A New Algorithm for Rachford-Rice for Multiphase Compositional Simulation
- Ryosuke Okuno (University of Texas at Austin) | Russell Johns (University of Texas at Austin) | Kamy Sepehrnoori (University of Texas at Austin)
- Document ID
- Society of Petroleum Engineers
- SPE Journal
- Publication Date
- June 2010
- Document Type
- Journal Paper
- 313 - 325
- 2010. Society of Petroleum Engineers
- 5.3.1 Flow in Porous Media, 5.5 Reservoir Simulation, 5.3.2 Multiphase Flow, 5.4 Enhanced Recovery, 5.2.2 Fluid Modeling, Equations of State, 5.6.4 Drillstem/Well Testing, 4.6 Natural Gas, 5.2.1 Phase Behavior and PVT Measurements, 5.4.2 Gas Injection Methods
- gasflooding, multiphase flash calculations, multiphase behavior, Rachford-Rice equations, compositional simulation
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- 1,370 since 2007
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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 gas floods, 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 non-monotonic convex function using the independent variables of (NP-1) phase mole fractions. The key to obtaining a robust algorithm is that we specify non-negative 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 super-linear 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 stand-alone flash code and in UTCOMP, a multiphase compositional simulator, and show that the algorithm is guaranteed to converge when a multiphase region exists as indicated by stability analysis.
|File Size||963 KB||Number of Pages||13|
Baker, L.E., Pierce, A.C., and Luks, K.D. 1982. Gibbs Energy Analysis of PhaseEquilibria. SPE J. 22 (5): 731-742. SPE-9806-PA. doi:10.2118/9806-PA.
Bertsekas, D.P. 1999. Nonlinear Programming, second edition. Nashua,New Hampshire: Athena Scientific.
Bünz, A.P., Dohrn, R., and Prausnitz, J.M. 1991. Three-Phase FlashCalculations for Multicomponent Systems. Computers & ChemicalEngineering 15 (1): 47-51. doi:10.1016/0098-1354(91)87005-T.
Chang, Y.-B. 1990. Development and Application of an Equation of StateCompositional Simulator. PhD dissertation, University of Texas at Austin,Austin, Texas.
Chang, Y.-B., Pope, G.A., and Sepehrnoori, K. 1990. A Higher-OrderFinite-Difference Compositional Simulator. J. Pet. Sci. Eng. 5 (1): 35-50. doi: 10.1016/0920-4105(90)90004-M.
Dennis, J.E. Jr. and Schnabel, R.B. 1996. Numerical Methods forUnconstrained Optimization and Nonlinear Equations, No. 16. Philadelphia,Pennsylvania: Classics in Applied Mathematics, SIAM.
Eubank, P.T. 2006. Equations and Proceduresfor VLLE Calculations. Fluid Phase Equilibria 241(1-2): 81-85. doi: 10.1016/j.fluid.2005.11.011.
Haugen, K.B., Sun, L., and Firoozabadi, A. 2007. Three-Phase Equilibrium Calculationsfor Compositional Simulation. Paper SPE 106045 presented at the SPEReservoir Simulation Symposium, Houston, 26-28 February. doi:10.2118/106045-MS.
Khan, S.A., Pope, G.A., and Sepehrnoori, K. 1992. Fluid Characterization of Three-PhaseCO2/Oil Mixtures. Paper SPE 24130 presented at the SPE/DOE Enhanced OilRecovery Symposium, Tulsa, 22-24 April. doi: 10.2118/24130-MS.
Leibovici, C.F. and Neoschil, J. 1995. A Solution ofRachford-Rice Equations for Multiphase Systems. Fluid PhaseEquilibria 112 (2): 217-221. doi:10.1016/0378-3812(95)02797-I.
Leibovici, C.F. and Nichita, D.V. 2008. A New Look at MultiphaseRachford-Rice Equations for Negative Flashes. Fluid Phase Equilibria 267 (2): 127-132. doi: 10.1016/j.fluid.2008.03.006.
Michelsen, M.L. 1982a. The Isothermal FlashProblem. Part II. Phase-Split Calculation. Fluid Phase Equilibria 9 (1): 21-40. doi: 10.1016/0378-3812(82)85002-4.
Michelsen, M.L. 1982b. The Isothermal FlashProblem. Part I. Stability. Fluid Phase Equilibria 9(1): 1-19. doi: 10.1016/0378-3812(82)85001-2.
Michelsen, M.L. 1994. Calculation of MultiphaseEquilibrium. Computers & Chemical Engineering 18(7): 545-550. doi: 10.1016/0098-1354(93)E0017-4.
Michelsen, M.L. and Mollerup, J.M. 2004. Thermodynamic Models:Fundamentals and Computational Aspects. Holte, Denmark: Tie-Line Press.
Nelson, P.A. 1987. Rapid Phase Determinationin Multiple-Phase Flash Calculations. Computers & ChemicalEngineering 11 (6): 581-591. doi:10.1016/0098-1354(87)87004-7.
Nghiem, L.X. and Li, Y.K. 1986. Effect of Phase Behavior on CO2Displacement Efficiency at Low Temperatures: Model Studies With an Equation ofState. SPE Res Eng 1 (4): 414-422. SPE-13116-PA. doi:10.2118/13116-PA.
Okuno, R, Johns, R.T., and Sepehrnoori, K. 2009. Three-Phase Flash inCompositional Simulation Using a Reduced Method. Accepted for SPE J.SPE-125226-PA.
Okuno, R. 2009. Modeling of Multiphase Behavior for Gas Flooding Simulation.PhD dissertation, University of Texas at Austin, Austin, Texas (August2009).
Peng, D-Y. and Robinson, D.B. 1976. A New Two-Constant Equation ofState. Ind. Eng. Chem. Fund. 15 (1): 59-64. doi:10.1021/i160057a011.
Perschke, D.R., Chang, Y.-B., Pope, G.A., and Sepehrnoori, K. 1989.Comparison of Phase Behavior Algorithms for an Equation-of-State CompositionalSimulator. Paper SPE 19443 available from SPE, Richardson, Texas.
Rachford, H.H. Jr. and Rice, J.D. 1952. Procedurefor Use of Electronic Digital Computers in Calculating Flash VaporizationHydrocarbon Equilibrium (Technical Note 136). J. Pet Tech 4 (10): 19; Trans., AIME, 195: 327-328.SPE-952327-G.
Wang, X. and Strycker, A. 2000. Evaluation of CO2 Injection WithThree Hydrocarbon Phases. Paper SPE 64723 presented at the InternationalOil and Gas Conference and Exhibition in China, Beijing, 7-10 November. doi:10.2118/64723-MS.
Whitson, C.H. and Michelsen, M.L. 1989. The Negative Flash.Fluid Phase Equilibria 53 (December): 51-71. doi:10.1016/0378-3812(89)80072-X.
Wilson, G.M. 1969. A Modified Redlich-Kwong Equation of State, Applicationto General Physical Data Calculations. Paper 15-C presented at the 65thNational AIChE Meeting, Cleveland, Ohio, USA, 4-7 May.
Young, L.C. and Stephenson, R.E. 1983. A Generalized Compositional Approachfor Reservoir Simulation. SPE J. 23 (4): 727-742.SPE-10516-PA. doi: 10.2118/10516-PA.
Yuan, H. and Johns, R.T. 2005. Simplified Method for Calculation ofMinimum Miscibility Pressure or Enrichment. SPE J. 10(4): 416-425. SPE-77381-PA. doi: 10.2118/77381-PA.