An Iterative Sequence for Phase-Equilibria Calculations Incorporating the Redlich-Kwong Equation of State
Authors: Fussell D.D. , Yanosik J.L. 
Published: December 2007

Abstract
Phase equilibria equations that incorporate the Redlich-Kwong equation of state are nonlinear and, therefore, must be solved by an iterative method. The method of successive substitutions commonly is used. This method, however, almost always diverges near the critical region for bubble point, dew point, and two-phase calculations. Iterative methods that converge for these calculations are presented. These iterative methods are called presented. These iterative methods are called "minimum variable Newton-Raphson" (MVNR) methods because they try to minimize the number of variable for which simultaneous iteration is required and use the Newton-Raphson method for the correction step. Procedures are given for obtaining starting values for the first iteration and several example problems are discussed.

Introduction
Reservoir performance predictions for gas condensate and volatile oil reservoirs require a knowledge of the vapor-liquid phase equilbria of the reservoir fluids. A similar knowledge also is required when studying multiple-contact, miscible oil recovery methods that involve injection of hydrocarbons and/or carbon dioxide. Such knowledge is obtained experimentally or calculated from physical properties of the components of the physical properties of the components of the reservoir fluid system. Calculation is desirable because experimental determination is both laborious and expensive.
A common basis for calculation of vapor-liquid phase equilibria is the single-stage separation unit. phase equilibria is the single-stage separation unit. This unit represents a PVT cell in which a fluid mixture of known over-all composition is equilibrated at the temperature and pressure of interest. Liquid and vapor compositions and moles of liquid and vapor per mole of fluid mixture are determined. Reliable estimates of other fluid properties (such as phase densities and viscosities) are obtained readily with these properties.
The Redlich-Kwong equation of state is used widely in the petroleum industry for phase equilibria calculations. The phase equilibria equations that incorporate this equation of state are nonlinear. As a result, they must be solved by an iterative method. The method of successive substitutions commonly is used. This method, however, almost always diverges for bubble point, dew point, and two-phase calculations near the critical region. This region is extremely important when studying multiple-contact, miscible oil recovery methods involving CO2 or rich-gas injection because the path of the over-all fluid mixture passes through path of the over-all fluid mixture passes through this region. The method of successive substitutions also will diverge for some fluid mixtures near their saturation (bubble point or dew point) pressure at conditions removed from the critical region.
This paper presents a reliable iterative sequence that can be used to predict phase equilibria of multiple-contact, miscible oil recovery methods. The method includes sequences for calculation of the saturation pressure and phase equilibria in the two phase region.
These MVNR methods rely on minimization of the number of unknowns for which simultaneous iteration is required and use the Newton-Raphson method for the correction step. Minimization is subject to the constraint that all additional unknowns can be calculated by using simple linear equations or, at most, an iteration method applied to one equation in one unknown.
MVNR is compared with the method of successive substitutions for a two-phase fluid mixture at various pressures for a fixed temperature. MVNR also is compared with the method of successive substitutions for saturation-envelope calculations near the critical region.

DESCRIPTION OF PHYSICAL SYSTEM
The single-stage separation unit is the basis for the phase equilibria calculations discussed in this study. This unit represents a PVT cell in which a fluid mixture of known over-all composition is equilibrated at the temperature and pressure of interest.
SPEJ
P. 173