Laboratory experiments have demonstrated theexistence of three hydrocarbon phases (two liquidphases and a vapor phase) for some reservoir fluid mixtures over certain ranges of pressure andtemperature. Minimum variable Newton-Raphson(MVNR) methods for calculating phase equilibriawhen three such phases exist are presented here.Methods for calculating conditions at which the thirdphase first appears also are presented. Examples of results are included. Introduction The study of multiple-contact, oil-recovery methodsinvolving injection of CO2 or hydrocarbon gasrequires a knowledge of vapor$liquid phaseequilibria. Phase-equilibria data may be obtained byexperimentation or by calculation. Calculationprocedures often are preferable because experimentaldata are costly, time consuming, and difficult tocorrelate. Laboratory experiments have shown thatthree hydrocarbon phases (two liquid phases and avapor phase) exist for some fluid mixtures (somemixtures of reservoir oil and CO2 or rich gas) overlimited ranges of pressure and temperature. These experimental results indicate that multiple phasesmay occur in the reservoir when enhancedoilrecovery methods are used. To predict the effects of multiple phases on the performance of thesemethods, one must calculate multiphase equilibria. This paper presents a technique for calculatingphase equilibria when three hydrocarbon phasesexist. This technique includes a system of nonlinearequations incorporating the Redlich-Kwong equationof state and an iterative sequence for solving theseequations. The iterative sequence is referred to asminimum variable Newton-Raphson (MVNR). Thissequence reduces the number of unknowns to becorrected and uses the Newton-Raphson method forthe correction step. Although the Redlich-Kwongequation of state was used here, theNewtonRaphson method can be used with any empirical equation of state. MVNR methods for calculating phase equilibriawhen two phases exist have been developed. Thesemethods have been shown to be superior to the method of successive substitutions in convergencerate and in calculations near the critical point.Therefore, these methods were used to develop a calculation technique for three phases. The system considered is a single-stage separationunit represented by a pressure/volume/temperaturecell in which a known fluid mixture is equilibrated ata particular pressure and temperature. Compositionsof the vapor phase and both liquid phases, the molesof each liquid phase, and the moles of the vaporphase then may be determined. From these results, other fluid properties, such as phase densities andviscosities, may be calculated. At a given temperature, three hydrocarbon phasesare known to exist for a particular fluid mixture overa relatively small range of pressure. The bounds ofthis pressure range are referred to as "upperconsolute pressure" and "lower consolute pressure." Atpressures below the lower consolute pressure, thefluid mixture is two-phase liquid/vapor. At pressuresabove the upper consolute pressure, the fluid mixtureis two-phase liquid/liquid. A technique also ispresented to calculate the upper and lower consolute pressures and the phase compositions that exist atthose pressures. SPEJ P. 203^