The Hyperbolic Nature of a System of Equations Describing Three-phase Flows in Wellbores

Abstract

Summary 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.