An Iterative Sequence for Phase-Equilibria Calculations Incorporating the Redlich-Kwong Equation of State

1978 ◽  
Vol 18 (03) ◽  
pp. 173-182 ◽  
Author(s):  
D.D. Fussell ◽  
J.L. Yanosik
Keyword(s):  
Equation Of State ◽  
Phase Equilibria ◽  
Oil Recovery ◽  
Critical Region ◽  
Single Stage ◽  
Dew Point ◽  
Two Phase ◽  
Fluid Mixture ◽  
Separation Unit ◽  
Stage Separation

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

An Iterative Technique for Compositional Reservoir Models

1979 ◽  
Vol 19 (04) ◽  
pp. 211-220 ◽  
Author(s):  
L.T. Fussell ◽  
D.D. Fussell
Keyword(s):  
Phase Equilibria ◽  
Iterative Methods ◽  
Oil Recovery ◽  
Single Stage ◽  
Flow Equations ◽  
Reservoir Models ◽  
Stage Separation ◽  
Grid Block ◽  
Correction Step ◽  
Recovery Methods

Original manuscript received in Society f Petroleum Engineers office Sept. 15, 1977. Paper accepted for publication July 11, 1978. Revised manuscript received March 26, 1979. Paper (SPE 6891) first presented at the SPE-AIME 52nd Annual Fall Technical Conference and Exhibition, held in Denver, Oct. 9-12, 1977. Abstract Compositional reservoir models are a class of phase equilibria models. The equations for these models, and especially those incorporating the Redlich-Kwong equation of state, are highly nonlinear and must be solved by an iterative method. The method of successive substitution commonly is used. This method, however, demonstrates linear convergence and almost always diverges or, at best, demonstrates poor convergence, for conditions where phase poor convergence, for conditions where phase equilibria calculations are required near the bubble point, dewpoint, or the critical point. Iterative point, dewpoint, or the critical point. Iterative methods that converge for these calculations are presented here. These iterative methods are called presented here. These iterative methods are called minimum-variable Newton-Raphson (MVNR) since they attempt to minimize the number of variables for which simultaneous iteration is required and use the Newton-Raphson method for the correction step. These methods demonstrate quadratic convergence near the solution. MVNR can be applied to one-, two-, and three-dimensional geometries. The hydrocarbon fluid is represented as an n component system. Flow equations for the water phase also are included; however, mass transfer between the hydrocarbon and waterphases is assumed negligible. The correction step uses a reordering technique that reduces the storage requirement and the computer cost per time step per grid block. An example problem is presented and discussed. problem is presented and discussed. Introduction Compositional reservoir models are used to predict the performance of oil-recovery methods when interphase mass transfer depends on phase composition as well as pressure. These methods include depletion or cycling of volatile oil, high-shrinkage oil, gas condensate reservoirs, and enhanced oil recovery (e.g., CO2 and rich-gas displacements). Most compositional models presented in the literature use table lookup K values (equilibrium ratios) or fits of the NGPA K values to describe the equilibrated distribution of components between phases. For complex oil-recovery methods, these phases. For complex oil-recovery methods, these techniques also require a convergence pressure correlation that accounts for the composition dependence on phase equilibria. One primary disadvantage of these techniques is that the predicted phase equilibria are not internally consistent - i.e. phase equilibria are not internally consistent - i.e. derivatives of thermodynamic quantities are not necessarily smooth or even continuous. This often leads to nonconvergence of the reservoir model. Using an equation of state removes this limitation and, furthermore, reduces the time and costs required for preparing data before reservoir simulation. Another significant advantage is that the phase-equilibria and flow equations can be used simultaneously to solve all variables. Table lookup models, presented in the literature, separate the flow equations from the phase-equilibria equations during the iteration toward the solution. Fussell and Yanosik presented MVNR iterative methods for solving the equations for another phase-equilibria model - the single-stage separation phase-equilibria model - the single-stage separation unit. This study extends MVNR methods to the more complex, fully compositional models. Our methods demonstrate the same advantages of MVNR methods for the single-stage separation equations. The correction step of MVNR methods for compositional models includes a set of n + 1 equations for each grid block of the reservoir system, where n is the number of components in the fluid system. SPEJ P. 211

Compositional Model Studies - CO2 Oil-Displacement Mechanisms

1981 ◽  
Vol 21 (01) ◽  
pp. 89-97 ◽  
Author(s):  
M.P. Leach ◽  
W.F. Yellig
Keyword(s):  
Equation Of State ◽  
Phase Equilibria ◽  
Oil Recovery ◽  
Petroleum Engineering ◽  
Contact Phase ◽  
Permeability Characteristics ◽  
Compositional Model ◽  
Oil Displacement ◽  
Synthetic Oil ◽  
Mechanistic Understanding

Abstract This paper presents matches, using a fully compositional model, of the performances of seven laboratory CO2 displacements of a 10-component synthetic oil. The criteria for achieving a match of laboratory performance include (1) comparisons of predicted and experimentally determined oil recovery and (2) effluent compositional profiles for each component as functions of hydrocarbon pore volumes (HCPV) of CO2 injected.An equation of state was tuned to predict single-contact (PVT) phase equilibria for CO2/synthetic-oil mixtures. The model incorporates this equation of state to predict the multiple-contact phase equilibria during a CO2 displacement test. Input to the model were independently determined gas/oil relative permeability characteristics and - for each laboratory displacement - injection rate, effluent pressure, pore volume, and temperature.The experimental displacements were conducted in linear Berea core systems using a synthetic (C1-C14) oil at 120 and 150 degrees F. Three displacements at 120 degrees F have been published by Metcalfe and Yarborough. Previously, it was thought that these displacements were conducted at selected pressures so that oil displacements encompassed immiscible, multiple-contact miscible (MCM), and contact miscible mechanisms. However, the model results show that only contact miscible and MCM displacement mechanisms were involved. To confirm the mechanistic understanding at 120 degrees F, three additional laboratory displacements were conducted at 150 degrees F. These encompassed pressures such that the displacement was controlled by an immiscible, an MCM, and a contact miscible mechanism, respectively. The model results at 150 degrees F match the experimental data and confirm the mechanistic understanding. The experimental and numerical results are in agreement with the minimum miscibility pressure theory of Yellig and Metcalfe.The results of this study confirm the importance of experimentally determined effluent compositional profiles and fully compositional models for CO2 mechanism studies. Introduction A mechanistic understanding of oil displacement by CO2 is basic to establishing the CO2 requirements and predicting performance for field projects. The petroleum engineering community relies heavily on fully compositional models and sophisticated laboratory experiments to acquire this mechanistic understanding.Currently, it is not deemed feasible to use fully compositional models to simulate performance of field-wide miscible floods. However, these models should be capable of predicting performances of laboratory-scale displacements. The results of such predictions will identify important variables that control oil recovery and that must be incorporated in mechanistically simpler field performance simulators. Further, confidence in field models will be enhanced greatly by the demonstrated ability to predict laboratory floods.Studies to improve the effectiveness and efficiency of multicomponent compositional simulators have been reported. A cell-to-cell flash model has been used to study mechanisms in rich-gas drives. However, no investigations have been reported previously that demonstrate that a fully compositional model can predict results of rich gas or CO2 laboratory displacements, including prediction of the phases and compositions developed in situ. SPEJ P. 89^

Critical Point and Saturation Pressure Calculations for Multipoint Systems (includes associated paper 8871 )

1980 ◽  
Vol 20 (01) ◽  
pp. 15-24 ◽  
Author(s):  
L.E. Baker ◽  
K.D. Luks
Keyword(s):  
Equation Of State ◽  
Critical Point ◽  
Critical Region ◽  
Equations Of State ◽  
Saturation Pressure ◽  
Direct Calculation ◽  
Iterative Solution ◽  
Dew Point ◽  
K Value ◽  
Bubble Point

Abstract Calculation of fluid properties and phase equilibria isimportant as a general reservoir engineering tool andfor simulation of the carbon dioxide or rich gasmultiple-contact-miscibility (MCM) mechanisms. Of particular interest in such simulations is thenear-critical region, through which the compositionalpath must go in an MCM process.This paper describes two mathematical techniquesthat enhance the utility of an equation of state forphase equilibrium calculations. The first is animproved method of estimating starting parameters(pressure and phase compositions) for the iterativesaturation pressure (bubble-point or dew-point)solution of the equation of state. Techniquespreviously have been presented for carrying out thisiterative solution; however, the previously describedprocedure for obtaining initial parameter values wasnot satisfactory in all cases. The improved methodutilizes the equation of state to estimate theparameter values. Since the same equation then isused to calculate the saturation pressure, the methodis self-consistent and results in improved reliability.The second development is the use of the equationof state to calculate directly the critical point of afluid mixture, based on the rigorous thermodynamic criteria set forth by Gibbs. The paper presents aniterative method for solving the highly nonlinearequations. Methods of obtaining initial estimates ofthe critical temperature and pressure also arepresented.The techniques described are illustrated withreference to a modified version of the Redlich-Kwong equation of state (R-K EOS); however, theyare applicable to other equations of state. They havebeen used successfully for a wide variety of reservoirfluid systems, from a simple binary to complexreservoir oils. Introduction MCM processes such as CO2 or rich gas miscibledisplacements (conducted at pressures below thecontact-miscible pressure) traverse a compositional path that goes through the near-critical region. Thishas been described in several papers. Simulationof an MCM process requires the use of an equation of state to describe the liquid- and vapor-phasesaturations and compositions. Fussell and Yanosikdescribed an MVNR (minimum-variableNewton-Raphson)method for solution of a version of theR-K EOS. They discussed some of the difficulties ofobtaining solutions to the equation of state in the near-critical region and showed that the MVNRmethod gave improved results.Experience with the MVNR method hasdemonstrated a need for an improved estimate ofinitial iteration parameters (pressure, phasecompositions)for an iterative solution of saturation(dew-point and bubble-point) pressures. It waslearned that the semitheoretical K-value correlationused for initial estimates usually gave satisfactoryresults when the fluid system contained significantamounts of heavy components (C7+) but was oftenunsatisfactory for fluid systems containing only lightcomponents. This type of system is exemplified bythe fluids in a dry gas-rich gas mixing zone or bymixtures rich in CH4, CO2, or N2.Experience also has demonstrated a need for direct calculation of the critical point. While the MVNRsolution technique discussed by Fussell and Yanosikexhibits improved convergence in the near-critical region, it often is difficult to obtain convergedsolutions of the equation of state at compositionswithin a few percent of the critical composition. SPEJ P. 15＾

Gibbs Energy Analysis of Phase Equilibria

1982 ◽  
Vol 22 (05) ◽  
pp. 731-742 ◽  
Author(s):  
L.E. Baker ◽  
A.C. Pierce ◽  
K.D. Luks
Keyword(s):  
Phase Equilibrium ◽  
Equation Of State ◽  
Gibbs Energy ◽  
Phase Equilibria ◽  
Oil Recovery ◽  
Equilibrium Problems ◽  
Equations Of State ◽  
Material Balance ◽  
Chemical Potentials ◽  
Temperature And Pressure

Abstract Equations of state are used to predict or to match equilibrium fluid phase behavior for systems as diverse as distillation columns and miscible gas floods of oil reservoirs. The success of such simulations depends on correct predictions of the number and the compositions of phases present at a given temperature, pressure, and overall fluid composition. For example, recent research has shown that three or more phases may exist in equilibrium in CO floods. This paper shows why an equation of state can predict the incorrect number of phases or incorrect phase compositions. The incorrect phase descriptions still satisfy the usual restrictions on equality of chemical potentials of components in each phase and conservation of moles in the system. A new method and its mathematical proof are presented for determining when a phase equilibrium solution is incorrect. Examples of instances where incorrect predictions may be made are described. These include a binary system in which a two-phase solution may be predicted for a single-phase fluid and a multicomponent CO /reservoir oil system in which three or more phases may coexist. Introduction Advances in reservoir oil recovery methods have necessitated advances in methods for prediction of phase equilibria associated with those methods. It was long considered sufficient to approximate the reservoir behavior of oil and gas systems with models in which compositions of the phases in equilibrium were unimportant. In such a model, the amounts and properties of the phases are dependent on pressure and temperature only. Later, experience in production from condensate and volatile oil reservoirs showed that models incorporating compositional effects were required to simulate the phase equilibria adequately. This led to the use of convergence pressure correlations and subsequently to the development of more sophisticated equation of state methods for modeling and predicting phase equilibria. For adequate description of the compositional effects that occur in enhanced oil recovery processes such as CO and rich gas flooding, an equation-of-state approach is a virtual necessity. equilibrium The use of equations of state for phase prediction is not limited to the petroleum industry. Such equations also find wide use in basic chemical and physical research, and in the refining and chemical processing industries. Solution techniques for phase equilibrium problems are varied and depend to some extent on the application and equation of state used however, there are three restrictions that all phase equilibrium solutions must satisfy. First, material balance must be preserved. Second, for phases in equilibrium there must be no driving force to cause a net movement of any component from one phase to any other phase. In thermodynamic parlance, the chemical potentials for each component must be the same in all phases. Third, the system of predicted phases at the equilibrium state must have the lowest possible Gibbs energy at the system temperature and pressure. The requirement that the Gibbs energy of a system. at a given temperature and pressure, must be a minimum is a statement of the second law of thermodynamics, equivalent to the more common version requiring the entropy of an isolated system to be a maximum. The equivalence is demonstrated formally in Ref. 1, for example. If the Gibbs energy of a predicted equilibrium state is greater than that of another state that also satisfies Requirements 1 and 2, the state with the greater Gibbs energy is not thermodynamically stable. Requirements 1 and 2, material balance and equality of chemical potentials, are used commonly as the sole criteria for solution of phase equilibrium problems. SPEJ P. 731^

Critical consolute point in hard-sphere binary mixtures: Effect of the value of the eighth and higher virial coefficients on its location

2010 ◽  
Vol 75 (3) ◽  
pp. 359-369 ◽  
Author(s):  
Mariano López De Haro ◽  
Anatol Malijevský ◽  
Stanislav Labík
Keyword(s):  
Equation Of State ◽  
Binary Mixtures ◽  
Hard Sphere ◽  
Hard Spheres ◽  
Virial Coefficients ◽  
Fluid Mixture ◽  
Binary Fluid ◽  
Virial Series ◽  
Consolute Point ◽  
Critical Constants

Various truncations for the virial series of a binary fluid mixture of additive hard spheres are used to analyze the location of the critical consolute point of this system for different size asymmetries. The effect of uncertainties in the values of the eighth virial coefficients on the resulting critical constants is assessed. It is also shown that a replacement of the exact virial coefficients in lieu of the corresponding coefficients in the virial expansion of the analytical Boublík–Mansoori–Carnahan–Starling–Leland equation of state, which still leads to an analytical equation of state, may lead to a critical consolute point in the system.

A Rational Equation of State for Water and Water Vapor in the Critical Region

1964 ◽  
Vol 86 (3) ◽  
pp. 320-326 ◽  
Author(s):  
E. S. Nowak
Keyword(s):  
Water Vapor ◽  
Equation Of State ◽  
Thermodynamic Properties ◽  
Specific Heat ◽  
Critical Point ◽  
Critical Region ◽  
Constant Pressure ◽  
Parametric Equation ◽  
Enthalpy And Entropy ◽  
Specific Volumes

A parametric equation of state was derived for water and water vapor in the critical region from experimental P-V-T data. It is valid in that part of the critical region encompassed by pressures from 3000 to 4000 psia, specific volumes from 0.0400 to 0.1100 ft3/lb, and temperatures from 698 to 752 deg F. The equation of state satisfies all of the known conditions at the critical point. It also satisfies the conditions along certain of the boundaries which probably separate “supercritical liquid” from “supercritical vapor.” The equation of state, though quite simple in form, is probably superior to any equation heretofore derived for water and water vapor in the critical region. Specifically, the deviations between the measured and computed values of pressure in the large majority of the cases were within three parts in one thousand. This coincides approximately with the overall uncertainty in P-V-T measurements. In view of these factors, the author recommends that the equation be used to derive values for such thermodynamic properties as specific heat at constant pressure, enthalpy, and entropy in the critical region.

Analysis of Single- and Two-Phase Flows in Turbopump Inducers

1967 ◽  
Vol 89 (4) ◽  
pp. 577-586 ◽  
Author(s):  
P. Cooper
Keyword(s):  
Equation Of State ◽  
Flow Field ◽  
Thermal Effects ◽  
Single Phase ◽  
Fluid Pressure ◽  
Three Dimensional ◽  
Pressure Rise ◽  
Recent Theory ◽  
Two Phase ◽  
Overall Efficiency

A model is developed for analytically determining pump inducer performance in both the single-phase and cavitating flow regimes. An equation of state for vaporizing flow is used in an approximate, three-dimensional analysis of the flow field. The method accounts for losses and yields internal distributions of fluid pressure, velocity, and density together with the resulting overall efficiency and pressure rise. The results of calculated performance of two sample inducers are presented. Comparison with recent theory for fluid thermal effects on suction head requirements is made with the aid of a resulting dimensionless vaporization parameter.

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