geoscientists and Engineers, EAGE
ISBN (Electronic)9781634391689
Publication statusPublished - 2014
Externally publishedYes
Event14th European Conference on the Mathematics of Oil Recovery 2014, ECMOR 2014 - Catania, Italy
Duration: 8 Sep 201411 Sep 2014

Publication series

Name14th European Conference on the Mathematics of Oil Recovery 2014, ECMOR 2014

Conference

Conference14th European Conference on the Mathematics of Oil Recovery 2014, ECMOR 2014
Country/TerritoryItaly
CityCatania
Period8/09/1411/09/14

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