Smooth Formulation for Three-Phase Black-Oil Simulation with Superior Nonlinear Convergence

Abstract

Black-oil simulations with phase changes are challenging, because of the complex interactions between the different components and the equilibrium behavior of the phases. The common method for solving this type of nonlinear problem is to use a fully-implicit approach. However, the conventional black-oil model can lead to difficulties with converging using Newton's method. Discontinuities in discrete system can occur when a phase transition happens, which can lead to oscillations or even failure of the Newton iterations. The goal is to design a smoothing formulation that eliminates any sudden changes in properties or discontinuities that occur during phase transitions. We first employ a compositional formulation based on K-values to describe the standard black-oil model. Next, the coupled system is reformulated such that the discontinuities are carried over to the phase equilibrium model. In this manner, a single, succinct non-smooth equation is obtained, which allows for deriving a smoothing approximation. A mixed complementarity problem (MCP) for phase-equilibrium in the area of chemical process modeling served as the foundation for the reformulation. The new formulation is non-intrusive and simple to implement, requiring minor changes to current black-oil simulator frameworks. We analyze and demonstrate that phase changes lead to the changes of fluid-properties and discrete system, under the conventional black-oil formulation. By comparison, the newly proposed formulation uses a smoothing parameter to ensure smooth transitions of variables between the phase regimes. It also generates unique solutions that are valid for all three phases. Several complex heterogeneous problems are tested. The conventional black-oil model experiences many time-step cuttings and wasting nonlinear iterates. On the contrary, the smoothing model exhibits excellent convergence behaviors. Overall, the new formulation addresses the issues with convergence caused by phase-changes, while barely affecting solution results.