Hybrid Flux Splitting Schemes for Numerical Resolution of Two-Phase Flows

Abstract

This thesis deals with the construction of numerical schemes for approximating solutions to a hyperbolix two-phase flow model. Numerical schemes for hyperbolic models are commonly divided in two main classes: Flux Vector Splitting (FVS) schemes which are based on scalar computations and Flux Difference Splitting (FDS) schemes which are based on matrix computations. FVS schemes are more efficient than FDS schemes, but FDS schemes are more accurate. The canonical FDS schemes are the approximate Riemann solvers which are based on a local decomposition of the system into its full wave structure.

In this thesis the mathematical structure of the model is exploited to construct a class of hybrid FVS/FDS schemes, denoted as Mixture Flux (MF) schemes. This approach is based on a splitting of the system in two components associated with the pressure and volume fraction variables respectively, and builds upon hybrid FVS/FDS schemes previously developed for one-phase flow models.

Through analysis and numerical experiments it is demonstrated that the MF approach provides several desirable features, including

- Improved efficiency compared to standard approximate Riemann solvers

- Robustness under stiff conditions

- Accuracy on linear and nonlinear phenomena

In particular it is demonstrated that the framework allows for an efficient weakly implicit implementation, focusing in an accurate resolution of slow transients relevant for the petroleum industry.