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Summary Efficient and robust phase equilibrium computation has become a prerequisite for successful large-scale compositional reservoir simulation. When knowledge of the number of phases is not available, the ideal strategy for phase-split calculation is the use of stability testing. Stability testing not only establishes whether a given state is stable, but also provides good initial guess for phase-split calculation. In this research, we present a general strategy for two- and three-phase split calculations based on reliable stability testing. Our strategy includes the introduction of systematic initialization of stability testing particularly for liquid/liquid and vapor/liquid/liquid equilibria. Powerful features of the strategy are extensively tested by examples including calculation of complicated phase envelopes of hydrocarbon fluids mixed with CO2 in single-, two-, and three-phase regions.
Zhao, Haining (China University of Petroleum, Beijing) | Jing, Hongbin (China University of Petroleum, Beijing) | Fang, Zhengbao (Xinjiang Oilfield Company, PetroChina) | Yu, Hongwei (Research Institute of Petroleum Exploration and Development, PetroChina)
Summary On the basis of a previously published reduced‐variables method, we demonstrate that using these reduced variables can substantially accelerate the conventional successive‐substitution iterations in solving two‐phase flash (TPF) problems. By applying the general dominant eigenvalue method (GDEM) to the successive‐substitution iterations in terms of the reduced variables, we obtained a highly efficient solution for the TPF problem. We refer to this solution as Reduced‐GDEM. The Reduced‐GDEM algorithm is then extensively compared with more than 10 linear‐acceleration and Newton‐Raphson (NR)‐type algorithms. The initial equilibrium ratio for flash calculation is generated from reliable phase‐stability analysis (PSA). We propose a series of indicators to interpret the PSA results. Two new insights were obtained from the speed comparison among various algorithms and the PSA. First, the speed and robustness of the Reduced‐GDEM algorithm are of the same level as that of the reduced‐variables NR flash algorithm, which has previously been proved to be the fastest flash algorithm. Second, two‐side phase‐stability‐analysis results indicate that the conventional successive‐substitution phase‐stability algorithm is time consuming (but robust) at pressures and temperatures near the stability‐test limit locus in the single‐phase region and near the spinodal in the two‐phase region.
Summary Flash calculations for use in compositional simulation are more difficult and time-consuming as the number of equilibrium phases increases beyond two. Because of its complexity, many simulators do not even attempt to incorporate three or more hydrocarbon phases, even though such cases are important in many low-temperature gasfloods or for high temperatures where hydrocarbons can partition into water. Multiphase flash algorithms typically use successive substitution (SS) followed by Newton's method. For NP-phase flash calculations, (NP–1) Rachford-Rice (RR) equations are solved in every iteration step in SS and, depending on the choice of independent variables, in Newton's method. Solution of RR equations determines both compositions and amounts of phases for a fixed overall composition and set of K-values. A robust algorithm for RR is critical to obtain convergence in multiphase compositional simulation and has not been satisfactorily developed, unlike the traditional two-phase flash. In this paper, we develop an algorithm for RR equations for multiphase compositional simulation that is guaranteed to converge to the correct solution independent of the number of phases for both positive and negative flash calculations. We derive a function whose gradient vector consists of RR equations. This correct solution to the RR equations is formulated as a minimization of the nonmonotonic convex function using the independent variables of (NP–1) phase mole fractions. The key to obtaining a robust algorithm is that we specify nonnegative constraints for the resulting equilibrium phase compositions, which are described by a very small region with no poles. The minimization uses Newton's direction with a line-search technique to exhibit superlinear convergence. We show a case in which a previously developed method cannot converge while our algorithm rapidly converges in a few iterations. We implement the algorithm both in a standalone flash code and in UTCOMP (Chang et al. 1990), a multiphase compositional simulator, and show that the algorithm is guaranteed to converge when a multiphase region exists as indicated by stability analysis.
Summary The conventional method for multiphase flash is the sequential usage of phase-stability and phase-split calculations. Multiphase flash requires the conventional method to obtain multiple false solutions in phase-split calculations and correct them in phase-stability analysis. Improvement of the robustness and efficiency of multiphase flash is important for compositional flow simulation with complex phase behavior. This paper presents a new algorithm that solves for stationary points of the tangent-plane-distance (TPD) function defined at an equilibrium-phase composition for isobaric-isothermal (PT) flash. A solution from the new algorithm consists of two groups of stationary points: tangent and nontangent stationary points of the TPD function. Hence, equilibrium phases, at which the Gibbs free energy is tangent to the TPD function, are found as a subset of the solution. Unlike the conventional method, the new algorithm does not require finding false solutions for robust multiphase flash. The advantage of the new algorithm in terms of robustness is more pronounced for more-complex phase behavior, for which multiple local minima of the Gibbs free energy are present. Case studies show that the new algorithm converges to a lower Gibbs free energy compared with the conventional method for the complex fluids tested. It is straightforward to implement the algorithm because of the simple formulation, which also allows for an arbitrary number of iterative compositions. It can be robustly initialized even when no K value correlation is available for the fluid of interest. Although the main focus of this paper is on robust solution of multiphase flash, the new algorithm can be used to initialize a second-order convergent method in the vicinity of a solution. Introduction A multiphase equilibrium calculation for a fixed pressure (P) and temperature (T) requires global minimization of the Gibbs free energy subject to material balance. The conventional algorithms after Michelsen (1982a, b) are based on the sequential usage of phase-stability and -split calculations. That is, a phase-stability calculation is performed for the overall composition specified or one of the phases from a multiphase solution, at which the plane tangent to the Gibbs free energy surface is defined.
Pan, Huanquan (Stanford University) | Imai, Motonao (Japan Oil, Gas and Metals National Corporation/Waseda University) | Connolly, Michael (Stanford University) | Tchelepi, Hamdi (Stanford University)
Abstract Robust and efficient multiphase flash calculations are crucial in compositional and thermal simulations for complex fluid systems in which three or four phase may co-exist. Solution of the Richford-Rice (RR) equations is an important operation in the multiphase flash. The Newton method generally does not converge during solution of the RR equations unless very good initial values are provided. In this paper, the solution of the RR equations is formulated as a minimization of a convex function problem. For the first time, we use a trust-region (TR) method to solve the RR equations through minimization of the convex function. The Hessian matrix of the convex function is always positive-definite, and the TR-based solver guarantees convergence. The key to successful implementation is to determine the relaxation parameter in the Newton update. We select this relaxation parameter to meet the boundary of the objective function and to ensure an adequate step length. We tested the RR solver for three and four phase RR problems in the construction of phase diagrams. The test cases are representative of complex fluid systems encountered in enhanced oil recovery, including injection of CO2 into low temperature reservoirs and steam injection into heavy oil reservoirs at elevated temperatures. We performed tens of millions of multiphase flash computations, the results of which reveal our RR solver to be robust, efficient, insensitive to initial values, and capable of handling negative phase amounts. We also evaluated the effect of the initial values on convergence and recommend methods to estimate the initial values in our RR solver. In summary, our RR solver greatly improves the multiphase flash calculations and strengthens the coupling of phase equilibrium calculations to the governing equations in multiphase compositional and thermal simulation.