Fast and Robust Algorithm for Compositional Modeling: Part II - Two-Phase Flash Computations
- Huanquan Pan (Reservoir Engineering Research Inst.) | Abbas Firoozabadi (Reservoir Engineering Research Inst.)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- December 2003
- Document Type
- Journal Paper
- 380 - 391
- 2003. Society of Petroleum Engineers
- 4.6 Natural Gas, 5.2.1 Phase Behavior and PVT Measurements, 5.2.2 Fluid Modeling, Equations of State, 5.5 Reservoir Simulation
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In Part I of our study, stability analysis testing in the reduced space was formulated, and its robustness and efficiency in comparison to the conventional approach was explored. In this paper, we present formulations including, first, direct solution of the nonlinear equations, and second, minimization of Gibbs free energy for two-phase flash computations in the reduced space. We use various algorithms including the successive substitution (SS), Newton's method, globally convergent modifications of Newton's method (line searches and trust region), and the dominant eigenvalue method (DEM) for direct solution of the nonlinear equations defining two-phase flash and the minimization of Gibbs free energy. We also suggest a criterion based on the tangent-plane-distance (TPD) for the initialization from the equilibrium ratios. The proposed criterion has a significant effect on reducing the number of iterations.
The results from various algorithms reveal that the direct solution of the nonlinear equations in the reduced space, combined with the use of the TPD criterion for initialization in the combined SS and Newton's method, can make flash computations extremely efficient. The efficiency and robustness of flash computations in the critical region are especially remarkable.
Two-phase vapor-liquid equilibrium computations are one important aspect of phase behavior computations in compositional modeling. In a recent study (to be published later) we have noticed that compositional modeling of a rich-retrograde gas condensate heterogeneous reservoir can be a real challenge. The main complexities in this particular example are that the appearance of two phases can occur in any part of the reservoir, and that there is no systematic pattern in the two-phase and single-phase regions.
In Part I of our work on fast and robust algorithm for compositional modeling, stability analysis testing in the reduction method was presented.1 In this paper, Part II of the work, flash computations in the two-phase gas-oil mixture are described.
To the best of our knowledge, the phase-split calculations in compositional modeling technology are presently based on the conventional method with dependent variables such as the mole numbers or mole fractions. Two approaches, direct solution of the nonlinear isofugacity equations and minimization of the Gibbs free energy, have been used for two-phase flash computations. The algorithms of SS,2 DEM,3 and Newton's method have been used to perform the direct solution of nonlinear equations. The SS algorithm keeps a descent direction and eventually converges.4 The major shortcoming of the SS algorithm is that it is extremely slow in the critical region; even thousands of iterations may not suffice for a single flash computation in the critical region. The DEM algorithm can accelerate the SS, and has at least double the speed of the SS (from our experience). However, the DEM algorithm may not have the robustness of the SS. Newton's algorithm converges quadratically, but only locally. In compositional reservoir simulators, often the SS and Newton's algorithms are combined (first SS is used) to solve the systems of nonlinear equation in the conventional method. Globally convergent modifications of Newton's method5 (the so-called quasi-Newton method), including the line search approach and the trust region approach, have also been used to minimize the Gibbs free energy6,7 with the conventional variables.
The numerical solution of the nonlinear equations in the reduced space has been addressed by the SS and Newton's methods in the literature.8-11 The central theme of papers on the reduction method is the demonstration that the number of variables can be reduced significantly without loss of accuracy. No attempt has yet been made to explore the numerical efficiency and robustness of various algorithms in the reduced space.
The main goal of this work is to study flash computations in the reduced space for vapor-liquid equilibria using various algorithms, and to propose ideas for robustness and efficiency of computations. Our work will include both the direct solution of the nonlinear isofugacity equations and the minimization of Gibbs free energy in the reduced space. The algorithms of SS, DEM, Newton's, and the globally convergent modifications of Newton's method will be implemented in our calculations.
In this paper, we will first provide a short review of the reduction method followed by the expression for the Gibbs free energy in the reduced space. Then, various algorithms are presented to minimize the Gibbs free energy and solve the system of nonlinear equations that define the vapor-liquid equilibrium in the reduced space. In the Results section, we compare various algorithms and then present the computations for five different mixtures. In the last section, in addition to concluding remarks, the key conclusions from the work are presented.
Brief Description of Reduction Method
The reduction method and the reduced-variable space are presented in Ref. 1 for stability analysis. Here, we will briefly review the reduced-variable space for the two-phase flash computations.
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