CLOSED-FORM VAN DER WAALS CRITICAL POINT

FOR PETROLEUM RESERVOIR FLUIDS

by

TALAL HUSSEIN HASSOUN, B.S., M.S.

A DISSERTATION

IN

PETROLEUM ENGINEERING

Submitted to the Graduate Faculty of Texas Tech University in

Partial Fulfillment of the Requirements for

the Degree of

DOCTOR OF PHILOSOPHY

Approved

Akanni S. Lawal Chairperson of the Committee

Lloyd R. Heinze

James F. Lea

Accepted

John Borrelli Dean of the Graduate School

May, 2005

ACKNOWLEDGEMENTS

I wish to express my sincere thanks for the advice, guidance, and encouragement

given by my supervisor, Dr. Akanni S. Lawal. I would also like to thank the members of

my committee, Dr. Lloyd R. Heinze, and Dr. James F. Lea for their time and efforts. A

special thanks is extended to Dr. U. Mann for his assistance and willingness to help.

Finally, I thank Mr. S. Andreas, Mr. N. Kumar, and Mr. Tarek Hassoun for their help and

discussion contributed to this dissertation. I would like to acknowledge the Petroleum

Engineering Department for providing the financial support during the course of my

doctoral studies.

This Dissertation is dedicated to my father, my mother, my wife, Majida, my

daughter Amani, and to my sons, Ashraf, Heitham, and Tarek for providing me with

inspiration and confidence.

ii

TABLE OF CONTENTS

ACKNOWLEDGEMENTS ii

ABSTRACT v

LIST OF TABLES vii

LIST OF FIGURES ix

NOMENCLATURE xi

CHAPTER

I. INTRODUCTION

1.1 Importance of Critical State

1.2 Approaches to Critical State Prediction

1.3 Retrograde Reservoir Fluids

1.4 Objectives of Work

1

7

12

14

16

II. CRITICAL PROPERTY CORRELATION METHODS

2.1 Criteria of the Critical State

2.2 Empirical Models

2.3 Corresponding States

2.4 Convergence Pressure

2.5 Equation of State Models

18

19

22

25

31

45

III. CLOSED-FORM VAN DER WAALS EXPRESSIONS

3.1 Van der Waals Equations of State Theory

3.2 Closed-Form Equations for Fluid Critical Point

54

54

61

iii

3.3 Closed-Form Critical Property Computation Methods 69

IV. CRITICAL PROPERTIES FOR RESERVOIR FLUIDS

4.1 Critical Pressure Data for Complex Hydrocarbon Mixtures

4.2 Calculation of Critical Properties

4.3 Results and Discussion

4.4 Comparison Between Calculated and Experimental Data

74

74

75

80

83

V. CONCLUSIONS AND RECOMMENDATIONS

5.1 Conclusions

5.2 Recommendations

86

86

88

REFERENCES 89

APPENDICES

A. ANALYTICAL SOLUTION FOR CUBIC EQUATIONS

B. VAN DER WAALS EXPRESSIONS FOR FLUID CRITICAL POINT

C. PREDICTION RESULTS OF CRITICAL PRESSURE, CRITICAL

TEMPERATURE, AND HEPTANE PLUS PROPERTIES

99

112

130

iv

ABSTRACT

The prediction of critical points is of great practical importance because the

classification of petroleum reservoir fluids as a dry gas, gas condensate, volatile oil, and

crude oil depends largely on the knowledge of the critical properties of the reservoir

fluid. Also, the critical pressure and critical temperature of reservoir fluids are important

properties for describing the reservoir fluid phase behavior, predicting volumetric

properties of reservoir fluids and designing supercritical fluid processes.

Previous work for determining critical pressure, and critical temperature for

reservoir fluids include, empirical correlations, corresponding states method, and pseudo-

critical property methods. The generality of these previous correlations is limited to the

range of conditions and parameters used in the establishment of the correlations. Methods

based on the Gibbs criteria have also been used with Redlich-Kwong and Peng-Robinson

equations for prediction of critical properties. However, the Gibbs criteria have not been

applied to predicting critical properties of reservoir fluids.

A closed-form equation is developed for predicting the critical properties (Tc, Pc)

of complex reservoir fluids by using the Lawal-Lake-Silberberg (LLS) equation of state

with the criticality criteria established by Nobel Laureates van der Waals (VDW) in 1873.

By inverting the parameters of the LLS EOS in terms of the mixing parameters that are

based on the constituent substances and composition of the reservoir fluids, experimental

critical pressures and temperatures are predicted with interaction parameters expressed in

terms of molecular weight ratios of the binary constituent of reservoir fluids.

v

The prediction results of critical pressures and temperatures based on the VDW

criticality criteria show that experimental data consisting of 85 reservoir fluid mixtures

are within average absolute percent deviation of 3% to 5% of the measured critical

pressures and temperatures. In contrast to the previous work, this research project

provides an accurate method for computing the critical properties of reservoir fluids and

it is easy to use because the parameters of the criticality equation are readily available.

This project is useful for unifying near-critical flash routine with phase equilibria of the

compositional reservoir models. The project is also very attractive for establishing

reservoir models that are based on the critical composition convergence pressure concept.

vi

LIST OF TABLES 2.1 Modification to the Attractive Term of van der Waals Equation of State ................52

2.2 Modification to the Repulsive Term of van der Waals Equation of State ................53

3.1 Parameter of Selected Equations of State ..................................................................62

3.2 Relationship of EOS Constants to Critical Parameters..............................................64

4.1 Sample of Experimental Data Used in Calculations of Mixture 1.............................76

4.2 Calculated Critical Data of Heptane-plus Fraction Correlation.................................77

4.3 Calculated Critical Data of Heptanes-Plus Fraction for Data Set 1...........................78

4.4 Calculated Results for Pure Component parameters .................................................78

4.5 Calculated Results for Mixtures Parameters..............................................................79

4.6 Predicted Critical Pressure, Pc, Critical Temperature, Tc for Mixtures .....................81

C.1 Critical Pressure Prediction for Complex Mixtures………………………………130

C.2 Critical Pressure Prediction for Complex Mixtures ……………………………...131

C.3 Critical Pressure Prediction for Complex Mixtures...….…………………………132

C.4 Critical Pressure Prediction for Complex Mixtures...… …………………………133

C.5 Critical Pressure Prediction for Complex Mixtures……………………………....134

C.6 Critical Pressure Prediction for ComplexMixture………………………………...135

C.7 Critical Pressure Prediction for Complex Mixtures ……………………………...136

C.8 Critical Pressure Prediction for Complex Mixtures ……………………………137

C.9 Critical Pressure Prediction for Complex Mixtures ……………………………138

C.10 Critical Temperature Prediction for Complex Mixtures ……………………….139

vii

C.11 Critical Temperature Prediction for Mixtures ………………………………….140

C.12 Critical Temperature Prediction for Complex Mixtures………………………..141

C.13 Critical Temperature Prediction for Complex Mixtures…………………….….142

C.14 Critical Temperature Prediction for Complex Mixtures ……………………….143

C.15 Critical Temperature Prediction for Complex Mixtures ……………………….144

C.16 Critical Temperature Prediction for Complex Mixtures ……………………….145

C.17 Critical Temperature Prediction for Complex Mixtures ……………………….146

C.18 Critical Temperature Prediction for Complex Mixtures ……………………….147

C.19 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus....….….….….….….….….….….….….….….….….….….….148

C.20 Critical Pressure, Temperature, and Properties Prediction

for Heptane Plus……………………………………………………………. ….149 C.21 Critical Pressure, Temperature, and Properties Prediction

for Heptane Plus…………………………………………………………...……150

C.22 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus…………………………………………………...……….. ….151

C.23 Critical Pressure, Temperature, and Properties Prediction

for Heptane Plus …………………………………………………………….….152 C.24 Critical Pressure, Temperature, and Properties Prediction

for Heptane Plus …………………………………………………………….….153

C.25 Critical Pressure, Temperature, and properties Prediction for Heptane Plus……………………………………………………………. ….154

C.26 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus……………………………………………………………. ….155

C.27 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus……………………………………………………………. ….156

viii

LIST OF FIGURES 1.1 Pressure-Temperature Phase Diagram of Petroleum Reservoir Fluids……………5

1.2 Pressure-Volume Diagram of Pure Components………………………………... 8

1.3 Pressure-Volume Diagram of Mixtures………………………………………….10

1.4 Specific Weight of Liquid and Gas for Propane in the Critical Region………....12

1.5 Pressure-Temperature Diagram of Retrograde Gas Condensate………………...14

2.1 Pressure-Volume Plot for a Single-Component System…………………………19

2.2 Critical Point Representation in a Multi-Component System……………………22

2.3 Compressibility Factors of Methane, Ethane, and Propane as a Function of Reduced Pressure………………………………………………………………...27

2.4 A Deviation Chart for Hydrocarbon Gases…………………………………… 28

2.5 Approximate Temperature of the Reduced Vapor Pressure…………………… 30

2.6 Equilibrium Ration for a Low-Shrinkage Oil……………………………………33

2.7 Equilibrium Ratio for a Condensate Fluid……………………………………….34

2.8 Illustration of Quasi-Convergence Pressure Concept……………………………38

2.9 Comparison of Equilibrium Ratios at 100°F for 1000 and 5000 psi Convergence Pressure……………………………………………………………40

2.10 Equilibrium Ratios of Heptanes-plus Fraction…………………………………. 42

2.11 K vs Pressure with C10+ Curve Required to Match Check Point Data…………...43

2.12 K vs Pressure with Curve Showing Effect of Choosing a Convergence Too High or Too Low for Condensate Depletion………………………………..44

3.1 Pressure-Volume Diagram for Pure Component………………………………. 58

3.2 Algorithm for Computation of Critical Parameters…………………………….. 73

ix

4.1 Predicted Critical Pressure of Complex Mixtures………………………….……82

4.2 Predicted Critical Temperature of Complex Mixtures…………………………...83

C.1 Critical Pressure Prediction for Complex Mixtures (All Data)…………………………………………………….…………………157

C.2 Critical Pressure Prediction for Complex Mixtures (Mixture 145 – 1)………………………………………………….....…………158

C.3 Critical Pressure Prediction for Complex Mixture (Mixture 145 – 10)…………………………………………………… ……… 159

C.4 Critical Pressure Prediction for Complex Mixture (Mixture 4 –6)…………………………………………………… ……………160

C.5 Critical Pressure Prediction for Complex Mixture (Mixture75 – 6)…………………………………………………… …………..161

C.6 Critical Pressure Prediction for Complex Mixture (Mixture141 – 7)…………………………………………………… ……….…162

C.7 Critical Pressure Prediction for Complex Mixture (Mixture 141 – 16)……………………………………………………………...163

C.8 Critical Pressure Prediction for Complex Mixture (Mixture 141 – 25)…………………………………………………… ………. 164

C.9 Critical Pressure Prediction for Complex Mixture (Mixture 58 – 1) ………………………………………………………………..165

C.10 Critical Pressure Prediction for Complex Mixture (Mixture 47 - 1)…...……………………………………………………………166

C.11 Critical Temperature Prediction for Complex Mixture (All Data)……………………………………………………………………….167

C.12 Critical Temperature Prediction for Complex Mixture

(Mixture 145 –1)..………………………………………………………………168

C.13 Critical Temperature Prediction for Complex Mixture (Mixture 145 –10)………………………………………………………………169

C.14 Critical Temperature Prediction for Complex Mixture

x

(Mixture 4 – 6).. …….………………………………………………………….170

C.15 Critical Temperature Prediction for Complex Mixture (Mixture 75 - 6).. ……………………………………………………………….171

C.16 Critical Temperature Prediction for Complex Mixture (Mixture 141 – 7).. ….………………………………………………………….172

C.17 Critical Temperature Prediction for Complex Mixture (Mixture 141 - 25)………………………………………………………………173

C.18 Critical Temperature Prediction for Complex Mixture (Mixture 58 - 1) ………………………………………………………………...174

C.19 Critical Temperature Prediction for Complex Mixture (Mixture 47 –1) …….…………………………………………………………..175

xi

NOMENCLATURE

Symbol Definition

a Attraction Parameter Term of EOS

A Dimensionless Constant ⎟⎠⎞

⎜⎝⎛

22TRaP

AD Absolute Deviation

b van der Waals co-volume

B Dimensionless Constant

API Oil Gravity

⎟⎠⎞

⎜⎝⎛

RTbP

C Characterization Factor

K Watson Characterization Factor

r Weight

re

ature

MW Molecula

P Pressure

Pc Critical Pressu

R Gas Constant

Sg Specific Gravity

T Absolute Temperature

Tc Critical Temperature

Tbp Boiling Temper

V Molar Volume

xii

x Mole Fraction

Z Compressibility Factor

Zc Critical Compressibility Factor

arameter

ω Acentric Factor

cr ts

cation

ij Component Identification

Greek Letter

α Parameter of LLS EOS

αij Binary Interaction

β Parameter of LLS EOS

βij Beta Binary Interaction

Ω Dimensionless P

Subs ip

c Critical Property

m Mixture Identifi

r Reduced State

VDW van der Waals Symbol

xiii

CHAPTER I

INTRODUCTION

Petroleum reservoir fluids contain a variety of substances of diverse chemical

nature that include hydrocarbons and non-hydrocarbons. Hydrocarbons (carbon-

hydrogen molecules) range from methane to asphalt. Non-hydrocarbons include

substances such as nitrogen, carbon dioxide, and sulfur compounds. The chemistry of

hydrocarbon reservoir fluids is very complex. However, the mixtures of these complex

hydrocarbons may be in gaseous or liquid state at the pressures and temperatures often

encountered in petroleum reservoirs.

In spite of the complexity of hydrocarbon fluids found in reservoir rocks, simple

cubic equations of state have shown surprising performance in the phase behavior

calculations for both the vapor-liquid and vapor-liquid-liquid equilibria of these reservoir

fluids.

Knowledge of the phase equilibria of two-phase system is very important in the

design of a separation process and petroleum reservoir studies. Our interests are

properties of the state at which two phases of vapor and liquid become indistinguishable.

This state is termed critical at which the intensive properties of the liquid and vapor

phases become identical. Hence, mixtures consist of two phases of identical composition

are called critical mixtures. Predicting critical properties such as critical pressure, c ,

critical temperature, , and critical volume, c of the composition of a system in which

two phases become indistinguishable is very difficult and costly to determine

P

cT V

1

experimentally. Also, it is particularly difficult to determine the critical state

experimentally for multi-component mixtures.

Many methods have been reported for predicting critical properties of fluid

mixtures. These methods have ranged from empirical correlations, 31,122,111, to rig

thermodynamic conditions. Currently, there are three fundamental approaches, namely,

van der Waals criterion, Gibbs free energy, and the Wilson renormalization-group (RG)

theory. The Gibbs, 36 Wilson, 20 and Jiang & Prausnitz approaches were highly

considered and gained the attention of researchers in the field but their techniques have

failed, 100, in determining the critical points of reservoir fluids that contain heptane-plu

(C

29 98 orous

s

11 46

46

7+) fractions because their method not based on ideal gas law. The van der Waals

approach proved to be practical in resolving critical states in binary systems

and therefore can be applicable for predicting the critical properties of multi-component

systems and the more complex reservoir fluids.

117,116,115,114

In order to predict adequately the critical properties ( c c c ) of complex

reservoir fluids, a background in phase behavior is needed to understand numerous

surface and subsurface aspects of petroleum engineering. A knowledge of reservoir fluid

properties and phase behavior is necessary to calculate fluid in place, fluid recovery by

primary depletion, and fluid recovery by enhanced oil recovery (EOR) techniques such as

gas cycling, hydrocarbon solvent injection, and carbon dioxide (CO

P ,T ,V

2) displacement. An

example is given for clarification is that, many reservoirs have problems need

compositional treatment to increase accuracy and obtain more realistic description of

their fluids. This compositional treatment is of two types: depletion and/or cycling of

2

volatile oil and gas condensate and the other is miscible flooding with Multi-contact

miscibility (MCM). The difference between these two types of treatment is that the

depletion of volatile oil and gas condensate involves the removal of composition from the

critical region, while the second type requires calculation of phase composition and

properties converging at the critical point. The compositional model is capable of solving

the problem of miscibility where the original reservoir fluid and injected fluid are

miscible on first contact. There is difficulty in modeling the MCM process in achieving

stable convergence of gas and oil phase compositions, densities, and viscosities near the

critical point. The use of an equation of state in the MCM process, the vapor liquid

equilibrium ratios (K-values), and phase densities can be calculated near the critical

point. Therefore, in light of the advantages of van der Waals equations of state in solving

reservoir problems and the fact that a priori knowledge of the location of mixtures critical

points is required for volatile oil and gas condensate, the van der Waals criticality

conditions is applied as a tool to develop a closed-form solution of equation of state for

the critical points of fluid mixtures.

The prediction of critical properties of petroleum reservoir fluids is an important

factor in understanding the overall phase behavior of EOR miscibility fluid injection

projects.10 In many chemical and petroleum processes, the knowledge of the critical

behavior of hydrocarbon mixtures is useful in settling the operating ranges in the process

equipment. Furthermore, in compositional reservoir simulation, the appearance and

disappearance of phases (single phase or two-phases) in the reservoir grid block is

important. The critical state is required in K- values correlations that use the

3

convergence pressure concept, because the convergence pressure is the critical pressure

under certain specified conditions. 94,66, Several methods, such as that of Hadden a

Grayson, for finding K-values depend on the use of convergence pressure as a

correlating parameter.

39 nd

39

At the present time, the major area of research study in petroleum and chemical

industries is the high-pressure phase equilibria, phase density and composition of fluid

mixtures, and the effects of changes in pressure and temperature in which they exist. The

most practical tool used by scientists and researchers in high-pressure phase equilibria

studies is the cubic equation of state. This equation is an expression relates pressure,

temperature, and volume of the fluid (PVT). Many papers and books have been

published on the study of phase behavior of single, binary and ternary systems, as well as

for multi-component and complex reservoir fluids under a wide variety of conditions of

pressures and temperatures. Majority of these scientific publications deals with the

determination and prediction of fluid critical point.

The critical state of multi-component mixture is important from theoretical and

practical point of view, and an ability to predict this condition is highly desirable. Even

though the van der Waals criterion for critical state was enunciated by van der Waals ,

no satisfactory analytical method for predicting the critical condition in multi-component

systems based on this criterion has ever been formulated. The object of the work

undertaken in this study was to develop a closed-form solution to the problem of

predicting the critical properties of defined multi-component mixtures from the van der

Waals criticality condition together with a four parameters Lawal-Lake-Silberberg (LLS)

116

4

cubic equation of state. The van der Waals on-fluid theory is employed in this project

and the application of mixing rules allows the pure component parameters of the LLS

equation of state to be extended to mixture parameters.

We can best illustrate what the effects various system compositions have on the

phase behavior of petroleum hydrocarbons by presenting schematically the phase

diagrams for particular systems. Figure 1.1 is a pressure-temperature diagram, which

shows the relative boundaries of the two phase-phase region for typical reservoir fluids.

These include a dry gas, a gas condensate, a volatile oil, and a crude oil.

Figure 1.1. Pressure-Temperature Phase-Diagram of Petroleum Reservoir Fluids. 7

Schematic phase diagrams for each of the four reservoirs fluid classifications are

shown in Figure 1.1, which relates the reservoir fluid state to the reservoir pressure and

temperature. The vertical dotted line on the Figure 1.1 represents pressure depletion in

5

the reservoir at a constant temperature. The area within the phase envelope for each type

of reservoir fluid represents the pressure and temperature conditions at which both liquid

and vapor phases can exist. Point C on each envelope represents the critical point,

where the properties of liquid and vapor become identical. The solid line to the left of the

critical point , represents 100 percent liquid (the bubble-point line); the solid line to the

right of the critical point C , represents 100 percent vapor (the dew-point line).

C

As the fluid composition becomes richer in the high molecular weight

hydrocarbons, the phase envelope is changed such that the critical point shifts toward

higher temperatures and lower pressures. The location of the reservoir temperature and

pressure with respect to the critical temperature and pressure on the phase diagram for

any given fluid dictates the phase state of the fluid in the reservoir. Generally speaking,

the fluid above and to the left of the critical is considered liquid; fluid above and to the

right of the critical point is considered gas. Refer again to Figure 1.1, the gas reservoir

envelope lies completely to the left of the reservoir temperature line largely because the

main gas constituent, methane, has a low critical temperature )116 Fo− . Therefore, o

one phase can exist at reservoir temperature regardless of pressure. Any liquid recove

from a gas reservoir is the result of surface condensation after the gas has left the

reservoir.

( nly

red

Criticality is an important concept in phase behavior that is closely related to

equilibrium and stability concepts. In this introductory chapter, the importance of critical

state criteria, and the background and approach to critical state predictions are introduced.

6

The retrograde reservoir fluid phenomena, is demonstrated, and the objective of this work

is defined.

1.1 Importance of Critical State

The prediction of the critical properties of hydrocarbon mixtures is an important

aspect of the general problem of predicting the overall phase behavior of petroleum

reservoir fluids. The critical state is the unique condition about which the liquid and

vapor phases are defined, and hence has theoretical and practical significance. In

hydrocarbon processing and producing operations, a knowledge of the critical condition

is necessary because many of these operations take place under conditions which are at or

near the bubble-point or upper dew point regions and are frequently accompanied by

isobaric (constant pressure) or isothermal (constant temperature) retrograde phenomena.

Fluid property predictions and design calculations in this region are often the most

difficult one to make, and knowledge of the precise location of the critical point for the

reservoir under study is of the utmost help.

From a theoretical point of view, the changes of many of the thermodynamics and

transport properties take on a special significance as the critical state is approached. In an

empirical method the critical state has formed an integral part of many useful generalized

correlations such as those based on the theorem of corresponding states or the

convergence pressure concept in vapor-liquid equilibrium calculations.

In many ways the characteristics of the critical state that make it theoretically and

practically important are also the characteristics that make it one of the more difficult

conditions to measure experimentally. The very fact that density differences between

7

phases disappear, that the rate of volume change with respect to pressure approaches

infinity, or that infinitesimal temperature gradients can be responsible for a transition

from 100 percent liquid to 100 percent vapor all make the critical condition that one of

the more difficult to measure or observe accurately. For obvious economic reasons, it is

a condition that cannot be obtained by experiment in any special way for the many

systems for which it is required. Consequently, many attempts have been made to

develop methods for predicting the critical properties based on generalized empirical or

semi-empirical procedures. Consider the plot of pressure versus volume of a pure

component shown in Figure 1.2.

Figure 1.2. Pressure-Volume Phase Diagram of Pure Components. 32

8

Figure 1.2 illustrates the variation of volume with pressure and temperature

throughout the critical region. It is of interest to note that cTT <1 ; c is the critical

temperature is tangent to the saturation line at the critical state. At point , the liquid is

compressed (this state is referred to as under-saturated liquid because more gas is

dissolved in it). As the pressure is decreased, the volume increases. At point , the

liquid is in the saturated and stable state. To the right of point ,A as the pressure is

lowered, the component might follow one of two routs. It might follow the line ,AD in

ich case point

T

wh

S

A

D represents the saturated and stable vapor, or it mi

,AB for which the fluid will be in a meta-stable condition. In this case, the limit of

stability is determined by the condition that

ght follow curve

1TVP⎟⎞

⎜⎛ ∂ vanishes (i.e., point B ). Similar

at point

⎠⎝ ∂y, l

R one can observe that as the pressure increases, condensation may no occur u

to poin here again

t p

t w ,C 1TV

P⎟⎞

⎜⎛ ∂ will vanish. Curve DC represents the locus of meta-

stability and point C is the limit of meta-stability at for the vapor. As the temperature

⎠⎝ ∂

ate there is a rapid increase in the slope

1T

is increased above that of the critical stpT

V⎛ ∂⎟⎠⎞

⎜⎝ ∂

of

the isobars at the critical volume. Both the isobaric thermal expansion and the isothermal

compressibility are infinite at the critical state.

The pressure-volume diagram of mixture differs from that of pure component.

pressure-volume diagram for mixtures is shown in Figure 1.3. The main differences

The pure component pressure-volume diagram can be seen in Figure 1.2, while the

9

between Figure 1.2 and Figure 1.3 are: (1) L and G of Figure 1.3 do not represent the

equilibrium states and (2) the critical points have different features.

V V

Figure 1.3. Pressure-Volume Diagram of Mixtures.

For a pure com

32

ponent, 02

2

=∂∂

=∂∂ V

PP , at the critical point. For a mixture, this V

nvelope. The Z-factors in Figure1.2 of the

equilibrium gas and liquid phases always meet the condition

does not occur at the top of the two-phase e

GL ZZ < . However, for

mixtures, when gas and liquid phases are at equilibrium, LZ might be smaller or larger

than GZ . At equilibrium, the density of the liquid phase is higher than the density of the

. Then from gas phase, GL ρρ >

ZRTPM

=ρ (2.1)

it follows that

10

G

GL

L ZM

ZM (2.2)

When Z-factors are less than one, then,

>

L GZ could be smaller or larger than Z .

Variation in density of the liquid phase and gas phase at the critical region is

shown in Figure 1.4. It has been observed, experimentally, that there is a nearly linear

relationship between the average density of the coexisting phases and the vapor pres

near to the critical state. It can be shown that

sure

Pavg

∂

∂ρ will reach a constant value as the

critical state is approached. This relation of the average density of the liquid and gas

phases to the prevailing pressure and temperature has been called the “Law of Rectilinear

Diameters”. This relation gives an acceptable basis for the experimental

determination of the critical volume of a pure component. It is necessary only to plot the

average specific weight of the coexisting phases as a function of pressure and to

extrapolate from the accurate experiment to the critical state. Such a plot for propane is

given in Figure 1.4. In this Figure 1.4, notice the slight curvature in the relation of the

average specific weight to pressure at the lower temperatures, but near the critical point

the relation becomes linear. In the light of this discussion it is seen that the critical point

is truly a state of the system.

102,101

11

1.2

102 Figure 1.4. Specific Weight of Liquid and Gas for Propane in the Critical Region.

Approaches to Critical State Prediction

al

ses.

a

estimated utilizing various correlations methods, which have been reviewed in terms of

Because of the difficulty of measuring the critical properties of hydrocarbon

mixtures experimentally, the ability to have reliable methods for correlating and

predicting these properties is highly desirable. A survey of the literature indicates many

correlations have been advanced for predicting the phase behavior, 77 predicting physic

properties, 49,69 developing equations of state, and designing supercritical fluid proces

For many pure components, these critical properties have been experimentally

determined. 91 However, experimental determinations of the critical properties of

mixtures are impractical because of the limitations in terms of time and costs. Even

though experimental data for some mixtures are vailable, but with less coverage of

composition range of data points.

In addition to the direct measurements, critical properties of mixtures are usually

12

their estimation procedure and accuracy108 . These correlation methods relied prim

on many approaches, namely, ap

arily

gr hical roach equation of state approach,

empiric

at is,

ect to

t constant temperature and pressure must be equal to zero. Determination

of , and for the mixtur o an extended form

of derivatives and an equation of state such as reported by Spear et al. There are two

setbacks in these approaches. First, most of these approaches are limited to estimating

critical properties of hydrocarbon mixtures. Even in the case of hydrocarbon mixtures,

most of the methods tend to yield higher order of errors when used to estimate the critical

properties of methane-containing mixtures.

In this work, a new concept for the development of a methodology for predicting

petroleum reservoir fluids critical properties for cubic equations of state consistent with

the criterion of van der Waals’ equation of state. The use of this concept is illustrated by

its application to the Lawal-Lake-Silberberg equation of state, and also the use of van der

Waals one-fluid theory and the application of mixing rules which allows the pure

component parameters of the LLS equation of state to be extended to mixtures

parameters.

app , 38 79,84,20

al procedures involving the use of excess property approach, 54 the use of

conformal solution theory (corresponding states principle) approach 58 based on the

concept that all thermodynamic properties of mixtures can be evaluated from pure

component properties if the components conform to certain postulates of statistical

mechanics, and rigorous thermodynamic potential approach for the critical state, th

the second and third partial derivatives of the molar Gibbs free energy with resp

composition a

cT , cP cV es involves a simultaneous soluti n of

107

107

13

1.3 Retrograde Reservoir Fluids

To illustrate retrograde reservoir fluid phenomena, Figure 1.5 shows the pressure-

temperature phase diagram for a mixture at fixed composition.

Figure 1.5. Pressure-Temperature Diagram of Retrograde Gas Condensatio

The solid thick line is the bubble- point curve (100 % liquid, 0 % vapor) and the thin l

is the dew-point curve (100 % vapor, 0 % liquid)

n

ine

; they meet at critical point where the

two ph i

tw

es

o

o

. 6

C

ases become identical. The basic cr terion for the critical point (point C ) is the

limiting condition where the system can exist in o phases. Near the critical point,

hydrocarbon mixtures exhibit a more complex behavior usually opposite to what would

be expected from observed behavior at low pressures. This reverse behavior compris

retrograde phenomena. Retrograde phenomena always exist when the critical p int of a

mixture is not at the highest pressure and temperature possible for the coexistence of tw

14

phases. Also, near the critical point, the density-dependent properties change wi sm

changes in temperature and pressure.

More than a century ago Kuenen 61,62 first observed the isothermal (constant

temperature) retrograde condensation shown in the dashed ABDE line, and isobaric

vaporization, shown in line AGH were observed by Duhen (1896, 1901). The defin

of these phenomena are:

1. Retrograde condensation occurs when a denser reservoir fluid phase is formed by

the isothermal decrease in pressure or the isobaric increase in temperature.

2. Retrograde vaporization occurs when a

th all

itions

less dense of reservoir fluid phase is

ndentherm These points represent the upper

bounds where phase separation can take place.

Refer again to Figure hase (vapor) exists. If we increase

the pre s

n to

C

formed by the isothermal increase in pressure or the isobaric decrease in

temperature.

The maximum pressure at point N is the cricondenbar, and the maximum

temperature at point M is called crico .102

1.5, at point a , a single p

ssure on this vapor isothermally to point ,E we encounter a dew-point state. A

pressure is increased beyond ,E more and more of the vapor will conde se until reach

point .D ontinue pressure increase from point D causes retrograde vaporization of

liquid that had previously be ondensed. This process continues until point where

an u

en c ,B

pp r dew point is reached. Continued pressure increase from point B to e A which

sses a single-phase fluid (vapor). If now reduce the temperature isobarically fromcompre

point A to point ,G the volume of the sin le- phas fluid will contr a g e act. At point G

15

bubble-poi is

dense fluid (va tinue to form until point

nt encountered. With continued decrease in temperature isobarically, less

por) will con H is reached. As conditions change

from

beyond re dense phase to increase until the

bub -

above-

point G to point ,H retrograde vaporization occurs. Reduction of temperature

H continue isobarically will cause a mo

ble point is reached. We could demonstrate retrograde condensation by reversing the

escribed procedure, that is, by proceeding from point d N to point Quantitative

und

ion of

.b

erstanding of these phase-equilibrium phenomena is useful for design of production,

storage, and transportation of crude products.

Several observations can be made from Figure 1.5. The bubble-point line

coincides with the dew-point line at the critical point C. The shaded areas represent

regions of retrograde phenomena. The region defined by points CBMD in the reg

isothermal retrograde condensation.

1.4 Objectives of Work

The objective of this work is to develop a robust computational technique

predicting the critical properties, critical pressure, ,cT critical pressure, ,cP and critical

volume, V for complex petroleum reservoir fluids. This objective consis

for

of three

major elements:

1. Developm prehensive closed-form solution to the criticality criteria

established by Nobel Laureates van der Waals in 1873. Utility of the concept is

illustrated by lication to:

c ts

ent of a com

its app

16

• Lawal-Lake-Silberberg (LLS) equation of state using van der Waals one-

fl

2. Establish interaction parameters for hydrocarbon and non-hydrocarbon and for

hydrocarbon with pseudo-components..

3. Develop an algorithm for calculating these critical properties for reservoir fluids

(gases, gas condensate, volatile oils, and crude oils).

In order to achieve these objectives, this research work has been organized into five

chapters. Thus, after an overview of the critical property correlation methods and

illustrate the criteria of the critical state in Chapter 2, a review of the critical models,

corresponding state theory, convergence pressure concept, and the equations of state is

introduced. The van der Waals equation of state theory and the resulting derived

equations for the closed form-solution are presented in Chapter 3 with computational

procedure is described. Chapter 4 presents the results of calculations and analysis of the

predicted critical properties and compared with experimental calculations. Finally, in

Chapter 5 conclusions and recommendations were made on the equation of state

approach to critical points predictions, the general level of accuracy and applicability, and

the implications for future work in this area of phase behavior research.

uid theory

17

CHAPTER II

CRITICAL PROPERTY CORRELATION METHODS

Several investigators have developed correlation techniques for predicting critical

properties of complex reservoir hydrocarbon mixtures. Many of these correlation

techniques were essentially both empirical and theoretical procedures in nature and were

aimed at predicting the critical properties of naturally hydrocarbon systems. The well

known of these was the method of Kurata and Katz for the critical properties of volatile

hydrocarbon mix

et al., (

ns

this chapter, a review of correlation methods for predicting critical properties

of com ir hase

behavio

given mi will be

corresponding states. Then the law of corresponding states and the convergence pressure

are presented in detail.

63

tures, and of Organic 80 for complex hydrocarbon systems. Later, Davis

1954) modified the original Kurata-Katz method to make it applicable to lighter

natural gas systems. All of these correlation methods make use of graphical correlatio

with parameters such as pseudo critical temperature and pressure, molal average boiling

point, or weight average equivalent molecular weight.

In

plex petroleum reservoir fluids is undertaken to help understand the p

r calculations. First, the criterion of the critical state is introduced, and the

empirical models of correlations are presented. If the critical state can be predicted for a

xture, the separation between the bubble-point and the dew point regions

defined, and physical properties of the mixture can be obtained by using the law of

18

2.1 Criteria of the Critical State

At a critical point, the fluid does not exist in a particular state, either gas or liquid,

but has characteristics of both. Hence, it is called a supercritical fluid. To see at what

m ature.

temperature, pressure, and volume, this supercritical behavior is observed, we use the fact

that at the critical point, the isother is both horizontal (zero slope) and has no curv

These two conditions are interpreted mathematically as follows:

0=∂∂VP (2.1)

02

2

=∂∂V

P (2.2)

The criteria of criticality can be analyzed by two approaches: a simple approac

that relies on geometrical presentation, and an alternative approach more reliable for

multi-component and complex mixtures. Consider a p-v diagram for a two-phase critical

point of a single-component system shown in Figure 2.1

h

.

Figure 2.1. Pressure-Volume Plot for a Single-component System

In this pressure-volume phase diagram, the dashed curve is the spinodal curve

and the solid curve is the binodal curve. The p-v isotherms at four different temperatures

. 32

19

,1T , ,cT and ,3T are shown. Points B an C represents the limit2T d s of stability at

temp Points B` and C` represent the limit of stability at temperature Based

on the criteria of the limits of stabi

erature .1T .2T

stability, 32 lity at T and T are obtained from 1 2

01

=⎟⎠⎞

⎜⎝ ∂v⎛ ∂

T

p and 02

=⎟⎠⎞

⎜⎝⎛∂∂

Tvp , respectively. Between points B and C, 0

1

>⎟⎠⎞

⎜⎝⎛∂∂

Tvp and

between B` and C` ,02

>⎟⎠

⎜⎝ ∂v

d these are therefore the unstab⎞⎛ ∂

T

p an le segment of the

isotherms. Note that the changes in curvature between B and C indicates an inflection

point where 01⎠⎝ T

unstable. As the temperature approaches the critical point (i.e., cT ), the limits of s

and un-stability points coincide, and since the inflection point is now located on the

bimodal curve, the inflection point

2

2

=⎟⎟⎞

⎜⎜⎛∂∂v

p exists between these two points. This inflection point is an

tability

02

2

=⎟⎠

⎞⎜⎝

⎛∂∂

cTvp is a stable point. Points A, D and A`

and D` represent equilibrium phases at 1T and 2T , respectively. Approaching toward the

critical point, the points A and A` and D and D` coincide also with the limit of stability.

At the critical point, the gas phase and liquid phase can be transformed into each other

without going through the two-phase region, and that the continu

⎟⎜

ity of gas and liquid

ate. The criteria of the critical state of a pure component are, therefore, st

0=⎜⎛ (2 )⎟

⎠⎞

⎝ ∂∂

cTvp .3

20

02

2

⎟⎠

⎞⎜⎝

⎛∂∂

cTvp

=⎟⎜ (2.4)

.03

3

<⎟⎠

⎞⎜⎝

⎛∂∂

cTvp (2.5)

Where

⎟⎜

03

<⎟⎠

⎞⎜⎝

⎛ ∂

cTvp indicates that the critical point is neither a maximum nor a

minimum.

3 ⎟⎜ ∂

ch for the

calculatio e

C

In 1980, Heidmann and Khalil 43 proposed an alternative approa

n of critical point that is mathematically different from the expressions of th

riticality conditions in Equation 2.6 , and Equation 2.7 32 32

0

)1(1,1

)1(3,1

)1(2,1

)1(1,

)1(3,

)1(2,

)1(1,2

)1(23

)1(22

) ==

++++

+

+

cccc

cccc

c

yyyyyy

yyy

L

L

L

λ (2.6)

an

1( +c MMMM

d

0

)1(1

)1(3

)1(2

)1(1,

)1(3,

)1(2,

1,22322

+

)1()1()1(

=

+

yyy

λλλ L

L

(2.7)

where

+++

+

cc

cccccc

c

yyy L

MLMM

231 ,,2

2)1(22

+

⎟⎠

⎞⎜⎝

⎛∂∂

cxxcXC

L

⎟⎜=y , 221 ,,,3

2)1(23 ⎟

⎠

⎞⎜⎝

⎛∂∂

xcXC

L +

⎟⎜=cx

y ) is the , cNNVTAy ,,,,()1( L=

Helmholtz free energy, and 0)1(1 =++ rent.

laborate of multi-

ent system of fixed composition sketched in Figure 2.2. The thick solid line on

ccλ , at the critical point but the concept is not diffe

As we have already seen, the critical point is a stable point at the limit of stability. To

e more on this concept, let us consider the pressure-volume diagram

compon

21

the left represents the bubble-points and the thin line represents the dew points. The

bubble-points and dew points are stable equilibrium states; a perturbation in pressure,

d

it

l point.

thus, results in a stable state. Critical point CP is the point at which bubble-point an

dew point converge and being a stable state, is at the limit of stability. These two

concepts of stable and limit of stability were used by Gibbs in 1876 to derive the

expressions for the critica

32

2.2

Figure 2.2. Critical Point Representation in a Multi-component System

.

Empirical Models

olving the use of excess properties, (2) the use of conformal

an

Many correlations have been developed for predicting critical properties of pure

components and mixtures. These correlations have relied on three approaches: (1) an

empirical approaches inv

solution theory based on the concept that all thermodynamic properties of mixtures c

be evaluated from component properties if the components conform to certain postulates

22

of statisti al mechanics, (3) a rigorous thermodynamic condition based on the sec

third partial derivatives of all molar Gibbs free energy with respect to composition.

The empirical approach involves calculations of the following model:

∑

c ond and

=

G lled the

cal

icted by

w se

relations

lts for systems containing ethane.

Prausnitz’ method for determining the pseudo-critical temperature can be

generalized for mixtures having ts. The generalized equation is

iciicm

+=n

icorrciic GGxG

1

(2.8)

wher c is the critical property desired and corr is a correction term which is ca

excess property of the mixture. Excess properties are usually estimated from empiri

relations. Many correlations have been proposed based on Equation 2.6 to predict the

critical properties of mixtures. The critical temperature of defined mixtures pred

e G

122

L as simplified by Chueh-Prausnitz, 21 for all empirical approaches. All of the

correlations yielded results with large errors and the chemical nature and sizes of

components limit the use of these results. Li’s 71 correlation gave the most accurate

predictions with the method of Prausnitz gave satisfactory results. All other cor

gave poor resu

i 71,

s 85

any number of componen

ijj

n

i

n

j

n

i

TT τθθθ ∑∑∑ +==1

(2.9)

where

∑

= ciii

Vxθ

ciiVx

23

(2.10)

23

and ijτ for each

with data set of six ternary system, two quaternary systems, and two quinary systems and

reported 0.4% deviation.

Li’s equation a pseudo-critical temperature for any mixture, has the following

form:

1)

interacting pair of molecules. 21 Equation 2.7 was tested by Prausnitz

cii

icm TT ∑= φ (2.1n

where

∑n

cii

i

Vx

Vx=

i

ciiφ (2.12)

The critical state correlation models for mixtures are much more important than is

the case for pure components. A wide variety of empirical correlation models, usually

with an average boiling temperature and composition of the mixture as parameters, have

been proposed. These types of correlations are of two types: 1) correlation models

which apply to simple mixtures of known composition and 2) correlation models which

apply to complex petroleum hydrocarbon fractions. In general, the correlations for both

types of mixtures are limited to hydrocarbons and often only to aliphatic and simple

aromatic hydrocarbons.

For simple hydrocarbon mixtures, several empirical correlations for critical

temperature and pressure have been proposed. The correlations suggested by Pawlewski,

and Kay is the most important from a historical point of view. More accurate

correlation models have been proposed by Joffe, Grieves, and Organick, Etter and

53

48 38 80

24

Kay. Edmister summarized the critical point correlations available for hydrocarbon

mixtures up to 1949. The best of these correlations are accurate within 1% for the critical

temperature and 3-5% for the critical pressure.

Correla odels for predicting the critical point of complex petroleum

hydrocarbon fractions also have been proposed. Significant correlations have been

suggested by Sm Kurata-Katz, Edmister, and Pollock. Organick also

proposed a correlation and introduced a comparison of his method with the Kurata-Katz.

it the critical point. The prediction of the

critical

2.3 Corresponding States

29 27

tion m

ith,105 63 27 80

Equations of state were involved in the critical state correlation in order to

ntegrate vapor-liquid equilibrium behavior w hi

point on the basis of van der Waals critical condition and the use of a suitable

equation of state is considered the first step toward the work carried out in this

investigation.

Van der Waals

on experime shows that compressibility factor for different

fluids exhibit similar behavior when correlated as a function of perature

116 in 1873 developed the theorem of corresponding states based

ntal observation, which Z

reduced tem rT

and reduced pressure rP . By definition,

cr T

T = (2.13) T

cPP

rP = (2.14)

25

and

cr V

VV = (2.15)

Where the subscript r represents the reduced state, and subscript represents

critical state. These dimensionless reduced conditions of temperature, pressure, and

olume p

“All flu ave

iat

d value of an

tem may be defined as the ratio of the value of that property in

a given state to the value of that property at the critical state. The theorem of

corresponding states does not hold for big ranges of pressure for real gases, and at the

same time is not perfect. However, when applied to gases with similar chemical structure

(as paraffin hydrocarbons), it offers a correlation with close agreement (satisfactory for

engineering work) permits the use of reduced properties as the basis for correlating

experimentally compressibility factor. This is illustrated by the reduced PVT data on

methane, ethane, and propane shown in Figure 2.3. In Figure 2.3 is a plotted of values of

for methane, ethane, and propane as a function of reduced pressure for reduced

temperatures to show the degree of correlation.

c

v rovide the basis for the simplest form of the theorem of corresponding states:

ids, when compared at the same reduced temperature and reduced pressure, h

approximately the same compressibility-factor, and all dev e from ideal-gas behavior to

about the same degree.”105 The term “reduced state” means that each reduce

intensive property of a yss

Z

26

3. Compressibility Factors of Methane, Ethane, and Propane as a FunctionFigure 2. of

u

The theory of corresponding states was extended to cover mixtures of gases.

Kay invented the concept of pseudo-critical Temperature and pseudo-critical pressure for

real ga

i 1

Reduced Pressure and Reduced Temperat re.109

ses. These pseudo-critical properties are obtained by using the Amagat’s law of

partial volumes for mixtures to the critical properties of the composition of the mixture.

These quantities are defined as

∑=n

ciipc TyT (2.16)

and

ci

n

iipc

=

PyP ∑==

(2.17)

where is the pseudo-critica

re th omponent,

mole fraction of ith component in

1

l pressure, pcT is the pseudo-critical temperature, ciT and pcP

ciP a iy the critical temperature and critical pressure respectively of i c

mixture, and n number of components. The physical

27

propert i ical

e ay that properties for pure gases are correlated with reduced

pressure and reduced temperature. Thus, these pseudo-critical properties are defined as

follows:

ies of gas m xtures are correlated with pseudo-critical pressure and pseudo-crit

temperature in the sam w

pcpr P

PP = (2.18)

pc

pr TTT = (2.19)

Compressibility factors, experimentally obtained, for natural gas have been

correlated with pseudo-reduced pressure and temperature. The petroleum industry has

universally adopted the correlations shown in Figures 2.4, to determine the

compressibility factor

85,14

Z .

109 Figure 2.4. A Deviation-Chart for Hydrocarbon Gases.

28

Correlations of Z - factor based on the theory of corresponding states are called

two-parameter correlations, because of the use of two reducing parameters rT and .rP

These c

-

tion t and ),

has been

orrelations are shown to be close for the simple fluids (argon, xenon), but

systematic deviations are observed for more complex fluids. A third corresponding

states parameter concept characteristic of molecular structure (in addi o cT cP

introduced by K. S. Pitzer 63 is the acentric factor .ω

The Pitzer acentric factor for a pure fluid is defined with reference to its vapor

pressure. Becau

reciprocal of absolute temperature,

se the logarithm of the vapor pressure of pure fluid is linear in the

mpr =1

log (2.20)

Tr

where rp is the reduced vapor pressure, rT is the reduced temperature, and m is the

sat

d

d sat

)(

vs. slope of the plot of rPlog .1

rT

It has been observed that

the slope would be the same for all pure fluids. This is not true ; because according

to Pitzer each fluid has its own characteristic value of Pitzer has noted that all vapor

a for simple fluids lie on the same line when plotted as vs.

if the two-parameters corresponding states were valid,

pressure dat

m 63

.m

satrPlog

rT1 and

the line passes through 0.1log −=satP at .7.0=rr T This is shown in Figure 2.5. Data

for other fluids define other lines whose locations can be fixed in relation to the line for

the simple fluids (SF) by the difference:

29

satrPSF log) − (2.21) sat

rP (log

The acentric factor is defined as the difference evaluated at :

satω (2.22)

7.0=rT

7.0)log(0.1 =rTrP −−≡

Therefore, ω can be determined for any fluid from cc PT , and vapor pressure

measurement made at .7.0=rT

Figure 2.5. Approximate Temperature Dependence of the reduced Vapor Pressure.105

30

2.4 Convergence Pressure

This section first discusses some fundamental thoughts on the equilibrium ratios

(K-values), and reviews the basic sources for obtaining these values. This followed by

providing a discussion about the convergence pressure (Hadden, 1953), and remarks on

the interdependency of K-values of heavy (plus) fractions and convergence pressures.

In order to evaluate the equilibrium behavior of multi-component two-phase

systems and obtain an expression for K-values; Dalton’s and Raoult’s laws can be

combined. Dalton’s law is defined by the Equations 2.23, and 2.24

(2.23)

and

40

∑=

=n

iipP

1

Pp

y ii = (2.24)

or

(2.25)

and Raoult’s equation is stated as “the partial pressure exerted by a constituent of liquid

phase is equal to the vapor pressure of that constituent times the mole fraction of that

constituent in the liquid phase. That is,

(2.26)

Where is the partial pressure of the ith component, is the mole fraction of the ith

ompo nt in por phase and

Pyp ii =

viii PxP =

ip ix

c ne liquid phase, iy is the mole fraction of ith component in va

31

vip

is vapor pressure of ith component. By combining Equation 2.24 and Equation 2.26

we obtain:

(2.27)

By definition,

iii xppy =

i

ii x

yK = (2.28)

Where K is defined as the distribution of a component, “i”, between vapor and liquid

phases is given by the equilibrium ratio, K, described by Equation 2.28.

The value of is dependent on the pressure, temperature, and composition of

rium ratios ( values) for low-shrinkage oil and a

onden te at F s re. The

iK

Kthe hydrocarbon system. Equilib

c sa are shown in Figures 2.6 6 and 2.7 6 as functions of pre su o200

equilibrium ratios ( K values) for both types of fluids are shown to converge to a point

value of 1. This point is called convergence pressure, defined on page 162 of the

NGSMA Data book. “Early high pressure experimental work revealed that if a

hydrocarbon system of fixed overall composition were held at constant temperature and

the pressure varied, the

K - values of various components converged toward a commo

value of unity at some high pressure. This pressure has been termed the convergence

pressure of the system “. If the temperature at which the

n

K - values were presented is the

critical temperature of the hydrocarbon mixture, then the convergence pressure will be

the critical pressure. For all temperature other than the l temperature, the

f sure less

critica

convergence o K - values is then an apparent convergence pressure. At a pres

than convergence pressure, the system will be at either dew point or bubble point, and

32

exists as a single-phase fluid at the conditions expressed by the point of apparent

convergence.

6 Figure 2.6. Equilibrium Ratio for Low-Shrinkage Oil.

33

Figure 2.7. Equilibrium Ratio for a condensate Fluid. 92

A widely accepted definition of convergence pressure by Hadden 39 in 1953 w

proposed. In fact, Hadden 39 defined the critical mixture, from which the convergence

pressure would be estimated, as that resulting from adding methane r nitrogen to the

equilibrium liquid. In such an addition always result in reaching the critical state, and

hence

as

o

a convergence pressure is defined, and satisfies the phase rule requirement. If

these two lightest components methane (or nitrogen) were in every system s

definition of convergence pressure is adequate. But, for purposes of general correlation

of data, m ny of which are from binary or ternary that have neither methane nor nitrogen,

the adequately of Hadden’s

of interest, thi

a

definition is questionable.

34

Ro n ased on what he called critical

com ositio vergence pressure is estimated from the

critical m e equilibrium phases. That means the critical

mi general and applicable to binaries or higher-

order systems without regard to which components may be present. Unfortunately, this

definition does not always define the convergence pressure. This limitation caused most

frequently at low pressure, where

we, 93 in 1964 proposed a definitio b

p n method. In this method, the con

ixture, which would give the sam

xture lies in the tie line. This definition is

K - values are not sensitive to convergence pressure,

but can occur at high pressure as well Lawal, 1981 .

The critical mixture is determined by the intersection of the tie line with the locus

m

for the isothermal locus of critical composition in hydrocarbon systems that calculated

the critical composition. The general Fair’s equation is stated as

ole

fraction of component in the critical mixture. In this notational form, j indicates the

component chosen as dependent; that is is determined from z .

e

66

of critical co positions at the equilibrium temperature. Fair has developed an equation

11,1

=∑≠=

i

j

ci

i zA (2.29)

Where is the critical composition locus coefficient for component i , is the m

31

N

iAicz

jc ic

For an equilibrium state with liquid mol fractions ix and vapor mole fraction

iy , the interaction of the equations for the tie line (actually 1

z

=N component balan

equations) with the isothermal critical composition locus Equation 2.28 is represented

ce

by

[ ]

iiii

iiiic y

yxAyAyx

zi

+−

−−=

∑∑

)(1)(

(2.30)

35

Where the summation is understood to be over all components except nitrogen. Equat

2.28 was used to calculate the critical m

ion

ixture for all ternary, multi-component, and

comple

Fair o

than the generalized locus functions. The critical compositions of the binaries were

calculated from the equilibrium temperature by the expression

iicL

1

1)

where z mole fraction of light component

x mixture equilibrium states.

, 30 presented correlations for binary data, which are considered m re accurate

∑ −+=m

icz )1(θθθ (2.3

Lc

=

=

LH

H

cc

c

TTTT

−

−=θ

Equilibrium temperature

= Critical temperature of the heavy component

u

eterm

t

(2.32)

=T

Hc

LcT = Critical temperature of the light component

The convergence pressure characteristic of a partic lar equilibrium sate is

T

d ined as the critical pressure of the critical mixture calculated either by Equation

2.26 or Equation 2.27. Methods of predicting critical pressures range from rigorous

thermodynamic o completely empirical, with methods having some degree of success.

The method we choose to present here is the one developed by Zais122 because of its

general applicability and convenience of calculation. Zais’ equation is written in the

form

36

∑∑∑−

=1 1i

i+== −+−+

+=1

21

2

)()(

N N

ij jiijjiijij

jiN

iicc wwcwwBA

wwwPP

m (2.33)

r ixture

= Binary interaction coefficient

s

g Equation 2.31 to binary data, Zais was able to obtain values for the

coefficients and for all hydrocarbon binary combinations from methane

through eicosane plus nitrogen and carbon dioxide. These coefficients and component

critical pressures for use in Equation 2.31 are tabulated. Binary coefficients for binary

heavy fractions in complex mixtures are obtained by interpolation on a molecular weight

basis. Zais predicted the critical pressures of 298 ternary, multi-component, and complex

mixtures with an average absolute deviation of 5.2%.

where

= Critical p essure for mmcP

icP = Critical pressure of component i

w = Weight fraction of component i

= Weight fraction of component j

ij

ijB = Binary interaction coefficient

ijC = binary interaction coefficient

After testing mole fraction, surface fraction, and volume fraction, Zais selected

weight fraction as the composition variable to use in Equation 2.31. To calculate

convergence pressure from Equation 2.31, it is necessary to convert the mole fraction

from Equation 2.27 or Equation 2.31to weight fraction.

i

jw

A

By fittin

,, ijij BA ijC

122

37

In hydrocarbon systems, there is no critical pressure at temperatures below the

critical temperature of the lightest component. For this reason, it is impossible to derive a

convergence pressure as we described. Lawal, in order to include such a data in the

new correlations he developed, a quasi-convergence pressure was defined which is

illustrated in Figure 2.8. The quasi- convergence pressure is read at the equilibrium

temperature from the binary critical locus reflected across an axis through the critical

temperature of the light component.

66

Figure 2.8. Illustration of Quasi-Convergence Pressure Concept. 66

38

In effect, for equilibrium at temperature T less than critical temperature of the

light componentT , the quasi-convergence pressure is defined as e critical pressure

T , where

TT − 2 (2

This definition of the quasi-convergence pressure is consistent with the

observation of Lenoir and White (1958) that quasi-convergence pressure should increase

with decreasing temperature. For systems higher than binaries, this definition of quasi-

Lc th at

LK cp − .34)

rmining the isothermal locus of critical

compositions at rather than at equilibrium temperature , according to Equation

2.32.

The fluid composition effects on the

Kp

T

convergence pressure is readily applied by dete

KpT T

K - values as shown in Figure 2.9, where

values for 1000 psia and 5000 psia convergence pressures are compared at F. The

differences in values for the two convergence pressures shown at pressures below 100

psia are not significant for the lighter hydrocarbons. The equilibrium ratios for fluids

with convergence pressures of 4000 psia or greater, are the same to fluids with 1000 psia.

Therefore, it is apparent that at low pressures and temperatures the equilibrium ratios are

closely independent of composition.

6

o100

K

39

Figure 2.9. Comparison of Equilibrium Ratios at 100 Fo for 1000- and 5000-psia convergence Pressure.

It is of practical value at this point to present brief remarks on the

interdependency of heavy (plus) fractions and convergence pressure concept. Because

e vapor-pressure curves and critical properties of hydrocarbon heavier than hexane

irly c -

values. Properties of heptanes plus fractions can be estimated from the properties of

a procedur r

characterizing the heptanes plus is to use correlated experimental data heptanes pl

fraction of fluids with similar properties to those predicted. For this purpose equilibrium

ratios for the heptanes-plus fraction reported by Katz and Hachmuth, and Roland,

6

th are

fa lose together, it is possible to characterize the mixture by an average set of K

heavier hydrocarbons such as decane. But, normally a more satisf ctory e fo

us

50

40

Smith, and Kaveller, are plotted in Figure 2.10. The data of Katz are preferred for crude

oil system, and the data of Roland et al densate fluids.

6 6

. are preferred for con

Figure 2.10. Equilibrium Ratios of Heptanes-Plus Fraction.6

lus) fraction controls the behavior of the system as liquid (mostly the

The heavy (p

plus-fraction) begins to drop out at the dew point. If a given set of K - values with

convergence pressure close enough to the correct convergence pressure is used m

through decane, the entire adjustment in the – values system can be made by adjusting the

ethane

K - value for the plus- fraction only. Then, it is easy to determine whether or not the K -

-

value system chosen for methane through decane is near enough to the correct

convergence pressure set by inspecting the decane K - values curve that was required to

match check-point data. If the selected convergence pressure is too low, the C +10 K

41

curve required to match checkpoint data will fall below the given published decane K -

curve. This shown in Figure 2.11 30 by the curve marked “ kP too low”. If the

ve will have the correct ecane

curve. This shown in the Figure 2.12 by the curve marked “ too high”. A set of

convergence pressure is too high (i.e., too close to the dew point pressure) the determined

KC +10 - values cur shape but will fall above the given d

30kP K

- values for the correct convergence pressure will result in the - values curve as

ome extent, cut across the

tion that

t as

KC +10

shown in Figure 2.11. Typically, the +10C K - value will, to s

constant temperature lines on the given Natural Gasoline Supply Men’s Association

(NGSMA) decane curve. This phenomena caused by the fact that the plus-frac

first drop at the dew point is generally less volatile than the plus-fraction that drops ou

pressure declines.

42

Figure 2.11. K vs Pressure with C10+ Curve Required to Match Check-Point Data.

43

Too High or ToFigure 2.12. K vs Pressure with Curve Showing Effect of Choosing a Convergence

o Low for Condensate Depletion.

30

44

2.5 Equation of State Models

com mixture. EOS may be used to describe the state of the fluid phase. The

volumetric phase behavior of a pure component and a multi-component mixture is

directly e

o to three

parameters. Basic parameters of these equations are the critical properties and the normal

boiling point or vapor pressure. For mixtures, the interaction coefficients between

ents should be included to account for highest accuracy. There are many

quations view

f several practical e

One of the first and simplest, perhaps the best known equation of state model

is the ideal gas law,

(2.35)

This law was derived by assuming that the molecules that make up the gas have

negligible sizes, that their collision with themselves and the wall are perfectly elastic, and

that the molecules have no interactions with each other. It has small applicability to

describe the volum reservoir fluids,

he petroleum industry

has adopted the concept of compressibility factor

An equation of state (EOS) can be defined as an algebraic equation that can

describe the relationship between pressure, temperature and volume for both a pure

ponent and a

given by the equation of stat . There are many families of EOS. The van der

Waals family is characterized by simple cubic form, and m st have two

constitu

e of state used for calculating vapor-liquid equilibrium, Reid et al 91 give re

o quations for chemical and petroleum industries. 52,51,12,11,10

RTPV =

etric phase behavior of petroleum because this law is

only valid for substances at low pressures and high temperatures. T

Z , or gas deviation factor for

45

describing the behavior of mixtures or gases at moderate high pressures. The

pressibility facto

(2.36)

and, by

com r Z is a correction factor for the ideal gas law, that is

ZRTPV =

definition

RTPVZ = (2.37)

itations in the use of Equation 2.34 to describe the behavior of natural

the earliest attempts to represent the behavior of real gases by an

equation was that of van der Waals equation of state mode.

Since the proposal of van der Waals equation (1873) , several

investigators have proposed many equations of state for representation of fluid

volumetric, thermodynamics, and phase equilibrium behavior. These equations, many of

them a modification of the van der Waals EOS, range in complexity from simple

equations containing two or three constants to complicated form having more than thirty

onstants. Even thoug

simplicity found in van der Waals cubic model while improving the accuracy through

modifications.

aals equation of state, is the first equation capable of representing

vapor-liq

The lim

gases gave the chance of

116

64,27,18,5

c h with this large number of EOS, not so many are considered by

engineers and researchers. Because of its computational problem, many prefer the

The van der W

uid coexistence

2Va

bVRTP −−

= (2.38)

46

The parameters a and b are constants that characterize the molecular properties

of Equation 2.36 represents

e i

of the substance in question. The first term on the right side

bVRT−

), and the other term ( 2Vathe repulsiv nteraction force ( ) is the attractive forces

rmined mathematically using Equation

e

Redlich-Kwong, Soave-Redlich-Kwong, and Peng-Robinson. Redlich-Kwong

n der

a temperature-dependent term. The R-K EOS has the form

between molecules. The parameters a and b can be obtained from the critical properties

of the fluid. Also, these parameters can be dete

2.1. In the development of cubic equations of state, modifications in the evaluation of the

param ter a in the attractive term by Soave is considered the most accurate results in

86 106 83 87

equation is the most important and successful model for the modification of the va

Waals equation of state.

Redlich-Kwong (1949) replaces the attractive term of the van der Waals EOS

with

5.0)(VVb + TbVP −

−= (2.39)

the Equation 2.37 constant parameters and usually

xpressed as

aRT

For pure components, a b

e

c

cTRa

2

Ω= a P

5.2

(2.40)

c

cb

RTb Ω=

P (2.41)

where and are dimensionless parameters with the following computed values:

aΩ bΩ

4278.0=Ωa and .0867.0=Ωb

47

Soave modified the R-K EOS and published the Soave-Redlich-Kwong (SRK) , 94

equation of state

)( bVV

aBV

RTP+

−−

=α (2.42)

nsionless factor αwhere the dime is a function of temperature:

(2.43)

2.41, is the slope and is the reduced temperature.

e slope, against the acentric factor,

[ ]25.0 )1(1 rTm −+=α

m rTIn the Equation

Soave correlated th ,m ,ω by the generalized form

(2.44)

of state, the constant parameters

are determined at the critical conditions in the form

2176.057.1480.0 ωω −+=m

a As the case in the Redlich-Kwong equation

band

cP c

aTR

a Ω= (2.45)

22

c

cb P

RTb Ω= (2.46)

where the valu less pure component parameters es of dimension aΩ and do not

hange as shown in Equations 2.43 and 2.44 due to the introduction of function

Pen oduced an improved Redlich-Kwong EOS capable of

redicting the liquid volumes and a critical compressibility factor of Their

equation is given by

bΩ

c ).(Ta

g and Robinson 83 intr

p .307.0=cZ

)()( bVbbVVa

bV −RTP

−++−=

α (2.47)

48

with

c

cc TRT 222

=c

a PPR

a 4724.0Ω= (2.48)

and

ccb P

RTP

RTb 07780.0=Ω=

Peng and Robinson adopted Soave’s- Redlich-Kwong approach for computin

cc (2.49)

g α

as shown in Equation 2.45, where they used ω as the correlating parameter for the slo

,m as given by

32 001667.01644.048503.1379642.0 ωωω +−+=m

Usdin and McAuliffe (UM),101 proposed a new parameter ,d to replace b in the

pe,

second term (attractive term) of the Soave-Redlich-Kwong106 equation:

)( dVVa

bVRTP

+−

−=

α (2.50)

where

c

ca P

TRa Ω= (2.51

22

)

c

cb P

b Ω=

RT(2.52)

c

cd P

RTd Ω=

ued that the two terms of the SRK equation are interconnected by

parameter concluding that all substances possess a value of critical compressibility

(2.53)

They arg

,b

49

factor .333.0=cZ They stated that severing the tie created by the shaping of the

parameter b and replacing it with d wo ld cause accurate liquu id density.

UM proved the dimensionle parameters ,, bass ΩΩ and Ωd can be related to the

critical compressibility parameter, by the following expression: ,cZ

[ ] 0))(38(3)(12)16(3 +Ω cd Z 222 =−+Ω−+Ω− ccdccd ZZZZ (2.54)

Ω cc Z (2.55)

,13 −+Ω=b

and

ba

ZΩ

3)(

Usdin and McAuliffe also adopted Soave’s formulation of

c=Ω (2.56)

α as Equation 2.45

with

[ ] ),7.0(02.0)35.0(67713.0516.448049.0 −−−++= rc TZm ωω

for 7.0≤rT

(2.57)

and

[ ] 23 )7.0(78662.0)(7846.37516.44049.0 −+++= rcc TZZm ωω , (2.58)

for 0.17.0 ≤≤ rT

Patel and Teja 81 introduced the following form of EOS:

)()( bVcbVVaRTP −=

bV −++−α (2.59)

where

50

cP

caa Ω= (2.60) TR 22

c

cb

RTb Ω=

P (2.61)

c

cc P

RTΩ= c (2.62)

where

cc ζ31−=Ω (2.63)

(2.64) )31()21(33 22cbbcca ζζζ −+Ω+Ω−+=Ω

c

ccc RT

VP=ζ (2.65)

is the smallest positive root of the cubic expression:

=cζ (2.66)

n by

.

and bΩ

03)32( 223 −Ω+Ω−+Ω bcbcb ζζ 3

A value of bΩ is give

−=Ω cb 0225.03243.0 ζ (2.67)

Finally, α is d termined by the following:

(2.68)

parameter constants

e

[ ]25.0 )1(1 rTF −+=α

The Patel-Teja equation of state therefore requires four

,,, ccc PT ζ and F for any

There are many other equations of state models with modifications of the

with

fluid desired.

attractive term, repulsive term, and combination of both of the van der Waals EOS

51

more th e param e

2.2.

a aals Eq

an two or thre eters. Some of these are summarized in Table 2.1 and Tabl

Table 2.1. Modifications to the Attractive Term of v n der W uation of State.117

Equation Attractive Term

Redlich-Kwong (RK, 1949)

)(5.1 bVRTa

+

Soave (SRK, 1972)

Peng-R

Kubic (1982)

Patel-T

)()(Ta

bVRT +

[ ])()()(

bVbbVVRTVTa

−++

obinson (PR, 1976)

Fuller, (1970)

Heyen, 1980-Sandler, (1994)

Schmidt-Knapp (1980)

)()(

cbVRTVTa+

[ ]cTbVcTbVRTVTa

)())(()(

−++ 2

)()(

22

VTa wbubVVRT ++

2)()(

cVRTVTa+

[ ]eja (PT)(1982)

Yu and Lu (1987)

Lawal-Lake-Silberberg (LLS)(1985)

)( VTa)()( bVcbVVRT −++

[ ]

)3()()(

cVbVVRTVa

+++

cT

22 bbVV βα ++ )(Ta

52

Table 2.2. Modifications to the Repulsive Term of the van der Waals EOS.117

Equation Repulsive Term

Reiss et al (1959)

Thiele (1963)

Guggenheim (1965)

Carnahan-Starling (1969)

3

2

)1(1

ηηη

−++

3

21 ηη ++ )1( η−

4)1( η− 1

3

32

)1(1

ηηηη

−−++

53

CHAPTER III

CLOSED-FORM VAN DER WAALS EXPRESSIONS

This chapter presents the derived equations, which form the basis of critical point

calculations. These include the van der Waals equations of state theory, the parameters

that characterize the individual components in the Lawal-Lake-Silberberg (LLS)

uids. Using the LLS equation of state as a

l resented. This is

plished by introducing the “VDW closure parameters” (that is, parameters

generalized cubic equation of state, and the algorithm developed for computing the

critical properties of petroleum reservoir fl

basis, a c osed-form solution for the van der Waals critical point is p

βα , accom

developed for the purpose of resolving critical point in fluids) into the original van der

Waals equation of state.

3.1 Van der Waals Equations of state Theory

The ideal gas law, nRTPV = , can be derived by assuming that the molecules

that make up the gas have negligible sizes, that their collision with themselves and the

wall of the vessel are perfectly elastic, and that the molecules have no interactions with

ach other.

An early attempt, to take these intermolecular forces into account was that of Van

aals (1873), who proposed that the idea

e

der W l gas equation of state be replaced by

RTbVaP MM

=−⎟⎟⎞

⎜⎜⎛

+ )(2 (3.1) V ⎠⎝

54

This equation differs from the ideal gas equation by the addition of the term 2Va to

pressure and the subtraction of the constant to from molar volume.

Here the par

V b

ameters a and b are constants particular to a given gas, where R is

the universal gas constant. The term 2Va represents an attempt to correct pressure for

forces of attraction between the molecules. The actual pressure exerted on the wall o

the

f the

vessel by real gas is ount less, by the am 2Va , than the pressure exerted by an ideal gas.

The para the size of each

se t th the

volume that the molecules have to move around in is not just the volume of the container

, but is reduced to . The parameter has a more difficult meaning and is

ted to the

intermolecular attractive forces is to r duce the pressure for a given volum

temperatu

meter b (or the so called co-volume parameter) is related to

molecule and repre n s e intermolecular repulsive forces in the sense that it is

V )( nbV − a

rela intermolecular attractive force between the molecules. The net effect of

e e and

re. When the density of the gas is low (i.e., when Vn is small and is small

compared to ) the Van der Waals equation reduces to that of the ideal gas law. The

e

nb

V

a and b parameters can b obtained from the critical properties of the fluid.

On the basis of the available volume ( bV − ), Van der Waals was able to

that Equation 3.1 is appropriate for the hard-sphere gas at low density. To see that this

leads to a pressure reduction, simply solve for P:

show

55

2VaRT

−= 3.2

Since a > 0 , and b << V , then will be a reduction in pressure by approximately

bVP

− ( )

2V.

It is clear that the Van der Waals equation predicts a deviation from ideal

behavior. This deviation can be analyzed

a

by defining a quantity , called the

ompre

Z

c ssibility factor, as

RTPVZ = (3.3)

For an ideal gas, it is clear that 1=Z .

To derive Z , start with Equation 3.2, then multiply both sides by V and divid

:RT

e by

RTVa

bVV

RTPVZ −

−==

or

(3.4)

RTV

a1

Vb

Z −−

=1

(3.5)

For very low density 1<<Vb , so we can e a Tayl eries to ap ate us or s proxim

Vb

−1. In

general, for

1

1<<X

XX

≈−

+1 1

1 (3.6)

56

Using this approximation, the compressibility factor becomes

RTVabZ −+= 1

V (3.7)

or

VRT

b )(1

−+=

a

Z (3.8)

The quantity

RTab −

ideal behavior

is an observable and calculable quantity, which measures

deviation from of a gas. Note that if 0>>−RTab , then a

RT

pressure is larger than ideal gas pressure. However, the condition

b > and the

RTb > tells that a

excluded volume effects, as measured by the constant b , so an increase in pressure is

what we would have predicted. On the other hand: if 0<−ab , then

RTab < , then

RT

the pressure is less than the ideal gas pressure.

Real gases can hav

ill lead to a low ring of

ideality, where as strongly repulsive forces can lead to a positive deviation. What do the

isotherm of the Van der Waals equation look like? Recall that the isotherms are curves

e both positive and negative deviations from ideal behavior,

depending on the pressure and temperature and the particular system. Strongly attractive

forces w e the pressure and hence a negative deviation from

corresponding to P vs. V at various fixed temperatures. For the Van der Waals

equation, some of the isotherms are shown in Figure 3.1.

57

Figure 3.1. Pressure-Volume Diagram for Pure Component.

For the isotherms where cTT > appear similar to those of an ideal gas, i.e., there

is a monotonic decrease of pres with increasing volume. The isotherm

where the curve is essentially horizontal (flat) with no curvature. At this point, there is

sure c

exhibits an unusual feature not present in any of the ideal gas isotherms – a small region

no change in pressure as the volume changes. Below this isotherm, the Van der Waals

starts to exhibit unphysical behavior. The cTT

TT =

< isotherm has a region where the

pressure decreases with decreasing volume, behavior that is not expected on physical

atic jum

that a phase transition has occurred, in this case, a change from a gaseous to a liquid state.

The isotherm just represents a boundary between those isotherms along which no

such phase transition occurs and those that exhibit phase transitions in the form of

grounds. What is observed experimentally, in fact, is that at a certain pressure, there is a

dramatic discontinuous change in the volume. This dram p in volume signifies

cTT =

58

d For this reason, the cTTiscontinuous changes in the volume. = isotherm is called the

critical isotherm, and the point at which the isotherm is flat the slope of the curve is zero,

se two

tuations correspond to zero values of the f

espect to

At the critical point, t

liquid, but has characteristics of both. Hence, it is called a supercritical fluid. To see at

is observed, we use

the fact that at the critical point, the isotherm is both horizontal (zero slope) and has no

curvature. These two conditions (i.e., criteria of criticalit r a pure component are:

and the isotherm has zero curvature at the critical point. Mathematically the

si irst and second derivatives of pressure with

r volume.

he system does not exist in a particular state, either gas or

what temperature, pressure and volume, this supercritical behavior

y) fo

0=⎟⎠∂ cpV

⎞⎜⎝⎛ ∂P (3.9)

02

2

=⎟⎟⎠

⎞⎜⎜⎝

⎛∂∂

cpVP (3.10)

S r 1 mole olving Van der Waals’ equation, Equation 3.1 for pressure yield fo

2Va (3.11)

bVRTP −−

=

To estimate , and a , b R at the critical point, it is necessary to obtain the first

and second derivatives with respect to volume of Equation B.1 in Appendix B and set

them equal to zero. The first of these conditions leads to

0)(3⎜

⎝ −−=⎟

⎠⎜⎝ ∂ ccp VVV

22 =⎟⎟⎠

⎞⎜⎛⎞⎛ ∂

cpc

c

bRTaP (3.12)

r o

59

2

RTc

− (3.13) 3 )(

2bVV

a

cc

=

The second condition leads to

0)(

26342

2 ⎞⎛ ∂ RTaP=

−−=⎟⎟

⎠⎜⎜⎝ ∂ bVV

c

cp

(3.14)

or

V cc

34 )(26

bVRT

Va

c

c

c −= (3.15)

Consider the Van der Waals equation of state at the critical state

ccc

c RTbVV

P =−⎜⎜⎝

+ )(2 (3.16)

The three equations Equation 3.13, throu

a⎟⎟⎠

⎞⎛

gh Equation 3.16 apply at the critical

point and by combination results in

c

c

Pa

64= TR27 22

(3.17)

nd a

c

c

Pb

8=

RT (3.18)

These equations for the c

inflection at the critical point.

onstant parameters will only work for pure component at

the critical point. For mixtures, the two equations, Equation 3.9 and Equation 3.10 do not

hold. The pressure-volume diagram for the mixtures does not exhibit horizontal

60

3.2 Closed-Form Equations for Fluid Critical Point

Another development of van der Waals modification is the single component

rm of t

this equa

fo he Lawal-Lake-Silberberg (LLS) equation of state. This generalized cubic

equation is used in this research project is presented in Equation 3.19. Related work to

tion of state has been given in the literature. 62 The LLS equation has the form

22 bbVVa

bVRTP

βα −+−=

− (3.19)

here parameters ,a ,b α and β are established for pure component as follows w

[ ]c

c

PTR 22

3cZa )1(1 −Ω+= ω : (3.20)

cc P

Zb )( ωΩ= cRT (3.21)

c

cc ZZω −Ω+=

31

Zω

αΩ

(3.22)

c

ccc

ZZ )3 ωΩ−ZZ

2

232 1(2)1(

ω

ωωβΩ

+Ω+−Ω= (3.23)

ωω 0274.01361.0

+=Ω (3.24)

where is the pressure, is the temperature, is the molar volume, , is the critical

compressibility factor and are the critical pressure and temperature of the

ponents,

P T V Z

, cP cT

c

R is the universal gas constant. The com Ω is a constant equals to 0.325, and ω

u work for the LLS EOS are psia for pressure, a degree Rankin, nits used in this Ro for

temperature, cu-ft/1b-mole for molar volume, and the constant R = 10.73 psia-cu-ft/1b-

mole The experimental data are reported in psia of pressure, mol fractions, and F, .Ro o

61

therefore, the data had to be converted to the units used in this work. The composition

and critical points obtained by the researchers, Sim ugh,98 Etter and Kay,29

and Zais for 85 hydrocarbon/non-hydrocarbon reservoir fluid mixtur

on-Yarboro

es are shown in

Tables

d en n m

C 1 through C 27 in Appendix C.

The Equation 3.19 has four in ep de t para eters ( ,a ,b ,α and β ) and i

generalized form and cubic in terms of molar volume. The LLS equation, Equati

can be viewed as a gene

t is a

on 3.19,

ralized cubic equation of state from

equatio s ed” cubic equations of

state is used when the equation can be reduced or m of any of the

known cubic equations of state by assigning specific integer to the parameters

which many known

n of state can be derived. Moreover, “the term generaliz

odified to the form

α and β .

It is interesting to find out that when 0== βα and 83

=Z the generalized form of LLS c

equation of state reduces to van der W quation of state. Similarly for the PR EOS

r

aals e

fo .1,2 == βα The resulting p

Table 3.1. Parameters of Selected Equations of State. E Ωc

arameters from the application of the criticality

conditions to various forms of cubic EOS are presented in Table 3.1. 75

quations of State Year Zc Ωa Ωb

v 0.333 an der Waals 1873 0.375 0.4218 0.1250

Dieterici 1896 0.271 0.6461 0.1355 0.500

Berthelot 1900 0.281 0.5365 0.936 0.333

Redlich-Kwong 1949 0.333 0.4275 0.0866 0.260

Peng-Robinson 1976 0.307 0.4572 0.0778 0.253 Harmens 1979 0.286 0.4831 0.0706 0.247

62

For example, when 0=α and 0=β in the LLS equation of state, the equation

reduced to the van der Waals equation of state previously shown in Equation 3.1.

Applying the criticality conditions to van der Waals equation of state yield the following

expressions: 98

02)( 32 =+

−=⎟

⎠⎞

⎜⎝⎛∂∂ c

Va

bVRT

VP (3−

ccTc

.25)

06)( 432

2

−⎟⎠

⎜⎝ ∂ ccT VbVV

c

Solving simultaneously Equations 3.25 and 3.26 for a and b parameters, yield

2=−=⎟

⎞⎜⎛ ∂ c aRTP (3.26)

3cV

b = (3.27)

89 ccVRT

a = (3.28)

Combining Equations 3.27and 3.28 with van der Waals Equation 3.1, a universal

compressibility factor is obtained and calculating the parameters and

from the following equations:

, 375.0=Z a b

c

c

Pa 4218.0= (3.29TR 22

)

cP

where 4218.=Ωa and 125.0

cb 125.0= (3.30) RT

0 =Ωb are constants at the critical point from the van der

Waals original equation. The parameter ωΩ is determined by dividing the parameter bΩ

by the critical comp . In the case of van der Waals, ressibility factor, cZ

63

375.0/125.0 ==Ωω .333.0 Solving Equation 3.28 and Equation 3.29 for the critical

properties for the original van der Waals and for those classical equations of state listed

in Table 3.1, the results are shown in Table 3.2.

Table 3.2. Relationship of EOS constants with Critical Parameters.

2m

mpc b

aη=

m

mTc Rb

aT

η= mvc bV η= P

EOS Year pη Tη vη

van ader W als 1973 1/27 8/27 3.0

Dietrich 1896 1/29.56 16.61/27 2.0

Berthelot 1900 1/27.04 10.67/27 3.0

Redlich-Kwong 1949 1/58.8 5.48/27 3.84

Peng-Robinson 1976 1/62.6 5.56/27 3.94

Harmens 1977 1/97.1 3.95/27 4.05

The criticality constraints for deriving the equations to predict the critical

properties for pure components or mixtures in thi s context of document are the

compressibility and vo

properties equations is presented in Appendix B. In this appendix, two methods for

deriving critical properties expressions are shown in terms of critical compressibility-

factor, , and critical volume, to the corresponding

lume forms of criticality conditions. The derivation of the critical

cZ cB 0)( =− cZZ .

The critical compressibility- factor form at critical condition is derived by the

expansion of the LLS cubic equation of state in term of and by comparison with the cZ

64

expansion of .0)( 3 =− cZZ This procedure is shown in A endix B and the resulting

cubic equation as Equation 3.30 where ),(

pp

βαfZc = .

3 =+θθcZ (3.31)

where,

(3.32)

θ (3.34

The critical volume form of the criticality condition is also determined by the

expansion of the LLS equation of state in terms of molar volume ( and by comparison

of the . This procedure is presented in Appendix B and the resulting cubic

equation is shown as Equation 3.36

(3.36)

where

)

) (3.38)

04322

1 ++ θθ cc ZZ

)6128( 321 αααθ +++=

)9912123( 22 βαβααθ −+++−= (3.33)

)6663( 2 βαβαα −++= )

)( 2 (3.35)

3

4 βααβθ −+−=

)cB

0)( 3 =− cZZ

04322

13 =+++ θθθθ ccc BBB

6128( 321 αααθ +++= (3.37)

3271515( 22 αβαθ −−+=

)36(3 αθ += (3.39)

14 −=θ (3.40)

65

The equations derived to pre ic , ,cP critical tempera urd t the critical pressure t e,

an

,cT d critical volume, cV pure components and mixtures are presented with the

following dimensionless parameter:

22TRaPA = (3.41)

RTbPB = (3.42)

RTPVZ = (3.43)

Applying Equation 3.19 at the critical Point we have

c

ccc RT

VPZ = (3.44)

22c

cc TR

aPA = (3.45)

c

cc RT

bPB = (3.46)

Where subscript c denotes the gas-liquid critical state. Solving for single component

critical pressure, critical Temperature, and critical volume, from Equations

3.43, 3.44, and 3.45, give the following expressions:

,cP ,cT cV

bRTB

P ccc = (3.47)

Solving for from Equation 3.44, obtain

cP

aTRA

P ccc

22

= (3.48)

66

Equating Equation 3.46 and Equation 3.47 yield

c

cc bRA

aBT = (3.49)

nd substituting Equation 3.48 into Equation 3.46 yield

a

c

cc Ab

aBP 2

2

= (3.50)

and expressions previously obtained in Equation 3.48 and

quation 3.49 in Equation 3.43, the critical volume equation is obtained as:

Replacing cP cT

E

c

cc B

bZV = (3.51)

The Equations 3.48, 3.49, and 3.50 can be expressed in terms of ,cA ,cB ,cZ ,a

, ,α and .β In the derivation of the criticality expressions in Appendix B Equation

.48, the parameter has been presented in term of

b

cA ,cB ,cZ α and .β B

ccccc BBBZA αβα +++= 2223 (3.52)

By substitu in the equations for and , the critical property

s of

ting cA ,cP cT cV

,cB ,cZ α and .β equations are expressed in term Then, Equations 3.48, 3.49 and

3.50 can be rewritten in the form

)( cccc

cc BBBZb

aBP

ααβ +++= 2222

2

3 (3.53)

)( cccc

cc BBBZbR

aBT

ααβ ++= 2223

(3.54)

67

c

cc B

bZV = (3.55)

r o

( )bbbcc Zb

PΩ+Ω+Ω+

Ω=

ααβ 2222

2

3 (

( )

ba3.56)

bbbc

c ZbRa

Ω+Ω+Ω+Ω

ααβ 2223

bT = (3.57)

b

cbZV

Ω= (3.58)

Where, bΩ is a constant parameter of the van der Waals equation of state. These

equations Equation 3.56 through Equation 3.58 to calculate ,cP ,cT and cV are practical

and directly obtained once the composition and the pure components are given.

For the mixtures, the express

c

ions for determining the critical properties are

expressed in terms of the parameters ,m ,m ,ma b α and mβ . These mixture parameters

require the LLS mixing rules to establish the following equations for the critical

properties for mixture:

( )cmcmcmcm

cmc

BaP = 222

2

(3.5

( )

BBBZb ααβ +++239)

cmcmcmcm

cmc BBBZRb

BaT

ααβ +++= 2223

(3.60)

c

mcc B

V = (3.61

By LLS mixing rules,

bZ)

68

ij

n

i

n

j

aaaxxa 21

21

∑∑=

3

1

3/1 ⎟⎠

⎞⎜⎝

⎛= ∑

ijim (3.62)

=

(3.63) n

i

bxb

iim

ijjijim xx αααα ∑∑= 21

21

(3.64)

n

i

n

j= =1 1

ijjij

n

i

n

jim xx ββββ 2

121

1 1∑∑= =

= (3.65)

The prediction of c tures can now be

achieve

critical temperature, and critical volume for multi-component systems.

The algorithm constructed for calculating critical properties for reservoir fluids

(gases, gas condensate, volatile oils, and crude oils) are discussed next.

ritical properties for hydrocarbon mix

d since all the necessary equations have been developed. In this project, the

iterative methods have been utilized to match the experimental critical pressure, ,Pc

cT cV

3.3 Closed-Form Critical Property Computation Methods

The type of data that are often available from laboratory work on reservoir fluid

samples for pure components are critical pressure, cP , cT , acentric factor, ω , Criti

compressibility factor, cZ , critical volume, cV , and molecular weight. Additional

information that may be available is the analysis of the equilibrium liquid and gas. The

data will permit critical properties to be calculated directly at reservoir conditions of

pressure and temperature. Usually, however, laboratory critical point values are

cal

se

69

furnished, and the critical volume of the reservoir fluids will be the most valuable d

available for reservoir study purposes.

ata

properties for pure compo losed-form Van der Waals

method. However, the fo owing procedure (algorithm) is included to show the

calculation procedure foll in program:

The following is t e the critical point of pure

components given the com

Computer program provides speed and accuracy in predicting the critical

nents and complex mixtures by the c

ll

owed by flowcharts of the ma

he step-by-step procedure to calculat

ponents measured ω,, cc PT .

Step1. From the single co pure components parameters

a, b,

mponent critical data, calculate the

α , and β using Equ 3.22, and Equation3.22,

Step 2. Calculate the dim using Equation 3.66.

ation 3.20, Equation 3.21, Equation

ensionless critical volume, Bc

c

cc RT

bPB = (3.66)

Step 3. Calculate the Dim eter fensionless Param c in terms oA ,,, αcc ZB and β .

ZA α ++= 2 (3 (3.67)

Step 4. Calculate the crit ture, , and the critical

using Equation 3.53, E

Step 5. Once the pure components parameters are calculated, the mixture parameters for

cBB αβ +2) ccc

ical pressure, P , the critical temperac cT

cV quation 3.54, and 3.55.

mmm ba α,, , and mβ are calculated using Equation 3.62, Equation 3.63, Equation 3.64,

and Equation 3.65.

70

Step 6. With the calculated values of mα and mβ , the coefficients of Equations 3.64 and

3.65 are calculated.

Step 7. Calculate the dimensionless critical volum for the mixture using Equation

3.68. The solution of this cubic equation consists of two imaginary roots and one real

root. The real root is chosen as the value for

e, cB

cB : ),( mmc fB βα= .

=++ θθθ ccc BB .68)

where

(3 += mmmαθ (3.69)

2 −+−−= mmm βααθ (3.70)

0012

23

3θ B + (3

)8+α 6 23 + α 12

)595(3 2

)2(31 += mαθ (3.71)

10 −=θ (3.72)

Step 8. Solve the cubic Equation 3.73 of the mixtures for cZ . The solution has two

imaginary roots and one real root. The real root is chosen as the value for cZ :

),( mmc fZ βα= ,

023 =++ θθθ ZZZ (3.73

where

)8126( 23

0123 +θccc )

(3.74)

(3.75)

3 +++= mmm αααθ

)33441(3 22 mmmmm βαβααθ −+++−=

71

6)

)

xtures using the equation

Bααβ +++= 2223 (3.78)

Step 10. Calculate the al t ica ssu cri

t

)222(3 21 mmmmm αβαβαθ +−+= (3.7

)( 20 mmmm αββαθ −−= (3.77

Step 9. Determine the constant cA for mi

mcmcc BBZA cmc

critic emperature, , critcT cP ,l pre re, and tical

volume, cV for the mix ures:

cBmcB +2mc α+2

m Bβ+cZT

α23(

mc Rb

cm Ba= (3.79)

)cB3( 22mcm Zb

P

and

22cm

2B

cB α+mc α+m Bβ+c =a

(3.80)

cc B

V mcbZ (3.81)

Step 11. ulate e a e d ons llo

Absolute D D

Calc the averag bsolut eviati as Fo ws:

eviation (A ) = )0 10(exp, ⎥

⎥⎦

⎤

⎢⎢⎣

⎡ −

c

dS 3.82)

Absolute D D

exp,cP, precPP

AB (

eviation (A ) = )0 10(exp,⎢

⎢⎣

⎡ −

c

dS (3.83)

arizes the step-by-step procedure presented above

exp,⎥

c, prec

⎥⎦

⎤TT

TAB

Figure 3.2 is an algorithm summ

to calculate the critical point of petroleum reservoir fluids.

72

Figure 3.2. Algorithm f arameters.

or Computation of Critical P

Input Tc Pc Vc Zc ω

Read parameters a b α β

Call equations of parameters a b α β calc. a b α β for pure components

Read coefficients θ3 θ2 θ1 θ0 and constant Bc

Call cubic e uation and calculate Bq c for pure components

Read coefficients θ3 θ2 θ1 θ0 and constant Zc

Call cubic equation and calculate Zc for pure components

Read constant Ac

Call equation Ac and calculate Ac for ts pure componen

Read Parameters am bm αm βm

Call equations of parameters am b α β and calculate a b α β

m m m m m m

m for mixtures

Read coefficients θ3 θ2 θ1 θ0 and constant Bc for mixtures

Call cubic equation and calculate Bc for mixtures

Read coefficients θ3 θ2 θ1 θ0 and cconstant Z for mixtures

Call cubic equation and calculate Zc for mixtures

Read constant Ac for mixtures

Call equations of Tc Pc Vc and calculate for Tc Pc Vc for pure components

Call equation Ac and calculate mixtures Ac for

Read critical properties Tc Pc Vc

Calcal

l equations of Tc Pc Vc culate for Tc Pc Vc for

mixtures

End

Start

73

CHAPTER IV

CRITICAL PROPERTIES FOR RESERVOIR FLUIDS

The critical point calculations using the modified Lawal-Lake-Silberberg (LLS)

equation of state introduced in Chapter 3 were carried for each pure component and

complex mixtures such as alkanes +, heptanes plus +, nitrogen and carbon dioxide. The

general behavior of the experimental parameters, and calculated critical properties for

these mixtures are presented in Tables C 1-C 27 and Figures C 1-C 19 in Appendix C. In

general, qualitative results were obtained for the critical pressure and critical temperature.

A detailed comparison and discussion of the results of each of the calculated critical

properties with respect to the corresponding experimental results, and with respect to the

other correlation predictions of Simon and Yarborough, Terry and Kats, and Zais are

presented in the following sections:

4.1 Critical Pressure Data for Complex Hydrocarbon Mixtures

In appendix C, tables and cross-plots are displayed for complex mixtures with th

results of critical pressures and critical temperatures. Each of the ta

e

bles corresponds to 9

omplex mixture. The predicted critical properties and acentric

ctor for C7 + of complex mixtures are also presented. The methane concentrations in

these

and 10 complex mixtures and each mixture is subdivided into different compositions.

Each table has the composition in mole fraction, the experimental values of the mixture

critical pressure, temperature, and the calculated values of critical pressure, critical

temperature for each c

fa

mixtures varied from 19 to 96.6 mol percent. The intermediate hydrocarbon

74

groups, consisting of ethane, propane, and bu a low of 3 to 59 mol

t, a e con ation pent nd he frac from 3 m ercen

e f he lu va m 2 to 14 mo nt on

ocarb xtu nsis trog d ca iox he tra no

car xtu ied to l percent. T sic ert

of the components fraction covered the range from light, paraffin to heavy, aromatic, and

asphalt.

4.2 Calculation of Critical Properties

tane varied from

percen nd th centr s of ane a xane tions 1 to ole p t,

and th raction ptane p s (C7+) ried fro l perce . The n -

hydr on mi res co t of ni en an rbon d ide. T concen tion of n-

hydro bon mi res var from less than 1 22 mo he phy al prop ies

In order to clarify the procedure to be followed when applying the methodology

of predicting the critical properties to an actual problem, an illustrative example of

calculation performed using the algorithm presented in Section 3.3 is given here.

To predict the critical pressure and critical temperature of a naturally occurring

mixture, it is given the following information for mixture 1 of the given data in Table 4.1.

As a st , it is

desirab

obtained from the previously determ and Lawal,

and the critical compressibility required as additional input for the equation of state

used in this work is obtained from Rowlinson correlation of ACS Symposium Series,

316, 1977. In this project, the critical properties of the fractio mplex

ep one, to compute the critical properties for complex mixtures

le first to determine these properties for the +7C fraction in the mixture. In

complex mixtures, components heavier than the n-heptane have been summed into a +7C

fraction. These fractions required for the parameters of equations of state are

59 62

+7C

ined correlations TR-4-99 Lawal,

cZ

+7C ns in the co

75

mixtures are calculated on the basis of the reported molecular weight of the fraction

. The empirical expressions shown in the Table 4.1 are used to estimate the

critical properties for the hydrocarbon fractions.

mple of Experimental Data Used for Calculations of mixture 145-1.

Hydrocarbon Mixture Composition

)(7+cMW

Table 4.1. A Sa

Component Mole Fraction Nitrogen 0.001 Carbon Dioxide 0.004 Methane 0.193 Ethane 0.032 Propane 0.585 i-Butane 0.007 n-Butane 0.012 i-Pentane 0.005 n-Pentane 0.007 Hexane 0.013 Heptane + 0.141 Heptanes + Properties: Mo. Wt. 243 Characterization Factor 11.6 Critical Temp., )(, RTc

o 725 Critical Temp. This work 725.39 Critical Pressure, )( psiaPc 2100 Predicted )( psiaPc : (S. Yarborough) 2002 (Etter and Kay) 2911 (Zais) 2175 This Work 2101

76

+7Table 4.2. Physical properties of Fractions Correlation.C 62

9.56084+=

MWAPI

5.1315.141

+=

APIS g 21 e

ge

obp SMWeT =

321 eee TSMWeC = 4322 eeee CTSMWeP = 4321 eeee CTSMWeT =

eeee

bpgo bpgc bpgc

bpg CTSMWe=ω

0 0

43210

Parameters 0e 1e 2e 3e 4e

bpT 108.701661 0.4224480 0.42682558 0.0000 0.0000

C 0.83282122 0.09255911 -0.0413045 0.12621158 0.0000

cP 237031780 -0.028484 2.755309 -1.374440 -2.947221

cT 6.206640 -0.059607 0.224357 0.968332 -0.802538

ω 1.5790E-13 -1.453063 -2.811708 4.883921 2.109476

The critical properties and acentric factors are estimated from the correlation

shown in Table 4.2. Bu using the given Mw of the as basis, the critical pressure,

as a lumped single pseudo-com onent for all

the pre

+7C

+7

critical temperature, critical compressibility factor, and acentric factor of +7C were

obtained. The heptane-plus +7C is treated

C

p

diction results shown in this work. Table 4.2 displays some of the predicted

critical properties for the heptane-plus fraction.

From the given single component critical data, the pure component parameters

77

,,, αba ,β ωΩ , the dimensionless critical parameters cA and cB are calculated fro

Equations 3.20, 3.18, 3.21, 3.22, 3.23, 3.24, 3.

m

67, and 3.66 respectively. The resulting

parameters are presented in Table 4.4.

Table 4.3. Calculated Critical Data of Heptane-Plus Fraction for Data Set 1.

Heptanes + Properties Mol. Wt. 243 243 243 243 191 191 191 191 191

Gravity(API) 33.31 33.31 33.31 33.31 39.98 39.98 39.98 39.98 39.98SG 0.86 0.86 0.86 0.86 0.83 0.83 0.83 0.83 0.83

Characterization Factor 11.6 11.6 11.6 11.6 11.9 11.9 11.9 11.9 11.9 Tb 1037.0 1037.0 1037.0 1037.0 920.9 920.9 920.9 920.9 920.9C 3.3475 3.3475 3.3475 3.3475 2304 3.2304 3.2304 3.2304 3.23043.

Pc, (psia) 271.0 271.0 271.0 271.0 319.8 319.8 319.8 319.8 319.8T , (c

oR) 1364.4 1364.4 1364.4 1364.4 1258.4 1258.4 1258.4 1258.4 1258.4ω 0.5673 0.5673 0.5673 0.5673 0.4676 0.4676 0.4676 0.4676 0.4676Z 0.2416 0.2416 0.2416 0.2416 0.2493 0.2493 0.2493 0.2493 0.2493c

Ωw 0.3555 0.3555 0.3555 0.3555 0.3564 0.3564 0.3564 0.3564 0.3564

Component

Table 4.4. Calculated Results of Pure Component Parameters.

a b α ωΩ cA cB β N2 6506.50 0.520 2.21 5.32 0.361 0.569 0.105

CO2 18107.70 0.540 2.81 6.02 0.360 0.587 0.098 C1 11111.20 0.567 2.42 5.56 0.361 0.576 0.103 C2 26880.60 0.854 2.43 5.58 0.360 0.576 0.102 C3 46045.90 1.140 2.75 5.96 0.355 0.583 0.098

i-C4 64482.90 1.513 2.49 5.66 0.359 0.577 0.102 n-C 109225.70 1.850 2.81 6.02 0.359 0.587 0.983 4

i-C5 90386.27 1.770 2.86 6.07 0.358 0.588 0.978 n-C5 139353.34 1.786 3.02 6.26 0.358 0.593 0.096 C6 130104.33 2.013 3.77 7.11 0.358 0.614 0.090 C7+ 475160.996 4.614 4.222 7.630 0.3535 0.6222 0.085

Once the pure components parameters are calculated, the mixture parameters for

mmm ba α,, and mβ can be determined by using Equations 3.62 through Equation 3.65.

78

Since equations of state are developed onents, the use of mixing rules is

necessary to make pr ures. Another

important parameter in every geometric and adratic mixing rule is the interaction

arameter. This binary interaction term is empirical and does not have any theoretical

basis but it is necessary in mixing rules application. A mixing rule is an algebraic

expression that relates the pure components’ parameters to the mixture composition when

the mixture parameter is established. In this work, the binary interaction terms,

for pure comp

oper application of the equations of state for mixt

qu

p

,, ijija α

and ijβ were assumed as constants, and assigned a value of 1. Then, using the Equation

3.52 the constant was estimated. The Equations 3.59 through 3.61, for calculating the

crit

cA

ical properties are expressed in terms of the mixture parameters, ,,, mmm ba α and mβ

after LLS mixing rules (Equations 3.62 through 3.65) are applied. Table 4.4 presents the

results of a sample calculations of a hydrocarbon mixture performed using the algorithm

developed in section 3.3. The prediction results are in agreement with the experimental

values. The absolute percent deviation of the results is in the neighborhood of 1.

Table 4.5. Calculation Results for Mixture Parameters (Mixture 1).

Parameter Mixture 1

ma 31177.9123

mb 0.11728247

mα 99.1126855

mβ 0.69748539

cB 0.269

cZ 0.2985

cA 34.15

79

4.3 Results and Discussion

The variation in behavior of the calculated critical pressure and temperature

agreed

5-4.6

e

ee

tion of state is selected for this work.

ead

thermore, the influence of the values of

interact

ifica y the

prediction of the critical propertie l pressure.

for

ment

iation of the results ranges from

0.03 to 0.13 percent.

both qualitatively and quantitatively with experimental data. Deviation errors in

the predictions of both critical pressure and temperature, as shown in Figures 4.

were in the range of 0.03 to 0.13 percent. In Figure 4.1, the predicted critical pressure

obtained from Simon and Yarborough, Etter and Kay, Zais, and this work using Lawal-

Lake-Silberberg (LLS) is plotted against Critical experimental data. As shown in Figur

4.1, this work shows better agreement with the experimental data than the other thr

correlations. Consequently, the LLS equa

A comparison of the accuracy of predicted critical properties of this work with the

other correlations indicates that the mixing rules used with the LLS equation of state l

to accurate predictions of mixture behavior. Fur

ion parameters on the accuracy of the prediction of critical properties was

adjusted. Adjustment of the interaction parameter associated with the constant b of the

LLS equation of state, had an effect on the critical point calculations. Also, adjustments

of the interaction parameter associated with the constant a , improved sign ntl

s, especially the critica

The results of the predicted critical pressure, ,cP and critical temperature,

complex mixtures are presented in Table 4.6. The prediction results are in agree

with the experimental values. The absolute percent dev

,cT

80

Ta

Properties M 9

ble 4.6. Predicted Critical Pressure, ,cP Critical Temperature, ,cT for Mixtures.

ix. 1 Mix. 2 Mix. 3 Mix. 4 Mix.5 Mix. 6 Mix. 7 Mix. 8 Mix.Tc, exp. 660 660 725 725 725 725 694 660 660 Tc, pred. 725. 13 659.22 39 725.45 724.58 725.81 694.91 694.44 659.17 661.

AD (%) 1 0.118 0.054 0.062 0.058 0.112 0.131 5.218 0.126 0.17Pc, exp. 2100 2500 3400 1920 2420 3430 4355 4295 4630 Pc, pred. 21 .4901.12 2501.55 3398.03 1922.15 1423.18 3432.17 4349.55 4302.33 4624

AD (%) 0.05 9 3 0.062 0.058 0.112 0.131 0.063 0.125 0.171 0.11

hty-five mixture sets of experimental data are analyzed for hydrocarbon

Table C.1 through C. 27 in Appendix C show the prediction results for

critical temperatures. Also, these results are displayed in Figures C. 1-through

Eig

mixtures. critical

pressures,

C. 10 c re in

Appendix

pressures

for ritical pressure, and in Figures C. 11 through C. 19 for critical temperatu

C. Figure 4.1 is a cross-plot shows the prediction results for the critical

for all data.

81

7500

6500ia)

4500

5500

ssu

3500d C

1500

2500Cal

500

8500

500 1500 2500 3500 4500 5500 6500 7500 8500

cula

terit

ical

Pre

re (p

s

Simon-Yarborough

Etter-Kay

Zais

This w ork

Experimental Critical Pressure (psia)

ssure against the experimental critical

e n

sults for the critical temperature are in agreement with the experimental data. Figure

Figure 4.1. Predicted Critical Pressure of Complex Mixtures.

The match of the predicted critical pre

pr ssure gives an absolute deviation between 0.03 and 0.13 %. Similarly, the predictio

re

4.2 is a cross-plot which displays the prediction results for critical temperatures for all

data.

82

800

750

This w ork

350

450

700

Cal

cula

ted

Crit

ical

Tem

ptu

re ( 650o R

)

550

600era

500

400

300300 350 400 450 500 550 600 650 700 750 800

Experimental Critical Temperature (oR)

Figure 4.2. Predicted Critical Temperature of Complex Mixtures.

ental Data

4.4 Comparison Between Calculated and Experim

To test the accuracy of the calculations of critical points of hydrocarbon mixtures,

mparison of calculated with experimental values of the pressure and temperature at a co

e critical points of mixtures of known compositions was made. For this purpose, the

crit

ere determined in the laboratory was employed. A total of about 85 mixtures of non-

hyd nts were studied. The results of these mixtures

are given in Table 4.6. Table 4.6 gives a summary of the results for all mixtures.

th

ical pressure and critical temperature on the hydrocarbon paraffin mixtures which

w

rocarbon and hydrocarbon compone

83

In general, the agreement between the calculated critical properties and the

experimental critical properties is very good. The over all deviation of the calculated

val about 0.03-0.13% of both the pressure and temperature.

res

nd experimental critical pressures for all data points. Also, Figure 4.2 is a cross-plot

com res

The comparison of the results of the critical point predictions using the LLS

equ arborough, Etter-Kay, and Zais empirical calculation

x

res hip between

ccuracy of the predictions of the critical properties. The critical point equations derived

fro e than was the

redictions proved to be in good agreement with the experimental critical points than the

imon-Yarborough, Etter-Kay, and Zais.

omprehensive comparison of the critical point predictions of the LLS

an be made for the critical pressure calculations for each class of mixtures. In this work,

for LS equation predicted more

ues from the experimental is

Figure 4.1 is a cross-plot shows the comparison between the calculated critical pressu

a

pares the calculated critical temperatures with the experimental critical temperatu

for all data.

ation of state and the Simon-Y

methods was necessary to provide guidelines for critical point predictions of comple

ervoir fluids. The most important factor to be considered was the relations

the complexity of the equations of state used in the critical point equations and the

a

m the LLS equation of state proved to be more simple and easy to deriv

case of many other equations of state. However, the results of the critical point

p

S

The most c

equation of state approach and the Simon-Yarborough, Etter-Kay, and Zais correlations

c

the hydrocarbon/non-hydrocarbon class of mixture the L

accurate critical pressure- mole fraction relationship than did the Simon-Yarborough,

84

Etter-Kay, and Zais correlations in all the mixtures in the basic calculations (with

adju

hat the

inary

mixing rules of the Redlich-Kwong and Benedict-Webb-Rubin (BWR) equations of state.

ssumption that the intermolecular energy can be described by the geometric means of

the t that the geometric mean

ly

equ

arameters will not equal to 1, which agrees with the results of this investigation.

LLS equation

thro tical pressure predictions

ot comparable to that obtained by this work with the equation of state approach using

Law vantages of the equation of state

ritical temperature, and critical volume are not restricted to any particular equation of

stat

stment of binary interaction parameters).

One of the most significant features of the critical point calculations is t

values of the interaction parameters are not equal to 1 as suggested in the original b

The assumption of a value of 1 to the binary interaction terms is equivalent to the

a

pure component energies. Chueh and Prausnitz 21 point ou

relationship is accurate only for simple, spherically, and symmetric molecules of near

al size. Thus, for most multi-component systems, the best value of the interaction

p

A comparison of the accuracy of the critical point prediction using

of state approach and three other empirical correlation methods appears in Tables C. 1

ugh C. 9 in Appendix C. In general, the accuracy of cri

from the empirical calculation methods of Simon-Yarborough, Etter-Kay, and Zais was

n

al-Lake-Silberberg equation of state. The ad

approach are that the critical properties are determined simultaneously, including the

c

e.

85

CHAPTER V

CONCLUSIONS AND RECOMMENDATION

5.1 Conclusions

The purpose of this work was to integrate the thermodynamic criteria of the

omplex petroleum reservoir fluids. The major conclusions drawn from each of the

objectives are presented along with recommendations for future study in the following

jective, to develop a closed-form solution to the van der Waals

ies

as achieved by using the Lawal-Lake-Silberberg (LLS) equation of state on complex

mix lations showed that

e average error levels in the predictions of the critical properties were comparable to

tho not comparable to those obtained from other

rawn from this result was that the ability of the equation of state to predict the critical

poi e

cor ies of pure components of the mixtures. Thus, because of

accurate and exact critical temperature and pressure for any pure component, the LLS

critical state criteria of mixtures with a LLS EOS to predict the critical properties of

c

sections:

The first ob

criticality conditions and perform numerical calculations to predict the critical propert

w

tures of hydrocarbons. Analysis of the results of the critical calcu

qualitative and quantitative agreement with experimental data was obtained. In general,

th

se obtained experimentally and

correlations (Simon and Yarborough, Etter and Kay, and Zais) 122,29,98 . The conclusion

d

nts of complex mixtures is directly related to the ability of the equations to predict th

responding critical propert

its simplicity and the fact that the Lawal-Lake-Silberberg equation applicability yields

86

equ cal

oint predictions of petroleum reservoir fluids.

for

ydrocarbon and non-hydrocarbon mixtures and for hydrocarbon with pseudo-

com d

by ion parameters to be equal to one in the mixing rules

e

tep

racy in the critical properties prediction carried out

5.2 Recommendations

ation proved by this work to be much more satisfactory equation of state for criti

p

The second objective, to determine the need to establish interaction parameters

h

ponents was achieved. The best estimations of the critical properties were obtaine

adjusting the value of the interact

of the constants ma and mb of the LLS equation. The binary interaction parameters are

expressed in terms of the ratio of molecular weights in the Equation B.91 through

Equation B.93 presented in Appendix B.

The third objective, to develop an algorithm for calculating the critical properties

of r servoir fluids was also achieved, and the procedural approach for computing the

-by-step method was efficiently performed. s

The equation of state approach to the prediction of critical points of mixtures

offers several advantages over the empirical and semi-empirical correlations methods in

use today. The general level of accu

in this work proved to be comparable to the experimental measurements.

The results obtained from this investigation are useful for PVT analysis of

res rvoir fluids especially in resolving retrograde behavior. Therefore, several

mmendations can be made regarding future wo

e

eco rk: r

87

1. Study should continue toward the implementation of this technique developed

in this search in a flash routine to resolve the convergence pressure problem for

sufficiently reliable for predicting the vapor-liquid equilibrium and volumetric

he use of pseudodization. Instead, utilize heptane-

predicting liquid dropout, constant volume depletion (CVD), constant

3. A possible use of the technique developed in this work into a reservoir

as- condensate simulation, and

near the critical region and retrograde behavior of reservoir fluids.

2. Based on the modified LLS equation of state used here in this work is

reservoir fluids without t

plus fraction as a lumped single-pseudo-component in equation of state by

composition expansion (CCE), and flash-differential liberation (FL-DL) tests.

simulation model to perform gas cycling, g

multi-contact miscibility studies.

88

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98

APPENDIX A

ANALYTICAL SOLUTION FOR CUBIC EQUATIONS

The general c

(A.1)

ivide the entire equation by

ubic equation is given by Dunham, 24

00123

3 =+++ aXaXaXa 2

D 3a ,

03

0

3

12

3

23 =+++aa

XaaX

aaX (A.2)

To find the roots of this equation, we first eliminate the quadratic term, 2X . To do this,

we make the substitution

3

2

3aayX −= (A.2)

then, by substituting in Equation A.1, to obtain

0)3

()3

()3

( 03

21

2

3

22

3

3

23 =+−+−+− a

aaya

aaya

aaya (A.3)

Expanding Equation A.3 and simplifying

33

32

23

222

3

233

3

2

273)

3(

aa

yaa

yaa

yaa

y −⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛−=− (A.4)

23

22

3

222

3

2

932

3 aa

yaa

yaa

y +⎟⎟⎠

⎞⎜⎜⎝

⎛−=⎟⎟

⎠

⎞⎜⎜⎝

⎛− (A.5)

Substituting in Equation A.3

99

03

932

273

03

21

23

2222

32

22223 ⎞⎛⎞⎛ aaaaa

323

32

333

=+⎟⎟⎠

⎞⎜⎜⎝

⎛−+

⎟⎟⎠

⎜⎜⎝

+−+⎟⎟⎠

⎜⎜⎝

−+−

aaaya

ay

aya

ay

ay

aya

(A.6)

or

0327

2 213

2 ⎟⎞aaa

3 32

301

223

3 =⎟⎠

⎜⎜⎝

⎛−++⎟

⎟⎠

⎞⎜⎜⎝

⎛+

−+

aaaya

aaya (A.7)

is

3

Equation A.7 is called the “depressed” cubic equation since the quadratic term, 2y

eliminated. Divide Equation A.7 by 3a , then

0327

213

1 223 ⎜

⎛−+

aay

33 ⎠a21

2

32

033

13

=⎟⎟⎞

⎜⎜⎝

⎛−++⎟

⎟⎠

⎞⎜⎝ a

aaaa

ay

aa (A.8)

Let

⎟⎟⎠

⎞⎜⎜⎝

⎛−=

3

22

13 3

1a

aaa

e

And

(A.9)

⎟⎟⎠

⎞⎜⎜⎝

−+= 22

03 27a

aa

f (A.10) ⎛

3

21

3

3

321

aaaa

Substituting Equations A.9 and A.10 into Equation A.8 and obtain

0 (A.11) 3 =++ feyy

Reducing Equation A.11 using Vieta’s Substitution

z

zy += s (A.12)

100

The constant s is an undefined constant for right now. Substituting Equation A.12 in

Equation A.11, yield

0=+⎟⎠

⎜⎝

++⎟⎠

+ fz

ez

z3

⎞⎛⎞⎜⎝⎛ szs (A.13)

xpand quation A.13 E ing E

3

323

3

323

3

33 zssszzsz −=−+−=⎟⎞+

33zs

zssz

zzz−+

⎠⎜⎝⎛ (A.14)

n t e right side of Equation A.14 by , yield

Multiplyi g h 3z

243

4322463 33 eszezz

szszszsszzzzsz +=+=−+−=⎟⎠⎞

⎜⎝⎛ +

Then

(A.15)

ow let

0)3()3( 33246 =++++++ sfzzesszesz

N 3es −= to simplify Equat

ion A.15 into a “tri-quadratic” equation, then

027

336 e

=++ fzz (A.16)

hich we can solve By substituting , then we have a general quadratic equation w3zw =

using the quadratic formula.

027

32 =−+

efww

S , hence using

r each of the two roots of hence will give

(A.17)

olve for the quadratic Equation A.17 and will give two roots for w

3zw = would then give three roots fo ,w

101

six root values for z . But the six root values of z would give only three values of y (for

zszy += ), and three values of x in Equation A.2.

lustrativ

4.203.0 23 +− xxx

(A.19)

Where

inate the quadratic term (i.e., depress the cubic

equation). Let

Il e Problem

Find the roots of the following cubic equation

00 6 =− (A.18) 1

Solution

For the general form of the cubic equation

023 =+++ dcxbxax

6104.2,0,03.0,1 −==−== xdcba

To find the roots of this equation, first elim

a

y3

−= bx

)1(303.0−

−= yx

(A.20)

ting the above value of

mplify, yield

(A.22)

01.0+= yx

Substitu x Equation A.20 into the cubic Equation A.18 and

si

0)4()103( 743 =+− −− xyxy (A.21)

Convert this depressed cubic equation into the form

03 =++ feyy

10

102

W e coefficients of Equation A.22 are 74 104,103 −− =−= xfxe here th

epressed equation by using Vieta’s substitution as,

Now solve the d

zszy +=

and obtain

(A.23)

Let

0)1033()104()1033( 32437446 =+−++−+ −−− szxsszxzxsz

44

103103

3−

−

=−

−=−=xes (A.24)

Substituting Equation A.24 into Equation A.24 to obtain the “tri-quadratic” equation

(A.25)

Convert Equation A.25 into a general quadratic equation by using

(A.26)

Using the quadratic equation, the values for are

0101)104( 12376 =++ −− xzxz

3zw =

0)101()104( 1272 =++ −− xwxw

w

)101(4)104(2104 1227

7

1−−

−

−±−= xxxw

) (A.27)

and

) (A.28)

The solution of gives three values of

771 10797958.9(102 −− +−= xixw

772 10797958.9(102 −− −−= xixw

3zw = z . These values in rectangular form are:

biaw +=

)sin(cos θθθ izzew i +==

103

Where πθ +=abarctan

Then,

⎥⎦

⎤⎢⎣

⎡⎟⎠⎞

⎜⎝⎛ ++⎟

⎠⎞

⎜⎝⎛ +=

32

3sin

32

3cos33 ninzw πθπθ

Then, the three values from 1w are in rectangular form

iz 00440.0008976.01 +−=

ix 009977.010707.6 42

−= z

iz 005569.0008305.03 +=

The values of z from w are in rectangular for2 m

Using V

iz 0044079.0008976.04 −−=

ixz 0099775.01070689228.6 45 −= −

i00556957.00083054.06 −= z

ieta’s substitution,

zszy +=

z

xzy4101 −

+= (A.29)

Substituting into Equation A.29 the value of z to find three values for y , choosin

0044079.0008976098.01 +−=z yields

g

01795.00044.0008976.0

101 4

1 +−

−

ix0044079.0008976.0 −=++−= iy

104

0013414.000997748.0706892.6

10100997748.0706892.64

2 =+

++=−

ixiy

Similarly, the other value of 3z gives

016611.03 = y

With the substitution in Equation A.20

The three roots of the given Equation A.19 are

Method 2

.30)

ide the entire equation by

(A.20) 01.0+= yx

0079522.01 −=x

0113414.02 =x

02661112.03 =x

The general cubic equation is given by

0012

23

3 =+++ aZaZaZ (Aa

Div 3a ,

03

01 a

3

2

3

23 =+++a

ZaaZ

aaZ (A.31)

Eliminate by making substitution of the form

2a

λ−= xZ (A.32)

By substituting Equation A.32 into Equation A.30,

23 =+−+−+− axaxax λλλ (A.33)

expanding Equation A.33,

0)()()( 012

By

105

0)()2 0122 =+−++− axax λλλ (A.34) ()33( 2

3223 +−+− xaxxx λλλ

(A.35)

Let

0)()32()3( 32210

221

22

3 =−+−++−+−+ λλλλλλ aaaxaaxax

3 in order to eliminate the , so

2a=λ 2x

231 axZ −≡ (A.36)

273

)3

( 22

3323 xaxxx −=−=3

22

2 axaa−+ (A.37)

32222 121222222 )( axaxaaxaZa +−=−= (A.38)

933

1212 31)

31 aaxaa −= (A.39) 11 (xaZa −=

Substituting back into Equation A.30, becomes

0)

39 0212

112733 (A.40)

1()21()(

3

321

22

22

222

3

=−+−

−+−++−+

aaaa

axaaaxaax

0)21()1( 03

2212

213 =−−−−+ aaaaxaax (A.41)

2733

027

22793

3 32021

2213 =

−−−

−+

aaaaaax (A.42)

et, L

3

3 221 aaP −

= (A.43)

and,

106

27

2279 32021 aaaa −−

q = (A.44)

then Equation A.42 can be written as

(A.45)

qPxx =+3

Make Vieta’s substitution to simplify the derivation by letting:

W3PWx −= (A.46)

Substituting for in Equation A.45, obtain

x

0)33 WW

or

()( 3 =−−+− qPWPPW

03

3 =−− q27 3W

P W (A.47)

lying through EquMultip ation A.47 by to obtain a quadratic equation in , 3W 3W

027

)()( 23 − WqW3

3 =−P (A.48)

Apply the quadratic formula (Birkhoff & Mclane 1996, P.106):

323

274(

21 PqqW +±=

32

271

41

21 Pqq +±=

32 QRR +±= (A.49)

Let:

107

27

41

21 qR =

33

22

PQ

qR

=

=

ieta’s “magic” substitution, first define the intermediate variables from Equation

.42

By V

A

Let:

9

3 21 aa −=

3

Q (A.50)

54

2279 32012 aaaa

R−−

= (

The cu

(A.52)

identity, which is sa sfied by perfect cubic

olynomial equations, is

) (A.53)

ince Q

33 +− xCBx

Regrouping terms, becomes

A.51)

bic Equation A.42 then becomes,

0233 =−+ RQxx

Let B and C be arbitrary constants. An ti

p

)(( 233 BBxxBxBx ++−=− 2

S 0≠ , add a multiple of (x-B) by C to both sides of Equation A.23 to give

0))(() 22 =+++−=− CBBxxBxB (A.54)

()(

( ) 0)()()( 2233 +x =+++−=+− CBBxxBxBCBCx (A.55)

Match th we must have e coefficients C and )( 3 BCB +− with those of Equation A.52, so

QC 3= (A.56)

108

( .57)

8)

nd a value for B and reduce Equation A.58 to a quadratic equation.

RBCB 23 =+ A

Then by substituting Equation A.56 into Equation A.57,

RQBB 233 =+ (A.5

Now, fi

( ) ( )312331

2 RQRRQRB +−+++= (A.59)

Taking the second and third powers of B gives

3

( ) ( ) ( )322331

23232

232 )(2 RQRRQRRQR +−++−+++= B (A.60)

( ) ( ) QRQRRQR 232

2332

23 −+−+++= (A.61)

( ) ( )

( ) ( )⎪⎭

⎪⎬⎫

⎪⎩

⎪⎨⎧

+−+++×

⎪⎭ (A.62)

⎪⎬⎫

⎪⎩

⎪⎨⎧

+−++++−=

31

2331

23

31

2331

233 2

RQRRQR

RQRRQRQBB

( ) ( ) ( ) ( )( ) ( ) QBR 222 −QRRQR

RQRRQRRQRR1

332

23

3233

123232

+−+++

+−++++−+ (A.63)

QR 3 ++=

2

( ) ( ) ( ) ⎟⎟⎠

⎞⎜⎝

−−++++−++−= 31

2331

23)(22 RQRRQRRQRRQB (A.64) ⎜⎛

31

232

RQBQBRQB 2322 +−=−+−= (A.65)

Plugging and B into the left side of Equation A.58 gives:

3B

RQBRQB 23)23( =++− (A.66)

109

Now, plugging C 3= Q into the quadratic part of Equation A.55 gives:

ch provides the solution.

0)3( 22 =+++ QBBxx (A.67)

whi

( ))3(421 22 QBBBx +−±−= (A 8) .6

QBB 12311 2 −−±−= (A.69) 22

QBiB 4321 2 −±−= (A.70)

These c

an be simplified by defining

( ) ( )312331

23 RQRRQRA +−−++≡ (A.71)

( ) ( ) ( )32332323232 )(2 RQRRQRRQRA +−++−−++= (A.72) 212

( ) ( ) QRQRRQR 232

23 ++−+++= (A.73)

(A.74)

tions to the quadratic por

32

23

QBA 422 +=

So the solu tion of Equation A.55 can be written as

AiB 322

±−= x 11 (A.75)

efining:

D

23 RQ +≡ (A.76) D

3 DRS +≡ (A.77)

3 DRT −≡ (A.78)

110

where

(A.79)

(A.80)

Therefore, at least, the roots of the original equation A.37 are given by:

TSB +=

TSA −=

)(3 211 TSaZ ++−= (A.81)

)(321)(

21

31

22 a−= TSiTSZ −++− (A.82)

)(321)(

21

31

23 TSiTSaZ −−+−−= (A.83)

With as the coefficient of2a 2Z in Equation A.1 and S and T as defined above; these

the cubic equation are sometimes known as

o’s formula. If the equation is in the standard form of Vieta,

(A.84)

three equations providing the three roots of

Candam

qPxx =+3

111

APPENDIX B

VAN DER WAALS EXPRESSIONS FOR FLUID CRITICAL POINT

The van der Waals equation of state (VDW-EOS), proposed in 1873, was the first

equatio

ation with pressure given by a cubic function of molar volume in the form

n to represent vapor-liquid coexistence. The VDR-EOS is a two-parameter

equ

2

)(TaT−=

VbVRP−

(B .1)

The first term on the right hand side is the repulsive term and the second term is

the attractive term is temperature dependent.

Multiplying both sides of Equation B.1 by RTV to obtain the VDW-EOS in Z

form

RTVa

bVV

−−

= Z (B.2)

n, where by definitio

RTPVZ = (B.3)

and

P

ZRT=V (B.4)

pressibility factor, T is tem olume, P is

pressure, and R is the molar universal gas constant. The parameter is a measure of the

Where Z is the com perature, V is v

a

112

attractive forces between molecules, and the parameter is the co-volume occupied by

the molecules (if the molecules are represented by hard-spheres of diameter d, then

b

3)2( 3σπNb = .

Substituting Equation B.4 into Equation B.2 leaves

22TZRaPZRTZ −=

PbZRT − (B.5)

r o

221

1TR

ap

ZRTPb

Z −−

= (B.6)

with

22TRaPA (B.7)

nd

=

a

RTbP

= B (B.8)

Substituting Equation B.7 and Equation B.8 into Equation B.6 leaves

ZA

ZB

Z −−

=1

1

or

(B.9)

ZBZ −AZZ −= (B.10)

or

113

ZBZ

BZAZZ)(

)(2

−−−

=

(B.12)

or

(B.11)

and )()( 2 BZAZZBZZ −−=−

ABAZZBZZ +−=− 223 (B.13)

or

(B.14)

then

(B.15)

Equation B.1 was expanded to Equation B.15 form, which is a cubic equation.

When specialized to the critical state, has three equal roots, that is, that it be of the form

(B.16)

.16), gives

( 3233 =− cc ZZZ (B.17)

g the coefficients of Equation B.15 to the coefficients of Equation B.17, leaves

0223 =−+−− ABAZZBZZ

0)1( 23 =−++− ABAZZBZ

0)( 3 =− cZZ

Expanding Equation (A

3) +−=− cc ZZZZZ 03

Equatin

cc BZ += 13 (B.18)

(B.19)

(B.20)

There are three equations B.18, B.19

Zc. To find these unknowns, substituting Ac of Equation B.19 into Equation B.20, obtain

cc AZ =23

and

ccc BAZ =3

, and B.20 with three unknowns Bc, Ac, and

114

ccc BZZ 23 3= (B.21)

en

th

33 2cZ

3cc

cZZB == (B.22)

Substituting Equation (B.22) into Equation (B.18), leaves

3

13 cc

ZZ +=

then

(B.23)

and

(B.24) cc ZZ += 39

83

=cZ (B.25)

Substituting the value of Zc in Equation B.25 into Equation B.19, obtain

642793 ==

64

Substi

cA (B.26)

tuting Equation B.25 into Equation B.18, gives

81

=cB (B.27)

Now, consider the generalized form of the Lawal-Lake-Silberberg (LLS) cubic

equation of state. That is,

22 bbVVbV βα −+−aRTP −= (B.28)

(B.29)

and the gas law is given,

ZRTPV =

115

where, by definition

RTPVZ = (B.30)

nd a

P

ZRTV =

th sides of Equation B.28 by the value

(B.31)

Multiplying boRTV

RTV

bbVVa

RTV

bVRTP

RTV

⋅−+

−⋅−

=⋅ 22 βα (B.32)

then

)( 22 bbVVRT

aVbV

VZβα −+

−−

=

Substituting Equation B.31 into Equation B.33, yields

(B.33)

⎥⎦⎤

⎢⎣⎡ −+

⋅ZRaZRT

−=22 )()( b

PZRTb

PZRT

PT

PZβα

(B.34)

− RTbP

ZRT

or

22 BBZZAZ

BZZZ

βα −+−

−= (B.35)

lying both sides of Equation B.35 by MultipZ1 , gives

22

1−=1

BBZZA

BZ βα −+− (B.36)

or

116

0)()())(( 2222 =−+−+−−+− BZABBZZBBZZBZ βαβα

Consider that at the critical point, the coefficients of the expanded form of the

cubic equation of state Equation B.37 can be compared to the coefficients of the

expansion shown in Equation B.38.

(B.38)

The coefficients of Equation B.37 are:

(B.39)

(B.40)

(B.41)

ubstituting these coef icients B.39, B .42 back in Equation B.37, leaves

(B.37)

0)( 3 =− cZZ

033 3223 =−+− ZZZZZZ cc

1:3Z

1:2 −− BBZ α

ABBBZ +−− ααβ 221 :

and

ABBBZ −+ 230 : ββ (B.42)

S f .40, B.41, and B

0)()()1( 232223 =−++−−−+−−+ ABBBZBBBAZBBZ ββααβα

(B.43)

Equate the coefficients of Equation B.43 to the coefficients of Equation B.38,

obtain

13 −−=− ccc BBZ α

ccc BBZ α−+= 13 (B.44)

BB αα −22 (B.45)

(B.46)

ccc BAZ β −−= 23 cc

ccccc BABBZ −+=− 233 ββ

117

where the subscript c represents the conditions at the critical state. Solve for Zc, Bc, and

c. From Equation B.44,

A

31−−

=− ccc

BBZ

α (B.47)

or

31 cc

cBB

Z+−

=α

(B.48)

uation B.45

(B.49)

ubstituting Equation B.48 into Equation B.46, gives

From Eq

ccccc BBBZA ααβ +++= 2223

S

( ) ccccccccc BBBBZBB

BBααβββ

α+++−+=⎟

⎠⎞

⎜⎝⎛ +−

− 222233

33

1 (B.50)

or

( ) ( ) ( )[ ] cccccccccc BBBBBBBBBB ααβαββα ++++−−+=+−− 222233 271927271 (B.51)

and

(B.52)

or

0272727

91818189927271

3336333

233

22332223

2222333233

=+++

++−−++−−−

−+−+−−+−

ccc

cccccccc

ccccccccc

BBB

BBBBBBBB

BBBBBBBBB

ααβ

αααββ

αααααα

118

( )( ) 0163)27181827 2 =−++++−− αααβ cB3

63(27927133 2323

−

+−+++−−+− αααββααα cB

(B.53)

271892 +−+ αα

and

( ) ( ) ( )6 cB 01=3+ α15 cB27 +β15 −α3 2α8126 223 −++−+++ αα

( .54)

3 +cBα

B

or

( ) ( ) ( ) 012359538126 22323 =++++ BBααα B

an be further simplif t l s m p

ng of the variables:

+++ θθθθ cc BBB .56)

here

++= αααθ (B

)2 αα − .58)

)

−cB+α+c+β−α−α+c ( .55)

Which c ied to he fol owing expres ion of the ter s and appro riate

groupi

0432

23

1 =c (B

w

)8+126( 231 .57)

3271515( 2βθ += − (B

36(3 αθ += (B

.60)

Critical it y ritical C s ty

.59)

14 −=θ

(B

Cond ion b the C ompre sibili Form

From equation B.44,

,1)1(3 −−=− αcc BZ (B.43)

119

the following expressions can be deduced:

)1()31(

−−

=α

cc

ZB

α−−

=1

13 cc

ZB

ting of Equati 1 E

(B.61)

Substitu on B.6 into quation B.55, yield

01 =))−cZ

)1(31(

)(6

))31(

()

(126 23

−+

+++

αα

ααZ

3

3)27153)1)(12

=−

−+

α

αα

c

c

Z

Z(B

Let

31)(1527153(

6

−=

−−+=

−−+−+−=

+

α

αα

αβαα

α

D

ZC

ZB

c

c (B.64)

And,

3(

)1−αc

()(1527 +β15 −α3− α) +

1( −α) 3c31− Z

+−

2)(8+α(

−

(B.62)

or

6( 23 + αα (3 −+ 2 +α ()2c3− Z −α 15+β8+ 1( − )1 +

.630)1 3−()1 2 −1)(63( −+α )(α

)

3

2

22

323

)1(

)1)(31)(63(

)1(

(= αA )3− Zc1)(8+12+ α

)

0)()()()( =−++ DCBA (B.65)

Expanding A, B, C, and D, then

120

812α672108549

216324+162 + 22627

)31)(86(

233

223233

33

++−−

−−−−

−++=

ααααα

αα

αα

cccc

cccc

c

ZZZ

ZZZ

ZA

62108

527(

13(

2

23

2

−−−−+−++−−−

−=

+−=

ββααβαβα

βα

α

cccc

cc

ZZZZ

Z

B

1)(63(3 +−

+=

α

α

cZ

C

)1( 3 −=−α

g A, , a in ua 65, ob ain

891283

1621818135243243

)6789

32427216162

23

23222222

222333

−+−+

+−−

−−+−−+

+

+++−

αα

ααβ

αβαβαβα

α

αα

ccc

cccccc

ccc

ZZ

ZZZZZZ

Z

ZZZZ

or

(B.67)

or

(B.68)

3cZ 3 +c

2 +α cZ324Z 7 3α Z 2α cZ(= 162 21−

Z2 − + +

)1527271518390901621

18α Z13−c243βZ243α162+ α

)1−α()2Z31)( −157 +β25 −α2

c23 2 2−cZ +cZ −cZ 3

Z 3 2α α αc

2

)69 +− α3 3+ α18− Z27α cZ9(

)1)(3− cZ −α

= c

=D ( 3α )133 2 +α −α

substitutin B, C nd D Eq tion B. t

0)1 =3+ α3− α() −693 3α

1−27()5 +7 −β272 −α12 +α33 +α90162 +− βZ

16227( 3− α

8+α122 +α2 cZ10−c54 2α216 2 −+ cZ

162324−27( −− α cZ 3 3 22 α cZ cZα αc

3 −cZ −α cZ 3 +α +

Z Zα−c c

−Z 3α Zc c

(B.66)

0)272727()16216281162(

)81243243324324()21632416227(22

22323

=−−+−−−−−

+−++−−−−−

βαβαβαβαα

αββααααα

c

cc

Z

ZZ

0)()6636(

)3991212()8126(22

22323

=−+−+−++

+−++−+++

αβαββαβαα

αββααααα

c

cc

Z

ZZ

121

Let

663(12123(

6128(

24

2

22

21

αβαβθ

ββαα

αθ

αα

−+−=

++

++−=

+++=

(B. 9)

e critical condition by t e criti al com ressibility form

(

tion No.2

rom E uatio B.44

)3αθ

)6α−

)

9β−9α +

3θ =

αβ 6

)(

Then th h c p is

0432

23

1 =+++ θθθθ ccc ZZZ B.70)

Deriva

F q n

)1()31(

−−

=B α

cc

Z (B

B+ 23 β (B

c BBBZA ααβ += 223 , obtain

.44)

From Equation B.46

BZ =− 3 β ccccc BA− .46)

Substituting for cc2+ +c c

[ ]cBαcBαcBβcZ +cB−cc BBZ ββ +++− 23 3 (B.71)

Substituting Equation B.61 into Equation B.68, then

c =3 2 2 2

⎥⎥⎦

⎤

⎢⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−+

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−=−

)1()31(

)1()31(

)1()31(

3

)1()31(

)1()31(

)1()31(

222

233

αα

αα

αβ

ααβ

αβ

cccc

cccc

ZZZZ

ZZZZ

(B.72)

Multiply both sides of Equation B.68 by , obtain 3)1( −α

122

33

)1()1(

)31()131()1(3

+−

−−−=−−

βα

βα

c

cccc ZZZ (B.73)

)(1()1(3

1)1()1(3222

33

−+−

−−−−

βα

βα

c

cc

Z

Z(B.74)

91)31 ccc ZZZ −=− (

1−=− ααα (

1 .77)

cc ZZ−= .74)

tions, B.75, , B.77 into Equation B.74, obtain

3

)961

3

)31()1

1)227933

2

2

2

32

=⎥⎢

⎡

−+

+−+

+

−+−

−−−+−

αα

α

βαα

α

αα cc

ZZ

ZZ

Z

Z

B

or

(B.76)

where,

(B.77)

[3Z

23 )(1() −−+ αβ Z3

]222 2 1(+α 2 1(+α)3− cZ 3− cZ 3− cZ )1)( −α

or

[ ](3 − β ()2 −α3− Z 3− Z )3Z1()1 + c

03 =)1−α3− cZ1(2 +α)3− cZ1(+α)3 cZc

where, ( 3 2 27−3 27Z+ c B.75)

3)1( 233 3 −+ α B.76)

2)1( 22 +−=− ααα (B

2)31( cZ− 2961 + (B

substituting Equa B.76

0

13+ α33 −)(Z1( −α

(

)⎥c9+c61( −)1 +2−(cZ

)(9 2+ cZ

6Z(− β7Z1( −β)1−( 33− αZ2 ⎤2

2Z+ c

⎢⎢

⎢

cc

( . 75)cc c

⎥⎥

⎣ ⎦c

0

)31)(12)(31(

)31)(961()31(

)961()31)(12(3

)1)(9

61()272791()133(

2

2

222

2

32233

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−+−−+

−+−+−

+−+−+−

+−+

−−−+−−−+−−

cc

cccc

cccc

c

ccccc

ZZ

ZZZZ

ZZZZ

Z

ZZZZZ

ααα

α

βαα

α

ββααα

222 96196)961)(1( cccccc ZZZZZZ −+−+−=+−− αααα

123

Z 36)12)(31( 22 −=+−− αααα (B.78)

27712)(91)(31 cccccc ZZZZZZ +−=++−− αα (B.79)

22

189

)(961(

cc

cc

ZZ

ZZ

−+

++−

αα

α

ting Equations B.77 , a 8 E o n

962

)27

91(

)3312(

)961

()27279()33

2

22

22

2

32233

=

⎥⎤

⎢⎡

−

++

+

+−−

+−+−

−−−−−−+−

c

cc

cccc

cc

ccc

ZZZ

Z

ZZZ

ZZZZ

ZZ

ZZZZ

αα

α

ααα

α

ααββαα

(B

123(

2273(

2 =−+−

++−

+

αββα

ααα c

Then,

(B.83)

Let,

cccc ZZZ312 2 +−+− αα

3222 29)16( −−

and,

2 Z +α2 2− α 12 Zα61−+)1 = α2− α c c (B.80) 222 29 cZ+

Substitu , B.78 B.79, nd B. 0 into quati n B.76, obtai

611(

0

)2⎦c92 +18α1⎢

⎣+

61−2− α(αα27⎢+

1( −α)3c272 −27+9− Z

63

9+

2cZ

32

2

2

⎥⎥⎥⎥c

⎢⎢+ β

+c−cZ

−c

−+

+c c Z

α

α−

cZ

cZ

.81)

and,

3

0)(

)αβ cZ66 −β6 2 +α3( +α) 2 +α cZ122 +α9 + βαβ

)7α Z9 ++18−9+β3− α 23 +α 2 −

(B.82)

0)()6636(

)9912123()86(22

2233

=−+−−+++

−+++−++

αββααββαα

αββαααα

c

cc

Z

ZZ

)6128( 321 αααθ +++=

124

3θ

)(

)β6α−62 βα +63(

912123(

24

22

αββαθ

α

βααθ

−+−=

+=

+++−=

(

+θθ cc ZZ (

Critica y r C

)9αβ−

B.84)

Then,

23

12

3+θ Z 0= 4+θc B.85)

l Condition b the C itical ompressibility Form For Mixtures

The generaliz a f f t f in ed cubic equ tion o state or mix ures is in the ollow g form:

22mmmm

m

m bVVa

bVP

α −+−

−= (B. )

der Waals mix ul e tur ram s m

bRT

β 86

By van ing r es, th mix e pa eter mmm ba βα ,,, take the

following forms:

∑∑=n a

ai j

m ijj 2iji axx 21

(B

xx (B. 8)

a1

a .87)

∑=i

mb (n

ji3) 8

∑∑=i j

m xαn n

x ααα 21

21

ijjiji (B.89)

∑∑=n

i

n

jijjijim xx ββββ 2

121

(B.90)

and the binary interaction parameters established in terms of Mw ratios of components

form

125

j i MW≤n

j

i MWMWMW

a

⎟⎟⎞

⎜⎜⎝

⎛= (B ija

⎠ .91)

j i MWn

j

iij MW

MWMW

⎟⎟⎠

⎞⎜⎜⎝

⎛ α

( ≤=α B.92)

j i

n

j

iij MW

MWMW

⎟⎟⎠

⎞⎜⎜⎝

⎛=

β

β (B

and are the exponents of the re a

xpand ng Eq ation .86 into Z term and b multiplying both sides of e equ tions by

MW≤ .93)

where ,, αnna an spective inter ction terms.

E i u B s y th a

RTV ,

⎟⎠

2 ⎟⎞

⎜⎜⎝

⎛

+−⎟⎟

⎠

⎞⎜⎜⎝

⎛− m

m

m ba

bVRT

− mβ=

RTRT 2V mm VbαRTVVVP (B.94)

⎟⎟⎞

⎠m⎜⎜

+−⎟⎟

⎞− 2

m

m

bVRTV

b ββα (B. 5)

by definitio

⎛

⎠⎝ m⎜⎜= V

Z⎛ V a

⎝V − 2mm

9

n,

PZRTV = ,

then,

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

−⎟⎠⎞

⎜⎝⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛⎟⎠⎞

⎜⎝⎛

−⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

−⎟⎠⎞

⎜⎝⎛

=2

2

mmmm

m

m bP

ZRTp

ZRT

aRTP

ZRT

bP

ZRTP

ZRT

Z

ββα

(B.96)

Multiplying by RTP and by 22

2

TRp , obtain,

126

⎟⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜⎜

⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛−⎟

⎠⎞

⎜⎝⎛+

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎠⎞

⎜⎝⎛−

⎟⎟⎟⎟

⎠

⎞

⎜⎜⎜⎜

⎝

⎛

⎟⎠⎞

⎜⎝⎛−

=

22

222

22

2

TRPb

RTZPZ

TRPa

RTZRT

RTPbZ

ZZ

mmmm

m

m ββα (B.97)

since ⎟⎟⎠

⎞⎜⎜⎝

⎛= 22

2

TRPa

A m and ⎟⎠⎞

⎜⎝⎛=

RTPb

B m , then

⎟⎟⎠

⎞⎜⎜⎝

⎛−+

−⎟⎠⎞

⎜⎝⎛

−= 22 BZBZ

ZABZ

ZZmm βα

(B.98)

or

(B.99)

(B.100)

(B.101)

(B.102)

At the critical point, the generalized cubic equation of state in terms of Zc takes the form:

(B.103)

Now, com sion

=0 shown in Equation B.100

(B.104)

)()())(( 2222 BZZABZBZZBZBZBZZ mmmm −−−+=−+− βαβα

0

22322223

=−++−−+−−−+

ABAZBZBZBZBBZZBBZZ mmmmmm βαβαβα

0)()1( 232223 =−++−+++−−+ ABBBABBBZBBZZ mmmmmm ββααβα

0()()1( 322223 =−−+−−−+−++ mmmmmm BBABBBBAZBBZZ βββααα

0(

)()1(32

2223

=−−+

−−−+−++

mcmccc

cmccmcmcccmcc

BBBA

ZBBBAZBBZ

ββ

βααα

paring the coefficients of Equation B.103 with the coefficients of the expan

( 3)cZZ −

0333 32223 =−++− cccc ZZZZZZZ

127

where,

cmcc BBZ α−+=13 (B.105)

(B.106)

and,

(B.107)

From Equation B.106 and Equation B.107, Bc and Ac can be solved to give,

mccmcmcc BBBAZ βαα 2223 −−−=

mcmcccc BBBAZ ββ 323 −−=

m

cc

ZB

α−−

=1

13 (B.105)

(B.106)

By substitution of Ac and Bc with Equation B.98 and simplifying the terms, the analytical

function of is obtained.

2223 cmcmcc BBZA βα ++=

∫= ),( mmcZ βα

32

2223

113

113

113

113

113

113

3

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−−⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−−

⎟⎟⎠

⎞⎜⎜⎝

⎛−

−⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−+⎟⎟

⎠

⎞⎜⎜⎝

⎛−

−+=

m

cm

m

cm

m

c

m

cm

m

cm

m

cmcc

ZZ

ZZZZZZ

αβ

αβ

ααβ

αα

αα

(B.107)

and can be further simplified to the following expression by expanding the terms by

grouping,

(B.108)

where:

0432

23

1 =+++ θθθθ ccc ZZZ

128

(B.109)

)(

)6663(

)9912123(

)6128(

24

23

22

321

mmmm

mmmmm

mmmmm

mmm

αβαβθ

βαβααθ

αββααθ

αααθ

−+−=

−++=

−+++−=

+++=

129

APPENDIX C

PREDICTION RESULTS OF CRITICAL PRESSURE, CRITICAL TEMPERATURE, AND HEPTANE PLUS PROPERTIES

Table C. 1. Critical Pressure Predictions for Complex Mixtures by Four Methods (Compositions in Mole Fractions) Mixture No. 145-1 145-2 145-3 145-4 145-5 145-6 145-7 145-8 145-9 Nitrogen 0.001 0.001 0.001 0.001 0.004 0.001 Carbon Dioxide 0.004 0.005 0.005 0.004 0.006 0.002 0.0149 0.014 0.0156Methane 0.193 0.271 0.363 0.229 0.365 0.482 0.5248 0.5335 0.5921Ethane 0.032 0.034 0.038 0.029 0.058 0.065 0.145 0.0679 0.066 Propane 0.585 0.482 0.374 0.407 0.242 0.167 0.157 0.2035 0.1043I-Butane 0.007 0.007 0.009 0.006 0.007 0.007 0.0055 0.0063 0.0058n-Butane 0.012 0.015 0.015 0.172 0.215 0.14 0.0179 0.204 0.0831I-Pentane 0.005 0.006 0.006 0.005 0.004 0.006 0.0055 0.0062 0.0054n-Pentane 0.007 0.007 0.008 0.006 0.007 0.008 0.0079 0.0092 0.0078Hexane 0.013 0.015 0.016 0.012 0.01 0.012 0.0082 0.0094 0.0081Heptanes + 0.141 0.157 0.165 0.129 0.082 0.11 0.1133 0.1296 0.1118 Avg. Mol. Wt. 67.3 68.5 67.5 66.1 48.8 48.8 45 48.4 44.9 Heptanes + Properties Mol. Wt. 243 243 243 243 191 191 191 191 191 Gravity(API) Characterization Factor 11.6 11.6 11.6 11.6 11.9 11.9 11.9 11.9 11.9

Critical Temp., Tc (F) 265 265 265 265 234 200 200 200 200 Critical Temp., Tc (o R) 725 725 725 725 694 660 660 660 660

Critical Pc (psia) 2100 2500 3400 1920 2420 3430 4355 4295 4630

Predicted Pc (psia) : Simon and Yarborough 2002 2550 3280 2113 2452 3490 3984 4261 4691

Etter and Kay 2911 3272 3658 2806 2509 3268 3909 3886 4040 Zais 2175 2613 3203 2298 2418 3480 4073 3982 4357

This work 2101.12 2501.55 3398.03 1922.15 2423.18 3432.17 4349.55 4302.33 4624.49

130

Table C. 2. (Continued)

Mixture No. 145 - 10 145 - 11 145 - 12 145 - 13 145 - 14 4 - 1 4 - 2 4 - 3 4 - 4 4 - 5 Nitrogen 0.0016 0.002 Carbon Dioxide 0.0153 0.0008 0.0139 0.0006 0.0008 0.0066 0.0067 0.0061 0.0061 0.0059Methane 0.5774 0.6298 0.6165 0.4786 0.5808 0.7243 0.7292 0.8227 0.8165 0.795 Ethane 0.0631 0.0858 0.0602 0.082 0.0562 0.0557 0.0556 0.0284 0.0284 0.0281Propane 0.1178 0.0672 0.0961 0.1359 0.0919 0.0308 0.0306 0.0124 0.0124 0.0128I-Butane 0.0055 0.023 0.0055 0.0076 0.0081 n-Butane 0.0976 0.0305 0.0773 0.1273 0.082 0.0241 0.0226 0.0091 0.0092 0.0098I-Pentane 0.005 0.0114 0.0053 0.0066 0.0071 n-Pentane 0.0073 0.0122 0.0077 0.0069 0.0073 0.015 0.0147 0.0067 0.007 0.0078Hexane 0.0075 0.0206 0.008 0.0139 0.0148 0.0179 0.0183 0.0125 0.0131 0.015 Heptanes + 0.1035 0.1187 0.1095 0.139 0.1488 0.1256 0.1223 0.1021 0.1073 0.1247 Avg. Mol. Wt. 44.3 44.4 43.4 54.7 53.1 39.9 39.3 33 33.8 36.5 Heptanes + Properties Mol. Wt. 191 167 191 205 205 167 167 158 158 158 Gravity(API) Characterization Fact. 11.9 11.7 11.9 11.6 11.6 11.8 11.8 11.9 11.9 11.9

Critical Temp., Tc (F) 200 292 200 180 180 285 251 100 160 212

Critical Temp., Tc (oR) 660 752 660 640 640 745 711 560 620 672

Critical Press., Pc (psia) 4364 4850 4745 3500 4800 5130 5350 6000 5820 5620

Predicted Pc: Simon and Yarborough 4413 4789 4930 3568 4735 5388 5412 5696 5695 5685

Etter and Kay 3865 4198 4169 3598 4209 4660 4679 5102 5094 5040 Zais 4150 4870 4592 3662 4488 5070 5083 5027 5980 5947

This work 4355.37 4848.37 4735.61 3495.07 4804.54 5134.45 5341.33 5988.62 5821.98 5608.11

131

Table C. 3. (Continued)

Mixture No. 4 - 6 4 - 7 4 - 8 4 - 9 4-10 75 - 1 75 - 2 75 - 3 75 - 4 75 - 5 Nitrogen 0.0128 0.0058 0.0053 0.0054 0.0038Carbon Dioxide 0.0035 0.0156 0.0008 0.0075 0.0074 0.0045Methane 0.6433 0.6865 0.657 0.7164 0.8213 0.788 0.724 0.728 0.597 0.83 Ethane 0.0638 0.0603 0.0869 0.0548 0.0637 0.059 0.0542 0.0546 0.089 0.0378Propane 0.0605 0.0232 0.0537 0.031 0.0409 0.0315 0.03 0.0302 0.05 0.0144I-Butane n-Butane 0.0565 0.0309 0.0303 0.0258 0.0235 0.0265 0.031 0.0207 0.049 0.0089I-Pentane n-Pentane 0.0404 0.0243 0.0195 0.0216 0.0122 0.0425 0.071 0.0688 0.093 0.0436Hexane 0.0366 0.0256 0.0173 0.0198 0.0103 0.0252 0.0456 0.0438 0.0308Heptanes + 0.0954 0.1245 0.1217 0.1231 0.0234 0.0214 0.0388 0.0375 0.122 0.0263 Avg. Mol. Wt. 35.2 41.5 45.7 40.1 25 29.5 29.1 36.2 24.5 Heptanes + Properties Mol. Wt. 114 171 207 167 114 106 106 106 100 106 Gravity (API) Characterization Fact. 11.7 12 12 11.8

Critical Temp., Tc (F) 195 239 145 243 55 109 109 169 54

Critical Temp., Tc (oR) 655 699 605 703 515 569 569 629 514

Critical Press., Pc (psia) 2720 5100 5570 5150 2270 2387 2574 2537 2515 2580

Predicted Pc (psia) Simon and Yarborough 2371 5557 6687 5375 2456 2467 2465 2821

Etter and Kay 3013 4490 4865 4560 2574 2725 2722 2020 2812 Zais 2641 5325 5623 4851 2272 2277 2441 2467 2356 2443

This work 2723.07 5096.27 5582.6 5147.51 2386.85 2572.96 2537.73 2514.61 2581.33

132

Table C. 4. (Continued)

Mixture No. 75 - 6 75 - 7 75 - 8 141 - 1 141 - 2 141 - 3 141 - 4 141 - 5 141 - 6

Nitrogen 0.0038 0.0036 0.003 0.1101 0.0477 0.0156 0.0123 Carbon Dioxide 0.0044 0.0043 0.0035 0.0008 0.0019 0.0047 0.0028 0.0172 0.011 Methane 0.815 0.784 0.643 0.3465 0.4675 0.4656 0.6173 0.6527 0.586 Ethane 0.0372 0.0355 0.0294 0.1561 0.1306 0.1645 0.1024 0.0725 0.1309Propane 0.0141 0.0136 0.0111 0.1499 0.105 0.1163 0.0673 0.0391 0.0699I-Butane 0.0121 0.0112 0.0141 0.0085 0.0157 0.0081n-Butane 0.0102 0.013 0.0252 0.0852 0.0472 0.0454 0.0352 0.0204 0.0349I-Pentane 0.0084 0.0169 0.0112 0.0106 0.016 0.0097n-Pentane 0.0501 0.0631 0.123 0.0109 0.0192 0.0182 0.0171 0.0123 0.0187Hexane 0.0354 0.0447 0.0871 0.02 0.0373 0.0215 0.0226 0.029 0.0194Heptanes + 0.0303 0.0382 0.0747 0.1 0.1155 0.1229 0.1039 0.1251 0.1114 Avg. Mol. Wt. 25.4 27.6 37.7 Heptanes + Properties Mol. Wt. 106 106 106 164 142 168 144 157 184 Gravity (API) 40.6 42.5 40.1 47.3 49.5 42.2 Characterization Fact. 11.6 11.55 11.75 11.75 12 11.85

Critical Temp., Tc (F) 65 90 189 110 169 180 159 262 185

Critical Temp., Tc (oR) 525 550 649 570 629 640 619 722 645

Critical Pressure, Pc (psia) 2675 2730 2900 2970 3010 3573 3470 4060 4335

Predicted Pc (psia) Simon and Yarborough 2817 2813 2788 2435 2574 3570 3249 4426 4505

Etter and Kay 2873 2930 2724 2932 3019 3528 3507 3959 3972 Zais 2513 2588 2355 2710 2668 3599 3410 4493 4680

This work 2671.26 2731.9 2897.64 2970.33 3012.3 3577.27 3469.13 4059.64 4342.19

133

Table C. 5. (Continued)

Mixture No. 141 - 7 141 - 8 141 - 9 141 - 10 141 - 11 141 - 12 141 - 13 141 - 14 141 - 15 Nitrogen 0.0009 0.0012 0.0013 Carbon Dioxide 0.0088 0.0041 0.0048 0.0056 0.0043 0.0038 0.0962 0.0776 0.0701Methane 0.722 0.7052 0.7175 0.6992 0.5564 0.4902 0.5431 0.4371 0.3954Ethane 0.0474 0.0725 0.0748 0.0638 0.0523 0.0464 0.1149 0.0932 0.0848Propane 0.0278 0.0465 0.0456 0.0404 0.2108 0.297 0.0768 0.216 0.2738I-Butane 0.0105 0.019 0.0177 0.0081 0.0069 0.0062 0.0147 0.0123 0.0115n-Butane 0.0158 0.0129 0.0136 0.0169 0.0147 0.0133 0.0257 0.0223 0.0207I=Pentane 0.0112 0.0064 0.0073 0.0079 0.007 0.0064 0.0082 0.0076 0.0072n-Pentane 0.008 0.008 0.0046 0.0058 0.0052 0.0047 0.0082 0.0078 0.0076Hexane 0.0181 0.01 0.0099 0.011 0.01 0.0093 0.0126 0.013 0.013 Heptanes + 0.1297 0.1142 0.1031 0.1413 0.1324 0.1227 0.0996 0.1131 0.1159 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 178 211 211 202 202 202 253 253 253 Gravity (API) 42.2 36.7 36.7 35 35 35 29 29 29 Characterization Fact. 11.8 11.7 11.7 11.55 11.55 11.55 11.55 11.55 11.55

Critical Temp., Tc (F) 265 226 216 202 202 202 280 280 280

Critical Temp., Tc (oR) 725 686 676 662 662 662 740 740 740

Critical Press., Pc (psia) 5420 6345 6560 8050 5130 4180 6715 4780 4310

Predicted Pc (psia) Simon and Yarborough 6035 6314 6304 5831 3972 3275 5241 3978 3542

Etter and Kay 4783 5201 5276 5095 4083 3659 4831 4051 3796 Zais 5470 7037 7299 5925 4436 3861 6250 4844 4380

This work 5416.47 6354.1 6569.65 8041.73 5129.18 4172.89 6708.13 4787.78 4315.78

134

Table C. 6. (Continued)

Mixture No. 141 - 16 141 - 17 141 - 18 141 - 19 141 - 20 141 - 21 141 - 22 141 - 23 141 - 24 Nitrogen 0.0052 0.0007 0.0008 0.0004 Carbon Dioxide 0.0871 0.0776 0.0778 0.0006 0.0011 0.0008 0.0008 0.0023 0.0022Methane 0.4911 0.4384 0.654 0.6091 0.5432 0.4789 0.3678 0.77 0.7247Ethane 0.1041 0.0935 0.0889 0.0865 0.0706 0.0624 0.0488 0.0316 0.0371Propane 0.0699 0.0632 0.061 0.0765 0.195 0.2786 0.4154 0.0228 0.0473I-Butane 0.0135 0.0123 0.0198 0.0183 0.0078 0.0075 0.007 0.0149 0.026 n-Butane 0.1063 0.1768 0.0317 0.0304 0.014 0.0132 0.0124 0.019 0.028 I-Pentane 0.0078 0.0074 0.0189 0.0134 0.0077 0.0072 0.0067 0.0084 0.0148n-Pentane 0.0077 0.0077 0.0172 0.0094 0.006 0.0057 0.0053 0.0137 0.0167Hexane 0.0123 0.0128 0.0307 0.0184 0.0165 0.0156 0.0145 0.0212 0.0202Heptanes + 0.1002 0.1103 0.0692 0.1374 0.1368 0.1293 0.1209 0.0961 0.083 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 253 253 126 229 229 229 229 152 152 Gravity (API) 29 29 60.4 30.2 30.2 30.2 30.2 51 51 Characterization Fact. 11.55 11.55 12.3 11.5 11.5 11.5 11.5 12.1 12.1

Critical Temp. Tc (F) 280 280 144 221 221 221 221 183 183

Critical Temp., Tc (oR) 740 740 604 681 681 681 681 643 643

Critical Press. Pc (psia) 5400 4130 2660 5700 5140 4040 2800 4120 3755

Predicted Pc (psia) Simon and Yarborough 4500 3959 2669 5164 4277 3566 2639 4851 4287

Etter and Kay 4219 3789 3009 4686 4333 3926 3315 4331 3845 Zais 5322 4527 2796 5497 4811 4202 3235 3963 3527

This work 5397.08 4137.16 2656.94 5693.09 5143.95 4043.34 2798.17 4118.03 3755.24

135

Table C. 7. (Continued)

Mixture No. 141 - 25 141 - 26 141 - 27 141 - 28 141 - 29 141 - 30 141 - 31 93 - 1 37 - 1 158 - 1 Nitrogen 0.0487 0.0249 0.0526 0.0115 0.0017 0.1898 0.0052 0.0076Carbon Dioxide 0.0014 0.0029 0.0149 0.2202 0.0073 0.003 0.0013

Hydrogen Sulfide .0068b Methane 0.4534 0.4957 0.6414 0.5325 0.2756 0.695 0.4004 0.5832 0.6849 0.5391Ethane 0.0705 0.1379 0.0733 0.1428 0.1144 0.0696 0.0767 0.1355 0.0971 0.142 Propane 0.1961 0.1015 0.0605 0.0926 0.2486 0.0426 0.0347 0.0761 0.0542 0.0964I-Butane 0.0936 0.0222 0.0154 0.0183 0.085 0.0109 0.0064 0.0084 0.0086 n-Butane 0.0825 0.0403 0.0298 0.0349 0.1294 0.0157 0.0204 0.0319 0.0223 0.0554I-Pentane 0.0542 0.0162 0.0142 0.0134 0.043 0.0102 0.0046 0.008 0.0088 n-Pentane 0.0343 0.0189 0.0178 0.0179 0.027 0.0086 0.0067 0.0161 0.0092 0.0299Hexane 0.0129 0.0234 0.0233 0.0318 0.0161 0.0184 0.0084 0.019 0.0142 0.0272

Heptanes + 0.0011 0.0952 0.0994 0.0632 0.0465 0.1124 0.0181 0.1145 .0925a 0.1011 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 114 161 161 117 192 180 115 193 147 178.5 Sp. Gravity or API 68.7 47.4 47.4 59.4 41 0.825 0.76 0.8135 Characterization Fact. 12.6 11.9 11.9 12.1 11.85 11.9

Critical Temp., Tc (F) 183 161 161 161 171 240 35 190 120 + 30 193

Critical Temp., Tc (oR) 643 621 621 621 631 700 495 650 #VALUE! 653

Critical Press., Pc (psia) 1410 3230 4090 2325 1980 4905 3454 4445 5500 3490

Predicted Pc (psia) Simon and Yarborough 1242 3157 4276 1899 2114 4862 3634

Etter and Kay 1478 3238 3861 2592 1843 4119 3430 Zais 1484 3364 4303 2263 1850 5136 1987 4528 3970 3839

This work 1409.62 3229.75 4084.59 2321.97 1977.91 4906.24 3450.08 4449.06 5503.58 3483.95(a) Heptanes plus treated as individual components through hexadecane (b) H2S treated as propane

136

Table C. 8. (Continued)

Mixture No. 58 - 1 11 - 1 6 - 1 6 - 2 6 - 3 30 - 1 52 - 2 52 - 3 52 - 6 Nitrogen 0.0066 0.0024 0.0021 0.002 0.0018 0.0124 0.0027Carbon Dioxide 0.018 0.0302 0.0084 0.0076 0.007 0.0065 0.0022 0.0003Methane 0.5966 0.6073 0.7504 0.6822 0.6253 0.5772 0.5706 0.5428 0.604 Ethane 0.1289 0.1177 0.0537 0.0489 0.0448 0.0414 0.1496 0.149 ..0517 propane 0.0653 0.0891 0.0136 0.0124 0.0114 0.0105 0.0825 0.0912 0.0757I-Butane 0.0173 0.0198 0.0021 0.0019 0.0018 0.0016 0.0092 0.0072 0.0455n-Butane 0.0219 0.0279 0.0127 0.1025 0.1773 0.2405 0.0367 0.0456 0.0389I-Pentane 0.01 0.0124 0.0106 0.0096 0.0088 0.0082 0.0064 0.0249n-Pentane 0.0096 0.008 0.0099 0.009 0.0082 0.0076 0.018 0.0186 0.0182Hexane 0.0132 0.0237 0.0215 0.0197 0.0182 0.0217 0.0255 0.0394

Heptanes + 0.1126 0.0878 .1125a .1023a .0937a .0865a 0.0907 0.1201 0.0897 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 155 172 118 118 118 118 173 166 133 Specific Gravity or API 0.7994 38 41.2 0.8126 51.9 Characterization Fact. 11.8 11.5 11.7 11.7 11.85

Critical Temp., Tc (F) 264 125 172.5 297.5 232.5 252.5 170 170 245

Critical Temp., Tc (oR) 724 585 632.5 757.5 692.5 712.5 630 630 705

Critical Pressure, Pc (psia) 3840 3940 4715 3950 3470 3140 4220 3875 2953

Predicted Pc (psia) Simon and Yarborough 3505 3574 3207 2609 2197 1900 3633 3349 2850

Etter and Kay 3720 3794 3794 3273 2901 2621 3642 3554 2987 Zais 3116 3905 4061 3501 3079 2753 4093 3768 2577

This work 3834.23 3945.43 4710.42 3947.13 3466.88 3140.71 4224.02 3871.53 2955.59

(a) Heptanes plus treated as individual components through hexadecane

137

Table C. 9. (Continued)

Mixture No. 47 - 1 26 - 1 26 - 2 26 - 3 47 - 2 26 - 4 26 - 5 26 - 6 26 - 7 26 - 8 Nitrogen 0.0884 0.1611 0.2441 0.113 0.24 0.1146 0.135 0.0705Carbon Dioxide 0.012 0.0109 0.01 0.0091 0.0044 0.003 0.002 0.0013 0.002 0.0025Methane 0.9089 0.8286 0.7625 0.887 0.858 0.7364 0.7364 0.7665 0.7515 0.8532Ethane 0.044 0.0401 0.0369 0.0333 0.016 0.015 0.012 0.0551 0.061 0.0411Propane 0.0191 0.0174 0.016 0.0144 0.007 0.006 0.0053 0.0335 0.0327 0.0198I-Butane 0.0033 0.003 0.0028 0.003 0.0014 0.0012 0.001 0.0035 0.0038 0.0037n-Butane 0.006 0.0055 0.0051 0.004 0.002 0.0018 0.0015 0.009 0.006 0.0039I-Pentane 0.0021 0.0019 0.0018 0.0016 0.0007 0.0006 0.0005 0.0017

n-Pentane 0.0013 0.0012 0.0011 0.001 0.0005 0.0004 0.0004 0.0015 .0020c .0022cHexane 0.0015 0.0014 0.0012 0.0011 0.0005 0.0004 0.0004

Heptanes + 0.0018 0.0016 0.0015 0.0014 0.0007 0.0006 0.0005 .0033d Helium 0.01 0.006 0.0031Avg. Mol. Wt. 18.4 19.25 19.94 20.75 16.9 18.11 19.56 20.02 19.98 18.6 Heptanes + Properties Mol. Wt. 115 115 115 115 115 115 115 100 Gravity (API) Characterization Fact.

Critical Temp., Tc (F) -79 -92 -104 -120 -101 -117 -131 -84 -90 -96

Critical Temp., Tc (oR) 381 368 356 340 359 343 329 376 370 364

Critical Pressure, Pc (psia) 925 955 968 973 765 790 815 1143 1107 918

Predicted Pc (psia) Simon and Yarborough 3285 3337 3380 3431 3681 3701 3740 3060 3042 3316

Etter and Kay 1385 1340 1303 1260 995 958 919 1519 1357 1265 Zais 1207 1273 1327 1340 907 958 978 1419 1328 1107

This work 924.78 955.88 967.83 973.21 765.06 790.12 815.1 1142.64 1107.03 918.27© Heavy fraction treated as n-pentane. (d) Heavy fraction treated as n-heptane. (e) Helium treated as methane.

138

Table C. 10. Critical Temperature Predictions for Complex Mixtures by LLS EOS Method (Compositions in Mole Fractions) Mixture No. 145-1 145-2 145-3 145-4 145-5 145-6 145-7 145-8 145-9 Nitrogen 0.001 0.001 0.001 0.001 0.004 0.001 Carbon Dioxide 0.004 0.005 0.005 0.004 0.006 0.002 0.0149 0.014 0.0156Methane 0.193 0.271 0.363 0.229 0.365 0.482 0.5248 0.5335 0.5921Ethane 0.032 0.034 0.038 0.029 0.058 0.065 0.145 0.0679 0.066 Propane 0.585 0.482 0.374 0.407 0.242 0.167 0.157 0.2035 0.1043I-Butane 0.007 0.007 0.009 0.006 0.007 0.007 0.0055 0.0063 0.0058n-Butane 0.012 0.015 0.015 0.172 0.215 0.14 0.0179 0.204 0.0831I-Pentane 0.005 0.006 0.006 0.005 0.004 0.006 0.0055 0.0062 0.0054n-Pentane 0.007 0.007 0.008 0.006 0.007 0.008 0.0079 0.0092 0.0078Hexane 0.013 0.015 0.016 0.012 0.01 0.012 0.0082 0.0094 0.0081Heptanes + 0.141 0.157 0.165 0.129 0.082 0.11 0.1133 0.1296 0.1118 Avg. Mol. Wt. 67.3 68.5 67.5 66.1 48.8 48.8 45 48.4 44.9 Heptanes + Properties Mol. Wt. 243 243 243 243 191 191 191 191 191 Gravity(API) Characterization Factor 11.6 11.6 11.6 11.6 11.9 11.9 11.9 11.9 11.9

Critical Temp., Tc (F) 265 265 265 265 234 200 200 200 200

Critical Temp., Tc (oR) 725 725 725 725 694 660 660 660 660 This work 725.39 725.45 724.58 725.81 694.91 694.44 659.17 661.13 659.22

139

Table C. 11. (Continued)

Mixture No. 145 - 10 145 - 11 145 - 12 145 - 13 145 - 14 4 - 1 4 - 2 4 - 3 4 - 4 4 - 5 Nitrogen 0.0016 0.002 Carbon Dioxide 0.0153 0.0008 0.0139 0.0006 0.0008 0.0066 0.0067 0.0061 0.0061 0.0059Methane 0.5774 0.6298 0.6165 0.4786 0.5808 0.7243 0.7292 0.8227 0.8165 0.795 Ethane 0.0631 0.0858 0.0602 0.082 0.0562 0.0557 0.0556 0.0284 0.0284 0.0281Propane 0.1178 0.0672 0.0961 0.1359 0.0919 0.0308 0.0306 0.0124 0.0124 0.0128I-Butane 0.0055 0.023 0.0055 0.0076 0.0081 n-Butane 0.0976 0.0305 0.0773 0.1273 0.082 0.0241 0.0226 0.0091 0.0092 0.0098I-Pentane 0.005 0.0114 0.0053 0.0066 0.0071 n-Pentane 0.0073 0.0122 0.0077 0.0069 0.0073 0.015 0.0147 0.0067 0.007 0.0078Hexane 0.0075 0.0206 0.008 0.0139 0.0148 0.0179 0.0183 0.0125 0.0131 0.015 Heptanes + 0.1035 0.1187 0.1095 0.139 0.1488 0.1256 0.1223 0.1021 0.1073 0.1247 Avg. Mol. Wt. 44.3 44.4 43.4 54.7 53.1 39.9 39.3 33 33.8 36.5 Heptanes + Properties Mol. Wt. 191 167 191 205 205 167 167 158 158 158 Gravity(API) Characterization Fact. 11.9 11.7 11.9 11.6 11.6 11.8 11.8 11.9 11.9 11.9

Critical Temp., Tc (F) 200 292 200 180 180 285 251 100 160 212

Critical Temp., Tc (oR) 660 752 660 640 640 745 711 560 620 672 This work 658.69 751.75 658.69 639.1 640.61 745.65 709.85 648.77 620.21 670.58

140

Table C. 12. (Continued)

Mixture No. 4 - 6 4 - 7 4 - 8 4 - 9 4-10 75 - 1 75 - 2 75 - 3 75 - 4 75 - 5 Nitrogen 0.0128 0.0058 0.0053 0.0054 0.0038Carbon Dioxide 0.0035 0.0156 0.0008 0.0075 0.0074 0.0045Methane 0.6433 0.6865 0.657 0.7164 0.8213 0.788 0.724 0.728 0.597 0.83 Ethane 0.0638 0.0603 0.0869 0.0548 0.0637 0.059 0.0542 0.0546 0.089 0.0378Propane 0.0605 0.0232 0.0537 0.031 0.0409 0.0315 0.03 0.0302 0.05 0.0144I-Butane n-Butane 0.0565 0.0309 0.0303 0.0258 0.0235 0.0265 0.031 0.0207 0.049 0.0089I-Pentane n-Pentane 0.0404 0.0243 0.0195 0.0216 0.0122 0.0425 0.071 0.0688 0.093 0.0436Hexane 0.0366 0.0256 0.0173 0.0198 0.0103 0.0252 0.0456 0.0438 0.0308Heptanes + 0.0954 0.1245 0.1217 0.1231 0.0234 0.0214 0.0388 0.0375 0.122 0.0263 Avg. Mol. Wt. 35.2 41.5 45.7 40.1 25 29.5 29.1 36.2 24.5 Heptanes + Properties Mol. Wt. 114 171 207 167 114 106 106 106 100 106 Gravity (API) Characterization Fact. 11.7 12 12 11.8

Critical Temp., Tc (F) 195 239 145 243 55 109 109 169 54

Critical Temp., Tc (oR) 655 699 605 703 515 569 569 629 514 This work 655.74 698.49 606.37 702.66 514.97 568.77 569.16 628.91 514.26

141

Table C. 13. (Continued)

Mixture No. 75 - 6 75 - 7 75 - 8 141 - 1 141 - 2 141 - 3 141 - 4 141 - 5 141 - 6 Nitrogen 0.0038 0.0036 0.003 0.1101 0.0477 0.0156 0.0123 Carbon Dioxide 0.0044 0.0043 0.0035 0.0008 0.0019 0.0047 0.0028 0.0172 0.011 Methane 0.815 0.784 0.643 0.3465 0.4675 0.4656 0.6173 0.6527 0.586 Ethane 0.0372 0.0355 0.0294 0.1561 0.1306 0.1645 0.1024 0.0725 0.1309Propane 0.0141 0.0136 0.0111 0.1499 0.105 0.1163 0.0673 0.0391 0.0699I-Butane 0.0121 0.0112 0.0141 0.0085 0.0157 0.0081n-Butane 0.0102 0.013 0.0252 0.0852 0.0472 0.0454 0.0352 0.0204 0.0349I-Pentane 0.0084 0.0169 0.0112 0.0106 0.016 0.0097n-Pentane 0.0501 0.0631 0.123 0.0109 0.0192 0.0182 0.0171 0.0123 0.0187Hexane 0.0354 0.0447 0.0871 0.02 0.0373 0.0215 0.0226 0.029 0.0194Heptanes + 0.0303 0.0382 0.0747 0.1 0.1155 0.1229 0.1039 0.1251 0.1114 Avg. Mol. Wt. 25.4 27.6 37.7 Heptanes + Properties Mol. Wt. 106 106 106 164 142 168 144 157 184 Gravity (API) 40.6 42.5 40.1 47.3 49.5 42.2 Characterization Fact. 11.6 11.55 11.75 11.75 12 11.85

Critical Temp., Tc (F) 65 90 189 110 169 180 159 262 185

Critical Temp., Tc (oR) 525 550 649 570 629 640 619 722 645 This work 524.27 540.38 648.47 570.06 629.48 640.76 618.84 721.94 646.07

142

Table C. 14. (Continued)

Mixture No. 141 - 7 141 - 8 141 - 9 141 - 10 141 - 11 141 - 12 141 - 13 141 - 14 141 - 15 Nitrogen 0.0009 0.0012 0.0013 Carbon Dioxide 0.0088 0.0041 0.0048 0.0056 0.0043 0.0038 0.0962 0.0776 0.0701Methane 0.722 0.7052 0.7175 0.6992 0.5564 0.4902 0.5431 0.4371 0.3954Ethane 0.0474 0.0725 0.0748 0.0638 0.0523 0.0464 0.1149 0.0932 0.0848Propane 0.0278 0.0465 0.0456 0.0404 0.2108 0.297 0.0768 0.216 0.2738I-Butane 0.0105 0.019 0.0177 0.0081 0.0069 0.0062 0.0147 0.0123 0.0115n-Butane 0.0158 0.0129 0.0136 0.0169 0.0147 0.0133 0.0257 0.0223 0.0207I=Pentane 0.0112 0.0064 0.0073 0.0079 0.007 0.0064 0.0082 0.0076 0.0072n-Pentane 0.008 0.008 0.0046 0.0058 0.0052 0.0047 0.0082 0.0078 0.0076Hexane 0.0181 0.01 0.0099 0.011 0.01 0.0093 0.0126 0.013 0.013 Heptanes + 0.1297 0.1142 0.1031 0.1413 0.1324 0.1227 0.0996 0.1131 0.1159 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 178 211 211 202 202 202 253 253 253 Gravity (API) 42.2 36.7 36.7 35 35 35 29 29 29 Characterization Fact. 11.8 11.7 11.7 11.55 11.55 11.55 11.55 11.55 11.55

Critical Temp., Tc (F) 265 226 216 202 202 202 280 280 280

Critical Temp., Tc (oR) 725 686 676 662 662 662 740 740 740 This work 724.53 686.98 676.99 661.32 661.89 660.87 739.24 741.2 740.99

143

Table C. 15. (Continued)

Mixture No. 141 - 16 141 - 17 141 - 18 141 - 19 141 - 20 141 - 21 141 - 22 141 - 23 141 - 24 Nitrogen 0.0052 0.0007 0.0008 0.0004 Carbon Dioxide 0.0871 0.0776 0.0778 0.0006 0.0011 0.0008 0.0008 0.0023 0.0022Methane 0.4911 0.4384 0.654 0.6091 0.5432 0.4789 0.3678 0.77 0.7247Ethane 0.1041 0.0935 0.0889 0.0865 0.0706 0.0624 0.0488 0.0316 0.0371Propane 0.0699 0.0632 0.061 0.0765 0.195 0.2786 0.4154 0.0228 0.0473I-Butane 0.0135 0.0123 0.0198 0.0183 0.0078 0.0075 0.007 0.0149 0.026 n-Butane 0.1063 0.1768 0.0317 0.0304 0.014 0.0132 0.0124 0.019 0.028 I-Pentane 0.0078 0.0074 0.0189 0.0134 0.0077 0.0072 0.0067 0.0084 0.0148n-Pentane 0.0077 0.0077 0.0172 0.0094 0.006 0.0057 0.0053 0.0137 0.0167Hexane 0.0123 0.0128 0.0307 0.0184 0.0165 0.0156 0.0145 0.0212 0.0202Heptanes + 0.1002 0.1103 0.0692 0.1374 0.1368 0.1293 0.1209 0.0961 0.083 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 253 253 126 229 229 229 229 152 152 Gravity (API) 29 29 60.4 30.2 30.2 30.2 30.2 51 51 Characterization Fact. 11.55 11.55 12.3 11.5 11.5 11.5 11.5 12.1 12.1

Critical Temp. Tc (F) 280 280 144 221 221 221 221 183 183

Critical Temp., Tc (oR) 740 740 604 681 681 681 681 643 643 This work 739.6 741.28 603.31 680.17 681.52 681.56 680.55 642.69 643.04

144

Table C. 16. (Continued)

Mixture No. 141 - 25

141 - 26

141 - 27

141 - 28

141 - 29

141 - 30

141 - 31

93 - 1 37 - 1 158 - 1

Nitrogen 0.0487 0.0249 0.0526 0.0115 0.0017 0.1898 0.0052 0.0076Carbon Dioxide 0.0014 0.0029 0.0149 0.2202 0.0073 0.003 0.0013Hydrogen Sulfide .0068b Methane 0.4534 0.4957 0.6414 0.5325 0.2756 0.695 0.4004 0.5832 0.6849 0.5391Ethane 0.0705 0.1379 0.0733 0.1428 0.1144 0.0696 0.0767 0.1355 0.0971 0.142 Propane 0.1961 0.1015 0.0605 0.0926 0.2486 0.0426 0.0347 0.0761 0.0542 0.0964I-Butane 0.0936 0.0222 0.0154 0.0183 0.085 0.0109 0.0064 0.0084 0.0086 n-Butane 0.0825 0.0403 0.0298 0.0349 0.1294 0.0157 0.0204 0.0319 0.0223 0.0554I-Pentane 0.0542 0.0162 0.0142 0.0134 0.043 0.0102 0.0046 0.008 0.0088 n-Pentane 0.0343 0.0189 0.0178 0.0179 0.027 0.0086 0.0067 0.0161 0.0092 0.0299Hexane 0.0129 0.0234 0.0233 0.0318 0.0161 0.0184 0.0084 0.019 0.0142 0.0272Heptanes + 0.0011 0.0952 0.0994 0.0632 0.0465 0.1124 0.0181 0.1145 .0925a 0.1011 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 114 161 161 117 192 180 115 193 147 178.5 Sp. Gravity or API 68.7 47.4 47.4 59.4 41 0.825 0.76 0.8135 Characterization Fact. 12.6 11.9 11.9 12.1 11.85 11.9 Critical Temp., Tc (F) 183 161 161 161 171 240 35 190 120 + 30 193

Critical Temp., Tc (oR) 643 621 621 621 631 700 495 650 #VALUE! 653

This work 642.83 620.95 620.18 620.19 630.34 700.18 494.44 650.59 580.38 651.87(a) Heptanes plus treated as individual components through hexadecane (b) H2S treated as propane

145

Table C. 17. (Continued)

Mixture No. 58 - 1 11 - 1 6 - 1 6 - 2 6 - 3 30 - 1 52 - 2 52 - 3 52 - 6 Nitrogen 0.0066 0.0024 0.0021 0.002 0.0018 0.0124 0.0027Carbon Dioxide 0.018 0.0302 0.0084 0.0076 0.007 0.0065 0.0022 0.0003Methane 0.5966 0.6073 0.7504 0.6822 0.6253 0.5772 0.5706 0.5428 0.604 Ethane 0.1289 0.1177 0.0537 0.0489 0.0448 0.0414 0.1496 0.149 ..0517 propane 0.0653 0.0891 0.0136 0.0124 0.0114 0.0105 0.0825 0.0912 0.0757I-Butane 0.0173 0.0198 0.0021 0.0019 0.0018 0.0016 0.0092 0.0072 0.0455n-Butane 0.0219 0.0279 0.0127 0.1025 0.1773 0.2405 0.0367 0.0456 0.0389I-Pentane 0.01 0.0124 0.0106 0.0096 0.0088 0.0082 0.0064 0.0249n-Pentane 0.0096 0.008 0.0099 0.009 0.0082 0.0076 0.018 0.0186 0.0182Hexane 0.0132 0.0237 0.0215 0.0197 0.0182 0.0217 0.0255 0.0394

Heptanes + 0.1126 0.0878 .1125a .1023a .0937a .0865a 0.0907 0.1201 0.0897 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 155 172 118 118 118 118 173 166 133 Specific Gravity or API 0.7994 38 41.2 0.8126 51.9 Characterization Fact. 11.8 11.5 11.7 11.7 11.85

Critical Temp., Tc (F) 264 125 172.5 297.5 232.5 252.5 170 170 245

Critical Temp., Tc (oR) 724 585 632.5 757.5 692.5 712.5 630 630 705 This work 722.91 585.81 631.89 667.01 691.88 712.66 630.6 629.44 705.62

(a) Heptanes plus treated as individual components through hexadecane

146

Table C. 18. (Continued)

Mixture No. 47 - 1 26 - 1 26 - 2 26 - 3 47 - 2 26 - 4 26 - 5 26 - 6 26 - 7 26 - 8 Nitrogen 0.0884 0.1611 0.2441 0.113 0.24 0.1146 0.135 0.0705Carbon Dioxide 0.012 0.0109 0.01 0.0091 0.0044 0.003 0.002 0.0013 0.002 0.0025Methane 0.9089 0.8286 0.7625 0.887 0.858 0.7364 0.7364 0.7665 0.7515 0.8532Ethane 0.044 0.0401 0.0369 0.0333 0.016 0.015 0.012 0.0551 0.061 0.0411Propane 0.0191 0.0174 0.016 0.0144 0.007 0.006 0.0053 0.0335 0.0327 0.0198I-Butane 0.0033 0.003 0.0028 0.003 0.0014 0.0012 0.001 0.0035 0.0038 0.0037n-Butane 0.006 0.0055 0.0051 0.004 0.002 0.0018 0.0015 0.009 0.006 0.0039I-Pentane 0.0021 0.0019 0.0018 0.0016 0.0007 0.0006 0.0005 0.0017

n-Pentane 0.0013 0.0012 0.0011 0.001 0.0005 0.0004 0.0004 0.0015 .0020c .0022cHexane 0.0015 0.0014 0.0012 0.0011 0.0005 0.0004 0.0004

Heptanes + 0.0018 0.0016 0.0015 0.0014 0.0007 0.0006 0.0005 .0033d Helium 0.01 0.006 0.0031Avg. Mol. Wt. 18.4 19.25 19.94 20.75 16.9 18.11 19.56 20.02 19.98 18.6 Heptanes + Properties Mol. Wt. 115 115 115 115 115 115 115 100 Gravity (API) Characterization Fact.

Critical Temp., Tc (F) -79 -92 -104 -120 -101 -117 -131 -84 -90 -96

Critical Temp., Tc (oR) 381 368 356 340 359 343 329 376 370 364 This work 380.91 368.34 355.94 340.07 359.03 343.05 329.04 375.88 370.01 364.11© Heavy fraction treated as n-pentane. (d) Heavy fraction treated as n-heptane. (e) Helium treated as methane.

147

Table C.19. Critical Pressure, Temperature and Properties Predictions for Heptane Plus

(Compositions in Mole Fractions) Mixture No. 145-1 145-2 145-3 145-4 145-5 145-6 145-7 145-8 145-9 Nitrogen 0.001 0.001 0.001 0.001 0.004 0.001 Carbon Dioxide 0.004 0.005 0.005 0.004 0.006 0.002 0.0149 0.014 0.0156Methane 0.193 0.271 0.363 0.229 0.365 0.482 0.5248 0.5335 0.5921Ethane 0.032 0.034 0.038 0.029 0.058 0.065 0.145 0.0679 0.066 Propane 0.585 0.482 0.374 0.407 0.242 0.167 0.157 0.2035 0.1043I-Butane 0.007 0.007 0.009 0.006 0.007 0.007 0.0055 0.0063 0.0058n-Butane 0.012 0.015 0.015 0.172 0.215 0.14 0.0179 0.204 0.0831I-Pentane 0.005 0.006 0.006 0.005 0.004 0.006 0.0055 0.0062 0.0054n-Pentane 0.007 0.007 0.008 0.006 0.007 0.008 0.0079 0.0092 0.0078Hexane 0.013 0.015 0.016 0.012 0.01 0.012 0.0082 0.0094 0.0081Heptanes + 0.141 0.157 0.165 0.129 0.082 0.11 0.1133 0.1296 0.1118 Avg. Mol. Wt. 67.3 68.5 67.5 66.1 48.8 48.8 45 48.4 44.9 Heptanes + Properties Mol. Wt. 243 243 243 243 191 191 191 191 191 Gravity(API) 33.31 33.31 33.31 33.31 39.98 39.98 39.98 39.98 39.98 SG 0.86 0.86 0.86 0.86 0.83 0.83 0.83 0.83 0.83 Characterization Factor 11.6 11.6 11.6 11.6 11.9 11.9 11.9 11.9 11.9 Tb 1037.0 1037.0 1037.0 1037.0 920.9 920.9 920.9 920.9 920.9 C 3.3475 3.3475 3.3475 3.3475 3.2304 3.2304 3.2304 3.2304 3.2304 Pc, psia 271.0 271.0 271.0 271.0 319.8 319.8 319.8 319.8 319.8

Tc, oR 1364.4 1364.4 1364.4 1364.4 1258.4 1258.4 1258.4 1258.4 1258.4 ω 0.5673 0.5673 0.5673 0.5673 0.4676 0.4676 0.4676 0.4676 0.4676

Zc 0.2416 0.2416 0.2416 0.2416 0.2493 0.2493 0.2493 0.2493 0.2493

Ωω 0.3555 0.3555 0.3555 0.3555 0.3564 0.3564 0.3564 0.3564 0.3564

148

Table C. 20. (Continued)

Mixture No. 145 - 10 145 - 11 145 - 12 145 - 13 145 - 14 4 - 1 4 - 2 4 - 3 4 - 4 4 - 5 Nitrogen 0.0016 0.002 Carbon Dioxide 0.0153 0.0008 0.0139 0.0006 0.0008 0.0066 0.0067 0.0061 0.0061 0.0059Methane 0.5774 0.6298 0.6165 0.4786 0.5808 0.7243 0.7292 0.8227 0.8165 0.795 Ethane 0.0631 0.0858 0.0602 0.082 0.0562 0.0557 0.0556 0.0284 0.0284 0.0281Propane 0.1178 0.0672 0.0961 0.1359 0.0919 0.0308 0.0306 0.0124 0.0124 0.0128I-Butane 0.0055 0.023 0.0055 0.0076 0.0081 n-Butane 0.0976 0.0305 0.0773 0.1273 0.082 0.0241 0.0226 0.0091 0.0092 0.0098I-Pentane 0.005 0.0114 0.0053 0.0066 0.0071 n-Pentane 0.0073 0.0122 0.0077 0.0069 0.0073 0.015 0.0147 0.0067 0.007 0.0078Hexane 0.0075 0.0206 0.008 0.0139 0.0148 0.0179 0.0183 0.0125 0.0131 0.015 Heptanes + 0.1035 0.1187 0.1095 0.139 0.1488 0.1256 0.1223 0.1021 0.1073 0.1247 Avg. Mol. Wt. 44.3 44.4 43.4 54.7 53.1 39.9 39.3 33 33.8 36.5 Heptanes + Properties Mol. Wt. 191 167 191 205 205 167 167 158 158 158 Gravity(API) 39.98 44.46 39.98 37.85 37.85 44.46 44.46 46.49 46.49 46.49 SG 0.83 0.80 0.83 0.84 0.84 0.80 0.80 0.79 0.79 0.79 Characterization Fact. 11.9 11.7 11.9 11.6 11.6 11.8 11.8 11.9 11.9 11.9 Tb 920.9 860.6 920.9 953.9 953.9 860.6 860.6 836.6 836.6 836.6 C 3.2304 3.1667 3.2304 3.2644 3.2644 3.1667 3.1667 3.1407 3.1407 3.1407 Pc,psia 319.8 348.0 319.8 305.1 305.1 348.0 348.0 359.7 359.7 359.7

Tc, oR 1258.4 1200.2 1258.4 1289.4 1289.4 1200.2 1200.2 1176.3 1176.3 1176.3 ω 0.4676 0.4209 0.4676 0.4946 0.4946 0.4209 0.4209 0.4033 0.4033 0.4033

Zc 0.2493 0.2531 0.2493 0.2472 0.2472 0.2531 0.2531 0.2545 0.2545 212

Ωω 0.3564 0.3569 0.3564 0.3562 0.3562 0.3569 0.3569 0.3571 0.3571 5620

149

Table C. 21. (Continued) Mixture No. 4 - 6 4 - 7 4 - 8 4 - 9 4-10 75 - 1 75 - 2 75 - 3 75 - 4 75 - 5 Nitrogen 0.0128 0.0058 0.0053 0.0054 0.0038Carbon Dioxide 0.0035 0.0156 0.0008 0.0075 0.0074 0.0045Methane 0.6433 0.6865 0.657 0.7164 0.8213 0.788 0.724 0.728 0.597 0.83 Ethane 0.0638 0.0603 0.0869 0.0548 0.0637 0.059 0.0542 0.0546 0.089 0.0378Propane 0.0605 0.0232 0.0537 0.031 0.0409 0.0315 0.03 0.0302 0.05 0.0144I-Butane n-Butane 0.0565 0.0309 0.0303 0.0258 0.0235 0.0265 0.031 0.0207 0.049 0.0089I-Pentane n-Pentane 0.0404 0.0243 0.0195 0.0216 0.0122 0.0425 0.071 0.0688 0.093 0.0436Hexane 0.0366 0.0256 0.0173 0.0198 0.0103 0.0252 0.0456 0.0438 0.0308Heptanes + 0.0954 0.1245 0.1217 0.1231 0.0234 0.0214 0.0388 0.0375 0.122 0.0263 Avg. Mol. Wt. 35.2 41.5 45.7 40.1 25 29.5 29.1 36.2 24.5 Heptanes + Properties Mol. Wt. 114 171 207 167 114 106 106 106 100 106 Gravity (API) 61.04 43.63 37.57 44.46 61.04 64.98 64.98 64.98 68.35 64.98 SG 0.73 0.81 0.84 0.80 0.73 0.72 0.72 0.72 0.71 0.72 Characterization Fact. 11.7 12 12 11.8 Tb 704.8 871.0 958.5 860.6 704.8 677.6 677.6 677.6 656.3 677.6 C 2.9917 3.1778 3.2691 3.1667 2.9917 2.9594 2.9594 2.9594 2.9337 2.9594 Pc,psia 427.1 343.0 303.1 348.0 427.1 441.1 441.1 441.1 452.0 441.1

Tc, oR 1038.0 1210.4 1293.6 1200.2 1038.0 1007.7 1007.7 1007.7 983.5 1007.7 ω 0.3158 0.4287 0.4985 0.4209 0.3158 0.2996 0.2996 0.2996 0.2874 0.2996

Zc 0.2620 0.2524 0.2469 0.2531 0.2620 0.2634 0.2634 0.2634 0.2645 212

Ωω 0.3579 0.3568 0.3561 0.3569 0.3579 0.3581 0.3581 0.3581 0.3582 5620

150

Table C. 22. (Continued) Mixture No. 75 - 6 75 - 7 75 - 8 141 - 1 141 - 2 141 - 3 141 - 4 141 - 5 141 - 6 Nitrogen 0.0038 0.0036 0.003 0.1101 0.0477 0.0156 0.0123 Carbon Dioxide 0.0044 0.0043 0.0035 0.0008 0.0019 0.0047 0.0028 0.0172 0.011 Methane 0.815 0.784 0.643 0.3465 0.4675 0.4656 0.6173 0.6527 0.586 Ethane 0.0372 0.0355 0.0294 0.1561 0.1306 0.1645 0.1024 0.0725 0.1309Propane 0.0141 0.0136 0.0111 0.1499 0.105 0.1163 0.0673 0.0391 0.0699I-Butane 0.0121 0.0112 0.0141 0.0085 0.0157 0.0081n-Butane 0.0102 0.013 0.0252 0.0852 0.0472 0.0454 0.0352 0.0204 0.0349I-Pentane 0.0084 0.0169 0.0112 0.0106 0.016 0.0097n-Pentane 0.0501 0.0631 0.123 0.0109 0.0192 0.0182 0.0171 0.0123 0.0187Hexane 0.0354 0.0447 0.0871 0.02 0.0373 0.0215 0.0226 0.029 0.0194Heptanes + 0.0303 0.0382 0.0747 0.1 0.1155 0.1229 0.1039 0.1251 0.1114 Avg. Mol. Wt. 25.4 27.6 37.7 Heptanes + Properties Mol. Wt. 106 106 106 164 142 168 144 157 184 Gravity (API) 64.98 64.98 64.98 40.6 42.5 40.1 47.3 49.5 42.2 SG 0.72 0.72 0.72 0.82 0.81 0.82 0.79 0.78 0.81 Characterization Fact. 11.6 11.55 11.75 11.75 12 11.85 Tb 677.6 677.6 677.6 862.2 807.5 872.1 802.9 828.4 901.6 C 2.9594 2.9594 2.9594 3.1592 3.0931 3.1704 3.0983 3.1372 3.2123 Pc,psia 441.1 441.1 441.1 371.8 421.8 364.8 392.3 349.4 323.4

Tc, oR 1007.7 1007.7 1007.7 1211.9 1164.0 1220.9 1148.0 1162.3 1237.5 ω 0.2996 0.2996 0.2996 0.4076 0.3599 0.4159 0.3715 0.4057 0.4559

Zc 0.2634 0.2634 0.2634 0.2542 0.2582 0.2535 0.2572 0.2543 0.2502

Ωω 0.3581 0.3581 0.3581 0.3570 0.3575 0.3569 0.3574 0.3570 0.3565

151

Table C. 23. (Continued) Mixture No. 141 - 7 141 - 8 141 - 9 141 - 10 141 - 11 141 - 12 141 - 13 141 - 14 141 - 15 Nitrogen 0.0009 0.0012 0.0013 Carbon Dioxide 0.0088 0.0041 0.0048 0.0056 0.0043 0.0038 0.0962 0.0776 0.0701Methane 0.722 0.7052 0.7175 0.6992 0.5564 0.4902 0.5431 0.4371 0.3954Ethane 0.0474 0.0725 0.0748 0.0638 0.0523 0.0464 0.1149 0.0932 0.0848Propane 0.0278 0.0465 0.0456 0.0404 0.2108 0.297 0.0768 0.216 0.2738i-Butane 0.0105 0.019 0.0177 0.0081 0.0069 0.0062 0.0147 0.0123 0.0115n-Butane 0.0158 0.0129 0.0136 0.0169 0.0147 0.0133 0.0257 0.0223 0.0207i=Pentane 0.0112 0.0064 0.0073 0.0079 0.007 0.0064 0.0082 0.0076 0.0072n-Pentane 0.008 0.008 0.0046 0.0058 0.0052 0.0047 0.0082 0.0078 0.0076Hexane 0.0181 0.01 0.0099 0.011 0.01 0.0093 0.0126 0.013 0.013 Heptanes + 0.1297 0.1142 0.1031 0.1413 0.1324 0.1227 0.0996 0.1131 0.1159 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 178 211 211 202 202 202 253 253 253 Gravity (API) 42.2 36.7 36.7 35 35 35 29 29 29 SG 0.81 0.84 0.84 0.85 0.85 0.85 0.88 0.88 0.88 Characterization Fact. 11.8 11.7 11.7 11.55 11.55 11.55 11.55 11.55 11.55 Tb 889.0 968.5 968.5 954.9 954.9 954.9 1066.8 1066.8 1066.8 C 3.1968 3.2784 3.2784 3.2581 3.2581 3.2581 3.3684 3.3684 3.3684 Pc,psia 334.8 300.5 300.5 321.3 321.3 321.3 274.9 274.9 274.9

Tc, oR 1228.0 1303.6 1303.6 1298.7 1298.7 1298.7 1400.4 1400.4 1400.4 ω 0.4423 0.5055 0.5055 0.4822 0.4822 0.4822 0.5778 0.5778 0.5778

Zc 0.2513 0.2463 0.2463 0.2481 0.2481 0.2481 0.2408 0.2408 0.2408

Ωω 0.3567 0.3561 0.3561 0.3563 0.3563 0.3563 0.3554 0.3554 0.3554

152

Table C. 24. (Continued) Mixture No. 141 - 16 141 - 17 141 - 18 141 - 19 141 - 20 141 - 21 141 - 22 141 - 23 141 - 24 Nitrogen 0.0052 0.0007 0.0008 0.0004 Carbon Dioxide 0.0871 0.0776 0.0778 0.0006 0.0011 0.0008 0.0008 0.0023 0.0022Methane 0.4911 0.4384 0.654 0.6091 0.5432 0.4789 0.3678 0.77 0.7247Ethane 0.1041 0.0935 0.0889 0.0865 0.0706 0.0624 0.0488 0.0316 0.0371Propane 0.0699 0.0632 0.061 0.0765 0.195 0.2786 0.4154 0.0228 0.0473I-Butane 0.0135 0.0123 0.0198 0.0183 0.0078 0.0075 0.007 0.0149 0.026 n-Butane 0.1063 0.1768 0.0317 0.0304 0.014 0.0132 0.0124 0.019 0.028 I-Pentane 0.0078 0.0074 0.0189 0.0134 0.0077 0.0072 0.0067 0.0084 0.0148n-Pentane 0.0077 0.0077 0.0172 0.0094 0.006 0.0057 0.0053 0.0137 0.0167Hexane 0.0123 0.0128 0.0307 0.0184 0.0165 0.0156 0.0145 0.0212 0.0202Heptanes + 0.1002 0.1103 0.0692 0.1374 0.1368 0.1293 0.1209 0.0961 0.083 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 253 253 126 229 229 229 229 152 152 Gravity (API) 29 29 60.4 30.2 30.2 30.2 30.2 51 51 SG 0.88 0.88 0.74 0.88 0.88 0.88 0.88 0.78 0.78 Characterization Fact. 11.55 11.55 12.3 11.5 11.5 11.5 11.5 12.1 12.1 Tb 1066.8 1066.8 736.3 1019.5 1019.5 1019.5 1019.5 814.3 814.3 C 3.3684 3.3684 3.0359 3.3194 3.3194 3.3194 3.3194 3.1221 3.1221 Pc,psia 274.9 274.9 387.6 300.1 300.1 300.1 300.1 355.0 355.0

Tc, oR 1400.4 1400.4 1064.6 1361.9 1361.9 1361.9 1361.9 1147.6 1147.6 ω 0.5778 0.5778 0.3454 0.5301 0.5301 0.5301 0.5301 0.3961 0.3961

Zc 0.2408 0.2408 0.2594 0.2444 0.2444 0.2444 0.2444 0.2551 0.2551

Ωω 0.3554 0.3554 0.3576 0.3558 0.3558 0.3558 0.3558 0.3571 0.3571

153

Table C. 25. (Continued) Mixture No. 141 - 25 141 - 26 141 - 27 141 - 28 141 - 29 141 - 30 141 - 31 93 - 1 37 - 1 158 - 1 Nitrogen 0.0487 0.0249 0.0526 0.0115 0.0017 0.1898 0.0052 0.0076Carbon Dioxide 0.0014 0.0029 0.0149 0.2202 0.0073 0.003 0.0013

Hydrogen Sulfide .0068b Methane 0.4534 0.4957 0.6414 0.5325 0.2756 0.695 0.4004 0.5832 0.6849 0.5391Ethane 0.0705 0.1379 0.0733 0.1428 0.1144 0.0696 0.0767 0.1355 0.0971 0.142Propane 0.1961 0.1015 0.0605 0.0926 0.2486 0.0426 0.0347 0.0761 0.0542 0.0964I-Butane 0.0936 0.0222 0.0154 0.0183 0.085 0.0109 0.0064 0.0084 0.0086 n-Butane 0.0825 0.0403 0.0298 0.0349 0.1294 0.0157 0.0204 0.0319 0.0223 0.0554I-Pentane 0.0542 0.0162 0.0142 0.0134 0.043 0.0102 0.0046 0.008 0.0088 n-Pentane 0.0343 0.0189 0.0178 0.0179 0.027 0.0086 0.0067 0.0161 0.0092 0.0299Hexane 0.0129 0.0234 0.0233 0.0318 0.0161 0.0184 0.0084 0.019 0.0142 0.0272

Heptanes + 0.0011 0.0952 0.0994 0.0632 0.0465 0.1124 0.0181 0.1145 .0925a 0.1011 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 114 161 161 117 192 180 115 193 147 178.5Sp. Gravity or API 68.7 47.4 47.4 59.4 41 0.825 0.76 0.8135

SG 0.71 0.79 0.79 0.74 0.82 0.825 0.76 0.8135 0.51 0.46 Characterization Fact. 12.6 11.9 11.9 12.1 11.85 11.9 Tb 693.2 841.5 841.5 715.2 920.6 898.1 717.6 919.4 671.4 697.4 C 2.9903 3.1492 3.1492 3.0034 3.2326 3.2025 2.9968 3.2347 3.0905 3.1752 Pc,psia 393.0 348.9 348.9 423.3 314.0 340.0 454.6 306.9 150.5 98.7

Tc, oR 1012.8 1177.7 1177.7 1049.9 1255.3 1241.1 1062.2 1250.3 875.5 858.5 ω 0.3245 0.4118 0.4118 0.3215 0.4718 0.4429 0.3109 0.4768 0.5151 0.6618

Zc 0.2612 0.2538 0.2538 0.2615 0.2490 0.2513 0.2624 0.2486 0.2456 0.2347

Ωω 0.3578 0.3570 0.3570 0.3578 0.3564 0.3567 0.3580 0.3563 0.3560 0.3546 (a) Heptanes plus treated as individual components through hexadecane (b) H2S treated as propane

154

Table C. 26. (Continued) Mixture No. 58 - 1 11 - 1 6 - 1 6 - 2 6 - 3 30 - 1 52 - 2 52 - 3 52 - 6 Nitrogen 0.0066 0.0024 0.0021 0.002 0.0018 0.0124 0.0027Carbon Dioxide 0.018 0.0302 0.0084 0.0076 0.007 0.0065 0.0022 0.0003Methane 0.5966 0.6073 0.7504 0.6822 0.6253 0.5772 0.5706 0.5428 0.604Ethane 0.1289 0.1177 0.0537 0.0489 0.0448 0.0414 0.1496 0.149 ..0517propane 0.0653 0.0891 0.0136 0.0124 0.0114 0.0105 0.0825 0.0912 0.0757I-Butane 0.0173 0.0198 0.0021 0.0019 0.0018 0.0016 0.0092 0.0072 0.0455n-Butane 0.0219 0.0279 0.0127 0.1025 0.1773 0.2405 0.0367 0.0456 0.0389I-Pentane 0.01 0.0124 0.0106 0.0096 0.0088 0.0082 0.0064 0.0249n-Pentane 0.0096 0.008 0.0099 0.009 0.0082 0.0076 0.018 0.0186 0.0182Hexane 0.0132 0.0237 0.0215 0.0197 0.0182 0.0217 0.0255 0.0394

Heptanes + 0.1126 0.0878 .1125a .1023a .0937a .0865a 0.0907 0.1201 0.0897 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 155 172 118 118 118 118 173 166 133 Specific Gravity or API 0.7994 38 59.27 59.27 59.27 59.27 41.2 0.8126 51.9

SG 0.7994 0.83 0.74 0.74 0.74 0.74 0.82 0.81 0.77 Characterization Fact. 11.8 11.5 11.7 11.7 11.85Tb 831.8 885.4 718.0 718.0 718.0 718.0 880.6 862.3 768.0 C 3.1322 3.1818 3.0072 3.0072 3.0072 3.0072 3.1838 3.1643 3.0616 Pc,psia 371.3 365.5 420.2 420.2 420.2 420.2 349.1 358.0 403.6

Tc, oR 1175.2 1237.2 1052.4 1052.4 1052.4 1052.4 1224.4 1206.4 1109.2 ω 0.3947 0.4213 0.3238 0.3238 0.3238 0.3238 0.4291 0.4156 0.3516

Zc 0.2552 0.2530 0.2613 0.2613 0.2613 0.2613 0.2524 0.2535 0.2589

Ωω 0.3571 0.3569 0.3578 0.3578 0.3578 0.3578 0.3568 0.3569 0.3576

(a) Heptanes plus treated as individual components through hexadecane

155

Table C. 27. (Continued) Mixture No. 47 - 1 26 - 1 26 - 2 26 - 3 47 - 2 26 - 4 26 - 5 26 - 6 26 - 7 26 - 8 Nitrogen 0.0884 0.1611 0.2441 0.113 0.24 0.1146 0.135 0.0705Carbon Dioxide 0.012 0.0109 0.01 0.0091 0.0044 0.003 0.002 0.0013 0.002 0.0025Methane 0.9089 0.8286 0.7625 0.887 0.858 0.7364 0.7364 0.7665 0.7515 0.8532Ethane 0.044 0.0401 0.0369 0.0333 0.016 0.015 0.012 0.0551 0.061 0.0411Propane 0.0191 0.0174 0.016 0.0144 0.007 0.006 0.0053 0.0335 0.0327 0.0198I-Butane 0.0033 0.003 0.0028 0.003 0.0014 0.0012 0.001 0.0035 0.0038 0.0037n-Butane 0.006 0.0055 0.0051 0.004 0.002 0.0018 0.0015 0.009 0.006 0.0039I-Pentane 0.0021 0.0019 0.0018 0.0016 0.0007 0.0006 0.0005 0.0017

n-Pentane 0.0013 0.0012 0.0011 0.001 0.0005 0.0004 0.0004 0.0015 .0020c .0022cHexane 0.0015 0.0014 0.0012 0.0011 0.0005 0.0004 0.0004

Heptanes + 0.0018 0.0016 0.0015 0.0014 0.0007 0.0006 0.0005 .0033d Helium 0.01 0.006 0.0031Avg. Mol. Wt. 18.4 19.25 19.94 20.75 16.9 18.11 19.56 20.02 19.98 18.6 Heptanes + Properties Mol. Wt. 115 115 115 115 115 115 115 100 Gravity (API) 60.58 60.58 60.58 60.58 60.58 60.58 60.58 68.35 #DIV/0! #DIV/0!

SG 0.74 0.74 0.74 0.74 0.74 0.74 0.74 0.71 #DIV/0! #DIV/0!Characterization Fact. Tb 708.1 708.1 708.1 708.1 708.1 708.1 708.1 656.3 #DIV/0! C 2.9956 2.9956 2.9956 2.9956 2.9956 2.9956 2.9956 2.9337 #DIV/0! Pc,psia 425.3 425.3 425.3 425.3 425.3 425.3 425.3 452.0 #DIV/0!

Tc, oR 1041.6 1041.6 1041.6 1041.6 1041.6 1041.6 1041.6 983.5 #DIV/0! ω 0.3178 0.3178 0.3178 0.3178 0.3178 0.3178 0.3178 0.2874 #DIV/0!

Zc 0.2618 0.2618 0.2618 0.2618 0.2618 0.2618 0.2618 0.2645 #DIV/0!

Ωω 0.3579 0.3579 0.3579 0.3579 0.3579 0.3579 0.3579 0.3582 #DIV/0! © Heavy fraction treated as n-pentane. (d) Heavy fraction treated as n-heptane. (e) Helium treated as methane.

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CRITICAL PRESSURE PREDICTIONS FOR COMPLEX MIXTURES BY FOUR METHODS

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Figure C. 1. Critical Pressure Prediction for Complex Mixtures – All Data.

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Figure C. 2. Critical Pressure Prediction for Complex Mixtures -Mixture 145-1.

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Figure C. 3. Critical Pressure Prediction for Complex Mixtures – Mixture 145 – 10.

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Figure C. 4. Critical Pressure Prediction for Complex Mixtures – Mixture 4 – 6.

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Figure C.5. Critical Pressure Prediction for a Complex Mixture 75 – 5.

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Figure C. 6. Critical Pressure Prediction for a Complex Mixture 141 – 7.

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Figure C. 7. Critical Pressure Prediction for a Complex Mixture 141 – 16.

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Figure C. 8. Critical Pressure Prediction for a Complex Mixture 141 – 25.

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Figure C. 9. Critical Pressure Prediction for a Complex Mixture 58 – 1.

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Figure C. 10. Critical Pressure Prediction for a Complex Mixture 47 – 1.

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CRITICAL TEMPERATURE PREDICTION FOR COMPLEX MIXTURES BY LLS EOS METHOD

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Figure C. 11. Critical Temperature Prediction for Complex Mixtures – All Data.

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Figure C. 12. Critical Temperature Prediction for a Complex Mixture 145 - 1

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Figure C. 13. Critical Temperature Prediction for a Complex Mixture 145 – 10

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Figure C.14. Critical Temperature Prediction for a Complex Mixture 4 – 6.

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Figure C. 15. Critical Temperature Prediction for a Complex Mixture 75 – 6.

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Figure C. 16. Critical Temperature Prediction for a Complex Mixture 141 – 7.

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Figure C. 17. Critical Temperature Prediction for a Complex Mixture 141 – 25.

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Figure C. 18. Critical Temperature Prediction for a Complex Mixture 58 – 1. P

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Figure C. 19. Critical Temperature Prediction for a Complex Mixture 47 – 1.