CLOSED-FORM VAN DER WAALS CRITICAL POINT
Transcript of CLOSED-FORM VAN DER WAALS CRITICAL POINT
CLOSED-FORM VAN DER WAALS CRITICAL POINT
FOR PETROLEUM RESERVOIR FLUIDS
by
TALAL HUSSEIN HASSOUN, B.S., M.S.
A DISSERTATION
IN
PETROLEUM ENGINEERING
Submitted to the Graduate Faculty of Texas Tech University in
Partial Fulfillment of the Requirements for
the Degree of
DOCTOR OF PHILOSOPHY
Approved
Akanni S. Lawal Chairperson of the Committee
Lloyd R. Heinze
James F. Lea
Accepted
John Borrelli Dean of the Graduate School
May, 2005
ACKNOWLEDGEMENTS
I wish to express my sincere thanks for the advice, guidance, and encouragement
given by my supervisor, Dr. Akanni S. Lawal. I would also like to thank the members of
my committee, Dr. Lloyd R. Heinze, and Dr. James F. Lea for their time and efforts. A
special thanks is extended to Dr. U. Mann for his assistance and willingness to help.
Finally, I thank Mr. S. Andreas, Mr. N. Kumar, and Mr. Tarek Hassoun for their help and
discussion contributed to this dissertation. I would like to acknowledge the Petroleum
Engineering Department for providing the financial support during the course of my
doctoral studies.
This Dissertation is dedicated to my father, my mother, my wife, Majida, my
daughter Amani, and to my sons, Ashraf, Heitham, and Tarek for providing me with
inspiration and confidence.
ii
TABLE OF CONTENTS
ACKNOWLEDGEMENTS ii
ABSTRACT v
LIST OF TABLES vii
LIST OF FIGURES ix
NOMENCLATURE xi
CHAPTER
I. INTRODUCTION
1.1 Importance of Critical State
1.2 Approaches to Critical State Prediction
1.3 Retrograde Reservoir Fluids
1.4 Objectives of Work
1
7
12
14
16
II. CRITICAL PROPERTY CORRELATION METHODS
2.1 Criteria of the Critical State
2.2 Empirical Models
2.3 Corresponding States
2.4 Convergence Pressure
2.5 Equation of State Models
18
19
22
25
31
45
III. CLOSED-FORM VAN DER WAALS EXPRESSIONS
3.1 Van der Waals Equations of State Theory
3.2 Closed-Form Equations for Fluid Critical Point
54
54
61
iii
3.3 Closed-Form Critical Property Computation Methods 69
IV. CRITICAL PROPERTIES FOR RESERVOIR FLUIDS
4.1 Critical Pressure Data for Complex Hydrocarbon Mixtures
4.2 Calculation of Critical Properties
4.3 Results and Discussion
4.4 Comparison Between Calculated and Experimental Data
74
74
75
80
83
V. CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusions
5.2 Recommendations
86
86
88
REFERENCES 89
APPENDICES
A. ANALYTICAL SOLUTION FOR CUBIC EQUATIONS
B. VAN DER WAALS EXPRESSIONS FOR FLUID CRITICAL POINT
C. PREDICTION RESULTS OF CRITICAL PRESSURE, CRITICAL
TEMPERATURE, AND HEPTANE PLUS PROPERTIES
99
112
130
iv
ABSTRACT
The prediction of critical points is of great practical importance because the
classification of petroleum reservoir fluids as a dry gas, gas condensate, volatile oil, and
crude oil depends largely on the knowledge of the critical properties of the reservoir
fluid. Also, the critical pressure and critical temperature of reservoir fluids are important
properties for describing the reservoir fluid phase behavior, predicting volumetric
properties of reservoir fluids and designing supercritical fluid processes.
Previous work for determining critical pressure, and critical temperature for
reservoir fluids include, empirical correlations, corresponding states method, and pseudo-
critical property methods. The generality of these previous correlations is limited to the
range of conditions and parameters used in the establishment of the correlations. Methods
based on the Gibbs criteria have also been used with Redlich-Kwong and Peng-Robinson
equations for prediction of critical properties. However, the Gibbs criteria have not been
applied to predicting critical properties of reservoir fluids.
A closed-form equation is developed for predicting the critical properties (Tc, Pc)
of complex reservoir fluids by using the Lawal-Lake-Silberberg (LLS) equation of state
with the criticality criteria established by Nobel Laureates van der Waals (VDW) in 1873.
By inverting the parameters of the LLS EOS in terms of the mixing parameters that are
based on the constituent substances and composition of the reservoir fluids, experimental
critical pressures and temperatures are predicted with interaction parameters expressed in
terms of molecular weight ratios of the binary constituent of reservoir fluids.
v
The prediction results of critical pressures and temperatures based on the VDW
criticality criteria show that experimental data consisting of 85 reservoir fluid mixtures
are within average absolute percent deviation of 3% to 5% of the measured critical
pressures and temperatures. In contrast to the previous work, this research project
provides an accurate method for computing the critical properties of reservoir fluids and
it is easy to use because the parameters of the criticality equation are readily available.
This project is useful for unifying near-critical flash routine with phase equilibria of the
compositional reservoir models. The project is also very attractive for establishing
reservoir models that are based on the critical composition convergence pressure concept.
vi
LIST OF TABLES 2.1 Modification to the Attractive Term of van der Waals Equation of State ................52
2.2 Modification to the Repulsive Term of van der Waals Equation of State ................53
3.1 Parameter of Selected Equations of State ..................................................................62
3.2 Relationship of EOS Constants to Critical Parameters..............................................64
4.1 Sample of Experimental Data Used in Calculations of Mixture 1.............................76
4.2 Calculated Critical Data of Heptane-plus Fraction Correlation.................................77
4.3 Calculated Critical Data of Heptanes-Plus Fraction for Data Set 1...........................78
4.4 Calculated Results for Pure Component parameters .................................................78
4.5 Calculated Results for Mixtures Parameters..............................................................79
4.6 Predicted Critical Pressure, Pc, Critical Temperature, Tc for Mixtures .....................81
C.1 Critical Pressure Prediction for Complex Mixtures………………………………130
C.2 Critical Pressure Prediction for Complex Mixtures ……………………………...131
C.3 Critical Pressure Prediction for Complex Mixtures...….…………………………132
C.4 Critical Pressure Prediction for Complex Mixtures...… …………………………133
C.5 Critical Pressure Prediction for Complex Mixtures……………………………....134
C.6 Critical Pressure Prediction for ComplexMixture………………………………...135
C.7 Critical Pressure Prediction for Complex Mixtures ……………………………...136
C.8 Critical Pressure Prediction for Complex Mixtures ……………………………137
C.9 Critical Pressure Prediction for Complex Mixtures ……………………………138
C.10 Critical Temperature Prediction for Complex Mixtures ……………………….139
vii
C.11 Critical Temperature Prediction for Mixtures ………………………………….140
C.12 Critical Temperature Prediction for Complex Mixtures………………………..141
C.13 Critical Temperature Prediction for Complex Mixtures…………………….….142
C.14 Critical Temperature Prediction for Complex Mixtures ……………………….143
C.15 Critical Temperature Prediction for Complex Mixtures ……………………….144
C.16 Critical Temperature Prediction for Complex Mixtures ……………………….145
C.17 Critical Temperature Prediction for Complex Mixtures ……………………….146
C.18 Critical Temperature Prediction for Complex Mixtures ……………………….147
C.19 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus....….….….….….….….….….….….….….….….….….….….148
C.20 Critical Pressure, Temperature, and Properties Prediction
for Heptane Plus……………………………………………………………. ….149 C.21 Critical Pressure, Temperature, and Properties Prediction
for Heptane Plus…………………………………………………………...……150
C.22 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus…………………………………………………...……….. ….151
C.23 Critical Pressure, Temperature, and Properties Prediction
for Heptane Plus …………………………………………………………….….152 C.24 Critical Pressure, Temperature, and Properties Prediction
for Heptane Plus …………………………………………………………….….153
C.25 Critical Pressure, Temperature, and properties Prediction for Heptane Plus……………………………………………………………. ….154
C.26 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus……………………………………………………………. ….155
C.27 Critical Pressure, Temperature, and Properties Prediction for Heptane Plus……………………………………………………………. ….156
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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
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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
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NOMENCLATURE
Symbol Definition
a Attraction Parameter Term of EOS
A Dimensionless Constant ⎟⎠⎞
⎜⎝⎛
22TRaP
AD Absolute Deviation
b van der Waals co-volume
B Dimensionless Constant
API Oil Gravity
⎟⎠⎞
⎜⎝⎛
RTbP
C Characterization Factor
K Watson Characterization Factor
r Weight
re
ature
MW Molecula
P Pressure
Pc Critical Pressu
R Gas Constant
Sg Specific Gravity
T Absolute Temperature
Tc Critical Temperature
Tbp Boiling Temper
V Molar Volume
xii
x Mole Fraction
Z Compressibility Factor
Zc Critical Compressibility Factor
arameter
ω Acentric Factor
cr ts
cation
ij Component Identification
Greek Letter
α Parameter of LLS EOS
αij Binary Interaction
β Parameter of LLS EOS
βij Beta Binary Interaction
Ω Dimensionless P
Subs ip
c Critical Property
m Mixture Identifi
r Reduced State
VDW van der Waals Symbol
xiii
CHAPTER I
INTRODUCTION
Petroleum reservoir fluids contain a variety of substances of diverse chemical
nature that include hydrocarbons and non-hydrocarbons. Hydrocarbons (carbon-
hydrogen molecules) range from methane to asphalt. Non-hydrocarbons include
substances such as nitrogen, carbon dioxide, and sulfur compounds. The chemistry of
hydrocarbon reservoir fluids is very complex. However, the mixtures of these complex
hydrocarbons may be in gaseous or liquid state at the pressures and temperatures often
encountered in petroleum reservoirs.
In spite of the complexity of hydrocarbon fluids found in reservoir rocks, simple
cubic equations of state have shown surprising performance in the phase behavior
calculations for both the vapor-liquid and vapor-liquid-liquid equilibria of these reservoir
fluids.
Knowledge of the phase equilibria of two-phase system is very important in the
design of a separation process and petroleum reservoir studies. Our interests are
properties of the state at which two phases of vapor and liquid become indistinguishable.
This state is termed critical at which the intensive properties of the liquid and vapor
phases become identical. Hence, mixtures consist of two phases of identical composition
are called critical mixtures. Predicting critical properties such as critical pressure, c ,
critical temperature, , and critical volume, c of the composition of a system in which
two phases become indistinguishable is very difficult and costly to determine
P
cT V
1
experimentally. Also, it is particularly difficult to determine the critical state
experimentally for multi-component mixtures.
Many methods have been reported for predicting critical properties of fluid
mixtures. These methods have ranged from empirical correlations, 31,122,111, to rig
thermodynamic conditions. Currently, there are three fundamental approaches, namely,
van der Waals criterion, Gibbs free energy, and the Wilson renormalization-group (RG)
theory. The Gibbs, 36 Wilson, 20 and Jiang & Prausnitz approaches were highly
considered and gained the attention of researchers in the field but their techniques have
failed, 100, in determining the critical points of reservoir fluids that contain heptane-plu
(C
29 98 orous
s
11 46
46
7+) fractions because their method not based on ideal gas law. The van der Waals
approach proved to be practical in resolving critical states in binary systems
and therefore can be applicable for predicting the critical properties of multi-component
systems and the more complex reservoir fluids.
117,116,115,114
In order to predict adequately the critical properties ( c c c ) of complex
reservoir fluids, a background in phase behavior is needed to understand numerous
surface and subsurface aspects of petroleum engineering. A knowledge of reservoir fluid
properties and phase behavior is necessary to calculate fluid in place, fluid recovery by
primary depletion, and fluid recovery by enhanced oil recovery (EOR) techniques such as
gas cycling, hydrocarbon solvent injection, and carbon dioxide (CO
P ,T ,V
2) displacement. An
example is given for clarification is that, many reservoirs have problems need
compositional treatment to increase accuracy and obtain more realistic description of
their fluids. This compositional treatment is of two types: depletion and/or cycling of
2
volatile oil and gas condensate and the other is miscible flooding with Multi-contact
miscibility (MCM). The difference between these two types of treatment is that the
depletion of volatile oil and gas condensate involves the removal of composition from the
critical region, while the second type requires calculation of phase composition and
properties converging at the critical point. The compositional model is capable of solving
the problem of miscibility where the original reservoir fluid and injected fluid are
miscible on first contact. There is difficulty in modeling the MCM process in achieving
stable convergence of gas and oil phase compositions, densities, and viscosities near the
critical point. The use of an equation of state in the MCM process, the vapor liquid
equilibrium ratios (K-values), and phase densities can be calculated near the critical
point. Therefore, in light of the advantages of van der Waals equations of state in solving
reservoir problems and the fact that a priori knowledge of the location of mixtures critical
points is required for volatile oil and gas condensate, the van der Waals criticality
conditions is applied as a tool to develop a closed-form solution of equation of state for
the critical points of fluid mixtures.
The prediction of critical properties of petroleum reservoir fluids is an important
factor in understanding the overall phase behavior of EOR miscibility fluid injection
projects.10 In many chemical and petroleum processes, the knowledge of the critical
behavior of hydrocarbon mixtures is useful in settling the operating ranges in the process
equipment. Furthermore, in compositional reservoir simulation, the appearance and
disappearance of phases (single phase or two-phases) in the reservoir grid block is
important. The critical state is required in K- values correlations that use the
3
convergence pressure concept, because the convergence pressure is the critical pressure
under certain specified conditions. 94,66, Several methods, such as that of Hadden a
Grayson, for finding K-values depend on the use of convergence pressure as a
correlating parameter.
39 nd
39
At the present time, the major area of research study in petroleum and chemical
industries is the high-pressure phase equilibria, phase density and composition of fluid
mixtures, and the effects of changes in pressure and temperature in which they exist. The
most practical tool used by scientists and researchers in high-pressure phase equilibria
studies is the cubic equation of state. This equation is an expression relates pressure,
temperature, and volume of the fluid (PVT). Many papers and books have been
published on the study of phase behavior of single, binary and ternary systems, as well as
for multi-component and complex reservoir fluids under a wide variety of conditions of
pressures and temperatures. Majority of these scientific publications deals with the
determination and prediction of fluid critical point.
The critical state of multi-component mixture is important from theoretical and
practical point of view, and an ability to predict this condition is highly desirable. Even
though the van der Waals criterion for critical state was enunciated by van der Waals ,
no satisfactory analytical method for predicting the critical condition in multi-component
systems based on this criterion has ever been formulated. The object of the work
undertaken in this study was to develop a closed-form solution to the problem of
predicting the critical properties of defined multi-component mixtures from the van der
Waals criticality condition together with a four parameters Lawal-Lake-Silberberg (LLS)
116
4
cubic equation of state. The van der Waals on-fluid theory is employed in this project
and the application of mixing rules allows the pure component parameters of the LLS
equation of state to be extended to mixture parameters.
We can best illustrate what the effects various system compositions have on the
phase behavior of petroleum hydrocarbons by presenting schematically the phase
diagrams for particular systems. Figure 1.1 is a pressure-temperature diagram, which
shows the relative boundaries of the two phase-phase region for typical reservoir fluids.
These include a dry gas, a gas condensate, a volatile oil, and a crude oil.
Figure 1.1. Pressure-Temperature Phase-Diagram of Petroleum Reservoir Fluids. 7
Schematic phase diagrams for each of the four reservoirs fluid classifications are
shown in Figure 1.1, which relates the reservoir fluid state to the reservoir pressure and
temperature. The vertical dotted line on the Figure 1.1 represents pressure depletion in
5
the reservoir at a constant temperature. The area within the phase envelope for each type
of reservoir fluid represents the pressure and temperature conditions at which both liquid
and vapor phases can exist. Point C on each envelope represents the critical point,
where the properties of liquid and vapor become identical. The solid line to the left of the
critical point , represents 100 percent liquid (the bubble-point line); the solid line to the
right of the critical point C , represents 100 percent vapor (the dew-point line).
C
As the fluid composition becomes richer in the high molecular weight
hydrocarbons, the phase envelope is changed such that the critical point shifts toward
higher temperatures and lower pressures. The location of the reservoir temperature and
pressure with respect to the critical temperature and pressure on the phase diagram for
any given fluid dictates the phase state of the fluid in the reservoir. Generally speaking,
the fluid above and to the left of the critical is considered liquid; fluid above and to the
right of the critical point is considered gas. Refer again to Figure 1.1, the gas reservoir
envelope lies completely to the left of the reservoir temperature line largely because the
main gas constituent, methane, has a low critical temperature )116 Fo− . Therefore, o
one phase can exist at reservoir temperature regardless of pressure. Any liquid recove
from a gas reservoir is the result of surface condensation after the gas has left the
reservoir.
( nly
red
Criticality is an important concept in phase behavior that is closely related to
equilibrium and stability concepts. In this introductory chapter, the importance of critical
state criteria, and the background and approach to critical state predictions are introduced.
6
The retrograde reservoir fluid phenomena, is demonstrated, and the objective of this work
is defined.
1.1 Importance of Critical State
The prediction of the critical properties of hydrocarbon mixtures is an important
aspect of the general problem of predicting the overall phase behavior of petroleum
reservoir fluids. The critical state is the unique condition about which the liquid and
vapor phases are defined, and hence has theoretical and practical significance. In
hydrocarbon processing and producing operations, a knowledge of the critical condition
is necessary because many of these operations take place under conditions which are at or
near the bubble-point or upper dew point regions and are frequently accompanied by
isobaric (constant pressure) or isothermal (constant temperature) retrograde phenomena.
Fluid property predictions and design calculations in this region are often the most
difficult one to make, and knowledge of the precise location of the critical point for the
reservoir under study is of the utmost help.
From a theoretical point of view, the changes of many of the thermodynamics and
transport properties take on a special significance as the critical state is approached. In an
empirical method the critical state has formed an integral part of many useful generalized
correlations such as those based on the theorem of corresponding states or the
convergence pressure concept in vapor-liquid equilibrium calculations.
In many ways the characteristics of the critical state that make it theoretically and
practically important are also the characteristics that make it one of the more difficult
conditions to measure experimentally. The very fact that density differences between
7
phases disappear, that the rate of volume change with respect to pressure approaches
infinity, or that infinitesimal temperature gradients can be responsible for a transition
from 100 percent liquid to 100 percent vapor all make the critical condition that one of
the more difficult to measure or observe accurately. For obvious economic reasons, it is
a condition that cannot be obtained by experiment in any special way for the many
systems for which it is required. Consequently, many attempts have been made to
develop methods for predicting the critical properties based on generalized empirical or
semi-empirical procedures. Consider the plot of pressure versus volume of a pure
component shown in Figure 1.2.
Figure 1.2. Pressure-Volume Phase Diagram of Pure Components. 32
8
Figure 1.2 illustrates the variation of volume with pressure and temperature
throughout the critical region. It is of interest to note that cTT <1 ; c is the critical
temperature is tangent to the saturation line at the critical state. At point , the liquid is
compressed (this state is referred to as under-saturated liquid because more gas is
dissolved in it). As the pressure is decreased, the volume increases. At point , the
liquid is in the saturated and stable state. To the right of point ,A as the pressure is
lowered, the component might follow one of two routs. It might follow the line ,AD in
ich case point
T
wh
S
A
D represents the saturated and stable vapor, or it mi
,AB for which the fluid will be in a meta-stable condition. In this case, the limit of
stability is determined by the condition that
ght follow curve
1TVP⎟⎞
⎜⎛ ∂ vanishes (i.e., point B ). Similar
at point
⎠⎝ ∂y, l
R one can observe that as the pressure increases, condensation may no occur u
to poin here again
t p
t w ,C 1TV
P⎟⎞
⎜⎛ ∂ will vanish. Curve DC represents the locus of meta-
stability and point C is the limit of meta-stability at for the vapor. As the temperature
⎠⎝ ∂
ate there is a rapid increase in the slope
1T
is increased above that of the critical stpT
V⎛ ∂⎟⎠⎞
⎜⎝ ∂
of
the isobars at the critical volume. Both the isobaric thermal expansion and the isothermal
compressibility are infinite at the critical state.
The pressure-volume diagram of mixture differs from that of pure component.
pressure-volume diagram for mixtures is shown in Figure 1.3. The main differences
The pure component pressure-volume diagram can be seen in Figure 1.2, while the
9
between Figure 1.2 and Figure 1.3 are: (1) L and G of Figure 1.3 do not represent the
equilibrium states and (2) the critical points have different features.
V V
Figure 1.3. Pressure-Volume Diagram of Mixtures.
For a pure com
32
ponent, 02
2
=∂∂
=∂∂ V
PP , at the critical point. For a mixture, this V
nvelope. The Z-factors in Figure1.2 of the
equilibrium gas and liquid phases always meet the condition
does not occur at the top of the two-phase e
GL ZZ < . However, for
mixtures, when gas and liquid phases are at equilibrium, LZ might be smaller or larger
than GZ . At equilibrium, the density of the liquid phase is higher than the density of the
. Then from gas phase, GL ρρ >
ZRTPM
=ρ (2.1)
it follows that
10
G
GL
L ZM
ZM (2.2)
When Z-factors are less than one, then,
>
L GZ could be smaller or larger than Z .
Variation in density of the liquid phase and gas phase at the critical region is
shown in Figure 1.4. It has been observed, experimentally, that there is a nearly linear
relationship between the average density of the coexisting phases and the vapor pres
near to the critical state. It can be shown that
sure
Pavg
∂
∂ρ will reach a constant value as the
critical state is approached. This relation of the average density of the liquid and gas
phases to the prevailing pressure and temperature has been called the “Law of Rectilinear
Diameters”. This relation gives an acceptable basis for the experimental
determination of the critical volume of a pure component. It is necessary only to plot the
average specific weight of the coexisting phases as a function of pressure and to
extrapolate from the accurate experiment to the critical state. Such a plot for propane is
given in Figure 1.4. In this Figure 1.4, notice the slight curvature in the relation of the
average specific weight to pressure at the lower temperatures, but near the critical point
the relation becomes linear. In the light of this discussion it is seen that the critical point
is truly a state of the system.
102,101
11
1.2
102 Figure 1.4. Specific Weight of Liquid and Gas for Propane in the Critical Region.
Approaches to Critical State Prediction
al
ses.
a
estimated utilizing various correlations methods, which have been reviewed in terms of
Because of the difficulty of measuring the critical properties of hydrocarbon
mixtures experimentally, the ability to have reliable methods for correlating and
predicting these properties is highly desirable. A survey of the literature indicates many
correlations have been advanced for predicting the phase behavior, 77 predicting physic
properties, 49,69 developing equations of state, and designing supercritical fluid proces
For many pure components, these critical properties have been experimentally
determined. 91 However, experimental determinations of the critical properties of
mixtures are impractical because of the limitations in terms of time and costs. Even
though experimental data for some mixtures are vailable, but with less coverage of
composition range of data points.
In addition to the direct measurements, critical properties of mixtures are usually
12
their estimation procedure and accuracy108 . These correlation methods relied prim
on many approaches, namely, ap
arily
gr hical roach equation of state approach,
empiric
at is,
ect to
t constant temperature and pressure must be equal to zero. Determination
of , and for the mixtur o an extended form
of derivatives and an equation of state such as reported by Spear et al. There are two
setbacks in these approaches. First, most of these approaches are limited to estimating
critical properties of hydrocarbon mixtures. Even in the case of hydrocarbon mixtures,
most of the methods tend to yield higher order of errors when used to estimate the critical
properties of methane-containing mixtures.
In this work, a new concept for the development of a methodology for predicting
petroleum reservoir fluids critical properties for cubic equations of state consistent with
the criterion of van der Waals’ equation of state. The use of this concept is illustrated by
its application to the Lawal-Lake-Silberberg equation of state, and also the use of van der
Waals one-fluid theory and the application of mixing rules which allows the pure
component parameters of the LLS equation of state to be extended to mixtures
parameters.
app , 38 79,84,20
al procedures involving the use of excess property approach, 54 the use of
conformal solution theory (corresponding states principle) approach 58 based on the
concept that all thermodynamic properties of mixtures can be evaluated from pure
component properties if the components conform to certain postulates of statistical
mechanics, and rigorous thermodynamic potential approach for the critical state, th
the second and third partial derivatives of the molar Gibbs free energy with resp
composition a
cT , cP cV es involves a simultaneous soluti n of
107
107
13
1.3 Retrograde Reservoir Fluids
To illustrate retrograde reservoir fluid phenomena, Figure 1.5 shows the pressure-
temperature phase diagram for a mixture at fixed composition.
Figure 1.5. Pressure-Temperature Diagram of Retrograde Gas Condensatio
The solid thick line is the bubble- point curve (100 % liquid, 0 % vapor) and the thin l
is the dew-point curve (100 % vapor, 0 % liquid)
n
ine
; they meet at critical point where the
two ph i
tw
es
o
o
. 6
C
ases become identical. The basic cr terion for the critical point (point C ) is the
limiting condition where the system can exist in o phases. Near the critical point,
hydrocarbon mixtures exhibit a more complex behavior usually opposite to what would
be expected from observed behavior at low pressures. This reverse behavior compris
retrograde phenomena. Retrograde phenomena always exist when the critical p int of a
mixture is not at the highest pressure and temperature possible for the coexistence of tw
14
phases. Also, near the critical point, the density-dependent properties change wi sm
changes in temperature and pressure.
More than a century ago Kuenen 61,62 first observed the isothermal (constant
temperature) retrograde condensation shown in the dashed ABDE line, and isobaric
vaporization, shown in line AGH were observed by Duhen (1896, 1901). The defin
of these phenomena are:
1. Retrograde condensation occurs when a denser reservoir fluid phase is formed by
the isothermal decrease in pressure or the isobaric increase in temperature.
2. Retrograde vaporization occurs when a
th all
itions
less dense of reservoir fluid phase is
ndentherm These points represent the upper
bounds where phase separation can take place.
Refer again to Figure hase (vapor) exists. If we increase
the pre s
n to
C
formed by the isothermal increase in pressure or the isobaric decrease in
temperature.
The maximum pressure at point N is the cricondenbar, and the maximum
temperature at point M is called crico .102
1.5, at point a , a single p
ssure on this vapor isothermally to point ,E we encounter a dew-point state. A
pressure is increased beyond ,E more and more of the vapor will conde se until reach
point .D ontinue pressure increase from point D causes retrograde vaporization of
liquid that had previously be ondensed. This process continues until point where
an u
en c ,B
pp r dew point is reached. Continued pressure increase from point B to e A which
sses a single-phase fluid (vapor). If now reduce the temperature isobarically fromcompre
point A to point ,G the volume of the sin le- phas fluid will contr a g e act. At point G
15
bubble-poi is
dense fluid (va tinue to form until point
nt encountered. With continued decrease in temperature isobarically, less
por) will con H is reached. As conditions change
from
beyond re dense phase to increase until the
bub -
above-
point G to point ,H retrograde vaporization occurs. Reduction of temperature
H continue isobarically will cause a mo
ble point is reached. We could demonstrate retrograde condensation by reversing the
escribed procedure, that is, by proceeding from point d N to point Quantitative
und
ion of
.b
erstanding of these phase-equilibrium phenomena is useful for design of production,
storage, and transportation of crude products.
Several observations can be made from Figure 1.5. The bubble-point line
coincides with the dew-point line at the critical point C. The shaded areas represent
regions of retrograde phenomena. The region defined by points CBMD in the reg
isothermal retrograde condensation.
1.4 Objectives of Work
The objective of this work is to develop a robust computational technique
predicting the critical properties, critical pressure, ,cT critical pressure, ,cP and critical
volume, V for complex petroleum reservoir fluids. This objective consis
for
of three
major elements:
1. Developm prehensive closed-form solution to the criticality criteria
established by Nobel Laureates van der Waals in 1873. Utility of the concept is
illustrated by lication to:
c ts
ent of a com
its app
16
• Lawal-Lake-Silberberg (LLS) equation of state using van der Waals one-
fl
2. Establish interaction parameters for hydrocarbon and non-hydrocarbon and for
hydrocarbon with pseudo-components..
3. Develop an algorithm for calculating these critical properties for reservoir fluids
(gases, gas condensate, volatile oils, and crude oils).
In order to achieve these objectives, this research work has been organized into five
chapters. Thus, after an overview of the critical property correlation methods and
illustrate the criteria of the critical state in Chapter 2, a review of the critical models,
corresponding state theory, convergence pressure concept, and the equations of state is
introduced. The van der Waals equation of state theory and the resulting derived
equations for the closed form-solution are presented in Chapter 3 with computational
procedure is described. Chapter 4 presents the results of calculations and analysis of the
predicted critical properties and compared with experimental calculations. Finally, in
Chapter 5 conclusions and recommendations were made on the equation of state
approach to critical points predictions, the general level of accuracy and applicability, and
the implications for future work in this area of phase behavior research.
uid theory
17
CHAPTER II
CRITICAL PROPERTY CORRELATION METHODS
Several investigators have developed correlation techniques for predicting critical
properties of complex reservoir hydrocarbon mixtures. Many of these correlation
techniques were essentially both empirical and theoretical procedures in nature and were
aimed at predicting the critical properties of naturally hydrocarbon systems. The well
known of these was the method of Kurata and Katz for the critical properties of volatile
hydrocarbon mix
et al., (
ns
this chapter, a review of correlation methods for predicting critical properties
of com ir hase
behavio
given mi will be
corresponding states. Then the law of corresponding states and the convergence pressure
are presented in detail.
63
tures, and of Organic 80 for complex hydrocarbon systems. Later, Davis
1954) modified the original Kurata-Katz method to make it applicable to lighter
natural gas systems. All of these correlation methods make use of graphical correlatio
with parameters such as pseudo critical temperature and pressure, molal average boiling
point, or weight average equivalent molecular weight.
In
plex petroleum reservoir fluids is undertaken to help understand the p
r calculations. First, the criterion of the critical state is introduced, and the
empirical models of correlations are presented. If the critical state can be predicted for a
xture, the separation between the bubble-point and the dew point regions
defined, and physical properties of the mixture can be obtained by using the law of
18
2.1 Criteria of the Critical State
At a critical point, the fluid does not exist in a particular state, either gas or liquid,
but has characteristics of both. Hence, it is called a supercritical fluid. To see at what
m ature.
temperature, pressure, and volume, this supercritical behavior is observed, we use the fact
that at the critical point, the isother is both horizontal (zero slope) and has no curv
These two conditions are interpreted mathematically as follows:
0=∂∂VP (2.1)
02
2
=∂∂V
P (2.2)
The criteria of criticality can be analyzed by two approaches: a simple approac
that relies on geometrical presentation, and an alternative approach more reliable for
multi-component and complex mixtures. Consider a p-v diagram for a two-phase critical
point of a single-component system shown in Figure 2.1
h
.
Figure 2.1. Pressure-Volume Plot for a Single-component System
In this pressure-volume phase diagram, the dashed curve is the spinodal curve
and the solid curve is the binodal curve. The p-v isotherms at four different temperatures
. 32
19
,1T , ,cT and ,3T are shown. Points B an C represents the limit2T d s of stability at
temp Points B` and C` represent the limit of stability at temperature Based
on the criteria of the limits of stabi
erature .1T .2T
stability, 32 lity at T and T are obtained from 1 2
01
=⎟⎠⎞
⎜⎝ ∂v⎛ ∂
T
p and 02
=⎟⎠⎞
⎜⎝⎛∂∂
Tvp , respectively. Between points B and C, 0
1
>⎟⎠⎞
⎜⎝⎛∂∂
Tvp and
between B` and C` ,02
>⎟⎠
⎜⎝ ∂v
d these are therefore the unstab⎞⎛ ∂
T
p an le segment of the
isotherms. Note that the changes in curvature between B and C indicates an inflection
point where 01⎠⎝ T
unstable. As the temperature approaches the critical point (i.e., cT ), the limits of s
and un-stability points coincide, and since the inflection point is now located on the
bimodal curve, the inflection point
2
2
=⎟⎟⎞
⎜⎜⎛∂∂v
p exists between these two points. This inflection point is an
tability
02
2
=⎟⎠
⎞⎜⎝
⎛∂∂
cTvp is a stable point. Points A, D and A`
and D` represent equilibrium phases at 1T and 2T , respectively. Approaching toward the
critical point, the points A and A` and D and D` coincide also with the limit of stability.
At the critical point, the gas phase and liquid phase can be transformed into each other
without going through the two-phase region, and that the continu
⎟⎜
ity of gas and liquid
ate. The criteria of the critical state of a pure component are, therefore, st
0=⎜⎛ (2 )⎟
⎠⎞
⎝ ∂∂
cTvp .3
20
02
2
⎟⎠
⎞⎜⎝
⎛∂∂
cTvp
=⎟⎜ (2.4)
.03
3
<⎟⎠
⎞⎜⎝
⎛∂∂
cTvp (2.5)
Where
⎟⎜
03
<⎟⎠
⎞⎜⎝
⎛ ∂
cTvp indicates that the critical point is neither a maximum nor a
minimum.
3 ⎟⎜ ∂
ch for the
calculatio e
C
In 1980, Heidmann and Khalil 43 proposed an alternative approa
n of critical point that is mathematically different from the expressions of th
riticality conditions in Equation 2.6 , and Equation 2.7 32 32
0
)1(1,1
)1(3,1
)1(2,1
)1(1,
)1(3,
)1(2,
)1(1,2
)1(23
)1(22
) ==
++++
+
+
cccc
cccc
c
yyyyyy
yyy
L
L
L
λ (2.6)
an
1( +c MMMM
d
0
)1(1
)1(3
)1(2
)1(1,
)1(3,
)1(2,
1,22322
+
)1()1()1(
=
+
yyy
λλλ L
L
(2.7)
where
+++
+
cc
cccccc
c
yyy L
MLMM
231 ,,2
2)1(22
+
⎟⎠
⎞⎜⎝
⎛∂∂
cxxcXC
L
⎟⎜=y , 221 ,,,3
2)1(23 ⎟
⎠
⎞⎜⎝
⎛∂∂
xcXC
L +
⎟⎜=cx
y ) is the , cNNVTAy ,,,,()1( L=
Helmholtz free energy, and 0)1(1 =++ rent.
laborate of multi-
ent system of fixed composition sketched in Figure 2.2. The thick solid line on
ccλ , at the critical point but the concept is not diffe
As we have already seen, the critical point is a stable point at the limit of stability. To
e more on this concept, let us consider the pressure-volume diagram
compon
21
the left represents the bubble-points and the thin line represents the dew points. The
bubble-points and dew points are stable equilibrium states; a perturbation in pressure,
d
it
l point.
thus, results in a stable state. Critical point CP is the point at which bubble-point an
dew point converge and being a stable state, is at the limit of stability. These two
concepts of stable and limit of stability were used by Gibbs in 1876 to derive the
expressions for the critica
32
2.2
Figure 2.2. Critical Point Representation in a Multi-component System
.
Empirical Models
olving the use of excess properties, (2) the use of conformal
an
Many correlations have been developed for predicting critical properties of pure
components and mixtures. These correlations have relied on three approaches: (1) an
empirical approaches inv
solution theory based on the concept that all thermodynamic properties of mixtures c
be evaluated from component properties if the components conform to certain postulates
22
of statisti al mechanics, (3) a rigorous thermodynamic condition based on the sec
third partial derivatives of all molar Gibbs free energy with respect to composition.
The empirical approach involves calculations of the following model:
∑
c ond and
=
G lled the
cal
icted by
w se
relations
lts for systems containing ethane.
Prausnitz’ method for determining the pseudo-critical temperature can be
generalized for mixtures having ts. The generalized equation is
iciicm
+=n
icorrciic GGxG
1
(2.8)
wher c is the critical property desired and corr is a correction term which is ca
excess property of the mixture. Excess properties are usually estimated from empiri
relations. Many correlations have been proposed based on Equation 2.6 to predict the
critical properties of mixtures. The critical temperature of defined mixtures pred
e G
122
L as simplified by Chueh-Prausnitz, 21 for all empirical approaches. All of the
correlations yielded results with large errors and the chemical nature and sizes of
components limit the use of these results. Li’s 71 correlation gave the most accurate
predictions with the method of Prausnitz gave satisfactory results. All other cor
gave poor resu
i 71,
s 85
any number of componen
ijj
n
i
n
j
n
i
TT τθθθ ∑∑∑ +==1
(2.9)
where
∑
= ciii
Vxθ
ciiVx
23
(2.10)
23
and ijτ for each
with data set of six ternary system, two quaternary systems, and two quinary systems and
reported 0.4% deviation.
Li’s equation a pseudo-critical temperature for any mixture, has the following
form:
1)
interacting pair of molecules. 21 Equation 2.7 was tested by Prausnitz
cii
icm TT ∑= φ (2.1n
where
∑n
cii
i
Vx
Vx=
i
ciiφ (2.12)
The critical state correlation models for mixtures are much more important than is
the case for pure components. A wide variety of empirical correlation models, usually
with an average boiling temperature and composition of the mixture as parameters, have
been proposed. These types of correlations are of two types: 1) correlation models
which apply to simple mixtures of known composition and 2) correlation models which
apply to complex petroleum hydrocarbon fractions. In general, the correlations for both
types of mixtures are limited to hydrocarbons and often only to aliphatic and simple
aromatic hydrocarbons.
For simple hydrocarbon mixtures, several empirical correlations for critical
temperature and pressure have been proposed. The correlations suggested by Pawlewski,
and Kay is the most important from a historical point of view. More accurate
correlation models have been proposed by Joffe, Grieves, and Organick, Etter and
53
48 38 80
24
Kay. Edmister summarized the critical point correlations available for hydrocarbon
mixtures up to 1949. The best of these correlations are accurate within 1% for the critical
temperature and 3-5% for the critical pressure.
Correla odels for predicting the critical point of complex petroleum
hydrocarbon fractions also have been proposed. Significant correlations have been
suggested by Sm Kurata-Katz, Edmister, and Pollock. Organick also
proposed a correlation and introduced a comparison of his method with the Kurata-Katz.
it the critical point. The prediction of the
critical
2.3 Corresponding States
29 27
tion m
ith,105 63 27 80
Equations of state were involved in the critical state correlation in order to
ntegrate vapor-liquid equilibrium behavior w hi
point on the basis of van der Waals critical condition and the use of a suitable
equation of state is considered the first step toward the work carried out in this
investigation.
Van der Waals
on experime shows that compressibility factor for different
fluids exhibit similar behavior when correlated as a function of perature
116 in 1873 developed the theorem of corresponding states based
ntal observation, which Z
reduced tem rT
and reduced pressure rP . By definition,
cr T
T = (2.13) T
cPP
rP = (2.14)
25
and
cr V
VV = (2.15)
Where the subscript r represents the reduced state, and subscript represents
critical state. These dimensionless reduced conditions of temperature, pressure, and
olume p
“All flu ave
iat
d value of an
tem may be defined as the ratio of the value of that property in
a given state to the value of that property at the critical state. The theorem of
corresponding states does not hold for big ranges of pressure for real gases, and at the
same time is not perfect. However, when applied to gases with similar chemical structure
(as paraffin hydrocarbons), it offers a correlation with close agreement (satisfactory for
engineering work) permits the use of reduced properties as the basis for correlating
experimentally compressibility factor. This is illustrated by the reduced PVT data on
methane, ethane, and propane shown in Figure 2.3. In Figure 2.3 is a plotted of values of
for methane, ethane, and propane as a function of reduced pressure for reduced
temperatures to show the degree of correlation.
c
v rovide the basis for the simplest form of the theorem of corresponding states:
ids, when compared at the same reduced temperature and reduced pressure, h
approximately the same compressibility-factor, and all dev e from ideal-gas behavior to
about the same degree.”105 The term “reduced state” means that each reduce
intensive property of a yss
Z
26
3. Compressibility Factors of Methane, Ethane, and Propane as a FunctionFigure 2. of
u
The theory of corresponding states was extended to cover mixtures of gases.
Kay invented the concept of pseudo-critical Temperature and pseudo-critical pressure for
real ga
i 1
Reduced Pressure and Reduced Temperat re.109
ses. These pseudo-critical properties are obtained by using the Amagat’s law of
partial volumes for mixtures to the critical properties of the composition of the mixture.
These quantities are defined as
∑=n
ciipc TyT (2.16)
and
ci
n
iipc
=
PyP ∑==
(2.17)
where is the pseudo-critica
re th omponent,
mole fraction of ith component in
1
l pressure, pcT is the pseudo-critical temperature, ciT and pcP
ciP a iy the critical temperature and critical pressure respectively of i c
mixture, and n number of components. The physical
27
propert i ical
e ay that properties for pure gases are correlated with reduced
pressure and reduced temperature. Thus, these pseudo-critical properties are defined as
follows:
ies of gas m xtures are correlated with pseudo-critical pressure and pseudo-crit
temperature in the sam w
pcpr P
PP = (2.18)
pc
pr TTT = (2.19)
Compressibility factors, experimentally obtained, for natural gas have been
correlated with pseudo-reduced pressure and temperature. The petroleum industry has
universally adopted the correlations shown in Figures 2.4, to determine the
compressibility factor
85,14
Z .
109 Figure 2.4. A Deviation-Chart for Hydrocarbon Gases.
28
Correlations of Z - factor based on the theory of corresponding states are called
two-parameter correlations, because of the use of two reducing parameters rT and .rP
These c
-
tion t and ),
has been
orrelations are shown to be close for the simple fluids (argon, xenon), but
systematic deviations are observed for more complex fluids. A third corresponding
states parameter concept characteristic of molecular structure (in addi o cT cP
introduced by K. S. Pitzer 63 is the acentric factor .ω
The Pitzer acentric factor for a pure fluid is defined with reference to its vapor
pressure. Becau
reciprocal of absolute temperature,
se the logarithm of the vapor pressure of pure fluid is linear in the
mpr =1
log (2.20)
Tr
where rp is the reduced vapor pressure, rT is the reduced temperature, and m is the
sat
d
d sat
)(
vs. slope of the plot of rPlog .1
rT
It has been observed that
the slope would be the same for all pure fluids. This is not true ; because according
to Pitzer each fluid has its own characteristic value of Pitzer has noted that all vapor
a for simple fluids lie on the same line when plotted as vs.
if the two-parameters corresponding states were valid,
pressure dat
m 63
.m
satrPlog
rT1 and
the line passes through 0.1log −=satP at .7.0=rr T This is shown in Figure 2.5. Data
for other fluids define other lines whose locations can be fixed in relation to the line for
the simple fluids (SF) by the difference:
29
satrPSF log) − (2.21) sat
rP (log
The acentric factor is defined as the difference evaluated at :
satω (2.22)
7.0=rT
7.0)log(0.1 =rTrP −−≡
Therefore, ω can be determined for any fluid from cc PT , and vapor pressure
measurement made at .7.0=rT
Figure 2.5. Approximate Temperature Dependence of the reduced Vapor Pressure.105
30
2.4 Convergence Pressure
This section first discusses some fundamental thoughts on the equilibrium ratios
(K-values), and reviews the basic sources for obtaining these values. This followed by
providing a discussion about the convergence pressure (Hadden, 1953), and remarks on
the interdependency of K-values of heavy (plus) fractions and convergence pressures.
In order to evaluate the equilibrium behavior of multi-component two-phase
systems and obtain an expression for K-values; Dalton’s and Raoult’s laws can be
combined. Dalton’s law is defined by the Equations 2.23, and 2.24
(2.23)
and
40
∑=
=n
iipP
1
Pp
y ii = (2.24)
or
(2.25)
and Raoult’s equation is stated as “the partial pressure exerted by a constituent of liquid
phase is equal to the vapor pressure of that constituent times the mole fraction of that
constituent in the liquid phase. That is,
(2.26)
Where is the partial pressure of the ith component, is the mole fraction of the ith
ompo nt in por phase and
Pyp ii =
viii PxP =
ip ix
c ne liquid phase, iy is the mole fraction of ith component in va
31
vip
is vapor pressure of ith component. By combining Equation 2.24 and Equation 2.26
we obtain:
(2.27)
By definition,
iii xppy =
i
ii x
yK = (2.28)
Where K is defined as the distribution of a component, “i”, between vapor and liquid
phases is given by the equilibrium ratio, K, described by Equation 2.28.
The value of is dependent on the pressure, temperature, and composition of
rium ratios ( values) for low-shrinkage oil and a
onden te at F s re. The
iK
Kthe hydrocarbon system. Equilib
c sa are shown in Figures 2.6 6 and 2.7 6 as functions of pre su o200
equilibrium ratios ( K values) for both types of fluids are shown to converge to a point
value of 1. This point is called convergence pressure, defined on page 162 of the
NGSMA Data book. “Early high pressure experimental work revealed that if a
hydrocarbon system of fixed overall composition were held at constant temperature and
the pressure varied, the
K - values of various components converged toward a commo
value of unity at some high pressure. This pressure has been termed the convergence
pressure of the system “. If the temperature at which the
n
K - values were presented is the
critical temperature of the hydrocarbon mixture, then the convergence pressure will be
the critical pressure. For all temperature other than the l temperature, the
f sure less
critica
convergence o K - values is then an apparent convergence pressure. At a pres
than convergence pressure, the system will be at either dew point or bubble point, and
32
exists as a single-phase fluid at the conditions expressed by the point of apparent
convergence.
6 Figure 2.6. Equilibrium Ratio for Low-Shrinkage Oil.
33
Figure 2.7. Equilibrium Ratio for a condensate Fluid. 92
A widely accepted definition of convergence pressure by Hadden 39 in 1953 w
proposed. In fact, Hadden 39 defined the critical mixture, from which the convergence
pressure would be estimated, as that resulting from adding methane r nitrogen to the
equilibrium liquid. In such an addition always result in reaching the critical state, and
hence
as
o
a convergence pressure is defined, and satisfies the phase rule requirement. If
these two lightest components methane (or nitrogen) were in every system s
definition of convergence pressure is adequate. But, for purposes of general correlation
of data, m ny of which are from binary or ternary that have neither methane nor nitrogen,
the adequately of Hadden’s
of interest, thi
a
definition is questionable.
34
Ro n ased on what he called critical
com ositio vergence pressure is estimated from the
critical m e equilibrium phases. That means the critical
mi general and applicable to binaries or higher-
order systems without regard to which components may be present. Unfortunately, this
definition does not always define the convergence pressure. This limitation caused most
frequently at low pressure, where
we, 93 in 1964 proposed a definitio b
p n method. In this method, the con
ixture, which would give the sam
xture lies in the tie line. This definition is
K - values are not sensitive to convergence pressure,
but can occur at high pressure as well Lawal, 1981 .
The critical mixture is determined by the intersection of the tie line with the locus
m
for the isothermal locus of critical composition in hydrocarbon systems that calculated
the critical composition. The general Fair’s equation is stated as
ole
fraction of component in the critical mixture. In this notational form, j indicates the
component chosen as dependent; that is is determined from z .
e
66
of critical co positions at the equilibrium temperature. Fair has developed an equation
11,1
=∑≠=
i
j
ci
i zA (2.29)
Where is the critical composition locus coefficient for component i , is the m
31
N
iAicz
jc ic
For an equilibrium state with liquid mol fractions ix and vapor mole fraction
iy , the interaction of the equations for the tie line (actually 1
z
=N component balan
equations) with the isothermal critical composition locus Equation 2.28 is represented
ce
by
[ ]
iiii
iiiic y
yxAyAyx
zi
+−
−−=
∑∑
)(1)(
(2.30)
35
Where the summation is understood to be over all components except nitrogen. Equat
2.28 was used to calculate the critical m
ion
ixture for all ternary, multi-component, and
comple
Fair o
than the generalized locus functions. The critical compositions of the binaries were
calculated from the equilibrium temperature by the expression
iicL
1
1)
where z mole fraction of light component
x mixture equilibrium states.
, 30 presented correlations for binary data, which are considered m re accurate
∑ −+=m
icz )1(θθθ (2.3
Lc
=
=
LH
H
cc
c
TTTT
−
−=θ
Equilibrium temperature
= Critical temperature of the heavy component
u
eterm
t
(2.32)
=T
Hc
LcT = Critical temperature of the light component
The convergence pressure characteristic of a partic lar equilibrium sate is
T
d ined as the critical pressure of the critical mixture calculated either by Equation
2.26 or Equation 2.27. Methods of predicting critical pressures range from rigorous
thermodynamic o completely empirical, with methods having some degree of success.
The method we choose to present here is the one developed by Zais122 because of its
general applicability and convenience of calculation. Zais’ equation is written in the
form
36
∑∑∑−
=1 1i
i+== −+−+
+=1
21
2
)()(
N N
ij jiijjiijij
jiN
iicc wwcwwBA
wwwPP
m (2.33)
r ixture
= Binary interaction coefficient
s
g Equation 2.31 to binary data, Zais was able to obtain values for the
coefficients and for all hydrocarbon binary combinations from methane
through eicosane plus nitrogen and carbon dioxide. These coefficients and component
critical pressures for use in Equation 2.31 are tabulated. Binary coefficients for binary
heavy fractions in complex mixtures are obtained by interpolation on a molecular weight
basis. Zais predicted the critical pressures of 298 ternary, multi-component, and complex
mixtures with an average absolute deviation of 5.2%.
where
= Critical p essure for mmcP
icP = Critical pressure of component i
w = Weight fraction of component i
= Weight fraction of component j
ij
ijB = Binary interaction coefficient
ijC = binary interaction coefficient
After testing mole fraction, surface fraction, and volume fraction, Zais selected
weight fraction as the composition variable to use in Equation 2.31. To calculate
convergence pressure from Equation 2.31, it is necessary to convert the mole fraction
from Equation 2.27 or Equation 2.31to weight fraction.
i
jw
A
By fittin
,, ijij BA ijC
122
37
In hydrocarbon systems, there is no critical pressure at temperatures below the
critical temperature of the lightest component. For this reason, it is impossible to derive a
convergence pressure as we described. Lawal, in order to include such a data in the
new correlations he developed, a quasi-convergence pressure was defined which is
illustrated in Figure 2.8. The quasi- convergence pressure is read at the equilibrium
temperature from the binary critical locus reflected across an axis through the critical
temperature of the light component.
66
Figure 2.8. Illustration of Quasi-Convergence Pressure Concept. 66
38
In effect, for equilibrium at temperature T less than critical temperature of the
light componentT , the quasi-convergence pressure is defined as e critical pressure
T , where
TT − 2 (2
This definition of the quasi-convergence pressure is consistent with the
observation of Lenoir and White (1958) that quasi-convergence pressure should increase
with decreasing temperature. For systems higher than binaries, this definition of quasi-
Lc th at
LK cp − .34)
rmining the isothermal locus of critical
compositions at rather than at equilibrium temperature , according to Equation
2.32.
The fluid composition effects on the
Kp
T
convergence pressure is readily applied by dete
KpT T
K - values as shown in Figure 2.9, where
values for 1000 psia and 5000 psia convergence pressures are compared at F. The
differences in values for the two convergence pressures shown at pressures below 100
psia are not significant for the lighter hydrocarbons. The equilibrium ratios for fluids
with convergence pressures of 4000 psia or greater, are the same to fluids with 1000 psia.
Therefore, it is apparent that at low pressures and temperatures the equilibrium ratios are
closely independent of composition.
6
o100
K
39
Figure 2.9. Comparison of Equilibrium Ratios at 100 Fo for 1000- and 5000-psia convergence Pressure.
It is of practical value at this point to present brief remarks on the
interdependency of heavy (plus) fractions and convergence pressure concept. Because
e vapor-pressure curves and critical properties of hydrocarbon heavier than hexane
irly c -
values. Properties of heptanes plus fractions can be estimated from the properties of
a procedur r
characterizing the heptanes plus is to use correlated experimental data heptanes pl
fraction of fluids with similar properties to those predicted. For this purpose equilibrium
ratios for the heptanes-plus fraction reported by Katz and Hachmuth, and Roland,
6
th are
fa lose together, it is possible to characterize the mixture by an average set of K
heavier hydrocarbons such as decane. But, normally a more satisf ctory e fo
us
50
40
Smith, and Kaveller, are plotted in Figure 2.10. The data of Katz are preferred for crude
oil system, and the data of Roland et al densate fluids.
6 6
. are preferred for con
Figure 2.10. Equilibrium Ratios of Heptanes-Plus Fraction.6
lus) fraction controls the behavior of the system as liquid (mostly the
The heavy (p
plus-fraction) begins to drop out at the dew point. If a given set of K - values with
convergence pressure close enough to the correct convergence pressure is used m
through decane, the entire adjustment in the – values system can be made by adjusting the
ethane
K - value for the plus- fraction only. Then, it is easy to determine whether or not the K -
-
value system chosen for methane through decane is near enough to the correct
convergence pressure set by inspecting the decane K - values curve that was required to
match check-point data. If the selected convergence pressure is too low, the C +10 K
41
curve required to match checkpoint data will fall below the given published decane K -
curve. This shown in Figure 2.11 30 by the curve marked “ kP too low”. If the
ve will have the correct ecane
curve. This shown in the Figure 2.12 by the curve marked “ too high”. A set of
convergence pressure is too high (i.e., too close to the dew point pressure) the determined
KC +10 - values cur shape but will fall above the given d
30kP K
- values for the correct convergence pressure will result in the - values curve as
ome extent, cut across the
tion that
t as
KC +10
shown in Figure 2.11. Typically, the +10C K - value will, to s
constant temperature lines on the given Natural Gasoline Supply Men’s Association
(NGSMA) decane curve. This phenomena caused by the fact that the plus-frac
first drop at the dew point is generally less volatile than the plus-fraction that drops ou
pressure declines.
42
Figure 2.11. K vs Pressure with C10+ Curve Required to Match Check-Point Data.
43
Too High or ToFigure 2.12. K vs Pressure with Curve Showing Effect of Choosing a Convergence
o Low for Condensate Depletion.
30
44
2.5 Equation of State Models
com mixture. EOS may be used to describe the state of the fluid phase. The
volumetric phase behavior of a pure component and a multi-component mixture is
directly e
o to three
parameters. Basic parameters of these equations are the critical properties and the normal
boiling point or vapor pressure. For mixtures, the interaction coefficients between
ents should be included to account for highest accuracy. There are many
quations view
f several practical e
One of the first and simplest, perhaps the best known equation of state model
is the ideal gas law,
(2.35)
This law was derived by assuming that the molecules that make up the gas have
negligible sizes, that their collision with themselves and the wall are perfectly elastic, and
that the molecules have no interactions with each other. It has small applicability to
describe the volum reservoir fluids,
he petroleum industry
has adopted the concept of compressibility factor
An equation of state (EOS) can be defined as an algebraic equation that can
describe the relationship between pressure, temperature and volume for both a pure
ponent and a
given by the equation of stat . There are many families of EOS. The van der
Waals family is characterized by simple cubic form, and m st have two
constitu
e of state used for calculating vapor-liquid equilibrium, Reid et al 91 give re
o quations for chemical and petroleum industries. 52,51,12,11,10
RTPV =
etric phase behavior of petroleum because this law is
only valid for substances at low pressures and high temperatures. T
Z , or gas deviation factor for
45
describing the behavior of mixtures or gases at moderate high pressures. The
pressibility facto
(2.36)
and, by
com r Z is a correction factor for the ideal gas law, that is
ZRTPV =
definition
RTPVZ = (2.37)
itations in the use of Equation 2.34 to describe the behavior of natural
the earliest attempts to represent the behavior of real gases by an
equation was that of van der Waals equation of state mode.
Since the proposal of van der Waals equation (1873) , several
investigators have proposed many equations of state for representation of fluid
volumetric, thermodynamics, and phase equilibrium behavior. These equations, many of
them a modification of the van der Waals EOS, range in complexity from simple
equations containing two or three constants to complicated form having more than thirty
onstants. Even thoug
simplicity found in van der Waals cubic model while improving the accuracy through
modifications.
aals equation of state, is the first equation capable of representing
vapor-liq
The lim
gases gave the chance of
116
64,27,18,5
c h with this large number of EOS, not so many are considered by
engineers and researchers. Because of its computational problem, many prefer the
The van der W
uid coexistence
2Va
bVRTP −−
= (2.38)
46
The parameters a and b are constants that characterize the molecular properties
of Equation 2.36 represents
e i
of the substance in question. The first term on the right side
bVRT−
), and the other term ( 2Vathe repulsiv nteraction force ( ) is the attractive forces
rmined mathematically using Equation
e
Redlich-Kwong, Soave-Redlich-Kwong, and Peng-Robinson. Redlich-Kwong
n der
a temperature-dependent term. The R-K EOS has the form
between molecules. The parameters a and b can be obtained from the critical properties
of the fluid. Also, these parameters can be dete
2.1. In the development of cubic equations of state, modifications in the evaluation of the
param ter a in the attractive term by Soave is considered the most accurate results in
86 106 83 87
equation is the most important and successful model for the modification of the va
Waals equation of state.
Redlich-Kwong (1949) replaces the attractive term of the van der Waals EOS
with
5.0)(VVb + TbVP −
−= (2.39)
the Equation 2.37 constant parameters and usually
xpressed as
aRT
For pure components, a b
e
c
cTRa
2
Ω= a P
5.2
(2.40)
c
cb
RTb Ω=
P (2.41)
where and are dimensionless parameters with the following computed values:
aΩ bΩ
4278.0=Ωa and .0867.0=Ωb
47
Soave modified the R-K EOS and published the Soave-Redlich-Kwong (SRK) , 94
equation of state
)( bVV
aBV
RTP+
−−
=α (2.42)
nsionless factor αwhere the dime is a function of temperature:
(2.43)
2.41, is the slope and is the reduced temperature.
e slope, against the acentric factor,
[ ]25.0 )1(1 rTm −+=α
m rTIn the Equation
Soave correlated th ,m ,ω by the generalized form
(2.44)
of state, the constant parameters
are determined at the critical conditions in the form
2176.057.1480.0 ωω −+=m
a As the case in the Redlich-Kwong equation
band
cP c
aTR
a Ω= (2.45)
22
c
cb P
RTb Ω= (2.46)
where the valu less pure component parameters es of dimension aΩ and do not
hange as shown in Equations 2.43 and 2.44 due to the introduction of function
Pen oduced an improved Redlich-Kwong EOS capable of
redicting the liquid volumes and a critical compressibility factor of Their
equation is given by
bΩ
c ).(Ta
g and Robinson 83 intr
p .307.0=cZ
)()( bVbbVVa
bV −RTP
−++−=
α (2.47)
48
with
c
cc TRT 222
=c
a PPR
a 4724.0Ω= (2.48)
and
ccb P
RTP
RTb 07780.0=Ω=
Peng and Robinson adopted Soave’s- Redlich-Kwong approach for computin
cc (2.49)
g α
as shown in Equation 2.45, where they used ω as the correlating parameter for the slo
,m as given by
32 001667.01644.048503.1379642.0 ωωω +−+=m
Usdin and McAuliffe (UM),101 proposed a new parameter ,d to replace b in the
pe,
second term (attractive term) of the Soave-Redlich-Kwong106 equation:
)( dVVa
bVRTP
+−
−=
α (2.50)
where
c
ca P
TRa Ω= (2.51
22
)
c
cb P
b Ω=
RT(2.52)
c
cd P
RTd Ω=
ued that the two terms of the SRK equation are interconnected by
parameter concluding that all substances possess a value of critical compressibility
(2.53)
They arg
,b
49
factor .333.0=cZ They stated that severing the tie created by the shaping of the
parameter b and replacing it with d wo ld cause accurate liquu id density.
UM proved the dimensionle parameters ,, bass ΩΩ and Ωd can be related to the
critical compressibility parameter, by the following expression: ,cZ
[ ] 0))(38(3)(12)16(3 +Ω cd Z 222 =−+Ω−+Ω− ccdccd ZZZZ (2.54)
Ω cc Z (2.55)
,13 −+Ω=b
and
ba
ZΩ
3)(
Usdin and McAuliffe also adopted Soave’s formulation of
c=Ω (2.56)
α as Equation 2.45
with
[ ] ),7.0(02.0)35.0(67713.0516.448049.0 −−−++= rc TZm ωω
for 7.0≤rT
(2.57)
and
[ ] 23 )7.0(78662.0)(7846.37516.44049.0 −+++= rcc TZZm ωω , (2.58)
for 0.17.0 ≤≤ rT
Patel and Teja 81 introduced the following form of EOS:
)()( bVcbVVaRTP −=
bV −++−α (2.59)
where
50
cP
caa Ω= (2.60) TR 22
c
cb
RTb Ω=
P (2.61)
c
cc P
RTΩ= c (2.62)
where
cc ζ31−=Ω (2.63)
(2.64) )31()21(33 22cbbcca ζζζ −+Ω+Ω−+=Ω
c
ccc RT
VP=ζ (2.65)
is the smallest positive root of the cubic expression:
=cζ (2.66)
n by
.
and bΩ
03)32( 223 −Ω+Ω−+Ω bcbcb ζζ 3
A value of bΩ is give
−=Ω cb 0225.03243.0 ζ (2.67)
Finally, α is d termined by the following:
(2.68)
parameter constants
e
[ ]25.0 )1(1 rTF −+=α
The Patel-Teja equation of state therefore requires four
,,, ccc PT ζ and F for any
There are many other equations of state models with modifications of the
with
fluid desired.
attractive term, repulsive term, and combination of both of the van der Waals EOS
51
more th e param e
2.2.
a aals Eq
an two or thre eters. Some of these are summarized in Table 2.1 and Tabl
Table 2.1. Modifications to the Attractive Term of v n der W uation of State.117
Equation Attractive Term
Redlich-Kwong (RK, 1949)
)(5.1 bVRTa
+
Soave (SRK, 1972)
Peng-R
Kubic (1982)
Patel-T
)()(Ta
bVRT +
[ ])()()(
bVbbVVRTVTa
−++
obinson (PR, 1976)
Fuller, (1970)
Heyen, 1980-Sandler, (1994)
Schmidt-Knapp (1980)
)()(
cbVRTVTa+
[ ]cTbVcTbVRTVTa
)())(()(
−++ 2
)()(
22
VTa wbubVVRT ++
2)()(
cVRTVTa+
[ ]eja (PT)(1982)
Yu and Lu (1987)
Lawal-Lake-Silberberg (LLS)(1985)
)( VTa)()( bVcbVVRT −++
[ ]
)3()()(
cVbVVRTVa
+++
cT
22 bbVV βα ++ )(Ta
52
Table 2.2. Modifications to the Repulsive Term of the van der Waals EOS.117
Equation Repulsive Term
Reiss et al (1959)
Thiele (1963)
Guggenheim (1965)
Carnahan-Starling (1969)
3
2
)1(1
ηηη
−++
3
21 ηη ++ )1( η−
4)1( η− 1
3
32
)1(1
ηηηη
−−++
53
CHAPTER III
CLOSED-FORM VAN DER WAALS EXPRESSIONS
This chapter presents the derived equations, which form the basis of critical point
calculations. These include the van der Waals equations of state theory, the parameters
that characterize the individual components in the Lawal-Lake-Silberberg (LLS)
uids. Using the LLS equation of state as a
l resented. This is
plished by introducing the “VDW closure parameters” (that is, parameters
generalized cubic equation of state, and the algorithm developed for computing the
critical properties of petroleum reservoir fl
basis, a c osed-form solution for the van der Waals critical point is p
βα , accom
developed for the purpose of resolving critical point in fluids) into the original van der
Waals equation of state.
3.1 Van der Waals Equations of state Theory
The ideal gas law, nRTPV = , can be derived by assuming that the molecules
that make up the gas have negligible sizes, that their collision with themselves and the
wall of the vessel are perfectly elastic, and that the molecules have no interactions with
ach other.
An early attempt, to take these intermolecular forces into account was that of Van
aals (1873), who proposed that the idea
e
der W l gas equation of state be replaced by
RTbVaP MM
=−⎟⎟⎞
⎜⎜⎛
+ )(2 (3.1) V ⎠⎝
54
This equation differs from the ideal gas equation by the addition of the term 2Va to
pressure and the subtraction of the constant to from molar volume.
Here the par
V b
ameters a and b are constants particular to a given gas, where R is
the universal gas constant. The term 2Va represents an attempt to correct pressure for
forces of attraction between the molecules. The actual pressure exerted on the wall o
the
f the
vessel by real gas is ount less, by the am 2Va , than the pressure exerted by an ideal gas.
The para the size of each
se t th the
volume that the molecules have to move around in is not just the volume of the container
, but is reduced to . The parameter has a more difficult meaning and is
ted to the
intermolecular attractive forces is to r duce the pressure for a given volum
temperatu
meter b (or the so called co-volume parameter) is related to
molecule and repre n s e intermolecular repulsive forces in the sense that it is
V )( nbV − a
rela intermolecular attractive force between the molecules. The net effect of
e e and
re. When the density of the gas is low (i.e., when Vn is small and is small
compared to ) the Van der Waals equation reduces to that of the ideal gas law. The
e
nb
V
a and b parameters can b obtained from the critical properties of the fluid.
On the basis of the available volume ( bV − ), Van der Waals was able to
that Equation 3.1 is appropriate for the hard-sphere gas at low density. To see that this
leads to a pressure reduction, simply solve for P:
show
55
2VaRT
−= 3.2
Since a > 0 , and b << V , then will be a reduction in pressure by approximately
bVP
− ( )
2V.
It is clear that the Van der Waals equation predicts a deviation from ideal
behavior. This deviation can be analyzed
a
by defining a quantity , called the
ompre
Z
c ssibility factor, as
RTPVZ = (3.3)
For an ideal gas, it is clear that 1=Z .
To derive Z , start with Equation 3.2, then multiply both sides by V and divid
:RT
e by
RTVa
bVV
RTPVZ −
−==
or
(3.4)
RTV
a1
Vb
Z −−
=1
(3.5)
For very low density 1<<Vb , so we can e a Tayl eries to ap ate us or s proxim
Vb
−1. In
general, for
1
1<<X
XX
≈−
+1 1
1 (3.6)
56
Using this approximation, the compressibility factor becomes
RTVabZ −+= 1
V (3.7)
or
VRT
b )(1
−+=
a
Z (3.8)
The quantity
RTab −
ideal behavior
is an observable and calculable quantity, which measures
deviation from of a gas. Note that if 0>>−RTab , then a
RT
pressure is larger than ideal gas pressure. However, the condition
b > and the
RTb > tells that a
excluded volume effects, as measured by the constant b , so an increase in pressure is
what we would have predicted. On the other hand: if 0<−ab , then
RTab < , then
RT
the pressure is less than the ideal gas pressure.
Real gases can hav
ill lead to a low ring of
ideality, where as strongly repulsive forces can lead to a positive deviation. What do the
isotherm of the Van der Waals equation look like? Recall that the isotherms are curves
e both positive and negative deviations from ideal behavior,
depending on the pressure and temperature and the particular system. Strongly attractive
forces w e the pressure and hence a negative deviation from
corresponding to P vs. V at various fixed temperatures. For the Van der Waals
equation, some of the isotherms are shown in Figure 3.1.
57
Figure 3.1. Pressure-Volume Diagram for Pure Component.
For the isotherms where cTT > appear similar to those of an ideal gas, i.e., there
is a monotonic decrease of pres with increasing volume. The isotherm
where the curve is essentially horizontal (flat) with no curvature. At this point, there is
sure c
exhibits an unusual feature not present in any of the ideal gas isotherms – a small region
no change in pressure as the volume changes. Below this isotherm, the Van der Waals
starts to exhibit unphysical behavior. The cTT
TT =
< isotherm has a region where the
pressure decreases with decreasing volume, behavior that is not expected on physical
atic jum
that a phase transition has occurred, in this case, a change from a gaseous to a liquid state.
The isotherm just represents a boundary between those isotherms along which no
such phase transition occurs and those that exhibit phase transitions in the form of
grounds. What is observed experimentally, in fact, is that at a certain pressure, there is a
dramatic discontinuous change in the volume. This dram p in volume signifies
cTT =
58
d For this reason, the cTTiscontinuous changes in the volume. = isotherm is called the
critical isotherm, and the point at which the isotherm is flat the slope of the curve is zero,
se two
tuations correspond to zero values of the f
espect to
At the critical point, t
liquid, but has characteristics of both. Hence, it is called a supercritical fluid. To see at
is observed, we use
the fact that at the critical point, the isotherm is both horizontal (zero slope) and has no
curvature. These two conditions (i.e., criteria of criticalit r a pure component are:
and the isotherm has zero curvature at the critical point. Mathematically the
si irst and second derivatives of pressure with
r volume.
he system does not exist in a particular state, either gas or
what temperature, pressure and volume, this supercritical behavior
y) fo
0=⎟⎠∂ cpV
⎞⎜⎝⎛ ∂P (3.9)
02
2
=⎟⎟⎠
⎞⎜⎜⎝
⎛∂∂
cpVP (3.10)
S r 1 mole olving Van der Waals’ equation, Equation 3.1 for pressure yield fo
2Va (3.11)
bVRTP −−
=
To estimate , and a , b R at the critical point, it is necessary to obtain the first
and second derivatives with respect to volume of Equation B.1 in Appendix B and set
them equal to zero. The first of these conditions leads to
0)(3⎜
⎝ −−=⎟
⎠⎜⎝ ∂ ccp VVV
22 =⎟⎟⎠
⎞⎜⎛⎞⎛ ∂
cpc
c
bRTaP (3.12)
r o
59
2
RTc
− (3.13) 3 )(
2bVV
a
cc
=
The second condition leads to
0)(
26342
2 ⎞⎛ ∂ RTaP=
−−=⎟⎟
⎠⎜⎜⎝ ∂ bVV
c
cp
(3.14)
or
V cc
34 )(26
bVRT
Va
c
c
c −= (3.15)
Consider the Van der Waals equation of state at the critical state
ccc
c RTbVV
P =−⎜⎜⎝
+ )(2 (3.16)
The three equations Equation 3.13, throu
a⎟⎟⎠
⎞⎛
gh Equation 3.16 apply at the critical
point and by combination results in
c
c
Pa
64= TR27 22
(3.17)
nd a
c
c
Pb
8=
RT (3.18)
These equations for the c
inflection at the critical point.
onstant parameters will only work for pure component at
the critical point. For mixtures, the two equations, Equation 3.9 and Equation 3.10 do not
hold. The pressure-volume diagram for the mixtures does not exhibit horizontal
60
3.2 Closed-Form Equations for Fluid Critical Point
Another development of van der Waals modification is the single component
rm of t
this equa
fo he Lawal-Lake-Silberberg (LLS) equation of state. This generalized cubic
equation is used in this research project is presented in Equation 3.19. Related work to
tion of state has been given in the literature. 62 The LLS equation has the form
22 bbVVa
bVRTP
βα −+−=
− (3.19)
here parameters ,a ,b α and β are established for pure component as follows w
[ ]c
c
PTR 22
3cZa )1(1 −Ω+= ω : (3.20)
cc P
Zb )( ωΩ= cRT (3.21)
c
cc ZZω −Ω+=
31
Zω
αΩ
(3.22)
c
ccc
ZZ )3 ωΩ−ZZ
2
232 1(2)1(
ω
ωωβΩ
+Ω+−Ω= (3.23)
ωω 0274.01361.0
+=Ω (3.24)
where is the pressure, is the temperature, is the molar volume, , is the critical
compressibility factor and are the critical pressure and temperature of the
ponents,
P T V Z
, cP cT
c
R is the universal gas constant. The com Ω is a constant equals to 0.325, and ω
u work for the LLS EOS are psia for pressure, a degree Rankin, nits used in this Ro for
temperature, cu-ft/1b-mole for molar volume, and the constant R = 10.73 psia-cu-ft/1b-
mole The experimental data are reported in psia of pressure, mol fractions, and F, .Ro o
61
therefore, the data had to be converted to the units used in this work. The composition
and critical points obtained by the researchers, Sim ugh,98 Etter and Kay,29
and Zais for 85 hydrocarbon/non-hydrocarbon reservoir fluid mixtur
on-Yarboro
es are shown in
Tables
d en n m
C 1 through C 27 in Appendix C.
The Equation 3.19 has four in ep de t para eters ( ,a ,b ,α and β ) and i
generalized form and cubic in terms of molar volume. The LLS equation, Equati
can be viewed as a gene
t is a
on 3.19,
ralized cubic equation of state from
equatio s ed” cubic equations of
state is used when the equation can be reduced or m of any of the
known cubic equations of state by assigning specific integer to the parameters
which many known
n of state can be derived. Moreover, “the term generaliz
odified to the form
α and β .
It is interesting to find out that when 0== βα and 83
=Z the generalized form of LLS c
equation of state reduces to van der W quation of state. Similarly for the PR EOS
r
aals e
fo .1,2 == βα The resulting p
Table 3.1. Parameters of Selected Equations of State. E Ωc
arameters from the application of the criticality
conditions to various forms of cubic EOS are presented in Table 3.1. 75
quations of State Year Zc Ωa Ωb
v 0.333 an der Waals 1873 0.375 0.4218 0.1250
Dieterici 1896 0.271 0.6461 0.1355 0.500
Berthelot 1900 0.281 0.5365 0.936 0.333
Redlich-Kwong 1949 0.333 0.4275 0.0866 0.260
Peng-Robinson 1976 0.307 0.4572 0.0778 0.253 Harmens 1979 0.286 0.4831 0.0706 0.247
62
For example, when 0=α and 0=β in the LLS equation of state, the equation
reduced to the van der Waals equation of state previously shown in Equation 3.1.
Applying the criticality conditions to van der Waals equation of state yield the following
expressions: 98
02)( 32 =+
−=⎟
⎠⎞
⎜⎝⎛∂∂ c
Va
bVRT
VP (3−
ccTc
.25)
06)( 432
2
−⎟⎠
⎜⎝ ∂ ccT VbVV
c
Solving simultaneously Equations 3.25 and 3.26 for a and b parameters, yield
2=−=⎟
⎞⎜⎛ ∂ c aRTP (3.26)
3cV
b = (3.27)
89 ccVRT
a = (3.28)
Combining Equations 3.27and 3.28 with van der Waals Equation 3.1, a universal
compressibility factor is obtained and calculating the parameters and
from the following equations:
, 375.0=Z a b
c
c
Pa 4218.0= (3.29TR 22
)
cP
where 4218.=Ωa and 125.0
cb 125.0= (3.30) RT
0 =Ωb are constants at the critical point from the van der
Waals original equation. The parameter ωΩ is determined by dividing the parameter bΩ
by the critical comp . In the case of van der Waals, ressibility factor, cZ
63
375.0/125.0 ==Ωω .333.0 Solving Equation 3.28 and Equation 3.29 for the critical
properties for the original van der Waals and for those classical equations of state listed
in Table 3.1, the results are shown in Table 3.2.
Table 3.2. Relationship of EOS constants with Critical Parameters.
2m
mpc b
aη=
m
mTc Rb
aT
η= mvc bV η= P
EOS Year pη Tη vη
van ader W als 1973 1/27 8/27 3.0
Dietrich 1896 1/29.56 16.61/27 2.0
Berthelot 1900 1/27.04 10.67/27 3.0
Redlich-Kwong 1949 1/58.8 5.48/27 3.84
Peng-Robinson 1976 1/62.6 5.56/27 3.94
Harmens 1977 1/97.1 3.95/27 4.05
The criticality constraints for deriving the equations to predict the critical
properties for pure components or mixtures in thi s context of document are the
compressibility and vo
properties equations is presented in Appendix B. In this appendix, two methods for
deriving critical properties expressions are shown in terms of critical compressibility-
factor, , and critical volume, to the corresponding
lume forms of criticality conditions. The derivation of the critical
cZ cB 0)( =− cZZ .
The critical compressibility- factor form at critical condition is derived by the
expansion of the LLS cubic equation of state in term of and by comparison with the cZ
64
expansion of .0)( 3 =− cZZ This procedure is shown in A endix B and the resulting
cubic equation as Equation 3.30 where ),(
pp
βαfZc = .
3 =+θθcZ (3.31)
where,
(3.32)
θ (3.34
The critical volume form of the criticality condition is also determined by the
expansion of the LLS equation of state in terms of molar volume ( and by comparison
of the . This procedure is presented in Appendix B and the resulting cubic
equation is shown as Equation 3.36
(3.36)
where
)
) (3.38)
04322
1 ++ θθ cc ZZ
)6128( 321 αααθ +++=
)9912123( 22 βαβααθ −+++−= (3.33)
)6663( 2 βαβαα −++= )
)( 2 (3.35)
3
4 βααβθ −+−=
)cB
0)( 3 =− cZZ
04322
13 =+++ θθθθ ccc BBB
6128( 321 αααθ +++= (3.37)
3271515( 22 αβαθ −−+=
)36(3 αθ += (3.39)
14 −=θ (3.40)
65
The equations derived to pre ic , ,cP critical tempera urd t the critical pressure t e,
an
,cT d critical volume, cV pure components and mixtures are presented with the
following dimensionless parameter:
22TRaPA = (3.41)
RTbPB = (3.42)
RTPVZ = (3.43)
Applying Equation 3.19 at the critical Point we have
c
ccc RT
VPZ = (3.44)
22c
cc TR
aPA = (3.45)
c
cc RT
bPB = (3.46)
Where subscript c denotes the gas-liquid critical state. Solving for single component
critical pressure, critical Temperature, and critical volume, from Equations
3.43, 3.44, and 3.45, give the following expressions:
,cP ,cT cV
bRTB
P ccc = (3.47)
Solving for from Equation 3.44, obtain
cP
aTRA
P ccc
22
= (3.48)
66
Equating Equation 3.46 and Equation 3.47 yield
c
cc bRA
aBT = (3.49)
nd substituting Equation 3.48 into Equation 3.46 yield
a
c
cc Ab
aBP 2
2
= (3.50)
and expressions previously obtained in Equation 3.48 and
quation 3.49 in Equation 3.43, the critical volume equation is obtained as:
Replacing cP cT
E
c
cc B
bZV = (3.51)
The Equations 3.48, 3.49, and 3.50 can be expressed in terms of ,cA ,cB ,cZ ,a
, ,α and .β In the derivation of the criticality expressions in Appendix B Equation
.48, the parameter has been presented in term of
b
cA ,cB ,cZ α and .β B
ccccc BBBZA αβα +++= 2223 (3.52)
By substitu in the equations for and , the critical property
s of
ting cA ,cP cT cV
,cB ,cZ α and .β equations are expressed in term Then, Equations 3.48, 3.49 and
3.50 can be rewritten in the form
)( cccc
cc BBBZb
aBP
ααβ +++= 2222
2
3 (3.53)
)( cccc
cc BBBZbR
aBT
ααβ ++= 2223
(3.54)
67
c
cc B
bZV = (3.55)
r o
( )bbbcc Zb
PΩ+Ω+Ω+
Ω=
ααβ 2222
2
3 (
( )
ba3.56)
bbbc
c ZbRa
Ω+Ω+Ω+Ω
ααβ 2223
bT = (3.57)
b
cbZV
Ω= (3.58)
Where, bΩ is a constant parameter of the van der Waals equation of state. These
equations Equation 3.56 through Equation 3.58 to calculate ,cP ,cT and cV are practical
and directly obtained once the composition and the pure components are given.
For the mixtures, the express
c
ions for determining the critical properties are
expressed in terms of the parameters ,m ,m ,ma b α and mβ . These mixture parameters
require the LLS mixing rules to establish the following equations for the critical
properties for mixture:
( )cmcmcmcm
cmc
BaP = 222
2
(3.5
( )
BBBZb ααβ +++239)
cmcmcmcm
cmc BBBZRb
BaT
ααβ +++= 2223
(3.60)
c
mcc B
V = (3.61
By LLS mixing rules,
bZ)
68
ij
n
i
n
j
aaaxxa 21
21
∑∑=
3
1
3/1 ⎟⎠
⎞⎜⎝
⎛= ∑
ijim (3.62)
=
(3.63) n
i
bxb
iim
ijjijim xx αααα ∑∑= 21
21
(3.64)
n
i
n
j= =1 1
ijjij
n
i
n
jim xx ββββ 2
121
1 1∑∑= =
= (3.65)
The prediction of c tures can now be
achieve
critical temperature, and critical volume for multi-component systems.
The algorithm constructed for calculating critical properties for reservoir fluids
(gases, gas condensate, volatile oils, and crude oils) are discussed next.
ritical properties for hydrocarbon mix
d since all the necessary equations have been developed. In this project, the
iterative methods have been utilized to match the experimental critical pressure, ,Pc
cT cV
3.3 Closed-Form Critical Property Computation Methods
The type of data that are often available from laboratory work on reservoir fluid
samples for pure components are critical pressure, cP , cT , acentric factor, ω , Criti
compressibility factor, cZ , critical volume, cV , and molecular weight. Additional
information that may be available is the analysis of the equilibrium liquid and gas. The
data will permit critical properties to be calculated directly at reservoir conditions of
pressure and temperature. Usually, however, laboratory critical point values are
cal
se
69
furnished, and the critical volume of the reservoir fluids will be the most valuable d
available for reservoir study purposes.
ata
properties for pure compo losed-form Van der Waals
method. However, the fo owing procedure (algorithm) is included to show the
calculation procedure foll in program:
The following is t e the critical point of pure
components given the com
Computer program provides speed and accuracy in predicting the critical
nents and complex mixtures by the c
ll
owed by flowcharts of the ma
he step-by-step procedure to calculat
ponents measured ω,, cc PT .
Step1. From the single co pure components parameters
a, b,
mponent critical data, calculate the
α , and β using Equ 3.22, and Equation3.22,
Step 2. Calculate the dim using Equation 3.66.
ation 3.20, Equation 3.21, Equation
ensionless critical volume, Bc
c
cc RT
bPB = (3.66)
Step 3. Calculate the Dim eter fensionless Param c in terms oA ,,, αcc ZB and β .
ZA α ++= 2 (3 (3.67)
Step 4. Calculate the crit ture, , and the critical
using Equation 3.53, E
Step 5. Once the pure components parameters are calculated, the mixture parameters for
cBB αβ +2) ccc
ical pressure, P , the critical temperac cT
cV quation 3.54, and 3.55.
mmm ba α,, , and mβ are calculated using Equation 3.62, Equation 3.63, Equation 3.64,
and Equation 3.65.
70
Step 6. With the calculated values of mα and mβ , the coefficients of Equations 3.64 and
3.65 are calculated.
Step 7. Calculate the dimensionless critical volum for the mixture using Equation
3.68. The solution of this cubic equation consists of two imaginary roots and one real
root. The real root is chosen as the value for
e, cB
cB : ),( mmc fB βα= .
=++ θθθ ccc BB .68)
where
(3 += mmmαθ (3.69)
2 −+−−= mmm βααθ (3.70)
0012
23
3θ B + (3
)8+α 6 23 + α 12
)595(3 2
)2(31 += mαθ (3.71)
10 −=θ (3.72)
Step 8. Solve the cubic Equation 3.73 of the mixtures for cZ . The solution has two
imaginary roots and one real root. The real root is chosen as the value for cZ :
),( mmc fZ βα= ,
023 =++ θθθ ZZZ (3.73
where
)8126( 23
0123 +θccc )
(3.74)
(3.75)
3 +++= mmm αααθ
)33441(3 22 mmmmm βαβααθ −+++−=
71
6)
)
xtures using the equation
Bααβ +++= 2223 (3.78)
Step 10. Calculate the al t ica ssu cri
t
)222(3 21 mmmmm αβαβαθ +−+= (3.7
)( 20 mmmm αββαθ −−= (3.77
Step 9. Determine the constant cA for mi
mcmcc BBZA cmc
critic emperature, , critcT cP ,l pre re, and tical
volume, cV for the mix ures:
cBmcB +2mc α+2
m Bβ+cZT
α23(
mc Rb
cm Ba= (3.79)
)cB3( 22mcm Zb
P
and
22cm
2B
cB α+mc α+m Bβ+c =a
(3.80)
cc B
V mcbZ (3.81)
Step 11. ulate e a e d ons llo
Absolute D D
Calc the averag bsolut eviati as Fo ws:
eviation (A ) = )0 10(exp, ⎥
⎥⎦
⎤
⎢⎢⎣
⎡ −
c
dS 3.82)
Absolute D D
exp,cP, precPP
AB (
eviation (A ) = )0 10(exp,⎢
⎢⎣
⎡ −
c
dS (3.83)
arizes the step-by-step procedure presented above
exp,⎥
c, prec
⎥⎦
⎤TT
TAB
Figure 3.2 is an algorithm summ
to calculate the critical point of petroleum reservoir fluids.
72
Figure 3.2. Algorithm f arameters.
or Computation of Critical P
Input Tc Pc Vc Zc ω
Read parameters a b α β
Call equations of parameters a b α β calc. a b α β for pure components
Read coefficients θ3 θ2 θ1 θ0 and constant Bc
Call cubic e uation and calculate Bq c for pure components
Read coefficients θ3 θ2 θ1 θ0 and constant Zc
Call cubic equation and calculate Zc for pure components
Read constant Ac
Call equation Ac and calculate Ac for ts pure componen
Read Parameters am bm αm βm
Call equations of parameters am b α β and calculate a b α β
m m m m m m
m for mixtures
Read coefficients θ3 θ2 θ1 θ0 and constant Bc for mixtures
Call cubic equation and calculate Bc for mixtures
Read coefficients θ3 θ2 θ1 θ0 and cconstant Z for mixtures
Call cubic equation and calculate Zc for mixtures
Read constant Ac for mixtures
Call equations of Tc Pc Vc and calculate for Tc Pc Vc for pure components
Call equation Ac and calculate mixtures Ac for
Read critical properties Tc Pc Vc
Calcal
l equations of Tc Pc Vc culate for Tc Pc Vc for
mixtures
End
Start
73
CHAPTER IV
CRITICAL PROPERTIES FOR RESERVOIR FLUIDS
The critical point calculations using the modified Lawal-Lake-Silberberg (LLS)
equation of state introduced in Chapter 3 were carried for each pure component and
complex mixtures such as alkanes +, heptanes plus +, nitrogen and carbon dioxide. The
general behavior of the experimental parameters, and calculated critical properties for
these mixtures are presented in Tables C 1-C 27 and Figures C 1-C 19 in Appendix C. In
general, qualitative results were obtained for the critical pressure and critical temperature.
A detailed comparison and discussion of the results of each of the calculated critical
properties with respect to the corresponding experimental results, and with respect to the
other correlation predictions of Simon and Yarborough, Terry and Kats, and Zais are
presented in the following sections:
4.1 Critical Pressure Data for Complex Hydrocarbon Mixtures
In appendix C, tables and cross-plots are displayed for complex mixtures with th
results of critical pressures and critical temperatures. Each of the ta
e
bles corresponds to 9
omplex mixture. The predicted critical properties and acentric
ctor for C7 + of complex mixtures are also presented. The methane concentrations in
these
and 10 complex mixtures and each mixture is subdivided into different compositions.
Each table has the composition in mole fraction, the experimental values of the mixture
critical pressure, temperature, and the calculated values of critical pressure, critical
temperature for each c
fa
mixtures varied from 19 to 96.6 mol percent. The intermediate hydrocarbon
74
groups, consisting of ethane, propane, and bu a low of 3 to 59 mol
t, a e con ation pent nd he frac from 3 m ercen
e f he lu va m 2 to 14 mo nt on
ocarb xtu nsis trog d ca iox he tra no
car xtu ied to l percent. T sic ert
of the components fraction covered the range from light, paraffin to heavy, aromatic, and
asphalt.
4.2 Calculation of Critical Properties
tane varied from
percen nd th centr s of ane a xane tions 1 to ole p t,
and th raction ptane p s (C7+) ried fro l perce . The n -
hydr on mi res co t of ni en an rbon d ide. T concen tion of n-
hydro bon mi res var from less than 1 22 mo he phy al prop ies
In order to clarify the procedure to be followed when applying the methodology
of predicting the critical properties to an actual problem, an illustrative example of
calculation performed using the algorithm presented in Section 3.3 is given here.
To predict the critical pressure and critical temperature of a naturally occurring
mixture, it is given the following information for mixture 1 of the given data in Table 4.1.
As a st , it is
desirab
obtained from the previously determ and Lawal,
and the critical compressibility required as additional input for the equation of state
used in this work is obtained from Rowlinson correlation of ACS Symposium Series,
316, 1977. In this project, the critical properties of the fractio mplex
ep one, to compute the critical properties for complex mixtures
le first to determine these properties for the +7C fraction in the mixture. In
complex mixtures, components heavier than the n-heptane have been summed into a +7C
fraction. These fractions required for the parameters of equations of state are
59 62
+7C
ined correlations TR-4-99 Lawal,
cZ
+7C ns in the co
75
mixtures are calculated on the basis of the reported molecular weight of the fraction
. The empirical expressions shown in the Table 4.1 are used to estimate the
critical properties for the hydrocarbon fractions.
mple of Experimental Data Used for Calculations of mixture 145-1.
Hydrocarbon Mixture Composition
)(7+cMW
Table 4.1. A Sa
Component Mole Fraction Nitrogen 0.001 Carbon Dioxide 0.004 Methane 0.193 Ethane 0.032 Propane 0.585 i-Butane 0.007 n-Butane 0.012 i-Pentane 0.005 n-Pentane 0.007 Hexane 0.013 Heptane + 0.141 Heptanes + Properties: Mo. Wt. 243 Characterization Factor 11.6 Critical Temp., )(, RTc
o 725 Critical Temp. This work 725.39 Critical Pressure, )( psiaPc 2100 Predicted )( psiaPc : (S. Yarborough) 2002 (Etter and Kay) 2911 (Zais) 2175 This Work 2101
76
+7Table 4.2. Physical properties of Fractions Correlation.C 62
9.56084+=
MWAPI
5.1315.141
+=
APIS g 21 e
ge
obp SMWeT =
321 eee TSMWeC = 4322 eeee CTSMWeP = 4321 eeee CTSMWeT =
eeee
bpgo bpgc bpgc
bpg CTSMWe=ω
0 0
43210
Parameters 0e 1e 2e 3e 4e
bpT 108.701661 0.4224480 0.42682558 0.0000 0.0000
C 0.83282122 0.09255911 -0.0413045 0.12621158 0.0000
cP 237031780 -0.028484 2.755309 -1.374440 -2.947221
cT 6.206640 -0.059607 0.224357 0.968332 -0.802538
ω 1.5790E-13 -1.453063 -2.811708 4.883921 2.109476
The critical properties and acentric factors are estimated from the correlation
shown in Table 4.2. Bu using the given Mw of the as basis, the critical pressure,
as a lumped single pseudo-com onent for all
the pre
+7C
+7
critical temperature, critical compressibility factor, and acentric factor of +7C were
obtained. The heptane-plus +7C is treated
C
p
diction results shown in this work. Table 4.2 displays some of the predicted
critical properties for the heptane-plus fraction.
From the given single component critical data, the pure component parameters
77
,,, αba ,β ωΩ , the dimensionless critical parameters cA and cB are calculated fro
Equations 3.20, 3.18, 3.21, 3.22, 3.23, 3.24, 3.
m
67, and 3.66 respectively. The resulting
parameters are presented in Table 4.4.
Table 4.3. Calculated Critical Data of Heptane-Plus Fraction for Data Set 1.
Heptanes + Properties Mol. Wt. 243 243 243 243 191 191 191 191 191
Gravity(API) 33.31 33.31 33.31 33.31 39.98 39.98 39.98 39.98 39.98SG 0.86 0.86 0.86 0.86 0.83 0.83 0.83 0.83 0.83
Characterization Factor 11.6 11.6 11.6 11.6 11.9 11.9 11.9 11.9 11.9 Tb 1037.0 1037.0 1037.0 1037.0 920.9 920.9 920.9 920.9 920.9C 3.3475 3.3475 3.3475 3.3475 2304 3.2304 3.2304 3.2304 3.23043.
Pc, (psia) 271.0 271.0 271.0 271.0 319.8 319.8 319.8 319.8 319.8T , (c
oR) 1364.4 1364.4 1364.4 1364.4 1258.4 1258.4 1258.4 1258.4 1258.4ω 0.5673 0.5673 0.5673 0.5673 0.4676 0.4676 0.4676 0.4676 0.4676Z 0.2416 0.2416 0.2416 0.2416 0.2493 0.2493 0.2493 0.2493 0.2493c
Ωw 0.3555 0.3555 0.3555 0.3555 0.3564 0.3564 0.3564 0.3564 0.3564
Component
Table 4.4. Calculated Results of Pure Component Parameters.
a b α ωΩ cA cB β N2 6506.50 0.520 2.21 5.32 0.361 0.569 0.105
CO2 18107.70 0.540 2.81 6.02 0.360 0.587 0.098 C1 11111.20 0.567 2.42 5.56 0.361 0.576 0.103 C2 26880.60 0.854 2.43 5.58 0.360 0.576 0.102 C3 46045.90 1.140 2.75 5.96 0.355 0.583 0.098
i-C4 64482.90 1.513 2.49 5.66 0.359 0.577 0.102 n-C 109225.70 1.850 2.81 6.02 0.359 0.587 0.983 4
i-C5 90386.27 1.770 2.86 6.07 0.358 0.588 0.978 n-C5 139353.34 1.786 3.02 6.26 0.358 0.593 0.096 C6 130104.33 2.013 3.77 7.11 0.358 0.614 0.090 C7+ 475160.996 4.614 4.222 7.630 0.3535 0.6222 0.085
Once the pure components parameters are calculated, the mixture parameters for
mmm ba α,, and mβ can be determined by using Equations 3.62 through Equation 3.65.
78
Since equations of state are developed onents, the use of mixing rules is
necessary to make pr ures. Another
important parameter in every geometric and adratic mixing rule is the interaction
arameter. This binary interaction term is empirical and does not have any theoretical
basis but it is necessary in mixing rules application. A mixing rule is an algebraic
expression that relates the pure components’ parameters to the mixture composition when
the mixture parameter is established. In this work, the binary interaction terms,
for pure comp
oper application of the equations of state for mixt
qu
p
,, ijija α
and ijβ were assumed as constants, and assigned a value of 1. Then, using the Equation
3.52 the constant was estimated. The Equations 3.59 through 3.61, for calculating the
crit
cA
ical properties are expressed in terms of the mixture parameters, ,,, mmm ba α and mβ
after LLS mixing rules (Equations 3.62 through 3.65) are applied. Table 4.4 presents the
results of a sample calculations of a hydrocarbon mixture performed using the algorithm
developed in section 3.3. The prediction results are in agreement with the experimental
values. The absolute percent deviation of the results is in the neighborhood of 1.
Table 4.5. Calculation Results for Mixture Parameters (Mixture 1).
Parameter Mixture 1
ma 31177.9123
mb 0.11728247
mα 99.1126855
mβ 0.69748539
cB 0.269
cZ 0.2985
cA 34.15
79
4.3 Results and Discussion
The variation in behavior of the calculated critical pressure and temperature
agreed
5-4.6
e
ee
tion of state is selected for this work.
ead
thermore, the influence of the values of
interact
ifica y the
prediction of the critical propertie l pressure.
for
ment
iation of the results ranges from
0.03 to 0.13 percent.
both qualitatively and quantitatively with experimental data. Deviation errors in
the predictions of both critical pressure and temperature, as shown in Figures 4.
were in the range of 0.03 to 0.13 percent. In Figure 4.1, the predicted critical pressure
obtained from Simon and Yarborough, Etter and Kay, Zais, and this work using Lawal-
Lake-Silberberg (LLS) is plotted against Critical experimental data. As shown in Figur
4.1, this work shows better agreement with the experimental data than the other thr
correlations. Consequently, the LLS equa
A comparison of the accuracy of predicted critical properties of this work with the
other correlations indicates that the mixing rules used with the LLS equation of state l
to accurate predictions of mixture behavior. Fur
ion parameters on the accuracy of the prediction of critical properties was
adjusted. Adjustment of the interaction parameter associated with the constant b of the
LLS equation of state, had an effect on the critical point calculations. Also, adjustments
of the interaction parameter associated with the constant a , improved sign ntl
s, especially the critica
The results of the predicted critical pressure, ,cP and critical temperature,
complex mixtures are presented in Table 4.6. The prediction results are in agree
with the experimental values. The absolute percent dev
,cT
80
Ta
Properties M 9
ble 4.6. Predicted Critical Pressure, ,cP Critical Temperature, ,cT for Mixtures.
ix. 1 Mix. 2 Mix. 3 Mix. 4 Mix.5 Mix. 6 Mix. 7 Mix. 8 Mix.Tc, exp. 660 660 725 725 725 725 694 660 660 Tc, pred. 725. 13 659.22 39 725.45 724.58 725.81 694.91 694.44 659.17 661.
AD (%) 1 0.118 0.054 0.062 0.058 0.112 0.131 5.218 0.126 0.17Pc, exp. 2100 2500 3400 1920 2420 3430 4355 4295 4630 Pc, pred. 21 .4901.12 2501.55 3398.03 1922.15 1423.18 3432.17 4349.55 4302.33 4624
AD (%) 0.05 9 3 0.062 0.058 0.112 0.131 0.063 0.125 0.171 0.11
hty-five mixture sets of experimental data are analyzed for hydrocarbon
Table C.1 through C. 27 in Appendix C show the prediction results for
critical temperatures. Also, these results are displayed in Figures C. 1-through
Eig
mixtures. critical
pressures,
C. 10 c re in
Appendix
pressures
for ritical pressure, and in Figures C. 11 through C. 19 for critical temperatu
C. Figure 4.1 is a cross-plot shows the prediction results for the critical
for all data.
81
7500
6500ia)
4500
5500
ssu
3500d C
1500
2500Cal
500
8500
500 1500 2500 3500 4500 5500 6500 7500 8500
cula
terit
ical
Pre
re (p
s
Simon-Yarborough
Etter-Kay
Zais
This w ork
Experimental Critical Pressure (psia)
ssure against the experimental critical
e n
sults for the critical temperature are in agreement with the experimental data. Figure
Figure 4.1. Predicted Critical Pressure of Complex Mixtures.
The match of the predicted critical pre
pr ssure gives an absolute deviation between 0.03 and 0.13 %. Similarly, the predictio
re
4.2 is a cross-plot which displays the prediction results for critical temperatures for all
data.
82
800
750
This w ork
350
450
700
Cal
cula
ted
Crit
ical
Tem
ptu
re ( 650o R
)
550
600era
500
400
300300 350 400 450 500 550 600 650 700 750 800
Experimental Critical Temperature (oR)
Figure 4.2. Predicted Critical Temperature of Complex Mixtures.
ental Data
4.4 Comparison Between Calculated and Experim
To test the accuracy of the calculations of critical points of hydrocarbon mixtures,
mparison of calculated with experimental values of the pressure and temperature at a co
e critical points of mixtures of known compositions was made. For this purpose, the
crit
ere determined in the laboratory was employed. A total of about 85 mixtures of non-
hyd nts were studied. The results of these mixtures
are given in Table 4.6. Table 4.6 gives a summary of the results for all mixtures.
th
ical pressure and critical temperature on the hydrocarbon paraffin mixtures which
w
rocarbon and hydrocarbon compone
83
In general, the agreement between the calculated critical properties and the
experimental critical properties is very good. The over all deviation of the calculated
val about 0.03-0.13% of both the pressure and temperature.
res
nd experimental critical pressures for all data points. Also, Figure 4.2 is a cross-plot
com res
The comparison of the results of the critical point predictions using the LLS
equ arborough, Etter-Kay, and Zais empirical calculation
x
res hip between
ccuracy of the predictions of the critical properties. The critical point equations derived
fro e than was the
redictions proved to be in good agreement with the experimental critical points than the
imon-Yarborough, Etter-Kay, and Zais.
omprehensive comparison of the critical point predictions of the LLS
an be made for the critical pressure calculations for each class of mixtures. In this work,
for LS equation predicted more
ues from the experimental is
Figure 4.1 is a cross-plot shows the comparison between the calculated critical pressu
a
pares the calculated critical temperatures with the experimental critical temperatu
for all data.
ation of state and the Simon-Y
methods was necessary to provide guidelines for critical point predictions of comple
ervoir fluids. The most important factor to be considered was the relations
the complexity of the equations of state used in the critical point equations and the
a
m the LLS equation of state proved to be more simple and easy to deriv
case of many other equations of state. However, the results of the critical point
p
S
The most c
equation of state approach and the Simon-Yarborough, Etter-Kay, and Zais correlations
c
the hydrocarbon/non-hydrocarbon class of mixture the L
accurate critical pressure- mole fraction relationship than did the Simon-Yarborough,
84
Etter-Kay, and Zais correlations in all the mixtures in the basic calculations (with
adju
hat the
inary
mixing rules of the Redlich-Kwong and Benedict-Webb-Rubin (BWR) equations of state.
ssumption that the intermolecular energy can be described by the geometric means of
the t that the geometric mean
ly
equ
arameters will not equal to 1, which agrees with the results of this investigation.
LLS equation
thro tical pressure predictions
ot comparable to that obtained by this work with the equation of state approach using
Law vantages of the equation of state
ritical temperature, and critical volume are not restricted to any particular equation of
stat
stment of binary interaction parameters).
One of the most significant features of the critical point calculations is t
values of the interaction parameters are not equal to 1 as suggested in the original b
The assumption of a value of 1 to the binary interaction terms is equivalent to the
a
pure component energies. Chueh and Prausnitz 21 point ou
relationship is accurate only for simple, spherically, and symmetric molecules of near
al size. Thus, for most multi-component systems, the best value of the interaction
p
A comparison of the accuracy of the critical point prediction using
of state approach and three other empirical correlation methods appears in Tables C. 1
ugh C. 9 in Appendix C. In general, the accuracy of cri
from the empirical calculation methods of Simon-Yarborough, Etter-Kay, and Zais was
n
al-Lake-Silberberg equation of state. The ad
approach are that the critical properties are determined simultaneously, including the
c
e.
85
CHAPTER V
CONCLUSIONS AND RECOMMENDATION
5.1 Conclusions
The purpose of this work was to integrate the thermodynamic criteria of the
omplex petroleum reservoir fluids. The major conclusions drawn from each of the
objectives are presented along with recommendations for future study in the following
jective, to develop a closed-form solution to the van der Waals
ies
as achieved by using the Lawal-Lake-Silberberg (LLS) equation of state on complex
mix lations showed that
e average error levels in the predictions of the critical properties were comparable to
tho not comparable to those obtained from other
rawn from this result was that the ability of the equation of state to predict the critical
poi e
cor ies of pure components of the mixtures. Thus, because of
accurate and exact critical temperature and pressure for any pure component, the LLS
critical state criteria of mixtures with a LLS EOS to predict the critical properties of
c
sections:
The first ob
criticality conditions and perform numerical calculations to predict the critical propert
w
tures of hydrocarbons. Analysis of the results of the critical calcu
qualitative and quantitative agreement with experimental data was obtained. In general,
th
se obtained experimentally and
correlations (Simon and Yarborough, Etter and Kay, and Zais) 122,29,98 . The conclusion
d
nts of complex mixtures is directly related to the ability of the equations to predict th
responding critical propert
its simplicity and the fact that the Lawal-Lake-Silberberg equation applicability yields
86
equ cal
oint predictions of petroleum reservoir fluids.
for
ydrocarbon and non-hydrocarbon mixtures and for hydrocarbon with pseudo-
com d
by ion parameters to be equal to one in the mixing rules
e
tep
racy in the critical properties prediction carried out
5.2 Recommendations
ation proved by this work to be much more satisfactory equation of state for criti
p
The second objective, to determine the need to establish interaction parameters
h
ponents was achieved. The best estimations of the critical properties were obtaine
adjusting the value of the interact
of the constants ma and mb of the LLS equation. The binary interaction parameters are
expressed in terms of the ratio of molecular weights in the Equation B.91 through
Equation B.93 presented in Appendix B.
The third objective, to develop an algorithm for calculating the critical properties
of r servoir fluids was also achieved, and the procedural approach for computing the
-by-step method was efficiently performed. s
The equation of state approach to the prediction of critical points of mixtures
offers several advantages over the empirical and semi-empirical correlations methods in
use today. The general level of accu
in this work proved to be comparable to the experimental measurements.
The results obtained from this investigation are useful for PVT analysis of
res rvoir fluids especially in resolving retrograde behavior. Therefore, several
mmendations can be made regarding future wo
e
eco rk: r
87
1. Study should continue toward the implementation of this technique developed
in this search in a flash routine to resolve the convergence pressure problem for
sufficiently reliable for predicting the vapor-liquid equilibrium and volumetric
he use of pseudodization. Instead, utilize heptane-
predicting liquid dropout, constant volume depletion (CVD), constant
3. A possible use of the technique developed in this work into a reservoir
as- condensate simulation, and
near the critical region and retrograde behavior of reservoir fluids.
2. Based on the modified LLS equation of state used here in this work is
reservoir fluids without t
plus fraction as a lumped single-pseudo-component in equation of state by
composition expansion (CCE), and flash-differential liberation (FL-DL) tests.
simulation model to perform gas cycling, g
multi-contact miscibility studies.
88
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APPENDIX A
ANALYTICAL SOLUTION FOR CUBIC EQUATIONS
The general c
(A.1)
ivide the entire equation by
ubic equation is given by Dunham, 24
00123
3 =+++ aXaXaXa 2
D 3a ,
03
0
3
12
3
23 =+++aa
XaaX
aaX (A.2)
To find the roots of this equation, we first eliminate the quadratic term, 2X . To do this,
we make the substitution
3
2
3aayX −= (A.2)
then, by substituting in Equation A.1, to obtain
0)3
()3
()3
( 03
21
2
3
22
3
3
23 =+−+−+− a
aaya
aaya
aaya (A.3)
Expanding Equation A.3 and simplifying
33
32
23
222
3
233
3
2
273)
3(
aa
yaa
yaa
yaa
y −⎟⎟⎠
⎞⎜⎜⎝
⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛−=− (A.4)
23
22
3
222
3
2
932
3 aa
yaa
yaa
y +⎟⎟⎠
⎞⎜⎜⎝
⎛−=⎟⎟
⎠
⎞⎜⎜⎝
⎛− (A.5)
Substituting in Equation A.3
99
03
932
273
03
21
23
2222
32
22223 ⎞⎛⎞⎛ aaaaa
323
32
333
=+⎟⎟⎠
⎞⎜⎜⎝
⎛−+
⎟⎟⎠
⎜⎜⎝
+−+⎟⎟⎠
⎜⎜⎝
−+−
aaaya
ay
aya
ay
ay
aya
(A.6)
or
0327
2 213
2 ⎟⎞aaa
3 32
301
223
3 =⎟⎠
⎜⎜⎝
⎛−++⎟
⎟⎠
⎞⎜⎜⎝
⎛+
−+
aaaya
aaya (A.7)
is
3
Equation A.7 is called the “depressed” cubic equation since the quadratic term, 2y
eliminated. Divide Equation A.7 by 3a , then
0327
213
1 223 ⎜
⎛−+
aay
33 ⎠a21
2
32
033
13
=⎟⎟⎞
⎜⎜⎝
⎛−++⎟
⎟⎠
⎞⎜⎝ a
aaaa
ay
aa (A.8)
Let
⎟⎟⎠
⎞⎜⎜⎝
⎛−=
3
22
13 3
1a
aaa
e
And
(A.9)
⎟⎟⎠
⎞⎜⎜⎝
−+= 22
03 27a
aa
f (A.10) ⎛
3
21
3
3
321
aaaa
Substituting Equations A.9 and A.10 into Equation A.8 and obtain
0 (A.11) 3 =++ feyy
Reducing Equation A.11 using Vieta’s Substitution
z
zy += s (A.12)
100
The constant s is an undefined constant for right now. Substituting Equation A.12 in
Equation A.11, yield
0=+⎟⎠
⎜⎝
++⎟⎠
+ fz
ez
z3
⎞⎛⎞⎜⎝⎛ szs (A.13)
xpand quation A.13 E ing E
3
323
3
323
3
33 zssszzsz −=−+−=⎟⎞+
33zs
zssz
zzz−+
⎠⎜⎝⎛ (A.14)
n t e right side of Equation A.14 by , yield
Multiplyi g h 3z
243
4322463 33 eszezz
szszszsszzzzsz +=+=−+−=⎟⎠⎞
⎜⎝⎛ +
Then
(A.15)
ow let
0)3()3( 33246 =++++++ sfzzesszesz
N 3es −= to simplify Equat
ion A.15 into a “tri-quadratic” equation, then
027
336 e
=++ fzz (A.16)
hich we can solve By substituting , then we have a general quadratic equation w3zw =
using the quadratic formula.
027
32 =−+
efww
S , hence using
r each of the two roots of hence will give
(A.17)
olve for the quadratic Equation A.17 and will give two roots for w
3zw = would then give three roots fo ,w
101
six root values for z . But the six root values of z would give only three values of y (for
zszy += ), and three values of x in Equation A.2.
lustrativ
4.203.0 23 +− xxx
(A.19)
Where
inate the quadratic term (i.e., depress the cubic
equation). Let
Il e Problem
Find the roots of the following cubic equation
00 6 =− (A.18) 1
Solution
For the general form of the cubic equation
023 =+++ dcxbxax
6104.2,0,03.0,1 −==−== xdcba
To find the roots of this equation, first elim
a
y3
−= bx
)1(303.0−
−= yx
(A.20)
ting the above value of
mplify, yield
(A.22)
01.0+= yx
Substitu x Equation A.20 into the cubic Equation A.18 and
si
0)4()103( 743 =+− −− xyxy (A.21)
Convert this depressed cubic equation into the form
03 =++ feyy
10
102
W e coefficients of Equation A.22 are 74 104,103 −− =−= xfxe here th
epressed equation by using Vieta’s substitution as,
Now solve the d
zszy +=
and obtain
(A.23)
Let
0)1033()104()1033( 32437446 =+−++−+ −−− szxsszxzxsz
44
103103
3−
−
=−
−=−=xes (A.24)
Substituting Equation A.24 into Equation A.24 to obtain the “tri-quadratic” equation
(A.25)
Convert Equation A.25 into a general quadratic equation by using
(A.26)
Using the quadratic equation, the values for are
0101)104( 12376 =++ −− xzxz
3zw =
0)101()104( 1272 =++ −− xwxw
w
)101(4)104(2104 1227
7
1−−
−
−±−= xxxw
) (A.27)
and
) (A.28)
The solution of gives three values of
771 10797958.9(102 −− +−= xixw
772 10797958.9(102 −− −−= xixw
3zw = z . These values in rectangular form are:
biaw +=
)sin(cos θθθ izzew i +==
103
Where πθ +=abarctan
Then,
⎥⎦
⎤⎢⎣
⎡⎟⎠⎞
⎜⎝⎛ ++⎟
⎠⎞
⎜⎝⎛ +=
32
3sin
32
3cos33 ninzw πθπθ
Then, the three values from 1w are in rectangular form
iz 00440.0008976.01 +−=
ix 009977.010707.6 42
−= z
iz 005569.0008305.03 +=
The values of z from w are in rectangular for2 m
Using V
iz 0044079.0008976.04 −−=
ixz 0099775.01070689228.6 45 −= −
i00556957.00083054.06 −= z
ieta’s substitution,
zszy +=
z
xzy4101 −
+= (A.29)
Substituting into Equation A.29 the value of z to find three values for y , choosin
0044079.0008976098.01 +−=z yields
g
01795.00044.0008976.0
101 4
1 +−
−
ix0044079.0008976.0 −=++−= iy
104
0013414.000997748.0706892.6
10100997748.0706892.64
2 =+
++=−
ixiy
Similarly, the other value of 3z gives
016611.03 = y
With the substitution in Equation A.20
The three roots of the given Equation A.19 are
Method 2
.30)
ide the entire equation by
(A.20) 01.0+= yx
0079522.01 −=x
0113414.02 =x
02661112.03 =x
The general cubic equation is given by
0012
23
3 =+++ aZaZaZ (Aa
Div 3a ,
03
01 a
3
2
3
23 =+++a
ZaaZ
aaZ (A.31)
Eliminate by making substitution of the form
2a
λ−= xZ (A.32)
By substituting Equation A.32 into Equation A.30,
23 =+−+−+− axaxax λλλ (A.33)
expanding Equation A.33,
0)()()( 012
By
105
0)()2 0122 =+−++− axax λλλ (A.34) ()33( 2
3223 +−+− xaxxx λλλ
(A.35)
Let
0)()32()3( 32210
221
22
3 =−+−++−+−+ λλλλλλ aaaxaaxax
3 in order to eliminate the , so
2a=λ 2x
231 axZ −≡ (A.36)
273
)3
( 22
3323 xaxxx −=−=3
22
2 axaa−+ (A.37)
32222 121222222 )( axaxaaxaZa +−=−= (A.38)
933
1212 31)
31 aaxaa −= (A.39) 11 (xaZa −=
Substituting back into Equation A.30, becomes
0)
39 0212
112733 (A.40)
1()21()(
3
321
22
22
222
3
=−+−
−+−++−+
aaaa
axaaaxaax
0)21()1( 03
2212
213 =−−−−+ aaaaxaax (A.41)
2733
027
22793
3 32021
2213 =
−−−
−+
aaaaaax (A.42)
et, L
3
3 221 aaP −
= (A.43)
and,
106
27
2279 32021 aaaa −−
q = (A.44)
then Equation A.42 can be written as
(A.45)
qPxx =+3
Make Vieta’s substitution to simplify the derivation by letting:
W3PWx −= (A.46)
Substituting for in Equation A.45, obtain
x
0)33 WW
or
()( 3 =−−+− qPWPPW
03
3 =−− q27 3W
P W (A.47)
lying through EquMultip ation A.47 by to obtain a quadratic equation in , 3W 3W
027
)()( 23 − WqW3
3 =−P (A.48)
Apply the quadratic formula (Birkhoff & Mclane 1996, P.106):
323
274(
21 PqqW +±=
32
271
41
21 Pqq +±=
32 QRR +±= (A.49)
Let:
107
27
41
21 qR =
33
22
PQ
qR
=
=
ieta’s “magic” substitution, first define the intermediate variables from Equation
.42
By V
A
Let:
9
3 21 aa −=
3
Q (A.50)
54
2279 32012 aaaa
R−−
= (
The cu
(A.52)
identity, which is sa sfied by perfect cubic
olynomial equations, is
) (A.53)
ince Q
33 +− xCBx
Regrouping terms, becomes
A.51)
bic Equation A.42 then becomes,
0233 =−+ RQxx
Let B and C be arbitrary constants. An ti
p
)(( 233 BBxxBxBx ++−=− 2
S 0≠ , add a multiple of (x-B) by C to both sides of Equation A.23 to give
0))(() 22 =+++−=− CBBxxBxB (A.54)
()(
( ) 0)()()( 2233 +x =+++−=+− CBBxxBxBCBCx (A.55)
Match th we must have e coefficients C and )( 3 BCB +− with those of Equation A.52, so
QC 3= (A.56)
108
( .57)
8)
nd a value for B and reduce Equation A.58 to a quadratic equation.
RBCB 23 =+ A
Then by substituting Equation A.56 into Equation A.57,
RQBB 233 =+ (A.5
Now, fi
( ) ( )312331
2 RQRRQRB +−+++= (A.59)
Taking the second and third powers of B gives
3
( ) ( ) ( )322331
23232
232 )(2 RQRRQRRQR +−++−+++= B (A.60)
( ) ( ) QRQRRQR 232
2332
23 −+−+++= (A.61)
( ) ( )
( ) ( )⎪⎭
⎪⎬⎫
⎪⎩
⎪⎨⎧
+−+++×
⎪⎭ (A.62)
⎪⎬⎫
⎪⎩
⎪⎨⎧
+−++++−=
31
2331
23
31
2331
233 2
RQRRQR
RQRRQRQBB
( ) ( ) ( ) ( )( ) ( ) QBR 222 −QRRQR
RQRRQRRQRR1
332
23
3233
123232
+−+++
+−++++−+ (A.63)
QR 3 ++=
2
( ) ( ) ( ) ⎟⎟⎠
⎞⎜⎝
−−++++−++−= 31
2331
23)(22 RQRRQRRQRRQB (A.64) ⎜⎛
31
232
RQBQBRQB 2322 +−=−+−= (A.65)
Plugging and B into the left side of Equation A.58 gives:
3B
RQBRQB 23)23( =++− (A.66)
109
Now, plugging C 3= Q into the quadratic part of Equation A.55 gives:
ch provides the solution.
0)3( 22 =+++ QBBxx (A.67)
whi
( ))3(421 22 QBBBx +−±−= (A 8) .6
QBB 12311 2 −−±−= (A.69) 22
QBiB 4321 2 −±−= (A.70)
These c
an be simplified by defining
( ) ( )312331
23 RQRRQRA +−−++≡ (A.71)
( ) ( ) ( )32332323232 )(2 RQRRQRRQRA +−++−−++= (A.72) 212
( ) ( ) QRQRRQR 232
23 ++−+++= (A.73)
(A.74)
tions to the quadratic por
32
23
QBA 422 +=
So the solu tion of Equation A.55 can be written as
AiB 322
±−= x 11 (A.75)
efining:
D
23 RQ +≡ (A.76) D
3 DRS +≡ (A.77)
3 DRT −≡ (A.78)
110
where
(A.79)
(A.80)
Therefore, at least, the roots of the original equation A.37 are given by:
TSB +=
TSA −=
)(3 211 TSaZ ++−= (A.81)
)(321)(
21
31
22 a−= TSiTSZ −++− (A.82)
)(321)(
21
31
23 TSiTSaZ −−+−−= (A.83)
With as the coefficient of2a 2Z in Equation A.1 and S and T as defined above; these
the cubic equation are sometimes known as
o’s formula. If the equation is in the standard form of Vieta,
(A.84)
three equations providing the three roots of
Candam
qPxx =+3
111
APPENDIX B
VAN DER WAALS EXPRESSIONS FOR FLUID CRITICAL POINT
The van der Waals equation of state (VDW-EOS), proposed in 1873, was the first
equatio
ation with pressure given by a cubic function of molar volume in the form
n to represent vapor-liquid coexistence. The VDR-EOS is a two-parameter
equ
2
)(TaT−=
VbVRP−
(B .1)
The first term on the right hand side is the repulsive term and the second term is
the attractive term is temperature dependent.
Multiplying both sides of Equation B.1 by RTV to obtain the VDW-EOS in Z
form
RTVa
bVV
−−
= Z (B.2)
n, where by definitio
RTPVZ = (B.3)
and
P
ZRT=V (B.4)
pressibility factor, T is tem olume, P is
pressure, and R is the molar universal gas constant. The parameter is a measure of the
Where Z is the com perature, V is v
a
112
attractive forces between molecules, and the parameter is the co-volume occupied by
the molecules (if the molecules are represented by hard-spheres of diameter d, then
b
3)2( 3σπNb = .
Substituting Equation B.4 into Equation B.2 leaves
22TZRaPZRTZ −=
PbZRT − (B.5)
r o
221
1TR
ap
ZRTPb
Z −−
= (B.6)
with
22TRaPA (B.7)
nd
=
a
RTbP
= B (B.8)
Substituting Equation B.7 and Equation B.8 into Equation B.6 leaves
ZA
ZB
Z −−
=1
1
or
(B.9)
ZBZ −AZZ −= (B.10)
or
113
ZBZ
BZAZZ)(
)(2
−−−
=
(B.12)
or
(B.11)
and )()( 2 BZAZZBZZ −−=−
ABAZZBZZ +−=− 223 (B.13)
or
(B.14)
then
(B.15)
Equation B.1 was expanded to Equation B.15 form, which is a cubic equation.
When specialized to the critical state, has three equal roots, that is, that it be of the form
(B.16)
.16), gives
( 3233 =− cc ZZZ (B.17)
g the coefficients of Equation B.15 to the coefficients of Equation B.17, leaves
0223 =−+−− ABAZZBZZ
0)1( 23 =−++− ABAZZBZ
0)( 3 =− cZZ
Expanding Equation (A
3) +−=− cc ZZZZZ 03
Equatin
cc BZ += 13 (B.18)
(B.19)
(B.20)
There are three equations B.18, B.19
Zc. To find these unknowns, substituting Ac of Equation B.19 into Equation B.20, obtain
cc AZ =23
and
ccc BAZ =3
, and B.20 with three unknowns Bc, Ac, and
114
ccc BZZ 23 3= (B.21)
en
th
33 2cZ
3cc
cZZB == (B.22)
Substituting Equation (B.22) into Equation (B.18), leaves
3
13 cc
ZZ +=
then
(B.23)
and
(B.24) cc ZZ += 39
83
=cZ (B.25)
Substituting the value of Zc in Equation B.25 into Equation B.19, obtain
642793 ==
64
Substi
cA (B.26)
tuting Equation B.25 into Equation B.18, gives
81
=cB (B.27)
Now, consider the generalized form of the Lawal-Lake-Silberberg (LLS) cubic
equation of state. That is,
22 bbVVbV βα −+−aRTP −= (B.28)
(B.29)
and the gas law is given,
ZRTPV =
115
where, by definition
RTPVZ = (B.30)
nd a
P
ZRTV =
th sides of Equation B.28 by the value
(B.31)
Multiplying boRTV
RTV
bbVVa
RTV
bVRTP
RTV
⋅−+
−⋅−
=⋅ 22 βα (B.32)
then
)( 22 bbVVRT
aVbV
VZβα −+
−−
=
Substituting Equation B.31 into Equation B.33, yields
(B.33)
⎥⎦⎤
⎢⎣⎡ −+
⋅ZRaZRT
−=22 )()( b
PZRTb
PZRT
PT
PZβα
(B.34)
− RTbP
ZRT
or
22 BBZZAZ
BZZZ
βα −+−
−= (B.35)
lying both sides of Equation B.35 by MultipZ1 , gives
22
1−=1
BBZZA
BZ βα −+− (B.36)
or
116
0)()())(( 2222 =−+−+−−+− BZABBZZBBZZBZ βαβα
Consider that at the critical point, the coefficients of the expanded form of the
cubic equation of state Equation B.37 can be compared to the coefficients of the
expansion shown in Equation B.38.
(B.38)
The coefficients of Equation B.37 are:
(B.39)
(B.40)
(B.41)
ubstituting these coef icients B.39, B .42 back in Equation B.37, leaves
(B.37)
0)( 3 =− cZZ
033 3223 =−+− ZZZZZZ cc
1:3Z
1:2 −− BBZ α
ABBBZ +−− ααβ 221 :
and
ABBBZ −+ 230 : ββ (B.42)
S f .40, B.41, and B
0)()()1( 232223 =−++−−−+−−+ ABBBZBBBAZBBZ ββααβα
(B.43)
Equate the coefficients of Equation B.43 to the coefficients of Equation B.38,
obtain
13 −−=− ccc BBZ α
ccc BBZ α−+= 13 (B.44)
BB αα −22 (B.45)
(B.46)
ccc BAZ β −−= 23 cc
ccccc BABBZ −+=− 233 ββ
117
where the subscript c represents the conditions at the critical state. Solve for Zc, Bc, and
c. From Equation B.44,
A
31−−
=− ccc
BBZ
α (B.47)
or
31 cc
cBB
Z+−
=α
(B.48)
uation B.45
(B.49)
ubstituting Equation B.48 into Equation B.46, gives
From Eq
ccccc BBBZA ααβ +++= 2223
S
( ) ccccccccc BBBBZBB
BBααβββ
α+++−+=⎟
⎠⎞
⎜⎝⎛ +−
− 222233
33
1 (B.50)
or
( ) ( ) ( )[ ] cccccccccc BBBBBBBBBB ααβαββα ++++−−+=+−− 222233 271927271 (B.51)
and
(B.52)
or
0272727
91818189927271
3336333
233
22332223
2222333233
=+++
++−−++−−−
−+−+−−+−
ccc
cccccccc
ccccccccc
BBB
BBBBBBBB
BBBBBBBBB
ααβ
αααββ
αααααα
118
( )( ) 0163)27181827 2 =−++++−− αααβ cB3
63(27927133 2323
−
+−+++−−+− αααββααα cB
(B.53)
271892 +−+ αα
and
( ) ( ) ( )6 cB 01=3+ α15 cB27 +β15 −α3 2α8126 223 −++−+++ αα
( .54)
3 +cBα
B
or
( ) ( ) ( ) 012359538126 22323 =++++ BBααα B
an be further simplif t l s m p
ng of the variables:
+++ θθθθ cc BBB .56)
here
++= αααθ (B
)2 αα − .58)
)
−cB+α+c+β−α−α+c ( .55)
Which c ied to he fol owing expres ion of the ter s and appro riate
groupi
0432
23
1 =c (B
w
)8+126( 231 .57)
3271515( 2βθ += − (B
36(3 αθ += (B
.60)
Critical it y ritical C s ty
.59)
14 −=θ
(B
Cond ion b the C ompre sibili Form
From equation B.44,
,1)1(3 −−=− αcc BZ (B.43)
119
the following expressions can be deduced:
)1()31(
−−
=α
cc
ZB
α−−
=1
13 cc
ZB
ting of Equati 1 E
(B.61)
Substitu on B.6 into quation B.55, yield
01 =))−cZ
)1(31(
)(6
))31(
()
(126 23
−+
+++
αα
ααZ
3
3)27153)1)(12
=−
−+
α
αα
c
c
Z
Z(B
Let
31)(1527153(
6
−=
−−+=
−−+−+−=
+
α
αα
αβαα
α
D
ZC
ZB
c
c (B.64)
And,
3(
)1−αc
()(1527 +β15 −α3− α) +
1( −α) 3c31− Z
+−
2)(8+α(
−
(B.62)
or
6( 23 + αα (3 −+ 2 +α ()2c3− Z −α 15+β8+ 1( − )1 +
.630)1 3−()1 2 −1)(63( −+α )(α
)
3
2
22
323
)1(
)1)(31)(63(
)1(
(= αA )3− Zc1)(8+12+ α
)
0)()()()( =−++ DCBA (B.65)
Expanding A, B, C, and D, then
120
812α672108549
216324+162 + 22627
)31)(86(
233
223233
33
++−−
−−−−
−++=
ααααα
αα
αα
cccc
cccc
c
ZZZ
ZZZ
ZA
62108
527(
13(
2
23
2
−−−−+−++−−−
−=
+−=
ββααβαβα
βα
α
cccc
cc
ZZZZ
Z
B
1)(63(3 +−
+=
α
α
cZ
C
)1( 3 −=−α
g A, , a in ua 65, ob ain
891283
1621818135243243
)6789
32427216162
23
23222222
222333
−+−+
+−−
−−+−−+
+
+++−
αα
ααβ
αβαβαβα
α
αα
ccc
cccccc
ccc
ZZ
ZZZZZZ
Z
ZZZZ
or
(B.67)
or
(B.68)
3cZ 3 +c
2 +α cZ324Z 7 3α Z 2α cZ(= 162 21−
Z2 − + +
)1527271518390901621
18α Z13−c243βZ243α162+ α
)1−α()2Z31)( −157 +β25 −α2
c23 2 2−cZ +cZ −cZ 3
Z 3 2α α αc
2
)69 +− α3 3+ α18− Z27α cZ9(
)1)(3− cZ −α
= c
=D ( 3α )133 2 +α −α
substitutin B, C nd D Eq tion B. t
0)1 =3+ α3− α() −693 3α
1−27()5 +7 −β272 −α12 +α33 +α90162 +− βZ
16227( 3− α
8+α122 +α2 cZ10−c54 2α216 2 −+ cZ
162324−27( −− α cZ 3 3 22 α cZ cZα αc
3 −cZ −α cZ 3 +α +
Z Zα−c c
−Z 3α Zc c
(B.66)
0)272727()16216281162(
)81243243324324()21632416227(22
22323
=−−+−−−−−
+−++−−−−−
βαβαβαβαα
αββααααα
c
cc
Z
ZZ
0)()6636(
)3991212()8126(22
22323
=−+−+−++
+−++−+++
αβαββαβαα
αββααααα
c
cc
Z
ZZ
121
Let
663(12123(
6128(
24
2
22
21
αβαβθ
ββαα
αθ
αα
−+−=
++
++−=
+++=
(B. 9)
e critical condition by t e criti al com ressibility form
(
tion No.2
rom E uatio B.44
)3αθ
)6α−
)
9β−9α +
3θ =
αβ 6
)(
Then th h c p is
0432
23
1 =+++ θθθθ ccc ZZZ B.70)
Deriva
F q n
)1()31(
−−
=B α
cc
Z (B
B+ 23 β (B
c BBBZA ααβ += 223 , obtain
.44)
From Equation B.46
BZ =− 3 β ccccc BA− .46)
Substituting for cc2+ +c c
[ ]cBαcBαcBβcZ +cB−cc BBZ ββ +++− 23 3 (B.71)
Substituting Equation B.61 into Equation B.68, then
c =3 2 2 2
⎥⎥⎦
⎤
⎢⎢⎣
⎡⎟⎟⎠
⎞⎜⎜⎝
⎛−
−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−+
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−=−
)1()31(
)1()31(
)1()31(
3
)1()31(
)1()31(
)1()31(
222
233
αα
αα
αβ
ααβ
αβ
cccc
cccc
ZZZZ
ZZZZ
(B.72)
Multiply both sides of Equation B.68 by , obtain 3)1( −α
122
33
)1()1(
)31()131()1(3
+−
−−−=−−
βα
βα
c
cccc ZZZ (B.73)
)(1()1(3
1)1()1(3222
33
−+−
−−−−
βα
βα
c
cc
Z
Z(B.74)
91)31 ccc ZZZ −=− (
1−=− ααα (
1 .77)
cc ZZ−= .74)
tions, B.75, , B.77 into Equation B.74, obtain
3
)961
3
)31()1
1)227933
2
2
2
32
=⎥⎢
⎡
−+
+−+
+
−+−
−−−+−
αα
α
βαα
α
αα cc
ZZ
ZZ
Z
Z
B
or
(B.76)
where,
(B.77)
[3Z
23 )(1() −−+ αβ Z3
]222 2 1(+α 2 1(+α)3− cZ 3− cZ 3− cZ )1)( −α
or
[ ](3 − β ()2 −α3− Z 3− Z )3Z1()1 + c
03 =)1−α3− cZ1(2 +α)3− cZ1(+α)3 cZc
where, ( 3 2 27−3 27Z+ c B.75)
3)1( 233 3 −+ α B.76)
2)1( 22 +−=− ααα (B
2)31( cZ− 2961 + (B
substituting Equa B.76
0
13+ α33 −)(Z1( −α
(
)⎥c9+c61( −)1 +2−(cZ
)(9 2+ cZ
6Z(− β7Z1( −β)1−( 33− αZ2 ⎤2
2Z+ c
⎢⎢
⎢
cc
( . 75)cc c
⎥⎥
⎣ ⎦c
0
)31)(12)(31(
)31)(961()31(
)961()31)(12(3
)1)(9
61()272791()133(
2
2
222
2
32233
=
⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢
⎣
⎡
−+−−+
−+−+−
+−+−+−
+−+
−−−+−−−+−−
cc
cccc
cccc
c
ccccc
ZZ
ZZZZ
ZZZZ
Z
ZZZZZ
ααα
α
βαα
α
ββααα
222 96196)961)(1( cccccc ZZZZZZ −+−+−=+−− αααα
123
Z 36)12)(31( 22 −=+−− αααα (B.78)
27712)(91)(31 cccccc ZZZZZZ +−=++−− αα (B.79)
22
189
)(961(
cc
cc
ZZ
ZZ
−+
++−
αα
α
ting Equations B.77 , a 8 E o n
962
)27
91(
)3312(
)961
()27279()33
2
22
22
2
32233
=
⎥⎤
⎢⎡
−
++
+
+−−
+−+−
−−−−−−+−
c
cc
cccc
cc
ccc
ZZZ
Z
ZZZ
ZZZZ
ZZ
ZZZZ
αα
α
ααα
α
ααββαα
(B
123(
2273(
2 =−+−
++−
+
αββα
ααα c
Then,
(B.83)
Let,
cccc ZZZ312 2 +−+− αα
3222 29)16( −−
and,
2 Z +α2 2− α 12 Zα61−+)1 = α2− α c c (B.80) 222 29 cZ+
Substitu , B.78 B.79, nd B. 0 into quati n B.76, obtai
611(
0
)2⎦c92 +18α1⎢
⎣+
61−2− α(αα27⎢+
1( −α)3c272 −27+9− Z
63
9+
2cZ
32
2
2
⎥⎥⎥⎥c
⎢⎢+ β
+c−cZ
−c
−+
+c c Z
α
α−
cZ
cZ
.81)
and,
3
0)(
)αβ cZ66 −β6 2 +α3( +α) 2 +α cZ122 +α9 + βαβ
)7α Z9 ++18−9+β3− α 23 +α 2 −
(B.82)
0)()6636(
)9912123()86(22
2233
=−+−−+++
−+++−++
αββααββαα
αββαααα
c
cc
Z
ZZ
)6128( 321 αααθ +++=
124
3θ
)(
)β6α−62 βα +63(
912123(
24
22
αββαθ
α
βααθ
−+−=
+=
+++−=
(
+θθ cc ZZ (
Critica y r C
)9αβ−
B.84)
Then,
23
12
3+θ Z 0= 4+θc B.85)
l Condition b the C itical ompressibility Form For Mixtures
The generaliz a f f t f in ed cubic equ tion o state or mix ures is in the ollow g form:
22mmmm
m
m bVVa
bVP
α −+−
−= (B. )
der Waals mix ul e tur ram s m
bRT
β 86
By van ing r es, th mix e pa eter mmm ba βα ,,, take the
following forms:
∑∑=n a
ai j
m ijj 2iji axx 21
(B
xx (B. 8)
a1
a .87)
∑=i
mb (n
ji3) 8
∑∑=i j
m xαn n
x ααα 21
21
ijjiji (B.89)
∑∑=n
i
n
jijjijim xx ββββ 2
121
(B.90)
and the binary interaction parameters established in terms of Mw ratios of components
form
125
j i MW≤n
j
i MWMWMW
a
⎟⎟⎞
⎜⎜⎝
⎛= (B ija
⎠ .91)
j i MWn
j
iij MW
MWMW
⎟⎟⎠
⎞⎜⎜⎝
⎛ α
( ≤=α B.92)
j i
n
j
iij MW
MWMW
⎟⎟⎠
⎞⎜⎜⎝
⎛=
β
β (B
and are the exponents of the re a
xpand ng Eq ation .86 into Z term and b multiplying both sides of e equ tions by
MW≤ .93)
where ,, αnna an spective inter ction terms.
E i u B s y th a
RTV ,
⎟⎠
2 ⎟⎞
⎜⎜⎝
⎛
+−⎟⎟
⎠
⎞⎜⎜⎝
⎛− m
m
m ba
bVRT
− mβ=
RTRT 2V mm VbαRTVVVP (B.94)
⎟⎟⎞
⎠m⎜⎜
+−⎟⎟
⎞− 2
m
m
bVRTV
b ββα (B. 5)
by definitio
⎛
⎠⎝ m⎜⎜= V
Z⎛ V a
⎝V − 2mm
9
n,
PZRTV = ,
then,
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
−⎟⎠⎞
⎜⎝⎛+⎟⎟
⎠
⎞⎜⎜⎝
⎛⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛⎟⎠⎞
⎜⎝⎛
−⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
−⎟⎠⎞
⎜⎝⎛
=2
2
mmmm
m
m bP
ZRTp
ZRT
aRTP
ZRT
bP
ZRTP
ZRT
Z
ββα
(B.96)
Multiplying by RTP and by 22
2
TRp , obtain,
126
⎟⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜⎜
⎝
⎛
⎟⎟⎠
⎞⎜⎜⎝
⎛−⎟
⎠⎞
⎜⎝⎛+
⎟⎟⎠
⎞⎜⎜⎝
⎛
⎟⎠⎞
⎜⎝⎛−
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
⎟⎠⎞
⎜⎝⎛−
=
22
222
22
2
TRPb
RTZPZ
TRPa
RTZRT
RTPbZ
ZZ
mmmm
m
m ββα (B.97)
since ⎟⎟⎠
⎞⎜⎜⎝
⎛= 22
2
TRPa
A m and ⎟⎠⎞
⎜⎝⎛=
RTPb
B m , then
⎟⎟⎠
⎞⎜⎜⎝
⎛−+
−⎟⎠⎞
⎜⎝⎛
−= 22 BZBZ
ZABZ
ZZmm βα
(B.98)
or
(B.99)
(B.100)
(B.101)
(B.102)
At the critical point, the generalized cubic equation of state in terms of Zc takes the form:
(B.103)
Now, com sion
=0 shown in Equation B.100
(B.104)
)()())(( 2222 BZZABZBZZBZBZBZZ mmmm −−−+=−+− βαβα
0
22322223
=−++−−+−−−+
ABAZBZBZBZBBZZBBZZ mmmmmm βαβαβα
0)()1( 232223 =−++−+++−−+ ABBBABBBZBBZZ mmmmmm ββααβα
0()()1( 322223 =−−+−−−+−++ mmmmmm BBABBBBAZBBZZ βββααα
0(
)()1(32
2223
=−−+
−−−+−++
mcmccc
cmccmcmcccmcc
BBBA
ZBBBAZBBZ
ββ
βααα
paring the coefficients of Equation B.103 with the coefficients of the expan
( 3)cZZ −
0333 32223 =−++− cccc ZZZZZZZ
127
where,
cmcc BBZ α−+=13 (B.105)
(B.106)
and,
(B.107)
From Equation B.106 and Equation B.107, Bc and Ac can be solved to give,
mccmcmcc BBBAZ βαα 2223 −−−=
mcmcccc BBBAZ ββ 323 −−=
m
cc
ZB
α−−
=1
13 (B.105)
(B.106)
By substitution of Ac and Bc with Equation B.98 and simplifying the terms, the analytical
function of is obtained.
2223 cmcmcc BBZA βα ++=
∫= ),( mmcZ βα
32
2223
113
113
113
113
113
113
3
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−−⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−−
⎟⎟⎠
⎞⎜⎜⎝
⎛−
−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛⎟⎟⎠
⎞⎜⎜⎝
⎛−
−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−+⎟⎟
⎠
⎞⎜⎜⎝
⎛−
−+=
m
cm
m
cm
m
c
m
cm
m
cm
m
cmcc
ZZ
ZZZZZZ
αβ
αβ
ααβ
αα
αα
(B.107)
and can be further simplified to the following expression by expanding the terms by
grouping,
(B.108)
where:
0432
23
1 =+++ θθθθ ccc ZZZ
128
(B.109)
)(
)6663(
)9912123(
)6128(
24
23
22
321
mmmm
mmmmm
mmmmm
mmm
αβαβθ
βαβααθ
αββααθ
αααθ
−+−=
−++=
−+++−=
+++=
129
APPENDIX C
PREDICTION RESULTS OF CRITICAL PRESSURE, CRITICAL TEMPERATURE, AND HEPTANE PLUS PROPERTIES
Table C. 1. Critical Pressure Predictions for Complex Mixtures by Four Methods (Compositions in Mole Fractions) Mixture No. 145-1 145-2 145-3 145-4 145-5 145-6 145-7 145-8 145-9 Nitrogen 0.001 0.001 0.001 0.001 0.004 0.001 Carbon Dioxide 0.004 0.005 0.005 0.004 0.006 0.002 0.0149 0.014 0.0156Methane 0.193 0.271 0.363 0.229 0.365 0.482 0.5248 0.5335 0.5921Ethane 0.032 0.034 0.038 0.029 0.058 0.065 0.145 0.0679 0.066 Propane 0.585 0.482 0.374 0.407 0.242 0.167 0.157 0.2035 0.1043I-Butane 0.007 0.007 0.009 0.006 0.007 0.007 0.0055 0.0063 0.0058n-Butane 0.012 0.015 0.015 0.172 0.215 0.14 0.0179 0.204 0.0831I-Pentane 0.005 0.006 0.006 0.005 0.004 0.006 0.0055 0.0062 0.0054n-Pentane 0.007 0.007 0.008 0.006 0.007 0.008 0.0079 0.0092 0.0078Hexane 0.013 0.015 0.016 0.012 0.01 0.012 0.0082 0.0094 0.0081Heptanes + 0.141 0.157 0.165 0.129 0.082 0.11 0.1133 0.1296 0.1118 Avg. Mol. Wt. 67.3 68.5 67.5 66.1 48.8 48.8 45 48.4 44.9 Heptanes + Properties Mol. Wt. 243 243 243 243 191 191 191 191 191 Gravity(API) Characterization Factor 11.6 11.6 11.6 11.6 11.9 11.9 11.9 11.9 11.9
Critical Temp., Tc (F) 265 265 265 265 234 200 200 200 200 Critical Temp., Tc (o R) 725 725 725 725 694 660 660 660 660
Critical Pc (psia) 2100 2500 3400 1920 2420 3430 4355 4295 4630
Predicted Pc (psia) : Simon and Yarborough 2002 2550 3280 2113 2452 3490 3984 4261 4691
Etter and Kay 2911 3272 3658 2806 2509 3268 3909 3886 4040 Zais 2175 2613 3203 2298 2418 3480 4073 3982 4357
This work 2101.12 2501.55 3398.03 1922.15 2423.18 3432.17 4349.55 4302.33 4624.49
130
Table C. 2. (Continued)
Mixture No. 145 - 10 145 - 11 145 - 12 145 - 13 145 - 14 4 - 1 4 - 2 4 - 3 4 - 4 4 - 5 Nitrogen 0.0016 0.002 Carbon Dioxide 0.0153 0.0008 0.0139 0.0006 0.0008 0.0066 0.0067 0.0061 0.0061 0.0059Methane 0.5774 0.6298 0.6165 0.4786 0.5808 0.7243 0.7292 0.8227 0.8165 0.795 Ethane 0.0631 0.0858 0.0602 0.082 0.0562 0.0557 0.0556 0.0284 0.0284 0.0281Propane 0.1178 0.0672 0.0961 0.1359 0.0919 0.0308 0.0306 0.0124 0.0124 0.0128I-Butane 0.0055 0.023 0.0055 0.0076 0.0081 n-Butane 0.0976 0.0305 0.0773 0.1273 0.082 0.0241 0.0226 0.0091 0.0092 0.0098I-Pentane 0.005 0.0114 0.0053 0.0066 0.0071 n-Pentane 0.0073 0.0122 0.0077 0.0069 0.0073 0.015 0.0147 0.0067 0.007 0.0078Hexane 0.0075 0.0206 0.008 0.0139 0.0148 0.0179 0.0183 0.0125 0.0131 0.015 Heptanes + 0.1035 0.1187 0.1095 0.139 0.1488 0.1256 0.1223 0.1021 0.1073 0.1247 Avg. Mol. Wt. 44.3 44.4 43.4 54.7 53.1 39.9 39.3 33 33.8 36.5 Heptanes + Properties Mol. Wt. 191 167 191 205 205 167 167 158 158 158 Gravity(API) Characterization Fact. 11.9 11.7 11.9 11.6 11.6 11.8 11.8 11.9 11.9 11.9
Critical Temp., Tc (F) 200 292 200 180 180 285 251 100 160 212
Critical Temp., Tc (oR) 660 752 660 640 640 745 711 560 620 672
Critical Press., Pc (psia) 4364 4850 4745 3500 4800 5130 5350 6000 5820 5620
Predicted Pc: Simon and Yarborough 4413 4789 4930 3568 4735 5388 5412 5696 5695 5685
Etter and Kay 3865 4198 4169 3598 4209 4660 4679 5102 5094 5040 Zais 4150 4870 4592 3662 4488 5070 5083 5027 5980 5947
This work 4355.37 4848.37 4735.61 3495.07 4804.54 5134.45 5341.33 5988.62 5821.98 5608.11
131
Table C. 3. (Continued)
Mixture No. 4 - 6 4 - 7 4 - 8 4 - 9 4-10 75 - 1 75 - 2 75 - 3 75 - 4 75 - 5 Nitrogen 0.0128 0.0058 0.0053 0.0054 0.0038Carbon Dioxide 0.0035 0.0156 0.0008 0.0075 0.0074 0.0045Methane 0.6433 0.6865 0.657 0.7164 0.8213 0.788 0.724 0.728 0.597 0.83 Ethane 0.0638 0.0603 0.0869 0.0548 0.0637 0.059 0.0542 0.0546 0.089 0.0378Propane 0.0605 0.0232 0.0537 0.031 0.0409 0.0315 0.03 0.0302 0.05 0.0144I-Butane n-Butane 0.0565 0.0309 0.0303 0.0258 0.0235 0.0265 0.031 0.0207 0.049 0.0089I-Pentane n-Pentane 0.0404 0.0243 0.0195 0.0216 0.0122 0.0425 0.071 0.0688 0.093 0.0436Hexane 0.0366 0.0256 0.0173 0.0198 0.0103 0.0252 0.0456 0.0438 0.0308Heptanes + 0.0954 0.1245 0.1217 0.1231 0.0234 0.0214 0.0388 0.0375 0.122 0.0263 Avg. Mol. Wt. 35.2 41.5 45.7 40.1 25 29.5 29.1 36.2 24.5 Heptanes + Properties Mol. Wt. 114 171 207 167 114 106 106 106 100 106 Gravity (API) Characterization Fact. 11.7 12 12 11.8
Critical Temp., Tc (F) 195 239 145 243 55 109 109 169 54
Critical Temp., Tc (oR) 655 699 605 703 515 569 569 629 514
Critical Press., Pc (psia) 2720 5100 5570 5150 2270 2387 2574 2537 2515 2580
Predicted Pc (psia) Simon and Yarborough 2371 5557 6687 5375 2456 2467 2465 2821
Etter and Kay 3013 4490 4865 4560 2574 2725 2722 2020 2812 Zais 2641 5325 5623 4851 2272 2277 2441 2467 2356 2443
This work 2723.07 5096.27 5582.6 5147.51 2386.85 2572.96 2537.73 2514.61 2581.33
132
Table C. 4. (Continued)
Mixture No. 75 - 6 75 - 7 75 - 8 141 - 1 141 - 2 141 - 3 141 - 4 141 - 5 141 - 6
Nitrogen 0.0038 0.0036 0.003 0.1101 0.0477 0.0156 0.0123 Carbon Dioxide 0.0044 0.0043 0.0035 0.0008 0.0019 0.0047 0.0028 0.0172 0.011 Methane 0.815 0.784 0.643 0.3465 0.4675 0.4656 0.6173 0.6527 0.586 Ethane 0.0372 0.0355 0.0294 0.1561 0.1306 0.1645 0.1024 0.0725 0.1309Propane 0.0141 0.0136 0.0111 0.1499 0.105 0.1163 0.0673 0.0391 0.0699I-Butane 0.0121 0.0112 0.0141 0.0085 0.0157 0.0081n-Butane 0.0102 0.013 0.0252 0.0852 0.0472 0.0454 0.0352 0.0204 0.0349I-Pentane 0.0084 0.0169 0.0112 0.0106 0.016 0.0097n-Pentane 0.0501 0.0631 0.123 0.0109 0.0192 0.0182 0.0171 0.0123 0.0187Hexane 0.0354 0.0447 0.0871 0.02 0.0373 0.0215 0.0226 0.029 0.0194Heptanes + 0.0303 0.0382 0.0747 0.1 0.1155 0.1229 0.1039 0.1251 0.1114 Avg. Mol. Wt. 25.4 27.6 37.7 Heptanes + Properties Mol. Wt. 106 106 106 164 142 168 144 157 184 Gravity (API) 40.6 42.5 40.1 47.3 49.5 42.2 Characterization Fact. 11.6 11.55 11.75 11.75 12 11.85
Critical Temp., Tc (F) 65 90 189 110 169 180 159 262 185
Critical Temp., Tc (oR) 525 550 649 570 629 640 619 722 645
Critical Pressure, Pc (psia) 2675 2730 2900 2970 3010 3573 3470 4060 4335
Predicted Pc (psia) Simon and Yarborough 2817 2813 2788 2435 2574 3570 3249 4426 4505
Etter and Kay 2873 2930 2724 2932 3019 3528 3507 3959 3972 Zais 2513 2588 2355 2710 2668 3599 3410 4493 4680
This work 2671.26 2731.9 2897.64 2970.33 3012.3 3577.27 3469.13 4059.64 4342.19
133
Table C. 5. (Continued)
Mixture No. 141 - 7 141 - 8 141 - 9 141 - 10 141 - 11 141 - 12 141 - 13 141 - 14 141 - 15 Nitrogen 0.0009 0.0012 0.0013 Carbon Dioxide 0.0088 0.0041 0.0048 0.0056 0.0043 0.0038 0.0962 0.0776 0.0701Methane 0.722 0.7052 0.7175 0.6992 0.5564 0.4902 0.5431 0.4371 0.3954Ethane 0.0474 0.0725 0.0748 0.0638 0.0523 0.0464 0.1149 0.0932 0.0848Propane 0.0278 0.0465 0.0456 0.0404 0.2108 0.297 0.0768 0.216 0.2738I-Butane 0.0105 0.019 0.0177 0.0081 0.0069 0.0062 0.0147 0.0123 0.0115n-Butane 0.0158 0.0129 0.0136 0.0169 0.0147 0.0133 0.0257 0.0223 0.0207I=Pentane 0.0112 0.0064 0.0073 0.0079 0.007 0.0064 0.0082 0.0076 0.0072n-Pentane 0.008 0.008 0.0046 0.0058 0.0052 0.0047 0.0082 0.0078 0.0076Hexane 0.0181 0.01 0.0099 0.011 0.01 0.0093 0.0126 0.013 0.013 Heptanes + 0.1297 0.1142 0.1031 0.1413 0.1324 0.1227 0.0996 0.1131 0.1159 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 178 211 211 202 202 202 253 253 253 Gravity (API) 42.2 36.7 36.7 35 35 35 29 29 29 Characterization Fact. 11.8 11.7 11.7 11.55 11.55 11.55 11.55 11.55 11.55
Critical Temp., Tc (F) 265 226 216 202 202 202 280 280 280
Critical Temp., Tc (oR) 725 686 676 662 662 662 740 740 740
Critical Press., Pc (psia) 5420 6345 6560 8050 5130 4180 6715 4780 4310
Predicted Pc (psia) Simon and Yarborough 6035 6314 6304 5831 3972 3275 5241 3978 3542
Etter and Kay 4783 5201 5276 5095 4083 3659 4831 4051 3796 Zais 5470 7037 7299 5925 4436 3861 6250 4844 4380
This work 5416.47 6354.1 6569.65 8041.73 5129.18 4172.89 6708.13 4787.78 4315.78
134
Table C. 6. (Continued)
Mixture No. 141 - 16 141 - 17 141 - 18 141 - 19 141 - 20 141 - 21 141 - 22 141 - 23 141 - 24 Nitrogen 0.0052 0.0007 0.0008 0.0004 Carbon Dioxide 0.0871 0.0776 0.0778 0.0006 0.0011 0.0008 0.0008 0.0023 0.0022Methane 0.4911 0.4384 0.654 0.6091 0.5432 0.4789 0.3678 0.77 0.7247Ethane 0.1041 0.0935 0.0889 0.0865 0.0706 0.0624 0.0488 0.0316 0.0371Propane 0.0699 0.0632 0.061 0.0765 0.195 0.2786 0.4154 0.0228 0.0473I-Butane 0.0135 0.0123 0.0198 0.0183 0.0078 0.0075 0.007 0.0149 0.026 n-Butane 0.1063 0.1768 0.0317 0.0304 0.014 0.0132 0.0124 0.019 0.028 I-Pentane 0.0078 0.0074 0.0189 0.0134 0.0077 0.0072 0.0067 0.0084 0.0148n-Pentane 0.0077 0.0077 0.0172 0.0094 0.006 0.0057 0.0053 0.0137 0.0167Hexane 0.0123 0.0128 0.0307 0.0184 0.0165 0.0156 0.0145 0.0212 0.0202Heptanes + 0.1002 0.1103 0.0692 0.1374 0.1368 0.1293 0.1209 0.0961 0.083 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 253 253 126 229 229 229 229 152 152 Gravity (API) 29 29 60.4 30.2 30.2 30.2 30.2 51 51 Characterization Fact. 11.55 11.55 12.3 11.5 11.5 11.5 11.5 12.1 12.1
Critical Temp. Tc (F) 280 280 144 221 221 221 221 183 183
Critical Temp., Tc (oR) 740 740 604 681 681 681 681 643 643
Critical Press. Pc (psia) 5400 4130 2660 5700 5140 4040 2800 4120 3755
Predicted Pc (psia) Simon and Yarborough 4500 3959 2669 5164 4277 3566 2639 4851 4287
Etter and Kay 4219 3789 3009 4686 4333 3926 3315 4331 3845 Zais 5322 4527 2796 5497 4811 4202 3235 3963 3527
This work 5397.08 4137.16 2656.94 5693.09 5143.95 4043.34 2798.17 4118.03 3755.24
135
Table C. 7. (Continued)
Mixture No. 141 - 25 141 - 26 141 - 27 141 - 28 141 - 29 141 - 30 141 - 31 93 - 1 37 - 1 158 - 1 Nitrogen 0.0487 0.0249 0.0526 0.0115 0.0017 0.1898 0.0052 0.0076Carbon Dioxide 0.0014 0.0029 0.0149 0.2202 0.0073 0.003 0.0013
Hydrogen Sulfide .0068b Methane 0.4534 0.4957 0.6414 0.5325 0.2756 0.695 0.4004 0.5832 0.6849 0.5391Ethane 0.0705 0.1379 0.0733 0.1428 0.1144 0.0696 0.0767 0.1355 0.0971 0.142 Propane 0.1961 0.1015 0.0605 0.0926 0.2486 0.0426 0.0347 0.0761 0.0542 0.0964I-Butane 0.0936 0.0222 0.0154 0.0183 0.085 0.0109 0.0064 0.0084 0.0086 n-Butane 0.0825 0.0403 0.0298 0.0349 0.1294 0.0157 0.0204 0.0319 0.0223 0.0554I-Pentane 0.0542 0.0162 0.0142 0.0134 0.043 0.0102 0.0046 0.008 0.0088 n-Pentane 0.0343 0.0189 0.0178 0.0179 0.027 0.0086 0.0067 0.0161 0.0092 0.0299Hexane 0.0129 0.0234 0.0233 0.0318 0.0161 0.0184 0.0084 0.019 0.0142 0.0272
Heptanes + 0.0011 0.0952 0.0994 0.0632 0.0465 0.1124 0.0181 0.1145 .0925a 0.1011 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 114 161 161 117 192 180 115 193 147 178.5 Sp. Gravity or API 68.7 47.4 47.4 59.4 41 0.825 0.76 0.8135 Characterization Fact. 12.6 11.9 11.9 12.1 11.85 11.9
Critical Temp., Tc (F) 183 161 161 161 171 240 35 190 120 + 30 193
Critical Temp., Tc (oR) 643 621 621 621 631 700 495 650 #VALUE! 653
Critical Press., Pc (psia) 1410 3230 4090 2325 1980 4905 3454 4445 5500 3490
Predicted Pc (psia) Simon and Yarborough 1242 3157 4276 1899 2114 4862 3634
Etter and Kay 1478 3238 3861 2592 1843 4119 3430 Zais 1484 3364 4303 2263 1850 5136 1987 4528 3970 3839
This work 1409.62 3229.75 4084.59 2321.97 1977.91 4906.24 3450.08 4449.06 5503.58 3483.95(a) Heptanes plus treated as individual components through hexadecane (b) H2S treated as propane
136
Table C. 8. (Continued)
Mixture No. 58 - 1 11 - 1 6 - 1 6 - 2 6 - 3 30 - 1 52 - 2 52 - 3 52 - 6 Nitrogen 0.0066 0.0024 0.0021 0.002 0.0018 0.0124 0.0027Carbon Dioxide 0.018 0.0302 0.0084 0.0076 0.007 0.0065 0.0022 0.0003Methane 0.5966 0.6073 0.7504 0.6822 0.6253 0.5772 0.5706 0.5428 0.604 Ethane 0.1289 0.1177 0.0537 0.0489 0.0448 0.0414 0.1496 0.149 ..0517 propane 0.0653 0.0891 0.0136 0.0124 0.0114 0.0105 0.0825 0.0912 0.0757I-Butane 0.0173 0.0198 0.0021 0.0019 0.0018 0.0016 0.0092 0.0072 0.0455n-Butane 0.0219 0.0279 0.0127 0.1025 0.1773 0.2405 0.0367 0.0456 0.0389I-Pentane 0.01 0.0124 0.0106 0.0096 0.0088 0.0082 0.0064 0.0249n-Pentane 0.0096 0.008 0.0099 0.009 0.0082 0.0076 0.018 0.0186 0.0182Hexane 0.0132 0.0237 0.0215 0.0197 0.0182 0.0217 0.0255 0.0394
Heptanes + 0.1126 0.0878 .1125a .1023a .0937a .0865a 0.0907 0.1201 0.0897 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 155 172 118 118 118 118 173 166 133 Specific Gravity or API 0.7994 38 41.2 0.8126 51.9 Characterization Fact. 11.8 11.5 11.7 11.7 11.85
Critical Temp., Tc (F) 264 125 172.5 297.5 232.5 252.5 170 170 245
Critical Temp., Tc (oR) 724 585 632.5 757.5 692.5 712.5 630 630 705
Critical Pressure, Pc (psia) 3840 3940 4715 3950 3470 3140 4220 3875 2953
Predicted Pc (psia) Simon and Yarborough 3505 3574 3207 2609 2197 1900 3633 3349 2850
Etter and Kay 3720 3794 3794 3273 2901 2621 3642 3554 2987 Zais 3116 3905 4061 3501 3079 2753 4093 3768 2577
This work 3834.23 3945.43 4710.42 3947.13 3466.88 3140.71 4224.02 3871.53 2955.59
(a) Heptanes plus treated as individual components through hexadecane
137
Table C. 9. (Continued)
Mixture No. 47 - 1 26 - 1 26 - 2 26 - 3 47 - 2 26 - 4 26 - 5 26 - 6 26 - 7 26 - 8 Nitrogen 0.0884 0.1611 0.2441 0.113 0.24 0.1146 0.135 0.0705Carbon Dioxide 0.012 0.0109 0.01 0.0091 0.0044 0.003 0.002 0.0013 0.002 0.0025Methane 0.9089 0.8286 0.7625 0.887 0.858 0.7364 0.7364 0.7665 0.7515 0.8532Ethane 0.044 0.0401 0.0369 0.0333 0.016 0.015 0.012 0.0551 0.061 0.0411Propane 0.0191 0.0174 0.016 0.0144 0.007 0.006 0.0053 0.0335 0.0327 0.0198I-Butane 0.0033 0.003 0.0028 0.003 0.0014 0.0012 0.001 0.0035 0.0038 0.0037n-Butane 0.006 0.0055 0.0051 0.004 0.002 0.0018 0.0015 0.009 0.006 0.0039I-Pentane 0.0021 0.0019 0.0018 0.0016 0.0007 0.0006 0.0005 0.0017
n-Pentane 0.0013 0.0012 0.0011 0.001 0.0005 0.0004 0.0004 0.0015 .0020c .0022cHexane 0.0015 0.0014 0.0012 0.0011 0.0005 0.0004 0.0004
Heptanes + 0.0018 0.0016 0.0015 0.0014 0.0007 0.0006 0.0005 .0033d Helium 0.01 0.006 0.0031Avg. Mol. Wt. 18.4 19.25 19.94 20.75 16.9 18.11 19.56 20.02 19.98 18.6 Heptanes + Properties Mol. Wt. 115 115 115 115 115 115 115 100 Gravity (API) Characterization Fact.
Critical Temp., Tc (F) -79 -92 -104 -120 -101 -117 -131 -84 -90 -96
Critical Temp., Tc (oR) 381 368 356 340 359 343 329 376 370 364
Critical Pressure, Pc (psia) 925 955 968 973 765 790 815 1143 1107 918
Predicted Pc (psia) Simon and Yarborough 3285 3337 3380 3431 3681 3701 3740 3060 3042 3316
Etter and Kay 1385 1340 1303 1260 995 958 919 1519 1357 1265 Zais 1207 1273 1327 1340 907 958 978 1419 1328 1107
This work 924.78 955.88 967.83 973.21 765.06 790.12 815.1 1142.64 1107.03 918.27© Heavy fraction treated as n-pentane. (d) Heavy fraction treated as n-heptane. (e) Helium treated as methane.
138
Table C. 10. Critical Temperature Predictions for Complex Mixtures by LLS EOS Method (Compositions in Mole Fractions) Mixture No. 145-1 145-2 145-3 145-4 145-5 145-6 145-7 145-8 145-9 Nitrogen 0.001 0.001 0.001 0.001 0.004 0.001 Carbon Dioxide 0.004 0.005 0.005 0.004 0.006 0.002 0.0149 0.014 0.0156Methane 0.193 0.271 0.363 0.229 0.365 0.482 0.5248 0.5335 0.5921Ethane 0.032 0.034 0.038 0.029 0.058 0.065 0.145 0.0679 0.066 Propane 0.585 0.482 0.374 0.407 0.242 0.167 0.157 0.2035 0.1043I-Butane 0.007 0.007 0.009 0.006 0.007 0.007 0.0055 0.0063 0.0058n-Butane 0.012 0.015 0.015 0.172 0.215 0.14 0.0179 0.204 0.0831I-Pentane 0.005 0.006 0.006 0.005 0.004 0.006 0.0055 0.0062 0.0054n-Pentane 0.007 0.007 0.008 0.006 0.007 0.008 0.0079 0.0092 0.0078Hexane 0.013 0.015 0.016 0.012 0.01 0.012 0.0082 0.0094 0.0081Heptanes + 0.141 0.157 0.165 0.129 0.082 0.11 0.1133 0.1296 0.1118 Avg. Mol. Wt. 67.3 68.5 67.5 66.1 48.8 48.8 45 48.4 44.9 Heptanes + Properties Mol. Wt. 243 243 243 243 191 191 191 191 191 Gravity(API) Characterization Factor 11.6 11.6 11.6 11.6 11.9 11.9 11.9 11.9 11.9
Critical Temp., Tc (F) 265 265 265 265 234 200 200 200 200
Critical Temp., Tc (oR) 725 725 725 725 694 660 660 660 660 This work 725.39 725.45 724.58 725.81 694.91 694.44 659.17 661.13 659.22
139
Table C. 11. (Continued)
Mixture No. 145 - 10 145 - 11 145 - 12 145 - 13 145 - 14 4 - 1 4 - 2 4 - 3 4 - 4 4 - 5 Nitrogen 0.0016 0.002 Carbon Dioxide 0.0153 0.0008 0.0139 0.0006 0.0008 0.0066 0.0067 0.0061 0.0061 0.0059Methane 0.5774 0.6298 0.6165 0.4786 0.5808 0.7243 0.7292 0.8227 0.8165 0.795 Ethane 0.0631 0.0858 0.0602 0.082 0.0562 0.0557 0.0556 0.0284 0.0284 0.0281Propane 0.1178 0.0672 0.0961 0.1359 0.0919 0.0308 0.0306 0.0124 0.0124 0.0128I-Butane 0.0055 0.023 0.0055 0.0076 0.0081 n-Butane 0.0976 0.0305 0.0773 0.1273 0.082 0.0241 0.0226 0.0091 0.0092 0.0098I-Pentane 0.005 0.0114 0.0053 0.0066 0.0071 n-Pentane 0.0073 0.0122 0.0077 0.0069 0.0073 0.015 0.0147 0.0067 0.007 0.0078Hexane 0.0075 0.0206 0.008 0.0139 0.0148 0.0179 0.0183 0.0125 0.0131 0.015 Heptanes + 0.1035 0.1187 0.1095 0.139 0.1488 0.1256 0.1223 0.1021 0.1073 0.1247 Avg. Mol. Wt. 44.3 44.4 43.4 54.7 53.1 39.9 39.3 33 33.8 36.5 Heptanes + Properties Mol. Wt. 191 167 191 205 205 167 167 158 158 158 Gravity(API) Characterization Fact. 11.9 11.7 11.9 11.6 11.6 11.8 11.8 11.9 11.9 11.9
Critical Temp., Tc (F) 200 292 200 180 180 285 251 100 160 212
Critical Temp., Tc (oR) 660 752 660 640 640 745 711 560 620 672 This work 658.69 751.75 658.69 639.1 640.61 745.65 709.85 648.77 620.21 670.58
140
Table C. 12. (Continued)
Mixture No. 4 - 6 4 - 7 4 - 8 4 - 9 4-10 75 - 1 75 - 2 75 - 3 75 - 4 75 - 5 Nitrogen 0.0128 0.0058 0.0053 0.0054 0.0038Carbon Dioxide 0.0035 0.0156 0.0008 0.0075 0.0074 0.0045Methane 0.6433 0.6865 0.657 0.7164 0.8213 0.788 0.724 0.728 0.597 0.83 Ethane 0.0638 0.0603 0.0869 0.0548 0.0637 0.059 0.0542 0.0546 0.089 0.0378Propane 0.0605 0.0232 0.0537 0.031 0.0409 0.0315 0.03 0.0302 0.05 0.0144I-Butane n-Butane 0.0565 0.0309 0.0303 0.0258 0.0235 0.0265 0.031 0.0207 0.049 0.0089I-Pentane n-Pentane 0.0404 0.0243 0.0195 0.0216 0.0122 0.0425 0.071 0.0688 0.093 0.0436Hexane 0.0366 0.0256 0.0173 0.0198 0.0103 0.0252 0.0456 0.0438 0.0308Heptanes + 0.0954 0.1245 0.1217 0.1231 0.0234 0.0214 0.0388 0.0375 0.122 0.0263 Avg. Mol. Wt. 35.2 41.5 45.7 40.1 25 29.5 29.1 36.2 24.5 Heptanes + Properties Mol. Wt. 114 171 207 167 114 106 106 106 100 106 Gravity (API) Characterization Fact. 11.7 12 12 11.8
Critical Temp., Tc (F) 195 239 145 243 55 109 109 169 54
Critical Temp., Tc (oR) 655 699 605 703 515 569 569 629 514 This work 655.74 698.49 606.37 702.66 514.97 568.77 569.16 628.91 514.26
141
Table C. 13. (Continued)
Mixture No. 75 - 6 75 - 7 75 - 8 141 - 1 141 - 2 141 - 3 141 - 4 141 - 5 141 - 6 Nitrogen 0.0038 0.0036 0.003 0.1101 0.0477 0.0156 0.0123 Carbon Dioxide 0.0044 0.0043 0.0035 0.0008 0.0019 0.0047 0.0028 0.0172 0.011 Methane 0.815 0.784 0.643 0.3465 0.4675 0.4656 0.6173 0.6527 0.586 Ethane 0.0372 0.0355 0.0294 0.1561 0.1306 0.1645 0.1024 0.0725 0.1309Propane 0.0141 0.0136 0.0111 0.1499 0.105 0.1163 0.0673 0.0391 0.0699I-Butane 0.0121 0.0112 0.0141 0.0085 0.0157 0.0081n-Butane 0.0102 0.013 0.0252 0.0852 0.0472 0.0454 0.0352 0.0204 0.0349I-Pentane 0.0084 0.0169 0.0112 0.0106 0.016 0.0097n-Pentane 0.0501 0.0631 0.123 0.0109 0.0192 0.0182 0.0171 0.0123 0.0187Hexane 0.0354 0.0447 0.0871 0.02 0.0373 0.0215 0.0226 0.029 0.0194Heptanes + 0.0303 0.0382 0.0747 0.1 0.1155 0.1229 0.1039 0.1251 0.1114 Avg. Mol. Wt. 25.4 27.6 37.7 Heptanes + Properties Mol. Wt. 106 106 106 164 142 168 144 157 184 Gravity (API) 40.6 42.5 40.1 47.3 49.5 42.2 Characterization Fact. 11.6 11.55 11.75 11.75 12 11.85
Critical Temp., Tc (F) 65 90 189 110 169 180 159 262 185
Critical Temp., Tc (oR) 525 550 649 570 629 640 619 722 645 This work 524.27 540.38 648.47 570.06 629.48 640.76 618.84 721.94 646.07
142
Table C. 14. (Continued)
Mixture No. 141 - 7 141 - 8 141 - 9 141 - 10 141 - 11 141 - 12 141 - 13 141 - 14 141 - 15 Nitrogen 0.0009 0.0012 0.0013 Carbon Dioxide 0.0088 0.0041 0.0048 0.0056 0.0043 0.0038 0.0962 0.0776 0.0701Methane 0.722 0.7052 0.7175 0.6992 0.5564 0.4902 0.5431 0.4371 0.3954Ethane 0.0474 0.0725 0.0748 0.0638 0.0523 0.0464 0.1149 0.0932 0.0848Propane 0.0278 0.0465 0.0456 0.0404 0.2108 0.297 0.0768 0.216 0.2738I-Butane 0.0105 0.019 0.0177 0.0081 0.0069 0.0062 0.0147 0.0123 0.0115n-Butane 0.0158 0.0129 0.0136 0.0169 0.0147 0.0133 0.0257 0.0223 0.0207I=Pentane 0.0112 0.0064 0.0073 0.0079 0.007 0.0064 0.0082 0.0076 0.0072n-Pentane 0.008 0.008 0.0046 0.0058 0.0052 0.0047 0.0082 0.0078 0.0076Hexane 0.0181 0.01 0.0099 0.011 0.01 0.0093 0.0126 0.013 0.013 Heptanes + 0.1297 0.1142 0.1031 0.1413 0.1324 0.1227 0.0996 0.1131 0.1159 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 178 211 211 202 202 202 253 253 253 Gravity (API) 42.2 36.7 36.7 35 35 35 29 29 29 Characterization Fact. 11.8 11.7 11.7 11.55 11.55 11.55 11.55 11.55 11.55
Critical Temp., Tc (F) 265 226 216 202 202 202 280 280 280
Critical Temp., Tc (oR) 725 686 676 662 662 662 740 740 740 This work 724.53 686.98 676.99 661.32 661.89 660.87 739.24 741.2 740.99
143
Table C. 15. (Continued)
Mixture No. 141 - 16 141 - 17 141 - 18 141 - 19 141 - 20 141 - 21 141 - 22 141 - 23 141 - 24 Nitrogen 0.0052 0.0007 0.0008 0.0004 Carbon Dioxide 0.0871 0.0776 0.0778 0.0006 0.0011 0.0008 0.0008 0.0023 0.0022Methane 0.4911 0.4384 0.654 0.6091 0.5432 0.4789 0.3678 0.77 0.7247Ethane 0.1041 0.0935 0.0889 0.0865 0.0706 0.0624 0.0488 0.0316 0.0371Propane 0.0699 0.0632 0.061 0.0765 0.195 0.2786 0.4154 0.0228 0.0473I-Butane 0.0135 0.0123 0.0198 0.0183 0.0078 0.0075 0.007 0.0149 0.026 n-Butane 0.1063 0.1768 0.0317 0.0304 0.014 0.0132 0.0124 0.019 0.028 I-Pentane 0.0078 0.0074 0.0189 0.0134 0.0077 0.0072 0.0067 0.0084 0.0148n-Pentane 0.0077 0.0077 0.0172 0.0094 0.006 0.0057 0.0053 0.0137 0.0167Hexane 0.0123 0.0128 0.0307 0.0184 0.0165 0.0156 0.0145 0.0212 0.0202Heptanes + 0.1002 0.1103 0.0692 0.1374 0.1368 0.1293 0.1209 0.0961 0.083 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 253 253 126 229 229 229 229 152 152 Gravity (API) 29 29 60.4 30.2 30.2 30.2 30.2 51 51 Characterization Fact. 11.55 11.55 12.3 11.5 11.5 11.5 11.5 12.1 12.1
Critical Temp. Tc (F) 280 280 144 221 221 221 221 183 183
Critical Temp., Tc (oR) 740 740 604 681 681 681 681 643 643 This work 739.6 741.28 603.31 680.17 681.52 681.56 680.55 642.69 643.04
144
Table C. 16. (Continued)
Mixture No. 141 - 25
141 - 26
141 - 27
141 - 28
141 - 29
141 - 30
141 - 31
93 - 1 37 - 1 158 - 1
Nitrogen 0.0487 0.0249 0.0526 0.0115 0.0017 0.1898 0.0052 0.0076Carbon Dioxide 0.0014 0.0029 0.0149 0.2202 0.0073 0.003 0.0013Hydrogen Sulfide .0068b Methane 0.4534 0.4957 0.6414 0.5325 0.2756 0.695 0.4004 0.5832 0.6849 0.5391Ethane 0.0705 0.1379 0.0733 0.1428 0.1144 0.0696 0.0767 0.1355 0.0971 0.142 Propane 0.1961 0.1015 0.0605 0.0926 0.2486 0.0426 0.0347 0.0761 0.0542 0.0964I-Butane 0.0936 0.0222 0.0154 0.0183 0.085 0.0109 0.0064 0.0084 0.0086 n-Butane 0.0825 0.0403 0.0298 0.0349 0.1294 0.0157 0.0204 0.0319 0.0223 0.0554I-Pentane 0.0542 0.0162 0.0142 0.0134 0.043 0.0102 0.0046 0.008 0.0088 n-Pentane 0.0343 0.0189 0.0178 0.0179 0.027 0.0086 0.0067 0.0161 0.0092 0.0299Hexane 0.0129 0.0234 0.0233 0.0318 0.0161 0.0184 0.0084 0.019 0.0142 0.0272Heptanes + 0.0011 0.0952 0.0994 0.0632 0.0465 0.1124 0.0181 0.1145 .0925a 0.1011 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 114 161 161 117 192 180 115 193 147 178.5 Sp. Gravity or API 68.7 47.4 47.4 59.4 41 0.825 0.76 0.8135 Characterization Fact. 12.6 11.9 11.9 12.1 11.85 11.9 Critical Temp., Tc (F) 183 161 161 161 171 240 35 190 120 + 30 193
Critical Temp., Tc (oR) 643 621 621 621 631 700 495 650 #VALUE! 653
This work 642.83 620.95 620.18 620.19 630.34 700.18 494.44 650.59 580.38 651.87(a) Heptanes plus treated as individual components through hexadecane (b) H2S treated as propane
145
Table C. 17. (Continued)
Mixture No. 58 - 1 11 - 1 6 - 1 6 - 2 6 - 3 30 - 1 52 - 2 52 - 3 52 - 6 Nitrogen 0.0066 0.0024 0.0021 0.002 0.0018 0.0124 0.0027Carbon Dioxide 0.018 0.0302 0.0084 0.0076 0.007 0.0065 0.0022 0.0003Methane 0.5966 0.6073 0.7504 0.6822 0.6253 0.5772 0.5706 0.5428 0.604 Ethane 0.1289 0.1177 0.0537 0.0489 0.0448 0.0414 0.1496 0.149 ..0517 propane 0.0653 0.0891 0.0136 0.0124 0.0114 0.0105 0.0825 0.0912 0.0757I-Butane 0.0173 0.0198 0.0021 0.0019 0.0018 0.0016 0.0092 0.0072 0.0455n-Butane 0.0219 0.0279 0.0127 0.1025 0.1773 0.2405 0.0367 0.0456 0.0389I-Pentane 0.01 0.0124 0.0106 0.0096 0.0088 0.0082 0.0064 0.0249n-Pentane 0.0096 0.008 0.0099 0.009 0.0082 0.0076 0.018 0.0186 0.0182Hexane 0.0132 0.0237 0.0215 0.0197 0.0182 0.0217 0.0255 0.0394
Heptanes + 0.1126 0.0878 .1125a .1023a .0937a .0865a 0.0907 0.1201 0.0897 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 155 172 118 118 118 118 173 166 133 Specific Gravity or API 0.7994 38 41.2 0.8126 51.9 Characterization Fact. 11.8 11.5 11.7 11.7 11.85
Critical Temp., Tc (F) 264 125 172.5 297.5 232.5 252.5 170 170 245
Critical Temp., Tc (oR) 724 585 632.5 757.5 692.5 712.5 630 630 705 This work 722.91 585.81 631.89 667.01 691.88 712.66 630.6 629.44 705.62
(a) Heptanes plus treated as individual components through hexadecane
146
Table C. 18. (Continued)
Mixture No. 47 - 1 26 - 1 26 - 2 26 - 3 47 - 2 26 - 4 26 - 5 26 - 6 26 - 7 26 - 8 Nitrogen 0.0884 0.1611 0.2441 0.113 0.24 0.1146 0.135 0.0705Carbon Dioxide 0.012 0.0109 0.01 0.0091 0.0044 0.003 0.002 0.0013 0.002 0.0025Methane 0.9089 0.8286 0.7625 0.887 0.858 0.7364 0.7364 0.7665 0.7515 0.8532Ethane 0.044 0.0401 0.0369 0.0333 0.016 0.015 0.012 0.0551 0.061 0.0411Propane 0.0191 0.0174 0.016 0.0144 0.007 0.006 0.0053 0.0335 0.0327 0.0198I-Butane 0.0033 0.003 0.0028 0.003 0.0014 0.0012 0.001 0.0035 0.0038 0.0037n-Butane 0.006 0.0055 0.0051 0.004 0.002 0.0018 0.0015 0.009 0.006 0.0039I-Pentane 0.0021 0.0019 0.0018 0.0016 0.0007 0.0006 0.0005 0.0017
n-Pentane 0.0013 0.0012 0.0011 0.001 0.0005 0.0004 0.0004 0.0015 .0020c .0022cHexane 0.0015 0.0014 0.0012 0.0011 0.0005 0.0004 0.0004
Heptanes + 0.0018 0.0016 0.0015 0.0014 0.0007 0.0006 0.0005 .0033d Helium 0.01 0.006 0.0031Avg. Mol. Wt. 18.4 19.25 19.94 20.75 16.9 18.11 19.56 20.02 19.98 18.6 Heptanes + Properties Mol. Wt. 115 115 115 115 115 115 115 100 Gravity (API) Characterization Fact.
Critical Temp., Tc (F) -79 -92 -104 -120 -101 -117 -131 -84 -90 -96
Critical Temp., Tc (oR) 381 368 356 340 359 343 329 376 370 364 This work 380.91 368.34 355.94 340.07 359.03 343.05 329.04 375.88 370.01 364.11© Heavy fraction treated as n-pentane. (d) Heavy fraction treated as n-heptane. (e) Helium treated as methane.
147
Table C.19. Critical Pressure, Temperature and Properties Predictions for Heptane Plus
(Compositions in Mole Fractions) Mixture No. 145-1 145-2 145-3 145-4 145-5 145-6 145-7 145-8 145-9 Nitrogen 0.001 0.001 0.001 0.001 0.004 0.001 Carbon Dioxide 0.004 0.005 0.005 0.004 0.006 0.002 0.0149 0.014 0.0156Methane 0.193 0.271 0.363 0.229 0.365 0.482 0.5248 0.5335 0.5921Ethane 0.032 0.034 0.038 0.029 0.058 0.065 0.145 0.0679 0.066 Propane 0.585 0.482 0.374 0.407 0.242 0.167 0.157 0.2035 0.1043I-Butane 0.007 0.007 0.009 0.006 0.007 0.007 0.0055 0.0063 0.0058n-Butane 0.012 0.015 0.015 0.172 0.215 0.14 0.0179 0.204 0.0831I-Pentane 0.005 0.006 0.006 0.005 0.004 0.006 0.0055 0.0062 0.0054n-Pentane 0.007 0.007 0.008 0.006 0.007 0.008 0.0079 0.0092 0.0078Hexane 0.013 0.015 0.016 0.012 0.01 0.012 0.0082 0.0094 0.0081Heptanes + 0.141 0.157 0.165 0.129 0.082 0.11 0.1133 0.1296 0.1118 Avg. Mol. Wt. 67.3 68.5 67.5 66.1 48.8 48.8 45 48.4 44.9 Heptanes + Properties Mol. Wt. 243 243 243 243 191 191 191 191 191 Gravity(API) 33.31 33.31 33.31 33.31 39.98 39.98 39.98 39.98 39.98 SG 0.86 0.86 0.86 0.86 0.83 0.83 0.83 0.83 0.83 Characterization Factor 11.6 11.6 11.6 11.6 11.9 11.9 11.9 11.9 11.9 Tb 1037.0 1037.0 1037.0 1037.0 920.9 920.9 920.9 920.9 920.9 C 3.3475 3.3475 3.3475 3.3475 3.2304 3.2304 3.2304 3.2304 3.2304 Pc, psia 271.0 271.0 271.0 271.0 319.8 319.8 319.8 319.8 319.8
Tc, oR 1364.4 1364.4 1364.4 1364.4 1258.4 1258.4 1258.4 1258.4 1258.4 ω 0.5673 0.5673 0.5673 0.5673 0.4676 0.4676 0.4676 0.4676 0.4676
Zc 0.2416 0.2416 0.2416 0.2416 0.2493 0.2493 0.2493 0.2493 0.2493
Ωω 0.3555 0.3555 0.3555 0.3555 0.3564 0.3564 0.3564 0.3564 0.3564
148
Table C. 20. (Continued)
Mixture No. 145 - 10 145 - 11 145 - 12 145 - 13 145 - 14 4 - 1 4 - 2 4 - 3 4 - 4 4 - 5 Nitrogen 0.0016 0.002 Carbon Dioxide 0.0153 0.0008 0.0139 0.0006 0.0008 0.0066 0.0067 0.0061 0.0061 0.0059Methane 0.5774 0.6298 0.6165 0.4786 0.5808 0.7243 0.7292 0.8227 0.8165 0.795 Ethane 0.0631 0.0858 0.0602 0.082 0.0562 0.0557 0.0556 0.0284 0.0284 0.0281Propane 0.1178 0.0672 0.0961 0.1359 0.0919 0.0308 0.0306 0.0124 0.0124 0.0128I-Butane 0.0055 0.023 0.0055 0.0076 0.0081 n-Butane 0.0976 0.0305 0.0773 0.1273 0.082 0.0241 0.0226 0.0091 0.0092 0.0098I-Pentane 0.005 0.0114 0.0053 0.0066 0.0071 n-Pentane 0.0073 0.0122 0.0077 0.0069 0.0073 0.015 0.0147 0.0067 0.007 0.0078Hexane 0.0075 0.0206 0.008 0.0139 0.0148 0.0179 0.0183 0.0125 0.0131 0.015 Heptanes + 0.1035 0.1187 0.1095 0.139 0.1488 0.1256 0.1223 0.1021 0.1073 0.1247 Avg. Mol. Wt. 44.3 44.4 43.4 54.7 53.1 39.9 39.3 33 33.8 36.5 Heptanes + Properties Mol. Wt. 191 167 191 205 205 167 167 158 158 158 Gravity(API) 39.98 44.46 39.98 37.85 37.85 44.46 44.46 46.49 46.49 46.49 SG 0.83 0.80 0.83 0.84 0.84 0.80 0.80 0.79 0.79 0.79 Characterization Fact. 11.9 11.7 11.9 11.6 11.6 11.8 11.8 11.9 11.9 11.9 Tb 920.9 860.6 920.9 953.9 953.9 860.6 860.6 836.6 836.6 836.6 C 3.2304 3.1667 3.2304 3.2644 3.2644 3.1667 3.1667 3.1407 3.1407 3.1407 Pc,psia 319.8 348.0 319.8 305.1 305.1 348.0 348.0 359.7 359.7 359.7
Tc, oR 1258.4 1200.2 1258.4 1289.4 1289.4 1200.2 1200.2 1176.3 1176.3 1176.3 ω 0.4676 0.4209 0.4676 0.4946 0.4946 0.4209 0.4209 0.4033 0.4033 0.4033
Zc 0.2493 0.2531 0.2493 0.2472 0.2472 0.2531 0.2531 0.2545 0.2545 212
Ωω 0.3564 0.3569 0.3564 0.3562 0.3562 0.3569 0.3569 0.3571 0.3571 5620
149
Table C. 21. (Continued) Mixture No. 4 - 6 4 - 7 4 - 8 4 - 9 4-10 75 - 1 75 - 2 75 - 3 75 - 4 75 - 5 Nitrogen 0.0128 0.0058 0.0053 0.0054 0.0038Carbon Dioxide 0.0035 0.0156 0.0008 0.0075 0.0074 0.0045Methane 0.6433 0.6865 0.657 0.7164 0.8213 0.788 0.724 0.728 0.597 0.83 Ethane 0.0638 0.0603 0.0869 0.0548 0.0637 0.059 0.0542 0.0546 0.089 0.0378Propane 0.0605 0.0232 0.0537 0.031 0.0409 0.0315 0.03 0.0302 0.05 0.0144I-Butane n-Butane 0.0565 0.0309 0.0303 0.0258 0.0235 0.0265 0.031 0.0207 0.049 0.0089I-Pentane n-Pentane 0.0404 0.0243 0.0195 0.0216 0.0122 0.0425 0.071 0.0688 0.093 0.0436Hexane 0.0366 0.0256 0.0173 0.0198 0.0103 0.0252 0.0456 0.0438 0.0308Heptanes + 0.0954 0.1245 0.1217 0.1231 0.0234 0.0214 0.0388 0.0375 0.122 0.0263 Avg. Mol. Wt. 35.2 41.5 45.7 40.1 25 29.5 29.1 36.2 24.5 Heptanes + Properties Mol. Wt. 114 171 207 167 114 106 106 106 100 106 Gravity (API) 61.04 43.63 37.57 44.46 61.04 64.98 64.98 64.98 68.35 64.98 SG 0.73 0.81 0.84 0.80 0.73 0.72 0.72 0.72 0.71 0.72 Characterization Fact. 11.7 12 12 11.8 Tb 704.8 871.0 958.5 860.6 704.8 677.6 677.6 677.6 656.3 677.6 C 2.9917 3.1778 3.2691 3.1667 2.9917 2.9594 2.9594 2.9594 2.9337 2.9594 Pc,psia 427.1 343.0 303.1 348.0 427.1 441.1 441.1 441.1 452.0 441.1
Tc, oR 1038.0 1210.4 1293.6 1200.2 1038.0 1007.7 1007.7 1007.7 983.5 1007.7 ω 0.3158 0.4287 0.4985 0.4209 0.3158 0.2996 0.2996 0.2996 0.2874 0.2996
Zc 0.2620 0.2524 0.2469 0.2531 0.2620 0.2634 0.2634 0.2634 0.2645 212
Ωω 0.3579 0.3568 0.3561 0.3569 0.3579 0.3581 0.3581 0.3581 0.3582 5620
150
Table C. 22. (Continued) Mixture No. 75 - 6 75 - 7 75 - 8 141 - 1 141 - 2 141 - 3 141 - 4 141 - 5 141 - 6 Nitrogen 0.0038 0.0036 0.003 0.1101 0.0477 0.0156 0.0123 Carbon Dioxide 0.0044 0.0043 0.0035 0.0008 0.0019 0.0047 0.0028 0.0172 0.011 Methane 0.815 0.784 0.643 0.3465 0.4675 0.4656 0.6173 0.6527 0.586 Ethane 0.0372 0.0355 0.0294 0.1561 0.1306 0.1645 0.1024 0.0725 0.1309Propane 0.0141 0.0136 0.0111 0.1499 0.105 0.1163 0.0673 0.0391 0.0699I-Butane 0.0121 0.0112 0.0141 0.0085 0.0157 0.0081n-Butane 0.0102 0.013 0.0252 0.0852 0.0472 0.0454 0.0352 0.0204 0.0349I-Pentane 0.0084 0.0169 0.0112 0.0106 0.016 0.0097n-Pentane 0.0501 0.0631 0.123 0.0109 0.0192 0.0182 0.0171 0.0123 0.0187Hexane 0.0354 0.0447 0.0871 0.02 0.0373 0.0215 0.0226 0.029 0.0194Heptanes + 0.0303 0.0382 0.0747 0.1 0.1155 0.1229 0.1039 0.1251 0.1114 Avg. Mol. Wt. 25.4 27.6 37.7 Heptanes + Properties Mol. Wt. 106 106 106 164 142 168 144 157 184 Gravity (API) 64.98 64.98 64.98 40.6 42.5 40.1 47.3 49.5 42.2 SG 0.72 0.72 0.72 0.82 0.81 0.82 0.79 0.78 0.81 Characterization Fact. 11.6 11.55 11.75 11.75 12 11.85 Tb 677.6 677.6 677.6 862.2 807.5 872.1 802.9 828.4 901.6 C 2.9594 2.9594 2.9594 3.1592 3.0931 3.1704 3.0983 3.1372 3.2123 Pc,psia 441.1 441.1 441.1 371.8 421.8 364.8 392.3 349.4 323.4
Tc, oR 1007.7 1007.7 1007.7 1211.9 1164.0 1220.9 1148.0 1162.3 1237.5 ω 0.2996 0.2996 0.2996 0.4076 0.3599 0.4159 0.3715 0.4057 0.4559
Zc 0.2634 0.2634 0.2634 0.2542 0.2582 0.2535 0.2572 0.2543 0.2502
Ωω 0.3581 0.3581 0.3581 0.3570 0.3575 0.3569 0.3574 0.3570 0.3565
151
Table C. 23. (Continued) Mixture No. 141 - 7 141 - 8 141 - 9 141 - 10 141 - 11 141 - 12 141 - 13 141 - 14 141 - 15 Nitrogen 0.0009 0.0012 0.0013 Carbon Dioxide 0.0088 0.0041 0.0048 0.0056 0.0043 0.0038 0.0962 0.0776 0.0701Methane 0.722 0.7052 0.7175 0.6992 0.5564 0.4902 0.5431 0.4371 0.3954Ethane 0.0474 0.0725 0.0748 0.0638 0.0523 0.0464 0.1149 0.0932 0.0848Propane 0.0278 0.0465 0.0456 0.0404 0.2108 0.297 0.0768 0.216 0.2738i-Butane 0.0105 0.019 0.0177 0.0081 0.0069 0.0062 0.0147 0.0123 0.0115n-Butane 0.0158 0.0129 0.0136 0.0169 0.0147 0.0133 0.0257 0.0223 0.0207i=Pentane 0.0112 0.0064 0.0073 0.0079 0.007 0.0064 0.0082 0.0076 0.0072n-Pentane 0.008 0.008 0.0046 0.0058 0.0052 0.0047 0.0082 0.0078 0.0076Hexane 0.0181 0.01 0.0099 0.011 0.01 0.0093 0.0126 0.013 0.013 Heptanes + 0.1297 0.1142 0.1031 0.1413 0.1324 0.1227 0.0996 0.1131 0.1159 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 178 211 211 202 202 202 253 253 253 Gravity (API) 42.2 36.7 36.7 35 35 35 29 29 29 SG 0.81 0.84 0.84 0.85 0.85 0.85 0.88 0.88 0.88 Characterization Fact. 11.8 11.7 11.7 11.55 11.55 11.55 11.55 11.55 11.55 Tb 889.0 968.5 968.5 954.9 954.9 954.9 1066.8 1066.8 1066.8 C 3.1968 3.2784 3.2784 3.2581 3.2581 3.2581 3.3684 3.3684 3.3684 Pc,psia 334.8 300.5 300.5 321.3 321.3 321.3 274.9 274.9 274.9
Tc, oR 1228.0 1303.6 1303.6 1298.7 1298.7 1298.7 1400.4 1400.4 1400.4 ω 0.4423 0.5055 0.5055 0.4822 0.4822 0.4822 0.5778 0.5778 0.5778
Zc 0.2513 0.2463 0.2463 0.2481 0.2481 0.2481 0.2408 0.2408 0.2408
Ωω 0.3567 0.3561 0.3561 0.3563 0.3563 0.3563 0.3554 0.3554 0.3554
152
Table C. 24. (Continued) Mixture No. 141 - 16 141 - 17 141 - 18 141 - 19 141 - 20 141 - 21 141 - 22 141 - 23 141 - 24 Nitrogen 0.0052 0.0007 0.0008 0.0004 Carbon Dioxide 0.0871 0.0776 0.0778 0.0006 0.0011 0.0008 0.0008 0.0023 0.0022Methane 0.4911 0.4384 0.654 0.6091 0.5432 0.4789 0.3678 0.77 0.7247Ethane 0.1041 0.0935 0.0889 0.0865 0.0706 0.0624 0.0488 0.0316 0.0371Propane 0.0699 0.0632 0.061 0.0765 0.195 0.2786 0.4154 0.0228 0.0473I-Butane 0.0135 0.0123 0.0198 0.0183 0.0078 0.0075 0.007 0.0149 0.026 n-Butane 0.1063 0.1768 0.0317 0.0304 0.014 0.0132 0.0124 0.019 0.028 I-Pentane 0.0078 0.0074 0.0189 0.0134 0.0077 0.0072 0.0067 0.0084 0.0148n-Pentane 0.0077 0.0077 0.0172 0.0094 0.006 0.0057 0.0053 0.0137 0.0167Hexane 0.0123 0.0128 0.0307 0.0184 0.0165 0.0156 0.0145 0.0212 0.0202Heptanes + 0.1002 0.1103 0.0692 0.1374 0.1368 0.1293 0.1209 0.0961 0.083 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 253 253 126 229 229 229 229 152 152 Gravity (API) 29 29 60.4 30.2 30.2 30.2 30.2 51 51 SG 0.88 0.88 0.74 0.88 0.88 0.88 0.88 0.78 0.78 Characterization Fact. 11.55 11.55 12.3 11.5 11.5 11.5 11.5 12.1 12.1 Tb 1066.8 1066.8 736.3 1019.5 1019.5 1019.5 1019.5 814.3 814.3 C 3.3684 3.3684 3.0359 3.3194 3.3194 3.3194 3.3194 3.1221 3.1221 Pc,psia 274.9 274.9 387.6 300.1 300.1 300.1 300.1 355.0 355.0
Tc, oR 1400.4 1400.4 1064.6 1361.9 1361.9 1361.9 1361.9 1147.6 1147.6 ω 0.5778 0.5778 0.3454 0.5301 0.5301 0.5301 0.5301 0.3961 0.3961
Zc 0.2408 0.2408 0.2594 0.2444 0.2444 0.2444 0.2444 0.2551 0.2551
Ωω 0.3554 0.3554 0.3576 0.3558 0.3558 0.3558 0.3558 0.3571 0.3571
153
Table C. 25. (Continued) Mixture No. 141 - 25 141 - 26 141 - 27 141 - 28 141 - 29 141 - 30 141 - 31 93 - 1 37 - 1 158 - 1 Nitrogen 0.0487 0.0249 0.0526 0.0115 0.0017 0.1898 0.0052 0.0076Carbon Dioxide 0.0014 0.0029 0.0149 0.2202 0.0073 0.003 0.0013
Hydrogen Sulfide .0068b Methane 0.4534 0.4957 0.6414 0.5325 0.2756 0.695 0.4004 0.5832 0.6849 0.5391Ethane 0.0705 0.1379 0.0733 0.1428 0.1144 0.0696 0.0767 0.1355 0.0971 0.142Propane 0.1961 0.1015 0.0605 0.0926 0.2486 0.0426 0.0347 0.0761 0.0542 0.0964I-Butane 0.0936 0.0222 0.0154 0.0183 0.085 0.0109 0.0064 0.0084 0.0086 n-Butane 0.0825 0.0403 0.0298 0.0349 0.1294 0.0157 0.0204 0.0319 0.0223 0.0554I-Pentane 0.0542 0.0162 0.0142 0.0134 0.043 0.0102 0.0046 0.008 0.0088 n-Pentane 0.0343 0.0189 0.0178 0.0179 0.027 0.0086 0.0067 0.0161 0.0092 0.0299Hexane 0.0129 0.0234 0.0233 0.0318 0.0161 0.0184 0.0084 0.019 0.0142 0.0272
Heptanes + 0.0011 0.0952 0.0994 0.0632 0.0465 0.1124 0.0181 0.1145 .0925a 0.1011 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 114 161 161 117 192 180 115 193 147 178.5Sp. Gravity or API 68.7 47.4 47.4 59.4 41 0.825 0.76 0.8135
SG 0.71 0.79 0.79 0.74 0.82 0.825 0.76 0.8135 0.51 0.46 Characterization Fact. 12.6 11.9 11.9 12.1 11.85 11.9 Tb 693.2 841.5 841.5 715.2 920.6 898.1 717.6 919.4 671.4 697.4 C 2.9903 3.1492 3.1492 3.0034 3.2326 3.2025 2.9968 3.2347 3.0905 3.1752 Pc,psia 393.0 348.9 348.9 423.3 314.0 340.0 454.6 306.9 150.5 98.7
Tc, oR 1012.8 1177.7 1177.7 1049.9 1255.3 1241.1 1062.2 1250.3 875.5 858.5 ω 0.3245 0.4118 0.4118 0.3215 0.4718 0.4429 0.3109 0.4768 0.5151 0.6618
Zc 0.2612 0.2538 0.2538 0.2615 0.2490 0.2513 0.2624 0.2486 0.2456 0.2347
Ωω 0.3578 0.3570 0.3570 0.3578 0.3564 0.3567 0.3580 0.3563 0.3560 0.3546 (a) Heptanes plus treated as individual components through hexadecane (b) H2S treated as propane
154
Table C. 26. (Continued) Mixture No. 58 - 1 11 - 1 6 - 1 6 - 2 6 - 3 30 - 1 52 - 2 52 - 3 52 - 6 Nitrogen 0.0066 0.0024 0.0021 0.002 0.0018 0.0124 0.0027Carbon Dioxide 0.018 0.0302 0.0084 0.0076 0.007 0.0065 0.0022 0.0003Methane 0.5966 0.6073 0.7504 0.6822 0.6253 0.5772 0.5706 0.5428 0.604Ethane 0.1289 0.1177 0.0537 0.0489 0.0448 0.0414 0.1496 0.149 ..0517propane 0.0653 0.0891 0.0136 0.0124 0.0114 0.0105 0.0825 0.0912 0.0757I-Butane 0.0173 0.0198 0.0021 0.0019 0.0018 0.0016 0.0092 0.0072 0.0455n-Butane 0.0219 0.0279 0.0127 0.1025 0.1773 0.2405 0.0367 0.0456 0.0389I-Pentane 0.01 0.0124 0.0106 0.0096 0.0088 0.0082 0.0064 0.0249n-Pentane 0.0096 0.008 0.0099 0.009 0.0082 0.0076 0.018 0.0186 0.0182Hexane 0.0132 0.0237 0.0215 0.0197 0.0182 0.0217 0.0255 0.0394
Heptanes + 0.1126 0.0878 .1125a .1023a .0937a .0865a 0.0907 0.1201 0.0897 Avg. Mol. Wt. Heptanes + Properties Mol. Wt. 155 172 118 118 118 118 173 166 133 Specific Gravity or API 0.7994 38 59.27 59.27 59.27 59.27 41.2 0.8126 51.9
SG 0.7994 0.83 0.74 0.74 0.74 0.74 0.82 0.81 0.77 Characterization Fact. 11.8 11.5 11.7 11.7 11.85Tb 831.8 885.4 718.0 718.0 718.0 718.0 880.6 862.3 768.0 C 3.1322 3.1818 3.0072 3.0072 3.0072 3.0072 3.1838 3.1643 3.0616 Pc,psia 371.3 365.5 420.2 420.2 420.2 420.2 349.1 358.0 403.6
Tc, oR 1175.2 1237.2 1052.4 1052.4 1052.4 1052.4 1224.4 1206.4 1109.2 ω 0.3947 0.4213 0.3238 0.3238 0.3238 0.3238 0.4291 0.4156 0.3516
Zc 0.2552 0.2530 0.2613 0.2613 0.2613 0.2613 0.2524 0.2535 0.2589
Ωω 0.3571 0.3569 0.3578 0.3578 0.3578 0.3578 0.3568 0.3569 0.3576
(a) Heptanes plus treated as individual components through hexadecane
155
Table C. 27. (Continued) Mixture No. 47 - 1 26 - 1 26 - 2 26 - 3 47 - 2 26 - 4 26 - 5 26 - 6 26 - 7 26 - 8 Nitrogen 0.0884 0.1611 0.2441 0.113 0.24 0.1146 0.135 0.0705Carbon Dioxide 0.012 0.0109 0.01 0.0091 0.0044 0.003 0.002 0.0013 0.002 0.0025Methane 0.9089 0.8286 0.7625 0.887 0.858 0.7364 0.7364 0.7665 0.7515 0.8532Ethane 0.044 0.0401 0.0369 0.0333 0.016 0.015 0.012 0.0551 0.061 0.0411Propane 0.0191 0.0174 0.016 0.0144 0.007 0.006 0.0053 0.0335 0.0327 0.0198I-Butane 0.0033 0.003 0.0028 0.003 0.0014 0.0012 0.001 0.0035 0.0038 0.0037n-Butane 0.006 0.0055 0.0051 0.004 0.002 0.0018 0.0015 0.009 0.006 0.0039I-Pentane 0.0021 0.0019 0.0018 0.0016 0.0007 0.0006 0.0005 0.0017
n-Pentane 0.0013 0.0012 0.0011 0.001 0.0005 0.0004 0.0004 0.0015 .0020c .0022cHexane 0.0015 0.0014 0.0012 0.0011 0.0005 0.0004 0.0004
Heptanes + 0.0018 0.0016 0.0015 0.0014 0.0007 0.0006 0.0005 .0033d Helium 0.01 0.006 0.0031Avg. Mol. Wt. 18.4 19.25 19.94 20.75 16.9 18.11 19.56 20.02 19.98 18.6 Heptanes + Properties Mol. Wt. 115 115 115 115 115 115 115 100 Gravity (API) 60.58 60.58 60.58 60.58 60.58 60.58 60.58 68.35 #DIV/0! #DIV/0!
SG 0.74 0.74 0.74 0.74 0.74 0.74 0.74 0.71 #DIV/0! #DIV/0!Characterization Fact. Tb 708.1 708.1 708.1 708.1 708.1 708.1 708.1 656.3 #DIV/0! C 2.9956 2.9956 2.9956 2.9956 2.9956 2.9956 2.9956 2.9337 #DIV/0! Pc,psia 425.3 425.3 425.3 425.3 425.3 425.3 425.3 452.0 #DIV/0!
Tc, oR 1041.6 1041.6 1041.6 1041.6 1041.6 1041.6 1041.6 983.5 #DIV/0! ω 0.3178 0.3178 0.3178 0.3178 0.3178 0.3178 0.3178 0.2874 #DIV/0!
Zc 0.2618 0.2618 0.2618 0.2618 0.2618 0.2618 0.2618 0.2645 #DIV/0!
Ωω 0.3579 0.3579 0.3579 0.3579 0.3579 0.3579 0.3579 0.3582 #DIV/0! © Heavy fraction treated as n-pentane. (d) Heavy fraction treated as n-heptane. (e) Helium treated as methane.
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CRITICAL PRESSURE PREDICTIONS FOR COMPLEX MIXTURES BY FOUR METHODS
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Figure C. 1. Critical Pressure Prediction for Complex Mixtures – All Data.
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Figure C. 2. Critical Pressure Prediction for Complex Mixtures -Mixture 145-1.
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Figure C. 3. Critical Pressure Prediction for Complex Mixtures – Mixture 145 – 10.
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Figure C. 4. Critical Pressure Prediction for Complex Mixtures – Mixture 4 – 6.
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Figure C.5. Critical Pressure Prediction for a Complex Mixture 75 – 5.
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Figure C. 6. Critical Pressure Prediction for a Complex Mixture 141 – 7.
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Figure C. 7. Critical Pressure Prediction for a Complex Mixture 141 – 16.
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Figure C. 8. Critical Pressure Prediction for a Complex Mixture 141 – 25.
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Figure C. 9. Critical Pressure Prediction for a Complex Mixture 58 – 1.
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Figure C. 10. Critical Pressure Prediction for a Complex Mixture 47 – 1.
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CRITICAL TEMPERATURE PREDICTION FOR COMPLEX MIXTURES BY LLS EOS METHOD
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Figure C. 11. Critical Temperature Prediction for Complex Mixtures – All Data.
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Figure C. 12. Critical Temperature Prediction for a Complex Mixture 145 - 1
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Figure C. 13. Critical Temperature Prediction for a Complex Mixture 145 – 10
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Figure C.14. Critical Temperature Prediction for a Complex Mixture 4 – 6.
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Figure C. 15. Critical Temperature Prediction for a Complex Mixture 75 – 6.
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Figure C. 16. Critical Temperature Prediction for a Complex Mixture 141 – 7.
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Figure C. 17. Critical Temperature Prediction for a Complex Mixture 141 – 25.
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Figure C. 18. Critical Temperature Prediction for a Complex Mixture 58 – 1. P
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Figure C. 19. Critical Temperature Prediction for a Complex Mixture 47 – 1.