(352v) Highly Robust and Efficient Reduction Method for Phase Stability Testing in a Minimization Framework

Phase stability testing is very important in process and compositional reservoir simulation, in which the stability routine is called repeatedly a very large number of times. The most difficult domains (unavoidable in reservoir simulation) for phase stability testing is located in the strict vicinity of the singularity that is, the stability test limit locus (STLL) and in a possibly wide region outside the STLL in the pressure-temperature plane, where the iterates have to pass a region of indefiniteness of the Hessian matrix and the number of iterations for any gradient-based method exhibits an erratic behavior, with significant local increases. This feature has adverse effects on implementations on a graphical processing unit (GPU) architecture, in which computations for different threads must follow the same execution path. Hence, a different approach that can guarantee a fast convergence in a fixed maximum number of iterations is required. GPU-based compositional reservoir simulators are currently developed for complex tertiary recovery processes and various mixed energy-environment related application such as CO₂ sequestration and geothermal production.

In reduction methods for phase equilibrium calculations, the dimensionality of the problem is significantly decreased and the number of independent variables does not depend on the number of components in the mixture, but on the rank of the matrix containing binary interaction parameters Reduction methods are particularly efficient for mixtures with many components and relatively few non-zero binary interaction parameters, as typically encountered in many chemical and process simulations. Small working arrays make these methods also highly attractive for implementation on a GPU platform. However, existing reduction methods are formulated as equation-solving rather than minimization problems and, unlike conventional methods, they take limited or not advantage on symmetry.

In this work, a minimization of the tangent plane distance function with respect to a specific set of variables (modified reduction parameters) and constraints is performed using either a modified Cholesky factorization with advanced line search procedures or a Trust-Region method; both methods guarantee a descent Newton direction and consist in adding some elements on the main diagonal of the Hessian matrix to restore its positive definiteness if some eigenvalues are negative. A proper scaling procedure improves condition numbers and convergence behavior. The new method requires about the same computational effort per iteration and follows a similar convergence path for most points as the best previous formulation (Nichita and Petitfrere, Phase stability testing using a reduction method, Fluid Phase Equilibria 358, 27-39, 2013); for difficult points or possible outliers, the computation time per iteration slightly increases, but the number of iterations is significantly reduced.

Numerical experiments carried out for a variety of CO₂-hydrocarbon mixtures, by spanning P-T, P-Z and T-Z planes using refined grids proved the robustness and speed of the proposed method, with no failure and, unlike previous methods, with no important locally increase in the number of iterations near the STLL (this feature makes it particularly suited for a GPU architecture). Several pathological situations are identified and described, in which the proposed method clearly outperforms all previous formulations. The proposed method is the first reduction method using powerful tools for unconstrained minimization and taking full advantage of symmetry in construction of key matrices, treatment of non-positive definiteness cases and resolution of the linear system.