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＾

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.

Determination of Bubble-Point and Dew-Point Pressure Without a Visual Cell

2014 ◽  
Author(s):  
R.. Hosein ◽  
R.. Mayrhoo ◽  
W. D. McCain
Keyword(s):  
Oil And Gas ◽  
Saturation Pressure ◽  
Volatile Oil ◽  
Phase Region ◽  
Gas Condensate ◽  
Dew Point ◽  
Bubble Point ◽  
Point Pressure ◽  
Dew Point Pressure

Abstract Bubble-point and dew-point pressures of oil and gas condensate reservoir fluids are used for planning the production profile of these reservoirs. Usually the best method for determination of these saturation pressures is by visual observation when a Constant Mass Expansion (CME) test is performed on a sample in a high pressure cell fitted with a glass window. In this test the cell pressure is reduced in steps and the pressure at which the first sign of gas bubbles is observed is recorded as bubble-point pressure for the oil samples and the first sign of liquid droplets is recorded as the dew-point pressure for the gas condensate samples. The experimental determination of saturation pressure especially for volatile oil and gas condensate require many small pressure reduction steps which make the observation method tedious, time consuming and expensive. In this study we have extended the Y-function which is often used to smooth out CME data for black oils below the bubble-point to determine saturation pressure of reservoir fluids. We started from the initial measured pressure and volume and by plotting log of the extended Y function which we call the YEXT function, with the corresponding pressure, two straight lines were obtained; one in the single phase region and the other in the two phase region. The point at which these two lines intersect is the saturation pressure. The differences between the saturation pressures determined by our proposed YEXT function method and the observation method was less than ± 4.0 % for the gas condensate, black oil and volatile oil samples studied. This extension of the Y function to determine dew-point and bubble-point pressures was not found elsewhere in the open literature. With this graphical method the determination of saturation pressures is less tedious and time consuming and expensive windowed cells are not required.

A Robust Iterative Method for Flash Calculations Using the Soave-Redlich-Kwong or the Peng-Robinson Equation of State

1983 ◽  
Vol 23 (03) ◽  
pp. 521-530 ◽  
Author(s):  
L.X. Nghiem ◽  
K. Aziz ◽  
Y.K. Li
Keyword(s):  
Equation Of State ◽  
Iterative Method ◽  
Critical Point ◽  
Newton’S Method ◽  
Newton's Method ◽  
Single Phase ◽  
Saturation Pressure ◽  
Phase Region ◽  
Single Phase Region ◽  
Poor Convergence

Abstract A robust algorithm for flash calculations that uses an equation of state(EOS) is presented. It first uses a special version of the successive substitution(SS) method and switches to Powell's method if poor convergence is observed. Criteria are established for an efficient switch from one method to the other. Experience shows that this method converges near the critical point and also detects the single-phase region without computing the saturation pressure. The Soave-Redlich-Kwong (SRK) and the Peng-Robinson (PR) EOS's are used in this work, but the method is general and applies to any EOS. Introduction The calculation of vapor/liquid equilibrium using an EOS in multicomponent systems yields a system of nonlinear equations that must be solved iteratively. The SS method is commonly used, but it exhibits poor rate of convergence near the critical point. To overcome convergence problems, Newton's method has been used by Fussell and Yanosik to solve the equations. The drawback of Newton's method is the necessity of computing a complicated Jacobian matrix and its inverse at every iteration. Hence, for systems removed from their critical point it involves more work to arrive at the solution than the SS method. Furthermore, the radius of convergence of Newton's method is relatively small when compared to that of the SS method; hence, a good initial guess is required before convergence can be achieved. The single-phase region usually is determined by computing the saturation pressure and comparing it with the pressure of the system. This requires additional work, pressure of the system. This requires additional work, and it is sometimes difficult to decide whether a dewpoint or bubblepoint pressure, which involve different equations, should be computed. This paper presents a robust iterative method for flash calculations using either the SRK or the PR EOS, both of which have received much interest in recent years. The proposed method combines SS with Powell's iteration, proposed method combines SS with Powell's iteration, which is a hybrid algorithm consisting of a quasi-Newton method and a steepest-descent method. The SS method is used initially and is replaced by Powell's method if it demonstrates poor convergence, thus taking advantage of the simplicity of the former method and the robustness of the latter. The SS method has been modified so that the single-phase region can be detected without having to compute the saturation pressure. The nonlinear equations to be solved by an iteration scheme could behave differently, depending on their form and the variables for which they are solved. In this paper three different approaches are considered with paper three different approaches are considered with Powell's method. One of the three approaches is based Powell's method. One of the three approaches is based on the minimization of the Gibbs free energy. The convergence properties of the proposed schemes are demonstrated by three example problems. SPEJ P. 521

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

Prediction of Liquid Molar Volume and Heat of Vaporization of Fatty Acids Using an Equation of State

2020 ◽  
Vol 10 (3) ◽  
pp. 189-198
Author(s):  
Michelle G. Gomes ◽  
Nattácia R. A.F. Rocha ◽  
Alex A. Moura ◽  
Nadine P. Merlo ◽  
Moilton R. Franco Júnior ◽  
...  
Keyword(s):  
Fatty Acids ◽  
Equation Of State ◽  
Molar Volume ◽  
Equations Of State ◽  
Saturation Pressure ◽  
Reference Value ◽  
Vaporization Enthalpy ◽  
Dodecanoic Acid ◽  
Good Reliability ◽  
Heat Of Vaporization

Background:: The liquid molar volume (V) and the heat of vaporization (ΔHVAP) of four fatty acids (n-Heptanoic acid, Hexadecanoic acid, n-Hexanoic acid and n- Dodecanoic acid) have been estimated. Objective:: This paper aims to calculate the liquid molar volume and the heat of vaporization of four fatty acids under the critical point using two traditional equations of state: Peng-Robinson (PR) [21] and Soave-Redlich-Kwong. Methods: The area rules method applicable to obtaining the saturation pressure of the compounds has been used. The properties of the acids investigated in this work have been compared with those provided by literature. For molar volumes, the equations of state have given improved predictions when compared to traditional equations such as Rackett equation and so on. According to the vapor enthalpy calculations, no reference value was required. Results: In general, the Clausius-Clapeyron equation provides a better estimation of the vaporization enthalpy of fatty acids when Soave-Redlich-Kwong (SRK) equation was used. The heat of vaporization for fatty acids can be calculated with good reliability in comparison with the Watson equation if suitable equation of state is used. Conclusion: Accurate results for heat of vaporization can be reached in comparison with the Watson equation if the reliable equation of state is used.

The DAN method for calculation of vapour-liquid equilibria using an equation of state; Bubble and dew point calculations

1985 ◽  
Vol 50 (1) ◽  
pp. 1-22 ◽  
Author(s):  
Josef P. Novák ◽  
Vlastimil Růžička ◽  
Anatol Malijevský ◽  
Jaroslav Matouš ◽  
Jan Linek
Keyword(s):  
Equation Of State ◽  
Equilibrium Point ◽  
Critical Point ◽  
Dew Point ◽  
Computational Technique ◽  
Equilibrium Conditions ◽  
Close Vicinity ◽  
Newton Raphson ◽  
Raphson Method ◽  
Phase Envelope

A modification of the computational technique for calculating bubble and dew points using an equation of state has been proposed. The procedure consists in the Double Application of the Newton-Raphson method (DAN) to the set of equilibrium conditions. The algorithm is very effective as it provides both values of equilibrium variables and a very qualified first estimate of the next equilibrium point. This enables to proceed along the phase envelope rather quickly and to achieve convergence within a few iterations except in the close vicinity of the critical point.

scholarly journals Structure and equation of state of metallic plasmas

1996 ◽  
Vol 14 (4) ◽  
pp. 685-697 ◽  
Author(s):  
M.A. Amato ◽  
N.H. March
Keyword(s):  
Electrical Resistivity ◽  
Equation Of State ◽  
Critical Point ◽  
Coordination Number ◽  
Structural Properties ◽  
Structure Factor ◽  
Equations Of State ◽  
Theoretical Discussion ◽  
The Way

Evidence will first be presented in favor of the usefulness of coordination number-dependent equations of state of metallic plasmas. After some theoretical discussion of the way such an equation of state may be related to dimer properties, with specific regard to Na and Al, critical point predictions for the alkalis will be brought into contact with experiments. In this context, some results on W will also be surveyed. Structural properties, static and dynamic, will then be considered in relation to transport in both Cu and H plasmas. This will embrace experiments on electrical resistivity in both examples. For H, a simple inference can be extracted about the structure factor, and the discussion will finally be generalized to treat the metalization of solid H.

SITE-SPECIFIC EQUATIONS OF STATE FOR COASTAL SEA AREAS AND INLAND WATER BODIES

2017 ◽  
Author(s):  
Natalia Andrulionis ◽  
Natalia Andrulionis ◽  
Ivan Zavialov ◽  
Ivan Zavialov ◽  
Elena Kovaleva ◽  
...  
Keyword(s):  
Equation Of State ◽  
Black Sea ◽  
Water Samples ◽  
Equations Of State ◽  
Sea Water ◽  
Ionic Composition ◽  
Water Bodies ◽  
Aral Sea ◽  
The Black Sea ◽  
Coastal Sea

This article presents a new method of laboratory density determination and construction equations of state for marine waters with various ionic compositions and salinities was developed. The validation of the method was performed using the Ocean Standard Seawater and the UNESCO thermodynamic equation of state (EOS-80). Density measurements of water samples from the Aral Sea, the Black Sea and the Issyk-Kul Lake were performed using a high-precision laboratory density meter. The obtained results were compared with the density values calculated for the considered water samples by the EOS-80 equation. It was shown that difference in ionic composition between Standard Seawater and the considered water bodies results in significant inaccuracies in determination of water density using the EOS-80 equation. Basing on the laboratory measurements of density under various salinity and temperature values we constructed a new equation of state for the Aral Sea and the Black Sea water samples and estimated errors for their coefficients.

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