Continuation of the Tie-simplex Space for Three-phase Thermal-compositional Systems

Modeling the phase behavior associated with compositional flow simulation of systems that form more than two phases (e.g., steam, or CO2 injection) is a challenging problem. In addition to the coupling of thermodynamics with the nonlinear equations of flow and transport, accurate phase-state identification for mixtures that form three (or more) phases, as function of pressure, temperature and composition, is the subject of active research. We describe a general negative-flash method for multi-component thermal systems that can form three, or more, fluid phases, including a convergence proof. This extended negative-flash approach integrates quite nicely with flow simulation based on adaptive tie-simplex compositional space parameterization. We prove that tie-simplexes change continuously with pressure, temperature, or along a continuous trajectory in compositional space. Continuation of the tie-simplex space provides theoretical justification for the tie-simplex based flow modeling, in which interpolation in pressure and temperature for a limited number of tie-simplexes is performed during a simulation run. We also show how a tie-simplex associated with a given composition can degenerate as a function of pressure and temperature. We focus on the complex behaviors of the tie-triangles and tie-lines associated with three-phase, steam-injection problems. The algorithms that capture the degeneration of the tie-triangles into tie-lines are described in detail. Interpolation in the parameterized compositional space is used to identify the phase-state and proves to be as reliable as a 'conventional' phase-stability test for three-phase mixtures. This extended negative-flash algorithm has been integrated into our tie-simplex based flow simulation framework. We demonstrate the effectiveness of the framework using several compositional steam-injection problems with complex behaviors.