thomas a. mullen, mech eng, may04

COMPRESSIBILITY FACTORS FOR NATURAL AND SOUR

RESERVOIR GASES BY CORRELATIONS AND

CUBIC EQUATIONS OF STATE

by

NEERAJ KUMAR, B.Tech.

A THESIS

IN

PETROLEUM ENGINEERING

Submitted to the Graduate Faculty

of Texas Tech University in Partial Fulfillment of the Requirements for

the Degree of

MASTER OF SCIENCE

IN

PETROLEUM ENGINEERING

Approved

Akanni Lawal Chairperson of the Committee

Paulus Adisoemarta

Accepted

John Borrelli Dean of the Graduate School

December, 2004

ii

ACKNOWLEDGEMENTS

There are many people who were associated with this thesis who deserve

recognition. I would like to thank Dr. Akanni S. Lawal for his direction, support and

training. Thanks to Dr. James F. Lea for helping me with industrial approach towards this

thesis. I would also like to thank Dr. Paulus Adisoemarta for serving on my committee

and for his guidance.

iii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii

ABSTRACT vi

LIST OF TABLES vii

LIST OF FIGURES viii

LIST OF ABBREVIATIONS xii

CHAPTER

1. INTRODUCTION 1

1.1 Background Information 1

1.2 Use of Compressibility Factors in Engineering Analysis 2

1.2.1 Z-Factor for Sour and Acid Gases 2

1.2.2 Z-Factor for Geologic CO2 Storage 2

1.3 Significance of the Project 3

1.4 Objective of the Project 4

2. COMPRESSIBILITY FACTOR PREDICTION TECHNIQUES 5

2.1 Theoretical Analysis of Gas Law-Based Z-Factors 5

2.2 Experimental Method for Compressibility Factors 5

2.3 Empirical Correlation Methods 5

2.3.1 Standing-Katz Compressibility Factor Chart 5

2.3.2 Hall-Yarborough Z-Factor Correlation 6

2.3.3 Wichert -Aziz Z-Factor Correlation 7

2.3.4 Dranchuk-Abou-Kassem Z-Factor Correlation 7

2.3.5 Beggs-Brill Equation for SK Z-Factor Chart 8

2.3.6 Amoco Company Equation for SK Z-Factor Chart 9

2.3.7 Gopal Best-Fit Equation for SK Z-Factor Chart 9

2.3.8 Shell Oil Company Equation for SK Z-Factor Chart 10

2.3.9 Physical Properties of C7+ Fractions Correlations 11

iv

2.4 Corresponding State Prediction Methods 12

2.5 Equations of State Prediction Methods 19

3. STANDING-KATZ Z-FACTOR CORRELATION 20

3.1 Standing-Katz Representation of Z-Factor Chart 20

3.2 Best-Fit Equations for SK Z-Factor Chart 20

3.3 Mixture Critical Property Prediction Methods 23

3.3.1 Heptane-Plus Fraction Correlation Methods 25

3.3.2 Pseudocritical Mixing Parameter Methods 27

3.3.3 Pseudocritical Gas Gravity Correlation Methods 33

3.3.4 van der Waals Theory of Pseudocritical Methods 37

3.3.5 Improved Theory for Pseudocritical Mixture Parameter 37

3.4 Designed Scaling Parameter for Standing-Katz Z-Factor Chart 38

3.4.1 Design Procedure for Scaling Parameter 38

3.5 Designed PR/Z Versus Z-Factor Chart 42

3.6 Prediction Results for Z-Factor of Natural Gases 43

3.7 Prediction Results for Z-Factor of Reservoir Gases 45

4. Z-FACTOR PREDICTIONS FROM CUBIC EQUATIONS OF STATE 53

4.1 Selection of Cubic Equations-of-State 53

4.2 Lawal-Lake-Silberberg Equation of State 54

4.3 van der Waals Equation of State 56

4.4 Redlich-Kwong Equation of State 57

4.5 Soave-Redlich-Kwong Equation of State 58

4.6 Peng-Robinson Equation of State 61

4.7 Schmidt-Wenzel Equation of State 62

4.8 Patel-Teja Equation of State 63

4.9 Trebble-Bishnoi Equation of State 65

v

4.10 Transformed Cubic Equations to the LLS EOS Form 66

4.11 Generalized Reduced State of Cubic Equations-of-State 67

4.12 Prediction Results for Z-Factor of Pure Substances 71

4.13 Development of Binary Interaction Parameters 74

4.14 Prediction Results of Z-Factor of Mixtures 75

4.15 Prediction Results for Z-Factor of Natural Gases 77

4.15. 1 Results for Excelsior Laboratory Data 78

4.15. 2 Results for TTU Laboratory Data 80

4.15. 3 Results for UCalgary Data 81

4.15. 4 Results for Elsharkawy Gas Data 88

4.15. 5 Results for Elsharkawy Miscellaneous Data 90

5. CONCLUSIONS AND RECOMMENDATIONS 94

5.1. Conclusions 94

5.2. Recommendations 95

REFERENCES 96

APPENDICES 106

A. REDUCED FORM OF CUBIC EQUATIONS OF STATE 106

B. PREDICTION RESULTS FOR PSEUDOCRITICAL PARAMETERS 117

C. SCALING FACTOR DEVELOPMENT AND RESULTS 122

D. PREDICTION OF Z-FACTOR FOR PURE SUBSTANCES 127

E. EXPERIMENTAL Z-FACTOR FOR MISCELLANEOUS GASES 137

F. PREDICTION OF Z-FACTOR FROM LLS EOS 171

G. FORTRAN PROGRAMS 174

vi

ABSTRACT

Compressibility factor (z-factor) values of natural gases are necessary in most

petroleum engineering calculations. The most common sources of z-factor values are

experimental measurement, equations of state method and empirical correlations.

Necessity arises when there is no available experimental data for the required

composition, pressure and temperature conditions. Presented here is a technique to

predict z-factor values of pure substances, natural gases and sour reservoir gases

regardless of the composition of the acid gases at all temperatures and pressures.

Eight equations of state have been thoroughly examined and the results suggest

that the Lawal-Lake-Silberberg (LLS-EOS) equation of state is capable of predicting z-

factor values of both pure substances and mixtures of gases. This equation of state

method allows the determination of reduced temperature (TR) and reduced pressure (PR)

instead of the pseudo-reduced temperature (TPR) and pseudo-reduced pressure (PPR) both

for pure substances and mixtures of gases. This EOS is robust and the results are accurate

even if of acid gases present in high concentration. A comparative z-factor prediction

result of the various EOS methods for different gas samples is presented fortifying the

capability of the LLS-EOS method. Another method of predicting z-factor values is

based on the famous Standing-Katz (S-K) Chart (empirical methods). Law of

Corresponding States principle has formed the basis to develop a universal adjustable

parameter. This developed adjustable parameter forms the basis for using LLS-EOS to

be able to use S-K Chart to predict accurate z-factor values of pure substances and

mixtures of gases regardless of the concentration of acid gases. In contrast to the existing

methods derived from other equations of states (EOS methods) and S-K Chart (empirical

methods), this project provides a simple and universal technique for predicting z-factor

values for pure substances, natural gases and sour reservoir gases.

vii

LIST OF TABLES

2.1 Heavy Fraction Property Correlations. 10

3.1 Coefficients of Cavett’s correlation. 25

3.2 Sources of Experimental Z-Factor for Pure Substances 36

3.3 Rich Gas Condensate Composition (Elsharkawy) 45

3.4 Highly Sour Gas Composition (Elsharkawy) 46

3.5 Carbon Dioxide Rich Composition (Elsharkawy) 47

3.6 Very Light Gas Composition (Elsharkawy) 48

3.7 Property Prediction for Gas Composition Data (Elsharkawy) 49

4.1 Common Specialization Cubic Equation of State 66

4.2 Sources of Experimental Z-Factor 77

4.3 Gas Composition Data for Excelsior 6 Laboratory Data. 78

4.4 Gold Creek Gas Composition. 81

4.5 Results of Elsharkawy Gas Data. 88

4.6 Z-Factor Results for Miscellaneous Gases. 90

B.1 Gas Composition Description. 118

E.1 UCalgary Z-Factor Data. 137

viii

LIST OF FIGURES

1.1 Critical compressibility factor for pure hydrocarbons (alkanes). 3

2.1 Z-Factor of Pure Substances at Reduced Conditions(TR=0.65). 22

2.3 Z-Factor of Pure Substances at Reduced Conditions (TR=0.85). 24








3.1 Comparison of Six Correlations for Pseudocritical Pressure Parameters. 35

3.2 Compare of Six Correlations for Pseudocritical Temperature Parameters. 35

3.3 Scaled Z-Factor for Buxton & Campbell Data (Mix-5) at 160 oF. 40

3.4 Scaled Z-Factor for Buxton and Campbell Data (Mix-5) at 130 oF. 40

3.5 Scaled Z-Factor for Buxton and Campbell Data (Mix-5) at 100 oF. 41

3.6 Scaled Z-Factor for Satter Data (Mix-E) at 160 oF. 41

3.7 SK Z-Chart Developed Based on Computation SK Technique. 42

3.8 Amount of gas produced. 43

3.9 Scaled Z-Factor Buxton & Campbell, Mix-2 Result, @ T = 130 oF. 43

3.10 Scaled Z-Factor Buxton & Campbell, Mix-2 Result, @ T = 100 oF 44

3.11 Scaled Z-Factor Buxton & Campbell, Mix-3 Result, @ T = 100 oF 44

3.12 Scaled Z-Factor for Very Light Gas Composition. 50

3.13 Scaled Z-Factor for Carbon Dioxide Rich Gas Composition. 50

3.14 Scaled Z-Factor for Rich Gas Condensate Composition. 51

3.15 Scaled Z-Factor for Highly Sour Gas Composition. 51

4.1 Z-Factor comparison for LLS-EOS for Methane. 71

4.2 Z-Factor comparison for LLS-EOS for Carbon dioxide. 72

ix

4.3 Z-Factor comparison for LLS-EOS for Nitrogen. 72

4.4 Z-Factor comparison for vdW-EOS for Methane. 73

4.5 Z-Factor comparison for vdW-EOS for Carbon dioxide. 73

4.6 Z-Factor comparison for CO2-C1 mixture at 49 oF. 75




4.10 Z-Factor for Sweet Natural Gas, Data from Excelsior 6 (FPP) at 581 oR. 79

4.11 Z-Factor Comparison Chart at 90 oF (Simon et. al.). 79

4.12 Z-Factor Comparison Chart at 120 oF (Simon et. al.). 80

4.13 75% CO2 - Dry Gas at 100 oF for CO2 Sequestration. 80

4.14 25% CO2 - Dry Gas at 160 oF for CO2 Sequestration. 81

4.15 Z-Factor for sour natural gas, data from Excelsior 6 (FPP) at 581 oR. 82

4.16 Z-Factor comparison for sour natural gas mixture at 84 oF. 82











B.1 Critical temperature prediction for Gore Data (Mix 47-1). 118

B.2 Critical pressure prediction for Gore Data (Mix 47-1). 119




x



C.1 Scaled z-factor result for Buxton & Campbell Data at 160 oF (Mix-4). 123







D.1 Z-Factor comparison for vdW-EOS for Nitrogen. 127

D.2 Z-Factor comparison for RK-EOS for Methane. 127

D.3 Z-Factor comparison for RK-EOS for Carbon dioxide. 128

D.4 Z-Factor comparison for RK-EOS for Nitrogen. 128

D.5 Z-Factor comparison for SRK-EOS for Methane. 129

D.6 Z-Factor comparison for SRK-EOS for Carbon dioxide. 129

D.7 Z-Factor comparison for SRK-EOS for Nitrogen. 130

D.8 Z-Factor comparison for PR-EOS for Methane. 130

D.9 Z-Factor comparison for PR-EOS for Carbon dioxide. 131

D.10 Z-Factor comparison for PR-EOS for Nitrogen. 131

D.11 Z-Factor comparison for SW-EOS for Methane. 132

D.12 Z-Factor comparison for SW-EOS for Carbon dioxide. 132

D.13 Z-Factor comparison for SW-EOS for Nitrogen. 133

D.14 Z-Factor comparison for PT-EOS for Methane. 133

D.15 Z-Factor comparison for PT-EOS for Carbon dioxide. 134

D.16 Z-Factor comparison for PT-EOS for Nitrogen. 134

D.17 Z-Factor comparison for TB-EOS for Methane. 135

D.18 Z-Factor comparison for TB-EOS for Carbon dioxide. 135

D.19 Z-Factor comparison for TB-EOS for Nitrogen. 136

F.1 Z-factor for pure substances (Methane). 171

xi

F.2 Z-factor for pure substances (n-Decane). 171

F.3 Z-factor for pure substances (Carbon Dioxide). 172

F.4 Z-factor for pure substances (Hydrogen Sulfide). 172

F.5 Z-factor for pure substances (Nitrogen). 173

xii

LIST OF ABBREVIATIONS

Symbol Definition

a Attraction Parameter in EOS

A Dimensionless Constant ⎟⎠⎞

⎜⎝⎛

22TRP)T(a

ACF Acentric Factor

AF Acentric Factor

API Oil Gravity

b van der Waals co-volume

B Dimensionless Constant ⎟⎠⎞

⎜⎝⎛

RTbP

BIN Binary Interaction Number

BIP Binary Interaction Parameter

EOS Equation of State

G Gibbs Free Energy

k Parameter of SRK EOS

LLS Lawal-Lake-Silberberg

m Parameter of SRK EOS

Mw Molecular Weight

Mw Molecular Weight

p Pressure in psia

P Pressure in psia

PR Peng Robinson

R Universal Gas Constant (10.73 psiD.ft3/ (lb-

mol. oR))

RK Redlick-Kwong

SRK Soave-Redlich-Kwong

xiii

SW Schmidt-Wenzel

t Inverse Absolute Temperature (1/T)

T Absolute Temperature

TB Trebble-Bishnoi

V Volume in cubic feet

vdW van der Waal

x Mole Fraction

z Compressibility Factor

Z Compressibility Factor

Greek Letter

α Parameter of LLS EOS

αij Binary Interaction Term

β Parameter of LLS EOS

Ω Dimensionless EOS Parameter

ω Acentric Factor

γg Specific Gravity

Subscripts

c Critical Property

pr Pseudo Reduced Property Identification

pc Pseudo Critical Property Identification

r Reduced Property Identification

m Mixture Definition

R Reduced State

i, j Component Identification

1, 2 Index for components 1 and 2

CHAPTER 1

INTRODUCTION

1.1 Background Information Compressibility Factor is a measure of the amount the gas deviates from perfect

behavior. It is more commonly called as the gas deviation factor, represented as z (or) Z.

It is a dimensionless quantity and by definition the ratio of the volume actually occupied

by a gas at a given pressure and temperature to the volume it would occupy if it behaved

ideally. Therefore, a value of z = 1 would represent an ideal gas condition.

p and T sameat molesn of volumeIdeal

p and Tat gas of molesn of volumeActualVVz

i

a ==

The kinetic theory of gases (basis for Ideal gas law) assumes that there are neither

attractive forces nor repulsive forces between the gas molecules.

In nature, ideal gases do not exist instead real gases exist. All molecules of real gases are

under two kinds of forces:

(a) to move apart from each other because of their constant kinetic motion, and

(b) to come together because of electrical attractive forces between the molecules.

At normal conditions, the molecules are quite far apart and the attractive forces are

negligible and same is the condition at high temperatures because of the greater kinetic

motion. Under these above mentioned conditions, the gas tends to approach ideal

behavior. While, at high pressures, the molecules come very close to each other resulting

in significant attractive forces. These theories qualitatively explain the behavior of non-

ideal (real) gases and a general representation of the gas law is as follows:

Ideal Gas Law: PV = nRT (1.1).

Real Gas Law: PV = znRT (1.2).

1

1.2 Use of Compressibility Factors in Engineering Analysis

Accurate information of compressibility factor values is necessary in engineering

applications like gas metering, pipeline design, estimating reserves, gas flow rate, and

material balance calculations. Some of the petroleum engineering applications which

involve use of z-factor values of gases are as follows:

1.2.1 Z-Factor for Sour and Acid Gases

If hydrogen sulfide is present in a natural gas mixture it is termed as sour natural

gas. The existing methods of calculating z-factor values when significant amounts of acid

gases like carbon dioxide (CO2) and hydrogen sulfide (H2S) are present in the natural gas

mixtures incur high deviations from the actual values.

1.2.2 Z-Factor for Geologic CO2 Storage

A high content of CO2 gas present in the atmosphere is the major cause for global

warming. A method to capture CO2 from the atmosphere or other sources of CO2

production and be able to store it into abandoned wells is called as CO2 sequestration.

CO2 gas in various concentrations can be required to be stored. Engineering this method

needs z-factor values.

Knowledge of accurate critical z-factor value for pure substances and mixtures is

essential in the determination of accurate z-factor values. Critical z-factor is unique for

each component and system. Figure 1.1 illustrates the capability of various equations-of-

states in predicting critical compressibility factor values.

2

0.24

0.26

0.28

0.30

0.32

0.34

0.36

0.38

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10Pure Substances

Crit

ical

Com

pres

sibi

lity

Fact

orVDW

RK

PR

SW

PT

LLSEXPTB

Twu

Figure 1.1: Critical compressibility factor for pure hydrocarbons (alkanes).

1.3 Significance of the Project

Today’s standard treatment of phase behavior in reservoir simulation is still based

on formation volume factors (FVF’s) and surface gas/oil ratios (GOR’s) which requires

the determination of z-factor and critical properties of mixtures, more importantly. As

more and more, sour environment reservoirs are discovered, it becomes a necessity to

have a simple and robust technique to be able to determine z-factor values accurately.

This project presents methods that allow accurate determination of z-factor values both

for pure components and gas mixtures including significant amounts of non-hydrocarbon

components for all ranges of pressures and temperatures.

3

1.4 Objective of the Project

This research project provides improved predictive techniques for z-factors based

on the approaches of cubic Equations-of-State (EOS) and empirical correlation of

Standing-Katz5 Chart. Eight EOS that are routinely used in reservoir calculations and

improved pseudocritical property methods for Standing & Katz (SK) Chart are utilized to

match experimentally determined z-factors for pure substances, natural and sour reservoir

gases. The experimental z-factors data for 3100 gas samples, including highly sour gases

(H2S), acid gases (CO2 and N2) and rich gas condensates (with significant amount of C7+)

are used to establish the improved predictive techniques for z-factors.

4

CHAPTER 2

COMPRESSIBILITY FACTOR PREDICTION TECHNIQUES

2.1 Theoretical Analysis of Gas Law-Based Z-Factors

The magnitude of deviation of real gases from the conditions of the ideal gas law

increases with increasing pressure and temperature and varies widely with the

composition of the gas.

Numerous equations-of-state have been developed in the attempt to correlate the

pressure-temperature-volume variables for real gases with experimental data. In order to

express a more exact relationship between the variables p, V, and T, z-factor must be

introduced into the ideal gas equation to account for the departure of gases from ideality.

This forms the basis for the real gas law and is represented as:

zRTpV = (1.2).

To account for this deviation factor (z-factor), numerous equations-of-state have

been proposed.

2.2 Experimental Method for Compressibility Factors

Among the existing method of determination of z-factors, experimental

measurement is one of the most accurate methods. It is hard to determine experimentally

measured z-factor values for all compositions of gases at all ranges of pressures and

temperatures. At the same time, this method is expensive and most of the time these

measurements are made at reservoir temperatures only.

2.3 Empirical Correlations Methods

2.3.1 Standing-Katz Compressibility Factor Chart Standing and Katz5 presented a generalized z-factor chart, which has become an

industry standard for predicting the volumetric behavior of natural gases. To be able to

5

use this chart, knowledge of reduced temperature and reduced pressure are required,

which further needs determination of critical properties (namely, critical pressure and

critical temperature of the system). Numerous methods have been suggested to predict

pseudocritical properties of the gases as a function of their specific gravity. The point to

be noted here is that these methods predict pseudo critical values which are evidently not

accurate values of the gas mixtures. The existing methods fail to predict accurate values

of pseudocritical values when non-hydrocarbon components are present in significant

amounts. Improved technique to predict critical properties have been discussed in the

Chapter 3 of this thesis report.

2.3.2 Hall-Yarborough Z-Factor Correlation

Hall and Yarborough8 (1973) presented an equation-of-state that accurately

determined the Standing and Katz z-factor chart. This is based on the Starling-Carnahan21

equation-of-state. Best fit mathematical expressions were determined based on the data

taken from Standing and Katz z-factor chart. The mathematical form of the equation is:

( )[ ]2pr t12.1EXPY

tp06125.0Z −−⎥

⎦

⎤⎢⎣

⎡=

(2.1).

where pPR = pseudo-reduced pressure

t = reciprocal of the pseudo-reduced temperature ⎟⎟⎠

⎞⎜⎜⎝

⎛=

TTpc

( )[ ]( )

( ) ( ) ( ) 0Yt4.42t2.242t7.90Yt58.4t76.9t76.14Y1

YYYYt12.1Tp06125.0)Y(F

t82.218.232232

3

4322

pr

=+−+−−

−−++

+−−−=

+ (2.2).

Hall and Yarborough pointed out that the method is not recommended for

application if the pseudo-reduced temperature is less than one.

6

2.3.3 Wichert-Aziz Z-Factor Correlation

Sour natural gases (containing H2S) and/or CO2 frequently exhibit different

compressibility factor behavior than do sweet natural gases. Wichert and Aziz22 (1972)

developed a calculation procedure to account for these differences. Wichert and Aziz

developed a pseudo-critcal temperature adjustment factor which is a function of the

concentration of CO2 and H2S in the sour gas. This correction factor is then used to adjust

the pseudo-critical temperature and pressure according to the following expressions:

ε−=′ pcpc TT (2.3).

( )ε−+

′=′

B1BTTp

ppc

pcpcpc

(2.4).

where = pseudo-critical temperature, pcT oR

ppc = pseudo-critical pressure, psia

pcT′ = corrected pseudo-critical temperature, oR

pcp = corrected pseudo-critical pressure, psia

B = mole fraction of H2S in the gas mixture

ε = pseudo-critical temperature adjustment factor and is defined mathematically

by the following expression

( ) ( )0.45.09.19.0 BB15AA120 −+−=ε (2.5).

where the coefficient A is the sum of the mole fraction of H2S and CO2 in the gas

mixture, or

22 COSH yyA += (2.6).

2.3.4 Dranchuk-Abu-Kassem Z-Factor Correlation

Dranchuk and Abu-Kassem23 (1975) proposed an eleven-constant equation-of-

state for calculating the gas compressibility factors. The equation is as follows:

7

( ) [ ] 1AEXPT

A1A

TA

TA

ATA

TA

A

TA

TA

TA

Az

2r113

pr

2r2

r1110

5r2

pr

8

pr

79

2r2

pr

8

pr

76

r5pr

53pr

3

pr

21

+ρ−ρ

ρ++

ρ⎥⎥⎦

⎤

⎢⎢⎣

⎡+−ρ

⎥⎥⎦

⎤

⎢⎢⎣

⎡+++

ρ⎥⎥⎦

⎤

⎢⎢⎣

⎡+++=

(2.7).

where rρ = reduced gas density and is defined by the following relationship:

pr

prr zT

p27.0=ρ

(2.8).

The constants A1 through A11 were determined by fitting the equation, using non-

linear regression models, to 1,500 points from the Standing and Katz z-factor chart. The

coefficients values:

A1 = 0.3262 A2 = -1.0700 A3 = -0.5339 A4 = 0.01569

A5 = -0.05165 A6 = 0.5475 A7 = -0.7361 A8 = 0.1884

A9 = 0.1056 A10 = 0.6134 A11 = 0.7210

This method is applicable over the ranges

0.3T0.1

30p2.0

pr

pr

≤<

<≤

with an average absolute error of 0.585 percent.

2.3.5 Beggs-Brill Equation for SK Z-Factor Chart

Beggs and Brill25 (1973) proposed a best-fit equation for the Standing and Katz z-

factor chart and is as follows:

( ) DprB Cp

eA1Az +

−+=

(2.9).

where

( ) 101.0T36.092.0T39.1A pr5.0

pr −−−=

8

( ) ( ) ( )6pr1T9

2pr

prprpr p

1032.0p037.0

86.0T066.0pT23.062.0B

pr −+⎥⎥⎦

⎤

⎢⎢⎣

⎡−

−+−=

( )( )prTlog32.0132.0C −=

( )2prpr T1824.0T49.03016.010D +−=

This method is cannot be used for reduced temperature ( Tpr ) values less than

0.92.

2.3.6 Amoco Company Equation for SK Z-Factor Chart

Amoco Company uses the Hall and Yarborough z-factor determination method

and can be defined as follows:

( )[ ]2pr t12.1EXPY

tp06125.0Z −−⎥

⎦

⎤⎢⎣

⎡=

(2.10).

where ppr = pseudo-reduced pressure


⎞⎜⎜⎝

⎛=

TTpc

( )[ ]( )

( ) ( ) ( ) 0Yt4.42t2.242t7.90Yt58.4t76.9t76.14Y1


t82.218.232232

3

4322

pr

=+−+−−

−−++

+−−−=

+ (2.11).

It should be noted that this method is not recommended for application if the

pseudo-reduced temperature is less than one.

2.3.7 Gopal Best-Fit Equation for SK Z-Factor Chart

Gopal’s33 correlation for z-factor estimation was developed by dividing the

Standing-Katz chart into two parts by drawing a line isobarically for PR up to 5.4. For

various Tr values, several z-factor values were tabulated isobarically for Pr up to 5.4

because, for any Pr,, the z-factor values show a uniform trend. His objective was to come

9

up with two noniterative equations, one for Pr less than or equal to 5.4, and the other for

Pr greater than 5.4. To describe the chart accurately, the chart was further divided into 12

parts24. A general equation was developed and is of the form:

( ) DCTBATPZ rrr +++= (2.12).

The values of constants A, B, C, and D for various combinations of PR and TR are

available in the reference33. For Pr greater than 5.4, harmonic equations are suggested to

be a good fit.

2.3.8 Shell Oil Company Equation for Z-Factor Chart

4pr

pr 10p

ZF)ZG(EXP)ZA1(pZBZAZ ⎟⎟⎠

⎞⎜⎜⎝

⎛×−−×−+×+=

(2.13).

where,

)919.0T(3868.1T36.0101.0ZA prpr −×+×−−=

)65.0T(04275.0021.0ZB

pr −+=

prT224.06222.0ZC ×−=

037.0)86.0T(

0657.0ZDpr

−−

=

))1T(53.19(EXP32.0ZE pr −×−×=

))1T(3.11(EXP122.0ZF pr −×−×=

)pZEpZDZC(*pZG 4prprpr ×+×+=

10

2.3.9 Physical Properties of C7+ Fractions Correlation

Table 2.1: Heavy Fraction Property Correlations.

9.5MW6084API +=

API5.1315.141SG

+=

( ) 21 ee0

obp SGMWeRT = 3e

bp2e1e

0 TSGMWeC =

4321 eebp

ee0c CTSGMWe)psi(,p =

( ) 4321 eebp

ee0

oc CTSGMWeRT =

4e3ebp

2e1e0 CTSGMWe=ω

( )ω375.01293.0

+=cZ ( )ω0274.01

361.0+

=Ωw

Parameters Property

e0 e1 e2 e3 e4

Tbp 108.701661 0.42244800 0.42682558 0.000000 0.000000

C 0.83282122 0.09255911 -0.0413045 0.12621158 0.000000

Pc 237031780 -0.028484 2.755309 -1.374444 -2.947221

Tc 6.206640 -0.059607 0.224357 0.968332 -0.802538

ω 1.5790E-13 -1.453063 -2.811708 4.883921 2.109476

ω 2.22065E-10 -0.45908 -2.25373 3.4452 0.000000

11

2.4 Corresponding States Prediction Methods

The theory of Corresponding States proposes that all gases will exhibit the same

behavior, e.g. z-factor, when viewed in terms of reduced pressure, reduced volume, and

reduced temperature. Mathematically, this principle can be defined as:

( RRc T,pzz Ψ= ) (2.14).

The mathematical derivation of the above expression is as follows:

Real gas law is,

zRTPV = (2.15).

Multiply and divide the LHS of the above equation by and RHS by zccVP cTc, we get,

cccccc

cc TzTz

zRTVP

PVVP = (2.16).

ccc

c

cc TT

VPT

zRVP

PV==

(2.17).

By definition

cR

cR T

TT&PPP ==

(2.18).

Rcc

cc

cRR T

VPTz

zzVP ==

(2.19).

we have from real gas law,

R1

VPTz

cc

cc = (2.20).

R

RR

cRR T

VPzzVP ==

(2.21).

RR

Rc VP

Tzz ==

( RRc T,pzz Ψ= ) (2.22).

Based on the above derivation, the following relationship can be established,

12

2R2R

2R2R

1R1R

1R1R

TZVp

TZVp

= (2.23).

0

1

2

3

4

5

0 5 10 15 20 25 30Reduced Pressure

Com

pres

sibi

lity

Fact

or

nC7nC9nC10vdWSRKLLSPTPR

Exp

TR = 0.65

Figure 2.1: Z-Factor of Pure Substances at Reduced Conditions (TR=0.65).

13

0

1

2

3

4

5


Com

pres

sibi

lity

Fact

or

nC4nC5nC6nC9vdWSRKLLSPTPR

Exp

TR = 0.75

Figure 2.2: Z-Factors of Pure Substances at Reduced Conditions (TR = 0.75).

14

0

1

2

3

4

5


Com

pres

sibi

lity

Fact

or

C3iC4nC4nC5nC6nC7nC9nC10vdWSRKLLSPTPR

Exp

TR = 0.85

Figure 2.3: Z- Factor of Pure Substances at Reduced Conditions (T =0.85). R

0.2

0.6

1

1.4

1.8

0 4 8 12Reduced Pressure

Com

pres

sibi

lity

Fact

or

16

CO2H2SC2C3nC5vdWSRKLLSPTPR

TR = 1.02

Exp


15

0.25

0.65

1.05

1.45

1.85

2.25

0 5 10 15 20Reduced Pressure

Com

pres

sibi

lity

Fact

or

C2C3nC4iC4nC5vdWSRKLLSPTPR

TR = 1.07

Exp


0.35

0.63

0.91

1.19

1.47

1.75

2.03


Com

pres

sibi

lity

Fact

or

CO2H2SC2C3nC4iC4vdWSRKLLSPTPR

TR = 1.13

Exp


16

0.5

0.85

1.2

1.55

1.9


Com

pres

sibi

lity

Fact

or

CO2H2SC2C3nC4iC4vdWSRKLLSPTPR

TR = 1.24

Exp


0.65

0.98

1.31

1.64


Com

pres

sibi

lity

Fact

or

CO2C1C2C3vdWSRKLLSPTPR

TR = 1.55

Exp


17

0.88

1.12

1.36

1.60


Com

pres

sibi

lity

Fact

or

CO2C1C2vdWSRKLLSPTPR

TR = 1.98

Exp


0.90

1.18

1.45

1.73

2.00


Com

pres

sibi

lity

Fact

or

N2C1vdWSRKLLSPTPR

TR = 2.03

Exp


18

2.5 Equations of State Prediction Methods

Cubic equations of state (EOS’s) are simple equations relating pressure, volume,

and temperature (PVT). They accurately describe the volumetric and phase behavior of

pure compounds and mixtures, requiring only critical properties and acentric factor of

each component. The same equation is used to calculate the properties of all phases,

thereby ensuring consistency in reservoir processes that approach critical conditions.

Multiple phase behavior, such as low-temperature CO2 flooding, can be treated with an

EOS, and even water-/hydrocarbon-phase behavior can be predicted accurately with a

cubic EOS.

Volumetric behavior is calculated by solving the simple cubic equation, usually

expressed in terms ofRTpVz = ,

0AzAzAz 0123 =+++ (2.24).

where constants A0, A1, and A2 are functions of pressure, temperature, and phase

composition. Chapter 4 presents a detailed use of equations-of-state method for solving z-

factors.

19

CHAPTER 3

STANDING-KATZ Z-FACTOR CORRELATION

3.1 Standing-Katz Representation of Z-Factor Chart

The z-factor in the Standing and Katz5 (SK) chart is a function of reduced

pressure and temperature. To be able to predict z-factor using the SK chart requires the

appropriate reduced temperature and pressure. Information on the composition of the gas

used to design the Standing-Katz z-factor chart is not provided. A close study and

comparison of the experimental data with that of SK chart z-factor values suggests that

the SK chart was developed based on the natural gas mixture without any significant

amounts of non-hydrocarbon components and C7+ in it.

3.2 Best-Fit Equations for SK Z-Factor Chart

Many empirical equations and EOSs have been fit to the original Standing-Katz z-

factor chart. Some of the commonly used methods in the petroleum industry are:

Hall & Yarborough11 Best Fit Equation:

( )[ 2pr t12.1expY

tp06125.0Z −−⎥

⎦

⎤⎢⎣

⎡= ] (3.1)

where ppr = pseudo-reduced pressure


⎞⎜⎜⎝

⎛=

TTpc

( )[ ]( )

( ) ( ) ( ) 0Yt4.42t2.242t7.90Yt58.4t76.9t76.14Y1


t82.218.232232

3

4322

pr

=+−+−−

−−++

+−−−=

+ (3.2).

20

Beggs-Brill25 Best-Fit Equation:

( ) DprB Cp

eA1Az +

−+=

(3.3)

where

( ) 101.0T36.092.0T39.1A pr5.0

pr −−−=

( ) ( ) ( )6pr1T9

2pr

prprpr p

1032.0p037.0

86.0T066.0pT23.062.0B

pr −+⎥⎥⎦

⎤

⎢⎢⎣

⎡−

−+−=

( )( )prTlog32.0132.0C −=

( )2prpr T1824.0T49.03016.010D +−=

Dranchuk-Abu-Kassem23 Best-Fit Equation:

( ) [ ] 11 2113

22

1110

5287

92

287

6

55

332

1

+−++

⎥⎥⎦

⎤

⎢⎢⎣

⎡+−

⎥⎥⎦

⎤

⎢⎢⎣

⎡+++

⎥⎥⎦

⎤

⎢⎢⎣

⎡+++=

rpr

rr

rprpr

rprpr

rprprpr

AEXPT

AA

TA

TA

ATA

TA

A

TA

TA

TAAz

ρρ

ρ

ρρ

ρ

(3.4)

where rρ = reduced gas density and is defined by the following relationship:

pr

prr zT

p27.0=ρ

The constants A1 through A11 were determined by fitting the equation,

using non-linear regression models, to 1,500 points from the Standing and Katz z-factor

chart. The coefficients values:

A1 = 0.3262 A2 = -1.0700 A3 = -0.5339 A4 = 0.01569

A5 = -0.05165 A6 = 0.5475 A7 = -0.7361 A8 = 0.1884

A9 = 0.1056 A10 = 0.6134 A11 = 0.7210

Dranchuk-Purvis-Robinson Method:

21

Dranchuk, Purvis, and Robinson’s (1974) correlation was developed based on

Benedict-Webb-Rubin57 type of equation of state. It consists of eight coefficients which

were obtained based on a best-fit of 1500 data points from Standing-Katz Z-factor chart.

The correlation is,

( ) ( )⎥⎥⎦

⎤

⎢⎢⎣

⎡ρ−ρ+ρ+

ρ⎥⎥⎦

⎤

⎢⎢⎣

⎡+ρ

⎥⎥⎦

⎤

⎢⎢⎣

⎡++ρ

⎥⎥⎦

⎤

⎢⎢⎣

⎡+++=

2r8

2r8

2r3

pr

7

5r

pr

652r

pr

54r3

pr

3

pr

21

AEXPA1TA

TAA

TA

ATA

TAA1Z

(3.5)

where pr

prr ZT

p27.0=ρ and the coefficients A1 to A8 have the following values:

A1 = 0.31506237 A2 = -1.0467099 A3 = -0.57832729 A4 = 0.53530771

A5 = -0.61232032 A6 = -0.10488813 A7 = 0.68157001 A8 = 0.68446549

This method is valid with in a temperature and pressure range of:

0

0

.3T05.1 pr <≤

. .3P2.0 pr ≤≤

Shell Oil Company Best-Fit Equation:

4pr

pr 10p

ZF)ZGexp()ZA1(pZBZAZ ⎟⎟⎠

⎞⎜⎜⎝

⎛×−−×−+×+= (3.6)

where,

)919.0T(3868.1T36.0101.0ZA RR −×+×−−=

)65.0T(04275.0021.0ZB

R −+=

RT224.06222.0ZC ×−=

037.0)86.0T(

0657.0ZDR

−−

=

))1T(53.19exp(32.0ZE R −×−×=

22

))1T(3.11exp(122.0ZF R −×−×=

)PZEPZDZC(*PZG 4RRR ×+×+=

3.3 Mixture Critical Property Prediction Methods

Numerous correlations and methods have been suggested in the past to predict

mixture critical properties. These methods predict pseudocritical properties and do not

represent a correct estimation of the properties for various ranges of composition. In most

cases, each of the correlations are designed for a limited reduced pressure, reduced

temperature and a fixed range of composition of gases (in Chapter 2 is discussed the

procedure to calculate properties of C7+). This calls for a need to have a generalized

method to calculate mixture critical properties and presented here is a generalized method

to calculated pure component and mixture critical properties. The expressions for mixture

critical point (Pc and Tc) are established in the following equations:

]BB)(Z3[B

baP

cm2cmm

2c

2c

2m

mc α+β+α+

= (3.7)

]BB)(Z3[RB

baT

cm2cmm

2c

c

m

mc α+β+α+

= (3.8).

The parameters going into the Equations 3.7 and 3.8 are calculated as shown

below:

The critical equation-of-state parameter Bc is obtained by solving the following cubic

equation:

0BBB 0c12c2

3c3 =φ+φ+φ+φ (3.9)

where

136

3271515

6128

0

m1

2mmm2

3m

2mm3

−=φα+=φ

α−β−α+=φ

α+α+α+=φ

Similarly, the expression for Zc (critical Z for mixtures) in terms of αm and βm are shown

in Equations 3.7 and 3.8 is obtained by solving the following cubic equation:

23

0ZZZ 0c12c2

3c3 =θ+θ+θ+θ (3.10)

where

)( 6663

)9912123(

6128

mmm2m0

mmm2mm1

mmm2mm2

3m

2mm3

βα−β+α−=θ

βα−β+α+α=θ

βα−β+α+α+−=θ

α+α+α+=θ

where the mixture parameters am, bm, αm and βm are prescribed as

ij

n

1i

n

1j

2/1j

2/1ijim a)T(a)T(axxa ∑∑

= =

= (3.11)

3n

1i

3/1iim bxb ⎟

⎠

⎞⎜⎝

⎛= ∑

= (3.12).

ij

n

1i

n

1j

2/1j

2/1ijim xx ααα=α ∑∑

= = (3.13)

ij

n

1i

n

1j

2/1j

2/1ijim xx βββ=β ∑∑

= = (3.14). The temperature function essential in the determination of the mixture equation of

state parameter (attractive term ‘a’) is defined as:

θ−−Ω+=Ω= Rc

2c

23

cwc

2c

2

a TPTR]Z)1(1[

PTR)T(a

(3.15) where . w

2 M03e59015.335714.008627.019708.0 ω−+ω+ω+=θ

The pure component parameters are defined as follows:

c

ccw

c

cb P

RTZ

PRT

b Ω=Ω= (3.16)

cw

ccw

ZZ3Z1

Ω−Ω+

=α (3.17)

c2w

wcc2w

3w

2c

Z)Z31(Z2)1(Z

ΩΩ−+Ω+−Ω

=β (3.18)

ω0274.01361.0

w +=Ω

(3.19).

24

A brief description of the previously used empirical correlations suggested is

given in the following paragraphs. It should be noted that only the commonly used

correlations are mentioned.

3.3.1 Heptane-Plus Fraction Correlation Methods

Cavett’s Correlation:

Cavett (1962) proposed correlations for estimating the critical pressure and

temperature of hydrocarbon fractions.

2b

26

2b5

3b4b3

2b2b10c

)T()API(a)T)(API(a

)T(a)T)(API(aTaTaaT

++

++++= (3.20)

( )2

b7b2

62

b5

3b4b3

2b2b10c

)T(b)T()API(b)T)(API(b

)T(b)T)(API(b)T(b)T(bbpLog

+++

++++= (3.21).

The coefficients in the above equations are tabulated below:

Table 3.1: Coefficients of Cavett’s correlation.

I ai bi

0 768.07121 2.8290406

1 1.7133693 0.94120109*10-3

2 -0.0010834003 -0.30474749*10-5

3 -0.0089212579 -0.20876110*10-4

4 0.38890584*10-6 0.15184103*10-8

5 0.53094290*10-5 0.11047899*10-7

6 0.32711600*10-7 -0.48271599*10-7

7 - 0.13949619*10-9

25

Kesler-Lee Correlations:

Kesler and Lee (1976) proposed a set of equations to estimate the critical

temperature, critical pressure, acentric factor, and molecular weight of petroleum

fractions.

Critical Pressure:

( )

3102

272

32

106977.142019.01047227.0648.34685.1

1011857.02898.224244.00566.03634.8ln

bb

bc

TT

Tp

−−

−

⎟⎟⎠

⎞⎜⎜⎝

⎛+−⎟⎟

⎠

⎞⎜⎜⎝

⎛+++

⎟⎟⎠

⎞⎜⎜⎝

⎛++−−=

γγγ

γγγ

(3.22)

Critical Temperature:

( ) ( )b

5

gbggc T1026238.34669.0T1174.04244.01.8117.341T γ−+γ++γ+= (3.23)

Molecular Weight:

( )

( )

( ) 3b

12

b

2gg

b

7

b

2gg

bggW

T10

T98.1818828.102226.080882.01

T10

T79.7203437.102058.077084.01

T3287.36523.44.94866.12272M

⎟⎟⎠

⎞⎜⎜⎝

⎛−γ+γ−+

⎟⎟⎠

⎞⎜⎜⎝

⎛−γ−γ−+

γ−+γ+−=

(3.24).

Winn-Sim-Daubert Correlation:

Sim and Daubert (1980) represented the critical pressure, critical temperature, and

molecular weight as follows:

4853.2g

3177.2b

9c T1048242.3p γ×= − (3.25)

[ ]04614.0g

08615.0bc T994718.3expT γ= (3.26)

9371.0g

3776.2b

5w T104350476.1M −− γ×= (3.27)

where Tb is the boiling point in oR.

26

Watansiri-Owens-Starling Correlation

Watansiri (1985) developed a set of correlations to estimate the critical properties

and acentric factor of coal compounds and other hydrocarbons and their derivatives.

Critical Temperature:

⎥⎦⎤

⎢⎣⎡ γ−γ−γ

++−−=

g3

1

g2

1

gw

bwbc

016943.0061061.0078154.0M

)Tln(11067.1)Mln(03905.0T0005217.00650504.0)Tln( (3.28)

Critical Volume:

)ln(1958.42)Mln(10108.1

175.131750.638038.129313887.76)Vln(

gw

3g

2ggc

γ++

γ−γ+γ−= (3.29)

Critical Pressure:

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎟⎠

⎞⎜⎜⎝

⎛+=

w

b

c

w8.0

c

cc M

T08843889.0

T

M712.8

V

T01617283.06418853.6)Pln(

(3.30)

Acentric Factor:

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

γ−−γ×+×+

γ+×−⎟⎟⎠

⎞⎜⎜⎝

⎛

γ×

++⎟⎟⎠

⎞⎜⎜⎝

⎛+×

=ω

−−

−−

−

w

b

2g

32

b

w

31

b2wg

42w

4

wgwb4

2

g

b5

ww

bb

4

M9T5

T00255452.0MT29959.66M102061.0M101265.0

M001261.0MT1012027778.0T10074691.0

M904.382

MT281826667.0T101236667.5

(3.31)

3.3.2 Pseudocritical Mixing Parameters Methods

Empirical Methods Gpc = Gideal + Gexcess (3.32)

27

( )(∑∑ Φ+==

iIDEALcii

n

1icipc xPPxP

i)

)

(3.33)

( )(∑∑ Φ+==

iIDEALcii

n

1icipc xTTxT

i (3.33)

Kay’s Rule (1936)

∑=n

iciipc PxP (3.34)

∑=n

iciipc TxT (3.35)

Joffe Method64 (1947)

i2

1

c

cn

ii

m2

1

pc

pc

P

Tx

P

T⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛∑

(3.36) 3

n

i

n

j

31

jc

c3

1

ic

cji

pc

pc

PT

PT

xx81

PT

∑∑ ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=⎟

⎟⎠

⎞⎜⎜⎝

⎛

(3.37) Prausnitz-Gunn (1958)

∑=

=n

1iciipc TxT

(3.38)

⎟⎠

⎞⎜⎝

⎛

⎟⎠

⎞⎜⎝

⎛⎟⎠

⎞⎜⎝

⎛

=

∑

∑∑

=

==

n

1icii

n

1icii

n

1icii

pc

Vx

TxRZxP

(3.39) Stewart-Burkhardt-Voo Method61 (1959)

∑ ∑

∑

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=c

i

22

1

ic

cnc

ii

ic

ci

2

i2

1

c

cc

ii

pc

PT

x32

PT

x31

P

Tx

T

(3.40)

28

2

nc

i

22

1

ic

cnc

ii

ic

ci

2

i2

1

c

cnc

ii

pc

PT

x32

PT

x31

P

Tx

P

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

∑ ∑

∑

(3.41) Leland-Mueller Method (1959)

[ ]γ

γ

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

=

∑∑

∑∑/1

33/1

jc

cc

3/1

ic

ccn

i

n

jji

33/1

jc

cc

3/1

ic

cc2/cc

n

i

n

jji

pc

PTZ

21

PTZ

21xx

PTZ

21

PTZ

21TTxx

Tji

(3.42)

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

∑∑

∑33/1

jc

cc

3/1

ic

ccn

i

n

jji

n

icic

pc

PTZ

21

PTZ

21xx

ZxTP

i

(3.43) where

( )

( ) ⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

Ψ=γ

∑

∑

=

=n

1iici

n

1iici

PxT

TxP

(3.44) Leland and co-workers later reported the following relationship,

44.2Px

Tx75.0 n

ici

n

ici

i

i

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−=γ

∑

∑

(3.45).

Satter-Campbell Method46 (1963)

∑∑= =

=n

1i

n

1i

21

cj2

1

cijipc TTxxT (3.46)

29

∑∑

∑

= = ⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

n

1i

n

1j

33

1

jc

cc3

1

ic

ccji

n

iciic

pc

PTZ

21

PTZ

21xx

ZxTP

(3.47). Lee-Kesler Method54 (1975)

ici 085.02905.0Z ω−= (3.48)

ci

cicici P

RTZV =

(3.49).

∑∑==

⎟⎠⎞⎜

⎝⎛ +=

n

1j

33

1

cj3

1

ciji

n

1ipc VVxx

81V

(3.50)

cjci

n

1j

33

1

cj3

1

ciji

n

1icpc TTVVxx

V81T ∑∑

==⎟⎠⎞⎜

⎝⎛ +=

(3.51)

c

b

TT

=Θ (3.52)

61

61c

i 43577.0ln4721.136875.152518.15169347.0ln28862.109648.692714.5)atm(Pln

Θ+Θ−Θ−Θ−Θ+Θ+−−

=ω −

−

(3.53).

∑=

ω=ωn

1iiix (3.54)

( )c

cpc V

RT085.02905.0P

ω−=

(3.55) Van Ness-Abbot Method63 (1982)

32

n

1i

n

1j jc

c

ic

cji

34

n

1i

n

1j

21

jc

25

c

21

ic

25

cji

2pc

PT

PT

xx

PT

PT

xx

T

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

∑∑

∑∑

= =

= =

(3.56)

30

35

n

1i

n

1j jc

c

ic

cji

34

n

1i

n

1j

21

jc

25

c

21

ic

25

cji

2pc

PT

PT

xx

PT

PT

xx

P

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

=

∑∑

∑∑

= =

= =

(3.57) Pedersen-Fredenslund-Christensen-Thomassen Method55,56 (1984)

cjci3n

i

n

j

31

jc

c3

1

ic

cji

3n

i

n

j

31

jc

c3

1

ic

cji

pc TT

PT

PT

xx

PT

PT

xx

T

∑∑

∑∑

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

=

(3.58)

cjci23n

i

n

j

31

jc

c3

1

ic

cji

3n

i

n

j

31

jc

c3

1

ic

cji

pc TT

PT

PT

xx

PT

PT

xx8

P

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

=

∑∑

∑∑

(3.59) Teja-Thurner-Pasumarti Method (1985)

∑∑= =

=n

1i

n

1j cij

cijji

pc

pc

PT

xxPT

(3.60)

∑∑= =

=n

ii

n

1j cij

2cij

jipc

2pc

PT

xxPT

(3.61) where

( ) 21

cjciijcij TTT ξ= (3.62)

33

1

jc

c3

1

ic

c

cijcij

PT

PT

T8P

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

(3.63) Sutton Method62 (1985)

31

∑ ∑

∑

−⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

=c

iJ

22

1

ic

cnc

ii

ic

ci

2

k

i2

1

c

cc

ii

pc

EPT

x32

PT

x31

EP

Tx

T

(3.64)

2

nc

iJ

22

1

ic

cnc

ii

ic

ci

2

k

i2

1

c

cnc

ii

pc

EPT

x32

PT

x31

EP

Tx

P

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡−

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛

⎥⎥

⎦

⎤

⎢⎢

⎣

⎡−⎟

⎟

⎠

⎞

⎜⎜

⎝

⎛

=

∑ ∑

∑

(3.65) Lawal-Lake-Silberberg65 (2002)

( ) ]BαBβαR[3ZbBa

Tcm

2cmm

2cm

cmc +++

= (3.66)

( ) ]BαBβα[3ZbBa

Pcm

2cmm

2c

2m

2cm

c +++=

(3.67)

∑∑=n

i

n

jij

0.5jijim a)a(axxa

(3.68)

3n

i

1/3iim bxb ⎟

⎠

⎞⎜⎝

⎛= ∑

(3.69)

∑∑=n

i

n

jij

0.5jijim α)α(αxxα

(3.70)

∑∑=n

i

n

jij

0.5jijim β)β(βxxβ

(3.71)

0ΘZΘZΘZΘ 0c12c2

3c3 =+++ (3.72)

where,

32

)αββ(α Θ)αβ2α2β3(2α Θ

)β3α3β4α4α3(1Θ

8)12α6α(α Θ

2mmmm0

mmmm2m1

mmmm2m2

m2m

3m3

−−=

+−+=

−+++−=

+++=

0θBθBθBθ 0c1

2c2

3c3 =+++ (3.73)

where,

1 θ2)3(α θ

5)9β5α3(αθ

8)12α6α(α θ

0

m1

mm2m2

m2m

3m3

−=+=

−+−−=

+++=

Redlich-Kwong-Abbott

32

i

n

i c

ci

34

n

i

21

ic

25

ci

pc

PTx

PTx

T

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞⎜

⎝⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

∑

∑ (3.74)

35

i

n

i c

ci

34

n

i

21

ic

25

ci

pc

PTx

PTx

P

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎠⎞⎜

⎝⎛

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

⎟⎟⎠

⎞⎜⎜⎝

⎛

=

∑

∑ (3.75)

3.3.3 Pseudocritical Gas Gravity Correlation Methods

The Standing90 gas gravity correlation is stated as: 2ggpc S1.11S7.51706)psia(P −−= (3.76)

2gg

opc S5.71S330187)R(T −+= (3.77)

33

The Sutton90 gas gravity correlation is stated as: 2ggpc S6.3S0.1318.756)psia(P −−= (3.78)

2gg

opc S0.74S5.3492.169)R(T −+= (3.79)

Elsharkawy-Hashem-Alikhan90 gas gravity correlation is stated as: 2ggpc S916.7S34.14706.787)psia(P −−= (3.80)

2gg

opc S976.66S14.35818.149)R(T −+= (3.81)

Hankinson-Thomas-Philips91 gas gravity correlation is stated as: gpc S718.58604.709)psia(P += (3.82)

go

pc S344.307491.170)R(T += (3.83)

Brill-Beggs102 gas gravity correlation is stated as: gpc S5.5775.708)psia(P −= (3.84)

go

pc S0.3140.169)R(T += (3.85)

This work: 2ggpc S012.63S306.5794.659)psia(P −+= (3.86)

2gg

opc S7924.0S82.32195.165)R(T ++= (3.87)

The R2 for Ppc

is 0.9821 and that for Tpc is 0.9999.

34

600

620

640

660

680

700

600 620 640 660 680 700Pc Experimental (psia)

Pc C

alcu

late

d (p

sia)

StanSuttElHAHaTPhBrBeThisWork

Figure 3.1: Comparison of Six Correlations for Pseudocritical Pressure Parameters.

330

370

410

450

490

530

330 370 410 450 490 530

Tc Experimental (oR)

Tc C

alcu

late

d (o R

)

StanSuttElHAHaTPhBrBeThisWork

Figure 3.2: Compare Six Correlations for Pseudocritical Temperature Parameters.

35

Table 3.2: Sources of Experimental Z-Factor for Pure Substances.

Authors Year System Reference No.

Sage-Lacey 1950 C1 8



Sage-Lacey 1950 iC4, nC4 8

Sage-Lacey 1950 iC5, nC5 8

Stewart-Sage-Lacey 1954 nC6 10

Sage-Reamer- Nichols

1955 nC7 9

Sage-Lacey 1950 H2S 8

Sage-Lacey 1950 N2 8

Sage-Lacey 1950 CO2 8

36

3.3.4 van der Waals Theory of Pseudocritical Methods

Criticality Theory

1. van der Waals Pseudocritical Theory3 (1873)

0

VP

T

=⎟⎠⎞

⎜⎝⎛

∂∂

at T = Tc and P = Pc (3.88)

0

VP

T2

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

at T = Tc and P = Pc (3.89)

2. Gibbs Criteria (1928)

0xG

P,T

=⎟⎠⎞

⎜⎝⎛

∂∂ (3.90)

0

xG

P,T2

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

(3.91)

3. Wilson Renormalization Theory (1982)

3.3.5 Improved Theory for Pseudocritical Mixture Parameter

Mixture rules significantly affect the accuracy of mixture property determination.

Weighted average based on mole fraction has been the general rule since Kay but this

method is an approximate method and more rigorous methods are required for accurate

determination of mixture properties. Numerous methods have been proposed but none of

them present a generalized method for critical property prediction with high accuracy.

Most of the methods are either statistical or empirical and therefore are bound by errors.

In this project a method is presented which is based on LLS EOS. This method is capable

of predicting critical properties of mixtures irrespective of the component and its

composition.

Prediction method of binary interaction number is an essential parameter in

mixture critical property determination. The BIN can be function of molecular weights,

acentric factor or product of molecular weight and acentric factor. Care must be taken in

37

applying these rules of predicting BIN with gas mixtures containing very light and heavy

components. Other occasions of concern could be when isomers are present in a gas

mixture. It should be noted that in BIN can be equal to 1 only in case of pure

composition.

3.4 Designed Scaling Parameter for Standing-Katz Z-Factor Chart

In order to extend the use of SK chart to the prediction of z-factors for sour and

acid gases without resorting to the Wichert-Aziz2 correction formula for pseudocritical

pressures and temperatures parameters, a universal scaling parameter has been

established. This scaling parameter is developed by overlaying the experimental z-factor

curves for the same range of pressures and temperatures as that of SK chart and

measuring the deviation from of the SK chart curves. A mathematical quantification of

this deviation for hydrocarbon compounds, non-hydrocarbon compounds, and sour

reservoir gases resulted in a similar modification (or scaling) parameter requirement.

This observation establishes the fact that the SK chart is designed perfectly but most of

the time, it is used wrongly. The error is in the method of calculation of the pseudocritical

pressures and temperatures.

3.4.1 Design Procedure for Scaling Parameter

The scaling parameter for hydrocarbon components is developed based on wide

range of available experimental data and measuring the deviation of SK computational

methods from experimental data. In the design of a scaling parameter, non-hydrocarbon

components that are commonly found in the reservoir gases like nitrogen, hydrogen

sulfide, and carbon dioxide were considered. A wide range of experimental z-factor data

for natural gases containing significant amounts of acid gases, sour gas, and C7+ fraction

were collected and used in this project to develop the scaling parameter.

38

The step-by-step procedure for obtaining the scaling factor is as follows:

1. c.Expt

SKSF z

zzz ×= .

2. A regression analysis of the reduced pressure and ZSK scaled z-factor (based on

step 1) is performed to obtain a quadratic expression for scaling factor at each

temperature of the mixture. The equation describing it is Equation 3.78.

3. The coefficients a0, a1 and a2 for each mixture is collected and is subjected to

linear regression analysis.

4. The general expressions for these coefficients are obtained by performing

regression analysis as functions of product of molecular weight and acentric

factor (ωMw).

5. The equations describing the final expressions are presented below. 2R2R10 PaPaaSF ++= (3.92)

where

)M(15675.004518.131.0a

w0 ω−

= (3.93)

)M(03E4852.402E2722.105E40.9a

w1 ω−−−

−=

(3.94)

)M(41702.083084.004E54.3a

w2 ω−

−−=

(3.95) The prediction results using the scaling factor technique is presented below. More

results on this can be seen in the Section C.1 of Appendix C.

39

0.72

0.82

0.92

1.02

1.12

1.22

1.32

1.42

0 2,000 4,000 6,000 8,000Pressure (Psia)

Z-Fa

ctor

Expt.

SK

Scaled

Figure 3.3: Scaled Z-Factor for Buxton & Campbell Data (Mix-5) at 160 oF.

0.71

0.81

0.91

1.01

1.11

1.21

1.31

1.41

0 2,000 4,000 6,000 8,000

Pressure (psia)

Z-Fa

ctor

Expt.

SK

Scaled

Figure 3.4: Scaled Z-Factor for Buxton and Campbell Data (Mix-5) at 130 oF.

40

0.65

0.75

0.85

0.95

1.05

1.15

1.25

1.35

0 2,000 4,000 6,000 8,000

Pressure (Psia)

Z-Fa

ctor

Expt.

SK

Scaled

Figure 3.5: Scaled Z-Factor for Buxton and Campbell Data (Mix-5) at 100 oF.

0.8

0.9

1

1.1

1.2

0 3 6 9Pressure (Psia)

Z-Fa

ctor

12

Expt.

Scaled

SK

Figure 3.6: Scaled Z-Factor for Satter Data (Mix-E) at 160 oF.

41

3.5Designed pR/z Versus Z-Factor Chart

This section provides a clear view of an ideal z-chart and the eventually the

capability of predicting amount of gas produced by a graphical means. Figure 3.5 is a z-

chart developed based on computation techniques developed based on a correlation

developed for SK Z-Chart.

0.1

0.3

0.5

0.7

0.9

1.1

0 2 4 6 8

Reduced Pressure

Z-Fa

ctor

Val

ues

Tr=1.0Tr=1.05Tr=1.1Tr=1.2Tr=1.3Tr=1.4Tr=1.5Tr=1.6Tr=1.7Tr=2.0Tr=2.2Tr=2.4Tr=2.6Tr=2.8Tr=3.0Tr=1.8Tr=1.9Tr=1.15Tr=1.25Tr=1.35Tr=1.45

TR = 1.0

Figure 3.7: SK Z-Chart Developed Based on Computation SK Technique.

42

0.00

0.20

0.40

0.60

0.80

1.00

0 2 4 6 8 10

PR/z

Z-Fa

ctor

12

Tr=1.0Tr=1.05Tr=1.1Tr=1.15Tr=1.2Tr=1.25Tr=1.30Tr=1.35Tr=1.40Tr=1.45Tr=1.50Tr=1.6Tr=1.7Tr=1.8Tr=1.9Tr=2.0Tr=2.2Tr=2.4Tr=2.6Tr=2.8Tr=3.0

TR=1.0

Figure 3.8: Amount of gas produced.

3.6 Prediction Results for Z-Factor of Natural Gases

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

0 2,000 4,000 6,000 8,000Pressure (psia)

Z-Fa

ctor

Exp

Standing-Katzl

Scaledt

Figure 3.9: Scaled Z-Factor Buxton & Campbell, Mix-2 Result, @ T = 130 oF.

43

0.8

0.9

1

1.1

1.2

1.3

0 2,000 4,000 6,000 8,000Pressure (psia)

Z-Fa

ctor

Exp

Standing-Katz

Scaled


0.76

0.86

0.96

1.06

1.16

1.26

0 2,000 4,000 6,000 8,000Pressure (psia)

Z-Fa

ctor

Exp

Standing-Katzl

Scaled


44

3.7 Prediction Results for Z-Factor of Reservoir Gases Scaling is done based on the law of corresponding states principle as follows:

RRRR T,PSF

SK

T,Pc

Scaled

ZZ

ZZ

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ (3.96).

Table 3.3: Rich Gas Condensate Composition (Elsharkawy71). Serial No. 39

Rich gas Condensate

IND 281 282 283 284 285 286 287 H2S 0 0 0 0 0 0 0 CO2 0.0231 0.0242 0.0248 0.0253 0.0258 0.0262 0.0266N2 0.0137 0.0155 0.0161 0.0166 0.0163 0.0155 0.0143C1 0.6583 0.7074 0.738 0.7559 0.7583 0.7485 0.7292C2 0.0803 0.0817 0.0821 0.0839 0.0863 0.0905 0.0944C3 0.0417 0.0411 0.0404 0.0402 0.0415 0.0447 0.0495iC4 0.0078 0.0073 0.007 0.0069 0.0073 0.0082 0.0091nC4 0.0184 0.017 0.0162 0.0159 0.0167 0.0186 0.0208iC5 0.0075 0.0067 0.0062 0.006 0.0062 0.007 0.008 nC5 0.0108 0.0097 0.0089 0.0084 0.0086 0.0096 0.0107nC6 0.0116 0.011 0.0103 0.0086 0.0078 0.0082 0.0092C7+ 0.1268 0.0784 0.05 0.0323 0.0252 0.023 0.0282Mw+ 191 154 139 128 120 115 113 Sg+ 0.831 0.804 0.789 0.778 0.77 0.765 0.763

Pc C7+, (psia) 324.6 378.4 404.3 427.5 447.2 461.0 466.9 Tc C7+, (oR) 1264.0 1177.7 1136.4 1105.1 1081.6 1066.6 1060.5

T(oF) 313 313 313 313 313 313 313 P (psia) 6010 5100 4100 3000 2000 1200 700 Z (Expt.) 1.212 1.054 0.967 0.927 0.93 0.952 0.97

ρ(lb/cu.ft.) 26.3 18.97 14.17 9.79 6.27 3.68 2.19 ZSK 1.0982 0.9806 0.9224 0.8964 0.9054 0.9311 0.9531

ZScaled(This Study) 1.0213 1.0931 0.5793 0.0725 0.4924 0.8660 0.9835

46

Table 3.4: Highly Sour Gas Composition (Elsharkawy71). Serial No. 57

Highly sour gas condensate

IND 439 440 441 442 443 444 445 H2S 0.282 0.277 0.272 0.27 0.273 0.289 0.318 CO2 0.0608 0.0644 0.0669 0.0685 0.0694 0.0699 0.0679 N2 0.0383 0.0455 0.0476 0.0473 0.0461 0.0434 0.0394 C1 0.4033 0.4382 0.4641 0.4807 0.4844 0.4688 0.4331 C2 0.0448 0.0471 0.0481 0.0487 0.0493 0.0496 0.0494 C3 0.0248 0.0243 0.0239 0.0237 0.0239 0.0252 0.0277 iC4 0.006 0.0055 0.0051 0.0049 0.0049 0.0055 0.0067 nC4 0.0132 0.012 0.0111 0.0106 0.0106 0.0114 0.014 iC5 0.0079 0.0068 0.006 0.0055 0.0053 0.0058 0.0074 nC5 0.0081 0.0069 0.006 0.0054 0.0052 0.0057 0.0071 nC6 0.0121 0.0096 0.0078 0.0066 0.006 0.0063 0.0077 C7+ 0.0991 0.063 0.0412 0.0286 0.0217 0.0192 0.0214 Mw+ 165 121 116 112 109 107 107 Sg+ 0.818 0.778 0.773 0.768 0.764 0.762 0.762

Pc C7+, psia 365.4 453.2 467.1 477.8 486.0 492.7 492.7 Tc C7+, oR 1209.7 1090.7 1075.7 1062.7 1052.5 1046.3 1046.3

T(oF) 250 250 250 250 250 250 250 P (psia) 4190 3600 3000 2400 1800 1200 700 Z (Expt.) 0.838 0.806 0.799 0.809 0.842 0.888 0.935

ρ(lb/cu.ft.) 27.34 19.52 15.06 11.3 7.95 5.06 2.91 ZSK 0.8295 0.9299 0.9615 0.9744 0.9755 0.9734 0.9742

ZScaled(This Study) 0.7981 0.8974 0.9347 0.9564 0.9709 0.9853 1.0020

47

Table 3.5: Carbon Dioxide Rich Gas Composition (Elsharkawy71). Serial No. 124 Carbon Dioxide Rich Gas

IND 926 927 928 929 930 931 932 H2S 0.003 0.003 0.003 0.003 0.003 0.003 0.004 CO2 0.6352 0.6395 0.6514 0.6579 0.6639 0.6706 0.6716N2 0.0386 0.0399 0.041 0.0417 0.0421 0.0411 0.0388C1 0.1937 0.1988 0.2008 0.207 0.2084 0.2037 0.1994C2 0.0303 0.0307 0.0308 0.0309 0.0313 0.0315 0.0318C3 0.0174 0.0172 0.017 0.0169 0.017 0.0175 0.0184iC4 0.0033 0.0032 0.0031 0.003 0.003 0.0032 0.0035nC4 0.0093 0.0088 0.0085 0.0082 0.0082 0.0088 0.0097iC5 0.0039 0.0036 0.0033 0.0031 0.003 0.0033 0.0039nC5 0.0047 0.0042 0.0038 0.0036 0.0035 0.0038 0.0046nC6 0.0051 0.0049 0.0046 0.0042 0.0036 0.003 0.0034C7+ 0.0551 0.0458 0.0324 0.0202 0.0127 0.0101 0.0113Mw+ 170 153 139 128 118 110 106 Sg+ 0.811 0.797 0.783 0.773 0.763 0.755 0.751

Pc C7+, psia 347.8 373.9 397.8 421.7 446.3 469.4 482.4 Tc C7+, oR 1211.6 1169.4 1131.0 1100.6 1071.2 1046.9 1034.5

T(oF) 219 219 219 219 219 219 219 P (psia) 4825 4100 3300 2600 1900 1200 700

Z (Expt.) 0.851 0.777 0.72 0.719 0.775 0.851 0.915 ρ(lb/cu.ft.) 34.88 30.9 25.58 19.39 12.87 7.38 4.03

ZSK 0.7935 0.7739 0.7884 0.8247 0.8653 0.9084 0.9434ZScaled(This Study) 0.7151 0.7028 0.7233 0.7661 0.8151 0.8685 0.9125

48

Table 3.6: Very Light Gas Composition (Elsharkawy71). 125 Very light gas

Serial No. IND 933 934 935 936 937 938 939 H2S 0 0 0 0 0 0 0 CO2 0.0033 0.0033 0.0034 0.0035 0.0035 0.0036 0.0038N2 0.0032 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033C1 0.942 0.9438 0.9451 0.9461 0.9468 0.9473 0.9467C2 0.0231 0.023 0.023 0.0231 0.0232 0.0233 0.0236C3 0.0082 0.0082 0.0082 0.0082 0.0082 0.0082 0.0083iC4 0.0023 0.0023 0.0023 0.0023 0.0023 0.0023 0.0023nC4 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0026iC5 0.0012 0.0012 0.0012 0.0012 0.0012 0.0012 0.0012nC5 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0009nC6 0.0014 0.0013 0.0013 0.0013 0.0013 0.0012 0.0013C7+ 0.012 0.0103 0.0089 0.0077 0.0069 0.0063 0.006 Mw+ 143 133 126 120 116 114 114 Sg+ 0.787 0.777 0.769 0.763 0.76 0.758 0.758

Pc C7+, psia 390.5 409.7 423.9 438.6 450.5 456.2 456.2 Tc C7+, oR 1142.1 1114.1 1093.0 1075.4 1064.3 1058.3 1058.3

T(oF) 209 209 209 209 209 209 209 P (psia) 4786 4000 3300 2600 1900 1300 700 Z (Expt.) 1.019 0.974 0.945 0.933 0.933 0.947 0.969

ρ(lb/cu.ft.) 12.13 10.42 8.76 6.92 5.03 3.37 1.78 ZSK 1.1235 1.0726 1.0375 1.0119 0.9955 0.9884 0.9883

ZScaled(This Work) 1.0007 0.9565 0.9310 0.9179 0.9170 0.9264 0.9458

49

Table 3.7: Property Prediction for Gas Composition Data (Elsharkawy71). Data No. T,oR P,psia TR PR Tc, oR Pc, psia Zc ZExpt

Rich Gas Condensate 281 773 6010 1.6602 9.545 465.6188 629.6467 0.3137 1.212282 773 5100 2.346 9.9547 329.4974 512.3187 0.3111 1.054283 773 4100 2.679 8.5429 288.5443 479.9311 0.3093 0.967284 773 3000 2.8195 6.3235 274.1664 474.4191 0.3079 0.927285 773 2000 2.8615 4.2224 270.1407 473.6642 0.3077 0.93286 773 1200 2.8533 2.5324 270.9140 473.8606 0.3083 0.952287 773 700 2.7923 1.4711 276.8299 475.8504 0.3095 0.97

Highly Sour Gas 439 710 4190 1.4904 5.6376 476.3853 743.2255 0.3176 0.838440 710 3600 1.8906 5.459 375.5364 659.4605 0.317 0.806441 710 3000 2.0593 4.7129 344.7844 636.5490 0.3167 0.799442 710 2400 2.156 3.8496 329.3079 623.4470 0.3163 0.809443 710 1800 2.1873 2.8831 324.5952 624.3326 0.3162 0.842444 710 1200 2.1462 1.8854 330.8097 636.4649 0.3161 0.888445 710 700 2.0284 1.0567 350.0302 662.4678 0.3161 0.935

CO2 Rich Gas

926 679 4825 1.3607 5.5574 499.0157 868.2062 0.3012 0.851927 679 4100 1.4475 4.8577 469.0735 844.0239 0.3014 0.777928 679 3300 1.5372 3.9986 441.7072 825.2834 0.3013 0.72929 679 2600 1.6176 3.2057 419.766 811.0509 0.3016 0.719930 679 1900 1.6622 2.3661 408.4864 803.0142 0.3016 0.775931 679 1200 1.6632 1.4895 408.2512 805.6412 0.3011 0.851932 679 700 1.6418 0.8669 413.5822 807.5066 0.3006 0.915

Very Light Gas 933 669 4786 2.3565 8.9693 283.8983 533.5994 0.2935 1.019934 669 4000 2.3535 7.4526 284.2594 536.7226 0.2933 0.974935 669 3300 2.3506 6.1209 284.603 539.1324 0.2932 0.945936 669 2600 2.3480 4.8043 284.9245 541.1855 0.2931 0.933937 669 1900 2.3460 3.5019 285.1668 542.5596 0.2930 0.933938 669 1300 2.3445 2.3914 285.3522 543.6069 0.2930 0.947939 669 700 2.3454 1.288 285.2353 543.4760 0.2930 0.969

50

0.9

0.95

1

1.05

1.1

1.15

0 2 4 6 8Pressure (Psia)

Z-Fa

ctor

10

Expt.SKScaled

Figure 3.12: Scaled Z-Factor for Very Light Gas Composition.

0.7

0.8

0.9

1

0 2 4Pressure (Psia)

Z-Fa

ctor

6

Expt.SKScaled

Figure 3.13: Scaled Z-Factor for Carbon Dioxide Rich Gas Composition.

51

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1 3 5 7 9 11Pressure (Psia)

Z-Fa

ctor

Expt.

SK

Scaled

Figure 3.14: Scaled Z-Factor for Rich Gas Condensate Composition.

0.8

0.85

0.9

0.95

1

1.05

0 2 4 6Pressure (Psia)

Z-Fa

ctor

Expt.

SK

Scaled

Figure 3.15: Scaled Z-Factor for Highly Sour Gas Composition.

52

CHAPTER 4

Z-FACTOR PREDICTION FROM CUBIC

EQUATIONS OF STATE

4.1 Selection of Cubic Equations-of-State

Some phase behavior applications require the use of an equation of state to predict

properties of petroleum reservoir fluids. Since the introduction of the van der Waals4

EOS, many cubic EOS’s have been proposed like the Redlich and Kwong13 EOS (RK

EOS) in 1949, the Peng and Robinson12 EOS (PR EOS) in 1976, to name only a few.

Most of these equations retain the original van der Waals repulsive term ( )bVRT−

,

modifying only the denominator in the attractive term. With the advent of simulation

techniques in petroleum engineering, accuracy was the priority but a universal equation-

of-state method was required as most of the EOSs worked best in a certain range of

pressures and temperatures and compositions.

Most petroleum engineering relied on the PR EOS or a modification of the RK

EOS. Soave’s19 modification (SRK EOS) was the simplest and most widely used. A

major drawback of the SRK EOS was the poor liquid density prediction. PR EOS

reported that their EOS predicts better liquid densities than the SRK EOS but not accurate

enough for all ranges of pressures and temperatures including other phases. Many other

proposed equations of states relied on complex temperature functions to represent the

highly nonlinear correction terms for EOS constants.

The critical properties, acentric factor, molecular weight, and binary-interaction

parameters (BIP’s) of components in mixture are required for EOS calculations. With the

existing chemical-separation techniques, we usually cannot identify the many hundreds

and thousands of components found in reservoir fluids. Another problem with the

53

existing EOS and other methods of predicting EOS parameters is that they cannot predict

properties of components heavier than approximately C20.

Eight equations of state have been chosen which are commonly used in the

reservoir simulation and calculation purposes in the petroleum industry. Each of these

equations of state has been thoroughly examined in their ability to be able to predict z-

factor both for pure substances and gas mixtures (including natural gases and sour natural

gases with significant amounts of C7+). It is observed that the prediction of z-factor is

significantly dependent on the accuracy of the critical properties supplied/predicted.

Based on this observation, LLS29 EOS was observed to be capable of predicting accurate

critical properties for gas mixtures and therefore, more accurate z-factor prediction is

possible with this method for a wide range of pressures and temperatures and for any gas

composition. Hence, LLS EOS method can be adopted as a universal method for z-factor

determination.

4.2 Lawal-Lake-Silberberg Equation of State

22 bbVV)T(a

bVRTP

β−α+−

−= (4.1)

where

cw

ccw

ZZ3Z1

Ω−Ω+

=α (4.2)

c2w

wcc2w

3w

2c

Z)Z31(Z2)1(Z

ΩΩ−+Ω+−Ω

=β (4.3)

3cwa )Z)1(1( −Ω+=Ω (4.4)

PTRac

2c

2

aΩ= (4.5)

c

cb

PRTb

Ω

= (4.6)

54

Z-Form of the EOS:

0ZZZ 012

23

3 =Φ+Φ+Φ+Φ (4.7)

where,

0.13 =Φ [ ]B)1(12 α−+−=Φ

[ ]21 B)(BA α+β−α−=Φ

⎡ ⎤)BB(AB 320 +β−−=Φ

where,

RTbPB ,

TRP)T(aA 22 ==

.

Mixing Rules:

∑∑=i j

ij2

1

j2

1

ijim aaaxxa (4.8)

3

i

31

iim bxb ⎥⎦

⎤⎢⎣

⎡= ∑ (4.9)

jij

iij for a ω≤ω

ωω

= (4.10)

jii

jij for a ω>ω

ω

ω= (4.11)

( ) 5.0

i jji

21

j2

1

ijim xx∑∑ αααα=α (4.12)

( ) 5.0

i jji

21

j2

1

ijim xx∑∑ ββββ=β (4.13)

( ) ( ) ( )jijijiij aaa ββ=αα== (4.14)

55

4.3 van der Waal Equation of State

van der Waals4 proposed the first cubic EOS in 1873. The van der Waals EOS

gives a simple, qualitatively accurate relation between pressure, temperature, and molar

volume. It can be mathematically expressed as:

2Va

bVRTp −−

= (4.15)

where a = attraction parameter

b = repulsion parameter

as compared to the ideal gas law, van der Waals EOS provides two important

improvements. First, the prediction of liquid behavior is more accurate because volume

approaches a limiting value, b, at high pressures,

b)p(Vlimp

=∞→

(4.16)

where be is referred to as the covolume.

a/V2 term in the vdW EOS represents the non-ideal gas behavior and is interpreted as the

attractive component of pressure.

van der Waals also stated that the critical criteria that are used to define the two

EOS constants a and b which are the first and second derivatives of pressure with respect

to volume equal to zero at the critical point of a pure component.

0V

pVp

cV,cT,cp2

2

cV,cT,cp

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

=⎟⎠⎞

⎜⎝⎛∂∂ (4.17)

Martin and Hou show that this constraint is equivalent to the condition at

the critical point. The constants a and b are given by:

( ) 0zz 3c =−

c

2c

2

pTR

6427a =

and c

c

pRT

81b = (4.18)

The critical compressibility results in 375.083zc == .

56

van der Waals EOS in terms of z can be written as:

( ) 0ABAzz1Bz 23 =−++− (4.19)

where ( ) 2

R

R2 T

p6427

RTpaA == (4.20)

R

R

Tp

81

RTpbB == (4.21)

vdW EOS has a fixed zc (=0.375) for all components which is not true and no

temperature function which is a drawback of vdW EOS.

4.4 Redlich-Kwong Equation of State

Redlich and Kwong13 (1948) developed an adjustment in the van der Waals’

attractive pressure term (a/V2), which could considerably improve the prediction of the

volumetric and physical properties of the vapor phase. This attractive pressure term has a

temperature dependence term and their equation can be represented as:

)bV(V)T(a

bVRTp

+−

−=

(4.22)

where T is the system temperature in oR.

The authors in their development of the equation, noted that as the system

pressure becomes very large, i.e., p → ∞, the molar volume V of the substance shrinks to

about 26% of its critical volume regardless of the system temperature. The Equation 2.17

was accordingly constructed to satisfy the following condition:

cV26.0b = (4.23)

Applying the critical point conditions (as expressed by Equation 4.17) on

Equation 4.19, and solving the resulting equations simultaneously, gives

)T(apTR

a Rc

5.2c

2

aΩ= (4.24)

c

cb p

RTb Ω= (4.25)

where

57

08664.042748.0

b

a

=Ω=Ω

Equating Equation 4.18 with Equation 4.21 gives

ccc RT333.0Vp = (4.26)

The above expression shows that Redlich-Kwong EOS produces a universal critical

compressibility factor (Zc) of 0.333 for all substances.

Replacing the molar volume V in Equation 4.20 with ZRT/p gives

22TRpaA =

(4.27)

RTbpB = (4.28).

Redlich and Kwong extended the application of their equation to hydrocarbon

liquid or gas mixtures by employing the following mixing rules: 2n

1i

5.0iim axa ⎥⎦

⎤⎢⎣

⎡= ∑

=

(4.29)

⎥⎦

⎤⎢⎣

⎡= ∑

=

n

1iiim bxb (4.30)

The Redlich-Kwong value of zc=1/3 is reasonable for lighter hydrocarbons but is

unsatisfactory for heavier components.

4.5 Soave-Redlich-Kwong Equation of State

A significant development of cubic equations of state was the publication by

Soave19 (1972) of a modification in the evaluation of the parameter a in the attractive

pressure term of the Redlich-Kwong equation of state (Equation 4.22). Soave replaced

the term (a/T0.5) in Equation 4.22 with a more general temperature-dependent term as

demonstrated by (aα), to give

)bV(V)T(a

bVRTp

+α

−−

= (4.31)

58

where α is a dimensionless factor which becomes unity at T = Tc. At temperatures

other than critical temperature, the parameter α is defined by the following expression: 25.0

r ))T1(m1( −+=α (4.32)

The parameter m is correlated with the acentric factor, to give 2176.0574.1480.0m ω−ω+= (4.33)

where ω is the acentric factor of the substance.

For any pure component, the constants a and b in Equation 4.31 are found by

imposing the classical van der Waals’ critical point constraints (Equation 4.17), on

Equation 4.31 and solving the resulting equations, to give

c

2c

2

a pTRa Ω=

(4.34)

c

cb p

RTb Ω=

(4. 35)

where Ωa and Ωb are the Soave-Redlich-Kwong (SRK) dimensionless pure component

parameters and have the following values:

Ωa = 0.42747 (4.36)

Ωa = 0.08664 (4.37)

The Z-Form of the Equation 4.31 is:

0ABZ)BBA(ZZ 223 =−−−+− (4.38)

where

( )2)RT(paA α

= (4.39)

RTbpB = (4.40)

To use the Equation 4.38 with mixtures, the following mixing rules were proposed by

Soave:

( ) ( ) ( )[ ]∑ ∑ −αα=αi j

ij5.0

jijijim 1kaaxxa (4.41)

59

[ ]∑=i

iim bxb (4.42)

with

( )2

m

)RT(paA α

= (4.43)

RTpb

B m= (4.44)

The parameter kij is an empirically determined correction factor called the binary

interaction coefficient, characterizing the binary formed by component i and component j

in the hydrocarbon mixture.

Modifications of the SRK EOS

Groboski and Daubert37 (1978) proposed a new expression for calculating the

parameter m of Equation 4.32 to improve the pure component vapor pressure predictions

by the SRK EOS. The proposed relationship has the following form: 215613.055171.148508.0m ω−ω+= (4.45)

Elliot and Daubert38 (1985) stated that the evaluation of optimal interaction

coefficients of asymmetric mixtures (components with significant difference in chemical

behavior), proposed the following set of expressions for calculating kij,

• For N2 systems: ∞+= ijij k9776.2107089.0k

• For CO2 systems:

( )2ijijij k8407.1k77215.008058.0k ∞∞ −−=

• For H2S systems: ∞+= ijij k017921.007654.0k

• For Methane systems with compounds of 10 or more:

( )2ijijij k10853k6958.217985.0k ∞∞ ++=

where, for the above expression:

60

)2/()(k ji2

jiij εεε−ε−=∞ (4.46)

and

i5.0

eii b/))2(loga(=ε (4.47).

The major drawback in the SRK EOS is that the critical compressibility factor

takes on the unrealistic universal critical compressibility of 0.333 for all substances.

Consequently, the molar volumes are typically overestimated, i.e., densities are

underestimated.

4.6 Peng-Robinson Equation of State

Peng and Robinson12 (1975) conducted a comprehensive study to evaluate the use

of SRK equation of state for predicting the behavior of naturally occurring hydrocarbon

systems. The authors showed emphasis on the ability of the equation to predict liquid

densities and other fluid properties particularly in the vicinity of the critical region. They

proposed the following expression:

22 cb)bV()T(a

bVRTp

−+α

−−

= (4. 48)

Equation 4.48 can be rewritten as:

)bV(b)bV(V)T(a

bVRTp

−++α

−−

= (4.49)

Imposing the classical critical point conditions (Equation 4.17) on Equation 4.48

and solving for the parameters a and b, yields

c

2c

2

a pTRa Ω= (4.50)

c

cb p

RTb Ω=

(4.51) where

07780.045724.0

b

a

=Ω=Ω

.

61

This equation predicts a universal critical gas compressibility factor of 0.307 compared to

0.333 for the SRK model. Peng and Robinson also adopted Soave’s approach for

calculating the parameter α: 25.0

R ))T1(m1( −+=α (4.52)

where (4.53) 22699.05423.13746.0m ω−ω+=

This was later expanded by the investigators (1978) to give the following

relationship: 32 016667.01644.048503.1379642.0m ω+ω−ω+= (4.54)

Rearranging Equation 2.37 into the compressibility factor form gives

0)BBAB(Z)B2B3A(Z)1B(Z 32223 =−−−−−+−+ (4.55)

The mixing rules for PR EOS are defined as follows:

∑∑= ij2/1

j2/1

ijim aaaxxa (4.56)

∑=i

iim bxb (4.57)

Although PR EOS is another widely used cubic EOSs in petroleum engineering

calculations, it underpredicts saturation pressure of reservoir fluids compared with SRK

EOS.

4.7 Schmidt-Wenzel Equation of State

Schmidt and Wenzel24 (1980) proposed an attractive pressure term that introduces

the acentric factor ω as a third parameter. The SW EOS has the following form:

23 b3bV)31(V)T(a

bVRTp

ω−ω++−

−= (4. 58)

with

α⎟⎟⎠

⎞⎜⎜⎝

⎛Ω=

c

2c

2

a pTRa (4.59)

62

⎟⎟⎠

⎞⎜⎜⎝

⎛Ω=

c

cb p

RTb (4.60)

where 3

cca ))1(1( β−ζ−=Ω (4.61)

ccb ξβ=Ω (4.62)

The βc is given by the smallest positive root of the following equation:

( ) 013316 c2c

3c =−β+β+β+ω (4. 63)

and

( )ωβ+=ξ

cc 13

1 (4.64)

4.8 Patel-Teja Equation of State

Patel and Teja20 (1982) proposed the following three-parameter cubic equation:

bcV)cb(V)T(a

bVRTp

2 −++−

−= (4.65)

In this equation “a” is a function of temperature, and b and c are constants

characteristic of each component. Equation 4.65 was constrained to satisfy the following

conditions:

0V

p

CT

=∂∂ (4.66)

0V

p2CT

2

=∂∂ (4.67)

cc

cc

RTVp

ξ= (4.68)

Patel and Teja pointed out that the third parameter c in the equation allows the

empirical parameter ξc to be chosen freely. Application of Equation 4.66 to Equation 4.67

yields:

63

25.0R

c

2c

2

a )]T1(m1[pTR

a −+Ω= (4.69)

c

cb p

RTb Ω= (4.70)

c

cc p

RTc Ω= (4.71)

where

cc 31 ξ−=Ω (4.72)

( ) ( )c2bbc

2ca 312133 ξ−+Ω+Ωξ−+ξ=Ω (4.73)

and Ωb is the smallest positive root of the following equation:

03)32( 3cb

2c

2bc

3b =ξ−Ωξ+Ωξ−+Ω (4.74)

Equation 4.74 can be solved for Ωb by using the Newton-Raphson iterative

method with an initial value for Ωb as given by

002005.0Z32429.0 cb −=Ω (4.75)

For non-polar fluids, the parameters m and, ξc are related to the acentric factor by

the following relationships: 2295937.030982.1452413.0m ω−ω+= (4.76)

2c 0211947.00767992.0329032.0 ω+ω−=ξ (4.77)

In terms of Z, Equation 4.65 can be rearranged to produce

( ) ( ) ( ) 0ABCBBCZBCBBC2AZ1CZ 2223 =−++−−−−+−+ (4.78)

where, for mixtures

( )2m

RTpaA = (4.79)

RTpbB m= (4.80)

RTpcC m= (4.81)

with

64

[ ]∑∑ −= )k1()aa(xxa ij5.0

jijim (4.82)

[∑=i

iim bxb ]

]

(4.83)

[∑=i

iim cxc (4.84)

An improved relationship for undefined components such as C7+, the parameters

m and ξc was proposed by Willman and Teja40 (1986) in terms of the boiling point Tb and

specific gravity γ. Therefore, a major drawback of PT EOS is that additional information

is required to be able to determine volumetric properties of composition involving C7+.

4.9 Trebble-Bishnoi-Salim Equation of State

Trebble and Bishnoi26 proposed a four parameter equation of state and it can be

represented as follows:

22 dbcV)cb(V)T(a

bVRTp

−−++−

−= (4.85)

Parameters “a” and “b” are temperature dependent while “c” and “d” are

independent of temperature. Therefore, new temperature functions for “a(T)” and “b(T)”

have been proposed. The value of “d” was determined for all the components available in

the database along the critical isotherm. “d” values for the remaining components were

calculated from a linear fit of optimized “d” values versus the critical volume. The value

of “c” was directly determined from the experimental value of the critical

compressibility. Once the parameters “c” and “d” are set, optimal values of “a” and “b”

are then calculated.

This TB EOS offers increased correlational flexibility and allows for significant

improvements in PVT predictions. Trebble and Bishnoi do mention that the quality of an

equation-of-state largely depends on the data used in its preparation

65

4.10 Transformed Cubic Equations to the LLS EOS Form

Lawal-Lake-Silberberg EOS is represented as:

22 bbVV)T(a

bVRTP

β−α+−

−=

(4.1)

It is the most general form of the EOSs described in this study. This can be

observed by substituting the values of α and β with numerical constants as described in

the Table 4.1.

Table 4.1: Common Specialization Cubic Equation of State

4.11 Generalized Reduced State of Cubic Equations-of-State

Described below is the derivation of reducing the general LLS EOS to the Z-

form:

66

22 bbVV)T(a

bVRTP

β−α+−

−=

(4.1)

Multiply on both sides of Equation 4.1 byRTV , we get

22 bbVVRTV)T(a

RTV

bVRT

RTVP

β−α+−

−=

(4.86)

Real Gas Equation: (4.87) ZRTPV =

Using the real gas law, Equation 4.86 becomes,

22 bbVVRT

V)T(a

bVVZ

β−α+−

−=

(4.88)

Simplifying Equation 4.88 using the real gas law: P

ZRTV = ,

( ) 22

2

PP

bZRTP

ZRTRT

PZRT)T(a

PZRT

b1

1Zβ−

α+

−−

=

(4.89)

Further simplifying Equation 4.89,

22

2

)bP(ZbPRT)ZRT(

PRT

PZRT)T(a

RTZbp1

1Zβ−α+

−−

=

(4.90)

Defining ( )2RTP)T(aA =

(4.91)

and RTbPB =

(4.92)

Using these definitions in Equation 4.90 and dividing the 2nd part of RHS by ( )2RT

1 ,we

get,

67

( )

( ) ( ) ( )2

2

22

2

2

RT)bP(

RTZbPRT

RT)ZRT(

ZRT

P)T(a

ZB1

1Zβ

−α

+−

−=

(4.93)

Simplifying Equation 4.93,

( )

( ) ( )2

22

2

RT)bP(

RTZbPZ

ZRT

P)T(a

ZB1

1Zβ

−α

+−

−=

(4.94)

Using the definitions of A and B and further simplifying Equation 4.94, we get

22 BBZZAZ

BZZZ

β−α+−

−=

(4.95)

Canceling Z on the numerators on both sides in Equation 4.95, we get

22 BBZZA

BZ11

β−α+−

−=

(4.96)

Simplifying Equation 4.96 by cross-multiplication, we get

( ) ( ) ABAZBBZZBZBBZBBZ 2232223 +−β−α+=β+α−β−+α+−+ (4.97)

= ( ) ( ) ( ) 0ABBBZBBBAZBB1Z 232223 =−β+β+α−β−α−+α+−−+ (4.98)

The reduced form of the real-gas law can be expressed as

) ,( RRc TPfZZ = (4.99)

In order to use Equation 4.99 to predict Z-factors of pure substances, natural and

sour gases by utilizing cubic equations of state, the computation of Z-factor from the

reduced form of Equation 4.1 requires composition-dependent critical compressibility

factor as well as mixture critical pressure, temperature and volume for the reduced

parameters (PR, νR, TR). The task is accomplished in the next paragraph

68

The reduced compressibility factor (ZR) equation for pure substances can be

derived from Equation 4.1 by dividing the expression of Equation 4.100 by Zc:

cw

ccw

ZZ3Z1

Ω−Ω+

=α (4.100)

0ZZZ 0R12R2

3R3 =θ+θ+θ+θ (4.101)

where

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧Ω

+⎭⎬⎫

⎩⎨⎧Ω

β−ΩΩ

−=θ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎭⎬⎫

⎩⎨⎧Ω

β+α−Ω

α−Ω

=θ

⎟⎟⎠

⎞⎜⎜⎝

⎛ Ωα−+−=θ

=θ

θ+

θ+

3

Rc

Rb

2

Rc

Rb

c3R

3c

2Rba

0

2

Rc

Rb

R2c

Rb2R

2c

Ra1

Rc

Rb

c2

3

TZP

TZP

Z1

TZP

TZP)(

TZP

TZP

TZP)1(

Z1

1

c

c

The reduced compressibility factor (ZR) equation for mixtures can be derived

from Equation 4.101 by replacing pure substance parameters with mixture parameters:

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎥⎥⎦

⎤

⎢⎢⎣

⎡

⎭⎬⎫

⎩⎨⎧

+⎭⎬⎫

⎩⎨⎧

β−−=θ

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

⎭⎬⎫

⎩⎨⎧

β+α−α−=θ

⎟⎟⎠

⎞⎜⎜⎝

⎛α−+−=θ

=θ

θ+

θ+

3

Rc

Rc

2

Rc

Rc

cm3

R3c

2Rcc

0

2

Rc

Rcmm

R2c

Rcm2

R2c

Rc1

Rc

Rcm

c2

3

TZPB

TZPB

Z1

TZPBA

TZPB

)(TZPB

TZPA

TZPB

)1(Z1

1

m

m

(4.102)

In Equations 4.102, the composition-dependent parameters Ac, Bc and Zc are

defined by Equations 4.103-4.105.

69

2ccm

2cmmc Z3BB)( A +α+β+α= (4.103)

0ZZZ 0c1

2c2

3c3 =θ+θ+θ+θ (4.104)

where

)( 6663

)9912123(

6128

mmm2m0

mmm2mm1

mmm2mm2

3m

2mm3

βα−β+α−=θ

βα−β+α+α=θ

βα−β+α+α+−=θ

α+α+α+=θ

0BBB 0c1

2c2

3c3 =φ+φ+φ+φ (4.105)

where

136

3271515

6128

0

m1

2mmm2

3m

2mm3

−=φα+=φ

α−β−α+=φ

α+α+α+=φ

The expressions for mixture critical pressure and temperature are thereby established in

Equations 4.99 and 4.102

]BB)(Z3[B

baP

cm2cmm

2c

2c

2m

mc α+β+α+=

(4.106)

]BB)(Z3[RB

baT

cm2cmm

2c

c

m

mc α+β+α+=

(4.107)

4.12 Prediction Results for Z-Factor of Pure Substances

A graphical comparative result of the eight EOSs is shown with the experimental

values for the components as shown below. More results on this can been seen in

Appendix D.

70

Z-Factor Comparison Graph (Expt. vs. LLS-EOS)

0.75

0.85

0.95

1.05

1.15

1.25

1.35

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Pressure, P (Psia)

Z-Fa

ctor

LLS 560 R

LLS 680 R

LLS 920 R

Expt. T=560

Expt. T=680

Expt. T=920

Figure 4.1: Z-Factor comparison for LLS-EOS for Methane.

Z-Factor Comparison Graph (Expt. vs. LLS)

0.23

0.33

0.43

0.53

0.63

0.73

0.83

0.93

1.03

1.13

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Pressure, P (Psia)

Z-Fa

ctor

LLS 560 R

LLS 680 R

LLS 920 R

Expt. T=560

Expt. T=680

Expt. T=920

Figure 4.2: Z-Factor comparison for LLS-EOS for Carbon dioxide.

71

Z-Factor Comparison Graph (Expt. vs. LLS-EOS)

0.95

1.05

1.15

1.25

1.35

1.45

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Pressure, P (Psia)

Z-Fa

ctor

LLS 560 R

LLS 680 R

LLS 920 R

Expt. T=560

Expt. T=680

Expt. T=920

Figure 4.3: Z-Factor comparison for LLS-EOS for Nitrogen.

Z-Factor Comparison (Expt. Vs. VdW-EOS)

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Pressure (psia)

Z-Fa

ctor

VdW-EOS T=100 FVdW-EOS T=220 FVdW-EOS T=460 F100 F EXP220 F EXP460 F EXP

Figure 4.4: Z-Factor comparison for vdW-EOS for Methane.

72

Z-Factor Comparison (Expt. Vs. vdW-EOS)

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Pressure, (psia)

Z-Fa

ctor

VdW-EOS T=100 F

VdW-EOS T=220 F

VdW-EOS T=460 F

100 F EXP

220 F EXP

460 F EXP

Figure 4.5: Z-Factor comparison for vdW-EOS for Carbon dioxide.

4.13 Development of Binary Interaction Parameters Binary Interaction Coefficient/Number (BIN):

These binary interaction coefficients are used to model the intermolecular

interaction through empirical adjustment of the (aα)m term as represented mathematically

by Equation 4.38. They are dependent on the difference in molecular size of components

in a binary system and they are characterized by the following properties, as summarized

by Slot-Petersen39 (1987):

• The interaction between hydrocarbon components increases as the relative

difference between their molecular weights increases:

k,j+1 > ki,j

• Hydrocarbon components with the same molecular weight have a binary

interaction coefficient of zero:

ki,j = 0

73

• The binary interaction coefficient matrix is symmetric:

ki,j = kj,i

4.14 Prediction Results for Z-Factor of Mixtures

Z-Factor Comparison Chart at 49 oF (Simon et. al.)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1000 1250 1500 1750 2000 2250 2500 2750 3000Pressure (Psia)

Z-Fa

ctor

Expt. MeasuredCorresponding statesBWR EOSVdWLLSPRPTRKSRKSWTB

Figure 4.6: Z-Factor comparison for CO2-C1 mixture at 49 oF.

74


0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

800 1000 1200 1400 1600 1800 2000 2200 2400 2600 2800Pressure (Psia)

Z-Fa

ctor




0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1250 1500 1750 2000 2250 2500 2750 3000Pressure (Psia)

Z-Fa

ctor


TB

VdW

LLS

PTPR

RKSRK


75


0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

500 1000 1500 2000 2500 3000Pressure (Psia)

Z-Fa

ctor


TB

SRK

VdW

PTLLS PR


76

4.15 Prediction Results for Z-Factor of Natural Gases

Table 4.2: Sources of Experimental Z-Factor. Authors Year System Reference

No.

Sage-Reamer-Lacey 1950 C1-C2 8

Sage-Lacey-Schfaasma 1934 C1-C3 79

Reamer-Olds-Sage-Lacey 1944 C1-CO2 75

Reamer-Sage-Lacey 1951 C1-H2S 80

Reamer-Olds-Sage-Lacey 1942 C1-nC10 82

Reamer-Olds-Sage-Lacey 1945 C2-CO2 76

Reamer-Selleck-Sage-Lacey

1952 C2-N2 83

Reamer-Sage 1962 C2-nC10 84

Sage-Reamer-Lacey 1951 C3-CO2 77

Reamer-Olds-Sage-Lacey 1949 nC4-CO2 78

Reamer-Selleck-Sage-Lacey

1953 nC10-H2S 81

Wichert 1970 CO2-H2S-N2-C1-C2-C3-iC4-nC4-iC5-nC5-nC6-C7+

86

Elsharkawy 2002, 2004 CO2-H2S-N2-C1-C2-C3-iC4-nC4-iC5-nC5-nC6-C7+

41, 71

Elsharkawy-Foda 1998 CO2-H2S-N2-C1-C2-C3-iC4-nC4-iC5-nC5-nC6-C7+

74

Satter-Campbell 1963 H2S-C1-C2 46

Buxton-Campbell CO2-N2-C1-C2-C3

Simon-Fesmire-Dicharry-Vorhis

1977 CO2-N2-C1-C2-C3-nC4-nC5-nC6 87

Fluid Prop. Package (Shell) 2003 CO2-N2-C1-C2-C3-iC4-nC4-iC5-nC5-nC6-C7+

Private

77

4.15.1 Results for Excelsior Laboratory Data

Table 4.3: Gas Composition for Excelsior 6 Laboratory Data.

COMPARISON OF

LABORATORIES AND FLUID

PROPERTIES PACKAGE

Pressure (Psia) 3317 2615

Core Lab. Fluid Prop.

Package Core Lab.

Fluid Prop. Package

BHT, oF 121 0.802 0.7942 0.768 0.7767

Z-Factor 0 0 0.171 0.2048 Produced

Fraction of Dew Point Gas 0 0 0.042 0.05

Liquid Saturation Vapor Phase Composition 0.0005 0.0005 0.0005 0.0005

Carbon Dioxide 0.0069 0.0069 0.0073 0.0071 Nitrogen 0.813 0.813 0.8321 0.8325 Methane 0.063 0.063 0.0625 0.0627 Ethane 0.0343 0.0343 0.0325 0.0334

Propane 0.0195 0.0195 0.0179 0.0185 iso-Butane 0.0153 0.0153 0.0137 0.0143 n-Butane 0.0102 0.0102 0.0085 0.0091

iso-Pentane 0.0052 0.0052 0.0042 0.0045 n-Pentane 0.0079 0.0079 0.0059 0.0063 Hexane (s) 0.0242 0.0242 0.0149 0.0111

Heptane plus 0 0 0 0

78

0.76

0.87

0.98

1.09

1.20

715 1235 1755 2275 2795 3315Pressure(psia)

Z-Fa

ctor

Lab. vdW

LLS PR

PT RK

SRK SW

Figure 4.10: Z-Factor for Sweet Natural Gas, Data from Excelsior 6 (FPP) at 581 oR


0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1250 1500 1750 2000 2250 2500 2750 3000Pressure (Psia)

Z-Fa

ctor


TB

VdW

LLS

PTPR

RKSRK

Figure 4.11: Z-Factor Comparison Chart at 90 oF (Simon et al.).

79


0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

500 1000 1500 2000 2500 3000Pressure (Psia)

Z-Fa

ctor


TB

SRK

VdW

PTLLS PR

Figure 4.12: Z-Factor Comparison Chart at 120 oF (Simon et al.).

4.15.2 Results for TTU Laboratory Data

0.5

0.6

0.7

0.8

0.9

1

0 1000 2000 3000 4000 5000Pressure, psia

Z-Fa

ctor

LLS EOS

PE Lab.

Figure 4.13: 75% CO2 - Dry Gas at 100 oF for CO2 Sequestration.

80

0.85

0.9

0.95

1

0 1000 2000 3000 4000 5000Pressure, psia

Z-Fa

ctor

LLS

PE Lab.

Figure 4.14: 25% CO2 - Dry Gas at 160 oF for CO2 Sequestration.

4.15.3 Results for UCalgary Data

Table 4.4: Gold Creek Gas Composition.

GOLD CREEK 10-5

P (Psia) T (210 oF)

4496 0.93 4815 0.948 4515 0.966 5015 0.984 5215 1.003 5515 1.032 6015 1.061

CO2 H2S N2

0.0318 0.0704 0.0401 Total Acid Gas 0.1022

C1 C2 C3 IC4 NC4 IC5 NC5 C6 C7+ 0.7069 0.0303 0.0209 0.0057 0.0109 0.006 0.0057 0.0093 0.046C7+ Fraction

Mole. Wt. 131

Sp. Gr. 0.785

81

0.9

1.1

1.3

1.5

1.7

4400 4700 5000 5300 5600 5900 6200Pressure (Psia)

Z-Fa

ctor

Expt.VdWLLSPRPTRKSRKSW

VdW

LLS

PT

PR

SW

RK SRK

Figure 4.15: Z-Factor for sour natural gas, data from Excelsior 6 (FPP) at 581 oR

Shell Marmattan 10-33 @ 84 oF

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

2014 2514 3014 3514 4014 4514 5014Pressure (Psia)

Z-Fa

ctor

Expt.VdW-EOSLLS-EOSPR-EOSPT-EOSRK-EOSSRK-EOSSW-EOSTB-EOS

LLSPTPR

SW

VdWTB

RKSRK

Figure 4.16: Z-Factor comparison for sour natural gas mixture at 84 oF.

82

Shell Marmattan 10-33 @ 73 oF

0.4

0.6

0.8

1.0

1.2

1.4

1.6

2014 2514 3014 3514 4014 4514 5014Pressure (Psia)

Z-Fa

ctor

Expt.VdW-EOSLLS-EOSPR-EOSPT-EOSRK-EOSSRK-EOSSW-EOSTB-EOS

LLSPT

PR

SW

VdW

TB

RKSRK


Sutte Plant, H,P Injection Line

0.83

0.88

0.93

0.98

1.03

1.08

1.13

1.18

1.23

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000Pressure (Psia)

Z-Fa

ctor

Expt.LLSPRPTRKSRKSWTBVdW

TBSW

VdW

LLS

PR

PTRK

SRK


83

Fina WindFall Processing Plant (510 oR)

0.42

0.52

0.62

0.72

0.82

0.92

1.02

1.12

1014 1514 2014 2514 3014 3514 4014 4514 5014Pressure (Psia)

Z-Fa

ctor

Expt.LLS-EOSVdW-EOSPR-EOSRK-EOSSRK-EOSPT-EOSSW-EOSTB-EOS

TB VdW

SRKSW

PR

PT

LLSRK



0.58

0.68

0.78

0.88

0.98

1.08

1.18

1014 1514 2014 2514 3014 3514 4014 4514 5014Pressure (Psia)

Z-Fa

ctor


TB VdW

SRKSW

PR PT

LLS

RK


84


0.59

0.69

0.79

0.89

0.99

1.09

1.19

1014 1514 2014 2514 3014 3514 4014 4514 5014Pressure (Psia)

Z-Fa

ctor


TB VdW

SRKSW

PR PTLLS

RK



0.70

0.75

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1014 1514 2014 2514 3014 3514 4014 4514 5014Pressure (Psia)

Z-Fa

ctor


TB VdW

SRKSW

PR

PTLLS

RK


85


0.74

0.79

0.84

0.89

0.94

0.99

1.04

1.09

1.14

1.19

1.24

1014 1514 2014 2514 3014 3514 4014 4514 5014Pressure (Psia)

Z-Fa

ctor


TBVdW

SRKSW

PR

PT

LLS

RK



0.74

0.79

0.84

0.89

0.94

0.99

1.04

1.09

1.14

1.19

1.24

1014 1514 2014 2514 3014 3514 4014 4514 5014Pressure (Psia)

Z-Fa

ctor


TBVdW

SRKSW

PR

PTLLS

RK


86


0.74

0.79

0.84

0.89

0.94

0.99

1.04

1.09

1.14

1.19

1.24

1014 1514 2014 2514 3014 3514 4014 4514 5014Pressure (Psia)

Z-Fa

ctor


TBVdW

SRKSW

PR

PTLLS

RK



0.84

0.89

0.94

0.99

1.04

1.09

1.14

1.19

1.24

1014 1514 2014 2514 3014 3514 4014 4514 5014Pressure (Psia)

Z-Fa

ctor


TB VdW

SRKSW

PR

PT

LLS

RK


87

4.15.4 Results for Elsharkawy Gas Data

Table 4.5: Results of Elsharkawy Gas Data. IND 1 84 416 439 504 752 817 H2S 0 0.0708 0.0383 0.2816 0 0.1693 0.1047 CO2 0.0017 0.0096 0.0058 0.0608 0.0097 0.0576 0.0163 N2 0.015 0.0064 0.002 0.0383 0.0041 0.0011 0.0244 C1 0.7284 0.6771 0.7564 0.4033 0.8616 0.6619 0.7352 C2 0.0847 0.0871 0.0706 0.0448 0.0355 0.0412 0.0498 C3 0.0418 0.0384 0.0336 0.0248 0.0154 0.0188 0.0181 IC4 0.011 0.005 0.0104 0.006 0.0046 0.0044 0.0059 NC4 0.0171 0.0156 0.0135 0.0132 0.0046 0.0076 0.0073 IC5 0.0088 0.0056 0.0072 0.0079 0.0026 0.0032 0.004 NC5 0.0084 0.0082 0.0055 0.0081 0.002 0.0036 0.0037 C6 0.0124 0.0083 0.0077 0.0121 0.0035 0.0052 0.0053

C7+ 0.0707 0.0656 0.049 0.0991 0.0564 0.0261 0.0253 Mw+ 152 154 158 165 253 144 132 Sg+ 0.81 0.776 0.783 0.818 0.85 0.788 0.774

Tc C7+, oR 1179.581 1151.75 1165.19 1209.73 1368.64 1144.9 1109.426Pc C7+,

psia 381.99 350.86 373.99 398.74 487.02 362.72 409.55

T (oF) 221 296 325 250 271 255 290 P (psia) 4973 4669 5095 4190 11830 4050 4255 Z (Expt.) 0.997 0.97 1.011 0.838 1.775 0.914 0.968 LLS (This

Study) 1.0052 0.9883 1.0651 0.7349 1.5917 0.9749 1.0172

88

Table 4.5 (Contd.) Component Mole Fraction

IND 1275 1277 1280 1714 1788 1866 H2S 0.068 0.1078 0.1826 0.2327 0.273 0.5137 CO2 0.0209 0.0616 0.0866 0.0287 0.0451 0.0319 N2 0.1019 0.004 0.0037 0.0304 0.0061 0.0258 C1 0.6857 0.7414 0.5213 0.5601 0.6459 0.4241 C2 0.059 0.0327 0.1165 0.082 0.0084 0.0024 C3 0.0282 0.0121 0.0142 0.0345 0.0093 0.0007 iC4 0.0047 0.0022 0.0039 0.0085 0.0027 0.0002 nC4 0.0116 0.0061 0.0083 0.011 0.002 0.0003 iC5 0.0085 0.0057 0.0095 0 0.002 0.0002 nC5 0 0 0 0.0071 0.001 0.0001 nC6 0.0035 0.0046 0.0103 0.0028 0.0012 0.0002 C7+ 0.008 0.0218 0.0431 0.0022 0.0032 0.0004 Mw+ 125 125 125 145 103 120 Sg+ 0.75 0.75 0.75 0.85 0.7 0.75

Tc C7+, oR 1074.0 1074.02 1074.0 1202.56 983.27 1063.8

Pc C7+, psia 405.26 405.26 405.26 394.76 272.93 422.82 T (oF) 157 189 216 120 250 230

P (psia) 2347 5065 5385 1000 5014 3514 Z (Expt.) 0.823 0.95 0.942 0.802 0.931 0.711

LLS (This Study) 0.9151 1.0328 0.9984 0.8768 1.0095 0.8330

89

4.15.5 Results for Elsharkawy Miscellaneous Data

Table 4.6: Z-Factor Results for Miscellaneous Gases. Rich Gas Condensate

Serial No. 281 282 283 284 285 286 287 H2S 0 0 0 0 0 0 0

CO2 0.0231 0.0242 0.0248 0.0253 0.0258 0.0262 0.0266

N2 0.0137 0.0155 0.0161 0.0166 0.0163 0.0155 0.0143

C1 0.6583 0.7074 0.738 0.7559 0.7583 0.7485 0.7292

C2 0.0803 0.0817 0.0821 0.0839 0.0863 0.0905 0.0944

C3 0.0417 0.0411 0.0404 0.0402 0.0415 0.0447 0.0495

iC4 0.0078 0.0073 0.007 0.0069 0.0073 0.0082 0.0091

nC4 0.0184 0.017 0.0162 0.0159 0.0167 0.0186 0.0208

iC5 0.0075 0.0067 0.0062 0.006 0.0062 0.007 0.008

nC5 0.0108 0.0097 0.0089 0.0084 0.0086 0.0096 0.0107

nC6 0.0116 0.011 0.0103 0.0086 0.0078 0.0082 0.0092

C7+ 0.1268 0.0784 0.05 0.0323 0.0252 0.023 0.0282

Mw+ 191 154 139 128 120 115 113 Sg+ 0.831 0.804 0.789 0.778 0.77 0.765 0.763

Pc C7+, psia

324.60 378.39 404.34 427.52 447.21 460.98 466.85

Tc C7+, oR 1264.023 1177.662 1136.436 1105.09 1081.6 1066.586 1060.503

T (oF) 313 313 313 313 313 313 313 P (psia) 6010 5100 4100 3000 2000 1200 700

Z (Expt.) 1.212 1.054 0.967 0.927 0.93 0.952 0.97 ρ (lb/cu.ft.) 26.3 18.97 14.17 9.79 6.27 3.68 2.19 LLS (This Study) 1.0614 1.0612 1.0239 0.9815 0.9595 0.9588 0.9675

90

Table 4.6 (Contd.) Highly Sour Gas Condensate

Serial No. 439 440 441 442 443 444 445 H2S 0.282 0.277 0.272 0.27 0.273 0.289 0.318 CO2 0.0608 0.0644 0.0669 0.0685 0.0694 0.0699 0.0679 N2 0.0383 0.0455 0.0476 0.0473 0.0461 0.0434 0.0394 C1 0.4033 0.4382 0.4641 0.4807 0.4844 0.4688 0.4331 C2 0.0448 0.0471 0.0481 0.0487 0.0493 0.0496 0.0494 C3 0.0248 0.0243 0.0239 0.0237 0.0239 0.0252 0.0277 iC4 0.006 0.0055 0.0051 0.0049 0.0049 0.0055 0.0067 nC4 0.0132 0.012 0.0111 0.0106 0.0106 0.0114 0.014 iC5 0.0079 0.0068 0.006 0.0055 0.0053 0.0058 0.0074 nC5 0.0081 0.0069 0.006 0.0054 0.0052 0.0057 0.0071 nC6 0.0121 0.0096 0.0078 0.0066 0.006 0.0063 0.0077 C7+ 0.0991 0.063 0.0412 0.0286 0.0217 0.0192 0.0214

Mw+ 165 121 116 112 109 107 107 Sg+ 0.818 0.778 0.773 0.768 0.764 0.762 0.762

Pc C7+, psia

365.42 453.28 467.1 477.79 486.02 492.70 492.70

Tc C7+, oR 1209.732 1090.741 1075.735 1062.661 1052.522 1046.28 1046.28 T (oF) 250 250 250 250 250 250 250

P (psia) 4190 3600 3000 2400 1800 1200 700 Z (Expt.) 0.838 0.806 0.799 0.809 0.842 0.888 0.935 ρ(lb/cu.ft.) 27.34 19.52 15.06 11.3 7.95 5.06 2.91 LLS (This Study) 0.8055 0.8698 0.8837 0.8920 0.9009 0.9156 0.9361

91

Table 4.6 (Contd.) Carbon Dioxide Rich Gas

Serial No. 926 927 928 929 930 931 932 H2S 0.003 0.003 0.003 0.003 0.003 0.003 0.004 CO2 0.6352 0.6395 0.6514 0.6579 0.6639 0.6706 0.6716 N2 0.0386 0.0399 0.041 0.0417 0.0421 0.0411 0.0388 C1 0.1937 0.1988 0.2008 0.207 0.2084 0.2037 0.1994 C2 0.0303 0.0307 0.0308 0.0309 0.0313 0.0315 0.0318 C3 0.0174 0.0172 0.017 0.0169 0.017 0.0175 0.0184 iC4 0.0033 0.0032 0.0031 0.003 0.003 0.0032 0.0035 nC4 0.0093 0.0088 0.0085 0.0082 0.0082 0.0088 0.0097 iC5 0.0039 0.0036 0.0033 0.0031 0.003 0.0033 0.0039 nC5 0.0047 0.0042 0.0038 0.0036 0.0035 0.0038 0.0046 nC6 0.0051 0.0049 0.0046 0.0042 0.0036 0.003 0.0034 C7+ 0.0551 0.0458 0.0324 0.0202 0.0127 0.0101 0.0113

Mw+ 170 153 139 128 118 110 106 Sg+ 0.811 0.797 0.783 0.773 0.763 0.755 0.751

Pc C7+, psia

347.8288 373.929 397.8203 421.6873 446.3173 469.4246 482.3509

Tc C7+, oR

1211.601 1169.445 1131.005 1100.627 1071.227 1046.943 1034.515

T (oF) 219 219 219 219 219 219 219 P (psia) 4825 4100 3300 2600 1900 1200 700

Z (Expt.) 0.851 0.777 0.72 0.719 0.775 0.851 0.915 ρ(lb/cu.ft.) 34.88 30.9 25.58 19.39 12.87 7.38 4.03 LLS (This Study) 0.7551 0.7276 0.7222 0.7483 0.7882 0.8437 0.8975

92

Table 4.6 (Contd.) Very light gas

Serial No. 933 934 935 936 937 938 939 H2S 0 0 0 0 0 0 0 CO2 0.0033 0.0033 0.0034 0.0035 0.0035 0.0036 0.0038 N2 0.0032 0.0033 0.0033 0.0033 0.0033 0.0033 0.0033 C1 0.942 0.9438 0.9451 0.9461 0.9468 0.9473 0.9467 C2 0.0231 0.023 0.023 0.0231 0.0232 0.0233 0.0236 C3 0.0082 0.0082 0.0082 0.0082 0.0082 0.0082 0.0083 iC4 0.0023 0.0023 0.0023 0.0023 0.0023 0.0023 0.0023 nC4 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0026 iC5 0.0012 0.0012 0.0012 0.0012 0.0012 0.0012 0.0012 nC5 0.0008 0.0008 0.0008 0.0008 0.0008 0.0008 0.0009 nC6 0.0014 0.0013 0.0013 0.0013 0.0013 0.0012 0.0013 C7+ 0.012 0.0103 0.0089 0.0077 0.0069 0.0063 0.006

Mw+ 143 133 126 120 116 114 114 Sg+ 0.787 0.777 0.769 0.763 0.76 0.758 0.758

Pc C7+, psia 390.4827 409.7106 423.93 438.5907 450.5163 456.162 456.162Tc C7+, oR 1142.133 1114.075 1093.042 1075.419 1064.345 1058.3 1058.3

T (oF) 209 209 209 209 209 209 209 P (psia) 4786 4000 3300 2600 1900 1300 700

Z (Expt.) 1.019 0.974 0.945 0.933 0.933 0.947 0.969 ρ(lb/cu.ft.) 12.13 10.42 8.76 6.92 5.03 3.37 1.78

LLS (This Study) 1.0018 0.9569 0.9252 0.9046 0.8995 0.9113 0.9411

93

CHAPTER 5

CONCLUSIONS AND RECOMMENDATIONS

5.1 Conclusions This project establishes the need and a solution for a simple and robust technique

of predicting z-factor values for sour reservoir gases and natural reservoir gases.

1. Z-factor from Equations of state has been established. Eight equations-of-state

routinely used in the reservoir simulators have been examined and the most

general EOS has been established.

2. LLS EOS is the most generalized EOS. Every other EOS can be derived from

LLS EOS by substituting for α and β.

3. Best-fit equations for Standing and Katz Z-Chart have been established. Eight

computational techniques available has been examined and Beggs and Brill

computation technique has been used in the development of the scaling factor.

4. A universal scaling factor has been developed for S-K Z-Chart which is capable

of predicting z-factors of

a. Natural gases

b. Sour reservoir gases

5. Determination of accurate critical parameters of mixtures is an essential step to

obtain accurate z-factor values.

6. Improved technique for mixture critical property has been established. 3100

experimental data from various sources were used in the development of scaling

factor and also used for comparison purposes.

94

5.2 Recommendations

The following points can be based for further studies:

1. design of a generalized chart for predicting the amount of gas produced,

2. improvement in the generalized scaling of z using Standing-Katz chart based on

law of corresponding principles.

95

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90. Elsharkawy, A. M., Hashem, Y. S. Kh. S., and Alikhan, A. A., “Compressibility Factor for Gas Condensates,” Energy & Fuels, 15 (4), 807-816 (2001).

91. Hankinson, R. W., Thomas, L. K., and Philips, K. A., “Predict Natural Gas

Properties,” Hydrocarbon Process., 106-108 (1969).

92. Matthews, T. A., Roland, C. H., and Katz, D. L., “High-Pressure Gas Measurement,” Pet. Refin., 21 (6), 58-70 (1942).

93. Corredor, J. H., Piper, L. D., and MacCain, W. D., Jr., “Compressibility Factors

for Naturally Occurring Petroleum Gases,” Paper SPE 24864, presented at the SPE Annual Technical Meeting and Exhibition, Washington, DC, Oct. 4-7 (1992).

94. Piper, L. D., MacCain W. D., Jr., and Corredor, J. H., “Compressibility Factors

for Naturally Occurring Petroleum Gases” Paper SPE 26668, Houston, TX, Oct. 3-6 (1993).

95. Rowe, A. M., Internally Consistent Correlation for Predicting Phase

Composition of Heptane and Heavier Fractions, Research Report 28, Tulsa, OK (1978).

96. Winn, F. W., “Simplified monograph presentation, characterization of petroleum

fraction,” Pet. Refin., 36 (2), 157 (1957).

97. Sim, W. J., and Daubert, T. E., “Prediction of Vapor-Liquid Equilibria of Undefined Mixtures,” Ind. Eng. Chem. Process. Des. Dev., 19 (3), 380-393 (1980).

98. Lin, H. M., and Chao, K. C., “Correlation of Critical Properties and Acentric

Factor of Hydrocarbon Derivatives,” AIChE J., 30 (6), 153-158 (1984).

103

99. Watansiri, S., Owens, V. H., and Starling, K. E., “Correlation for Estimating

Critical Constants, Accentric Factor, and Dipole Moment for Undefined Coal – Fluid Fractions,” Ind. Eng. Chem. Process. Des. Dev., 24, 294-296 (1985).

100. Riazi, M. R., and Daubert, T. E., “Simplify Property Prediction,” Hydrocarbon

Process., 115-116 (1980).

101. Riazi, M. R., and Daubert, T. E., “Characterization Parameters for Petroleum Fractions,” Ind. Eng. Chem. Res., 26 (24), 755-759 (1987).

102. Brill, J. P., and Beggs, H. D., “Two-Phase flow in pipes,” INTERCOMP

Course, The Huge (1974).

103. Katz, D. L., “High Pressure Gas Measurement,” Pet. Refin., 21 (6) 64-70 (1942).

104. Ahmed, T., Hydrocarbon Phase Behavior, Gulf Publishing Co., Houston

(1989).

105. Buxton, T. S., and Campbell, J. M., “Compressibility Factors for Naturally Occurring Petroleum Gases,” SPEJ, 80-86, March (1967).

106. Dranchuk, P. M., Purvis, R. A., and Robinson, D. B., “Computer Calculation of

Natural Gas Compressibility Factors Using the Standing and Katz Correlations,” Inst. of Petroleum Technical Institute Series, No. IP74-008, 1-13 (1974).

107. Elsharkawy, A. M., Hashem, Y. S. K. S, and Alikhan, A. A., “Compressibility

Factor for Gas Condensates,” Paper SPE 59702 presented at the SPE 2000 Permian Basin Oil and Gas Recovery Conference, Midland, TX, March 21-23 (2000).

108. Kay, W. B., “Density of Hydrocarbon Gases and Vapor at High Temperature

and Pressure,” Ind. Eng. Chem., 1014-1019, Sept. (1936).

109. MacCain, W. D. Jr., The Properties of Petroleum Fluids, (2nd Ed.), PennWell Books: Tulsa (1990).

110. McLeod, W. R., Application of Molecular Refraction to the Principle of

Corresponding States, Ph.D Thesis, University of Oklahoma (1968).

111. Pedersen, K. S., Thomassen, P., and Fredenslund, Aa., Advances in Thermodynamics, 1, 137 (1989).

104

112. Robinson, R. L., Jr., and Jacoby, R. H., “Better Compressibility Factors,”

Hydro. Proc., 44 (4), April, 141-145 (1965).

113. Simon, R., and Briggs, J. E., “Application of Benedict-Webb-Rubin Equation of State to Hydrogen Sulfide Mixtures,” AIChE J., 10 (4), July, 548-550 (1964).

114. Sutton, R. P., “Compressibility Factors for High Molecular Weight Reservoir

Gases,” Paper SPE 14265 presented at the SPE Annual Technical Meeting and Exhibition, Las Vegas, Sept. 22-25 (1985).

115. Takacs, G., “Comparison Made for Computer Z-Factor Calculation,” Oil & Gas

J., 20, December, 64-66 (1976).

116. Whitson, C. H., “Evaluating Constant-Volume Depletion Data,” J. Pet. Tech., 83, March, 610-620 (1972).

105

APPENDIX A

REDUCED FORM OF CUBIC EQUATIONS OF STATE

A.1 Lawal-Lake-Silberberg Reduced Equation of State

22 bbVV)T(a

bVRTP

β−α+−

−=

(A.1)

cw

ccw

ZZ3Z1

Ω−Ω+

=α (A.2)

c2w

wcc2w

3w

2c

Z)Z31(Z2)1(Z

ΩΩ−+Ω+−Ω

=β (A.3)

3cwa )Z)1(1( −Ω+=Ω (A.4)

c

2c

2

a PTRa Ω=

(A.5)

c

cb

PRT

bΩ

= (A.6)

2R

c

aT)T(aθ−

= (A.7) where,

w

32c M

005783.4363589.1720661.0763758.1309833.0 ω−ω−ω+ω+=θ

(A.8) Z-Form of the LLS-EOS is as follows:

0ZZZ 012

23

3 =Φ+Φ+Φ+Φ (A.9) where,

0.13 =Φ (A.10) [ B)1(12 α−+−=Φ ] (A.11)

[ ]21 B)(BA α+β−α−=Φ (A.12)

⎡ ⎤)BB(AB 320 +β−−=Φ (A.13)

106

22TRP)T(aA =

(A.14)

RTbPB =

(A.15) Mixing Rules

∑∑=i j

ij2

1

j2

1

ijim aaaxxa (A.16)

3

i

31

iim bxb ⎥⎦

⎤⎢⎣

⎡= ∑

(A.17)

jij

iij for a ω≤ω

ωω

= (A.18)

jii

jij for a ω>ω

ω

ω=

(A.19)

( ) 5.0

i jji

21

j2

1

ijim xx∑∑ αααα=α (A.20)

( ) 5.0

i jji

21

j2

1

ijim xx∑∑ ββββ=β (A.21)

( ) ( ) ( )jijijiij aaa ββ=αα== (A.22)

A.2 van der Waals Reduced Equation of State

2V)T(a

bVRTP −−

= (A.23)

0=α (A.24) 0=β (A.25)

3w

3cwa )375.0)1(1()Z)1(1( −Ω+=−Ω+=Ω (A.26)

a)T(a = (A.27)

PTRac

2c

2

aΩ= (A.28)

c

ccw

PRTZΩb =

(A.29) Z-Form of the vdW-EOS is as follows:

107

0ZZZ 012

23


0.13 =Φ (A.31) [ ] ]B1[B)1(12 +−=α−+−=Φ (A.32)

[ ] ]A[ B)(BA 21 =α+β−α−=Φ (A.33)

⎡ ⎤ ]AB[ )BB(AB 320 −=+β−−=Φ (A.34)

where

22TRa(T)PA =

(A.35)

RTbPB =

(A.36)

Mixing Rules:

∑∑=i j

ij2

1

j2

1

ijim aaaxxa (A.37)

∑=i

iim bxb (A.38)

jij

iij for a ω≤ω

ωω

= (A.39)

jii

jij for a ω>ω

ω

ω=

(A.40)

( ) 5.0

i jji

21

j2

1


( ) 5.0

i jji

21

j2

1



A.3 Redlich-Kwong Reduced Equation of State

bVV)T(a

bVRTP 2 +

−−

= (A.44)

108

0.1=α (A.45) 0.0=β (A.46)

42751.0a =Ω (A.47)

PTR42747.0ac

2c

2

= (A.48)

c

c

PRT08664.0b =

(A.49)

aT1)T(a

R

= (A.50)

Z-Form of the RK-EOS is as follows:

0ZZZ 012

23


0.13 =Φ (A.52) 0.12 −=Φ (A.53)

]BBA[ 21 −−=Φ (A.54)

]AB[0 −=Φ (A.55) where,

22TRP)T(aA =

(A.56)

RTbPB =

(A.57) Mixing Rules

∑∑=i j

ij2

1

j2

1

ijim aaaxxa (A.58)

∑=i

iim bxb (A.59)

jij

iij for a ω≤ω

ωω

= (A.60)

jii

jij for a ω>ω

ω

ω=

(A.61)

109

( ) 5.0

i jji

21

j2

1


( ) 5.0

i jji

21

j2

1



A.4 Soave-Redlich-Kwong Reduced Equation of State

bVV)T(a

bVRTP 2 +

−−

= (A.63)

0.1=α (A.64) 0.0=β (A.65)

42751.0a =Ω (A.66)

PTR42747.0ac

2c

2

= (A.67)

c

c

PRT08664.0b =

(A.68) a)]T0.1)(176.0574.148.0(0.1[)T(a 25.0

R2 −ω−ω++= (A.69)

Z-Form of the SRK-EOS is as follows

0ZZZ 012

23

3 =Φ+Φ+Φ+Φ (A.70) 0.13 =Φ

RTbPB ,

TRP)T(aA

where,]AB[

]BBA[

0.1

22

0

21

2

==

−=Φ−−=Φ

−=Φ

Mixing Rules:

∑∑=i j

ij2

1

j2

1

ijim aaaxxa (A.71)

∑=i

iim bxb (A.72)

110

jij

iij for a ω≤ω

ωω

= (A.73)

jii

jij for a ω>ω

ω

ω=

(A.74)

( ) 5.0

i jji

21

j2

1


( ) 5.0

i jji

21

j2

1



A.5 Peng-Robinson Reduced Equation of State

22 bbV2V)T(a

bVRTP

−+−

−=

(A.78) 0.2=α (A.79)

0.1=β (A.80) 45724.0a =Ω (A.81) 07780.0b =Ω (A.82)

PTR

45724.0ac

2c

2

= (A.83)

c

c

PRT07780.0b =

(A.84) a )]T0.1)(26992.054226.137464.0(0.1[)T(a 25.0

R2 −ω−ω++= (A.85)

Z-Form of the PR-EOS

0ZZZ 012

23

3 =Φ+Φ+Φ+Φ (A.86)

0.13 =Φ

]B1[ 2 −−=Φ

]B3B2A[ 21 −−=Φ

)]BB(AB[ 320 +−−=Φ

RTbPB ,

TRP)T(aA 22 ==

111

Mixing Rules

∑∑=i j

ij2

1

j2

1

ijim aaaxxa (A.87)

∑=i

iim bxb (A.88)

jij

iij for a ω≤ω

ωω

= (A.89)

jii

jij for a ω>ω

ω

ω=

(A.90)

( ) 5.0

i jji

21

j2

1



A.6 Schmidt-Wenzel Reduced Equation of State

22 b3bV)31(V)T(a

bVRTP

ω−ω++−

−=

(A.93) 0133)16( c

2c

3c =−β+β+β+ω (A.94)

equation. above theofroot positivesmallest c =β

)1(31

cc ωβ+=ζ

(A.95) ccb βζ=Ω (A.96)

3cca ))1(1( β−ξ−=Ω (A.97)

c

2c

2

a PTRa Ω=

(A.98)

PT.Rbc

cbΩ=

(A.99) )k,T(a)T(a Rα= (A.100)

2R0RR ))T1)(k,T(k1()k,T( −+=α (A.101)

where, 2

0 528.0347.1465.0k ω−ω+= (A.102)

112

0.1Tfor ,70

)1k3T5(k)k,T(k R

20R

00R ≤−−

+= (A.103)

0.1Tfor )k,1(k)k,T(k R00R >= (A.104) Z-Form of SW-EOS:

0)]BB(3AB[ ]B)61(B)30.1(A[Z]B))30.1(0.1(0.1[Z

32

223

=+ω−−

ω+−ω+−+ω+−+−

(A.105) 0.1 : 1Φ

]B))30.1(0.1(0.1[ : 2 ω+−+−Φ

B)61(B)30.1(A : 23 ω+−ω+−Φ

:0Φ )]BB(3AB[ 32 +ω−−

)RT(P)T(aA 2=

RTbPB =

Mixing Rules:

∑∑=i j

ij2

1

j2

1

ijim aaaxxa (A.106)

∑=i

iim bxb (A.107)

jij

iij for a ω≤ω

ωω

= (A.108)

jii

jij for a ω>ω

ω

ω=

(A.109) mm 31 ω+=α (A.110)

mm 3ω−=β (A.111) ( )

( )iwi

iiwii

m Mx

Mx∑ ω=ω

(A.112) where Mwi is the component’s molecular weight.

113

A.7 Patel-Teja Reduced Equation of State

cbVcbVTa

bVRTP

−++−

−=

)()(

2 (A.113)

Z-Form of PT-EOS

0]C)BB(AB[Z)CBBBC2A(Z)C0.1(Z 2223 =+−−−−−−+−− (A.114)

)CB(BAB: - Φ

CBBBC2: AΦ

-C)0.1(: -Φ0.1 :

20

23

2

1

++

−−−−

Φ

where, 22/1

RR )]T1(F1[)(T −+=α (A.115) 2295937.030982.1452413.0F ω−ω+= (A.116)

2c 0211947.0076799.0329032.0Z ω+ω−= (A.117)

cc Z31−=Ω (A.118) 0ZZ3)Z32( solve, 3

cb2c

2bc

3bb =−Ω+Ω−+ΩΩ (A.119)

c2bbc

2ca Z31)Z21(3Z 3 −+Ω+Ω−+=Ω (A.120)

broot positive smallest the pick Ω=

)T(a)T(a Rα= (A.121)

c

cc

c

cb

c

2ca

2

PRTc

RTcPC

PRTb

RTbPB

P)RT(a

)RT(P)T(aA

Ω==

Ω==

Ω==

Mixing Rules:

∑∑=i j

ij2

1

j2

1

ijim aaaxxa (A.122)

∑=i

iim bxb (A.123)

∑=i

iim cxc (A.124)

114

jij

iij for a ω≤ω

ωω

= (A.125)

jii

jij for a ω>ω

ω

ω=

(A.126)

( ) 5.0

i jji

21

j2

1


( ) 5.0

i jji

21

j2

1



A.8 Trebble-Bishnoi-Salim Reduced Equation of State

22 dbcV)cb(V)T(a

bVRTP

−−++−

−=

(A.130)

Z-Form of the TB-EOS Equation 0])D1(B)CB(B[AB)ZDCBBBC2(AC)Z1(Z 222223 =+−+−−−−−−−+−− (A.131)

0.1 : Φ1 C)1( : Φ2 −−

) DCBBBC2(A : Φ 223 −−−−−

] )D1(B)CB(B[AB : Φ 220 +−+−−

RTdP; D

RTcP

; CRTbP

; BTR

P)T(aA

where,

22 ====

c

cd

c

cc

c

cb

c

2c

2

Ra PRT

d;PRT

c;PRT

b;PTR

)T()T(a

,whereΩ

=Ω

=Ω

=αΩ=

)3.0Z(3.9854.012.3662.0m C

2

−+ω−ω+= (A.132) -1mol g 128Mfor 0.2475.0p ≤ω+= (A.133)

-12 mol g 128 Mfor 06.462.0613.0p >ω+ω+= (A.134) ])T(1)T()7.0(p)]T1(m1[[)(T 2/1

R2/1

R2/12/1

RR −−+−+=α (A.135) cc Z063.1 ×=ξ (A.136)

115

0.30.1 cc ξ−=Ω (A.137) 0)(0.3)0.30.2( 3

c2db

2c

2bc

3b =ξ+Ω−Ωξ+Ωξ−+Ω (A.138)

equation. above in theroot positivesmallest theb =Ω 2d

2bcbcb

2ca 23 Ω+Ω+Ω+Ω+ΩΩ+ξ=Ω

0.3

Vcd =Ω

Mixing Rules

∑∑=i j

ij2

1

j2

1

ijim aaaxxa (A.139)

∑∑=i j

ij2

1

j2

1

ijim bbbxxb

∑∑=i j

ij2

1

j2

1

ijim cccxxc

jij

iij for a ω≤ω

ωω

=

jii

jij for a ω>ω

ω

ω=

( ) 5.0

i jji

21

j2

1

ijim xx∑∑ αααα=α

( ) 5.0

i jji

21

j2

1

ijim xx∑∑ ββββ=β

( ) ( ) ( )jijijiijijij aacba ββ=αα====

116

APPENDIX B

PREDICTION RESULTS FOR

PSEUDOCRITICAL PARAMETERS

B.10 Pseduocritical Parameter Results

Table B.1: Gas Composition Description.

Mix No. 47-1 26-1 26-2 26-3 47-2 26-4 26-5 CO2 0.0120 0.0109 0.0100 0.0091 0.0044 0.0030 0.0020 N2 0.0000 0.0884 0.1611 0.2441 0.0000 0.1130 0.2400 C1 0.9089 0.8286 0.7625 0.6870 0.9668 0.8580 0.7364 C3 0.0191 0.0174 0.0160 0.0144 0.0070 0.0060 0.0053 iC4 0.0033 0.0030 0.0028 0.0030 0.0014 0.0012 0.0010 nC4 0.0060 0.0055 0.0051 0.0040 0.0020 0.0018 0.0015 iC5 0.0021 0.0019 0.0018 0.0016 0.0007 0.0006 0.0005 nC5 0.0013 0.0012 0.0011 0.0010 0.0005 0.0004 0.0004 nC6 0.0015 0.0014 0.0012 0.0011 0.0005 0.0004 0.0004 C7+ 0.0018 0.0016 0.0015 0.0014 0.0007 0.0006 0.0005

He 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Table B.1 (Contd.)

Mix No. 26-6 26-7 26-8 Mix-1 Mix-2 Mix-3 Mix-4 CO2 0.0013 0.0020 0.0025 0.0069 0.0069 0.0079 0.0079 N2 0.1146 0.1350 0.0705 0.0150 0.0148 0.0149 0.0143 C1 0.7665 0.7515 0.8532 0.9027 0.8906 0.8313 0.7961 C3 0.0335 0.0327 0.0198 0.0134 0.0139 0.0365 0.0399 iC4 0.0035 0.0038 0.0037 0.0037 0.0040 0.0080 0.0101 nC4 0.0090 0.0060 0.0039 0.0034 0.0039 0.0108 0.0149 iC5 0.0017 0.0000 0.0000 0.0019 0.0023 0.0033 0.0065 nC5 0.0015 0.0020 0.0022 0.0012 0.0019 0.0023 0.0052 nC6 0.0000 0.0000 0.0000 0.0020 0.0038 0.0028 0.0087 C7+ 0.0033 0.0000 0.0000 0.0029 0.0110 0.0029 0.0171 He 0.0100 0.0060 0.0031 0.0000 0.0000 0.0000 0.0000

117

Table B.1 (Contd.)

Mix No. Mix-5 Mix-6 Mix-7 CO2 0.0079 0.0079 0.0014 N2 0.0138 0.0135 0.0000 C1 0.7644 0.7507 0.4534 C3 0.0430 0.0443 0.1961 iC4 0.0120 0.0128 0.0936 nC4 0.0186 0.0202 0.0825 iC5 0.0094 0.0106 0.0542 nC5 0.0078 0.0089 0.0343 nC6 0.0140 0.0164 0.0129 C7+ 0.0299 0.0355 0.0011 He 0.0000 0.0000 0.0000

Mixture 47-1 (Gore Data)

Kay Joffe PG LK SBV

TTP

VNA Pedersen LMSC Sutton

LLS Expt.

0

100

200

300

400

500

Kay Joffe PG LK SBV TTP VNA Pedersen LM SC Sutton LLS Expt.Critical Property Methods

Cri

tical

Tem

pera

ture

(o R)

Figure B.1: Critical temperature prediction for Gore Data (Mix 47-1).

118


Kay Joffe PGLK SBV

TTP


LLS Expt.

0

200

400

600

800

1000


Cri

tical

Pre

ssur

e (p

sia)

Figure B.2: Critical pressure prediction for Gore Data (Mix 47-1).


Kay Joffe PG LK SBV

TTP


LLS Expt.

0

200

400

600

800

1000


Cri

tical

Pre

ssur

e (p

sia)


119


Kay Joffe PG LK SBV

TTP

VNA PedersenLM

SCSutton

LLS Expt.

0

100

200

300

400


Cri

tical

Tem

pera

ture

(o R)



Kay Joffe PG LK SBV

TTP

VNA PedersenLM

SC Sutton

LLS Expt.

0

200

400

600

800

1000


Cri

tical

Pre

ssur

e (p

sia)

Figure B.5: Critical pressure prediction for Gore Data (Mix 26-2.

120


Kay Joffe PG LK SBV

TTP

VNA PedersenLM

SCSutton LLS Expt.

0

100

200

300

400


Cri

tical

Tem

pera

ture

(o R)



Kay Joffe PG LK SBV

TTP


LLS Expt.

0

215

430

645

860

1075


Cri

tical

Pre

ssur

e (p

sia)


121

APPENDIX C

SCALING FACTOR DEVELOPMENT

AND RESULTS

The following three forms of the scaling parameter were tested and the

exponential form was selected based on its prediction and matching capability:

([ 2

RSF T1k1Z −+= )]

)

… (3.18).

θ= RSF TZ … (3.19).

( RSF bTaEXPZ = … (3.20).

The step-by-step procedure for obtaining the scaling factor is as follows:

1. c.Expt

SKSF z

zzz ×=

2. Plot zSF vs. TR graph; obtain the best fit-curve and the corresponding equation in

the form of for each pure component. Obtain a and b for each

component.

RbTSF eaz −×=

3. Plot a vs. ωMw graph; obtain the best fit-curve and the corresponding equation of

the form . 2coeffw1 )M(coeffa −ω=

4. Plot b vs. ω graph; obtain the best fit-curve and the corresponding equation of the

form ]CBA[b 2 +ω+ω=

where A, B, and C are constants.

5. The scaling factor expressions for:

a. Pure Components

]CBA[b

)M(coeffa,where

)bTexp(az

2

2coeffw1

RSF

+ω+ω=

ω=

=

b. Mixtures follow the following Mixing rule:

122

[ ] [ ]2n

i

5.0iwiw

2n

i

5.0iim MxM x ⎟

⎠

⎞⎜⎝

⎛ω=ω⎟

⎠

⎞⎜⎝

⎛ω=ω ∑∑

where a, b, A, B, and C belong to the above described procedure only.

C.1 Scaled Z-Factor Results Buxton & Campbell at 160 oF (Mix-4)

0.75

0.85

0.95

1.05

1.15

1.25

1.35

1.45

0 2,000 4,000 6,000 8,000Pressure (Psia)

Z-Fa

ctor

Expt.

SK

Scaled

Figure C.1: Scaled z-factor result for Buxton & Campbell Data at 160 oF (Mix-4).

123

Buxton & Campbell, Mix-4, T = 130 F, Quadratic

0.72

0.82

0.92

1.02

1.12

1.22

1.32

1.42

0 2,000 4,000 6,000 8,000Pressure (psia)

Z-Fa

ctor

Exp

SK

Scaled


Buxton & Campbell, Mix-3, T = 160 F

0.8

0.9

1

1.1

1.2

1.3

1.4

0 2,000 4,000 6,000 8,000Pressure (psia)

Z-Fa

ctor

ExpSKScaled


124

Buxton & Campbell, Mix-3, T = 130 F

0.76

0.86

0.96

1.06

1.16

1.26

1.36

0 2,000 4,000 6,000 8,000Pressure (psia)

Z-Fa

ctor

Exp

SK

Scaled


Buxton & Campbell, Mix-3, T = 100 F, Quadratic

0.76

0.91

1.06

1.21

1.36

0 2,000 4,000 6,000 8,000Pressure (psia)

Z-Fa

ctor

Exp SK

Scaled


125

Buxton & Campbell, Mix-2, T = 130 F,Quadratic

0.8

0.9

1

1.1

1.2

0 2,000 4,000 6,000 8,000Pressure (psia)

Z-Fa

ctor

Exp

SK

Scaled


Buxton and Campbell Data, Mix-1, Quadratic


160 oF1.35

Expt. SK

1.25 Scaled

1.15 Z-Factor

1.05

0.95

0.85 0 6,000 2,000 4,000 8,000

Pressure (Psia)

126

APPENDIX D

PREDICTION OF Z-FACTOR FOR PURE SUBSTANCES

D.1 Prediction Results by Equations of State Method Z-Factor Comparison (Expt. Vs. vdW-EOS)

0.85

0.95

1.05

1.15

1.25

1.35

1.45

1.55

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Pressure (psia)

Z-Fa

ctor

VdW-EOS T=100 FVdW-EOS T=220 FVdW-EOS T=460 F100 F EXPT.220 F EXPT.460 F EXPT.

Figure D.1: Z-Factor comparison for vdW-EOS for Nitrogen.

Z-Factor Comparison Graph (Exp. vs. RK-EOS)

0.84

0.92

0.99

1.07

1.14

1.22

1.29

1.37

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Pressure, P (Psia)

Z-Fa

ctor

RK-EOS 560 R

RK-EOS 680 R

RK-EOS 920 R

Expt. T=560

Expt. T=680

Expt. T=920

Figure D.2: Z-Factor comparison for RK-EOS for Methane.

127

Z-Factor Comparison Graph (Expt. vs. RK-EOS)

0.23

0.38

0.53

0.68

0.83

0.98

1.13

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Pressure, P (Psia)

Z-Fa

ctor

RK-EOS 560 R

RK-EOS 680 R

RK-EOS 920 R

Expt. T=560

Expt. T=680

Expt. T=920

Figure D.3: Z-Factor comparison for RK-EOS for Carbon dioxide.

Z-Factor Comparison Graph (Expt. vs. RK-EOS)

0.98

1.06

1.13

1.21

1.28

1.36

1.43

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Pressure, P (Psia)

Z-Fa

ctor

RK-EOS 560 R

RK-EOS 680 R

RK-EOS 920 R

Expt. T=560

Expt. T=680

Expt. T=920

Figure D.4: Z-Factor comparison for RK-EOS for Nitrogen.

128

SOAVE-REDLICH-KWONG(C1) vs. Experimental Z-Factor Plot

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 2000 4000 6000 8000 10000Pressure(psia)

Z-Fa

ctor

T=100 F

T=220 F

T=460 F

100 F EXP

220 F EXP

460 F EXP

Figure D.5: Z-Factor comparison for SRK-EOS for Methane.

SOAVE-REDLICH-KWONG(CO2) vs. Experimental Z-Factor Comparison Plot

0.2

0.42

0.64

0.86

1.08

1.3

0 2000 4000 6000 8000 10000Pressure(psia)

Z-Fa

ctor

T=100 F

T=220 F

T=460 F

100 F EXP

220 F EXP

460 F EXP

Figure D.6: Z-Factor comparison for SRK-EOS for Carbon dioxide.

129

SOAVE-REDLICH-KWONG(N2) vs. Experimental Z-Factor Comparison Plot

0.95

1.05

1.15

1.25

1.35

1.45

1.55

0 2000 4000 6000 8000 10000

Pressure(psia)

Z-Fa

ctor

T=100 F

T=220 F

T=460 F

100 F EXP

220 F EXP

460 F EXP

Figure D.7: Z-Factor comparison for SRK-EOS for Nitrogen.

Z-Factor Comparison Graph (Expt. vs. PR-EOS)

0.80

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Pressure (Psia)

Z-Fa

ctor

PR-EOS 560 R

PR-EOS 680 R

PR-EOS 920 R

Expt. T=560

Expt. T=680

Expt. T=920

Figure D.8: Z-Factor comparison for PR-EOS for Methane.

130


0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Pressure, P (Psia)

Z-Fa

ctor

T = 560 R

T = 680 R

T = 920 R

Expt. T=560

Expt. T=680

Expt. T=920

Figure D.9: Z-Factor comparison for PR-EOS for Carbon dioxide.


1.0

1.0

1.1

1.1

1.2

1.2

1.3

1.3

1.4

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Pressure (Psia)

Z-Fa

ctor

PR-EOS 560 RPR-EOS 680 RPR-EOS 920 RExpt. T=560Expt. T=680Expt. T=920

Figure D.10: Z-Factor comparison for PR-EOS for Nitrogen.

131

Z-Factor Comparison (Expt. Vs. SW-EOS)

0.8

0.9

1

1.1

1.2

1.3

1.4

1.5

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000P (psia)

Z-Fa

ctor

SW-EOS 100 F

SW-EOS 220 F

SW-EOS 460 F

100 F EXP

220 F EXP

460 F EXP

Figure D.11: Z-Factor comparison for SW-EOS for Methane.


0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Pressure (psia)

Z-Fa

ctor

SW-EOS 100 F

SW-EOS 220 F

SW-EOS 460 F

100 F EXP

220 F EXP

460 F EXP

Figure D.12: Z-Factor comparison for SW-EOS for Carbon dioxide.

132


0.95

1.05

1.15

1.25

1.35

1.45

1.55

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000P (psia)

Z-Fa

ctor

SW-EOS 100 F

SW-EOS 220 F

SW-EOS 460 F

100 F EXP

220 F EXP

460 F EXP

Figure D.13: Z-Factor comparison for SW-EOS for Nitrogen.

Z-Factor Comparison Graph (Exp. vs. PT-EOS)

0.83

0.93

1.03

1.13

1.23

1.33

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Pressure, P (Psia)

Z-Fa

ctor

PT-EOS 560 R

PT-EOS 680 R

PT-EOS 920 R

Expt. T=560

Expt. T=680

Expt. T=920

Figure D.14: Z-Factor comparison for PT-EOS for Methane.

133

Z-Factor Comparison Graph (Exp. vs. PT-EOS)

0.23

0.33

0.43

0.53

0.63

0.73

0.83

0.93

1.03

1.13

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Pressure, P (Psia)

Z-Fa

ctor

PT-EOS 560 R

PT-EOS 680 R

PT-EOS 920 R

Expt. T=560

Expt. T=680

Expt. T=920

Figure D.15: Z-Factor comparison for PT-EOS for Carbon dioxide.

Z-Factor Comparison Graph (Expt. vs. PT-EOS)

0.96

1.06

1.16

1.26

1.36

1.46

0 1000 2000 3000 4000 5000 6000 7000 8000 9000Pressure, P (Psia)

Z-Fa

ctor

PT-EOS 560 R

PT-EOS 680 R

PT-EOS 920 R

Expt. T=560

Expt. T=680

Expt. T=920

Figure D.16: Z-Factor comparison for PT-EOS for Nitrogen.

134

TB-EOS (METHANE)

0.75

0.85

0.95

1.05

1.15

1.25

1.35

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Pressure (psia)

Z-Fa

ctor

100 F

220 F

460 F

100 F EXP

220 F EXP

460 F EXP

Figure D.17: Z-Factor comparison for TB-EOS for Methane.

TB-EOS (CO2)

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000P(psia)

Z-Fa

ctor

T=100 F

T=160 F

T=220 F

100 F EXP

220 F EXP

460 F EXP

Figure D.18: Z-Factor comparison for TB-EOS for Carbon dioxide.

135

TB-EOS (NITROGEN)

0.9

1

1.1

1.2

1.3

1.4

1.5

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000Pressure (psia)

Z-Fa

ctor

100 F

220 F

460 F

100 F EXP

220 F EXP

460 F EXP

Figure D.19: Z-Factor comparison for TB-EOS for Nitrogen.

136

APPENDIX E

EXPERIMENTAL Z-FACTOR FOR

MISCELLANEOUS GASES

Table E.1: UCalgary Z-Factor Data. Thesi

s T (oF) Cmpnt.

CO2 H2S N2 C1 C2 C3 iC4

nC4 iC5

nC5

nC6

C7+

Mcleod-

Mix-1 40 Mole

% 0.50 22.60

0.46 75.61

0.71

0.08

0.02

0.02

0.00

0.00

0.00

0.00

P

(psia) Zexpt

. 600 0.847 1000 0.748 1500 0.639 2000 0.586 2500 0.595 3000 0.632 4000 0.732 5000 0.845

Thesis T (oF)

Compone

nt CO

2 H2S N2 C1 C2 C3 iC4

nC4 iC5

nC5

nC6

C7+

Mcleod-

Mix-1 100 Mole

% 0.50 22.60

0.46 75.61

0.71

0.08

0.02

0.02

0.00

0.00

0.00

0.00

P

(psia) Zexpt

. 600 0.895 1000 0.836 1500 0.779 2000 0.731 2500 0.713 3000 0.722 4000 0.783 5000 0.876

137

Table E.1 (Contd.)

t

t

t

t

Thesis T (oF) Cmpnt. CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+

Mcleod-Mix-1 40 Mole % 0.50 22.60 0.46 75.61 0.71 0.08 0.02 0.02 0.00 0.00 0.00 0.00P (psia) Zexpt.

600 0.8471000 0.7481500 0.6392000 0.5862500 0.5953000 0.6324000 0.7325000 0.845

Thesis T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+


600 0.8951000 0.8361500 0.7792000 0.7312500 0.7133000 0.7224000 0.7835000 0.876



600 0.9331000 0.91500 0.8652000 0.8392500 0.8263000 0.8254000 0.8565000 0.914



626 0.866848 0.82

1022 0.7871521 0.7062021 0.6622521 0.6633021 0.694021 0.7815021 0.887


Mcleod-Mix-2 65 Mole % 0.30 14.38 0.46 84.14 0.59 0.08 0.03 0.02 0.00 0.00 0.00 0.00

138

Table E.1 (Contd.)

t

t

t

P (psia) Zexpt.624 0.889823 0.859

1022 0.8281521 0.7632021 0.7222522 0.7163022 0.7324022 0.8075022 0.903



607 0.911625 0.909824 0.886

1023 0.8641522 0.8152021 0.7832521 0.7733021 0.7813521 0.8054021 0.8374521 0.8765021 0.919



655 0.926824 0.911

1023 0.8941522 0.8582021 0.8342521 0.8263021 0.834021 0.8735021 0.941



565 0.946823 0.93

1023 0.9181522 0.8912021 0.8742521 0.8673021 0.8694021 0.9035021 0.958

139

Table E.1 (Contd.)

t

t

t

t

t



600 0.8921000 0.8191500 0.7512000 0.7112500 0.7073000 0.734000 0.8145000 0.918



600 0.9131000 0.8831500 0.8432000 0.8162500 0.8083000 0.8154000 0.8675000 0.945



600 0.9131000 0.8831500 0.8432000 0.8162500 0.8083000 0.8154000 0.8675000 0.945



600 0.9511000 0.9291500 0.9092000 0.8962500 0.8923000 0.8974000 0.9325000 0.986



500 0.902

140

Table E.1 (Contd.)

t

t

t

t

t

1000 0.8231500 0.7562000 0.7252500 0.7283000 0.7553500 0.787



500 0.9251000 0.8621500 0.8112000 0.7752500 0.7723000 0.7973500 0.821



500 0.961000 0.9271500 0.92000 0.8842500 0.8823000 0.8913500 0.91



500 0.9181000 0.8421500 0.7772000 0.7422500 0.7393000 0.7653500 0.802



500 0.9471000 0.9031500 0.8682000 0.8482500 0.8433000 0.8533500 0.877


141

Table E.1 (Contd.)

t

t

t

t

t

t


500 0.961000 0.9291500 0.9062000 0.8922500 0.893000 0.8893500 0.916



500 0.9291000 0.875



500 0.9391000 0.895



500 0.9581000 0.925

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+S & B T4 160 Mole % 0.00 10.00 0.00 90.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00

P (psia) Zexpt.600 0.955

1000 0.9291500 0.8992000 0.8793000 0.8744000 0.9115000 0.969



1000 0.9531500 0.9372000 0.9263000 0.9244000 0.9555000 1.002


142

Table E.1 (Contd.)

t

t

t

t

t-


1000 0.9721500 0.9632000 0.9573000 0.964000 0.9895000 1.027



1000 0.9091500 0.872000 0.8423000 0.8254000 0.8635000 0.925



1000 0.9411500 0.9172000 0.8993000 0.8874000 0.9125000 0.959



1000 0.9631500 0.9482000 0.9373000 0.9314000 0.9535000 0.99

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+Mix-2-API-PRJ-37 157 Mole % 2.09 6.80 10.19 68.57 5.90 2.82 0.47 1.16 0.85 0.00 0.35 0.80

P (psia) Zexpt.2115 0.829 MC7+ 1252347 0.823 SgC7+ 0.7500

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+Mix-17-API-PRJ37 189 Mole % 6.16 10.78 0.4 74.14 3.27 1.21 0.22 0.61 0.57 0 0.46 2.18


143

Table E.1 (Contd.)

t-

t

t

t

t

t

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+Mix-25-API-PRJ37 191 Mole % 4.16 9.13 0 78.77 2.97 1.27 0.27 0.6 0.43 0 0.43 1.97

P (psia) Zexpt.4945 0.955 MC7+ 125

SgC7+ 0.7500Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+

Mix-2-API-PRJ-37 216 Mole % 8.66 18.26 0.37 52.13 11.65 1.42 0.39 0.83 0.95 0.00 1.03 4.31



P (psia) Zexpt.3000 0.903 MC7+ 1253270 0.91 SgC7+ 0.75003400 0.9143600 0.9183800 0.9254000 0.9344200 0.9464400 0.9544600 0.974800 0.9815130 1.006




P (psia) Zexpt.4720 0.948 MC7+ 1254743 0.95 SgC7+ 0.75004774 0.9554815 0.9594915 0.9695015 0.9815115 0.9915315 1.0145515 1.039

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+Mix-707-API-PRJ-37 218 Mole % 3.17 18.5 2.18 56.22 4.83 2.5 0.56 1.49 1.48 0 1.15 7.82

P (psia) Zexpt.4475 0.884 MC7+ 1254515 0.887 SgC7+ 0.75004565 0.893

144

Table E.1 (Contd.)

t

t

t

t

4615 0.8994715 0.914915 0.9335215 0.975515 1.0066015 1.067








P (psia) Zexpt.4430 0.987 MC7+ 1254515 0.994 SgC7+ 0.75004715 1.011

145

Table E.1 (Contd.)

t

t

t

t

t

4915 1.0295215 1.0565515 1.083

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+Shell-Et.Al-Marmattan-10-33 73 Mole % 3.19 51.37 2.58 42.41 0.24 0.07 0.02 0.03 0.02 0.01 0.02 0.04

P (psia) Zexpt.2114 0.402 MC7+ 1202514 0.438 SgC7+ 0.75003014 0.4953514 0.5534014 0.6124514 0.675014 0.728







Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+

146

Table E.1 (Contd.)

t

t

t

t

Shell-Et.Al-Marmattan-10-33 147 Mole % 3.19 51.37 2.58 42.41 0.24 0.07 0.02 0.03 0.02 0.01 0.02 0.04







147-Sutte-Plant 198 Mole % 4.2 3.32 1.06 77.91 7.74 2.99 0.58 1.45 0.25 0.23 0.16 0.11P (psia) Zexpt.

200 0.991 MC7+ 125500 0.968 SgC7+ 0.7340

1000 0.9241500 0.8952000 0.8782500 0.8693000 0.8763500 0.8934000 0.9184500 0.9515000 0.988

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+147-Waterton 156 Mole % 8.03 29.66 1.04 52.75 3.48 0.82 0.15 0.6 0.22 0.23 0.45 2.57

P (psia) Zexpt.

147

Table E.1 (Contd.)

t

t

t

t

3914 0.749 MC7+ 1304014 0.757 SgC7+ 0.83404214 0.7744414 0.7924714 0.824914 0.8395064 0.8535114 0.8585214 0.8685414 0.8885714 0.9176014 0.948

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+Gold Creek 210 Mole % 3.18 7.04 4.81 70.69 3.83 2.09 0.57 1.09 0.60 0.57 0.93 4.60


Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+HOME-ET.AL-6-8(L) 77 Mole % 3.08 49.35 2.66 44.47 0.23 0.06 0.02 0.03 0.02 0.01 0.03 0.04





P (psia) Zexpt.

148

Table E.1 (Contd.)

t

t

-

t

-

t

-

t

1014 0.802 MC7+ 1251514 0.692 SgC7+ 0.75002014 0.6122514 0.5843014 0.5963514 0.6264014 0.6654514 0.7075014 0.752



Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+DICK-LAKE-A-15(SMOOTHEDDATA) 93 Mole % 1.23 1.62 2.52 77.48 10.32 3.94 0.54 1.30 0.27 0.24 0.54 0.00

P (psia) Zexpt.615 0.882715 0.872815 0.861915 0.849


P (psia) Zexpt.615 0.888715 0.878815 0.868915 0.858


P (psia) Zexpt.615 0.896715 0.888815 0.879915 0.87

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+RIMBEY-GAS-PLANT 84 Mole % 1.21 1.50 2.08 78.14 10.29 4.05 0.62 1.23 0.29 0.29 0.30 0.00

149

Table E.1 (Contd.)

-

t

-

t

-

t

-

t

-

t

DICK-LAKE-A-15(SMOOTHEDDATA) P (psia) Zexpt.

500 0.926750 0.885

1000 0.8411250 0.8051400 0.787

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+RIMBEY-GAS-PLANT 95 Mole % 1.86 3.29 2.28 80.34 6.56 3.02 0.52 1.07 0.37 0.34 0.35 0.00DICK-LAKE-A-23(SMOOTHEDDATA) P (psia) Zexpt.

500 0.945750 0.911

1000 0.8731250 0.8421500 0.816

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+HOMEGLEN-RIMBEY-A-25(SMOOTHEDDATA) 83 Mole % 1.61 3.26 2.75 80.52 6.61 2.92 0.42 0.99 0.21 0.21 0.50 0.00

P (psia) Zexpt.615 0.881715 0.872815 0.862915 0.853

1015 0.844


P (psia) Zexpt.615 0.892715 0.883815 0.874915 0.865

1015 0.856


P (psia) Zexpt.615 0.905715 0.896815 0.887915 0.878

1015 0.869


150

Table E.1 (Contd.)

t

t

t

SHELL-LOW-WATERTON-NO.5-17 156 Mole % 3.48 16.03 0.97 65.49 3.93 1.53 0.32 0.92 0.52 0.50 1.12 5.19


Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+SHELL-NO.3-BURNT-TIMBER 196 Mole % 5.78 6.42 0.33 81.87 3.64 0.74 0.22 0.19 0.10 0.07 0.13 0.51

P (psia) Zexpt.936 0.935 MC7+ 118

1058 0.929 SgC7+ 0.75801203 0.9231363 0.9151557 0.9061733 0.8991912 0.8942275 0.8862772 0.8853199 0.8933506 0.9013781 0.9123827 0.9143851 0.9154014 0.9234514 0.955014 0.9815514 1.0146014 1.051

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+SHELL-NO.4-W.J.P.2-23 120 Mole % 6.17 5.40 0.49 81.07 3.74 0.94 0.32 0.34 0.18 0.12 0.23 1.00

P (psia) Zexpt.4279 0.882 MC7+ 1274295 0.883 SgC7+ 0.80504314 0.8854330 0.8864346 0.8874414 0.8934514 0.9025014 0.9485514 0.9956014 1.0446514 1.094


151

Table E.1 (Contd.)

t

t

t

WIMBORNE-NO.6-11-(UPPER) 162 Mole % 2.06 12.96 9.63 66.34 3.11 1.80 0.34 0.94 0.33 0.45 0.56 1.48

P (psia) Zexpt.2899 0.82 MC7+ 1152916 0.82 SgC7+ 0.76102945 0.8212984 0.8233014 0.8233064 0.8253114 0.8273214 0.8313314 0.8363514 0.8464014 0.8784514 0.9165014 0.9585514 1.0036014 1.048

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+ROYALITE-EDSON-6-4-52-17 160 Mole % 4.34 1.04 0.17 90.31 2.70 0.66 0.16 0.18 0.09 0.07 0.11 0.17

P (psia) Zexpt.514 0.959 MC7+ 125

1014 0.925 SgC7+ 0.75001514 0.8982014 0.882514 0.8733014 0.8783514 0.8964014 0.918

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+ROYALITE-EDSON-6-4-52-17 220 Mole % 4.34 1.04 0.17 90.31 2.70 0.66 0.16 0.18 0.09 0.07 0.11 0.17

P (psia) Zexpt.514 0.975 MC7+ 125

1014 0.955 SgC7+ 0.75001514 0.9392014 0.9292514 0.9263014 0.9313514 0.9434014 0.961

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+HB-UNION-KAYBOS-S-7-14 250 Mole % 3.13 16.82 1.12 60.09 7.72 3.12 1.00 1.44 0.41 0.63 1.10 3.42

P (psia) Zexpt.3542 0.94 MC7+ 1244014 0.971 SgC7+ 0.79404514 1.0095014 1.051

152

Table E.1 (Contd.)

t

t

t

t

t

t

t

5514 1.0966014 1.144

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+HB-UNION-KAYBOS-S-11-1 230 Mole % 3.12 15.62 1.01 60.12 7.85 3.28 0.82 1.56 0.67 0.75 1.11 4.09


Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+PANTHER-RIVER-5-23 50 Mole % 12.86 35.99 1.54 49.53 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00







R&J-Mixture-17 200 Mole % 1.18 20.27 0.23 76.30 1.29 0.73 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

246 0.979363 0.97532 0.956776 0.938

1125 0.9151623 0.846



252 0.967369 0.952536 0.931772 0.902

1101 0.8641556 0.821


153

Table E.1 (Contd.)

t

t

t

t

t


194 0.968 MC7+

285 0.953 SgC7+

414 0.932597 0.903849 0.863

1188 0.8121640 0.854



400 0.912 MC7+

600 0.867 SgC7+

800 0.821000 0.874



400 0.922600 0.883800 0.842

1000 0.902



250 0.975360 0.964538 0.949783 0.927

1130 0.91622 0.869



233 0.962344 0.974505 0.963739 0.947

1076 0.9271559 0.903



231 0.972339 0.959

154

Table E.1 (Contd.)

t

t

t

t

t

495 0.941717 0.916

1029 0.8831464 0.845



258 0.962377 0.945547 0.921785 0.889

1111 0.8471558 0.798



197 0.965288 0.95418 0.927600 0.894848 0.851

1178 0.7941609 0.829



255 0.972375 0.959547 0.941791 0.916

1135 0.8841617 0.846



248 0.965363 0.949526 0.926754 0.895

1074 0.8541511 0.808



352 0.964515 0.948749 0.926

155

Table E.1 (Contd.)

t

t

t

t

t

1000 0.8981550 0.867



342 0.963500 0.946726 0.924

1049 0.8941496 0.86



256 0.97376 0.956548 0.937792 0.911

1134 0.8771614 0.84



254 0.967371 0.952540 0.931779 0.902

1111 0.8651570 0.822



154 0.97225 0.956328 0.935473 0.907672 0.866937 0.811

1275 0.8431703 0.867


B&C-Mixture-1 100 Mole % 5.06 0.00 0.53 89.77 4.64 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

1026 0.8611526 0.8442026 0.8162526 0.8113026 0.822

156

Table E.1 (Contd.)

t

t

t

t

3526 0.8464026 0.8784526 0.9175026 0.9596026 1.057026 1.146



1026 0.9041526 0.8752026 0.8552526 0.8513026 0.8573526 0.8774026 0.9044526 0.9375026 0.9746026 1.0567026 1.143



1026 0.9251526 0.8992026 0.8852526 0.8743026 0.9053526 0.9014026 0.9234526 0.9545026 1.0096026 1.0647026 1.143



1026 0.8811526 0.8382026 0.8122526 0.8643026 0.8753526 0.9084026 0.9214526 0.965026 1.0026026 1.0957026 1.19


157

Table E.1 (Contd.)

t

t

t


1026 0.9041526 0.8722026 0.852526 0.8443026 0.8513526 0.8794026 0.8974526 0.935026 0.9696026 1.0527026 1.139



1026 0.9231526 0.8992026 0.8822526 0.8773026 0.8833526 0.8994026 0.9214526 0.9495026 0.9836026 1.0587026 1.139



1026 0.8651526 0.8142026 0.8782526 0.8623026 0.8783526 0.9044026 0.9384526 0.9795026 0.9236026 1.1147026 1.214



1026 0.8891526 0.8522026 0.9252526 0.9143026 0.92

158

Table E.1 (Contd.)

t

t

t

t

3526 0.9394026 0.9674526 1.0025026 1.0416026 1.1267026 1.214



1026 0.911526 0.8822026 0.862526 0.953026 0.9553526 0.974026 0.9934526 1.0245026 1.0566026 1.1347026 1.215



1026 0.9131526 0.852026 0.8142526 0.8143026 0.8373526 0.8754026 0.8214526 0.9715026 1.0236026 1.1317026 1.241



1026 0.9511526 0.8992026 0.872526 0.8633026 0.8783526 0.9064026 0.9454526 0.9895026 1.0366026 1.0357026 1.136


159

Table E.1 (Contd.)

t

t

t


1026 0.9741526 0.9342026 0.912526 0.9043026 0.9133526 0.9364026 0.9674526 1.0035026 1.0456026 1.1367026 1.228



1026 0.8931526 0.8152026 0.7762526 0.7783026 0.8083526 0.8514026 0.9014526 0.9555026 1.016026 1.1247026 1.138



1026 0.931526 0.8722026 0.8362526 0.833026 0.8483526 0.8824026 0.9244526 0.9715026 1.0216026 1.1257026 1.231



1026 0.9611526 0.9132026 0.8832526 0.8753026 0.886

160

Table E.1 (Contd.)

t

t

t

t

3526 0.9114026 0.9474526 0.9895026 1.0336026 1.1297026 1.227

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+FINA-ET.AL-Creek-10-7 236 Mole % 3.4 16 1.15 59.09 7.59 3.09 0.78 1.69 0.67 0.78 1.2 4.56

P (psia) Zexpt.3514 0.9144014 0.9554514 0.9995014 1.0425514 1.0926014 1.1426514 1.193


FINA-WINDFALL-T-3 219 Mole % 7.74 11.83 1.62 63.00 4.20 2.69 0.69 1.80 0.70 0.79 0.92 4.02

P (psia) Zexpt.3814 0.843 MC7+ 139.03854 0.845 SgC7+ 0.78803914 0.8494014 0.8574514 0.9015014 0.946

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+FINA-WINDFALL-PILOT-PLANT-INJECTION-GAS-T-3 150 Mole % 5.13 15.67 2.68 66.84 4.55 3.01 0.47 0.83 0.22 0.23 0.15 0.22

P (psia) Zexpt.3014 0.764 MC7+ 125.03514 0.787 SgC7+ 0.75004014 0.824514 0.8595014 0.901



161

Table E.1 (Contd.)

t

t

t

t

t





Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+FINA-WINDFALL-PROCESSING-PLANT-INJECTION-GAS-T-3 50 Mole % 4.51 27.30 0.61 64.59 0.84 0.93 0.27 0.20 0.20 0.11 0.12 0.32





162

Table E.1 (Contd.)

t

t

t

FINA-WINDFALL-PROCESSING-PLANT-INJECTION-GAS-T-3 125 Mole % 4.51 27.30 0.61 64.59 0.84 0.93 0.27 0.20 0.20 0.11 0.12 0.32








163

Table E.1 (Contd.)

t

t

t

t


P (psia) Zexpt.1014 0.925 MC7+ 103.02014 0.865 SgC7+ 0.70002514 0.848 0.7483014 0.841 0.7414014 0.864 0.7645014 0.914 0.814



Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+CHEVRON-CLARK-LAKE-11-33 228 Mole % 3.44 17.6 1.02 57.41 7.55 3.24 0.87 1.63 0.63 0.79 1.31 4.51

P (psia) Zexpt.3366 0.79 MC7+ 150.03414 0.793 SgC7+ 0.80003469 0.7973514 0.83814 0.8224214 0.8574514 0.8834672 0.8994714 0.9024814 0.9125014 0.9315514 0.9826014 1.0346514 1.0867014 1.139

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+NEVIS-NO.110-30 143 Mole % 2.06 6.21 10.15 70.52 5.38 2.8 0.37 0.95 0.29 0.35 0.3 0.62

P (psia) Zexpt.2354 0.832 MC7+ 125.0

164

Table E.1 (Contd.)

t

t

t

t

t

t

t

t

2404 0.832 SgC7+ 0.75002414 0.8322514 0.8322614 0.8343014 0.8443514 0.8694014 0.9034514 0.9445014 0.987

Reservoir T (oF) Componen CO2 H2S N2 C1 C2 C3 iC4 nC4 iC5 nC5 nC6 C7+Panther River5-

23 50 Mole % 10.67 31.08 3.37 54.78 0.10 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

864 0.759


23 50 Mole % 9.67 27.72 4.34 58.21 0.06 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

1114 0.702


23 50 Mole % 9.29 26.77 4.63 59.24 0.07 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

1189 0.674


23 100 Mole % 11.27 63.57 1.09 23.90 0.17 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

664 0.779


23 100 Mole % 10.53 50.44 1.98 36.86 0.19 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

984 0.7


23 100 Mole % 11.10 49.80 1.80 37.18 0.12 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

1014 0.697


23 100 Mole % 10.22 45.47 2.56 41.64 0.11 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

1344 0.613


23 100 Mole % 10.13 44.74 2.75 42.26 0.12 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

1454 0.589

165

Table E.1 (Contd.)

t

t

t

t

t

t

t

-

t


23 150 Mole % 7.96 73.85 0.75 17.27 0.17 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

714 0.758


23 150 Mole % 8.74 69.95 0.89 20.27 0.15 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

1064 0.635


23 150 Mole % 9.14 67.16 1.04 22.53 0.13 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

1374 0.543


23 150 Mole % 9.12 66.84 1.06 22.85 0.13 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

1414 0.518


23 150 Mole % 9.14 65.87 1.08 23.73 0.18 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

1594 0.452


23 175 Mole % 8.65 70.03 0.92 20.24 0.16 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

664 0.8291014 0.7261364 0.606


Jeff Lake Et.Al Cross 11-25-25

29 W4M (Smoothed

Data) 100 Mole % 10.16 33.16 2.16 53.41 1.11 0.00 0.00 0.00 0.00 0.00 0.00 0.00P (psia) Zexpt.

200 0.958600 0.871

1000 0.7861500 0.6812000 0.6112500 0.5983000 0.6193500 0.6564000 0.7


166

Table E.1 (Contd.)

-

t

t

t

Jeff Lake Et.Al Cross 11-25-25

29 W4M (Smoothed


200 0.972600 0.92

1000 0.8731500 0.8162000 0.772500 0.7443000 0.743500 0.7524000 0.776


Jeff Lake Et.Al Cross 6-32-25-

28 W4M (Smoothed


200 0.956600 0.876

1000 0.8041500 0.7192000 0.662500 0.6453000 0.663500 0.6954000 0.735



28 W4M (Smoothed


200 0.972600 0.924

1000 0.8831500 0.8372000 0.8012500 0.783000 0.783500 0.7954000 0.816


167

Table E.1 (Contd.)

t

t

t


28 W4M (Smoothed


200 0.963600 0.902

1000 0.8511500 0.7952000 0.7552500 0.743000 0.7493500 0.7754000 0.809



28 W4M (Smoothed


200 0.976600 0.942

1000 0.9151500 0.8842000 0.862500 0.8463000 0.8493500 0.8634000 0.884


Imperial FINA Wildcat Hills 6-35-27-6 W5M (Smoothed


513 0.9131013 0.835 7+ Fraction Mole Wt. 1391513 0.772 Sp. Gr. 0.78602013 0.7362513 0.733013 0.7483513 0.7814013 0.825


168

Table E.1 (Contd.)

t

t

t













Imperial Mobil Brazeau River

6-11 (Smoothed


169

Table E.1 (Contd.)

813 0.9481063 0.934 7+ Fraction Mole Wt. 1391516 0.913 Sp. Gr. 0.78602085 0.9042513 0.8982828 0.93013 0.9013429 0.9063743 0.9194013 0.935

170

APPENDIX F

PREDICTION OF Z-FACTOR FROM LLS EOS

Methane Z-Factor (LLS EOS)

0.758

0.813

0.868

0.923

0.978

1.033

1.088

1.5 1.7 1.9 2.1 2.3 2.5 2.7Reduced Temperature

Z-Fa

ctor

400 Expt.

LLS 400 psia

1500 Expt.

LLS 1500 psia

2000 Expt

LLS 2000 psia

3000 Expt

LLS 3000 psia

4000 Expt

LLS 4000 Psia

Figure F.1: Z-factor for pure substances (Methane).

n-Decane Z- Factor (LLS EOS)

0.0

0.5

1.0

1.5

2.0

2.5

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85Reduced Temperature

Z-Fa

ctor

400 Expt.

LLS 400 psia

1500 Expt.

LLS 1500 psia

2000 Expt

LLS 2000 psia

3000 Expt

LLS 3000 psia

4000 Expt

LLS 4000 Psia

Figure F.2: Z-factor for pure substances (n-Decane).

171

Carbon Dioxide Z-Factor (LLS EOS)

0.2

0.4

0.6

0.8

1

0.9 1.1 1.3 1.5 1.7Reduced Temperature

Z-Fa

ctor

400 Expt.

LLS 400 psia

1500 Expt.

LLS 1500 psia

2000 Expt

LLS 2000 psia

3000 Expt

LLS 3000 psia

4000 Expt

LLS 4000 Psia

Figure F.3: Z-factor for pure substances (Carbon Dioxide).

Hydrogen Sulfide Z-Factor (LLS EOS)

0

0.1875

0.375

0.5625

0.75

0.9375

0.74 0.83 0.92 1.01 1.10 1.19Reduced Temperature

Z-Fa

ctor

400 Expt.

LLS 400 psia

1500 Expt.

LLS 1500 psia

2000 Expt

LLS 2000 psia

3000 Expt

LLS 3000 psia

4000 Expt

LLS 4000 Psia

Figure F.4: Z-factor for pure substances (Hydrogen Sulfide).

172

Nitrogen Z-Factor (LLS EOS)

0.97

1.005

1.04

1.075

1.11

1.145

2.40 3.00 3.60 4.20 4.80Reduced Temperature

Z-Fa

ctor

400 Expt.

LLS 400 psia

1500 Expt.

LLS 1500 psia

2000 Expt

LLS 2000 psia

3000 Expt

LLS 3000 psia

4000 Expt

LLS 4000 Psia

Figure F.5: Z-factor for pure substances (Nitrogen).

173

APPENDIX G

FORTRAN PROGRAMS

! Z-FACTOR PROGRAM BY LLS EOS METHOD FOR MIXTURES DIMENSION root(3),coeff(4),XC(20),P(30), ac(20), bc(20), zc(20), omgw(20) Dimension PPR(40),TTR(20),PP(30),Zexpt(30,30),Corr(20),BWR(20),XCMP(20),BIJA(15,15),& BIJB(15,15),BIJC(15,15),BIJD(15,15) Dimension Alp(20), Bet(20), AF(20), wm(20), tc(20), pc(20),APDB(5),ZZV(5,30) Data PPR/0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9,2.0& ,2.5,3.0,3.5,4.0,4.5,5.0,5.5,6.0,6.5,7.0,7.5,8.0,8.5,9.0,9.5,10,15,20,25,30/ Data TTR/0.1,0.5,1,1.5,2,3,4,5/ OPEN (UNIT=5,FILE='Input_SageLaceyC1-C2_ZFactData.TXT',STATUS='old') OPEN (UNIT=6,FILE='OUTPUT_SageLaceyC1-C2_ZFactData.TXT',STATUS='unknown') WRITE (6,*) 'LLS-MIXTURE-Sage and LaceyC1-C2_ZFactData' Read (5,*)NData do 20 I = 1,2 Read (5,*)wm(I),zc(I),AF(I),omgw(I),pc(I),tc(I) 20 Continue R = 10.73 ! BIJA = Average Acentric Factor ! BIJB = Average Molecular Weight ! BIJC = Average of Acentric Factor X Molecular Weight ! BIJD = Average of Tc / Sqrt(Pc) Do 100 K=1,NData Read (5,*)Ncomp,TT,NDataP Read (5,*) (XCMP(J),J=1,9) Do 220 I=1,2 ! Write (6,*)wm(I),zc(I),AF(I),omgw(I),pc(I),tc(I) Call Para(AF(I),zc(I),wm(I),tc(I),pc(I),R,TT,Alp(I),Bet(I),ac(I),bc(I)) ! write(6,*)I,TT,AF(I),zc(I),wm(I),tc(I),pc(I),Alp(I),Bet(I),ac(I),bc(I) ! write(6,*)I,TT,PP,Alp(I),Bet(I),ac(I),bc(I) 220 continue ! Computation of Binary Interaction Parameter Call BinIJ(Ncomp,AF,wm,tc,pc, BIJA,BIJB,BIJC,BIJD) ! If( K .GT. 1) GO TO 122 Do 121 I=1,Ncomp Do 121 J=1,Ncomp write(6,*) BIJA(I,J),BIJB(I,J),BIJC(I,J),BIJD(I,J) 121 Continue 122 Continue Do 120 J=1,NDataP Read (5,*)PP(J),(Zexpt(J,I),I=1,9) ! Write (6,*) TT,PP(J),(Zexpt(J,I),I=1,9) 120 Continue Do 160 J=1,9 AAPD=0.0 IT=0 XC(1)=XCMP(J) XC(2)=1.0-XC(1) Write (6,125) Write (6,*)' C1 Mole Fraction = ',XC(1),' C2 Mole Fraction = ',XC(2) 125 Format(/) Do 145 IB=1,4 IT=0 AAPD=0.0 Do 140 I=1,NDataP

174

If(IB .EQ. 1)Call Mixrule(Ncomp, XC, ac, bc,tc,pc, Alp, Bet, AF, wm,BIJA, am, bm, Alpm, Betm,wmix,AFmix) If(IB .EQ. 2)Call Mixrule(Ncomp, XC, ac, bc,tc,pc, Alp, Bet, AF, wm,BIJB, am, bm, Alpm, Betm,wmix,AFmix) If(IB .EQ. 3)Call Mixrule(Ncomp, XC, ac, bc,tc,pc, Alp, Bet, AF, wm,BIJC, am, bm, Alpm, Betm,wmix,AFmix) If(IB .EQ. 4)Call Mixrule(Ncomp, XC, ac, bc,tc,pc, Alp, Bet, AF, wm,BIJD, am, bm, Alpm, Betm,wmix,AFmix) call Zfactr (Alpm, Betm, am, bm, PP(I), TT, R, ZV, ZL) call TcPcMix(am,bm,Alpm,Betm,R,Tcm,Pcm) DenV=PP(I)*wmix/(ZV*R*TT) APD=((ZV-Zexpt(I,J))/Zexpt(I,J))*100.0 AAPD=AAPD+ABS(APD) ! write (6,*) ! write (6,15) TT,PP(I),ZV,ZL,Zexpt(I,J),APD ZZV(IB,I)=ZV 15 Format(2F8.1,5F8.4) 140 Continue DatP=NDataP AAPD=AAPD/DatP APDB(IB)=AAPD ! write(6,*)'End of Data = ',K ! write(6,*)'Average Absolute Percent Deviation = ',AAPD 145 Continue Do 146 I=1,NDataP write (6,15) TT,PP(I),(ZZV(N1,I),N1=1,4),Zexpt(I,J) 146 Continue write(6,51)(APDB(N1),N1=1,4) 51 Format(16X,5F8.2) write(6,25) 160 Continue 25 Format(/) 100 Continue close(5) close(6) STOP END Subroutine Zfactr (Alp, Bet, AT, BC, P, T, R, ZV, ZL) Dimension Coef(4),RT(3) AA = AT*P/(R**2*T**2) BB = BC*P/(R*T) Coef(1) = 1. Coef(2) = -(1.+(1-Alp)*BB) Coef(3) = AA-(Alp*BB)-(Bet+Alp)*BB**2 Coef(4) = -(AA*BB-Bet*(BB**2+BB**3)) Call Cubic (MTYPE, Coef, RT) ! write(6,*) 'Root=',RT Rmin = 1.E+10 Rmax = 1.E-10 Do 70 I=1,3 if (RT(I).LE. 0.0) Go to 70 if (RT(I).LT. BB) Go to 70 if (RT(I).GT. Rmax) Rmax = RT(I) if (RT(I).LT. Rmin) Rmin = RT(I) 70 Continue ZV=Rmax ZL=Rmin Return End Subroutine ZRedMix (Alp, Bet,AF,wm, Zc,Pr,tr, ZV, ZL) Dimension Coef(4),RT(3) w=AF theta = 0.309833 + 1.763758*w + 0.720661*w*w - 1.363589*w**3 - 4.005783*w/(sqrt(wm)) tmp = tr ** (-theta/2.0) omgW=0.361/(1.0+0.0274*w) omga=(1.0+(omgW-1.0)*Zc)**3 omgb = omgW*Zc

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Coef(1) = 1.0 Coef(2) = -(1.0/Zc+(1.0-Alp)*omgb*Pr/(Zc*tr)) Coef(3) = omga*Pr/(Zc**2*tr**(2+theta))-Alp*omgb*Pr/(Zc**2*tr)-(Alp+Bet)*(omgb*Pr/(Zc*tr))**2 Coef(4) = -(omga*omgb*Pr**2/(Zc**3*tr**(3+theta))-Bet*(1.0/Zc*(omgb*Pr/(Zc*tr))**2+(omgb*Pr/(Zc*tr))**3)) Call Cubic (MTYPE, Coef, RT) ! write(6,*) 'Root=',RT Rmin = 1.E+10 Rmax = 1.E-10 Do 70 I=1,3 if (RT(I).LE. 0.0) Go to 70 if (RT(I).LT. BB) Go to 70 if (RT(I).GT. Rmax) Rmax = RT(I) if (RT(I).LT. Rmin) Rmin = RT(I) 70 Continue ZV=Rmax*Zc ZL=Rmin*Zc Return End Subroutine Zcfactr (Alp, Bet, Zc, RT) Dimension Coef(4),RT(3) Coef(1) = Alp**3 + 6*Alp**2+ 12*Alp + 8. Coef(2) = -(12*Alp**2+12 *Alp + 9*Bet- 9*Alp*Bet+ 3.) Coef(3) = 6*Alp**2 + 3*Alp + 6*Bet - 6*Alp*Bet Coef(4) = -(Alp**2+Bet-Alp*Bet) Call Cubic (MTYPE, Coef, RT) Rmin = 1.E+10 Rmax = 1.E-10 Do 70 I=1,3 if (RT(I).LE. 0.0) Go to 70 ! if (RT(I).LT. BB) Go to 70 if (RT(I).GT. Rmax) Rmax = RT(I) if (RT(I).LT. Rmin) Rmin = RT(I) 70 Continue ZV=Rmax ZL=Rmin Zc = Rmax Return End Subroutine Bcfact (Alp, Bet, Bc,RT) Dimension Coef(4),RT(3) Coef(1) = Alp**3 + 6*Alp**2+ 12*Alp + 8. Coef(2) = -(3*Alp**2-15 *Alp+27*Bet-15.) Coef(3) = 3*Alp+6. Coef(4) = -1. Call Cubic (MTYPE, Coef, RT) Rmin = 1.E+10 Rmax = 1.E-10 Do 70 I=1,3 if (RT(I).LE. 0.0) Go to 70 ! if (RT(I).LT. BB) Go to 70 if (RT(I).GT. Rmax) Rmax = RT(I) if (RT(I).LT. Rmin) Rmin = RT(I) 70 Continue ZV = Rmax ZL = Rmin Bc = Rmax Return End Subroutine TcPcMix(am,bm,alpm,betm,R,Tcm,Pcm) Dimension RTB(3), RTZ(3) call Bcfact (alpm, betm, Bc,RTB) call Zcfactr (alpm, betm, Zc, RTZ) denom = 3*zc**2+(alpm+betm)*Bc**2+alpm*Bc

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Pcm = (am*Bc**2)/(bm**2*denom) Tcm = (am*Bc)/(bm*R*denom) return End Subroutine Para(AF,zc,wm,tc,pc,R,TT,Alp,Bet,ac,bc) omgww = 0.361/(1.0+0.0274*AF) Alp = (1.0+omgww*zc-3.0*zc)/(omgww*zc) Bet = (zc*zc*(omgww-1.0)**3.0+(2.0*zc*omgww**2)+& omgww*(1.0-3.0*zc))/(omgww**2*zc) omga = (1.0+(omgww-1.0)*zc)**3.0 omgb = omgww*zc tr = TT/tc theta = 0.309833 + 1.763758*AF + 0.720661*AF*AF - 1.363589*AF**3 - 4.005783*AF/sqrt(wm) tmp = tr ** (-theta/2.0) ! theta = 0.19708+0.08627*AF+0.35714*AF**2+3.59015E-03*AF*wm ! tmp = tr**(-theta) CB = omgb*R*tc/pc CA = omga*R**2*tc**2/pc ac = CA*tmp bc = CB Return End Subroutine Mixrule(Ncomp, x, ac, bc,tc,pc, Alp, Bet, AF, wm,BIN, Sumam,bmLLS, SumAlpm, SumBetm,wmmix,AFmix) Dimension x(20), ac(20), bc(20), Alp(20), Bet(20), AF(20), wm(20),tc(20),pc(20),BIN(15,15) Sumam = 0.0 SumbmLLS = 0.0 SumAlpm = 0.0 SumBetm = 0.0 wmmix = 0.0 AFmix = 0.0 Do 10 I = 1,Ncomp Sumbm = Sumbm + x(I)*bc(I) AFmix=AFmix+x(I)*sqrt(AF(I)) wmmix=wmmix+x(I)*sqrt(wm(I)) SumbmLLS = SumbmLLS + x(I)*bc(I)**(1.0/3.0) Do 10 J = 1,Ncomp Sumam = Sumam + x(I)*x(J)*sqrt(ac(I))*sqrt(ac(J))*BIN(I,J) SumAlpm = SumAlpm + x(I)*x(J)*sqrt(Alp(I))*sqrt(Alp(J))*BIN(I,J) SumBetm = SumBetm + x(I)*x(J)*sqrt(Bet(I))*sqrt(Bet(J))*BIN(I,J) 10 Continue bmLLS = (SumbmLLS)**3.0 AFmix=AFmix**2 wmmix=wmmix**2 Return End Subroutine BinIJ(Ncomp,AF,wm,tc,pc, BIJA,BIJB,BIJC,BIJD) Dimension AF(20),wm(20),tc(20),pc(20) Dimension BIJA(15,15),BIJB(15,15),BIJC(15,15),BIJD(15,15) Do 10 I = 1,Ncomp Do 10 J = 1,Ncomp AFI=AF(I) AFJ=AF(J) WMI=wm(I) WMJ=wm(J) AFWI=wm(I)*AF(I) AFWJ=wm(J)*AF(J) TPCI=tc(I)/sqrt(pc(I)) TPCJ=tc(J)/sqrt(pc(J)) ! If(AFI.LE.AFJ) BIJA(I,J) = (AFI/AFJ)**0.5 If(AFI.GT.AFJ) BIJA(I,J)= (AFJ/AFI)**0.5 !

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If(WMI.LE.WMJ) BIJB(I,J) = (WMI/WMJ)**0.5 If(WMI.GT.WMJ) BIJB(I,J) = (WMJ/WMI)**0.5 ! If(AFWI.LE.AFWJ) BIJC(I,J) = (AFWI/AFWJ)**0.5 If(AFWI.GT.AFWJ) BIJC(I,J) = (AFWJ/AFWI)**0.5 ! If(TPCI.LE.TPCJ) BIJD(I,J) = (TPCI/TPCJ)**0.5 If(TPCI.GT.TPCJ) BIJD(I,J) = (TPCJ/TPCI)**0.5 10 Continue Return End Subroutine Cubic(MTYPE,A,Z) DIMENSION B(3), A(4), Z(3) B(1)=A(2)/A(1) B10V3=B(1)/3.0 B(2)=A(3)/A(1) B(3)=A(4)/A(1) ALF=B(2)-B(1)*B10V3 BBT=2.0*B10V3**3-B(2)*B10V3+B(3) BETOV=BBT/2.0 ALFOV=ALF/3.0 CUAOV=ALFOV**3 SQBOV=BETOV**2 DEL=SQBOV+CUAOV IF (DEL) 90,10,40 10 MTYPE = 0 ! Three Equal Roots GAM=SQRT(-ALFOV) IF (BBT) 30,30,20 20 Z(1) = -2.0*GAM-B10V3 Z(2) = GAM-B10V3 Z(3) = Z(2) GO TO 130 30 Z(1) = 2.0*GAM-B10V3 Z(2) = -GAM-B10V3 Z(3) = Z(2) GO TO 130 40 MTYPE = 1 ! One Real Root & 2 Imaginary Conjugate Roots EPS=SQRT(DEL) TAU=-BETOV RCU=TAU+EPS SCU=TAU-EPS SIR=1.0 SIS=1.0 IF (RCU) 50,60,60 50 SIR=-1.0 60 IF (SCU) 70,80,80 70 SIS=-1.0 80 R=SIR*(SIR*RCU)**0.3333333333 S=SIS*(SIS*SCU)**0.3333333333 Z(1)=R+S-B10V3 Z(2)=-(R+S)/2.0-B10V3 Z(3)=0.86602540*(R-S) GO TO 130 90 MTYPE = -1 ! Three Dissimilar and Real Roots QUOT=SQBOV/CUAOV RCOT=SQRT(-QUOT) IF (BBT) 110,100,100 100 PEI=(1.5707963+ATAN(RCOT/SQRT(1.0-RCOT**2)))/3.0 GO TO 120 110 PEI=ATAN(SQRT(1.0-RCOT**2)/RCOT)/3.0 120 FACT=2.0*SQRT(-ALFOV) Z(1)=FACT*COS(PEI)-B10V3

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PEI=PEI+2.0943951 Z(2)=FACT*COS(PEI)-B10V3 PEI=PEI+2.0943951 Z(3)=FACT*COS(PEI)-B10V3 130 CONTINUE IF (MTYPE .EQ. 1) Z(2) = -99.99 IF (MTYPE .EQ. 1) Z(3) = -99.99 RETURN END ! Z-FACTOR PROGRAM BY LLS EOS METHOD FOR MIXTURES DIMENSION root(3),coeff(4),XC(20),P(20), ac(20), bc(20), zc(20), omgw(20) Dimension PPR(40),TTR(20),Zexpt(20),BIJA(15,15),BIJB(15,15),BIJC(15,15),BIJD(15,15) Dimension APDB(5,20),ZZV(5,30),DEV(5),TT(30),PP(30) Dimension Alp(20), Bet(20), AF(20), wm(20), tc(20), pc(20) Data PPR/0.1,0.2,0.3,0.4,0.5,0.6,0.7,0.8,0.9,1.0,1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.9,2.0& ,2.5,3.0,3.5,4.0,4.5,5.0,5.5,6.0,6.5,7.0,7.5,8.0,8.5,9.0,9.5,10,15,20,25,30/ Data TTR/0.1,0.5,1,1.5,2,3,4,5/ OPEN (UNIT=5,FILE='Input-Elsh_SPE74369.TXT',STATUS='old') OPEN (UNIT=6,FILE='OUTPUT-LLS-Elsh-SPE74369.TXT',STATUS='unknown') WRITE (6,*) 'LLS-Elsh-SPE74369' Read (5,*)NData Read (5,*)(Zexpt(I),I=1,NData) R = 10.73 AAPD=0.0 ! BIJA = Average Acentric Factor ! BIJB = Average Molecular Weight ! BIJC = Average of Acentric Factor X Molecular Weight ! BIJD = Average of Tc / Sqrt(Pc) IT=1 Do 100 K=1,NData Read (5,*)Ncomp,PP(K),TT(K) ! Write (6,*) Ncomp,TT(K),PP(K) do 20 I = 1,Ncomp Read (5,*)XC(I),wm(I),zc(I),AF(I),omgw(I),pc(I),tc(I) ! Write (6,*)XC(I),wm(I),zc(I),AF(I),omgw(I),pc(I),tc(I) Call Para(AF(I),zc(I),wm(I),tc(I),pc(I),R,TT,Alp(I),Bet(I),ac(I),bc(I)) ! write(6,*)I,TT(K),AF(I),zc(I),wm(I),tc(I),pc(I),Alp(I),Bet(I),ac(I),bc(I) ! write(6,*)I,TT(K),PP(K),Alp(I),Bet(I),ac(I),bc(I) 20 continue ! Computation of Binary Interaction Parameter Call BinIJ(Ncomp,AF,wm,tc,pc, BIJA,BIJB,BIJC,BIJD) ! If(IT .GT. 1) GO TO 122 Do 121 II=1,Ncomp 121 write(6,126) (BIJA(II,JJ),JJ=II,Ncomp) write(6,127) Do 123 II=1,Ncomp 123 write(6,126) (BIJB(II,JJ),JJ=II,Ncomp) write(6,127) Do 124 II=1,Ncomp 124 write(6,126) (BIJC(II,JJ),JJ=II,Ncomp) write(6,127) Do 125 II=1,Ncomp 125 write(6,126) (BIJD(II,JJ),JJ=II,Ncomp) write(6,127) 126 Format(12F6.3) 127 Format(/) IT=IT+1 122 Continue ! write(6,*)'End of Data = ',K Do 145 IB=1,4 AAPD=0.0

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If(IB .EQ. 1)Call Mixrule(Ncomp, XC, ac, bc,tc,pc, Alp, Bet, AF, wm,BIJA, am, bm, Alpm, Betm,wmix,AFmix) If(IB .EQ. 2)Call Mixrule(Ncomp, XC, ac, bc,tc,pc, Alp, Bet, AF, wm,BIJB, am, bm, Alpm, Betm,wmix,AFmix) If(IB .EQ. 3)Call Mixrule(Ncomp, XC, ac, bc,tc,pc, Alp, Bet, AF, wm,BIJC, am, bm, Alpm, Betm,wmix,AFmix) If(IB .EQ. 4)Call Mixrule(Ncomp, XC, ac, bc,tc,pc, Alp, Bet, AF, wm,BIJD, am, bm, Alpm, Betm,wmix,AFmix) ! call Mixrule(Ncomp, XC, ac, bc,tc,pc,Alp, Bet, AF, wm, am, bm, Alpm, Betm,wmix,AFmix) call Zfactr (Alpm, Betm, am, bm, PP(K), TT(K), R, ZV, ZL) call TcPcMix(am,bm,Alpm,Betm,R,Tcm,Pcm) ! DenV=PP*wmix/(ZV*R*TT(K)) APD=((ZV-Zexpt(K))/Zexpt(K))*100.0 AAPD=AAPD+ABS(APD) APDB(IB,K)=AAPD ! write (6,*);write (6,*) ! write (6,15) TT(K),PP(K),ZV,ZL,Zexpt(K),APD ZZV(IB,K)=ZV 15 Format(2F8.1,5F8.4) 145 Continue 100 continue Dat=NData AAPD=AAPD/Dat ! write(6,*)'Average Absolute Percent Deviation = ',AAPD Do 146 I=1,NData write (6,15) TT(I),PP(I),(ZZV(N1,I),N1=1,4),Zexpt(I) 146 Continue APDA2=0.0 APDB2=0.0 APDC2=0.0 APDD2=0.0 Do 147 I=1,NData APDA2=APDA2+APDB(1,I) APDB2=APDB2+APDB(2,I) APDC2=APDC2+APDB(3,I) APDD2=APDD2+APDB(4,I) 147 Continue DEV(1)=APDA2/Dat DEV(2)=APDB2/Dat DEV(3)=APDC2/Dat DEV(4)=APDD2/Dat write(6,51)(DEV(N1),N1=1,4) 51 Format(16X,5F8.2) close(5) close(6) STOP END Subroutine Zfactr (Alp, Bet, AT, BC, P, T, R, ZV, ZL) Dimension Coef(4),RT(3) AA = AT*P/(R**2*T**2) BB = BC*P/(R*T) Coef(1) = 1. Coef(2) = -(1.+(1-Alp)*BB) Coef(3) = AA-(Alp*BB)-(Bet+Alp)*BB**2 Coef(4) = -(AA*BB-Bet*(BB**2+BB**3)) Call Cubic (MTYPE, Coef, RT) ! write(6,*) 'Root=',RT Rmin = 1.E+10 Rmax = 1.E-10 Do 70 I=1,3 if (RT(I).LE. 0.0) Go to 70 if (RT(I).LT. BB) Go to 70 if (RT(I).GT. Rmax) Rmax = RT(I) if (RT(I).LT. Rmin) Rmin = RT(I) 70 Continue ZV=Rmax ZL=Rmin Return End

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Subroutine ZRedMix (Alp, Bet,AF,wm, Zc,Pr,tr, ZV, ZL) Dimension Coef(4),RT(3) w=AF theta = 0.309833 + 1.763758*w + 0.720661*w*w - 1.363589*w**3 - 4.005783*w/(sqrt(wm)) tmp = tr ** (-theta/2.0) omgW=0.361/(1.0+0.0274*w) omga=(1.0+(omgW-1.0)*Zc)**3 omgb = omgW*Zc Coef(1) = 1.0 Coef(2) = -(1.0/Zc+(1.0-Alp)*omgb*Pr/(Zc*tr)) Coef(3) = omga*Pr/(Zc**2*tr**(2+theta))-Alp*omgb*Pr/(Zc**2*tr)-(Alp+Bet)*(omgb*Pr/(Zc*tr))**2 Coef(4) = -(omga*omgb*Pr**2/(Zc**3*tr**(3+theta))-Bet*(1.0/Zc*(omgb*Pr/(Zc*tr))**2+(omgb*Pr/(Zc*tr))**3)) Call Cubic (MTYPE, Coef, RT) ! write(6,*) 'Root=',RT Rmin = 1.E+10 Rmax = 1.E-10 Do 70 I=1,3 if (RT(I).LE. 0.0) Go to 70 if (RT(I).LT. BB) Go to 70 if (RT(I).GT. Rmax) Rmax = RT(I) if (RT(I).LT. Rmin) Rmin = RT(I) 70 Continue ZV=Rmax*Zc ZL=Rmin*Zc Return End Subroutine Zcfactr (Alp, Bet, Zc, RT) Dimension Coef(4),RT(3) Coef(1) = Alp**3 + 6*Alp**2+ 12*Alp + 8. Coef(2) = -(12*Alp**2+12 *Alp + 9*Bet- 9*Alp*Bet+ 3.) Coef(3) = 6*Alp**2 + 3*Alp + 6*Bet - 6*Alp*Bet Coef(4) = -(Alp**2+Bet-Alp*Bet) Call Cubic (MTYPE, Coef, RT) Rmin = 1.E+10 Rmax = 1.E-10 Do 70 I=1,3 if (RT(I).LE. 0.0) Go to 70 ! if (RT(I).LT. BB) Go to 70 if (RT(I).GT. Rmax) Rmax = RT(I) if (RT(I).LT. Rmin) Rmin = RT(I) 70 Continue ZV=Rmax ZL=Rmin Zc = Rmax Return End Subroutine Bcfact (Alp, Bet, Bc,RT) Dimension Coef(4),RT(3) Coef(1) = Alp**3 + 6*Alp**2+ 12*Alp + 8. Coef(2) = -(3*Alp**2-15 *Alp+27*Bet-15.) Coef(3) = 3*Alp+6. Coef(4) = -1. Call Cubic (MTYPE, Coef, RT) Rmin = 1.E+10 Rmax = 1.E-10 Do 70 I=1,3 if (RT(I).LE. 0.0) Go to 70 ! if (RT(I).LT. BB) Go to 70 if (RT(I).GT. Rmax) Rmax = RT(I) if (RT(I).LT. Rmin) Rmin = RT(I) 70 Continue ZV = Rmax ZL = Rmin Bc = Rmax

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Return End Subroutine TcPcMix(am,bm,alpm,betm,R,Tcm,Pcm) Dimension RTB(3), RTZ(3) call Bcfact (alpm, betm, Bc,RTB) call Zcfactr (alpm, betm, Zc, RTZ) denom = 3*zc**2+(alpm+betm)*Bc**2+alpm*Bc Pcm = (am*Bc**2)/(bm**2*denom) Tcm = (am*Bc)/(bm*R*denom) return End Subroutine Para(AF,zc,wm,tc,pc,R,TT,Alp,Bet,ac,bc) omgww = 0.361/(1.0+0.0274*AF) Alp = (1.0+omgww*zc-3.0*zc)/(omgww*zc) Bet = (zc*zc*(omgww-1.0)**3.0+(2.0*zc*omgww**2)+& omgww*(1.0-3.0*zc))/(omgww**2*zc) omga = (1.0+(omgww-1.0)*zc)**3.0 omgb = omgww*zc tr = TT/tc theta = 0.309833 + 1.763758*AF + 0.720661*AF*AF - 1.363589*AF**3 - 4.005783*AF/sqrt(wm) tmp = tr ** (-theta/2.0) ! theta = 0.19708+0.08627*AF+0.35714*AF**2+3.59015E-03*AF*wm ! tmp = tr**(-theta) CB = omgb*R*tc/pc CA = omga*R**2*tc**2/pc ac = CA*tmp bc = CB Return End Subroutine Mixrule(Ncomp, x, ac, bc,tc,pc, Alp, Bet, AF, wm,BIN, Sumam,bmLLS, SumAlpm, SumBetm,wmmix,AFmix) Dimension x(20), ac(20), bc(20), Alp(20), Bet(20), AF(20), wm(20),& tc(20),pc(20),BIN(15,15) Sumam = 0.0 SumbmLLS = 0.0 SumAlpm = 0.0 SumBetm = 0.0 wmmix = 0.0 AFmix = 0.0 Do 10 I = 1,Ncomp Sumbm = Sumbm + x(I)*bc(I) AFmix=AFmix+x(I)*sqrt(AF(I)) wmmix=wmmix+x(I)*sqrt(wm(I)) SumbmLLS = SumbmLLS + x(I)*bc(I)**(1.0/3.0) Do 10 J = 1,Ncomp Sumam = Sumam + x(I)*x(J)*sqrt(ac(I))*sqrt(ac(J))*BIN(I,J) SumAlpm = SumAlpm + x(I)*x(J)*sqrt(Alp(I))*sqrt(Alp(J))*BIN(I,J) SumBetm = SumBetm + x(I)*x(J)*sqrt(Bet(I))*sqrt(Bet(J))*BIN(I,J) 10 Continue bmLLS = (SumbmLLS)**3.0 AFmix=AFmix**2 wmmix=wmmix**2 Return End Subroutine BinIJ(Ncomp,AF,wm,tc,pc, BIJA,BIJB,BIJC,BIJD) Dimension AF(20),wm(20),tc(20),pc(20) Dimension BIJA(15,15),BIJB(15,15),BIJC(15,15),BIJD(15,15) Do 10 I = 1,Ncomp Do 10 J = 1,Ncomp AFI=AF(I) AFJ=AF(J) WMI=wm(I)

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WMJ=wm(J) AFWI=wm(I)*AF(I) AFWJ=wm(J)*AF(J) TPCI=tc(I)/sqrt(pc(I)) TPCJ=tc(J)/sqrt(pc(J)) ! If(AFI.LE.AFJ) BIJA(I,J) = (AFI/AFJ)**0.5 If(AFI.GT.AFJ) BIJA(I,J)= (AFJ/AFI)**0.5 ! If(WMI.LE.WMJ) BIJB(I,J) = (WMI/WMJ)**0.5 If(WMI.GT.WMJ) BIJB(I,J) = (WMJ/WMI)**0.5 ! If(AFWI.LE.AFWJ) BIJC(I,J) = (AFWI/AFWJ)**0.5 If(AFWI.GT.AFWJ) BIJC(I,J) = (AFWJ/AFWI)**0.5 ! If(TPCI.LE.TPCJ) BIJD(I,J) = (TPCI/TPCJ)**0.5 If(TPCI.GT.TPCJ) BIJD(I,J) = (TPCJ/TPCI)**0.5 10 Continue Return End Subroutine Cubic(MTYPE,A,Z) DIMENSION B(3), A(4), Z(3) B(1)=A(2)/A(1) B10V3=B(1)/3.0 B(2)=A(3)/A(1) B(3)=A(4)/A(1) ALF=B(2)-B(1)*B10V3 BBT=2.0*B10V3**3-B(2)*B10V3+B(3) BETOV=BBT/2.0 ALFOV=ALF/3.0 CUAOV=ALFOV**3 SQBOV=BETOV**2 DEL=SQBOV+CUAOV IF (DEL) 90,10,40 10 MTYPE = 0 ! Three Equal Roots GAM=SQRT(-ALFOV) IF (BBT) 30,30,20 20 Z(1) = -2.0*GAM-B10V3 Z(2) = GAM-B10V3 Z(3) = Z(2) GO TO 130 30 Z(1) = 2.0*GAM-B10V3 Z(2) = -GAM-B10V3 Z(3) = Z(2) GO TO 130 40 MTYPE = 1 ! One Real Root & 2 Imaginary Conjugate Roots EPS=SQRT(DEL) TAU=-BETOV RCU=TAU+EPS SCU=TAU-EPS SIR=1.0 SIS=1.0 IF (RCU) 50,60,60 50 SIR=-1.0 60 IF (SCU) 70,80,80 70 SIS=-1.0 80 R=SIR*(SIR*RCU)**0.3333333333 S=SIS*(SIS*SCU)**0.3333333333 Z(1)=R+S-B10V3 Z(2)=-(R+S)/2.0-B10V3 Z(3)=0.86602540*(R-S) GO TO 130 90 MTYPE = -1

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! Three Dissimilar and Real Roots QUOT=SQBOV/CUAOV RCOT=SQRT(-QUOT) IF (BBT) 110,100,100 100 PEI=(1.5707963+ATAN(RCOT/SQRT(1.0-RCOT**2)))/3.0 GO TO 120 110 PEI=ATAN(SQRT(1.0-RCOT**2)/RCOT)/3.0 120 FACT=2.0*SQRT(-ALFOV) Z(1)=FACT*COS(PEI)-B10V3 PEI=PEI+2.0943951 Z(2)=FACT*COS(PEI)-B10V3 PEI=PEI+2.0943951 Z(3)=FACT*COS(PEI)-B10V3 130 CONTINUE IF (MTYPE .EQ. 1) Z(2) = -99.99 IF (MTYPE .EQ. 1) Z(3) = -99.99 RETURN END

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thomas a. mullen, mech eng, may04

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