Closed-Form Van Der Waals Critical Point

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CLOSED-FORM VAN DER WAALS CRITICAL POINT

FOR PETROLEUM RESERVOIR FLUIDS

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

A DISSERTATION

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

1.1 Importance of Critical State 7

1.2 Approaches to Critical State Prediction 12

1.3 Retrograde Reservoir Fluids 14

1.4 Objectives of Work 16

II. CRITICAL PROPERTY CORRELATION METHODS 18

2.1 Criteria of the Critical State 19

2.2 Empirical Models 22

2.3 Corresponding States 25

2.4 Convergence Pressure 31

2.5 Equation of State Models 45

III. CLOSED-FORM VAN DER WAALS EXPRESSIONS 54

3.1 Van der Waals Equations of State Theory 54

3.2 Closed-Form Equations for Fluid Critical Point 61

iii 3.3 Closed-Form Critical Property Computation Methods 69

IV. CRITICAL PROPERTIES FOR RESERVOIR FLUIDS 74

4.1 Critical Pressure Data for Complex Hydrocarbon Mixtures 74

4.2 Calculation of Critical Properties

4.3 Results and Discussion 75

4.4 Comparison Between Calculated and Experimental Data 80

V. CONCLUSIONS AND RECOMMENDATIONS 86

5.1 Conclusions 86

5.2 Recommendations 88

REFERENCES 89

APPENDICES

A. ANALYTICAL SOLUTION FOR CUBIC EQUATIONS 99

B. VAN DER WAALS EXPRESSIONS FOR FLUID CRITICAL POINT 112

C. PREDICTION RESULTS OF CRITICAL PRESSURE, CRITICAL 130

TEMPERATURE, AND HEPTANE PLUS PROPERTIES

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

⎛ aP ⎞ A Dimensionless Constant ⎜ ⎟ ⎝ R 2T 2 ⎠

AD Absolute Deviation

API Oil Gravity

b van der Waals co-volume

⎛ bP ⎞ B Dimensionless Constant ⎜ ⎟ ⎝ RT ⎠

C Characterization Factor

K Watson Characterization Factor

MW Molecular Weight

P Pressure

Pc Critical Pressure

R Gas Constant

Sg Specific Gravity

T Absolute Temperature

Tc Critical Temperature

Tbp Boiling Temperature

V Molar Volume

xii x Mole Fraction

Z Compressibility Factor

Zc Critical Compressibility Factor

Greek Letter

α Parameter of LLS EOS

αij Binary Interaction

β Parameter of LLS EOS

βij Beta Binary Interaction

Ω Dimensionless Parameter

ω Acentric Factor

Subscripts

c Critical Property

m Mixture Identification

r Reduced State

VDW van der Waals Symbol

ij Component Identification

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, Pc ,

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

two phases become indistinguishable is very difficult and costly to determine

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, 29 98,111,122,31 to rigorous 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, 1120 and Jiang & Prausnitz 46 approaches were highly

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

failed, 46,100 in determining the critical points of reservoir fluids that contain heptane-plus

(C7+) 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 114,115,116,117

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

systems and the more complex reservoir fluids.

In order to predict adequately the critical properties ( Pc ,Tc ,Vc ) 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 (CO2) 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. 39,66,94 Several methods, such as that of Hadden and

Grayson, 39 for finding K-values depend on the use of convergence pressure as a correlating parameter.

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 116 , 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)

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 C , 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).

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 (−116o F) . Therefore, only

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

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

reservoir.

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 T1 < Tc ; Tc is the critical

temperature is tangent to the saturation line at the critical state. At point S , 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 A, 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 which case point D represents the saturated and stable vapor, or it might follow curve

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

⎛ ∂P ⎞ stability is determined by the condition that ⎜ ⎟ vanishes (i.e., point B ). Similarly, ∂V ⎝ ⎠T1

at point R one can observe that as the pressure increases, condensation may not occur up

⎛ ∂P ⎞ to point C, where again ⎜ ⎟ will vanish. Curve DC represents the locus of meta- ∂V ⎝ ⎠T1

stability and point C is the limit of meta-stability at T1 for the vapor. As the temperature

⎛ ∂V ⎞ is increased above that of the critical state there is a rapid increase in the slope ⎜ ⎟ of ⎝ ∂T ⎠ p 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.

The pure component pressure-volume diagram can be seen in Figure 1.2, while the pressure-volume diagram for mixtures is shown in Figure 1.3. The main differences

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

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

Figure 1.3. Pressure-Volume Diagram of Mixtures. 32

∂P ∂ 2 P For a pure component, = = 0 , at the critical point. For a mixture, this ∂V ∂V 2

does not occur at the top of the two-phase envelope. The Z-factors in Figure1.2 of the

equilibrium gas and liquid phases always meet the condition Z L < Z G . However, for mixtures, when gas and liquid phases are at equilibrium, Z L might be smaller or larger than Z G . At equilibrium, the density of the liquid phase is higher than the density of the gas phase, ρ L > ρ G . Then from

PM ρ = (2.1) ZRT

it follows that

10 M L M G > (2.2) Z L Z G

When Z-factors are less than one, then, Z L could be smaller or larger than Z G .

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 pressure

∂ρ near to the critical state. It can be shown that avg will reach a constant value as the ∂P 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”. 101,102 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.

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

1.2 Approaches to Critical State Prediction

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 physical properties, 69,49 developing equations of state, and designing supercritical fluid processes.

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 available, but with less coverage of composition range of data points.

In addition to the direct measurements, critical properties of mixtures are usually estimated utilizing various correlations methods, which have been reviewed in terms of

12 their estimation procedure and accuracy 108 . These correlation methods relied primarily on many approaches, namely, graphical approach, 38 equation of state approach, 20,84,79 empirical 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, that is, the second and third partial derivatives of the molar Gibbs free energy with respect to composition at constant temperature and pressure must be equal to zero. Determination

of Tc , Pc , and Vc for the mixtures involves a simultaneous solution of an extended form

of derivatives and an equation of state such as reported by Spear et al. 107 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. 107

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.

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 Condensation. 6

The solid thick line is the bubble- point curve (100 % liquid, 0 % vapor) and the thin line is the dew-point curve (100 % vapor, 0 % liquid); they meet at critical point C where the

two phases become identical. The basic criterion for the critical point (point C ) is the limiting condition where the system can exist in two 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 comprises retrograde phenomena. Retrograde phenomena always exist when the critical point of a mixture is not at the highest pressure and temperature possible for the coexistence of two

14 phases. Also, near the critical point, the density-dependent properties change with small

changes in temperature and pressure.

More than a century ago Kuenen 62,61 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 definitions 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 less dense of reservoir fluid phase is

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 cricondentherm. 102 These points represent the upper bounds where phase separation can take place.

Refer again to Figure 1.5, at point a , a single phase (vapor) exists. If we increase the pressure on this vapor isothermally to point E, we encounter a dew-point state. As

pressure is increased beyond E, more and more of the vapor will condense until reach to point D. Continue pressure increase from point D causes retrograde vaporization of liquid that had previously been condensed. This process continues until point B, where an upper dew point is reached. Continued pressure increase from point B to A which compresses a single-phase fluid (vapor). If now reduce the temperature isobarically from point A to point G, the volume of the single- phase fluid will contract. At point G a

15 bubble-point is encountered. With continued decrease in temperature isobarically, less

dense fluid (vapor) will continue to form until point H is reached. As conditions change from point G to point H, retrograde vaporization occurs. Reduction of temperature beyond H continue isobarically will cause a more dense phase to increase until the

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

above-described procedure, that is, by proceeding from point N to point b. Quantitative

understanding 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 region of

isothermal retrograde condensation.

1.4 Objectives of Work

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

predicting the critical properties, critical pressure, Tc, critical pressure, Pc , and critical

volume, Vc for complex petroleum reservoir fluids. This objective consists of three

major elements:

1. Development of a comprehensive closed-form solution to the criticality criteria

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

illustrated by its application to:

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

fluid theory

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.

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 63 for the critical properties of volatile hydrocarbon mixtures, and of Organic 80 for complex hydrocarbon systems. Later, Davis et al., (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 correlations with parameters such as pseudo critical temperature and pressure, molal average boiling point, or weight average equivalent molecular weight.

In this chapter, a review of correlation methods for predicting critical properties of complex petroleum reservoir fluids is undertaken to help understand their phase behavior 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 given mixture, the separation between the bubble-point and the dew point regions will be defined, and physical properties of the mixture can be obtained by using the law of corresponding states. Then the law of corresponding states and the convergence pressure are presented in detail.

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

temperature, pressure, and volume, this supercritical behavior 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 are interpreted mathematically as follows:

∂P = 0 (2.1) ∂V

∂ 2 P = 0 (2.2) ∂V 2

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

Figure 2.1. Pressure-Volume Plot for a Single-component System. 32

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

19 T1 , T2 , Tc , and T3 , are shown. Points B and C represents the limits of stability at

temperature T1. Points B` and C` represent the limit of stability at temperature T2 . Based

32 on the criteria of stability, the limits of stability at T1 and T2 are obtained from

⎛ ∂p ⎞ ⎛ ∂p ⎞ ⎛ ∂p ⎞ ⎜ ⎟ = 0 and ⎜ ⎟ = 0 , respectively. Between points B and C, ⎜ ⎟ > 0 and ∂v ∂v ∂v ⎝ ⎠T1 ⎝ ⎠T2 ⎝ ⎠T1

⎛ ∂p ⎞ between B` and C` ⎜ ⎟ > 0, and these are therefore the unstable segment of the ∂v ⎝ ⎠T2

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

⎛ ∂ 2 p ⎞ point where ⎜ ⎟ = 0 exists between these two points. This inflection point is an ⎜ ∂v 2 ⎟ ⎝ ⎠T1

unstable. As the temperature approaches the critical point (i.e., Tc ), the limits of stability

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

⎛ ∂ 2 p ⎞ bimodal curve, the inflection point ⎜ ⎟ = 0 is a stable point. Points A, D and A` ⎜ ∂v 2 ⎟ ⎝ ⎠Tc

and D` represent equilibrium phases at T1 and T2 , 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 continuity of gas and liquid state. The criteria of the critical state of a pure component are, therefore,

⎛ ∂p ⎞ ⎜ ⎟ = 0 (2.3) ∂v ⎝ ⎠Tc

20 ⎛ ∂ 2 p ⎞ ⎜ ⎟ = 0 (2.4) ⎜ ∂v 2 ⎟ ⎝ ⎠Tc

⎛ ∂ 3 p ⎞ ⎜ ⎟ < 0. (2.5) ⎜ ∂v 3 ⎟ ⎝ ⎠Tc

⎛ ∂ 3 p ⎞ Where ⎜ ⎟ < 0 indicates that the critical point is neither a maximum nor a ⎜ ∂v 3 ⎟ ⎝ ⎠Tc minimum.

In 1980, Heidmann and Khalil 43 proposed an alternative approach for the calculation of critical point that is mathematically different from the expressions of the

Criticality conditions in Equation 2.6 32 , and Equation 2.7 32

(1) (1) (1) y22 y23 L y2,c+1

(c+1) M M M M λ = (1) (1) (1) = 0 (2.6) yc,2 yc,3 L yc,c+1 (1) (1) (1) yc+1,2 yc+1,3 L yc+1,c+1

and (1) (1) (1) y22 y23 L y2,c+1 M M L M (1) (1) (1) = 0 (2.7) yc,2 yc,3 L yc,c+1 (c+1) (c+1) (c+1) λ2 λ3 L λc+1 ⎛ ∂C ⎞ ⎛ ∂C ⎞ (1) ⎜ 2 ⎟ (1) ⎜ 2 ⎟ (1) where y22 = ⎜ ⎟ , y23 = ⎜ ⎟ , y = A(T,V , N ,L, N c ) is the ∂X 2 ∂X 3 ⎝ ⎠c1 ,x3L,xc+2 ⎝ ⎠c1 ,x2 ,L,xc +2

(c+1) Helmholtz free energy, and λc+1 = 0 , at the critical point but the concept is not different.

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

elaborate more on this concept, let us consider the pressure-volume diagram of multi-

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

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, thus, results in a stable state. Critical point CP is the point at which bubble-point and dew point converge and being a stable state, it 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 critical point.

Figure 2.2. Critical Point Representation in a Multi-component System. 32

2.2 Empirical Models

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 involving the use of excess properties, (2) the use of conformal solution theory based on the concept that all thermodynamic properties of mixtures can be evaluated from component properties if the components conform to certain postulates

22 of statistical mechanics, (3) a rigorous thermodynamic condition based on the second and third partial derivatives of all molar Gibbs free energy with respect to composition.

The empirical approach involves calculations of the following model:

n Gc = ∑ xiGci + Gcorr (2.8) i=1

whereGc is the critical property desired and Gcorr is a correction term which is called the excess property of the mixture. Excess properties are usually estimated from empirical relations. 122 Many correlations have been proposed based on Equation 2.6 to predict the critical properties of mixtures. The critical temperature of defined mixtures predicted by

Li 71 , was simplified by Chueh-Prausnitz, 21 for all empirical approaches. All of these 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 correlations gave poor results for systems containing ethane.

Prausnitz’s 85 method for determining the pseudo-critical temperature can be generalized for mixtures having any number of components. The generalized equation is

n n n Tcm = ∑θ iTci + ∑∑θ iθ jτ ij (2.9) i=1 i j where

2 3 xiVci θ i = (2.10) ∑ xiVci

23 21 andτ ij for each interacting pair of molecules. Equation 2.7 was tested by Prausnitz 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:

n Tcm = ∑φiTci (2.11) i where

xiVci φi = n (2.12) ∑ xiVci i

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 53 is the most important from a historical point of view. More accurate correlation models have been proposed by Joffe, 48 Grieves, 38 and Organick, 80 Etter and

24 Kay. 29 Edmister 27 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.

Correlation models for predicting the critical point of complex petroleum hydrocarbon fractions also have been proposed. Significant correlations have been suggested by Smith, 105 Kurata-Katz, 63 Edmister, and Pollock. 27 Organick 80 also proposed a correlation and introduced a comparison of his method with the Kurata-Katz.

Equations of state were involved in the critical state correlation in order to integrate vapor-liquid equilibrium behavior with the critical point. The prediction of the critical 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.

2.3 Corresponding States

Van der Waals 116 in 1873 developed the theorem of corresponding states based on experimental observation, which shows that compressibility factor Z for different

fluids exhibit similar behavior when correlated as a function of reduced temperature Tr

and reduced pressure Pr . By definition,

T Tr = (2.13) Tc

P Pr = (2.14) Pc

25 and

V Vr = (2.15) Vc

Where the subscript r represents the reduced state, and subscript c represents critical state. These dimensionless reduced conditions of temperature, pressure, and volume provide the basis for the simplest form of the theorem of corresponding states:

“All fluids, when compared at the same reduced temperature and reduced pressure, have approximately the same compressibility-factor, and all deviate from ideal-gas behavior to about the same degree.” 105 The term “reduced state” means that each reduced value of an intensive property of a system 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

Z for methane, ethane, and propane as a function of reduced pressure for reduced temperatures to show the degree of correlation.

Figure 2.3. Compressibility Factors of Methane, Ethane, and Propane as a Function of Reduced Pressure and Reduced Temperature. 109

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 gases. 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 T pc = ∑ yiTci (2.16) i=1

and

n Ppc = ∑ yi Pci (2.17) i=1

where Ppc is the pseudo-critical pressure, T pc is the pseudo-critical temperature, Tci and

th Pci are the critical temperature and critical pressure respectively of i component, yi mole fraction of ith component in mixture, and n number of components. The physical

27 properties of gas mixtures are correlated with pseudo-critical pressure and pseudo-critical temperature in the same way that properties for pure gases are correlated with reduced pressure and reduced temperature. Thus, these pseudo-critical properties are defined as follows:

P Ppr = (2.18) Ppc

T Tpr = (2.19) Tpc

Compressibility factors, experimentally obtained, for natural gas have been correlated with pseudo-reduced pressure and temperature. 14,85 The petroleum industry has universally adopted the correlations shown in Figures 2.4, to determine the compressibility factor Z .

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

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 Tr and Pr .

These correlations 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 addition to Tc and Pc ), has been 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. Because the logarithm of the vapor pressure of pure fluid is linear in the reciprocal of absolute temperature,

d log p sat r = m (2.20) 1 d( ) Tr

where pr is the reduced vapor pressure, Tr is the reduced temperature, and m is the

sat 1 slope of the plot of log Pr vs. . Tr

It has been observed that if the two-parameters corresponding states were valid, the slope m would be the same for all pure fluids. This is not true 63 ; because according to Pitzer each fluid has its own characteristic value of m. Pitzer has noted that all vapor

sat 1 pressure data for simple fluids lie on the same line when plotted as log Pr vs. and Tr

sat the line passes through log Pr = −1.0 at Tr = 0.7. 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 sat sat log Pr (SF ) − log Pr (2.21)

The acentric factor is defined as the difference evaluated at Tr = 0.7 :

ω ≡ −1.0 − log(P sat ) (2.22) r Tr =0.7

Therefore, ω can be determined for any fluid from Tc , Pc and vapor pressure

measurement made at Tr = 0.7.

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), 40 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

n P = ∑ pi (2.23) i=1

and

p y = i (2.24) i P

pi = yi P (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,

Pi = xi Pvi (2.26)

th th Where pi is the partial pressure of the i component, xi is the mole fraction of the i

th component in liquid phase, yi is the mole fraction of i component in vapor phase and

31 th pvi is vapor pressure of i component. By combining Equation 2.24 and Equation 2.26 we obtain:

pyi = pi xi (2.27)

By definition,

yi K i = (2.28) xi

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 K i is dependent on the pressure, temperature, and composition of the hydrocarbon system. Equilibrium ratios ( K values) for low-shrinkage oil and a condensate at 200o F are shown in Figures 2.6 6 and 2.7 6 as functions of pressure. The 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 common value of unity at some high pressure. This pressure has been termed the convergence pressure of the system “. If the temperature at which the 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 critical temperature, the convergence of K - values is then an apparent convergence pressure. At a pressure less 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.

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

Figure 2.7. Equilibrium Ratio for a condensate Fluid. 92

A widely accepted definition of convergence pressure by Hadden 39 in 1953 was proposed. In fact, Hadden 39 defined the critical mixture, from which the convergence pressure would be estimated, as that resulting from adding methane or nitrogen to the equilibrium liquid. In such an addition always result in reaching the critical state, and hence a convergence pressure is defined, and satisfies the phase rule requirement. If these two lightest components methane (or nitrogen) were in every system of interest, this definition of convergence pressure is adequate. But, for purposes of general correlation of data, many of which are from binary or ternary that have neither methane nor nitrogen, the adequately of Hadden’s definition is questionable.

34 Rowe, 93 in 1964 proposed a definition based on what he called critical composition method. In this method, the convergence pressure is estimated from the critical mixture, which would give the same equilibrium phases. That means the critical mixture lies in the tie line. This definition is 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 K - values are not sensitive to convergence pressure, but can occur at high pressure as well Lawal, 1981 66 .

The critical mixture is determined by the intersection of the tie line with the locus of critical compositions at the equilibrium temperature. Fair 31 has developed an equation for the isothermal locus of critical composition in hydrocarbon systems that calculated the critical composition. The general Fair’s equation is stated as

N A z = 1 (2.29) ∑ i ci i=1, j ≠1

Where A is the critical composition locus coefficient for component i , z is the mole i ci fraction of component in the critical mixture. In this notational form, j indicates the component chosen as dependent; that is z is determined from z . c j ci

For an equilibrium state with liquid mole fractions xi and vapor mole fraction

yi , the interaction of the equations for the tie line (actually N = 1 component balance equations) with the isothermal critical composition locus Equation 2.28 is represented by

(x − y )[1− A y ] z = i i ∑ i i + y (2.30) ci i ∑ Ai (xi − yi )

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

2.28 was used to calculate the critical mixture for all ternary, multi-component, and complex mixture equilibrium states.

Fair, 30 presented correlations for binary data, which are considered more accurate than the generalized locus functions. The critical compositions of the binaries were calculated from the equilibrium temperature by the expression

m z = θ + c θ (θ i −1) (2.31) cL ∑ i i=1 where z = mole fraction of light component cL

T − T θ = cH (2.32) T − T cH cL

T = Equilibrium temperature

T = Critical temperature of the heavy component cH

T = Critical temperature of the light component cL

The convergence pressure characteristic of a particular equilibrium sate is determined 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 to completely empirical, with methods having some degree of success.

The method we choose to present here is the one developed by Zais 122 because of its general applicability and convenience of calculation. Zais’ equation is written in the form

36 N N −1 N w w P = P w 2 + i j (2.33) cm ∑ ci i ∑∑ 2 i=1 i=11j=i+Aij + Bij (wi − w j ) + cij (wi − w j )

where

P = Critical pressure for mixture cm

P = Critical pressure of component i ci

wi = Weight fraction of component i

w j = Weight fraction of component j

Aij = Binary interaction coefficient

Bij = Binary interaction coefficient

C ij = 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 fractions from Equation 2.27 or Equation 2.31to weight fraction.

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

coefficients Aij , Bij , and C ij 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. 122 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%.

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, 66 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.

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 the critical pressure at cL

T , where pK

T − 2T − T (2.34) pK cL

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- convergence pressure is readily applied by determining the isothermal locus of critical compositions at T rather than at equilibrium temperature T , according to Equation pK

2.32.

The fluid composition effects on the K - values as shown in Figure 2.9, 6 where values for 1000 psia and 5000 psia convergence pressures are compared at 100 o F. The differences in K 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.

Figure 2.9. Comparison of Equilibrium Ratios at 100 o F for 1000- and 5000-psia convergence Pressure. 6

It is of practical value at this point to present brief remarks on the interdependency of heavy (plus) fractions and convergence pressure concept. Because the vapor-pressure curves and critical properties of hydrocarbon heavier than hexane are fairly close together, it is possible to characterize the mixture by an average set of K - values. Properties of heptanes plus fractions can be estimated from the properties of heavier hydrocarbons such as decane. But, normally a more satisfactory procedure for characterizing the heptanes plus is to use correlated experimental data heptanes plus fraction of fluids with similar properties to those predicted. For this purpose equilibrium ratios for the heptanes-plus fraction reported by Katz and Hachmuth, 50 and Roland,

40 Smith, and Kaveller, 6 are plotted in Figure 2.10. 6 The data of Katz are preferred for crude oil system, and the data of Roland et al. are preferred for condensate fluids.

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

The heavy (plus) fraction controls the behavior of the system as liquid (mostly the 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 methane through decane, the entire adjustment in the – values system can be made by adjusting the

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 C10+ K -

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

30 curve. This shown in Figure 2.11 by the curve marked “ Pk too low”. If the convergence pressure is too high (i.e., too close to the dew point pressure) the determined

C10+ K - values curve will have the correct shape but will fall above the given decane

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

- values for the correct convergence pressure will result in the C10+ K - values curve as

shown in Figure 2.11. Typically, the C10+ K - value will, to some extent, cut across the constant temperature lines on the given Natural Gasoline Supply Men’s Association

(NGSMA) decane curve. This phenomena caused by the fact that the plus-fraction that first drop at the dew point is generally less volatile than the plus-fraction that drops out as pressure declines.

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

Figure 2.12. K vs Pressure with Curve Showing Effect of Choosing a Convergence Too High or Too Low for Condensate Depletion. 30

44 2.5 Equation of State Models

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 component and a 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 given by the equation of state. There are many families of EOS. The van der

Waals family is characterized by simple cubic form, and most have two 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 constituents should be included to account for highest accuracy. There are many equations of state used for calculating vapor-liquid equilibrium, Reid et al 91 give review of several practical equations for chemical and petroleum industries. 10,11,12,51,52

One of the first and simplest, perhaps the best known equation of state model is the ideal gas law,

PV = RT (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 volumetric phase behavior of petroleum reservoir fluids, because this law is only valid for substances at low pressures and high temperatures. The petroleum industry has adopted the concept of compressibility factor Z , or gas deviation factor for

45 describing the behavior of mixtures or gases at moderate high pressures. The compressibility factor Z is a correction factor for the ideal gas law, that is

PV = ZRT (2.36) and, by definition

PV Z = (2.37) RT

The limitations in the use of Equation 2.34 to describe the behavior of natural gases gave the chance of 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) 116 , several investigators 5,18,27,64 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 constants. Even though with this large number of EOS, not so many are considered by engineers and researchers. Because of its computational problem, many prefer the simplicity found in van der Waals cubic model while improving the accuracy through modifications.

The van der Waals equation of state, is the first equation capable of representing vapor-liquid coexistence

RT a P = − (2.38) V − b V 2

46 The parameters a and b are constants that characterize the molecular properties of the substance in question. The first term on the right side of Equation 2.36 represents

RT a the repulsive interaction force ( ), and the other term ( ) is the attractive forces V − b V 2 between molecules. The parameters a and b can be obtained from the critical properties of the fluid. Also, these parameters can be determined mathematically using Equation

2.1. In the development of cubic equations of state, modifications in the evaluation of the parameter a in the attractive term by Soave is considered the most accurate results in

Redlich-Kwong, 86 Soave-Redlich-Kwong, 106 and Peng-Robinson. 83 Redlich-Kwong 87 equation is the most important and successful model for the modification of the van der

Waals equation of state.

Redlich-Kwong (1949) replaces the attractive term of the van der Waals EOS with a temperature-dependent term. The R-K EOS has the form

RT a P = − (2.39) V − b V (V + b)T 0.5

For pure components, the Equation 2.37 constant parameters a and b usually expressed as

2 2.5 R Tc a = Ω a (2.40) Pc

RTc b = Ωb (2.41) Pc

where Ωa and Ω b are dimensionless parameters with the following computed values:

Ωa = 0.4278 and Ωb = 0.0867.

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

RT aα P = − (2.42) V − B V (V + b) where the dimensionless factor α is a function of temperature:

0.5 2 α = []1+ m(1− Tr ) (2.43)

In the Equation 2.41, m is the slope and Tr is the reduced temperature.

Soave correlated the slope, m, against the acentric factor, ω, by the generalized form

m = 0.480 +1.57ω − 0.176ω 2 (2.44)

As the case in the Redlich-Kwong equation of state, the constant parameters a and b are determined at the critical conditions in the form

2 2 R Tc a = Ω a (2.45) Pc

RTc b = Ωb (2.46) Pc

where the values of dimensionless pure component parameters Ωa and Ω b do not change as shown in Equations 2.43 and 2.44 due to the introduction of function a(T ).

Peng and Robinson 83 introduced an improved Redlich-Kwong EOS capable of

predicting the liquid volumes and a critical compressibility factor of Z c = 0.307. Their equation is given by

RT aα P = − (2.47) V − b V (V + b) + b(V − b)

48 with

2 2 2 R Tc R Tc a = Ω a = 0.4724 (2.48) Pc Pc and

RTc RTc b = Ωb = 0.07780 (2.49) Pc Pc

Peng and Robinson adopted Soave’s- Redlich-Kwong approach for computing α as shown in Equation 2.45, where they used ω as the correlating parameter for the slope, m, as given by

m = 0.379642 +1.48503ω − 0.1644ω 2 + 0.001667ω 3

Usdin and McAuliffe (UM), 101 proposed a new parameter d, to replace b in the second term (attractive term) of the Soave-Redlich-Kwong 106 equation:

RT aα P = − (2.50) V − b V (V + d) where

2 2 R Tc a = Ω a (2.51) Pc

RTc b = Ωb (2.52) Pc

RTc d = Ωd (2.53) Pc

They argued that the two terms of the SRK equation are interconnected by parameter b, concluding that all substances possess a value of critical compressibility

49 factor Z c = 0.333. They stated that severing the tie created by the shaping of the parameter b and replacing it with d would cause accurate liquid density.

UM proved the dimensionless parameters Ωa ,Ωb , and Ωd can be related to the

critical compressibility parameter, Z c , by the following expression:

3 2 2 2 Ωd + (6Z c −1)Ωd + [12(Z c ) − 3Z c ]Ωd + (8Z c − 3)(Z c ) = 0 (2.54)

Ωb = Ωc + 3Z c −1, (2.55) and

3 (Z c ) Ωa = (2.56) Ωb

Usdin and McAuliffe also adopted Soave’s formulation of α as Equation 2.45 with

m = 0.48049 + 4.516ωZ c + [0.67713(ω − 0.35) − 0.02](Tr − 0.7), (2.57)

for Tr ≤ 0.7 and

3 2 m = 0.4049 + 4.516ωZ c + [37.7846ω(Z c ) + 0.78662](Tr − 0.7) , (2.58)

for 0.7 ≤ Tr ≤ 1.0

Patel and Teja 81 introduced the following form of EOS:

RT aα P = − (2.59) V − b V (V + b) + c(V − b) where

50 2 2 R Tc a = Ω a (2.60) Pc

RTc b = Ωb (2.61) Pc

RTc c = Ωc (2.62) Pc where

Ωc = 1− 3ζ c (2.63)

2 2 Ωa = 3ζ c + 3(1− 2ζ c )Ωb + Ωb + (1− 3ζ c ) (2.64)

PcVc ζ c = (2.65) RTc

and Ωb is the smallest positive root of the cubic expression:

3 2 2 3 Ωb + (2 − 3ζ c )Ωb + 3ζ c Ωb −ζ c = 0 (2.66)

A value of Ω b is given by

Ωb = 0.3243ζ c − 0.0225. (2.67)

Finally, α is determined by the following:

0.5 2 α = []1+ F(1− Tr ) (2.68)

The Patel-Teja equation of state therefore requires four parameter constants

Tc , Pc ,ζ c , and F for any fluid desired.

There are many other equations of state models with modifications of the attractive term, repulsive term, and combination of both of the van der Waals EOS with

51 more than two or three parameters. Some of these are summarized in Table 2.1 and Table

2.2.

Table 2.1. Modifications to the Attractive Term of van der Waals Equation of State. 117

Equation Attractive Term

Redlich-Kwong (RK, 1949) a

RT 1.5 (V + b)

a(T) Soave (SRK, 1972) RT(V + b)

a(T)V

Peng-Robinson (PR, 1976) RT[V (V + b) + b(V − b)]

a(T)V

RT(V + cb) Fuller, (1970) a(T)V

RT[]V 2 + (b(T) + c)V − b(T)c Heyen, 1980-Sandler, (1994) a(T)V

RT (V 2 + ubV + wb 2 ) Schmidt-Knapp (1980) a(T)V 2 Kubic (1982) RT (V + c)

a(T)V

RT[V (V + b) + c(V − b)] Patel-Teja (PT)(1982) a(T)V Yu and Lu (1987) RT[V (V + c) + b(3V + c)]

a(T)

Lawal-Lake-Silberberg (LLS)(1985) V 2 + αbV + βb 2

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

Equation Repulsive Term

Reiss et al (1959) 1+η +η 2

(1−η)3

1+η +η 2 Thiele (1963) (1−η)3

1 Guggenheim (1965) (1−η) 4

1+η +η 2 −η 3

Carnahan-Starling (1969) (1−η)3

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) generalized cubic equation of state, and the algorithm developed for computing the critical properties of petroleum reservoir fluids. Using the LLS equation of state as a basis, a closed-form solution for the van der Waals critical point is presented. This is accomplished by introducing the “VDW closure parameters” (that is, parameters α, β 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, PV = nRT , 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 each other.

An early attempt, to take these intermolecular forces into account was that of Van der Waals (1873), who proposed that the ideal gas equation of state be replaced by

⎛ a ⎞ ⎜ P + ⎟(V − b) = RT (3.1) ⎜ 2 ⎟ M ⎝ VM ⎠

54 a This equation differs from the ideal gas equation by the addition of the term to V 2 pressure and the subtraction of the constant V to b from molar volume.

Here the parameters a and b are constants particular to a given gas, where R is

a the universal gas constant. The term represents an attempt to correct pressure for the V 2 forces of attraction between the molecules. The actual pressure exerted on the wall of the

a vessel by real gas is less, by the amount , than the pressure exerted by an ideal gas. V 2

The parameter b (or the so called co-volume parameter) is related to the size of each molecule and represents the intermolecular repulsive forces in the sense that it is the volume that the molecules have to move around in is not just the volume of the container

V , but is reduced to (V − nb) . The parameter a has a more difficult meaning and is related to intermolecular attractive force between the molecules. The net effect of the intermolecular attractive forces is to reduce the pressure for a given volume and

n temperature. When the density of the gas is low (i.e., when is small and nb is small V compared toV ) the Van der Waals equation reduces to that of the ideal gas law. The a and b parameters can be obtained from the critical properties of the fluid.

On the basis of the available volume (V − b ), Van der Waals was able to show 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:

55 RT a P = − (3.2) V − b V 2

a Since a > 0 , and b << V , then will be a reduction in pressure by approximately . V 2

It is clear that the Van der Waals equation predicts a deviation from ideal behavior. This deviation can be analyzed by defining a quantity Z , called the compressibility factor, as

PV Z = (3.3) RT

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

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

RT :

PV V a Z = = − (3.4) RT V − b RTV

1 a Z = − (3.5) b RTV 1− V

b 1 For very low density <<1, so we can use a Taylor series to approximate . In V b 1− V general, for X << 1

1 ≈ 1+ X (3.6) 1− X

56 Using this approximation, the compressibility factor becomes

b a Z = 1+ − (3.7) V RTV or

a (b − ) Z = 1+ RT (3.8) V

a The quantity b − is an observable and calculable quantity, which measures RT

a a deviation from ideal behavior of a gas. Note that if b − >> 0 , then b > and the RT RT

a pressure is larger than ideal gas pressure. However, the condition b > tells that RT excluded volume effects, as measured by the constant b , so an increase in pressure is

a a what we would have predicted. On the other hand: if b − < 0 , then b < , then RT RT the pressure is less than the ideal gas pressure.

Real gases can have both positive and negative deviations from ideal behavior, depending on the pressure and temperature and the particular system. Strongly attractive forces will lead to a lowering of the pressure and hence a negative deviation from 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 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.

Figure 3.1. Pressure-Volume Diagram for Pure Component.

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

is a monotonic decrease of pressure with increasing volume. The T = Tc isotherm exhibits an unusual feature not present in any of the ideal gas isotherms – a small region where the curve is essentially horizontal (flat) with no curvature. At this point, there is no change in pressure as the volume changes. Below this isotherm, the Van der Waals

starts to exhibit unphysical behavior. The T < Tc isotherm has a region where the pressure decreases with decreasing volume, behavior that is not expected on physical grounds. What is observed experimentally, in fact, is that at a certain pressure, there is a dramatic discontinuous change in the volume. This dramatic jump in volume signifies that a phase transition has occurred, in this case, a change from a gaseous to a liquid state.

The T = Tc 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

58 discontinuous changes in the volume. For this reason, the T = Tc isotherm is called the critical isotherm, and the point at which the isotherm is flat the slope of the curve is zero, and the isotherm has zero curvature at the critical point. Mathematically these two situations correspond to zero values of the first and second derivatives of pressure with respect to volume.

At the critical point, the system 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 temperature, pressure and volume, this supercritical behavior 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 criticality) for a pure component are:

⎛ ∂P ⎞ ⎜ ⎟ = 0 (3.9) ⎝ ∂V ⎠cp

⎛ ∂ 2 P ⎞ ⎜ ⎟ = 0 (3.10) ⎜ 2 ⎟ ⎝ ∂V ⎠cp

Solving Van der Waals’ equation, Equation 3.1 for pressure yield for 1 mole

RT a P = − (3.11) V − b V 2

To estimate a , b , and 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

⎛ ∂P ⎞ ⎛ 2a RT ⎞ ⎜ ⎟ = ⎜ − c ⎟ = 0 (3.12) ∂V ⎜V 3 (V − b) 2 ⎟ ⎝ ⎠cp ⎝ c c ⎠cp or

59 2a RTc 3 = 2 (3.13) Vc (Vc − b)

The second condition leads to

⎛ ∂ 2 P ⎞ 6a 2RT ⎜ ⎟ = − c = 0 (3.14) ⎜ 2 ⎟ 4 3 ⎝ ∂V ⎠cp Vc (Vc − b) or

6a 2RTc 4 = 3 (3.15) Vc (Vc − b)

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

⎛ a ⎞ ⎜ P + ⎟(V − b) = RT (3.16) ⎜ c 2 ⎟ c c ⎝ Vc ⎠

The three equations Equation 3.13, through Equation 3.16 apply at the critical point and by combination results in

27R 2T 2 a = c (3.17) 64Pc and

RT b = c (3.18) 8Pc

These equations for the constant 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 inflection at the critical point.

60 3.2 Closed-Form Equations for Fluid Critical Point

Another development of van der Waals modification is the single component form of the 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 this equation of state has been given in the literature. 62 The LLS equation has the form

RT a P = − (3.19) V − b V 2 + αbV − βb 2 where parameters a, b, α and β are established for pure component as follows

2 2 3 R Tc a = []1+ (Ωω −1)Z c : (3.20) Pc

RTc b = (Ωω Z c ) (3.21) Pc 1+ Ω Z − 3Z α = ω c c (3.22) Ωω Z c

2 3 2 Z c (Ωω −1) + 2Ωω Z c + (1− 3Z c )Ωω β = 2 (3.23) Ωω Z c 0.361 Ω = (3.24) ω 1+ 0.0274ω

where P is the pressure, T is the temperature, V is the molar volume, Z c , is the critical

compressibility factor, Pc and Tc are the critical pressure and temperature of the

components, Ωω is a constant equals to 0.325, and R is the universal gas constant. The units used in this work for the LLS EOS are psia for pressure, a degree Rankin, o R for temperature, cu-ft/1b-mole for molar volume, and the constant R = 10.73 psia-cu-ft/1b- mole o R. The experimental data are reported in psia of pressure, mol fractions, and o F,

61 therefore, the data had to be converted to the units used in this work. The composition and critical points obtained by the researchers, Simon-Yarborough,98 Etter and Kay,29 and Zais for 85 hydrocarbon/non-hydrocarbon reservoir fluid mixtures are shown in

Tables C 1 through C 27 in Appendix C.

The Equation 3.19 has four independent parameters ( a, b, α, and β ) and it is a generalized form and cubic in terms of molar volume. The LLS equation, Equation 3.19, can be viewed as a generalized cubic equation of state from which many known equations of state can be derived. Moreover, “the term generalized” cubic equations of state is used when the equation can be reduced or modified to the form of any of the known cubic equations of state by assigning specific integer to the parameters α and β .

3 It is interesting to find out that when α = β = 0 and Z = the generalized form of LLS c 8 equation of state reduces to van der Waals equation of state. Similarly for the PR EOS for α = 2, β = 1. The resulting parameters from the application of the criticality conditions to various forms of cubic EOS are presented in Table 3.1. 75

Table 3.1. Parameters of Selected Equations of State.

Equations of State Year Zc Ωa Ωb Ωc van der Waals 1873 0.375 0.4218 0.1250 0.333 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 α = 0 and β = 0 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

⎛ ∂P ⎞ − RT 2a ⎜ ⎟ = c + = 0 (3.25) ∂V (V − b) 2 3 ⎝ ⎠Tc c Vc

⎛ ∂ 2 P ⎞ 2RT 6a ⎜ ⎟ = c − = 0 (3.26) ⎜ 2 ⎟ 3 4 ∂V (Vc − b) V ⎝ ⎠Tc c

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

V b = c (3.27) 3

9RT V a = c c (3.28) 8

Combining Equations 3.27and 3.28 with van der Waals Equation 3.1, a universal compressibility factor, Z = 0.375 is obtained and calculating the parameters a and b from the following equations:

R 2T 2 a = 0.4218 c (3.29) Pc

RT b = 0.125 c (3.30) Pc

where Ωa = 0.4218 and Ωb = 0.125 are constants at the critical point from the van der

Waals original equation. The parameter Ωω is determined by dividing the parameter Ω b

by the critical compressibility factor, Z c . In the case of van der Waals,

63 Ωω = 0.125/ 0.375 = 0.333. 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.

η p a m ηT am Pc = 2 Tc = Vc = ηv bm b m Rbm

EOS Year η p ηT ηv van der Waals 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 this context of document are the compressibility and volume forms of criticality conditions. The derivation of the critical 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, Z c , and critical volume, Bc to the corresponding (Z − Z c ) = 0 .

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

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

64 3 expansion of (Z − Z c ) = 0. This procedure is shown in Appendix B and the resulting

cubic equation as Equation 3.30 where Z c = f (α, β ) .

3 2 Z c θ1 + Z c θ 2 + Z cθ 3 +θ 4 = 0 (3.31) where,

2 3 θ1 = (8 +12α + 6α + α ) (3.32)

2 θ 2 = −(3 +12α +12α + 9β − 9βα ) (3.33)

2 θ 3 = (3α + 6α + 6β − 6βα ) (3.34)

2 θ 4 = −(β + α − βα ) (3.35)

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 ( Bc ) and by comparison

3 of the (Z − Z c ) = 0 . This procedure is presented in Appendix B and the resulting cubic equation is shown as Equation 3.36

3 2 Bc θ1 + Bc θ 2 + Bcθ 3 +θ 4 = 0 (3.36) where

2 3 θ1 = (8 +12α + 6α + α ) (3.37)

2 θ 2 = (15 +15α − 27β − 3α ) (3.38)

θ 3 = (6 + 3α) (3.39)

θ 4 = −1 (3.40)

65 The equations derived to predict the critical pressure, Pc , critical temperature,

Tc , and critical volume, Vc pure components and mixtures are presented with the following dimensionless parameter:

aP A = (3.41) R 2T 2

bP B = (3.42) RT

PV Z = (3.43) RT

Applying Equation 3.19 at the critical Point we have

PcVc Z c = (3.44) RTc

aP A = c (3.45) c 2 2 R Tc

bPc Bc = (3.46) RTc

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

critical pressure, Pc , critical Temperature, Tc , and critical volume, Vc from Equations

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

B RT P = c c (3.47) c b

Solving for Pc from Equation 3.44, obtain

A R 2T 2 P = c c (3.48) c a

66 Equating Equation 3.46 and Equation 3.47 yield

aBc Tc = (3.49) bRAc and substituting Equation 3.48 into Equation 3.46 yield

2 aBc Pc = 2 (3.50) b Ac

Replacing Pc and Tc expressions previously obtained in Equation 3.48 and

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

Z c b Vc = (3.51) Bc

The Equations 3.48, 3.49, and 3.50 can be expressed in terms of Ac , Bc , Z c , a, b, α, and β . In the derivation of the criticality expressions in Appendix B Equation

B.48, the parameter Ac has been presented in term of Bc , Z c , α and β .

2 2 2 Ac = 3Z c + αBc + βBc +αBc (3.52)

By substituting Ac in the equations for Pc , Tc and Vc , the critical property

equations are expressed in terms of Bc , Z c , α and β . Then, Equations 3.48, 3.49 and

3.50 can be rewritten in the form

aB 2 P = c (3.53) c 2 2 2 2 b (3Z c + βBc + αBc + αBc )

aBc Tc = 2 2 2 (3.54) bR(3Z c βBc + αBc + αBc )

67 Z c b Vc = (3.55) Bc or

aΩ 2 P = b (3.56) c 2 2 2 2 b ()3Z c + βΩb + αΩb + αΩb

aΩb Tc = 2 2 2 (3.57) bR()3Z c + βΩb + αΩb + αΩb

Z c b Vc = (3.58) Ωb

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

equations Equation 3.56 through Equation 3.58 to calculate Pc, Tc, and Vc are practical and directly obtained once the composition and the pure components are given.

For the mixtures, the expressions for determining the critical properties are

expressed in terms of the parameters am , bm , α m , and β m . These mixture parameters require the LLS mixing rules to establish the following equations for the critical properties for mixture:

a B 2 P = m c (3.59) c 2 2 2 2 bm ()3Z c +β m Bc + α m Bc + α m Bc

am Bc Tc = 2 2 2 (3.60) bm R()3Z c + β m Bc + α m Bc + α m Bc

Z c bm Vc = (3.61) Bc

By LLS mixing rules,

68 n n 1 1 2 2 am = ∑∑xi x j ai a aij (3.62) i j

n 3 ⎛ 1/ 3 ⎞ bm = ⎜∑ xibi ⎟ (3.63) ⎝ i=1 ⎠

n n 1 1 2 2 α m = ∑∑xi x jα i α j α ij (3.64) i==11j

n n 1 1 2 2 β m = ∑∑xi x j β i β j β ij (3.65) i==11j

The prediction of critical properties for hydrocarbon mixtures can now be achieved since all the necessary equations have been developed. In this project, the

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

critical temperature, Tc and critical volume Vc 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.

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, Pc , Tc , acentric factor, ω , Critical

compressibility factor, Z c , critical volume, Vc , and molecular weight. Additional information that may be available is the analysis of the equilibrium liquid and gas. These data will permit critical properties to be calculated directly at reservoir conditions of pressure and temperature. Usually, however, laboratory critical point values are

69 furnished, and the critical volume of the reservoir fluids will be the most valuable data available for reservoir study purposes.

Computer program provides speed and accuracy in predicting the critical properties for pure components and complex mixtures by the closed-form Van der Waals method. However, the following procedure (algorithm) is included to show the calculation procedure followed by flowcharts of the main program:

The following is the step-by-step procedure to calculate the critical point of pure

components given the components measured Tc , Pc ,ω .

Step1. From the single component critical data, calculate the pure components parameters a, b, α , and β using Equation 3.20, Equation 3.21, Equation 3.22, and Equation3.22,

Step 2. Calculate the dimensionless critical volume, Bc using Equation 3.66.

bPc Bc = (3.66) RTc

Step 3. Calculate the Dimensionless Parameter Ac in terms of Bc , Z c ,α, and β .

2 2 Ac = 3Zc + (α + β )Bc + αBc (3.67)

Step 4. Calculate the critical pressure, Pc , the critical temperature, Tc , and the critical

Vc using Equation 3.53, Equation 3.54, and 3.55.

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

am ,bm ,α m , and β m are calculated using Equation 3.62, Equation 3.63, Equation 3.64, and Equation 3.65.

70 Step 6. With the calculated values of α m and β m , the coefficients of Equations 3.64 and

3.65 are calculated.

Step 7. Calculate the dimensionless critical volume, Bc 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 Bc : Bc = f (α m , β m ).

3 2 θ3 Bc +θ 2 Bc +θ1Bc +θ 0 = 0 (3.68) where

3 2 θ 3 = (α m + 6α m +12α m + 8) (3.69)

2 θ 2 = −3(α m − 5α m + 9β m − 5) (3.70)

θ1 = 3(α m + 2) (3.71)

θ 0 = −1 (3.72)

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

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

Z c = f (α m , β m ) ,

3 2 θ3 Zc +θ 2 Zc +θ1Zc +θ 0 = 0 (3.73) where

3 2 θ 3 = (α m + 6α m +12α m + 8) (3.74)

2 θ 2 = −3(1+ 4α m + 4α m + 3β m − 3α m β m ) (3.75)

71 2 θ1 = 3(2α m + 2β m − 2α m β m +α m ) (3.76)

2 θ 0 = (α m β m − β m −α m ) (3.77)

Step 9. Determine the constant Ac for mixtures using the equation

2 2 2 Ac = 3Z c + β m Bc + α m Bc + α m Bc (3.78)

Step 10. Calculate the critical temperature, Tc , critical pressure, Pc , and critical

volume,Vc for the mixtures:

am Bc Tc = 2 2 2 (3.79) bm R(3Zc + β m Bc +α m Bc +α m Bc

2 am Bc Pc = 2 2 2 2 (3.80) bm (3Zc + β m Bc +α m Bc +α m Bc ) and

Z c bm Vc (3.81) Bc

Step 11. Calculate the average absolute deviations as Follows:

⎡ Pc, pred − Pc,exp ⎤ Absolute Deviation (AD) = ABS ⎢ ⎥(100) (3.82) ⎣⎢ Pc,exp ⎦⎥

⎡Tc, pred − Tc,exp ⎤ Absolute Deviation (AD) = ABS ⎢ ⎥(100) (3.83) ⎣⎢ Tc,exp ⎦⎥

Figure 3.2 is an algorithm summarizes the step-by-step procedure presented above to calculate the critical point of petroleum reservoir fluids.

72 Start

Input Tc Pc Vc Zc ω Read Parameters am bm αm βm

Read parameters a b α β Call equations of parameters am bm αm βm and calculate am bm αm β for mixtures Call equations of parameters a b m α β calc. a b α β for pure components

Read coefficients θ3 θ2 θ1 θ0 and constant Bc for mixtures Read coefficients θ3 θ2 θ1 θ0 and constant Bc

Call cubic equation and calculate Bc for mixtures Call cubic equation and calculate Bc for pure components

Read coefficients θ3 θ2 θ1 θ0 and constant Zc for mixtures Read coefficients θ3 θ2 θ1 θ0 and constant Zc

Call cubic equation and calculate Zc for mixtures Call cubic equation and calculate Zc for pure components

Read constant Ac for mixtures Read constant A c

Call equation Ac and calculate Ac for mixtures

Call equation Ac and calculate Ac for pure components

Read critical properties Tc Pc Vc

Call equations of Tc Pc Vc calculate for Tc Pc Vc for Call equations of Tc Pc Vc and calculate mixtures for Tc Pc Vc for pure components

End

Figure 3.2. Algorithm for Computation of Critical Parameters.

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 the results of critical pressures and critical temperatures. Each of the tables corresponds to 9 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 complex mixture. The predicted critical properties and acentric factor for C7 + of complex mixtures are also presented. The methane concentrations in these mixtures varied from 19 to 96.6 mol percent. The intermediate hydrocarbon

74 groups, consisting of ethane, propane, and butane varied from a low of 3 to 59 mol percent, and the concentrations of pentane and hexane fractions from 1 to 3 mole percent, and the fraction heptane plus (C7+) varied from 2 to 14 mol percent. The non- hydrocarbon mixtures consist of nitrogen and carbon dioxide. The concentration of non- hydrocarbon mixtures varied from less than 1 to 22 mol percent. The physical properties of the components fraction covered the range from light, paraffin to heavy, aromatic, and asphalt.

4.2 Calculation of Critical Properties

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 step one, to compute the critical properties for complex mixtures, it is

desirable first to determine these properties for the C7+ fraction in the mixture. In

complex mixtures, components heavier than the n-heptane have been summed into a C7+

fraction. These C7+ fractions required for the parameters of equations of state are obtained from the previously determined correlations TR-4-99 Lawal, 59 and Lawal, 62

and the critical compressibility Z c 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 C7+ fractions in the complex

75 mixtures are calculated on the basis of the reported molecular weight of the fraction

(MW ) . The empirical expressions shown in the Table 4.1 are used to estimate the c7+ critical properties for the hydrocarbon fractions.

Table 4.1. A Sample of Experimental Data Used for Calculations of mixture 145-1.

Hydrocarbon Mixture Composition 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 o Critical Temp., Tc ,( R) 725 Critical Temp. This work 725.39

Critical Pressure, Pc ( psia) 2100

Predicted Pc ( psia) : (S. Yarborough) 2002 (Etter and Kay) 2911 (Zais) 2175 This Work 2101

62 Table 4.2. Physical properties of C7+ Fractions Correlation.

6084 141.5 e e API = + 5.9 S = T = e MW 1 S 2 MW g API +131.5 bp o g

e1 e2 e3 e2 e2 e3 e4 e1 e2 e3 e4 C = eo MW S g Tbp Pc = e0 MW S g Tbp C Tc = e0 MW S g Tbp C

e1 e2 e3 e4 ω = e0 MW S g Tbp C

Parameters e0 e1 e2 e3 e4

108.701661 0.4224480 0.42682558 0.0000 0.0000 Tbp

C 0.83282122 0.09255911 -0.0413045 0.12621158 0.0000

237031780 -0.028484 2.755309 -1.374440 -2.947221 Pc

6.206640 -0.059607 0.224357 0.968332 -0.802538 Tc

ω 1.5790E-13 -1.453063 -2.811708 4.883921 2.109476

The C7+ critical properties and acentric factors are estimated from the correlation

shown in Table 4.2. Bu using the given Mw of the C7+ as basis, the critical pressure,

critical temperature, critical compressibility factor, and acentric factor of C7+ were

obtained. The heptane-plus C7+ is treated as a lumped single pseudo-component for all the prediction 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 a,b,α, β , Ωω , the dimensionless critical parameters Ac and Bc are calculated from

Equations 3.20, 3.18, 3.21, 3.22, 3.23, 3.24, 3.67, 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 39.98 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 o Tc, ( R) 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

Ωw 0.3555 0.3555 0.3555 0.3555 0.3564 0.3564 0.3564 0.3564 0.3564

Table 4.4. Calculated Results of Pure Component Parameters.

Component a b α β Ωω Ac Bc 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-C4 109225.70 1.850 2.81 6.02 0.359 0.587 0.983

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

am ,bm ,α m and β m can be determined by using Equations 3.62 through Equation 3.65.

78 Since equations of state are developed for pure components, the use of mixing rules is necessary to make proper application of the equations of state for mixtures. Another important parameter in every geometric and quadratic mixing rule is the interaction parameter. 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, aij ,α ij ,

and β ij were assumed as constants, and assigned a value of 1. Then, using the Equation

3.52 the constant Ac was estimated. The Equations 3.59 through 3.61, for calculating the

critical properties are expressed in terms of the mixture parameters, am ,bm ,α m , and β m 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

am 31177.9123

bm 0.11728247

α m 99.1126855

β m 0.69748539

Bc 0.269

Z c 0.2985

Ac 34.15

79 4.3 Results and Discussion

The variation in behavior of the calculated critical pressure and temperature agreed both qualitatively and quantitatively with experimental data. Deviation errors in the predictions of both critical pressure and temperature, as shown in Figures 4.5-4.6 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 Figure

4.1, this work shows better agreement with the experimental data than the other three correlations. Consequently, the LLS equation of state is selected for this work.

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 lead to accurate predictions of mixture behavior. Furthermore, the influence of the values of interaction 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 significantly the prediction of the critical properties, especially the critical pressure.

The results of the predicted critical pressure, Pc , and critical temperature, Tc , for complex mixtures are presented in Table 4.6. The prediction results are in agreement with the experimental values. The absolute percent deviation of the results ranges from

0.03 to 0.13 percent.

80 Table 4.6. Predicted Critical Pressure, Pc , Critical Temperature, Tc , for Mixtures.

Properties Mix. 1 Mix. 2 Mix. 3 Mix. 4 Mix.5 Mix. 6 Mix. 7 Mix. 8 Mix. 9

Tc, exp. 725 725 725 725 694 660 660 660 660

Tc, pred. 725.39 725.45 724.58 725.81 694.91 694.44 659.17 661.13 659.22 AD (%) 0.054 0.062 0.058 0.112 0.131 5.218 0.126 0.171 0.118

Pc, exp. 2100 2500 3400 1920 2420 3430 4355 4295 4630

Pc, pred. 2101.12 2501.55 3398.03 1922.15 1423.18 3432.17 4349.55 4302.33 4624.49 AD (%) 0.053 0.062 0.058 0.112 0.131 0.063 0.125 0.171 0.119

Eighty-five mixture sets of experimental data are analyzed for hydrocarbon

mixtures. Table C.1 through C. 27 in Appendix C show the prediction results for critical

pressures, critical temperatures. Also, these results are displayed in Figures C. 1-through

C. 10 for critical pressure, and in Figures C. 11 through C. 19 for critical temperature in

Appendix C. Figure 4.1 is a cross-plot shows the prediction results for the critical

pressures for all data.

8500

7500 )

a 6500 i s p (

e r

u 5500 s s e r P

a 4500 c i t i r C

e 3500 t a l u c l a 2500 C Simon-Yarborough

1500 Etter-Kay Zais This w ork 500 500 1500 2500 3500 4500 5500 6500 7500 8500 Experimental Critical Pressure (psia)

Figure 4.1. Predicted Critical Pressure of Complex Mixtures.

The match of the predicted critical pressure against the experimental critical pressure gives an absolute deviation between 0.03 and 0.13 %. Similarly, the prediction results for the critical temperature are in agreement with the experimental data. Figure

4.2 is a cross-plot which displays the prediction results for critical temperatures for all data.

82 800

750

This w ork 700 ) R o 650 e ( r u t a r

e 600 mp e 550

Critical T 500 ated l 450 Calcu

400

350

300 300 350 400 450 500 550 600 650 700 750 800

Experimental Critical Temperature (oR)

Figure 4.2. Predicted Critical Temperature of Complex Mixtures.

4.4 Comparison Between Calculated and Experimental Data

To test the accuracy of the calculations of critical points of hydrocarbon mixtures, a comparison of calculated with experimental values of the pressure and temperature at the critical points of mixtures of known compositions was made. For this purpose, the critical pressure and critical temperature on the hydrocarbon paraffin mixtures which were determined in the laboratory was employed. A total of about 85 mixtures of non- hydrocarbon and hydrocarbon components 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.

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 values from the experimental is about 0.03-0.13% of both the pressure and temperature.

Figure 4.1 is a cross-plot shows the comparison between the calculated critical pressures and experimental critical pressures for all data points. Also, Figure 4.2 is a cross-plot compares the calculated critical temperatures with the experimental critical temperatures for all data.

The comparison of the results of the critical point predictions using the LLS equation of state and the Simon-Yarborough, Etter-Kay, and Zais empirical calculation methods was necessary to provide guidelines for critical point predictions of complex reservoir fluids. The most important factor to be considered was the relationship between the complexity of the equations of state used in the critical point equations and the accuracy of the predictions of the critical properties. The critical point equations derived from the LLS equation of state proved to be more simple and easy to derive than was the case of many other equations of state. However, the results of the critical point predictions proved to be in good agreement with the experimental critical points than the

Simon-Yarborough, Etter-Kay, and Zais.

The most comprehensive comparison of the critical point predictions of the LLS equation of state approach and the Simon-Yarborough, Etter-Kay, and Zais correlations can be made for the critical pressure calculations for each class of mixtures. In this work, for the hydrocarbon/non-hydrocarbon class of mixture the LLS equation predicted more 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 adjustment of binary interaction parameters).

One of the most significant features of the critical point calculations is that the values of the interaction parameters are not equal to 1 as suggested in the original binary mixing rules of the Redlich-Kwong and Benedict-Webb-Rubin (BWR) equations of state.

The assumption of a value of 1 to the binary interaction terms is equivalent to the assumption that the intermolecular energy can be described by the geometric means of the pure component energies. Chueh and Prausnitz 21 point out that the geometric mean relationship is accurate only for simple, spherically, and symmetric molecules of nearly equal size. Thus, for most multi-component systems, the best value of the interaction parameters will not equal to 1, which agrees with the results of this investigation.

A comparison of the accuracy of the critical point prediction using LLS equation of state approach and three other empirical correlation methods appears in Tables C. 1 through C. 9 in Appendix C. In general, the accuracy of critical pressure predictions from the empirical calculation methods of Simon-Yarborough, Etter-Kay, and Zais was not comparable to that obtained by this work with the equation of state approach using

Lawal-Lake-Silberberg equation of state. The advantages of the equation of state approach are that the critical properties are determined simultaneously, including the critical temperature, and critical volume are not restricted to any particular equation of state.

85 CHAPTER V

CONCLUSIONS AND RECOMMENDATION

5.1 Conclusions

The purpose of this work was to integrate the thermodynamic criteria of the critical state criteria of mixtures with a LLS EOS to predict the critical properties of complex petroleum reservoir fluids. The major conclusions drawn from each of the objectives are presented along with recommendations for future study in the following sections:

The first objective, to develop a closed-form solution to the van der Waals criticality conditions and perform numerical calculations to predict the critical properties was achieved by using the Lawal-Lake-Silberberg (LLS) equation of state on complex mixtures of hydrocarbons. Analysis of the results of the critical calculations showed that qualitative and quantitative agreement with experimental data was obtained. In general, the average error levels in the predictions of the critical properties were comparable to those obtained experimentally and not comparable to those obtained from other correlations (Simon and Yarborough, Etter and Kay, and Zais) 98,29,122 . The conclusion drawn from this result was that the ability of the equation of state to predict the critical points of complex mixtures is directly related to the ability of the equations to predict the corresponding critical properties of pure components of the mixtures. Thus, because of its simplicity and the fact that the Lawal-Lake-Silberberg equation applicability yields accurate and exact critical temperature and pressure for any pure component, the LLS

86 equation proved by this work to be much more satisfactory equation of state for critical point predictions of petroleum reservoir fluids.

The second objective, to determine the need to establish interaction parameters for hydrocarbon and non-hydrocarbon mixtures and for hydrocarbon with pseudo- components was achieved. The best estimations of the critical properties were obtained by adjusting the value of the interaction parameters to be equal to one in the mixing rules

of the constants am and bm 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 reservoir fluids was also achieved, and the procedural approach for computing the step-by-step method was efficiently performed.

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 accuracy in the critical properties prediction carried out in this work proved to be comparable to the experimental measurements.

5.2 Recommendations

The results obtained from this investigation are useful for PVT analysis of reservoir fluids especially in resolving retrograde behavior. Therefore, several recommendations can be made regarding future work:

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

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

sufficiently reliable for predicting the vapor-liquid equilibrium and volumetric

reservoir fluids without the use of pseudodization. Instead, utilize heptane-

plus fraction as a lumped single-pseudo-component in equation of state by

predicting liquid dropout, constant volume depletion (CVD), constant

composition expansion (CCE), and flash-differential liberation (FL-DL) tests.

3. A possible use of the technique developed in this work into a reservoir

simulation model to perform gas cycling, gas- condensate simulation, and

multi-contact miscibility studies.

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98 APPENDIX A

ANALYTICAL SOLUTION FOR CUBIC EQUATIONS

The general cubic equation is given by Dunham, 24

3 2 a3 X + a2 X + a1 X + a0 = 0 (A.1)

Divide the entire equation by a3 ,

a a a X 3 + 2 X 2 + 1 X + 0 = 0 (A.2) a3 a3 a3

To find the roots of this equation, we first eliminate the quadratic term, X 2 . To do this, we make the substitution

a X = y − 2 (A.2) 3a3 then, by substituting in Equation A.1, to obtain

a2 3 a2 2 a2 a3 ( y − ) + a2 ( y − ) + a1 ( y − ) + a0 = 0 (A.3) 3a3 3a3 3a3

Expanding Equation A.3 and simplifying

a ⎛ a ⎞ ⎛ a 2 ⎞ a 3 ( y − 2 )3 = y 3 − ⎜ 2 ⎟ y 2 + ⎜ 2 ⎟ y − 2 (A.4) 3a ⎜ a ⎟ ⎜ 2 ⎟ 3 3 ⎝ 3 ⎠ ⎝ 3a3 ⎠ 27a3

2 ⎛ a ⎞ ⎛ a ⎞ a 2 ⎜ y − 2 ⎟ = y 2 − 2⎜ 2 ⎟ y + 2 (A.5) ⎜ ⎟ ⎜ ⎟ 2 ⎝ 3a3 ⎠ ⎝ 3a3 ⎠ 9a3

Substituting in Equation A.3

99 ⎛ a a 2 a 3 ⎞ ⎛ a a 2 ⎞ ⎜ 3 2 2 2 2 ⎟ ⎜ 2 2 2 ⎟ a3 y − y + y − + a2 y − 2 y + ⎜ a 3a 2 27a 3 ⎟ ⎜ 3a 9a 2 ⎟ ⎝ 3 3 3 ⎠ ⎝ 3 3 ⎠ (A.6) ⎛ a ⎞ ⎜ 2 ⎟ + a1 ⎜ y − ⎟ + a0 = 0 ⎝ 3a3 ⎠ or

⎛ − a 2 ⎞ ⎛ 2a 3 a a ⎞ a y 3 + ⎜ 2 + a ⎟ y + ⎜a + 2 − 1 2 ⎟ = 0 (A.7) 3 ⎜ 3a 1 ⎟ ⎜ 0 2 3a ⎟ ⎝ 3 ⎠ ⎝ 27a3 3 ⎠

Equation A.7 is called the “depressed” cubic equation since the quadratic term, y 2 is

eliminated. Divide Equation A.7 by a3 , then

1 ⎛ a 2 ⎞ 1 ⎛ 2a 3 a a ⎞ y 3 + ⎜a − 2 ⎟ y + ⎜a + 2 − 1 2 ⎟ = 0 (A.8) a ⎜ 1 3a ⎟ a ⎜ 0 2 3a ⎟ 3 ⎝ 3 ⎠ 3 ⎝ 27a3 3 ⎠

Let

1 ⎛ a 2 ⎞ e = ⎜a − 2 ⎟ (A.9) ⎜ 1 ⎟ a3 ⎝ 3a3 ⎠

And

1 ⎛ 2a 3 a a ⎞ f = ⎜a + 2 − 1 2 ⎟ (A.10) a ⎜ 0 2 3a ⎟ 3 ⎝ 27a3 3 ⎠

Substituting Equations A.9 and A.10 into Equation A.8 and obtain

y 3 + ey + f = 0 (A.11)

Reducing Equation A.11 using Vieta’s Substitution

s y = z + (A.12) z

100 The constant s is an undefined constant for right now. Substituting Equation A.12 in

Equation A.11, yield

3 ⎛ s ⎞ ⎛ s ⎞ ⎜ z + ⎟ + e⎜ z + ⎟ + f = 0 (A.13) ⎝ z ⎠ ⎝ z ⎠

Expanding Equation A.13

3 ⎛ s ⎞ s 2 s 3 3s 2 s 3 ⎜ z + ⎟ = z 3 − 3sz + 3 − = z 3 − 3sz + − (A.14) ⎝ z ⎠ z z 3 z z 3

Multiplying the right side of Equation A.14 by z 3 , yield

⎛ s ⎞ sz 3 ⎜ z + ⎟z 3 = z 6 − 3sz 4 + 3s 2 z 2 − s 3 = z 4 + s = ez 4 + esz 2 ⎝ z ⎠ z

Then

z 6 + (3s + e)z 4 + s(3s + e)z 2 + fz 3 + s 3 = 0 (A.15)

e Now let s = − to simplify Equation A.15 into a “tri-quadratic” equation, then 3

e 3 z 6 + fz 3 + = 0 (A.16) 27

By substituting w = z 3 , then we have a general quadratic equation which we can solve using the quadratic formula.

e 3 w 2 + fw − = 0 (A.17) 27

Solve for the quadratic Equation A.17 and will give two roots for w , hence using w = z 3 would then give three roots for each of the two roots of w, hence will give

101 six root values for z . But the six root values of z would give only three values of y (for

s y = z + ), and three values of x in Equation A.2. z

Illustrative Problem

Find the roots of the following cubic equation

x 3 − 0.03x 2 + 2.4x10 −6 = 0 (A.18)

Solution

For the general form of the cubic equation

ax 3 + bx 2 + cx + d = 0 (A.19)

Where a = 1,b = −0.03,c = 0,d = 2.4x10−6

To find the roots of this equation, first eliminate the quadratic term (i.e., depress the cubic equation). Let

b x = y − 3a

− 0.03 x = y − 3(1)

x = y + 0.01 (A.20)

Substituting the above value of x Equation A.20 into the cubic Equation A.18 and simplify, yield

y 3 − (3x10−4 ) y + (4x10−7 ) = 0 (A.21)

Convert this depressed cubic equation into the form

3 y + ey + f = 0 (A.22)

102 Where the coefficients of Equation A.22 are e = −3x10−4 , f = 4x10−7

Now solve the depressed equation by using Vieta’s substitution as,

s y = z + z and obtain

z 6 + (3s − 3x10−4 )z 4 + (4x10−7 )z 3 + s(3s − 3x10−4 )z 2 + s 3 = 0 (A.23)

Let

e − 3x10−4 s = − = − = 10−4 (A.24) 3 3

Substituting Equation A.24 into Equation A.24 to obtain the “tri-quadratic” equation

z 6 + (4x10−7 )z 3 +1x10−12 = 0 (A.25)

Convert Equation A.25 into a general quadratic equation by using w = z 3

w 2 + (4x10−7 )w + (1x10−12 ) = 0 (A.26)

Using the quadratic equation, the values for w are

4x10−7 w = − ± (4x10−7 ) 2 − 4(1x10−12 ) 1 2

−7 −7 w1 = −2x10 + i(9.797958x10 ) (A.27) and

−7 −7 w2 = −2x10 − i(9.797958x10 ) (A.28)

The solution of w = z 3 gives three values of z . These values in rectangular form are:

w = a + bi

w = ze iθ = z(cosθ + i sinθ )

103 b Where θ = arctan + π a

Then,

3 3 ⎡ ⎛θ 2πn ⎞ ⎛θ 2πn ⎞⎤ w = z ⎢cos⎜ + ⎟ + i sin⎜ + ⎟⎥ ⎣ ⎝ 3 3 ⎠ ⎝ 3 3 ⎠⎦

Then, the three values from w1 are in rectangular form

z1 = −0.008976 + 0.00440i

−4 z2 = 6.707x10 0.009977i

z3 = 0.008305 + 0.005569i

The values of z from w2 are in rectangular form

z4 = −0.008976− 0.0044079i

−4 z5 = 6.70689228 x10 − 0.0099775i

z6 = 0.0083054 − 0.00556957i

Using Vieta’s substitution,

s y = z + z

1x10 −4 y = z + (A.29) z

Substituting into Equation A.29 the value of z to find three values for y , choosing

z1 = −0.008976098+ 0.0044079 yields

1x10−4 y = −0.008976 + 0.0044079i + = −0.01795 1 − 0.008976 + 0.0044i

104 1x10−4 y = 6.706892 + 0.00997748i + = 0.0013414 2 6.706892 + 0.00997748i

Similarly, the other value of z3 gives

y3 = 0.016611

With the substitution in Equation A.20

x = y + 0.01 (A.20)

The three roots of the given Equation A.19 are

x1 = −0.0079522

x2 = 0.0113414

x3 = 0.02661112

Method 2

The general cubic equation is given by

3 2 a3 Z + a2 Z + a1 Z + a0 = 0 (A.30)

Divide the entire equation by a3 ,

a a a Z 3 + 2 Z 2 + 1 Z + 0 = 0 (A.31) a3 a3 a3

Eliminate a2 by making substitution of the form

Z = x − λ (A.32)

By substituting Equation A.32 into Equation A.30,

3 2 (x − λ) + a2 (x − λ) + a1 (x − λ) + a0 = 0 (A.33)

By expanding Equation A.33,

105 3 2 2 3 2 2 (x − 3λx + 3λ x − λ ) + a2 (x − 2λx + λ ) + a1 (x − λ) + a0 = 0 (A.34)

3 2 2 2 3 x + (a2 − 3λ)x + (a1 − 2a2λ + 3λ )x + (a0 − a1λ + a 2λ − λ ) = 0 (A.35)

a Let λ = 2 in order to eliminate the x 2 , so 3

1 Z ≡ x − a (A.36) 3 2

a a 2 x a 3 x 3 = (x − 2 )3 = x 3 − a x 2 + 2 − 2 (A.37) 3 2 3 27

1 2 1 a Z 2 = a (x − a ) 2 = a x 2 − a 2 x + a 3 (A.38) 2 2 3 2 2 3 2 9 2

1 1 a Z = a (x − a ) = a x − a a (A.39) 1 1 3 2 1 3 2 1

Substituting back into Equation A.30, becomes

1 2 1 x 3 + (−a + a )x 2 + ( a 2 − a 2 + a )x − ( a 3 2 2 3 2 3 2 1 27 2 (A.40) 1 1 − a 3 + a a − a ) = 0 9 2 3 1 2 0

1 1 2 x 3 + (a − a 2 )x − ( a a − a 3 − a ) = 0 (A.41) 1 3 2 3 1 2 27 2 0

3a − a 2 9a a − 27a − 2a 3 x 3 + 1 2 − 1 2 0 2 = 0 (A.42) 3 27

Let,

3a − a 2 P = 1 2 (A.43) 3 and,

106 9a a − 27a − 2a 3 q = 1 2 0 2 (A.44) 27 then Equation A.42 can be written as

x 3 + Px = q (A.45)

Make Vieta’s substitution to simplify the derivation by letting:

P x = W − (A.46) 3W

Substituting for x in Equation A.45, obtain

P P (W − )3 + P(W − ) − q = 0 3W 3W or

P 3 W 3 − − q = 0 (A.47) 27W 3

Multiplying through Equation A.47 by W 3 to obtain a quadratic equation in W 3 ,

P 3 (W 3 ) 2 − q(W 3 ) − = 0 (A.48) 27

Apply the quadratic formula (Birkhoff & Mclane 1996, P.106):

1 4 W 3 = (q ± q 2 + P 3 2 27

1 1 1 = q ± q 2 + P 3 2 4 27

= R ± R 2 + Q 3 (A.49)

Let:

107 1 R = q 2 1 R 2 = q 2 4 P 3 Q 3 = 27

By Vieta’s “magic” substitution, first define the intermediate variables from Equation

A.42

Let:

3a − a 3 Q = 1 2 (A.50) 9

9a a − 27a − 2a 3 R = 2 1 0 2 (A.51) 54

The cubic Equation A.42 then becomes,

x 3 + 3Qx − 2R = 0 (A.52)

Let B and C be arbitrary constants. An identity, which is satisfied by perfect cubic polynomial equations, is

x 3 − B 3 = (x − B)(x 2 + Bx + B 2 ) (A.53)

Since Q ≠ 0 , add a multiple of (x-B) by C to both sides of Equation A.23 to give

(x 3 − B)3 + C(x − B) = (x − B)(x 2 + Bx + B 2 + C) = 0 (A.54)

Regrouping terms, becomes

x 3 + Cx − (B 3 + BC) = (x − B)(x 2 + Bx + (B 2 + C)) = 0 (A.55)

Match the coefficients C and − (B 3 + BC)with those of Equation A.52, so we must have

C = 3Q (A.56)

108 B 3 + BC = 2R (A.57)

Then by substituting Equation A.56 into Equation A.57,

B 3 + 3QB = 2R (A.58)

Now, find a value for B and reduce Equation A.58 to a quadratic equation.

1 1 B = (R + Q 3 + R 2 )3 + (R − Q 3 + R 2 )3 (A.59)

Taking the second and third powers of B gives

2 1 2 B 2 = (R + Q 3 + R 2 )3 + 2()R 2 − (Q 3 + R 2 ) 3 + (R − Q 3 + R 2 )3 (A.60)

2 2 = (R + Q 3 + R 2 )3 + (R − Q 3 + R 2 )3 − 2Q (A.61)

⎧ 1 1 ⎫ 3 ⎪ 3 2 3 2 ⎪ B = −2QB + ⎨(R + Q + R )3 + (R − Q + R )3 ⎬ ⎩⎪ ⎭⎪ (A.62) ⎧ 1 1 ⎫ ⎪ 3 2 3 2 ⎪ × ⎨(R + Q + R )3 + (R − Q + R )3 ⎬ ⎩⎪ ⎭⎪

2 1 3 3 2 3 2 3 2 3 3 2 = (R + Q + R )+ (R − Q + R )+ (R + Q + R ) (R − Q + R ) (A.63) 2 1 + (R + Q 3 + R 2 )3 (R − Q 3 + R 2 )2 − 2QB

1 ⎛ 1 1 ⎞ = −2QB + 2R + R 2 − (Q 3 + R 2 ) 3 ⎜ R + Q 3 + R 2 3 + R − Q 3 − R 2 3 ⎟ (A.64) ()⎜( ) ( ) ⎟ ⎝ ⎠

= −2QB + 2R − QB = −3QB + 2R (A.65)

Plugging B 3 and B into the left side of Equation A.58 gives:

(−3QB + 2R) + 3QB = 2R (A.66)

109 Now, plugging C = 3Q into the quadratic part of Equation A.55 gives:

x 2 + Bx + (B 2 + 3Q) = 0 (A.67) which provides the solution.

1 x = (− B ± B 2 − 4(B 2 + 3Q) ) (A.68) 2

1 1 = − B ± − 3B 2 −12Q (A.69) 2 2

1 = − B ± 3i B 2 − 4Q (A.70) 2

These can be simplified by defining

1 1 A ≡ (R + Q 3 + R 2 )3 − (R − Q 3 + R 2 )3 (A.71)

2 1 2 A2 = (R + Q 3 + R 2 )3 − 2()R 2 − (Q 3 + R 2 ) 3 + (R − Q 3 + R 2 )3 (A.72)

2 2 = (R + Q 3 + R 2 )3 + (R − Q 3 + R 2 )3 + 2Q (A.73)

A2 = B 2 + 4Q (A.74)

So the solutions to the quadratic portion of Equation A.55 can be written as

1 1 x = − B ± 3i A (A.75) 2 2

Defining:

D ≡ Q 3 + R 2 (A.76)

S ≡ 3 R + D (A.77)

T ≡ 3 R − D (A.78)

110 where

B = S + T (A.79)

A = S −T (A.80)

Therefore, at least, the roots of the original equation A.37 are given by:

1 Z = − a + (S + T ) (A.81) 1 3 2

1 1 1 Z = − a − (S + T ) + i 3(S − T ) (A.82) 2 3 2 2 2

1 1 1 Z = − a − (S + T ) − i 3(S − T ) (A.83) 3 3 2 2 2

2 With a2 as the coefficient of Z in Equation A.1 and S and T as defined above; these three equations providing the three roots of the cubic equation are sometimes known as

Candamo’s formula. If the equation is in the standard form of Vieta,

x 3 + Px = q (A.84)

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 equation to represent vapor-liquid coexistence. The VDR-EOS is a two-parameter equation with pressure given by a cubic function of molar volume in the form

RT a(T ) P = − (B .1) V − b V 2

The first term on the right hand side is the repulsive term and the second term is the attractive term is temperature dependent.

V Multiplying both sides of Equation B.1 by to obtain the VDW-EOS in Z RT form

V a Z = − (B.2) V − b RTV

by definition, where

PV Z = (B.3) RT and

ZRT V = (B.4) P

Where Z is the compressibility factor, T is temperature, V is volume, P is pressure, and R is the molar universal gas constant. The parameter a is a measure of the

112 attractive forces between molecules, and the parameter b is the co-volume occupied by the molecules (if the molecules are represented by hard-spheres of diameter d, then

(2πNσ 3 ) b = . 3

Substituting Equation B.4 into Equation B.2 leaves

ZRT aP Z = − (B.5) ZRT − Pb ZR 2T 2 or

1 ap Z = − (B.6) Pb R 2T 2 1− ZRT with

aP A = (B.7) R 2T 2 and

bP B = (B.8) RT

Substituting Equation B.7 and Equation B.8 into Equation B.6 leaves

1 A Z = − (B.9) B Z 1− Z or

Z A Z = − (B.10) Z − B Z or

113 Z 2 − A(Z − B) Z = (B.11) (Z − B)Z and Z(Z − B)Z = Z 2 − A(Z − B) (B.12) or

Z 3 − BZ 2 = Z 2 − AZ + AB (B.13) or

Z 3 − BZ 2 − Z 2 + AZ − AB = 0 (B.14) then

Z 3 − (1+ B)Z 2 + AZ − AB = 0 (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

3 (Z − Z c ) = 0 (B.16)

Expanding Equation (A.16), gives

3 3 2 3 (Z − Z c ) = Z − 3Z c Z + 3Z c Z − Z c = 0 (B.17)

Equating the coefficients of Equation B.15 to the coefficients of Equation B.17, leaves

3Z c = 1+ Bc (B.18)

2 3Z c = Ac (B.19) and

3 Z c = Ac Bc (B.20)

There are three equations B.18, B.19, and B.20 with three unknowns Bc, Ac, and

Zc. To find these unknowns, substituting Ac of Equation B.19 into Equation B.20, obtain

114 3 2 Z c = 3Zc Bc (B.21) then

3 Z c Z c Bc = 2 = (B.22) 3Z c 3

Substituting Equation (B.22) into Equation (B.18), leaves

Z 3Z = 1+ c (B.23) c 3 and

9Z c = 3 + Z c (B.24) then

3 Z = (B.25) c 8

Substituting the value of Zc in Equation B.25 into Equation B.19, obtain

9 27 A = 3 = (B.26) c 64 64

Substituting Equation B.25 into Equation B.18, gives

1 B = (B.27) c 8

Now, consider the generalized form of the Lawal-Lake-Silberberg (LLS) cubic equation of state. That is,

RT a P = − (B.28) V − b V 2 + αbV − βb 2 and the gas law is given,

PV = ZRT (B.29)

115 where, by definition

PV Z = (B.30) RT and

ZRT V = (B.31) P

V Multiplying both sides of Equation B.28 by the value RT

V RT V a V ⋅ P = ⋅ − ⋅ (B.32) RT V − b RT V 2 + αbV − βb 2 RT then

V aV Z = − (B.33) V − b RT (V 2 + αbV − βb 2 )

Substituting Equation B.31 into Equation B.33, yields

ZRT ZRT a ⋅ Z = P − P (B.34) ZRT ⎡ ZRT 2 ZRT 2 ⎤ − b RT ⎢( ) + αb( ) − βb ⎥ P ⎣ P P ⎦ or

Z AZ Z = − (B.35) Z − B Z 2 + αBZ − βB 2

1 Multiplying both sides of Equation B.35 by , gives Z

1 A 1 = − (B.36) Z − B Z 2 + αBZ − βB 2 or

116 (Z − B)(Z 2 + αBZ − βB 2 ) − (Z 2 + αBZ − βB 2 ) + A(Z − B) = 0 (B.37)

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

3 expansion (Z − Z c ) = 0 shown in Equation B.38.

3 2 2 3 Z − 3Z c Z + 3Z c Z − Z = 0 (B.38)

The coefficients of Equation B.37 are:

Z 3 :1 (B.39)

Z 2 :αB − B −1 (B.40)

Z 1 :βB 2 −αB 2 −αB + A (B.41)

and

Z 0 : βB 3 + βB 2 − AB (B.42)

Substituting these coefficients B.39, B.40, B.41, and B.42 back in Equation B.37, leaves

Z 3 + (αB − B −1)Z 2 + (A − βB 2 −αB 2 −αB)Z + (βB 3 + βB 2 − AB) = 0

(B.43)

Equate the coefficients of Equation B.43 to the coefficients of Equation B.38, obtain

− 3Z c = αBc − Bc −1

3Z c = 1+ Bc −αBc (B.44)

2 2 2 3Z c = Ac − βBc −αBc −αBc (B.45)

3 3 2 − Z c = βBc + βBc − Ac Bc (B.46)

117 where the subscript c represents the conditions at the critical state. Solve for Zc, Bc, and

Ac. From Equation B.44,

αB − B −1 − Z = c c (B.47) c 3 or

1−αB + B Z = c c (B.48) c 3

From Equation B.45

2 2 2 Ac = 3Z c + βBc +αBc +αBc (B.49)

Substituting Equation B.48 into Equation B.46, gives

3 ⎛1−αBc + Bc ⎞ 3 2 2 2 2 − ⎜ ⎟ = βBc + βBc − ()3Z c + βBc + αBc + αBc Bc (B.50) ⎝ 3 ⎠ or

3 3 2 2 2 2 − ()1−αBc + Bc = 27βBc + 27βBc − [9(1−αBc + Bc ) + 27(βBc + αBc + αBc )]Bc (B.51) and

3 3 2 3 3 3 2 2 2 2 α Bc − 3α Bc + 3αBc − Bc − 3α Bc + 6αBc − 3Bc + 3αBc − 3Bc 3 2 2 2 3 3 2 2 −1− 27βBc − 27βBc + 9α B c +9Bc −18αBc −18αBc +18Bc + 9Bc 3 3 2 + 27βBc + 27αBc + 27αBc = 0

(B.52) or

118 ()α 3 − 3α 2 + 3α −1− 27β + 9α 2 + 9 −18α + 27β + 27α B 3 + (−3α 2 + 6α c 2 − 3 − 27β −18α +18 + 27α)Bc + ()3α + 6 −1 = 0

(B.53) and

3 2 3 2 2 (α + 6α +12α + 8)Bc + (− 3α +15α − 27β +15)Bc + (3α + 6)Bc −1 = 0

(B.54) or

3 2 3 2 2 (α + 6α +12α + 8)Bc + 3(−α + 5α − 9β + 5)Bc + 3(α + 2)Bc −1 = 0 (B.55)

Which can be further simplified to the following expression of the terms and appropriate grouping of the variables:

3 2 θ1Bc +θ 2 Bc +θ 3 Bc +θ 4 = 0 (B.56) where

3 2 θ1 = (α + 6α +12α + 8) (B.57)

2 θ 2 = (15 +15α − 27β − 3α ) (B.58)

θ 3 = (6 + 3α) (B.59)

θ = −1 4 (B.60)

Critical Condition by the Critical Compressibility Form

From equation B.44,

− 3Z c = Bc (α −1) −1, (B.43)

119 the following expressions can be deduced:

(1− 3Z ) B = c c (α −1)

3Z −1 B = c (B.61) c 1−α

Substituting of Equation B.61 into Equation B.55, yield

(1− 3Z ) (1− 3Z ) (α 3 + 6α +12α + 8)( c )3 + (−3α 2 +15α − 27β +15)( c ) 2 + (α −1) (α −1)

(1− 3Z ) (3α + 6)( c ) −1 = 0 (α −1)

(B.62) or

(α 3 + 6α 2 +12α + 8)(1− 3Z )3 + (−3α 2 +15α − 27β +15)(1− 3Z )2 (α −1) + c c (B.63 2 3 (3α + 6)(1− 3Zc )(α −1) − (α −1) = 0

)

Let

3 2 3 A = (α + 6α +12α + 8)(1− 3Z c ) B = (−3α 2 +15α − 27β +15)(1− 3Z ) 2 (α −1) c (B.64) 2 C = (3α + 6)(1− 3Z c )(α −1) D = (α −1)3

And,

(A) + (B) + (C) − (D) = 0 (B.65)

Expanding A, B, C, and D, then

120 3 3 A = (α + 6α + 8)(1− 3Z c ) 3 3 2 3 3 3 3 2 2 2 2 2 = (−27α Z c −162α Z c − 324Z c − 216Z c + 27α Z c +162α Z c + 324αZ c + 216Z c − 3 2 3 2 9α Z c − 54α Z c −108αZ c − 72Z c + α + 6α +12α + 8

2 2 B = (−3α +15α − 27β +15)(1− 3Z c ) (α −1) = (−27α 3 Z 2 +162α 3 Z 2 − 243αβZ 2 + 243βZ 2 −135Z 2 −18α 3 Z c c c c c c 2 3 2 −108α Z c −162αβZ c −162βZ c + 90αZ c + 90Z c − 3α +18α −15α − 27αβ − 27β −15)

C = (3α + 6)(1− 3Z )(α −1) 2 c 3 3 = (−9α Z c + 27αZ c −18Z c + 3α − 9α + 6)

D = (α −1)3 = (α 3 − 3α 2 + 3α −1)

substituting A, B, C, and D in Equation B.65, obtain

3 2 3 3 3 3 2 2 2 2 (−27αZ c −162α Z c − 324αZ c − 216Z c + 27α Z c +162α Z c + 324αZ c 2 3 2 3 2 + 216Z c − 9α Z c − 54α Zc −108αZ c − 72Z c + α + 6α +12α + 8) + 3 2 2 2 2 2 2 3 2 (−27α Z c +162α Z c − 243αβZ c − 243βZ c −135Z c +18α Z c −18α Z c −162αβZ c 3 2 2 3 −162βZ c + 90Z c − 3α + 3α +18α − 27αβ − 27β −15) + (−9α Z c + 27αZ c −18Z c + 3α 3 − 9α + 6) − (α 3 − 3α 2 + 3α −1) = 0 (B.66) or

(−27α 3 −162α 2 − 324α − 216)Z 3 − (324α 2 + 324α + 243β − 243αβ + 81)Z 2 c c 2 2 − (−162α − 81α −162αβ −162β )Z c + (27α − 27αβ − 27β ) = 0

(B.67) or

(α 3 + 6α 2 +12α + 8)Z 3 − (12α 2 +12α + 9β − 9αβ + 3)Z 2 c c (B.68) 2 2 + (6α + 3α − 6αβ + 6β )Z c − (β + α −αβ ) = 0

121 Let

2 3 θ1 = (8 +12α + 6α + α ) θ = −(3 +12α +12α 2 + 9β − 9βα ) 2 (B.69) 2 θ 3 = (3α + 6α + 6β − 6αβ ) 2 θ 4 = −(β + α −αβ )

Then the critical condition by the critical compressibility form is

3 2 θ1 Z c +θ 2 Z c +θ 3 Z c +θ 4 = 0 (B.70)

Derivation No.2

From Equation B.44

(1− 3Z ) B = c (B.44) c (α −1)

From Equation B.46

3 3 2 − Z c = βBc + βBc − Ac Bc (B.46)

2 2 2 Substituting for Ac = 3Z c + βBc +αBc + αBc , obtain

3 3 2 2 2 2 − Z c = βBc + βBc − Bc [3Z c + βBc +αBc +αBc ] (B.71)

Substituting Equation B.61 into Equation B.68, then

3 2 3 ⎛ (1 − 3Z c ) ⎞ ⎛ (1 − 3Z c ) ⎞ ⎛ (1 − 3Z c ) ⎞ − Z c = β ⎜ ⎟ + β ⎜ ⎟ − ⎜ ⎟ ⎝ (α − 1) ⎠ ⎝ (α − 1) ⎠ ⎝ (α − 1) ⎠ 2 2 ⎡ ⎛ (1 − 3Z ) ⎞ ⎛ (1 − 3Z ) ⎞ ⎛ (1 − 3Z ) ⎞⎤ ⎢3Z 2 + β ⎜ c ⎟ + α ⎜ c ⎟ + α ⎜ c ⎟⎥ (B.72) c ⎜ (α − 1) ⎟ ⎜ (α − 1) ⎟ ⎜ (α − 1) ⎟ ⎣⎢ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎦⎥

Multiply both sides of Equation B.68 by (α −1)3 , obtain

122 − 3Z 3 (α −1)3 = β (1− 3Z )3 + β (1− 3Z 2 )(α −1) − (1− 3Z ) c c c c (B.73) 2 2 2 2 2 [3Z c (α −1) + β (1− 3Z c ) + α(1− 3Z c ) + α(1− 3Z c )(α −1) ] or

− 3Z 3 (α −1)3 − β (1− 3Z )3 − β (1− 3Z ) 2 (α −1) + (1− 3Z ) c c c c (B.74) 2 2 2 2 3 []3Z c (α −1) + β (1− 3Z c ) + α(1− 3Z c ) + α(1− 3Z c )(α −1) = 0

3 2 3 where, (1− 3Z c ) = 1− 9Z c + 27Z c − 27Z c (B.75)

(α −1)3 = α 3 − 3α 2 + 3α −1 (B.76)

(α −1) 2 = α 2 − 2α +1 (B.77)

2 2 (1− 3Z c ) = 1− 6Z c + 9Z c (B.74) substituting Equations, B.75, B.76, B.77 into Equation B.74, obtain

3 3 2 2 3 − Z c (α − 3α + 3α −1) − β (1− 9Z c + 27Z c − 27Z c ) − β (1− 6Z c 2 2 2 ⎡3Z c (α − 2α +1) + β (1− 6Z c + 9Z c )⎤ 2 ⎢ 2 ⎥ (B. 75) + 9Z c )(α −1) + (1− 3Z c )⎢+ α(1− 6Z c + 9Z c ) ⎥ = 0 ⎢ ⎥ + α(1− 3Z )(α 3 − 3α 2 + 3α −1 ⎣⎢ c ⎦⎥ or

3 3 2 2 3 − Z c (α − 3α + 3α −1) − β (1− 9Z c + 27Z c − 27Z c ) − β (1− 6Z c 2 2 2 ⎡3Z c (α − 2α +1)(1− 3Z c ) + β (1− 6Z c + 9Z c )⎤ 2 ⎢ 2 ⎥ (B.76) + 9Z c )(α −1) + ⎢(1− 3Z c ) + α(1− 6Z c + 9Z c )(1− 3Z c ) ⎥ = 0 ⎢ ⎥ + α(1− 3Z )(α 2 − 2α +1)(1− 3Z ) ⎣⎢ c c ⎦⎥ where,

2 2 2 (α −1)(1− 6Z c + 9Z c ) = α − 6αZ c + 9αZ c −1+ 6Z c − 9Z c (B.77)

123 2 2 2 (1− 3Z c )(α − 2α +1) = α − 2α +1− 3α Z c + 6αZ c − 3Z c (B.78)

2 2 2 3 (1− 3Zc )(1− 6Z c + 9Z c )(α − 2α +1) = 1− 9Z c + 27Z c − 27Z c (B.79) and,

(1− 6Z + 9Z 2 )(α 2 − 2α +1) = α 2 − 2α +1− 6α 2 Z +12αZ c c c c (B.80) 2 2 2 2 + 9α Z c −18αZ c + 9Z c

Substituting Equations B.77, B.78, B.79, and B.80 into Equation B.76, obtain

3 3 2 2 3 − Z c (α − 3α + 3α −1) − β (1− 9Z c + 27Z c − 27Z c ) − β (α − 6αZ c ⎡3Z 2 (α 2 − 2α +1− 3α 2 Z + 6αZ − 3Z ) ⎤ ⎢ c c c c ⎥ ⎢+ β (1− 9Z + 27Z 2 − 27Z 3 ) + α(1− 9Z ⎥ + 9αZ 2 −1+ 6Z − 9Z 2 ) + c c c c = 0 c c c ⎢ 2 3 2 2 ⎥ ⎢+ 27Z c − 27Z c ) + α(α − 2α +1− 6α ⎥ ⎢ 2 2 2 2 ⎥ ⎣+12αZ c − 6Z c + 9α Z c −18αZ c + 9Z c )⎦

(B.81) and,

3 2 2 3 (α − 3α + 3α + 27β + 9α −18α + 9 + 27α)Z c − 2 2 2 (−9αβ + β + 3 +12α +12α)Z c + (3α + 6α + 6β − 6αβ )Z c (B.82) − (α 2 + β −αβ ) = 0

Then,

(α 3 + 6α + 8)Z 3 − (3 +12α +12α 2 + 9β − 9αβ )Z 2 c c (B.83) 2 2 + (6α + 3α + 6β − 6αβ )Z c − (α + β −αβ ) = 0

Let,

2 3 θ1 = (8 +12α + 6α +α )

124 2 θ 2 = −(3 +12α +12α + 9β − 9αβ ) 2 θ 3 = (3α + 6α + 6β − 6αβ ) (B.84) 2 θ 4 = −(α + β − αβ )

Then,

3 2 θ1 Z c +θ 2 Z c +θ 3 Z c +θ 4 = 0 (B.85)

Critical Condition by the Critical Compressibility Form For Mixtures

The generalized cubic equation of state for mixtures is in the following form:

RT a P = − m (B.86) 2 2 V − bm V + α m bmV − β m bm

By van der Waals mixing rules, the mixture parameters am ,bm ,α m , β m take the following forms:

n a 1 1 2 2 am = ∑∑xi x j ai a j aij (B.87) i j

n 3 bm = (∑ xi x j ) (B.88) i

n n 1 1 2 2 α m = ∑∑xi x jα i α j α ij (B.89) i j

n n 1 1 2 2 β m = ∑∑xi x j β i β j β ij (B.90) i j and the binary interaction parameters established in terms of Mw ratios of components form

125 na ⎛ MW ⎞ a = ⎜ i ⎟ MW ≤ MW (B.91) ij ⎜ ⎟ i j ⎝ MW j ⎠

nα ⎛ MW ⎞ α = ⎜ i ⎟ MW ≤ MW (B.92) ij ⎜ ⎟ i j ⎝ MW j ⎠

nβ ⎛ MW ⎞ β = ⎜ i ⎟ MW ≤ MW (B.93) ij ⎜ ⎟ i j ⎝ MW j ⎠

where na ,nα , and na are the exponents of the respective interaction terms.

Expanding Equation B.86 into Z terms and by multiplying both sides of the equations by

V , RT

VP V ⎛ RT ⎞ V ⎛ a ⎞ = ⎜ ⎟ − ⎜ m ⎟ (B.94) RT RT ⎜V − b ⎟ RT ⎜ 2 2 ⎟ ⎝ m ⎠ ⎝V + α m bmV − β m bm ⎠

⎛ V ⎞ V ⎛ a ⎞ Z = ⎜ ⎟ − ⎜ m ⎟ (B.95) ⎜V − b ⎟ RT ⎜ 2 2 ⎟ ⎝ m ⎠ ⎝V + α m β mV − β m bm ⎠ by definition,

ZRT V = , P then,

⎛ ⎞ ⎛ ⎛ ZRT ⎞ ⎞⎛ ⎞ ⎜ ZRT ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ P ⎟ ⎜ ⎝ P ⎠ ⎟ am Z = − ⎜ ⎟ (B.96) ⎜ ZRT ⎟ ⎜ RT ⎟ 2 ⎛ ⎞ ⎜ ⎛ ZRT ⎞ ⎛ ZRT ⎞ 2 ⎟ ⎜ ⎜ ⎟ − bm ⎟ ⎜ ⎟ ⎜ ⎟ + α β − β b P ⎜ ⎜ ⎟ m m ⎜ ⎟ m m ⎟ ⎝ ⎝ ⎠ ⎠ ⎝ ⎠⎝ ⎝ p ⎠ ⎝ P ⎠ ⎠

P p 2 Multiplying by and by , obtain, RT R 2T 2

126 ⎛ ⎞ ⎛ ⎛ P 2 ⎞ ⎞ ⎜ a ⎜ ⎟ ⎟ ⎜ ⎟ m ⎜ 2 2 ⎟ ⎜ Z ⎟ ⎛ ZRT ⎞⎜ ⎝ R T ⎠ ⎟ Z = − ⎜ ⎟ (B.97) ⎜ ⎟ ⎜ 2 ⎟ ⎛ P ⎞ ⎝ RT ⎠ ⎛ ZP ⎞ 2 ⎛ P ⎞ ⎜ Z − b ⎜ ⎟ ⎟ ⎜ Z 2 + α β − β b ⎜ ⎟ ⎟ m ⎜ m m ⎜ ⎟ m m ⎜ 2 2 ⎟ ⎟ ⎝ ⎝ RT ⎠ ⎠ ⎝ ⎝ RT ⎠ ⎝ R T ⎠ ⎠

⎛ a P 2 ⎞ ⎛ b P ⎞ since A = ⎜ m ⎟ and B = m , then ⎜ 2 2 ⎟ ⎜ ⎟ ⎝ R T ⎠ ⎝ RT ⎠

⎛ Z ⎞ ⎛ ZA ⎞ Z = − ⎜ ⎟ (B.98) ⎜ ⎟ ⎜ 2 2 ⎟ ⎝ Z − B ⎠ ⎝ Z + α m ZB − β m B ⎠ or

2 2 2 2 Z(Z − B)(Z + α m ZB − β m B ) = Z(Z + α m ZB − β m B ) − ZA(Z − B) (B.99)

Z 3 + α Z 2 B − ZB 2 β − BZ 2 −α ZB 2 + B 3 β − Z 2 −α ZB + B 2 β m m m m m m (B.100) + AZ − AB = 0

3 2 2 2 3 2 Z + Z (α m B − B −1) + Z(B β m + α m B + α m B − A) + B β m + B β m − AB = 0 (B.101)

3 2 2 2 2 3 Z + Z (1+ B −α m B) + Z(A −α m B −α m B − B β m ) + (AB − B β m − B β m = 0

(B.102)

At the critical point, the generalized cubic equation of state in terms of Zc takes the form:

Z 3 + (1+ B −α B )Z 2 + (A −α B − α B 2 − B 2 β )Z c c m c c c m c m c c m c (B.103) 2 3 + (Ac Bc − Bc β m − Bc β m = 0

Now, comparing the coefficients of Equation B.103 with the coefficients of the expansion

3 ( Z − Z c ) =0 shown in Equation B.100

3 2 2 2 3 Z − 3Z c Z + 3Z c + 3Z c Z − Z c = 0 (B.104)

127 where,

3Z c = 1+ Bc −α m Bc (B.105)

2 2 2 3Z c = Ac −α m Bc −α m Bc − Bc β m (B.106) and,

3 2 3 Z c = Ac Bc − Bc β m − Bc β m (B.107)

From Equation B.106 and Equation B.107, Bc and Ac can be solved to give,

3Z c −1 Bc = (B.105) 1−α m

2 2 2 Ac = 3Z c +α m Bc + β m Bc (B.106)

By substitution of Ac and Bc with Equation B.98 and simplifying the terms, the analytical function of Z = (α , β ) is obtained. c ∫ m m

2 2 ⎛ ⎛ 3Z −1⎞ ⎛ 3Z −1⎞ ⎛ 3Z −1⎞ ⎞⎛ 3Z −1⎞ Z 3 = ⎜3Z 2 + α ⎜ c ⎟ + α ⎜ c ⎟ + β ⎜ c ⎟ ⎟⎜ c ⎟ c ⎜ c m ⎜ ⎟ m ⎜ ⎟ m ⎜ ⎟ ⎟⎜ ⎟ ⎝ 1−α m ⎠ ⎝ 1−α m ⎠ ⎝ 1−α m ⎠ ⎝ 1−α m ⎠ ⎝ ⎠ (B.107) 2 3 ⎛ 3Z −1⎞ ⎛ 3Z −1⎞ ⎜ c ⎟ ⎜ c ⎟ − β m ⎜ ⎟ − β m ⎜ ⎟ ⎝ 1−α m ⎠ ⎝ 1−α m ⎠ and can be further simplified to the following expression by expanding the terms by grouping,

3 2 θ1 Z c +θ 2 Z c +θ 3 Z c +θ 4 = 0 (B.108) where:

128 2 3 θ1 = (8 +12α m + 6α m + α m ) θ = −(3 +12α +12α 2 + 9β − 9β α ) 2 m m m m m (B.109) 2 θ 3 = (3α m + 6α m + 6β m − 6α m β m ) 2 θ 4 = −(β m + α m − β mα m )

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 0.0156 Methane 0.193 0.271 0.363 0.229 0.365 0.482 0.5248 0.5335 0.5921 Ethane 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 0.1043 I-Butane 0.007 0.007 0.009 0.006 0.007 0.007 0.0055 0.0063 0.0058 n-Butane 0.012 0.015 0.015 0.172 0.215 0.14 0.0179 0.204 0.0831 I-Pentane 0.005 0.006 0.006 0.005 0.004 0.006 0.0055 0.0062 0.0054 n-Pentane 0.007 0.007 0.008 0.006 0.007 0.008 0.0079 0.0092 0.0078 Hexane 0.013 0.015 0.016 0.012 0.01 0.012 0.0082 0.0094 0.0081 Heptanes + 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 0.0059 Methane 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 0.0281 Propane 0.1178 0.0672 0.0961 0.1359 0.0919 0.0308 0.0306 0.0124 0.0124 0.0128 I-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 0.0098 I-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 0.0078 Hexane 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 0.0038 Carbon Dioxide 0.0035 0.0156 0.0008 0.0075 0.0074 0.0045 Methane 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 0.0378 Propane 0.0605 0.0232 0.0537 0.031 0.0409 0.0315 0.03 0.0302 0.05 0.0144 I-Butane n-Butane 0.0565 0.0309 0.0303 0.0258 0.0235 0.0265 0.031 0.0207 0.049 0.0089 I-Pentane n-Pentane 0.0404 0.0243 0.0195 0.0216 0.0122 0.0425 0.071 0.0688 0.093 0.0436 Hexane 0.0366 0.0256 0.0173 0.0198 0.0103 0.0252 0.0456 0.0438 0.0308 Heptanes + 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 0.1309 Propane 0.0141 0.0136 0.0111 0.1499 0.105 0.1163 0.0673 0.0391 0.0699 I-Butane 0.0121 0.0112 0.0141 0.0085 0.0157 0.0081 n-Butane 0.0102 0.013 0.0252 0.0852 0.0472 0.0454 0.0352 0.0204 0.0349 I-Pentane 0.0084 0.0169 0.0112 0.0106 0.016 0.0097 n-Pentane 0.0501 0.0631 0.123 0.0109 0.0192 0.0182 0.0171 0.0123 0.0187 Hexane 0.0354 0.0447 0.0871 0.02 0.0373 0.0215 0.0226 0.029 0.0194 Heptanes + 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 0.0701 Methane 0.722 0.7052 0.7175 0.6992 0.5564 0.4902 0.5431 0.4371 0.3954 Ethane 0.0474 0.0725 0.0748 0.0638 0.0523 0.0464 0.1149 0.0932 0.0848 Propane 0.0278 0.0465 0.0456 0.0404 0.2108 0.297 0.0768 0.216 0.2738 I-Butane 0.0105 0.019 0.0177 0.0081 0.0069 0.0062 0.0147 0.0123 0.0115 n-Butane 0.0158 0.0129 0.0136 0.0169 0.0147 0.0133 0.0257 0.0223 0.0207 I=Pentane 0.0112 0.0064 0.0073 0.0079 0.007 0.0064 0.0082 0.0076 0.0072 n-Pentane 0.008 0.008 0.0046 0.0058 0.0052 0.0047 0.0082 0.0078 0.0076 Hexane 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 0.0022 Methane 0.4911 0.4384 0.654 0.6091 0.5432 0.4789 0.3678 0.77 0.7247 Ethane 0.1041 0.0935 0.0889 0.0865 0.0706 0.0624 0.0488 0.0316 0.0371 Propane 0.0699 0.0632 0.061 0.0765 0.195 0.2786 0.4154 0.0228 0.0473 I-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 0.0148 n-Pentane 0.0077 0.0077 0.0172 0.0094 0.006 0.0057 0.0053 0.0137 0.0167 Hexane 0.0123 0.0128 0.0307 0.0184 0.0165 0.0156 0.0145 0.0212 0.0202 Heptanes + 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 0.0076 Carbon 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 0.5391 Ethane 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 0.0964 I-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 0.0554 I-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 0.0299 Hexane 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 0.0027 Carbon Dioxide 0.018 0.0302 0.0084 0.0076 0.007 0.0065 0.0022 0.0003 Methane 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 0.0757 I-Butane 0.0173 0.0198 0.0021 0.0019 0.0018 0.0016 0.0092 0.0072 0.0455 n-Butane 0.0219 0.0279 0.0127 0.1025 0.1773 0.2405 0.0367 0.0456 0.0389 I-Pentane 0.01 0.0124 0.0106 0.0096 0.0088 0.0082 0.0064 0.0249 n-Pentane 0.0096 0.008 0.0099 0.009 0.0082 0.0076 0.018 0.0186 0.0182 Hexane 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 0.0705 Carbon Dioxide 0.012 0.0109 0.01 0.0091 0.0044 0.003 0.002 0.0013 0.002 0.0025 Methane 0.9089 0.8286 0.7625 0.887 0.858 0.7364 0.7364 0.7665 0.7515 0.8532 Ethane 0.044 0.0401 0.0369 0.0333 0.016 0.015 0.012 0.0551 0.061 0.0411 Propane 0.0191 0.0174 0.016 0.0144 0.007 0.006 0.0053 0.0335 0.0327 0.0198 I-Butane 0.0033 0.003 0.0028 0.003 0.0014 0.0012 0.001 0.0035 0.0038 0.0037 n-Butane 0.006 0.0055 0.0051 0.004 0.002 0.0018 0.0015 0.009 0.006 0.0039 I-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 .0022c Hexane 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 0.0031 Avg. 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

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

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

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

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

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

144 Table C. 16. (Continued)

Mixture No. 141 - 141 - 141 - 141 - 141 - 141 - 141 - 93 - 1 37 - 1 158 - 1 25 26 27 28 29 30 31

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

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 0.0705 Carbon Dioxide 0.012 0.0109 0.01 0.0091 0.0044 0.003 0.002 0.0013 0.002 0.0025 Methane 0.9089 0.8286 0.7625 0.887 0.858 0.7364 0.7364 0.7665 0.7515 0.8532 Ethane 0.044 0.0401 0.0369 0.0333 0.016 0.015 0.012 0.0551 0.061 0.0411 Propane 0.0191 0.0174 0.016 0.0144 0.007 0.006 0.0053 0.0335 0.0327 0.0198 I-Butane 0.0033 0.003 0.0028 0.003 0.0014 0.0012 0.001 0.0035 0.0038 0.0037 n-Butane 0.006 0.0055 0.0051 0.004 0.002 0.0018 0.0015 0.009 0.006 0.0039 I-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 .0022c Hexane 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 0.0031 Avg. 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

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

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

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

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 0.0701 Methane 0.722 0.7052 0.7175 0.6992 0.5564 0.4902 0.5431 0.4371 0.3954 Ethane 0.0474 0.0725 0.0748 0.0638 0.0523 0.0464 0.1149 0.0932 0.0848 Propane 0.0278 0.0465 0.0456 0.0404 0.2108 0.297 0.0768 0.216 0.2738 i-Butane 0.0105 0.019 0.0177 0.0081 0.0069 0.0062 0.0147 0.0123 0.0115 n-Butane 0.0158 0.0129 0.0136 0.0169 0.0147 0.0133 0.0257 0.0223 0.0207 i=Pentane 0.0112 0.0064 0.0073 0.0079 0.007 0.0064 0.0082 0.0076 0.0072 n-Pentane 0.008 0.008 0.0046 0.0058 0.0052 0.0047 0.0082 0.0078 0.0076 Hexane 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

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

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 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

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 11.85

Tb 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

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.

156 CRITICAL PRESSURE PREDICTIONS FOR COMPLEX MIXTURES BY FOUR METHODS

8500

7500

6500

5500

4500

3500 Calculated Critical Pressure (psia)

2500

Simon-Yarborough Etter-Kay 1500 Zais This work

500 500 1500 2500 3500 4500 5500 6500 7500 8500 Experimental Critical Pressure (psia)

Figure C. 1. Critical Pressure Prediction for Complex Mixtures – All Data.

157 5000

Simon-Yarborough 4500 Etter-Kay Zais This work

4000

3500

3000 Calculated Critical Pressure (psia) 2500

2000

1500 1500 2000 2500 3000 3500 4000 4500 5000 Experimental Critical Pressure (psia)

Figure C. 2. Critical Pressure Prediction for Complex Mixtures -Mixture 145-1.

158 6300

Simon-Yarborough Etter-Kay 5800 Zais This work

5300

4800

4300 Calculated Critical Pressure (psia)

3800

3300 3300 3800 4300 4800 5300 5800 6300 Experimental Critical Pressure (psia)

Figure C. 3. Critical Pressure Prediction for Complex Mixtures – Mixture 145 – 10.

159 5800

5300 Simon-Yarborough Etter-Kay Zais This work

4800

4300 cal Pressure (psia)

3800 Calculated Criti

3300

2800

2300 2300 2800 3300 3800 4300 4800 5300 5800 Experimental Critical Pressure (psia)

Figure C. 4. Critical Pressure Prediction for Complex Mixtures – Mixture 4 – 6.

160 4500

4300 Simon-Yarborough Etter-Kay 4100 Zais This work

3900

3700

3500

3300

Calculated Critical Pressure (psia) 3100

2900

2700

2500 2500 2700 2900 3100 3300 3500 3700 3900 4100 4300 4500 Experimental Critical Pressure (psia)

Figure C.5. Critical Pressure Prediction for a Complex Mixture 75 – 5.

161 8200

7700 Simon-Yarborough Etter-Kay Zais This work 7200

6700

6200

5700 Calculated Critical Pressure (psia)

5200

4700

4200 4200 4700 5200 5700 6200 6700 7200 7700 8200 Experimental Critical Pressure (psia)

Figure C. 6. Critical Pressure Prediction for a Complex Mixture 141 – 7.

162 6000

Simon-Yarborough 5500 Etter-Kay Zais This work

5000 sia) re (p

4500

Critical Pressu 4000 ated l

Calcu 3500

3000

2500 2500 3000 3500 4000 4500 5000 5500 6000 Experimental Critical Pressure (psia)

Figure C. 7. Critical Pressure Prediction for a Complex Mixture 141 – 16.

163 5900

Simon-Yarborough 5400 Etter-Kay Zais This w ork 4900

4400

3900

3400

2900 Calculated Critical Pressure (psia)

2400

1900

1400 1400 1900 2400 2900 3400 3900 4400 4900 5400 5900 Experimental Critical Pressure (psia)

Figure C. 8. Critical Pressure Prediction for a Complex Mixture 141 – 25.

164 5000

Simon-Yarborough Etter-Kay 4500 Zais This w ork

4000

3500 Calculated Critical Pressure (psia)

3000

2500 2500 3000 3500 4000 4500 5000 Experimental Critical Pressure (psia)

Figure C. 9. Critical Pressure Prediction for a Complex Mixture 58 – 1.

165 4000

3500

3000

2500

2000

Simon-Yarborough Etter-Kay Calculated Critical Pressure (psia) 1500 Zais This work

1000

500 500 1000 1500 2000 2500 3000 3500 4000 Experimental Critical Pressure (psia)

Figure C. 10. Critical Pressure Prediction for a Complex Mixture 47 – 1.

166

CRITICAL TEMPERATURE PREDICTION FOR COMPLEX MIXTURES BY LLS EOS METHOD

800

750

This work 700

650 R) o

600

550

500

Calculated Critical Temperature ( 450

400

350

300 300 350 400 450 500 550 600 650 700 750 800 Experimental Critical Temperature (oR)

Figure C. 11. Critical Temperature Prediction for Complex Mixtures – All Data.

167 750

740

This w ork

730

720 R) o

710

700

690

Calculated Critical Temperature ( 680

670

660

650 650 660 670 680 690 700 710 720 730 740 750

Experimental Critical Temperature (oR)

Figure C. 12. Critical Temperature Prediction for a Complex Mixture 145 - 1

168 800

This work

750 R) o

700

650 Calculated Critical Temperature (

600

550 550 600 650 700 750 800 Experimental Critical Temperature (oR)

Figure C. 13. Critical Temperature Prediction for a Complex Mixture 145 – 10

169 P 700

680

This w ork 660

640

620

600

580

560

540

520

500 500 520 540 560 580 600 620 640 660 680 700 Experimental Critical Temperature ( o R)

Figure C.14. Critical Temperature Prediction for a Complex Mixture 4 – 6.

170 750

This w ork

700 R) o

650

600 Calculated Critical Temperature (

550

500 500 550 600 650 700 750

Experimental Critical Temperature (oR)

Figure C. 15. Critical Temperature Prediction for a Complex Mixture 75 – 6.

171 750

740

This w ork 730

720 R) o

710

700

690

Calculated Critical Temperature ( 680

670

660

650 650 660 670 680 690 700 710 720 730 740 750

Experimental Critical Temperature (oR)

Figure C. 16. Critical Temperature Prediction for a Complex Mixture 141 – 7.

172 750

This work 700 R) o 650

600

550 Calculated Critical Temperature (

500

450 450 500 550 600 650 700 750 Experimental Critical Temperature (oR)

Figure C. 17. Critical Temperature Prediction for a Complex Mixture 141 – 25.

173 750

730

This w ork 710

690 R) o

670

650

630

Calculated Critical Temperature ( 610

590

570

550 550 570 590 610 630 650 670 690 710 730 750

Experimental Critical Temperature (oR)

Figure C. 18. Critical Temperature Prediction for a Complex Mixture 58 – 1. P

174 400

390

This work 380

370 R) o

360

350

340 Calculated Critical Temperature (

330

320

310

300 300 310 320 330 340 350 360 370 380 390 400 Experimental Critical Temperature (oR)

Figure C. 19. Critical Temperature Prediction for a Complex Mixture 47 – 1.

175