COEFFICIENT OF ISOTHERMAL OIL COMPRESSIBILITY
FOR RESERVOIR FLUIDS BY CUBIC
EQUATION-OF-STATE
by
OLAOLUWA OPEOLUWA ADEPOJU, B.Tech. Chem Engr.
A THESIS
IN
PETROLEUM ENGINEERING
Submitted to the Graduate Faculty
of Texas Tech University in
Partial Fulfillment of
the Requirements for
the Degree of
MASTER OF SCIENCE
IN
PETROLEUM ENGINEERING
Approved
Lloyd Heinze
Chairperson of the Committee
Shameem Siddiqui
Accepted
John Borrelli
Dean of the Graduate School
December, 2006
ACKNOWLEDGEMENTS
I extend my profound gratitude to Dr. Akanni Lawal for inspiring me into phase behavior
and into this research. Special thanks to Dr. Lloyd Heinze, the chair of my Masters committee
and Dr. Shameem Siddiqui for their support.
I would like to acknowledge the African Development Bank for awarding me the 20052006 ADB/Japan Fellowship award.
I extend my profound appreciation to my family for their support and encouragement. I
also acknowledge my colleagues in Texas Tech University for their support.
To God be the glory.
ii
TABLE OF CONTENTS
ACKNOWLEDGEMENTS
ii
ABSTRACT
v
LIST OF TABLES
vi
LIST OF FIGURES
viii
LIST OF ABBREVIATIONS
x
CHAPTER
1
2
3
4
INTRODUCTION
1.1 Background Information
1
1.2 Importance of isothermal Oil Compressibility
4
1.3 Scope of the Project
9
1.4 Objectives of the Project
9
COEFFIECIENT OF ISOTHERMAL OIL COMPRESSIBILITY
2.1 Defining Equations for Isothermal Oil Compressibility
10
2.2 Isothermal Oil Compressibility Correlation and Computation Methods
24
DESIGN OF CUBIC EQUATION OF STATE
3.1 Generalized Cubic Equation of State
35
3.2 Characterization of Heavy Petroleum Fractions
43
3.3 Cubic EOS Based Isothermal Compressibility Equation
43
ANALYSIS OF PREDICTION RESULTS
4.1 Computing Isothermal Oil Compressibility from Reservoir Fluid Study Report
56
4.2 Predicted Molar Volume from Cubic Equation of State
56
4.3 Predicted Coefficient of Isothermal Oil Compressibility from Cubic
iii
5
Equation of State
66
4.4 Discussion of Results
74
CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusions
77
5.2 Recommendations
78
REFERENCES
79
APPENDIX
A. DERIVATION OF PR EOS COMPRESSIBILITY FACTOR AND MOLAR
VOLUME EQUATION.
83
B. DERIVATION OF PR EOS BASED ISOTHERMAL COMPRESSIBILITY
EQUATION.
86
C. DERIVATION OF THE PR EOS ISOTHERMAL GAS COMPRESSIBILITY
EQUATION.
88
D. DERIVATION OF THE FLASH CALCULATION ALGORITHM
90
E. MATHEMATICAL CONSISITENCY OF ISOTHERMAL COMPRESSIBILITY
ADDITION BELOW BUBBLE POINT PRESSURE.
94
F. RESERVOIR FLUID STUDY REPORT AND ANALYSIS
97
G. RESERVOIR FLUIDS CRITICAL PROPERTIES TABLE
108
H. AVERAGE ABSOLUTE PERCENT DEVIATION (AAPD)
109
I.
FORTRAN CODE DOCUMENTATION
116
J.
DEVELOPED FORTRAN CODE
120
K. VITA
162
iv
ABSTRACT
Calculations of reservoir performance for petroleum reservoirs require accurate
knowledge of the volumetric behavior of hydrocarbon mixtures, both liquid and gaseous.
Coefficient of Isothermal oil compressibility is required in transient fluid flow problems,
extension of fluid properties from values at the bubble point pressure to higher pressures of
interest and in material balance calculations38, 35. Coefficient of Isothermal oil compressibility is a
measure of the fractional change in volume as pressure is changed at constant temperature29.
Coefficients of isothermal oil compressibility are usually obtained from reservoir fluid
analysis. Reservoir fluid analysis is an expensive and time consuming operation that is not always
available when the volumetric properties of reservoir fluids are needed. For this reason
correlations have been developed and are being developed for predicting fluid properties
including the coefficient of isothermal oil compressibility.
This project developed a mathematical model for predicting the coefficient of isothermal
oil compressibility based on Peng-Robinson Equation of State (PR EOS). A computer program
was developed to predict the coefficient of isothermal compressibility using the developed model.
The predicted coefficient of isothermal oil compressibility closely matches the experimentally
derived coefficient of isothermal compressibility.
v
LIST OF TABLES
2.1 Coefficients for the co Correlations
34
4.1 Predicted Molar Volume for Oil Well No. 4, Good Oil Company, Samson,
Texas, Bubble Point = 2619.7 psia
57
4.2 Predicted molar volume for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver,
Oklahoma. Bubble Point = 2666.7 psia
60
4.3 Predicted molar volume for Jacques Unit #5603, Rough Ride Field, A.C.T.
Operating Company, Fisher County, Texas. Bubble Point = 1689.7 psia
63
4.4 Predicted Coefficient of Isothermal Oil Compressibility for Oil Well No. 4,
Good Oil Company. Samson, Texas. Bubble Point = 2619.7 psia
66
4.5 Predicted Coefficient of Isothermal Oil Compressibility for Jehlicka 1A,
Wilshire Oil Co. of Texas, Beaver, Oklahoma. Bubble Point = 2666.7 psia.
69
4.6 Predicted Coefficient of Isothermal Oil Compressibility for Jacques
Unit #5603, Rough Ride Field, A.C.T. Operating Company, Fisher County,
Texas. Bubble Point = 1689.7 psia
72
F.1 Reservoir Fluid Composition for Oil Well No. 4
98
F.2 Molar Volume Determination from Pressure-Volume Relations for Oil Well
No. 4
99
F.3 Differential Vaporization data for Oil Well No. 4
100
F.4 Reservoir Fluid Composition for Jehlicka 1A
101
F.5 Molar Volume Determination from Pressure-Volume Relations for Jehlicka 1A
102
F.6 Differential Vaporization data for Jehlicka 1A
103
F.7 Reservoir Fluid Composition for Jacques Unit #5603
104
F.8 Molar Volume Determination from Pressure-Volume Relations for Jacques
Unit #5603
105
F.9 Differential Vaporization data for Jacques Unit #5603
106
G.1 Reservoir Fluids Critical Properties
107
vi
H.1 The AAPD for the Predicted Molar Volume for Oil Well No. 4, Good Oil
Company. Samson, Texas
108
H.2 The AAPD for the Predicted Coefficient of Isothermal Oil Compressibility
For Oil Well No. 4, Good Oil Company. Samson, Texas.
109
H.3 The AAPD for the Predicted Molar Volume for Jacques Unit #5603, Rough
Ride Field, A.C.T. Operating Company, Fisher County, Texas.
110
H.4 The AAPD for the Predicted Coefficient of Isothermal Oil Compressibility
for Jacques Unit #5603, Rough Ride Field, A.C.T. Operating Company,
Fisher County, Texas (Using 11 Components).
111
H.5 The AAPD for the Predicted Molar Volume for Jehlicka 1A, Wilshire Oil
Co. of Texas, Beaver, Oklahoma.
112
H.6 The AAPD for the Predicted Coefficient of Isothermal Oil Compressibility
for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver, Oklahoma.
113
H.7 The AAPD for the Predicted Coefficient of Isothermal Oil Compressibility
for Jacques Unit #5603, Rough Ride Field, A.C.T. Operating Company,
Fisher County, Texas (Using 11 Components).
114
vii
LIST OF FIGURES
1.1 Pressure-Volume/Compressibility Relationship
7
2.1 Typical Shape of the Isothermal Gas Compressibility of Gas as a Function
of Pressure at Constant Reservoir Temperature
11
2.2 Standing and Katz compressibility factors chart
15
2.3 Variation of Reduced Compressibility with Reduced Pressures for Various
Fixed values of Reduced Temperature for Natural Gases (0.1< cr <1.0)
18
2.4 Variation of Reduced Compressibility with Reduced Pressures for Various
Fixed values of Reduced Temperature for Natural Gases (0.01< cr < 0.1)
18
2.5 Typical Shape of the Isothermal Oil Compressibility as a Function of
Reservoir Pressure at Constant Temperatures at Pressures above the
Bubble Point Pressure
20
2.6 Typical Shape of the Coefficient of Isothermal Oil Compressibility as a
Function of Pressure at Constant Reservoir Temperature
24
2.7 Trube’s Pseudo Reduced Compressibility of Undersaturated Crude Oils
25
2.8 Coefficient of Isothermal Compressibility of Undersaturated Black Oils
27
2.9 Coefficient of Isothermal Compressibility of Saturated Black Oils
29
3.1 Volumetric Behavior of Pure Compounds by Van der Waal Cubic EOS
37
3.2 Reservoir Oil at Pressure above the Bubble Point Pressure
44
3.3 Reservoir Oil at Pressure below the Bubble Point Pressure
47
3.4 Schematic diagram of Flash Liberation Experiment
49
4.1 Predicted Molar Volume for Oil Well No- 4, Good Oil Company. Samson,
Texas (Model Based on PR EOS).
58
4.2 Predicted Molar Volume for Oil Well No- 4, Good Oil Company. Samson,
Texas (Model Based on MPR EOS).
59
4.3 Predicted Molar Volume for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver,
Oklahoma. (Model Based on PR EOS).
61
viii
4.4 Predicted Molar Volume for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver,
Oklahoma. (Model Based on MPR EOS).
62
4.5 Predicted Molar Volume for Jacques Unit #5603, Rough Ride Field, Fisher
County, Texas. (Model Based on PR EOS).
64
4.6 Predicted Molar Volume for Jacques Unit #5603, Rough Ride Field, Fisher
County, Texas. (Model Based on MPR EOS).
65
4.7 Predicted Coefficient of Isothermal Oil Compressibility for Oil Well No. 4,
Good Oil Company. Samson, Texas. (Model Based on PR EOS)
67
4.8 Predicted Coefficient of Isothermal Oil Compressibility for Oil Well No. 4,
Oil Company. Samson, Texas. (Model Based on MPR EOS).
68
4.9 Predicted Coefficient of Isothermal Oil Compressibility for Jehlicka 1A,
Wilshire Oil Co. of Texas, Beaver, Oklahoma. (Model Based on PR EOS).
70
4.10 Predicted Coefficient of Isothermal Oil Compressibility for Jehlicka 1A,
Wilshire Oil Co. of Texas, Beaver, Oklahoma. (Model Based on MPR EOS)
71
4.11 Predicted Coefficient of Isothermal Oil Compressibility for Jacques
Unit #5603, Rough Ride Field, A.C.T. Operating Company, Fisher
County, Texas. (Model Based on PR EOS).
73
4.12 Predicted Coefficient of Isothermal Oil Compressibility for Jacques
Unit #5603, Rough Ride Field, A.C.T. Operating Company, Fisher
County, Texas. (Model Based on MPR EOS).
74
D.1 Schematic of the Flash Vaporization Process
90
E.1 Schematic Diagram of Volume and Isothermal Compressibility at Pressures
above and below the Bubble Point Pressure.
94
ix
LIST OF ABBREVIATIONS
Symbol
a
A
AAPD
API
b
B
Bg
Bo
Bw
cf
cg
cL
co
ct
C
Ki
Mw
p
ppc
ppr
R
Rs
Sg
So
Sw
T
Tpc
Tpr
V
z
Definition
Attraction Parameter Term of EOS
aP
2 2
Dimensionless Constant R T
1
Expt − Predicted
Average Absolute Percent Deviation AAPD = ∑
N
Exptal
Oil Gravity
Van der Waals co-volume
bP
2 2
Dimensionless Constant R T
Formation Gas Volume Factor
Formation Oil Volume Factor
Formation Water Volume Factor
Formation Compressibility
Isothermal Gas Compressibility
Isothermal Liquid Compressibility
Isothermal Oil Compressibility
Total System Isothermal Compressibility
Characterization Factor
Equilibrium Ratio
Molecular Weight
Pressure
Pseudo-Critical Pressure
Pseudo-Reduced Pressure
Gas Constant
Solution Gas-Oil Ratio
Gas Saturation
Oil Saturation
Water Saturation
Absolute Temperature
Pseudo-Critical Temperature
Pseudo-Reduced Temperature
Molar Volume
Compressibility Factor
Greek Letter
α
β
Parameter of LLS EOS
Parameter of LLS EOS
x
δ ij
γ
ρ
ω
ψi
Binary Interaction Parameter
Specific Gravity
Density
Acentric Factor
Fugacity Parameter
ψi =
∑ [x (a a )
ψ = ∑∑ [x x (a a ) ]
0.5
j
i
j
j
ψ
Fugacity Parameter
0.5
i
i
Ω
j
i
j
Dimensionless Parameter
Subscript
c
g
o
r
i
Critical Property
Gas
Oil
Reduced
Component i
xi
j
]
CHAPTER I
INTRODUCTION
1.1.Background Information
The coefficient of isothermal compressibility is defined as the fractional change
of fluid volume per unit change in pressure at constant reservoir temperature1. The
coefficient of isothermal oil compressibility, co, is usually determined from the pressurevolume measurements of reservoir fluids. These data are usually obtained from reservoir
fluid analysis. A convenient method of estimating the coefficient of isothermal
compressibility for reservoir fluids for a finite change in pressure and volume is to use
this simple equation41.
co = −
V1 − V2
V1 (p 2 − p1 )
(1.1)
Normally, an increase in fluid pressure (p2 > p1) causes the fluid volume to
decrease (V1 > V2). Hence the negative sign, to make the value of the isothermal
compressibility positive.
Isothermal compressibility is usually recorded for gas, oil, water and rock. In
order to have the number on the same basis, compressibility is recorded in 10-6 for
convenience. The unit of isothermal compressibility is the reciprocal of psi, psi-1,
sometimes called sip. A value of 10-6 psi-1 is called a microsip.
Equation 1.1 is useful in approximating the isothermal compressibility of single
phase gases and liquids undergoing small pressure changes. This assumption has a
1
definite limitation when the compressibility varies during small pressure changes. It is
further limited over large pressure changes by confusion over whether the denominator
should be V1 or V2, or some value in between41. To overcome this limitation, equation
1.1 is usually expressed in partial differential form at constant temperature, as follows
co = −
1 ⎛ ∂V ⎞
⎜
⎟
V ⎜⎝ ∂p ⎟⎠ T
(1.2)
Equation 1.2, defines the instantaneous coefficient of isothermal compressibility
of a substance at a point on an isothermal pressure-volume curve for that substance.
Based on equation 1.2, the coefficient of isothermal compressibility is defined as the
fractional change of volume as pressure is changed at constant temperature.
The isothermal oil compressibility is a point function and it can be calculated
from the slope of a pressure versus specific volume curve, or from the differentiation of
an equation of state, or a correlation involving compressibility factor z, density or
formation volume factor
26, 10
. There are also a number of correlations that are available
to calculate the isothermal compressibility.
To construct a curve of pressure versus specific volume, the pressure and volume
data required are acquired from laboratory studies on reservoir samples collected from
the bottom of the wellbore and from the surface. Such experimental data are not always
available because of one or more of the following reasons: (i) reservoir samples collected
are not reliable. (ii) samples have not been taken because of cost savings. (iii) PVT
analyses are not available when data are needed. (iv) obtaining an accurate PVT behavior
of each reservoir fluid encountered will be costly and time-consuming. In such cases
2
when the experimental data are not available, PVT properties such as the isothermal oil
compressibility are determined from empirically derived correlations or equation of state.
There has been a lot of work in the last 50 years on the derivation of PVT
correlations. However, each of these correlations is applicable to a good degree of
reliability only in a well-defined range of reservoir fluid characteristics. This is due to the
fact that each correlation has been developed based on fluid samples from a restricted
geographical area, with similar fluid compositions and API gravity16. Some correlations
are also applicable only at a well-defined range of temperature and pressure.
Methods of predicting reservoir properties and performance, particularly those
based on the compositional-material-balance depend on the capability of accurately
expressing reservoir fluid properties as functions of pressure, temperature and
composition10. An equation of state has been defined as an analytical expression relating
the pressure to the volume and temperature. Equation of state (EOS) is used to describe
the volumetric behavior, the vapor/liquid equilibra (VLE) and thermal properties of pure
substances and mixtures3.
The equation of state has to be expressed in the form of the defining equation for
the particular reservoir properties it is describing. For the isothermal compressibility, the
equation of state is expressed in volume and partially differentiated with respect to
pressure at constant temperature. Also, since the evaluation of the isothermal
compressibility involves the derivatives of the volumetric data, the possible deviation in
these quantities is much greater than is encountered with the volumetric data only.
3
Accordingly, if the derived quantities are to be obtained from an equation for volume, it
is essential that the error be reduced to a minimum10.
Since the introduction of the equation of state by Van der Waals, many equation
of state (EOS) having the form of Van der Waal’s equation have been proposed. Two of
the most common and popularly accepted EOS is the Redlich-Kwong (RK EOS) and
Peng Robinson (PR EOS) EOS. The Redlich-Kwong equation has been the most popular
basis for developing new EOS’s. Several modifications of the Redlich-Kwong equation
have found acceptance, with Soave’s modification (SRK EOS) being the simplest and the
most widely used. Another trend has been to propose generalized three-, four-, and five –
constant cubic equations that can be simplified into the PR EOS, RK EOS and other
familiar forms44,
pp 47
. The Lawal-Lake-Siberberg (LLS EOS) EOS is a four parameter
EOS, which can be reduced to both the PR EOS and the RK EOS. The equation of state
used in this study to generate the isothermal compressibility for reservoir fluids are the
PR EOS and modified PR EOS.
1.2 Importance of Isothermal Oil Compressibility
Isothermal compressibility is used in a wide range of calculations involving
production and exploration of hydrocarbon reservoirs. Some of the reservoir
engineering applications for the isothermal compressibility include well testing analysis
and metering18.
Isothermal compressibility is required in all solutions of transient flow problems.
Solutions of pressure buildup and drawdown problems contain a parameter called the
4
total system isothermal compressibility. Calculations of the total system isothermal
compressibility involve the evaluation of the isothermal compressibility coefficient for
reservoir fluids as well as the compressibility of the formation. The total system
isothermal compressibility is expressed as,
c t = c oSo + c w S w + c g Sg + c f
(1.3)
ct= Total Isothermal Compressibility, psi-1
co= Coefficient of Isothermal Oil Compressibility, psi-1
cg= Coefficient of Isothermal Gas Compressibility, psi-1
cw= Coefficient of Isothermal Water Compressibility, psi-1
cf= Formation Compressibility, psi-1
So= Oil Saturation, fraction
Sw= Water Saturation, fraction
Sg= Gas Saturation, fraction
The use of accurate total system compressibility in a properly-executed buildup
and drawdown analyses includes35:
o Better planning of pressure build-ups may be achieved to avoid unnecessary
loss of revenue due to excessively long shut-in periods, or to shut-in periods too
short to yield useable data. For example the equation to estimate the time at
which the boundary effect can be felt in a pressure drawdown test is given as:
⎡ Φμc t A ⎤
t eia = ⎢
⎥ (t DA )eia
⎣ 0.0002637k ⎦
(1.4)
teia = Time to end the infinite-acting period, hours
A = Well drainage area, ft2
5
ct = Total Compressibility, psi-1
(tDA)eia = Dimensionless time to the end of the infinite-acting period
k = Permeability, md
Φ = Porosity, fraction
μ = Viscosity, cp
o Better and more reliable estimates of the static reservoir pressure for reserves
estimate and rate performance estimates.
o Reliable information for evaluation of well completion effectiveness, and
planning and interpretation of well stimulation efforts.
The isothermal compressibility coefficient is also essentially the controlling factor
in identifying the type of reservoir fluid. Reservoir fluids are generally classified into
three groups:
o Incompressible fluids – These are fluids whose volume does not change with
pressure i.e. the values of the coefficient of isothermal compressibility for these
fluids are constant. Incompressible fluid does not exist; however, this behavior
may be assumed in some cases to simplify the derivation and the final form of
flow equations.
o Slightly Compressible fluids – These “slightly” compressible fluids exhibit
small changes in volume, or density, with changes in pressure i.e. there is a
slight change in the coefficient of isothermal compressibility with changes in
pressure. Crude oil and formation water are slightly compressible fluids.
o Compressible fluids – These fluids experiences a large change in volume as a
function of a change in pressure i.e. the coefficient of isothermal compressibility
changes drastically with a change in pressure5.
6
Incompressible
Volume / Coefficient of
Isothermal Compressibility
Slightly Compressible
Compressible
Pressure
Figure 1.1: Pressure – Volume/ Compressibility Relationship5.
Isothermal oil compressibility is also used in the material balance equation for
undersaturated oil reservoirs, i.e. reservoirs that the initial reservoir pressure is greater
than the bubble point pressure. This material balance equation can be used to estimate
the initial reserve and predict future production. The material balance equation for
undersaturated oil reservoir is expressed as:
NB oi c e (p i − p ) = N p B o − We + B w Wp
ce =
So c o + S w c w + c f
B oi (p i − p )
(1.5)
(1.6)
N = Initial Oil in Place, STB
Boi = Initial Oil Formation Volume Factor, bbl/STB
ce= Effective Compressibility, psi-1
pi = Initial Reservoir Pressure, psi
7
p = Reservoir Pressure, psi
Np = Cumulative Oil Produced, STB
Bo = Formation Volume Factor, bbl/STB
Bw = Water Formation Volume Factor, bbl/STB
We = Water Influx, bbl
Wp = Water Produced, STB
The isothermal compressibility is used also for the extension of fluid properties
from correlations starting at the bubble point pressure to pressures above the bubble
point pressure. This application is used for black oil reservoir simulation38. The
coefficient of isothermal compressibility can be used to estimate the formation volume
factor and oil density at higher pressures from their calculated values at bubble point.
This can be shown by the following equations:
B o = B ob e (co (p b − p ))
(1.7)
ρ o = ρ ob e (co (p − p b ))
(1.8)
The isothermal compressibility is an important physical property in the design of
high-pressure surface equipments7. Compressibility is important in the design of
process equipment handling liquids at high pressure. The determination of the
isothermal compressibility has specific applications in the transfer of crude oil
containing dissolved gas components from a reservoir to a gas-oil separation plant and
the pipeline transportation of degassed liquids such as crude oils and refined products.
8
Experimental data for the isothermal compressibility of such mixtures are seldom
available11.
1.3 Scope of the Project
The scope of this project is to develop a mathematical model based on two cubic
equations of states (PR EOS and Modified PR EOS) that can predict the isothermal oil
compressibility with a good degree of accuracy at pressures below and above the
bubble point.
1.4 Objectives of the Project
The objective of this project is to design an equation of state based mathematical
model that be used to predict the coefficient of isothermal compressibility for reservoir
fluids. This equation will allow the coefficient of isothermal compressibility to vary
with both pressure and composition.
9
CHAPTER II
COEFFICIENT OF ISOTHERMAL OIL COMPRESSIBILITY
2.1 Defining Equations for Isothermal Oil Compressibility
In
petroleum
engineering
calculations,
the
coefficient
of
isothermal
compressibility is required for oil, gas, water and the formation. These coefficients are
usually combined as the total system compressibility. The coefficient of isothermal
compressibility for oil, gas and water is defined as the fractional change in volume per
unit change in pressure at constant temperature.
2.1.1 The Coefficient of Isothermal Compressibility for Reservoir Gases
Fluid compressibility varies with pressure and temperature and the knowledge of
these variations is essential in reservoir engineering calculations. Liquid phase
compressibility is usually small and usually assumed constant. The coefficient of
isothermal compressibility for reservoir liquids tends to be pressure sensitive but not
nearly so much as the reservoir gas41. Therefore the gas phase compressibility is neither
small nor constant.
The isothermal gas compressibility is the change in volume per unit volume for a
unit change in pressure at constant temperature2 p. 106. This is expressed as29 p. 170,
cg = −
1 ⎛ ∂V ⎞
⎟
⎜
V ⎜⎝ ∂p ⎟⎠ T
(2.1)
10
1 ⎛ ∂v ⎞
c g = − ⎜⎜ ⎟⎟
v ⎝ ∂p ⎠ T
cg = −
1
Vm
⎛ ∂Vm
⎜⎜
⎝ ∂p
(2.2)
⎞
⎟⎟
⎠T
(2.3)
cg = Isothermal Gas Compressibility, psi-1
V = Gas Volume, cubic feet or bbl
v = Specific Volume, cubic feet/kg or bbl/lb
Vm = Molar Volume, bbl/lb-mole
The isothermal gas compressibility increases as the reservoir pressure decreases.
The typical relationship of the isothermal gas compressibility with reservoir pressure is
shown in the Figure 2.1.
Isothermal Gas
Compressibility, psi-1
Reservoir Pressure
Figure 2.1 – Typical Shape of the Isothermal Gas Compressibility of gas as a function
of pressure at constant reservoir temperature29 p. 170.
11
The partial derivative is used in equation 2.1, 2.2 and 2.3 rather than the ordinary
derivative because the volume varies only with the pressure while the temperature is
held constant.
2.1.1.1 Isothermal Gas Compressibility of an Ideal Gas29 p. 172
Equation of state for ideal gas is usually combined with the equation for the
isothermal gas compressibility to increase its usefulness. The ideal gas equation is
given as:
pV = nRT
(2.4)
In terms of volume,
V=
nRT
p
(2.5)
Differentiating equation 2.5, with respect to pressure yields,
⎛ ∂V ⎞
nRT
⎟⎟ = − 2
⎜⎜
p
⎝ ∂p ⎠ T
(2.6)
Therefore the isothermal gas compressibility equation for ideal gases can be expressed
as;
⎛ 1 ⎞⎛ nRT ⎞
c g = ⎜ − ⎟⎜⎜ − 2 ⎟⎟
⎝ V ⎠⎝ p ⎠
(2.7)
Substituting for V in equation 2.7, yields;
12
p ⎞⎛ nRT ⎞ 1
⎛
cg = ⎜ −
⎟⎜⎜ − 2 ⎟⎟ =
⎝ nRT ⎠⎝ p ⎠ p
(2.8)
where
n= Number of Moles, lb-mole
R = Universal Gas Constant, (psi, cu ft)/ (lb mole oR)
T = Temperature, oR
p = Pressure, psi
Equation 2.8 shows that the isothermal gas compressibility for ideal gas is
inversely proportional to pressure. Though the ideal gas equation does not adequately
describe the behavior of gases at the temperatures and pressures normally encountered
in reservoir engineering, equation 2.8 can be used to estimate the order of magnitude of
the isothermal gas compressibility.
2.1.1.2 Isothermal Gas Compressibility of Real Gas29, p. 173
The real gas equation of state is commonly used in petroleum engineering
calculations. The real gas equation can be used to model the isothermal gas
compressibility for real gases. The gas deviation factor or the compressibility factor (z)
varies with pressure, therefore it is considered as a variable.
The real gas equation of state is given as;
pV = znRT
(2.9)
13
This real gas equation can be expressed in terms of volume, to yield;
V = nRT
z
p
(2.10)
Differentiating equation 2.10 in terms of pressure at constant temperature yields;
⎡ 1 ⎛ ∂z ⎞
⎛ ∂V ⎞
z⎤
⎜⎜
⎟⎟ = nRT ⎢ ⎜⎜ ⎟⎟ − 2 ⎥
⎢⎣ p ⎝ ∂p ⎠ T p ⎥⎦
⎝ ∂p ⎠ T
(2.11)
Substituting equation 2.11 into equation 2.1 yields;
⎡ 1 ⎛ ∂z ⎞
z ⎤ ⎫⎪
p ⎤ ⎧⎪
⎡
⎜
⎟
−
nRT
c g = ⎢−
⎢
⎨
2 ⎥⎬
⎜ ⎟
⎣ znRT ⎥⎦ ⎪⎩
⎢⎣ p ⎝ ∂p ⎠ T p ⎥⎦ ⎪⎭
(2.12)
Simplifying equation 2.12 yields;
cg =
1 1 ⎛ ∂z ⎞
− ⎜ ⎟
p z ⎜⎝ ∂p ⎟⎠ T
(2.13)
⎛ ∂z ⎞
The partial derivative ⎜⎜ ⎟⎟ can be determined from the slope of z-factor plotted
⎝ ∂p ⎠ T
against pressure at constant temperature. It can be observed from Figure 2.2 that at low
pressures, the z-factor decreases as pressure increases, causing the slope to be negative
thus making the cg larger than the case of an ideal gas. At high pressures, the z-factor
increases with increasing pressure, therefore the slope of the z-factor chart will be
positive; thus the values of cg will be smaller than the case of an ideal gas.
14
Figure 2.2: Standing and Katz compressibility factors chart. Courtesy of the Gas
Processors Suppliers Association. Published in the GPSA Engineering Data Book,
Tenth Edition, 1987.2 p. 92
15
For an ideal gas, z-factor is a constant with a value of 1.0. Therefore the value of
the partial derivative of z-factor with respect to pressure at constant temperature equal
to zero. Therefore equation 2.13 reduces to equation 2.8.
2.1.1.3 Isothermal Pseudo-reduced Compressibility29 p. 175
The law of corresponding state states that all pure gases have the same z-factor at
the same values of reduced pressure and reduced temperature. The reduced property of
a substance is the value of that property divided by the critical property for the
substance.
The law of corresponding states can be used to express the isothermal gas
compressibility equation in reduced form. The chain rule is used to express the partial
derivative of the z-factor with respect to pressure in reduced form.
⎛ ∂z ⎞
⎛ ∂p
⎜⎜ ⎟⎟ = ⎜⎜ Pr
⎝ ∂p ⎠ T ⎝ ∂p
⎞⎛ ∂z
⎟⎟⎜⎜
⎠⎝ ∂p Pr
⎞
⎟⎟
⎠
(2.14)
The pseudo-reduced pressure is expressed as;
p = p pc p pr
(2.15)
The derivative of the pseudo-reduced pressure with respect to pressure is
expressed as;
⎛ ∂p pr
⎜⎜
⎝ ∂p
⎞
1
⎟⎟ =
⎠ p pc
(2.16)
Substituting equation 2.16 into equation 2.14 gives;
16
⎛ ∂z ⎞
1 ⎛⎜ ∂z
⎜⎜ ⎟⎟ =
⎝ ∂p ⎠ T p pc ⎜⎝ ∂p pr
⎞
⎟
⎟
⎠
(2.17)
Substituting equation 2.17 and equation 2.15 into equation 2.13 gives;
cg =
1
1 ⎛⎜ ∂z
−
p pc p pr zp pc ⎜⎝ ∂p pr
⎞
⎟
⎟
⎠ Tpr
(2.18)
This can be better expressed in dimensionless pseudo-reduced compressibility cpr
as;
c pr = c g p pc =
1 1 ⎛⎜ ∂z
−
p pr z ⎜⎝ ∂p pr
⎞
⎟
⎟
⎠ Tpr
(2.19)
Where
cpr = Isothermal Pseudo-reduced Compressibility
ppc = Pseudo-critical pressure, psi
ppr = Pseudo-reduced pressure
Tpr = Pseudo-reduced temperature
Trube41 (1957) presented graphs (Figures 2.3 and 2.4) from which the isothermal
compressibility of natural gases may be obtained. The graphs give the isothermal
pseudo-reduced compressibility as a function of pseudo-reduced temperature and
pressure.
17
Figure 2.3: Variation of reduced compressibility with reduced pressures for various
fixed values of reduced temperature for natural gases. (For cr between 1.0 and 0.1)
Trube41 (1957).
Figure 2.4: Variation of reduced compressibility with reduced pressures for various
fixed values of reduced temperature for natural gases. (For cr between 0.1 and 0.01)
Trube41 (1957)
18
2.1.2 The Coefficient of Isothermal Oil Compressibility29 p. 231
The isothermal oil compressibility is required in the determination of the physical
properties of the undersaturated crude oil. Isothermal oil compressibility is usually
determined from a laboratory reservoir fluid study. At pressures above the bubble point,
the isothermal oil compressibility is defined exactly as the coefficient of isothermal
compressibility of a gas. At pressures below the bubble point, there is the presence of
free gas in the reservoir; therefore an additional term must be added to account for the
volume of gas which evolves.
2.1.2.1 Isothermal Oil Compressibility at Pressures above the Bubble point
The isothermal oil compressibility, co at pressures above the bubble point is
defined as the fractional change in volume of oil as pressure is changed at constant
temperature. The isothermal oil compressibility is expressed as29 p. 231;
co = −
1 ⎛ ∂V ⎞
⎜
⎟
V ⎜⎝ ∂p ⎟⎠ T
(2.20)
where
co = Isothermal Oil Compressibility of crude oil, psi-1
The isothermal oil compressibility at pressures above the bubble point is virtually
constant except at pressures near the bubble point. The typical relationship of co with
pressure above the bubble point pressure is shown in Figure 2.5.
19
Isothermal Oil Compressibility, Co psi-1
pb
Reservoir Pressure
Figure 2.5: Typical shape of the isothermal oil compressibility as a function of reservoir
pressure at constant temperature at pressures above the bubblepoint29 p. 232.
Equation 2.20 can be expressed as;
⎛ ∂ ln V ⎞
⎟⎟
c o = −⎜⎜
⎝ ∂p ⎠ T
(2.21)
Formation volume factor, Bo can be substituted directly into equation 2.20 as;
co = −
1
Bo
⎛ ∂B o
⎜⎜
⎝ ∂p
⎞
⎟⎟
⎠T
(2.22)
Equation 2.22 can be integrated as co is assumed to be constant at pressures above
the bubble point.
p
Bo
dB o
Bo
Bob
c o ∫ dp = − ∫
pb
(2.23)
These results in;
B o = B ob e [c o (p b − p )]
(2.24)
20
Equation 2.24 is used to calculate the formation volume factor at pressures above
the bubble point.
The isothermal oil compressibility can also be written in terms of density. The
isothermal oil compressibility in terms of specific volume v is written as;
1 ⎛ ∂v ⎞
c o = − ⎜⎜ ⎟⎟
v ⎝ ∂p ⎠ T
(2.25)
The density of oil, ρo is related to the specific volume, v by;
v=
1
ρo
(2.26)
Chain rule can be used to obtain the partial derivative of the specific volume with
respect to pressure at constant temperature.
⎛ ∂v ⎞
∂v
⎜⎜ ⎟⎟ =
⎝ ∂p ⎠ T ∂ρ o
⎛ ∂ρ o
⎜⎜
⎝ ∂p
⎞
⎟⎟
⎠T
(2.27)
Differentiating equation 2.26 with respect to density yields;
∂v
1
=− 2
∂ρ o
ρo
(2.28)
Substituting equations 2.28, 2.27 and 2.26 into equation 2.25 gives;
⎡ 1 ⎤⎡ 1
c o = −⎢
⎥ ⎢− 2
⎣1 ρ o ⎦ ⎣⎢ ρ o
⎛ ∂ρ o
⎜⎜
⎝ ∂p
⎞ ⎤
⎟⎟ ⎥
⎠ T ⎦⎥
(2.29)
Thus,
co =
1
ρo
⎛ ∂ρ o
⎜⎜
⎝ ∂p
⎞
⎟⎟
⎠T
(2.30)
21
Assuming constant co at pressures above the bubble point, equation 2.30 can be
integrated to give 29 p. 234;
ρ o = ρ ob e [c o (p − p b )]
(2.31)
Equation 2.31 can be used to calculate the density of oil at pressures above the
bubble point. The density at the bubble point is the starting point.
2.1.2.2. Isothermal Oil Compressibility at Pressures below the Bubble Point
When reservoir pressure is below the bubble point pressure, there is the presence
of free gas in the reservoir. These types of reservoir are called saturated oil reservoirs.
The total change in volume as pressure is changed must account for the effect of the
solution gas.
The fractional change in volume per unit pressure change at constant temperature
is expressed as:
co = −
1 ⎛ ∂V ⎞
1
⎟⎟ = −
⎜⎜
V ⎝ ∂p ⎠ T
Bo
⎛ ∂B o
⎜⎜
⎝ ∂p
⎞
⎟⎟
⎠T
(2.32)
The oil formation volume factor (Bo) contains the effect of the solution gas (Rs)
on the change in liquid phase volume35. The effect of the change in the solution gas on
the liquid volume as pressure is reduced below the bubble point must be added to
equation 2.32.
The change in the solution gas oil ratio is expressed as:
B g ⎛ ∂R s
⎜
B o ⎜⎝ ∂p
⎞
⎟⎟
⎠T
(2.33)
22
Thus the isothermal oil compressibility at pressures below the bubble point
pressure is expressed as27.
co = −
1
Bo
⎛ ∂B o
⎜⎜
⎝ ∂p
Bg
⎞
⎟⎟ +
⎠ T Bo
⎛ ∂R s
⎜⎜
⎝ ∂p
⎞
⎟⎟
⎠T
(2.34)
⎞ ⎤
⎟⎟ ⎥
⎠ T ⎦⎥
(2.35)
This is better expressed as;
co = −
1
Bo
⎡⎛ ∂B o
⎢⎜⎜
⎣⎢⎝ ∂p
⎞
⎛ ∂R
⎟⎟ − B g ⎜⎜ s
⎠T
⎝ ∂p
where
Rs= Solution gas-oil ratio, Scf/STB
Bg = Gas formation volume factor, res bbl/Scf
Bo = Oil formation volume factor, res bbl/STB
Equation 2.35 is consistent with the equation for calculating the isothermal
compressibility at pressures above the bubble point; because at pressures above the
bubble point pressure the gas solubility is constant and the derivative of Rs with
pressure is zero.
The typical relationship of the isothermal oil compressibility as a function of
reservoir pressure shows a discontinuity at the bubble point pressure (Figure 2.6). This
discontinuity is caused by the large increase in the value of the isothermal
compressibility at the evolution of the first bubble of gas.
23
Coefficient of Isothermal Oil Compressibility, psi-1
Pb
Figure 2.6: Typical shape of the coefficient ofReservoir
isothermal
Pressure, oil
psi compressibility as a
29 pp 236
.
function of pressure at constant reservoir temperature
2.2 Isothermal Compressibility Correlation and Computation Methods
Laboratory PVT analysis is used to acquire data that are used in the estimation of
the isothermal compressibility from the equations given above. Values of the fluid
properties including the isothermal compressibility are often required when laboratory
PVT data are not available. Thus, there are a number of developed correlations for the
estimation of the isothermal compressibility using readily available fluid properties.
2.2.1 Trube’s Correlation42
As shown in figures 2.3, 2.4 and 2.7, Trube42 (1957) presented a correlation for
estimating the pseudo reduced compressibility cr, for natural gases and undersaturated
crude oils. The pseudo reduced compressibility was correlated with the pseudo reduced
temperature and pressure, ppr and Tpr. Trube’s graphical correlation can be used to
24
estimate the isothermal compressibility at any pseudo reduced temperature and
pressure. The relationship between the pseudo reduced isothermal compressibility and
the isothermal compressibility is expressed as:
c r = c o p pc
(2.36)
where
cr = Pseudo reduced compressibility, dimensionless
co = Isothermal Oil Compressibility, psi-1
ppc = Pseudo-critical pressure, psi
Figure 2.7 – Trube’s pseudo reduced compressibility of undersaturated crude oils42
25
2.2.2 Vasquez-Beggs’ Correlation43
Vasquez and Beggs43 (1980) developed a correlation for the isothermal oil
compressibility with the gas solubility Rs, reservoir temperature T, APIo gravity, gas
specific gravity γg and reservoir pressure p. They proposed the following expression:
co =
− 1433 + 5R s + 17.2(T − 460) − 1180γ gs + 12.61γ API
10 5 p
(2.37)
Vasquez and Beggs proposed that the value of the gas specific gravity γg obtained
at the separator pressure of 100 psig be used for the above equation. This reference
pressure was chosen because it represents the average reservoir field separator pressure.
They also proposed the relationship for adjustment of the gas gravity γg to the reference
separator pressure γgs.
⎡
⎛ p sep ⎞⎤
⎟⎟⎥
γ gs = γ g ⎢1 + 5.912(10 −5 )(γ API )(Tsep − 460)Log⎜⎜
⎢⎣
⎝ 114.7 ⎠⎥⎦
(2.38)
where
γgs = gas gravity at the reference separator pressure
γg = gas gravity at the actual separator conditions of psep and Tsep
psep = actual separator pressure, psi
Tsep = actual separator temperature, oR
This correlation is also represented in a graphical form (Figure 2.8). This is the
best available correlation considering both accuracy and ease of use. The results are
generally low by as much as 50 percent at high pressures29. Accuracy is increased as
the bubble point pressure is approached.
26
Figure 2.8: Coefficients of isothermal compressibility of undersaturated black
oils29p. 327
27
2.2.3 McCain et al Correlation for Isothermal Oil Compressibility at Pressures above
the Bubble Point
The work of McCain30 (1988) presented a correlation for estimating the
isothermal oil compressibility at pressures below the bubble point. The correlation is
expressed as:
ln(c o ) = −7.633 − 1.497ln(p ) + 1.115ln(T ) + 0.533ln(γ API ) + 0.184ln(R sb )
(2.39)
The results are accurate to within 10% at pressures above 500 psia. Below 500
psia, the accuracy is within 20%. If the bubble point pressure is known, the accuracy of
the correlation can be improved by using this expression proposed by McCain28 :
ln(c o ) = −7.573 − 1.450ln(p ) − 0.383ln(p b ) + 1.402ln(T ) + 0.256ln(γ API ) + 0.449ln(R sb )
(2.40)
This correlation is also expressed graphically in Figure 2.9:
28
Figure 2.9 – Coefficient of Isothermal Compressibility of Saturated Black Oil.
McCain30 (1988)
29
2.2.4 Ahmed’s Correlation
Ahmed4 (1985) used 245 experimental data points to propose a mathematical
expression for the isothermal oil compressibility using the gas solubility Rs as the only
correlating parameter. Other correlating parameter such as γo, γg, and T are
implemented in the equation through the gas solubility Rs5.
The correlation is expressed as:
co =
1
e (a 3 p )
a1 + a 2R s
(2.41)
where
a1 = 24841.0822
a2 = 14.07428745
a3 = -0.00018473
The average absolute error of this expression is given as 3.9 percent when tested
against the experimental data used in developing the equation. The isothermal oil
compressibility can also be determined from this expression:
1.175
⎤
⎡
⎡ ⎛ γ g ⎞ 0.5
⎤
⎥
⎢ a + a ⎢R ⎜ ⎟ + 1.25(T − 460)⎥
2
s⎜
⎟
⎥
⎢ 1
⎢⎣ ⎝ γ o ⎠
⎥⎦
⎥ e (a 3 p )
⎢
co =
⎥
⎢
a 4 γO + a 5R s γg
⎥
⎢
⎥
⎢
⎦
⎣
where
a1 = 1.026638
a2 = 0.0001553
30
(2.42)
a3 = -0.0001847272
a4 = 62400
a5 = 13.6
This correlation only applies for isothermal oil compressibility at pressures above
the bubble point pressure.
2.2.5 De Ghetto et al Correlations15
De Ghetto et al15 (1994) evaluated the reliability of some PVT correlations and
came up with some modified correlations which they reported as being more accurate.
They characterized the fluid samples used in their studies as extra-heavy oils
(oAPI ≤ 10), heavy oils (10 < oAPI ≤ 22.3), medium oils (22.3 < oAPI ≤ 31.1) and light
oils (oAPI > 31.1). They reported that the errors on the correlations were decreased by
about five percentage points. The most significant improvements for the entire sample
of the oils were from 24.5% to 19.8%. The modified correlations were given as:
Extra-heavy oils: Modified Vasquez-Beggs correlation
co =
− 889.6 + 3.1374R s + 20T − 627.3γ gcorr − 81.4476γ API
p10 5
(2.43)
Heavy oils: Modified Vasquez-Beggs correlation
co =
− 2841.8 + 2.9646R s + 25.5439T − 1230.5γ gcorr − 41.91γ API
p10 5
31
(2.44)
Medium oils: Modified Vasquez-Beggs correlation
co =
− 705.288 + 2.2246R s + 26.0644T − 2080.823γ gcorr − 9.6807γ API
p10 5
(2.45)
Light oils: Modified Labedi’s Correlation
(
c o = 10 −6.1646 B o
1.8789
)
⎛ p
0.1966
− ⎜⎜1 − b
γ 0.3646
API T
p
⎝
⎞ −8.98 3.9392 1.349
⎟⎟ 10
Bo T
⎠
(
)
(2.46)
Where
⎡
⎛ p sp ⎞ − 4 ⎤
⎟⎟10 ⎥
γ gcorr = γ g p sp ⎢1 + 0.5912γ API Tsp Log⎜⎜
⎝ 114.7 ⎠
⎦⎥
⎣⎢
(2.47)
T = Reservoir temperature, oF
p = Reservoir pressure, psia
psp = Separator pressure, psia
2.2.6 Dinodruk-Christman Correlation for the Gulf of Mexico16
Dinodruk and Christman16 (2001) proposed a set of PVT correlations for the Gulf
of Mexico. The proposed oil compressibility correlation predicts the oil compressibility
values with an average relative error of -0.85% and average absolute relative error of
6.21%. The proposed isothermal oil compressibility is for undersaturated oil reservoirs.
(c o )bp
(
)
= 4.487462368 + 0.005197040A + 0.000012580A 2 × 10 −6
32
(2.48)
where
1.759732076
7
⎛ R s0.980922372 γ 0.02100307
⎞
g
0.300001059
⎜
⎟
(
)
+
−
−
20.0000635
8
T
60
0.87681362
2R
s
0.338486128
⎜
⎟
γ
0
⎝
⎠
A=
2
−1.713572145 ⎞
⎛
⎜ 2.749114986 + (T − 60 ) 2R s
⎟
9.999932841 ⎟
⎜
γ
g
⎝
⎠
(2.49)
2.2.7 Spivey et al. Correlation38
Spivey et al.38 (2005) proposed a set of correlations for estimating the isothermal
oil compressibility for three applications in reservoir engineering. The applications and
the correlation for each application are given as follows:
o Correlation for Coefficient of Isothermal Compressibility from the Bubble point
to a Pressure of Interest
This correlation is used to estimate the isothermal oil compressibility when
the values of the isothermal oil compressibility will be used to extend the values
of some fluid properties from the bubble point pressure of the oil to higher
pressures.
ln (c ofb ) = 2.434 + 0.475z + 0.048z 2
(2.50)
6
z = ∑ zn
(2.51)
n =1
z n = C 0,n + C1,n x n + C 2,n x 2n
(2.52)
33
Table 2.1 Coefficients for the co correlation Equation. 2.52
n
x
C0,n
C1,n
C2,n
1
ln oAPI
3.011
-2.6254
0.497
2
ln γgsp
-0.0835
-0.259
0.382
3
ln pb
3.51
-0.0289
-0.0584
4
ln p/pb
0.327
-0.608
0.0911
5
ln Rsb
-1.918
-0.642
0.154
6
ln TR
2.52
-2.73
0.429
o Correlation for Coefficient of Isothermal Compressibility from Initial Pressure
to a Pressure of Interest.
The value of the isothermal oil compressibility at initial pressure, cofb, can be
used to estimate the isothermal compressibility at lower pressure, cofi.
c ofi =
(p − p b )c ofb p − (p i − p b )c ofb p i
p − pi
(2.53)
o Correlation for Coefficient of Isothermal Compressibility Tangent at Some
Pressure of Interest.
This gives the tangent or instantaneous compressibility, co.
c o = c ofb + (p − p b )c ofb (0.475 + 0.096z )
34
− 0.608 + 0.1822ln
p
p
pb
(2.54)
CHAPTER III
DESIGN OF CUBIC EQUATION OF STATE
3.1 Generalized Cubic Equation of State
3.1.1 Equation of state
Equation of state can be defined as a mathematical expression that relates pressure
p, volume V and Temperature T. The simplest form of an equation of state (EOS) is the
ideal gas equation.
p=
RT
V
(3.1)
where V = gas volume in ft3 per mole of gas
The ideal gas equation is only used at pressures close to the atmospheric pressure
where real gas behavior can be assumed to be ideal.
The behavior of most real gases cannot be predicted by the ideal gas equation;
therefore a correction factor, called the compressibility factor or the gas deviation factor
is inserted into the ideal gas equation.
p=
zRT
V
(3.2)
where, z = gas deviation factor.
35
3.1.2 Van der Waals’ (VDW) EOS
In developing the ideal gas equation of state, the following assumptions were
made;
o The volume occupied by the molecules of the gas is insignificant compared to
the volume occupied by the gas.
o There are no attractive and repulsive forces between the molecules of the gas or
between the molecules of the gas and the walls of the container.
o All collisions of the gas molecules are perfectly elastic.
In 1873, Van der Waal improved the ideal gas equation by attempting to eliminate
the first two assumptions of the ideal gas equation. Van der Wall introduced the
parameter
a
and b to account for the intermolecular attractive and repulsive forces
V2
respectively.
p=
a
RT
− 2
V−b V
(3.3)
The Van der Waal EOS written in terms of volume or the compressibility factor
takes a cubic form and is often referred to as a cubic EOS.
⎛
RT ⎞ 2 ⎛ a ⎞
ab
⎟⎟V + ⎜⎜ ⎟⎟V −
V 3 − ⎜⎜ b +
=0
p ⎠
p
⎝
⎝p⎠
(3.4)
z 3 − (1 + B)z 2 + Az − AB = 0
(3.5)
where the dimensionless parameter A and B are defined as,
A=
aP
(RT )2
(3.6)
36
B=
bp
RT
(3.7)
The typical volumetric behavior of Van der Waal EOS is shown in the Figure 3.1.
T=Tc
T2
T2 > T1
Single Phase
T1
C
Pressure
P1
Two Phase
Region
V3
V2
V1
Volume
Fig. 3.1: Volumetric Behavior of Pure Compounds by Van der Waal Cubic EOS13 pp 133.
The Van der Waal cubic EOS may give three real roots for volume V, or the
compressibility factor z, at pressure P1, as shown in Figure 3.1. The highest value, V1 or
z1, corresponds to the volume or the compressibility factor of the vapor, while the
lowest value, V3 or z3 corresponds to the volume or compressibility factor of the liquid.
The middle root within the two phase region, V2 or z2, is of no physical significance.
37
It can also be observed from Figure 3.1 that at the critical point C, the critical
temperature isotherm Tc has a horizontal point of inflection as it passes through the
critical pressure. Therefore at the critical point of a pure substance,
⎛ ∂p ⎞
⎜
⎟ =0
⎝ ∂V ⎠ Tc
(3.8)
⎛ ∂ 2p
⎜⎜ 2
⎝ ∂V
(3.9)
⎞
⎟⎟ = 0
⎠ Tc
Applying this condition at the critical point, the constant “a” and “b” of the Van
der Waal cubic EOS can be determined.
27 ⎛ R 2 Tc2
⎜
a=
64 ⎜⎝ p c
1 ⎛ RT
b = ⎜⎜ c
8 ⎝ pc
⎞
⎟
⎟
⎠
(3.10)
⎞
⎟⎟
⎠
(3.11)
where the subscript c, refers to the values at the critical point.
3.1.3 Peng-Robinson (PR) Equation of State32.
Peng and Robinson proposed a slightly different form of EOS to predict liquid
densities and other volumetric properties of the fluid at the vicinity of the critical
region. Peng and Robinson proposed a slightly different attractive term compared to the
VDW EOS.
38
p=
RT
a
−
V − b V(V + b ) + b(V − b )
(3.12)
The PR EOS in terms of volume is given as,
⎛
⎛ RT
⎞
⎛a
RTb 2 ab ⎞
2bRT ⎞
⎟⎟V + ⎜⎜ b 3 +
− b ⎟⎟V 2 + ⎜⎜ − 3b 2 −
− ⎟⎟ = 0
V 3 − ⎜⎜
p
p⎠
p ⎠
⎝ p
⎠
⎝p
⎝
(3.13)
Or, in terms of the compressibility factor,
(
) (
)
z 3 − (1 − B)z 2 + A − 3B 2 − 2B z − AB − B 2 − B 3 = 0
(3.14)
The PR EOS constants are given as,
a = Ωa
R 2 Tc2
α
pc
(3.15)
where Ω a = 0.45724
b = Ωb
RTc
pc
Ω b = 0.07780
where
[
(
α = 1 + m 1 - Tr0.5
and
(3.16)
)]
2
(3.17)
m = 0.374964 + 1.54226ω − 0.26992ω 2
(3.18)
The value of “m” was later expanded by Robinson et al. for heavier components
(ω > 0.49) 44 p. 50.
m = 0.3796 + 1.485ω − 0.1644ω 2 + 0.01667ω 3
Parameters A and B are as defined in equation 3.6 and 3.7 respectively.
39
(3.19)
3.1.4 Lawal-Lake-Silberberg (LLS) EOS
The LLS EOS is a four parameter cubic EOS defined by Lawal et al.25. The term
α and β were introduced to account for the shape of the components on the attractive
term. The general form of the LLS EOS is written as,
P=
RT
a
− 2
V − b V + αβV − βb 2
(3.20)
The LLS parameters a, b, α and β are defined as,
a = Ωa k
where
(RTc )2
pc
(3.21)
Ω a = (1 + (Ω w − 1)z c )
3
[
(
k = 1 + m 1 − Tr0.5
(3.22)
)]
2
(3.23)
m = 0.14443 + 1.06624ω + 0.02756ω 2 − 0.1807ω 3
b = Ωb
and
RTc
pc
Ωb = zcΩ w
(3.25)
(3.26)
Ωw =
where
α=
(3.24)
0.361
1 + 0.274ω
1 + z c (Ω w − 3)
zcΩ w
(3.27)
(3.28)
z c2 (Ω w − 1) + 2Ω 2w z c + Ω w (1 − 3z c )
βi =
z c Ω 2w
3
40
(3.29)
The LLS EOS can be written in terms of molar volume as,
⎛ RT
⎞
⎛a
RTαβ
+ b − αβ ⎟⎟V 2 + ⎜⎜ − βb 2 − bαβ −
V − ⎜⎜
p
⎝ p
⎠
⎝p
3
⎛ 3
⎞
RTβT2 ab ⎞
⎟⎟V + ⎜⎜ b β +
− ⎟⎟ = 0
p
p⎠
⎠
⎝
(3.30)
or, in terms of the compressibility factor
[
] [
(
)]
z 3 + z 2 [B(α − 1) − 1] − z B 2 (α + β ) + αB − A − AB − β B 2 − B 3 = 0
(3.31)
3.1.5 Modified Peng-Robinson Equation of State
The parameters “a” and “b” in the Peng-Robinson cubic EOS is constant, as
shown in equation 3.15 and equation 3.16. To account for the effect of each component
in the mixture on the parameters, the parameters “a” and “b” in the Peng-Robinson
cubic EOS is replaced with the parameters “a” and “b” in the LLS EOS to create a
Modified PR EOS.
3.1.6 Mixing Rules.
Equation of state are applied to multi-component systems by the use of mixing
rules to calculate the constant terms of the EOS (a, b, β, α) that will represent the multicomponent system. The mixture parameters used in the study are defined as,
a m = ∑∑ x i x j (a i a j ) δ ij
n
n
i
j
1 ⎤
⎡n
b m = ⎢∑ x i b i 3 ⎥
⎦
⎣ i
0.5
(3.32)
3
(3.33)
41
α m = ∑∑ x i x j (α i α j ) δ ij
n
n
i
j
0.5
(3.34)
β m = ∑∑ x i x j (β i β j ) δ ij
n
n
i
j
0.5
The binary interaction parameter
(3.35)
δ ij
is generally determined by minimizing the
difference between predicted and experimental data. A binary interaction parameter
should therefore be considered as a fitting parameter and not a rigorous physical term13.
The interaction parameters between hydrocarbon systems with little difference in size
are generally considered to be unity, but values of the binary interaction parameters for
non-hydrocarbon–hydrocarbon system may not be 1. For this study the binary
interaction parameter is taken as unity, because the reservoir fluids used in the study
does not have significant amount of non-hydrocarbons.
For multi- component systems that have significant amount of non-hydrocarbons
Lawal1 expressed the interaction parameter as,
⎡⎛ T
⎞⎤
⎢ ⎜ ci 0.5 ⎟ ⎥
⎢ ⎝ Pci ⎠ ⎥
⎢⎛T
⎞⎥
⎢ ⎜⎜ cj 0.5 ⎟⎟ ⎥
⎣⎢ ⎝ Pcj ⎠ ⎦⎥
where,
0.5
(3.36)
⎞
⎛ Tci
⎞ ⎛ Tcj
⎜
0.5 ⎟ < ⎜
0.5 ⎟
⎜
⎟
⎝ Pci ⎠ ⎝ Pcj ⎠
42
3.2 Characterization of Heavy Petroleum Fractions
To determine the properties of the heptane-plus (C7+), which account for the
heavy component in the multi-component reservoir fluid system, we use a correlation
proposed by Lawal-Tododo-Heinze40.
The correlation expressed the critical pressure (Pc), critical temperature (Tc),
acentric factor (ω), boiling point (Tb) and critical z-factor (zc) as a function of the
pseudo-component as a function of apparent molecular weight (M) and specific gravity
(Sg) of the pseudo-component.
Pc (psia ) = 3.1839 * 10 4 M −0.93426S1.64074
Tb0.49447 C −2.39909
g
Tc
( R ) = 66.3775M
o
0.12286
S 0.47926
Tb0.41286 C −0.35734
g
ω = 4.54949 * 10 −9 M 0.02445S g−2.08511Tb2.903798 C −1.54424
Tb
( R ) = 108.7017M
o
zc =
C=
0.4225
S 0.4268
g
(3.37)
(3.38)
(3.39)
(3.40)
0.293
1 + 0.375ω
(3.41)
3.8501
1.54057 − 0.02494 M
(3.42)
3.3 Cubic EOS Based Isothermal Compressibility Equation
Coefficient of isothermal compressibility can be obtained by differentiating an
equation of state or any correlation involving compressibility factor z, density or
formation volume factor10. In 1968 Kennedy H. T. and Avasthi S. M10 proposed a
43
method to develop an equation for predicting the molar volume and the coefficient of
isothermal compressibility by differentiation a cubic EOS in terms of volume. They
developed a correlation to express the differentiated volume.
The approach in this research is to develop a mathematical model that can predict
the coefficient of isothermal compressibility based on a Peng-Robinson EOS.
3.3.1 Coefficient of Isothermal Compressibility at Pressures above the Bubble Point
Pressure.
The coefficient of Isothermal Compressibility oil compressibility is defined as the
fractional change in oil volume per unit change in pressure at constant temperature. At
pressures above the bubble point pressure, the reservoir fluid is undersaturated. At point
A in Figure 3.2 there is no free gas in the system.
A
Pressure
C
Bubble Point Curve
Dew Point Curve
Temperature
Figure 3.2: Reservoir Oil at Pressure above the Bubble Point Pressure
44
The coefficient of isothermal compressibility at this point is given as stated in
equation 2.20,
co = −
1 ⎛ ∂V ⎞
⎜
⎟
V ⎜⎝ ∂p ⎟⎠ T
(2.20)
The approach to this research is to express the EOS in the volume form (which is
cubic) to determine the molar volume of the reservoir oil at any given pressure. This
will enable us to determine “ −
1
” part of equation 2.20. If the cubic EOS in terms of
V
volume gives three real roots, as explained in section 3.1.2 and Figure 3.1, the smallest
root corresponds to the volume of the oil.
⎛ ∂V ⎞
⎟⎟ ” term of
The second step is to differentiate the EOS to obtain the “ ⎜⎜
⎝ ∂p ⎠ T
equation 2.20. That is we differentiate the EOS in terms of volume with respect to
pressure at constant temperature. The two terms “ −
⎛ ∂V ⎞
1
⎟⎟ ” are then
” and “ ⎜⎜
V
⎝ ∂p ⎠ T
multiplied to give the coefficient of isothermal compressibility at pressures above the
bubble point.
3.3.1.1 Peng-Robinson (PR) EOS based Isothermal Compressibility Equation
The PR EOS is given in equation 3.12 as,
p=
RT
a
−
V − b V(V + b ) + b(V − b )
(3.12)
The PR EOS in terms of volume is given as,
45
⎛
⎛ RT
⎞
⎛a
2bRT ⎞
RTb 2 ab ⎞
⎟⎟V + ⎜⎜ b 3 +
− b ⎟⎟V 2 + ⎜⎜ − 3b 2 −
− ⎟⎟ = 0
V 3 − ⎜⎜
p ⎠
p
p⎠
⎝ p
⎠
⎝p
⎝
(3.13)
A computer algorithm was written to solve for the volume of the undersaturated
oil using equation 3.13. The procedure for calculating the volume is stated below.
o Read in the temperature T (oR), pressure p (psia) and the universal gas constant
R (psia cu ft/lb mole/oR).
o Calculate the EOS parameter “a” and “b” for the multi-component system.
o Determine the coefficient of the cubic equation (equation 3.13).
o Determine the roots equation 3.13. If there are three real roots, the smallest root
corresponds to the root of the multi-component oil system.
The differential of the volume with respect to pressure at constant temperature for
the PR EOS is given as,
(
)
2
⎛ ∂V ⎞
(
V − b ) V 2 + 2bV − b 2
⎜⎜
⎟⎟ =
2
⎝ ∂p ⎠ T 2a (V + b )(V − b ) − RT V 2 + 2bV − b 2
2
(
)
2
(3.43)
There are no unknowns in equation 3.43. The volume has already been
determined from equation 3.13. The universal gas constant R and the reservoir
temperature T are known. The PR EOS parameters “a” and “b” can be calculated using
the critical properties and the mixing rules.
Therefore the PR EOS based mathematical model for predicting the coefficient of
isothermal compressibility at pressures above the bubble point can be written as,
(
)
(V − b ) V 2 + 2bV − b 2
⎛ 1⎞
co = ⎜ − ⎟
⎝ V ⎠ 2a (V + b )(V − b )2 − RT V 2 + 2bV − b 2
2
2
(
46
)
2
(3.44)
3.3.1.2 Modified Peng-Robinson (PR) EOS based Isothermal Compressibility Equation
In order to modify the PR EOS, the constant parameters in the PR EOS, that is the
parameters “a” and “b” are calculated using the LLS EOS definition for the parameters
“a” and “b”. The definitions for these constant parameters are given in equations 3.21,
3.22, 3.23, 3.24, 3.25, 3.26 and 3.27.
3.3.2 Coefficient of Isothermal Compressibility at Pressures below the Bubble Point
Pressure.
At pressures below the bubble point curve (point B on Figure 3.3), there are free
gases in the reservoir fluids and the system is described as a saturated system.
C
Pressure
Bubble Point Curve
B
Dew Point Curve
Temperature
Figure 3.3: Reservoir Oil at Pressure below the Bubble Point Pressure
For saturated system the total volume of the system is the sum of the volume of
the gas and the volume of the liquid. Therefore the total system isothermal
compressibility has to account for the free gas in the system.
47
At pressures below the bubble point pressure the volume of liquid decreases with
reduction in pressure, contrary to the behavior at pressures above the bubble. Therefore
the coefficient for isothermal oil compressibility at pressure below the bubble point
pressure can be expressed as point29 p. 235,
co = −
1
Bo
⎛ ∂B o
⎜⎜
⎝ ∂p
B g ⎛ ∂R s
⎞
⎟⎟ +
⎜⎜
⎠ T B o ⎝ ∂p
⎞
⎟⎟
⎠T
(3.45)
Equation 3.45 was used to analyze the experimental differential liberation data to
calculate the coefficient of isothermal compressibility at pressures below the bubble
point pressure.
3.3.2.1 Molar Volume Prediction below the Bubble Point Pressure
At pressures below the bubble point pressure there is a distinct gas phase in the
system as shown by Figure 3.4.
Gas
Oil
Oil
Oil
Gas
Gas
Oil
Oil
Gas
Oil
P1>Pb
P2=Pb
P3<Pb
P4<<Pb
Pi<<Pb
Figure 3.4: Schematic Diagram of Flash Liberation Experiment
48
Pn<<<Pb
To determine the volume of the gas and the volume of the liquid at pressures
below the bubble point, flash calculations were conducted to simulate the flash
liberation (or constant composition experiment).
The flash liberation experiment involves putting a sample of reservoir fluid in a
laboratory cell. The pressure and temperature of the cell is set at the initial condition of
the reservoir. The pressure in the laboratory cell is reduced in increments and the
volume of the cell is measured. The measured volume is then plotted against pressure.
The bubble point pressure is the pressure at which the slope of the graph changes. The
volume at the bubble point pressure is the saturation volume Vsat. Flash calculations
were used to determine the amount (in moles) and the composition of hydrocarbon
liquid and gas co-existing at the same temperature and pressure.
The criterion for thermodynamic equilibrium between two phases is determined
by the fugacities of the components in each of the phases. The fugacity of a component
in one phase with respect to the fugacity of the component in another is a measure of
the potential for the transfer of the component from one phase to the other. The
component moves from the phase with the lower fugacity to the phase with the higher
fugacity. When the fugacities of the component in the two phases are equal, there is no
net transfer of the component between the two phases. The condition for
thermodynamic equilibrium is achieved when there is zero net transfer between the two
phases2 pp 305.
The procedure for calculating the equilibrium amount (in moles) and the
equilibrium composition of the vapor and liquid phase at temperatures below the
49
bubble point is stated below. The procedure is also referred to a two phase flash
calculations.
o Calculate the initial equilibrium ratio values (K-values) using Wilson45 (1968)
correlation.
⎡
⎛
Tci ⎞ ⎤
⎟⎥
⎠⎦
p ⎢ 5.37 (1+ ωi )⎜⎝ 1− T
K i = ci e ⎣
p
(3.46)
o Calculate Nvmin (minimum number of moles of the vapor phase) and Nvmax
(maximum number of moles of the vapor phase). Nvmin should be less than zero,
while Nvmax should be greater than one.
N vmin =
1
1 − K imax
(3.47)
N vmax =
1
1 − K imin
(3.48)
o Solve the Rachford-Rice flash algorithm to calculate Nv (total number of moles
in the vapor phase), limited between Nvmin and Nvmax.
o Assume an initial value for Nv (say Nv = 0.5)
o Evaluate Nv by using Newton-Raphson iteration technique.
N nv +1 = N nv −
f (n v ) = ∑
i
f (n v )
f ' (n v )
(3.49)
z i (K i − 1)
n v (K i − 1) + 1
(3.50)
⎡ z i (K i − 1)2 ⎤
f ' (n v ) = −∑ ⎢
2 ⎥
i ⎢
⎣ [n v (K i − 1) + 1] ⎥⎦
(3.51)
Where zi is the mole fraction component “i” in the original mixture and n, is the
iteration counter. If f’(nv) is equal to zero use bisection method to evaluate Nv.
50
o Calculate the mole fraction of component i in the liquid phase “xi”
xi =
zi
1 + n v (K i − 1)
(3.52)
o Calculate the mole fraction of the component in the vapor phase “yi”.
yi =
ziK i
1 + n v (K i − 1)
(3.53)
o Calculate the compressibility factor (z-factor) of the vapor and the liquid phase
using the compressibility factor form of the PR EOS. Using equation 3.14
3
z − (1 − B)z 2 + A − 3B 2 − 2B z − AB − B 2 − B 3 = 0
(3.14)
(
) (
)
The mole fraction of the vapor is used in the calculation of the z-factor of the
vapor. If three real roots are obtained from equation 3.14, the largest root
corresponds to the root of the vapor. The mole fraction of the liquid is used in the
calculation of the z-factor of the liquid. It three real roots are obtained for the zfactor of the liquid; the smallest real root corresponds to the root of the liquid.
o The component fugacities of the liquid “fLi” and the vapor phase “fVi” are
calculated from the EOS.
⎡ 2ψ i Bi ⎤ ⎛ z L + 2.414B ⎞
f Li B i L
A
L
⎟
ln
=
z − 1 − ln z − B −
− ⎥ ln⎜⎜ L
xip B
2.82843B ⎢⎣ ψ
B ⎦ ⎝ z − 0.414B ⎟⎠
(3.54)
⎡ 2ψ i Bi ⎤ ⎛ z v + 2.414B ⎞
f vi Bi v
A
⎟
=
z − 1 − ln z v − B −
− ⎥ ln⎜⎜ v
yi p B
2.82843B ⎢⎣ ψ
B ⎦ ⎝ z − 0.414B ⎟⎠
(3.55)
(
ln
(
) (
) (
)
)
The parameters of the PR EOS are calculated using the appropriate mole
fractions.
o Check for convergence of the fugacities
51
2
⎛ f Li
⎞
⎜⎜
− 1⎟⎟ < ε
∑
i =1 ⎝ f Vi
⎠
n
(3.56)
where ε is a convergence tolerance.
o If the convergence criteria is satisfied, the mole fraction of the liquid “xi”, the
mole fraction of the vapor “yi”, the total number in the vapor phase “Nv” and the
z-factor for both the liquid and vapor are returned as the equilibrium properties
the multi-component system.
o If the convergence criterion is not satisfied, the equilibrium ratios (K-values) are
updated.
(n )
( n +1)
( n ) f Li
Ki
= K i (n )
f Vi
(3.57)
where the superscripts (n) and (n+1) indicate the iteration level.
o Check for convergence into a trivial solution (Ki is tending towards 1) using,
N
∑ (ln K )
i =1
2
i
< 10 − 4
(3.58)
o If a trivial solution is not detected, return to step 2. Otherwise, confirm the
trivial solution with a stability test 44, 2.
o When the convergence criterion is satisfied, the volumes of the liquid and gas
phases can then be calculated from the expressions
VL =
Vg =
(1 − N v )z L RT
p
(3.59)
(N v )z v RT
p
(3.60)
o The total volume VT is calculated using,
VT = VL + Vg
(3.61)
52
3.3.2.2 Coefficient of Isothermal Gas compressibility below Bubble Point Pressure
The defining equation for the coefficient of isothermal gas compressibility is
given in equation 2.13,
cg =
1 1 ⎛ ∂z ⎞
− ⎜ ⎟
p z ⎜⎝ ∂p ⎟⎠ T
(2.13)
The partial differential of compressibility factor with respect to pressure at
constant temperature for the PR EOS is derived as,
bp
⎛ ∂z ⎞
(RT )
⎜⎜ ⎟⎟ =
⎝ ∂p ⎠ T
3
(2a − 3pb ) +
2
(6zpb
1
2
zb
(2 − z )
) (RT
)
− 2pb 2 − za +
(RT )
p
2bp
(z − 1) + z(3z − 2)
a − 3pb 2 +
2
(RT )
(RT )
(
2
)
(3.62)
Equation 2.13 can thus be used to calculate the coefficient of isothermal gas
compressibility, since the pressure “p” is known and the z-factor “z” has already been
calculated.
6zpb 2 2zb 2abp z 2 b
za
2pb 2 3p 2 b 3
+
+
−
−
−
−
1 1 (RT )2 RT (RT )3 RT (RT )2 (RT )2 (RT )3
c = −
g p z
⎡
bp
ap
3(bp )2 2bp ⎤
2
⎢3z − 2z + 2z
⎥
+
−
−
RT (RT )2 (RT )2 RT ⎥
⎢⎣
⎦
53
(3.63)
3.3.2.3 Coefficient of Isothermal Oil compressibility below Bubble Point Pressure
Below the bubble point the coefficient of isothermal oil compressibility can be
calculated using,
co = −
1 ⎛ ∂V ⎞
⎜
⎟
V ⎜⎝ ∂p ⎟⎠ T
(1.2)
Experimental data shows that at pressures below the bubble point pressure, the
volume of the liquid decreases as the pressure decreases. This behavior is also noticed
in the result of the flash calculations. It was therefore suggested that there is no need for
the negative sign, because the purpose of the negative sign is to make the isothermal
compressibility positive. The observation is not used in this report because it has not
been rigorously proved.
Using the actual volume of liquid calculated form the flash calculations, the liquid
part of the coefficient of isothermal compressibility at pressures below the bubble point
can be calculated as,
z L RT
V=
p
(3.64)
(
)
(V − b ) V 2 + 2bV − b 2
⎛1⎞
c o = −⎜ ⎟
⎝ V ⎠ 2a (V + b )(V − b )2 − RT V 2 + 2bV − b 2
2
2
(
)
2
(3.43)
The constant parameters of the PR EOS are calculated using the mole fraction of
the liquid “xi”.
54
3.3.2.4 Total Coefficient of Isothermal Compressibility below the Bubble Point
Pressure
The total coefficient of isothermal compressibility below the bubble point will be
the combination of the coefficient of isothermal gas compressibility below the bubble
point pressure and the coefficient of the isothermal oil compressibility below the bubble
point pressure.
This combination is achieved by using the volume fraction of the gas to multiple
the isothermal compressibility of the gas and the volume fraction of the oil to multiply
the isothermal compressibility of the oil.
co =
(V
L
Vg
+ Vg )
cg +
VL
(VL + Vg ) c L
55
(3.66)
CHAPTER IV
ANALYSIS OF PREDICTION RESULTS
4.1 Computing Isothermal Oil Compressibility from Reservoir Fluid Study Report
The coefficient of isothermal compressibility is not directly measured in the
laboratory. They are usually obtained from reservoir fluid study data. The constant
composition expansion experiment (also called Pressure-Volume relations) gives the
behavior of the reservoir fluid at pressures above the bubble point pressure.
The relative volume data from the constant composition expansion is used to
calculate the coefficient of isothermal oil compressibility at pressures above the bubble
point.
At pressures below the bubble point pressure, the reservoir behavior is simulated
by the differential liberation experiment. The solution gas-oil ratio RsD, the relative oil
volume BoD and the gas formation volume factor from the differential vaporization
experiment are used in the coefficient of isothermal compressibility at pressures below
the bubble point pressure.
4.2 Predicted Molar Volume from Cubic Equation of State
The predicted molar volumes from the Peng-Robinson cubic EOS and the
modified Peng-Robinson cubic EOS are reported in Table 4.1, 4.2 and 4.3. A good
degree of accuracy are observed between the experimental molar volumes and the
predicted molar volumes as shown in Figures 4.1, 4.2 4.3 4.4 4.5 and 4.6
56
Table 4.1: Predicted molar volume for Oil Well No. 4, Good Oil Company. Samson,
Texas. Bubble Point = 2619.7 psia 29 pp 259-269
Pressure(psia)
5014.7
4514.7
4014.7
3514.7
3014.7
2914.7
2814.7
2714.7
2634.7
2619.7
2605.7
2530.7
2415.7
2267.7
2104.7
1911.7
1712.7
1491.7
1306.7
1054.7
844.7
654.7
486.7
Volume (ft3/lb mole)
Exptal
PR EOS
2.2058
2.2047
2.2204
2.2201
2.2360
2.2371
2.2531
2.2559
2.2721
2.2770
2.2760
2.2815
2.2801
2.2861
2.2845
2.2909
2.2884
2.2948
2.2934
2.2033
2.2978
2.2083
2.3236
2.2427
2.3685
2.2897
2.4360
2.3597
2.5264
2.4519
2.6621
2.5878
2.8435
2.7688
3.1163
3.0482
3.4353
3.3596
4.0738
3.9944
4.9482
4.8609
6.2960
6.1895
8.5187
8.3257
57
MPR EOS
2.2672
2.2816
2.2974
2.3148
2.3341
2.3382
2.3424
2.3468
2.3503
2.6633
2.6655
2.6804
2.7081
2.7531
2.8157
2.9152
3.0547
3.2799
3.5454
4.1111
4.9103
6.1702
8.2369
Molar Volume Prediction for Oil Well No-4,Good Oil Company, Samson,TX. Pb =2619.7 psia
9.0
8.0
Molar Volume (ft3/lbmol)
7.0
6.0
5.0
Exptal
Model Based on PR EOS
4.0
3.0
2.0
1.0
0.0
0
1000
2000
3000
4000
5000
6000
Pressure (psia)
Figure 4.1: Predicted Molar Volume for Oil Well No- 4, Good Oil Company. Samson,
Texas (Model Based on PR EOS).
58
Molar Volume Predictionfor Oil Well No-4,Good Oil Company, Samson,TX. Pb =2619.7 psia
9.0
8.0
Molar volume (ft3/lbmol)
7.0
6.0
5.0
Exptal
Model Based on MPR EOS
4.0
3.0
2.0
1.0
0.0
0
1000
2000
3000
4000
5000
6000
Pressure (psia)
Figure 4.2: Predicted Molar Volume for Oil Well No- 4, Good Oil Company. Samson,
Texas (Model Based on MPR EOS).
59
Table 4.2: Predicted molar volume for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver,
Oklahoma. Bubble Point = 2666.7 psia 37
Volume (ft3/lb mole)
Pressure
(psia)
Exptal
PR EOS
MPR EOS
5014.7
2.2110
2.2956
2.2779
4514.7
2.2226
2.3087
2.2892
4014.7
2.2351
2.3229
2.3015
3514.7
2.2487
2.3386
2.3149
3014.7
2.2633
2.3560
2.3296
2914.7
2.2662
2.3596
2.3327
2814.7
2.2694
2.3634
2.3359
2714.7
2.2726
2.3673
2.3391
2677.7
2.2737
2.3688
2.3403
2653.7
2.2794
2.2131
2.5787
2591.7
2.2949
2.2308
2.5899
2507.7
2.3181
2.2569
2.6056
2365.7
2.3645
2.3070
2.6378
2192.7
2.4575
2.3949
2.6874
2012.7
2.5282
2.4780
2.7570
1811.7
2.6644
2.6147
2.8618
1604.7
2.8558
2.8039
3.0107
1392.7
3.1289
3.0740
3.2342
1193.7
3.4998
3.4411
3.5483
949.7
4.2155
4.1399
4.1746
760.7
5.1473
5.0465
5.0142
591.7
6.5507
6.4199
6.3198
435.7
8.8958
8.7522
8.5807
60
Molar Volume Prediction for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver, Oklahoma. (MPR
EOS) Bubble Point = 2666.7 psia.
10.0
9.0
8.0
Molar Volume (ft3/lbmol)
7.0
6.0
Exptal
Model Based on PR EOS
5.0
4.0
3.0
2.0
1.0
0.0
0
1000
2000
3000
4000
5000
6000
Pressure (psia)
Figure 4.3: Predicted Molar Volume for Jehlicka 1A, Wilshire Oil Co. of Texas,
Beaver, Oklahoma. (Model Based on PR EOS).
61
Molar Volume Prediction for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver, Oklahoma. (MPR
EOS) Bubble Point = 2666.7 psia.
10.0
9.0
8.0
Molar Volume (ft3/lbmol)
7.0
6.0
Exptal
Model Based on MPR EOS
5.0
4.0
3.0
2.0
1.0
0.0
0
1000
2000
3000
4000
5000
6000
Pressure (psia)
Figure 4.4: Predicted Molar Volume for Jehlicka 1A, Wilshire Oil Co. of Texas,
Beaver, Oklahoma. (Model Based on MPR EOS).
62
Table 4.3: Predicted molar volume for Jacques Unit #5603, Rough Ride Field, A.C.T.
Operating Company, Fisher County, Texas. Bubble Point = 1689.7 psia 36
Volume (ft3/lb mole)
Pressure (psia)
Exptal
PR EOS
MPR EOS
5014.7
2.2175
2.2330
2.3248
4514.7
2.2269
2.2426
2.3345
4014.7
2.2368
2.2528
2.3451
3514.7
2.2474
2.2640
2.3565
3014.7
2.2586
2.2761
2.3689
2514.7
2.2707
2.2893
2.3824
2014.7
2.2838
2.3039
2.3974
1914.7
2.2866
2.3070
2.4006
1814.7
2.2893
2.3102
2.4038
1714.7
2.2921
2.3134
2.4071
1689.7
2.2928
2.3142
2.4079
1680.7
2.2969
2.1812
2.4891
1670.7
2.3015
2.1860
2.4929
1660.7
2.3063
2.1909
2.4968
1650.7
2.3111
2.1959
2.5008
1627.7
2.3223
2.2076
2.5106
1541.7
2.3693
2.2565
2.5506
1402.7
2.4636
2.3549
2.6325
1244.7
2.6080
2.5060
2.7595
1064.7
2.8460
2.7572
2.9765
900.7
3.1745
3.0966
3.2800
747.7
3.6485
3.5832
3.7261
631.7
4.1962
4.1440
4.2506
501.7
5.1681
5.1419
5.2026
390.7
6.5871
6.6222
6.6328
287.7
8.9929
9.1869
9.1431
63
Molar Volume Prediction for Jacques Unit #5603, Rough Ride Field, A.C.T. Operating
Company, Fisher County, Texas. Bubble Point = 1689.7 psia
10.0
9.0
8.0
Molar Volume (ft3/lbmol)
7.0
6.0
Exptal
Model based on PR EOS
5.0
4.0
3.0
2.0
1.0
0.0
0
1000
2000
3000
4000
5000
6000
Pressure (psia)
Figure 4.5: Predicted Molar Volume for Jacques Unit #5603, Rough Ride Field, Fisher
County, Texas. (Model Based on PR EOS).
64
Molar Volume Prediction for Jacques Unit #5603, Rough Ride Field, A.C.T. Operating
Company, Fisher County, Texas. Bubble Point = 1689.7 psia
10.0
9.0
8.0
Molar Volume (ft3/lbmol)
7.0
6.0
Exptal
Model Based on MPR EOS
5.0
4.0
3.0
2.0
1.0
0.0
0
1000
2000
3000
4000
5000
6000
Presssure (psia)
Figure 4.6: Predicted Molar Volume for Jacques Unit #5603, Rough Ride Field, Fisher
County, Texas.(Model based on MPR EOS).
65
4.3 Predicted Coefficient of Isothermal Oil Compressibility from Cubic Equation of
State
The developed mathematical models predicted the isothermal oil compressibility
with a good degree of accuracy as shown in Tables 4.4, 4.5 and 4.6. It can be observed
from Figures 4.7, 4.8, 4.9, 4.10, 4.11 and 4.12 that the developed model gave a very
good prediction at pressures above the bubble point pressure. At pressures below the
bubble point pressure, the predicted values also gave a good degree of accuracy
considering the difference between the experimental and the predicted values.
Table 4.4: Predicted Coefficient of Isothermal Oil Compressibility for Oil Well No. 4,
Good Oil Company. Samson, Texas. Bubble Point = 2619.7 psia 29 pp 259 - 269
Pressure(psia)
5014.7
4514.7
4014.7
3514.7
3014.7
2914.7
2814.7
2714.7
2634.7
2364.7
2114.7
1864.7
1614.7
1364.7
1114.7
864.7
614.7
364.7
173.7
Isothermal Compressibility (psia-1)
Exptal
PR EOS
MPR EOS
1.33E-05
1.21E-05
1.32E-05
1.45E-05
1.32E-05
1.40E-05
1.59E-05
1.44E-05
1.53E-05
1.76E-05
1.58E-05
1.68E-05
1.96E-05
1.75E-05
1.71E-05
2.01E-05
1.79E-05
1.81E-05
2.05E-05
1.83E-05
1.91E-05
2.10E-05
1.87E-05
2.13E-05
2.14E-05
1.90E-05
1.55E-04
2.35E-04
2.06E-04
1.83E-04
2.74E-04
2.44E-04
2.09E-04
3.26E-04
2.95E-04
2.53E-04
3.96E-04
3.65E-04
3.15E-04
4.95E-04
4.65E-04
4.17E-04
6.44E-04
6.16E-04
5.76E-04
8.84E-04
8.63E-04
8.87E-04
1.33E-03
1.32E-03
1.83E-03
2.42E-03
2.44E-03
5.20E-03
5.42E-03
5.47E-03
66
Coefficient of Isothermal Oil Compressibility for Oil Well No-4,Good Oil Company,
Samson,TX. Pb =2619.7 psia
0.006
0.005
Co (1/psi)
0.004
Exptal
Model Based on PR EOS
0.003
0.002
0.001
0.000
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Pressure (psia)
Figure 4.7: Predicted Coefficient of Isothermal Oil Compressibility for Oil Well No. 4,
Good Oil Company. Samson, Texas. (Model Based on PR EOS).
67
Coefficient of Isothermal Oil Compressibility for Oil Well No-4,Good Oil Company,
Samson,TX. Pb =2619.7 psia
0.006
0.005
Co (1/psia)
0.004
Exptal MPR
Model Based on MPR EOS
0.003
0.002
0.001
0.000
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Pressure (psia)
Figure 4.8: Predicted Coefficient of Isothermal Oil Compressibility for Oil Well No. 4,
Good Oil Company. Samson, Texas. (Model Based on MPR EOS).
68
Table 4.5: Predicted Coefficient of Isothermal Oil Compressibility for Jehlicka 1A,
Wilshire Oil Co. of Texas, Beaver, Oklahoma. Bubble Point = 2666.7 psia. 37
Isothermal Compressibility (psia-1)
Pressure
(psia)
Exptal
PR EOS
MPR EOS
5014.7
1.09E-05
9.56E-06
4514.7
1.05E-05
1.18E-05
1.03E-05
4014.7
1.12E-05
1.29E-05
1.11E-05
3514.7
1.22E-05
1.41E-05
1.21E-05
3014.7
1.29E-05
1.55E-05
1.32E-05
2914.7
1.31E-05
1.59E-05
1.35E-05
2814.7
1.40E-05
1.62E-05
1.37E-05
2714.7
1.40E-05
1.65E-05
1.40E-05
2677.7
1.35E-05
1.67E-05
1.41E-05
2414.7
1.34E-04
2.12E-04
1.88E-04
1914.7
1.89E-04
2.96E-04
2.69E-04
1664.7
2.27E-04
3.60E-04
3.33E-04
1414.7
3.07E-04
4.50E-04
4.23E-04
1164.7
4.09E-04
5.82E-04
5.58E-04
914.7
5.57E-04
7.95E-04
7.76E-04
664.7
8.69E-04
1.18E-03
1.17E-03
414.7
1.61E-03
2.05E-03
2.07E-03
197.7
4.58E-03
4.66E-03
4.72E-03
69
Coefficient of isothermal Oil Compressibility for Jehlicka 1A, Wilshire Oil Co. of Texas,
Beaver, Oklahoma. (PR EOS) Bubble Point = 2666.7 psia.
5.00E-03
4.50E-03
4.00E-03
3.50E-03
Co (1/psia)
3.00E-03
Exptal
Model Based on PR EOS
2.50E-03
2.00E-03
1.50E-03
1.00E-03
5.00E-04
0.00E+00
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Pressure (psia)
Figure 4.9: Predicted Coefficient of Isothermal Oil Compressibility for Jehlicka 1A,
Wilshire Oil Co. of Texas, Beaver, Oklahoma. (Model Based on PR EOS).
70
Coefficient of Isothermal Oil Compressibility for Jehlicka 1A, Wilshire Oil Co. of Texas,
Beaver, Oklahoma. (MPR EOS) Bubble Point = 2666.7 psia.
0.005
0.005
0.004
0.004
Co (1/psia)
0.003
Exptal
Model Based on MPR EOS
0.003
0.002
0.002
0.001
0.001
0.000
0
500
1000
1500
2000
2500
3000
3500
4000
4500
5000
Pressure (psia)
Figure 4.10: Predicted Coefficient of Isothermal Oil Compressibility for Jehlicka 1A,
Wilshire Oil Co. of Texas, Beaver, Oklahoma. (Model Based on MPR EOS).
71
Table 4.6: Predicted Coefficient of Isothermal Oil Compressibility for Jacques Unit
#5603, Rough Ride Field, A.C.T. Operating Company, Fisher County, Texas. Bubble
Point = 1689.7 psia 36
Isothermal Compressibility (psia-1)
Pressure
(psia)
Exptal
PR EOS
MPR EOS
5014.7
8.24E-06
8.10E-06
4514.7
8.46E-06
8.83E-06
8.68E-06
4014.7
8.83E-06
9.49E-06
9.34E-06
3514.7
9.41E-06
1.03E-05
1.01E-05
3014.7
9.97E-06
1.11E-05
1.09E-05
2514.7
1.07E-05
1.21E-05
1.19E-05
2014.7
1.15E-05
1.33E-05
1.31E-05
1914.7
1.20E-05
1.36E-05
1.34E-05
1814.7
1.20E-05
1.38E-05
1.36E-05
1714.7
1.20E-05
1.41E-05
1.39E-05
1689.7
1.20E-05
1.42E-05
1.40E-05
1514.7
2.39E-04
4.32E-04
4.00E-04
1364.7
2.72E-04
4.98E-04
4.65E-04
1214.7
3.32E-04
5.82E-04
5.49E-04
1064.7
3.99E-04
6.91E-04
6.58E-04
914.7
4.98E-04
8.38E-04
8.08E-04
614.7
9.05E-04
1.36E-03
1.34E-03
464.7
1.38E-03
1.88E-03
1.87E-03
314.7
2.48E-03
2.91E-03
2.91E-03
174.7
5.98E-03
5.47E-03
5.50E-03
72
Coefficient of Isothermal Oil compressibility for Jacques Unit #5603, Rough Ride Field, A.C.T.
Operating Company, Fisher County, Texas. Bubble Point = 1689.7 psia
0.007
0.006
Co (1/psia)
0.005
0.004
Exptal
Model Based on PR EOS
0.003
0.002
0.001
0.000
0.0
500.0
1000.0
1500.0
2000.0
2500.0
3000.0
3500.0
4000.0
4500.0
5000.0
Pressure (psia)
Figure 4.11: Predicted Coefficient of Isothermal Oil Compressibility for Jacques Unit
#5603, Rough Ride Field, A.C.T. Operating Company, Fisher County, Texas. (Model
Based on PR EOS).
73
Coefficient of Isothermal Oil Compressibility for Jacques Unit #5603, Rough Ride Field, A.C.T.
Operating Company, Fisher County, Texas. Bubble Point = 1689.7 psia
0.007
0.006
Co (1/psia)
0.005
0.004
Exptal
Model Based on MPR EOS
0.003
0.002
0.001
0.000
0.0
500.0
1000.0
1500.0
2000.0
2500.0
3000.0
3500.0
4000.0
4500.0
5000.0
Pressure (psia)
Figure 4.12: Predicted Coefficient of Isothermal Oil Compressibility for Jacques Unit
#5603, Rough Ride Field, A.C.T. Operating Company, Fisher County, Texas. (Model
Based on MPR EOS).
4.4 Discussion of Result
The reservoir fluid used in this project can be classified as black oil because their API
gravity was less than 450API.29, p. 151
The developed model based on the PR EOS and the Modified PR EOS predicted
the coefficient of isothermal oil compressibility above the bubble point with a near
perfect match. At pressures below the bubble point the developed model also gave a good
prediction of the coefficient of isothermal oil compressibility.
74
The average absolute percent deviation (AAPD) as shown in Appendix H, for the
developed model based on the MPR EOS was smaller than the model based on the PR
EOS. The AAPD for the predicted models above the bubble point was smaller than the
AAPD for the total system. The smallest AAPD (of 1.9% deviation) above the bubble
point pressure was obtained by the model based on the MPR EOS for the Jehlicka 1A
well of Wilshire Oil Co. of Texas. The largest AAPD (of 12.39% deviation) above the
bubble point pressure was obtained by the model based on the PR EOS for the Jacques
Unit #5603 of the A.C.T Operating Company.
The smallest AAPD (of 19.10% deviation) for the predicted coefficient of
isothermal oil compressibility for the whole system (i.e. at pressures above and below the
bubble point pressure) was obtained from the model based on the MPR EOS for the
Jehlicka 1A well of Wilshire Oil Co. of Texas. The largest AAPD (of 32.44% deviation)
for the predicted coefficient of isothermal oil compressibility for the whole system (i.e. at
pressures above and below the bubble point pressure) was obtained from the model based
on the PR EOS for the Jacques Unit #5603 of the A.C.T Operating Company.
The suggested reason for the larger deviation in the predicted result for the
Jacques Unit #5603 of the A.C.T Operating Company is because the reservoir fluid
composition used was a recombination of the well stream production. The error acquired
through the recombination could be magnified by the EOS model.
This study used the components given in the fluid study report and lumped
together the heptane-plus (C7+). In order to quantify the effect of using more components,
the developed model was run using 24 components for the Jacques Unit #5603 of the
75
A.C.T Operating Company. The AAPD above the bubble point pressure was 2.84%
deviation above the bubble point pressure and 22.30% deviation for the total system for
the model based on the MPR EOS. This was smaller than the 10.66% deviation above the
bubble point pressure and 28.75% deviation for the total system for the model based on
MPR EOS when the C7+ was lumped together. For the model based on the PR EOS, the
AAPD increased from 12.39% to 49.92% deviation at pressures above the bubble point
and from 32.44% to 48.05% deviation for the total system at pressures above and below
the bubble point pressure.
The increase in the error in the model based on the PR EOS could be due to the
correlation used to calculate the critical parameter of the Eicosanes-plus (C20+). The
effect of this correlation was believed to be minimized in the model based on the MPR
EOS because the constant parameter “a” which is dependent of the components critical
parameter.
The number of iterations recorded in predicting the coefficient of isothermal oil
compressibility using 11 components of the Jacques Unit #5603 of the A.C.T. Operating
Company was 9,791. When the same reservoir fluid data was used to predict the
isothermal oil compressibility using the 24 components, the number of iterations recorded
was 24,961. This shows a 155 % increase in the number of iterations. This increase in the
number of iterations will translate into a higher computing cost when calculating the
isothermal compressibility for each grid cell in a compositional reservoir simulation.
76
CHAPTER V
CONCLUSIONS AND RECOMMENDATIONS
5.1 Conclusions
The following conclusions can be drawn from this project:
1. A mathematical model based on Peng-Robinson equation of state for calculating the
coefficient of isothermal oil compressibility was developed.
2. The mathematical model was extended to pressures below the bubble point, using the
two phase flash calculations.
3. A mathematical model based on Peng-Robinson equation of state was also developed
to predict the coefficient of isothermal gas compressibility at pressures below the
bubble point.
4. A mathematically consistent additive technique was developed to add the resultant
coefficient of isothermal oil compressibility and the coefficient of isothermal gas
compressibility at pressures below the bubble point.
5. A good degree of accuracy was obtained between the predicted coefficient of
isothermal oil compressibility from the developed EOS based mathematical model
and the experimentally derived coefficient of isothermal compressibility.
6. The predicted molar volume from the two phase calculation was reported and
compared with the experimental measured molar volume from Pressure-Volume
relations; a good degree of accuracy was achieved.
77
7. A computer algorithm (using FORTRAN) was developed to compute the coefficient
of isothermal oil compressibility using the developed mathematical model.
5.2 Recommendations
1. Research is very active in the area of two phase calculation to determine the
equilibrium quantity and composition of liquid and vapor in the two phase region. A
more robust two phase flash algorithm is recommended to increase the accuracy of
the predicted coefficient of isothermal oil compressibility at pressures below the
bubble point pressure.
2. A further work is recommended in tuning the developed mathematical model, using
the constant parameters of the Peng-Robinson EOS (i.e. “a” and “b”) and the binary
interaction parameter in order to achieve a higher level of accuracy.
3. In order to predict the isothermal compressibility of gas condensate reservoirs, it is
recommended that the developed mathematical model be coupled with the simulation
of constant volume depletion experiment.
4. Observations bothering on the analysis of the differential vaporization experiment
suggested that the accuracy of equation 3.45 be validated. The reason for this
validation is the slope change observed in the graph of the formation volume factor
and the pressure, at pressures above and below the bubble point pressure.
5. The developed model can also be extended to gas fields to predict the coefficient of
isothermal gas compressibility.
78
REFERENCES
1. Adetunji, L. A., Thermodynamically Equivalent Pseudo-components validated for
Compositional Reservoir Models. Ph.D. Dissertation, Texas Tech University,
Lubbock, Texas (May 2006).
2. Ahmed, T. H., Hydrocarbon Phase Behavior, Gulf Publishing Company, Vol. 7,
(1989).
3. Ahmed, T., “A Practical Equation of State,” paper SPE 18532, SPE Reservoir
Engineering, pp 137, February (1991).
4. Ahmed. T., “Compositional Modeling of Tyler and Mission Canyon Formation Oils
with Co2 and Lean Gases,” final report submitted to Montanans on a New Track
Science (MONTS) (Montana National Science Foundation Gran Program), 19851988.
5. Ahmed, T. and McKinney, P. D., Advanced Reservoir Engineering, Gulf Professional
Publishing,pp1-4 (2005).
6. Alani, G. H. and Kennedy, H. T., “Volumes of Liquid Hydrocarbons at High
Temperatures and Pressures,” JPT, pp 272, November (1960).
7. Alani, G. H. and Kennedy, H. T., “Volumes of Liquid Hydrocarbons at High
Temperatures and Pressures,” Petroleum Transactions, AIME, Vol. 219, 288, (1960).
8. Al-Marhoun, M. A., “The Coefficient of Isothermal Compressibility of Black Oils,”
paper SPE 81432 presented at the SPE 13th Middle East Oil Show & Conference,
Bahrain, April 5-8 (2003).
9. Arbabi, S. and Firoozabadi, A., “Near-Critical Phase Behavior of Reservoir Fluids
Using Equation of State,” paper SPE 24491, SPE Advanced Technology Series,Vol 3,
No.1, pp 139, (1995).
10. Avasthi, S. M. and Kennedy, H. T., “The Prediction of Volumes, Compressibilities
and Thermal Expansion Coefficients of Hydrocarbon Mixtures,” Society of
Petroleum Journals, Vol. 234, pp 95, June (1958).
11. Brelvi, S. W., “Correlations developed for compressibility,” Oil & Gas Journal, pp
60, January 29 (2001).
12. Cameron, A., “The Isothermal and Adiabatic Compressibility of Oil,” Journal of the
Institute of Petroleum, Vol. 31, No. 263, November (1945).
79
13. Danesh, A., PVT and Phase Behavior of Petroleum Reservoir Fluids., Elsevier
Science, 2nd Edition, (2001).
14. De Ghetto, G., Paone, F. and Villa, M., “Pressure-Volume-Temperature Correlations
for Heavy and Extra Heavy Oils,” paper SPE 303316 presented at the 1995
International Heavy Oil Symposium, Calgary, Alberta, Canada, June 19-21.
15. De Ghetto, G., Paone, F. and Villa, M., “Reliability Analysis on PVT Correlations,”
paper SPE 28904 presented at the 1994 European Petroleum Conference, London,
U.K, October 25-27.
16. Dindoruk, B. and Christman, P. G., “PVT Properties and Viscosity Correlations for
Gulf of Mexico Oils,” SPEREE, pp 427-437, December, (2004).
17. Downer, L. and Gardiner, E. S., “The Compressibility of Crude Oils,” Journal of the
Institute of Petroleum, Vol. 58, No. 559, January (1972).
18. Firoozabadi, A. and Huanquan, P., “Two-Phase Isentropic Compressibility and TwoPhase Sonic Velocity for Multi component-Hydrocarbon Mixtures,” paper SPE 38844
presented at the 1997 Annual Technical Conference and Exhibition, San Antonio,
Texas, October 5-8.
19. Firoozabadi, A. and Kashchiev, D., “Pressure and Volume Evolution During Gas
Phase Formation in Solution Gas Drive Process,” paper SPE 26286, SPE Journal, pp
219, September (1996).
20. Firoozabadi, A., “Reservoir-Fluid Phase Behavior and Volumetric Prediction with
Equations of State,” JPT, 397, (April, 1988).
21. Firoozabadi, A., Thermodynamics of Hydrocarbon Reservoirs., McGraw-Hill, (1999).
22. Glasǿ, Ǿ., “Generalized Pressure-Volume-Temperature Correlations,” JPT, pp 785,
May (1980).
23. Hawkins, M. F., “Material Balance in Expansion Type Reservoirs above Bubble
Point,” paper SPE 499-G, Petroleum Transactions, AIME ,204, pp 267 (1955).
24. Lawal, A. S., “Revival of the van der Waals Classical Theory via Silberberg
Constant,” paper SPE 22712 presented at the 1991 Annual Technical Conference and
Exhibition, Dallas, Texas, October 6-9.
80
25. Lawal, A. S., Van der Laan, E. T., and Thambynayagam, R. K. M., “Four-Parameter
Modification of the Lawal-Lake-Silberberg Equation of State for Calculating GasCondensate Phase Equilibria,“ paper SPE 14269 presented at the 1985 Annual
Technical Conference and Exhibition, Las Vegas, Nevada, September 22-25.
26. Macias, L. C.’ “Multiphase, Multicomponent Compressibility in Petroleum Reservoir
Engineering,” paper SPE 15538, presented at the 1986 Annual Technical Conference
and Exhibition, New Orleans, LA, October 5-8.
27. Martin, J. C., “Simplified Equations of Flow in Gas Drive Reservoirs and the
Theoretical Foundation of Multiphase Buildup Analyses,” paper SPE 1235-G,
Petroleum Transactions, Vol. 216, 321 (1959).
28. McCain, W. D., “Reservoir-Fluid Property Correlations-State of the Art,” SPE
Reservoir Engineering, pp 266, May (1991).
29. McCain, W. D., Jr., The Properties of Petroleum Fluids., PennWell Books, 2nd
Edition, (1990).
30. McCain, W. D., Rollins, J. B. and Villena Lanzi, A. J., “The Coefficient of Isothermal
Compressibility of Black Oils at Pressures Below the Bubble Point,” SPE Formation
Evaluation, pp 659, September (1988).
31. Nghiem, L. X. and Li, Y. K., “Approximate Flash Calculations for Equation-of-State
Compositional Models”, SPE Reservoir Engineering, pp 107, February (1990).
32. Peng, D. Y. and Robinson, D. B.,”A New Two-Constant Equation of State,” Ind. Eng.
Chem. Fundamentals, 15, 59 (1976).
33. Petrosky, G. E., Jr. “Pressure-Volume-Temperature Correlations for Gulf of Mexico
Crude Oils,” SPE Reservoir Evaluation & Engineering, pp 416, October (1998).
34. Rachford, H. H., Jr. and Rice, J. D., “Procedure for Use of Electronic Digital
Computers in Calculating Flash vaporization Hydrocarbon Equilibrium,” Petroleum
Transactions, AIME, Vol. 195, 327, (1952).
35. Ramey, H. J., “Rapid Methods for Estimating Reservoir Compressibilites,” paper
SPE 772, JPT, 447, April (1964).
36. Reservoir Fluid Study on Jacques Unit #5603, Rough Ride Field, Core Laboratories,
Inc. Report (Aug 1999).
81
37. Reservoir Fluid Study on Jehlicka 1A Well, Core Laboratories, Inc. Report
1984).
(May
38. Spivey, J. P., Valko, P. P. and McCain, W. D., “Applications of the Coefficient of the
Isothermal Compressibility to Various Reservoir Situations with New Correlation for
Each Situation,” paper SPE 96415 presented at the 2005 Annual Technical
Conference and Exhibition, Dallas, Texas, October 9-12.
39. Standing, M. B., Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems. ,
Society of Petroleum Engineers of AIME, Dallas, Texas (1977).
40. Tododo, A. O., Thermodynamically Equivalent Pseudocomponents for Compositional
Reservoir Simulation Models. Ph.D. Dissertation, Texas Tech University, Lubbock,
Texas (May 2005).
41. Trube, A. S., “Compressibility of Natural Gases,” paper SPE-697-G, Technical Note,
398, pp 69, January (1957).
42. Trube, A. S., “Compressibility of Undersaturated Hydrocarbon Reservoir Fluids,”
paper SPE 899-G, Petroleum Transactions, Vol. 210, pp 341, (1957).
43. Vasquez, M. and Beggs, H. D., “Correlations for Fluid Physical Property Prediction,”
JPT, pp. 968-970, June (1980).
44. Whitson, C. H. and Brule, M. R., Phase Behavior., Monograph Series, SPE,
Richardson, Texas (2000).
45. Wilson, G., “A Modified Redlich-Kwong EOS, Application to General Physical Data
Calculations,” paper 15C, presented at the Annual AIChE National Meeting held in
Cleveland, Ohio, May 4-7 (1968).
46. Yale, D. P., Nabor, G. W. and Russell, J. A., “Application of Variable Formation
Compressibility for Improved Reservoir Analysis,” paper SPE 26647 presented at the
1993 Annual Technical Conference and Exhibition, Houston, Texas, October 3-6.
47. Young, L. C., “Continuous Compositional Volume-Balance Equations,” paper 66346
presented at the 2001 SPE Reservoir Simulation Symposium, Houston, Texas,
February 11-14 (2001).
82
APPENDIX A
DERIVATION OF PR EOS COMPRESSIBILITY
FACTOR AND MOLAR VOLUME
EQUATION
A.1 Derivation of PR EOS Compressibility Factor Equation
The PR EOS is expressed as (ref),
P=
RT
a
−
V − b V(V + b) + b(V − b)
(A.1)
Multiplying both sides by (V – b),
P(V − b) = RT −
a(V − b)
V(V + b) + b(V − b)
(A.2)
a(V − b)
V + 2Vb − b 2
(A.3)
Simplifying,
P(V − b) = RT −
2
P(V − b)(V 2 + 2Vb − b 2 ) = RT(V 2 + 2Vb − b 2 ) − a(V − b)
(A.4)
Expanding,
P(V 3 + 2bV 2 − Vb 2 − bV 2 − 2Vb 2 + b 3 ) = RT(V 2 + 2Vb − b 2 ) − a(V − b)
(A.5)
Dividing by RT,
P
a
(V 3 + bV 2 − 3Vb 2 + b 3 ) = (V 2 + 2Vb − b 2 ) −
(V − b)
RT
RT
83
(A.6)
But
V=
ZRT
P (Real gas equation), hence (A.6) becomes,
2
P ⎡ Z3 (RT)3
Z 2 (RT)2
Z 2 (RT)
Z(RT) 2 a ⎛ Z(RT) ⎞
2 Z(RT)
3⎤
+
b
−
3b
+
b
=
+ 2b
−b −
− b⎟
⎜
⎢
⎥
3
2
2
RT ⎣ P
RT ⎝ P
P
P
P
P
⎠
⎦
(A.7)
Simplifying equation A.7 gives,
Z 3 (RT) 2
Z 2 (RT)
Z 2 (RT) 2
Z(RT)
Z ba
2
3 P
+
b
−
3b
Z
+
b
−
− 2b
+ b2 + a −
=0
2
2
RT
P RT
P
P
P
P
(A.8)
Multiplying by P2 gives,
Z 3 (RT) 2 + bZ 2 (RT)P − 3P 2 b 2 Z + b 3
aPbP
P3
− Z 2 (RT) 2 − 2bZ(RT)P + P 2 b 2 + aZP −
=0
RT
RT
(A.9)
Dividing by (RT)2 gives,
Z3 + Z 2
aP
aP bP
Pb Pb Pb
P3
Pb Pb
bP
− 3Z
+ b3
− Z 2 − 2Z
+
+Z
−
=0
3
2
RT RT RT
RT RT
RT
(RT)
(RT) 2 RT
(RT)
(A.10)
But
B=
aP
bP
A= 2 2
RT and
R T , therefore,
Z 3 + Z 2 B − 3ZB 2 + B 3 − Z 2 − 2ZB + B 2 + AZ − AB = 0
(A.13)
Z 3 − (1 − B)Z 2 + (A − 3B 2 − 2B)Z − (AB − B 2 − B 3 ) = 0
(A.14)
84
A.2 Derivation of PR EOS Molar Volume Equation
The PR EOS is expressed as,
P=
RT
a
−
V − b V(V + b ) + b(v − b)
(A.15)
Multiplying through by (V – b) gives,
P(V − b ) = RT −
a (V − b )
V(V + b ) + b(v − b)
(A.16)
Simplifying,
P(V − b )(V(V + b ) + b(v − b) ) = RT (V(V + b ) + b(v − b) ) − a (v − b )
(
)
(
)
P V 3 + 2V 2 b − Vb 2 − bV 2 − 2Vb 2 + b 3 = RT V 2 + 2Vb − b 2 − a (v − b )
(
)
(
)
P V 3 + V 2 b − 3Vb 2 + b 3 − RT V 2 + 2Vb − b 2 + a (v − b ) = 0
(A.17)
(A.18)
(A.19)
Dividing by p and simplifying gives,
RT ⎞ 2 ⎛ a
RT ⎞
⎛
⎛ 3 RT 2 aP ⎞
2
V3 + ⎜ b −
b − ⎟=0
⎟V + ⎜ b +
⎟V + ⎜ − 3b − 2b
P ⎠
P ⎠
P
b ⎠
⎝
⎝P
⎝
85
(A.20)
APPENDIX B
DERIVATION OF PR EOS BASED ISOTHERMAL
COMPRESSIBILITY EQUATION
The coefficient of isothermal compressibility (c) is defined by
c=−
1 ⎛ ∂V ⎞
⎜
⎟
V ⎜⎝ ∂p ⎟⎠ T
(B.1)
The Peng-Robinson EOS is expressed as,
p=
RT
a
−
V − b V(V + b ) + b(V − b )
(B.2)
Differentiating equation (B.2) gives,
− RT
2a (V + b )
⎛ ∂p ⎞
+
⎜
⎟ =
2
2
⎝ ∂V ⎠ T (V − b )
V + 2bV − b 2
(
)
2
(B.3)
Simplifying equation B.3 gives,
(
2a (V + b )(V − b ) − RT V 2 + 2bV − b 2
⎛ ∂p ⎞
⎜
⎟ =
⎝ ∂V ⎠ T
(V − b )2 V 2 + 2bV − b 2 2
2
(
)
)
2
(B.4)
Inverting equation B.4 gives,
(
)
2
⎛ ∂V ⎞
(
V − b ) V 2 + 2bV − b 2
⎜⎜
⎟⎟ =
2
⎝ ∂p ⎠ T 2a (V + b )(V − b ) − RT V 2 + 2bV − b 2
2
(
)
2
(B.5)
Therefore the PR EOS based coefficient of isothermal compressibility equation is
expressed as,
86
(
)
(V − b ) V 2 + 2bV − b 2
1⎡
co = − ⎢
V ⎢⎣ 2a (V + b )(V − b )2 − RT V 2 + 2bV − b 2
2
2
(
)
2
⎤
⎥
⎥⎦
(B.6)
The molar volume V in equation B.6 is determined from the cubic molar volume
equation of the PR EOS. The cubic molar volume equation of the PR EOS is expressed
as,
⎛
⎛ RT
⎞
⎛a
RT ⎞
RTb 2 ap ⎞
⎟⎟V + ⎜⎜ b 3 +
V 3 − ⎜⎜
− b ⎟⎟V 2 + ⎜⎜ − 3b 2 − 2b
− ⎟⎟ = 0
p ⎠
p
b⎠
⎝ p
⎠
⎝p
⎝
87
(B.7)
APPENDIX C
DERIVATION OF THE PR EOS ISOTHERMAL
GAS COMPRESSIBILITY EQUATION
The compressibility factor equation of the PR EOS is given as,
Z 3 − (1 − B)Z 2 + (A − 3B 2 − 2B)Z − (AB − B 2 − B 3 ) = 0
where
B=
(C.1)
aP
bP
A= 2 2
RT and
R T
Replacing “A” and “B” in equation C.1 gives,
2
⎡⎛ ap
bp ⎤
bp ⎞ 2 ⎡ ap
⎛
⎛ bp ⎞
−
−2
3
Z 3 − ⎜1 −
⎟Z + ⎢
⎜
⎟
⎥ Z − ⎢⎜⎜
2
2
RT ⎥⎦
⎝ RT ⎠
⎝ RT ⎠
⎢⎣ (RT )
⎣⎢⎝ (RT )
⎞ bp ⎛ bp ⎞ 2 ⎛ bp ⎞ 3 ⎤
⎟
⎟ RT − ⎜⎝ RT ⎟⎠ − ⎜⎝ RT ⎟⎠ ⎥ = 0
⎠
⎦⎥
(C.2)
The coefficient of isothermal gas compressibility is expressed as,
cg =
1 1 ⎛ ∂z ⎞
− ⎜ ⎟
p z ⎜⎝ ∂p ⎟⎠ T
(C.3)
Differentiating equation C.2 implicitly gives,
⎡ bp ⎛ ∂z ⎞
⎛ ∂z ⎞
⎛ ∂z ⎞
z 2 b ⎤ ⎡ ap
⎜
⎜⎜ ⎟⎟ +
3z ⎜ ⎟⎟ − 2z⎜⎜ ⎟⎟ + ⎢2z
⎥+⎢
2
⎝ ∂p ⎠ T
⎝ ∂p ⎠ T ⎢⎣ RT ⎝ ∂p ⎠ T RT ⎥⎦ ⎢⎣ (RT )
2
⎡ 3(bp )2
−⎢
2
⎣⎢ (RT )
⎛ ∂z ⎞
za ⎤
⎜⎜ ⎟⎟ +
2 ⎥
⎝ ∂p ⎠ T (RT ) ⎥⎦
⎛ ∂z ⎞
6zpb 2 ⎤ ⎡ 2bp ⎛ ∂z ⎞
2zb ⎤ ⎡ 2abp
2pb 2 3p 2 b 3 ⎤
⎜⎜ ⎟⎟ +
⎜
⎟
−
+
−
−
−
=0
⎢
⎥ ⎢
2 ⎥
3
2
⎜ ⎟
(RT )3 ⎥⎦
⎝ ∂p ⎠ T (RT ) ⎦⎥ ⎣⎢ RT ⎝ ∂p ⎠ T RT ⎥⎦ ⎣ (RT ) (RT )
(C.4)
88
Simplifying equation C.4 gives,
⎛ ∂z ⎞ ⎡ 2
bp
ap
3(bp )2 2bp ⎤
⎜⎜ ⎟⎟ ⎢3z − 2z + 2z
⎥=
+
−
−
RT (RT )2 (RT )2 RT ⎥
⎝ ∂p ⎠ ⎢⎣
⎦
6zpb 2 2zb 2abp z 2 b
za
2pb 2 3p 2 b 3
+
+
−
−
−
−
(RT )2 RT (RT )3 RT (RT )2 (RT )2 (RT )3
(C.5)
6zpb 2 2zb 2abp z 2 b
za
2pb 2 3p 2 b 3
+
+
−
−
−
−
2
3
2
2
RT
RT
⎛ ∂z ⎞ (RT )
(RT )
(RT ) (RT ) (RT )3
⎜⎜ ⎟⎟ =
⎡
⎝ ∂p ⎠
bp
ap
3(bp )2 2bp ⎤
⎢3z 2 − 2z + 2z
⎥
+
−
−
RT (RT )2 (RT )2 RT ⎥
⎢⎣
⎦
(C.6)
Therefore the PR EOS based equation for the isothermal gas compressibility is
expressed as,
6zpb 2 2zb 2abp z 2 b
za
2pb 2 3p 2 b 3
+
+
−
−
−
−
1 1 (RT )2 RT (RT )3 RT (RT )2 (RT )2 (RT )3
c = −
g p z
⎡
bp
ap
3(bp )2 2bp ⎤
⎢3z 2 − 2z + 2z
⎥
+
−
−
RT (RT )2 (RT )2 RT ⎥
⎢⎣
⎦
89
(C.7)
APPENDIX D
DERIVATION OF THE FLASH CALCULATION ALGORITHM 2
pp 244-247
Flash calculation is used to calculate the equilibrium quantity and composition of
vapor and liquid below the bubble point pressure. Flash calculation is a simulation of
the flash vaporization process.
nv (y i)
n (z i)
nL (x i)
Figure D.1: Schematic of the Flash Vaporization Process
where,
n = total number of moles of the hydrocarbon mixture, lb-mole
nL = total number of moles of the liquid phase, lb-mole
nV = total number of moles in the vapor (gas) phase, lb-mole
zi = mole fraction of component i in the entire hydrocarbon mixture
xi = mole fraction of component i in the liquid phase
yi = mole fraction of component i in the vapor phase
90
Considering the mole balance,
n = nL + nV
(D.1)
A component based material balance gives,
zi n = xi nL + yi nV
(D.2)
By definition of mole fraction,
∑x
i
=1
∑y
i
=1
i
=1
∑z
(D.3)
(D.4)
(D.5)
For simplicity phase-equilibra calculations are based on one mole of the
hydrocarbon mixture, i.e., n =1. Therefore,
nL + nV = 1
(D.6)
zi = xi nL + yi nV
(D.7)
Mathematically, the equilibrium ration K for component i is defined as,
Ki =
yi
xi
(D.8)
Combining equation D.8 and equation D.7, gives,
z =x n +x K n
i
i L
i i V
(D.9)
Solving for xi from equation D.9 gives,
xi =
zi
nL + KinV
(D.10)
Solving for yi, gives,
91
yi =
ziKi
nL + KinV
(D.11)
Combining equation D.10 and D.3 gives,
∑x = ∑ n
i
i
i
L
zi
=1
+ nVKi
(D.12)
Combining equation D.11 and D.4 gives,
∑y = ∑ n
i
i
i
∑x
But
Hence,
∑n
i
i
ziKi
=1
L + nVKi
=1
and
∑y − ∑x
i
∑y
i
i
(D.13)
=1
=0
ziKi
zi
−∑
=0
i nL + nVKi
L + nVKi
(D.14)
Simplifying equation D.14,
z i (K i − 1)
=0
L + nVKi
∑n
i
(D.15)
From equation D.6
nL = 1 - nV
(D.16)
Combining equations D.15 and D.16 gives,
z i (K i − 1)
=0
V
i − 1) + 1
∑ n (K
i
(D.17)
92
In order to determine the moles and the compositions of the vapor and liquid
phases in the reservoir at temperatures below the bubble point pressure, equation D.17
is solved numerically to obtain nV. The calculated mole fraction of the vapor phase can
then be used to calculate the mole fraction of the individual component in the vapor and
liquid phase using equations D.10 and D.11.
In this study equation D.15 is solved using the Newton Raphson’s method. The
mole fraction of the vapor phase is calculated by an iterative process using,
(n v )
f' (n v )
(n V )n +1 = (n V )n − f
(D.18)
where n and n+1 indicates the iterative steps and the functions are defined as,
f (n V ) = ∑
i
z i (K i − 1)
n V (K i − 1) + 1
(D.19)
⎡ z i (K i − 1)2 ⎤
f' (n V ) = −∑ ⎢
2 ⎥
i ⎣
⎢ [n V (K i − 1) + 1] ⎦⎥
(D.20)
This iteration is carried performed till the desired convergence is achieved.
93
APPENDIX E
MATHEMATICAL CONSISTENCY OF ISOTHERMAL
COMPRESSIBILITY ADDITION BELOW
BUBBLE POINT PRESSURE
At pressures below the bubble point pressure the hydrocarbon mixture is in the
two phase region (Figure E.1), containing hydrocarbon liquid and vapor. Each of these
phases has an isothermal compressibility effect on the total system.
Vt
co
Oil
Gas
Vg; cg
Oil
VL; cL
Vt
co
P > Pb
P < Pb
Figure E.1: Schematic Diagram of Volume and Isothermal Compressibility at pressures
above and below the bubble point pressure.
From figure E.1, the total volume of the system can be represented by,
Vt = VL + Vg
(E.1)
Differentiating equation E.1 in terms of pressure as constant temperature gives,
94
⎛ ∂Vt
⎜⎜
⎝ ∂p
⎞
⎛ ∂V
⎟⎟ = ⎜⎜ L
⎠ T ⎝ ∂p
⎛ ∂Vg ⎞
⎞
⎟⎟
⎟⎟ + ⎜⎜
⎠ T ⎝ ∂p ⎠ T
(E.2)
The coefficients of isothermal compressibility are defined as,
ct = −
cL = −
cg = −
⎛ ∂Vt
⎜⎜
⎝ ∂p
1
Vt
⎞
⎟⎟
⎠T
(E.3)
1
VL
⎛ ∂VL
⎜⎜
⎝ ∂p
⎞
⎟⎟
⎠T
(E.4)
1
Vg
⎛ ∂Vg
⎜⎜
⎝ ∂p
⎞
⎟⎟
⎠T
(E.5)
The proposed method for adding the coefficient of isothermal oil compressibility
and the coefficient of isothermal gas compressibility is described as,
ct =
Vg
Vt
cg +
VL
cL
Vt
(E.6)
Combining equations E.3, E.4, E.5 and E.6 gives,
1
Vt
⎛ ∂Vt
⎜⎜
⎝ ∂p
⎞
V 1 ⎛ ∂VL
⎟⎟ = L
⎜⎜
⎠ T Vt VL ⎝ ∂p
⎛ ∂V
Substituting for ⎜⎜ t
⎝ ∂p
1
Vt
⎡⎛ ∂VL
⎢⎜⎜
⎣⎢⎝ ∂p
Vg 1 ⎛ ∂Vg ⎞
⎞
⎜⎜
⎟⎟
⎟⎟ +
⎠ T Vt Vg ⎝ ∂p ⎠ T
(E.7)
⎞
⎟⎟ in equation E.7, using equation E.2 gives,
⎠T
⎛ ∂Vg ⎞ ⎤ VL 1 ⎛ ∂VL
⎞
⎟⎟ ⎥ =
⎟⎟ + ⎜⎜
⎜⎜
⎠ T ⎝ ∂p ⎠ T ⎦⎥ Vt VL ⎝ ∂p
Vg 1 ⎛ ∂Vg ⎞
⎞
⎜⎜
⎟⎟
⎟⎟ +
⎠ T Vt Vg ⎝ ∂p ⎠ T
Simplifying,
95
(E.8)
1
Vt
⎛ ∂VL
⎜⎜
⎝ ∂p
⎞
1 ⎛ ∂Vg
⎜⎜
⎟⎟ +
⎠ T Vt ⎝ ∂p
⎞
1 ⎛ ∂VL
⎟⎟ =
⎜⎜
⎠ T Vt ⎝ ∂p
⎞
1 ⎛ ∂Vg
⎜⎜
⎟⎟ +
⎠ T Vt ⎝ ∂p
⎞
⎟⎟
⎠T
This shows that equation E.6 is mathematically consistent.
96
(E.9)
APPENDIX F
RESERVOIR FLUID STUDY REPORT AND ANALYSIS
The coefficient of isothermal oil compressibility of not measured directly in the
laboratory. It is calculated from the experimental data reported in the reservoir fluid
study report.
The coefficient of isothermal oil compressibility is calculated from the PressureVolume Relations at pressures above the bubble point pressure. The expressions used to
calculate the coefficient of isothermal oil compressibility at pressures above the bubble
point pressure is given as29 pp 288 ,
co =
⎡ ⎛ Vt ⎞ ⎤
⎟⎟ ⎥
⎢ ⎜⎜
⎝ Vb ⎠ 2 ⎥
⎢
ln
⎢⎛ V ⎞ ⎥
⎢ ⎜⎜ t ⎟⎟ ⎥
⎢⎣ ⎝ Vb ⎠1 ⎥⎦
(F.1)
p1 − p 2
⎛V
where ⎜⎜ t
⎝ Vb
⎞
⎟⎟ are Pressure-Volume Relations data at different pressures of interest.
⎠
At pressures below the bubble point pressure, the coefficient of isothermal oil
compressibility is calculated from the experimental Differential Vaporization data. This
expression is given as 27, 35
co = −
1
Bo
⎛ ∂B o
⎜⎜
⎝ ∂p
Bg
⎞
⎟⎟ +
⎠ T Bo
⎛ ∂R s
⎜⎜
⎝ ∂p
⎞
⎟⎟
⎠T
(F.2)
97
Equation F.2 is further simplified by McCain 29 p. 290, to give
co =
1
B oD
⎛ ∂R SD
⎜⎜
⎝ ∂p
⎞ ⎡
⎛ ∂B
⎟⎟ ⎢B g − ⎜⎜ OD
⎠ T ⎣⎢
⎝ ∂p
⎞ ⎤
⎟⎟ ⎥
⎠ T ⎦⎥
(F.3)
The subscript D shows that the data are from the differential vaporization
experiment.
The experimental molar volume and isothermal oil compressibility are determined
in the Tables below.
98
F.1 Reservoir Fluid Study Field Report 129 pp 259-269
Company: Good Oil Company
Well: Oil Well No. 4
Field: Productive
County and State: Samson, Texas.
Bubble Point Pressure: 2620 psig @ 220 oF
Specific Volume at saturation Pressure: 0.02441 (ft 3/lb) @ 220 oF
Table F.1: Reservoir Fluid Composition for Oil Well No. 4
1
2
Molecular
Weight
Component
Mol %
(lbm/lbm mol)
H2S
0.00
34.08
CO2
0.91
44.01
N2
0.16
28.02
CH4
36.47
16.04
C2H6
9.67
30.07
C3H8
6.95
44.09
i-C4H10
1.44
58.12
n-C4
3.93
58.12
i-C5
1.44
72.15
n-C5
1.41
72.15
C6
4.33
86.17
C7+
33.29
218.00
Total
100
Sp Gr C7+
=
0.8515
Source: McCain29 p. 260
99
(1/100) x 2
App Mol Weight
0.0000
0.4005
0.0448
5.8498
2.9078
3.0643
0.8369
2.2841
1.0390
1.0173
3.7312
72.5722
93.7478
Table F.2: Molar Volume Determination from Pressure-Volume Relations for Oil Well No. 4
Pressure
(psig)
5000
4500
4000
3500
3000
2900
2800
2700
2620
2605
2591
2516
2401
2253
2090
1897
1698
1477
1292
1040
830
640
472
(1) Rel Vol
(Vt/Vo)
0.9639
0.9703
0.9771
0.9846
0.9929
0.9946
0.9964
0.9983
1.0000
1.0022
1.0041
1.0154
1.0350
1.0645
1.1040
1.1633
1.2426
1.3618
1.5012
1.7802
2.1623
2.7513
3.7226
(2) Vsat * Rel
Vol ( ft3/lb)
0.0235
0.0237
0.0239
0.0240
0.0242
0.0243
0.0243
0.0244
0.0244
0.0245
0.0245
0.0248
0.0253
0.0260
0.0269
0.0284
0.0303
0.0332
0.0366
0.0435
0.0528
0.0672
0.0909
Vt *ft3/lbmol)
= App Mol
Wt * (2)
2.2058
2.2204
2.2360
2.2531
2.2721
2.2760
2.2801
2.2845
2.2884
2.2934
2.2978
2.3236
2.3685
2.4360
2.5264
2.6621
2.8435
3.1163
3.4353
4.0738
4.9482
6.2960
8.5187
100
co(psi-1)
1.32E-05
1.40E-05
1.53E-05
1.68E-05
1.71E-05
1.81E-05
1.91E-05
2.13E-05
Table F.3: Differential Vaporization data for Oil Well No. 4
Pressure
Bg (res cu
(psig)
Rsd
BoD
dBod/dRsd
ft/ bbl)
dRsd/dp
2620
854
1.6000
2350
763
1.5540
0.0069
0.3370
0.0005
2100
684
1.5150
0.0077
0.3160
0.0005
1850
612
1.4790
0.0088
0.2880
0.0005
1600
544
1.4450
0.0103
0.2720
0.0005
1350
479
1.4120
0.0125
0.2600
0.0005
1100
416
1.3820
0.0155
0.2520
0.0005
850
354
1.3510
0.0204
0.2480
0.0005
600
292
1.3200
0.0293
0.2480
0.0005
350
223
1.2830
0.0507
0.2760
0.0005
159
157
1.2440
0.1083
0.3455
0.0006
101
Bg(res
bbl/scf)
co(p<pb)
0.0012
0.0014
0.0016
0.0018
0.0022
0.0028
0.0036
0.0052
0.0090
0.0193
1.55E-04
1.83E-04
2.09E-04
2.53E-04
3.15E-04
4.17E-04
5.76E-04
8.87E-04
1.83E-03
5.20E-03
F.2 Reservoir Fluid Study Report 2 37
Company: Wilshire Oil Co. of Texas
Well: Jehlicka 1A
Field: S. Elmwood
County and State: Beaver, Oklahoma.
Bubble Point Pressure: 2663 psig @ 162 oF
Specific Volume at saturation Pressure: 0.02380 (ft 3/lb) @ 162 oF
Table F.4: Reservoir Fluid Composition for Jehlicka 1A
Molecular
Weight
App. Mol.
Component
Mol %
(lbm/lbm mol)
Weight
H2S
0.00
34.08
0.00
CO2
0.49
44.01
0.22
N2
0.53
28.02
0.15
CH4
38.83
16.04
6.23
C2H6
9.86
30.07
2.96
C3H8
9.53
44.09
4.20
i-C4H10
1.23
58.12
0.71
n-C4
4.31
58.12
2.50
i-C5
1.20
72.15
0.87
n-C5
1.87
72.15
1.35
C6
2.82
86.17
2.43
C7+
29.33
252.00
73.91
Total
100.00
95.54
Sp Gr C7+
=
0.8413
102
Table F.5: Molar Volume Determination from Pressure-Volume Relations for Jehlicka 1A
Vsat *
Pressure
RelVol
Rel
Vt (ft3 /
(psig)
(Vt/Vo)
Vol
lb mol)
co(psi-1)
5000
0.9724
0.0231
2.2110
4500
0.9775
0.0233
2.2226
1.05E-05
4000
0.9830
0.0234
2.2351
1.12E-05
3500
0.9890
0.0235
2.2487
1.22E-05
3000
0.9954
0.0237
2.2633
1.29E-05
2900
0.9967
0.0237
2.2662
1.31E-05
2800
0.9981
0.0238
2.2694
1.40E-05
2700
0.9995
0.0238
2.2726
1.40E-05
2663
1.0000
0.0238
2.2737
1.35E-05
2639
1.0025
0.0239
2.2794
2577
1.0093
0.0240
2.2949
2493
1.0195
0.0243
2.3181
2351
1.0399
0.0247
2.3645
2178
1.0808
0.0257
2.4575
1998
1.1119
0.0265
2.5282
1797
1.1718
0.0279
2.6644
1590
1.2560
0.0299
2.8558
1378
1.3761
0.0328
3.1289
1179
1.5392
0.0366
3.4998
935
1.8540
0.0441
4.2155
746
2.2638
0.0539
5.1473
577
2.8810
0.0686
6.5507
421
3.9124
0.0931
8.8958
103
Table F.6: Differential Vaporization data for Jehlicka 1A
Pressure
Bg (res cu
(PSIG)
Rsd
BoD
ft/ bbl)
dRsd/dp
2663
928
1.5150
2400
840
1.4750
0.0059
0.3346
2150
765
1.4410
0.0065
0.3000
1900
690
1.4080
0.0075
0.3000
1650
619
1.3760
0.0087
0.2840
1400
547
1.3450
0.0105
0.2880
1150
476
1.3150
0.0130
0.2840
900
407
1.2850
0.0170
0.2760
650
336
1.2530
0.0241
0.2840
400
262
1.2180
0.0398
0.2960
183
183
1.1760
0.0861
0.3641
104
dBod/dRsd
Bg(res
bbl/scf)
co(1/psia)
0.0005
0.0005
0.0004
0.0005
0.0004
0.0004
0.0004
0.0005
0.0005
0.0005
0.0010
0.0012
0.0013
0.0015
0.0019
0.0023
0.0030
0.0043
0.0071
0.0153
1.34E-04
1.48E-04
1.89E-04
2.27E-04
3.07E-04
4.09E-04
5.57E-04
8.69E-04
1.61E-03
4.58E-03
F.3 Reservoir Fluid Study Report 3 36
Company: A.C.T. Operating Company
Well: Jacques Unit #5603
Field: Rough Ride Field
County and State: Fisher County, Texas.
Bubble Point Pressure: 1675 psig @ 138 oF
Specific Volume at saturation Pressure: 0.022584 (ft 3/lb) @ 138 oF
Table F.7: Reservoir Fluid Composition for Jacques Unit #5603
Molecular
Weight
(lbm/lbm
App. Mol.
Component
Mol %
mol)
Weight
H2S
0.00
34.08
0.00
CO2
0.16
44.01
0.07
N2
1.94
28.02
0.54
CH4
25.50
16.04
4.09
C2H6
7.37
30.07
2.22
C3H8
11.21
44.09
4.94
i-C4H10
3.49
58.12
2.03
n-C4
4.17
58.12
2.42
i-C5
2.18
72.15
1.57
n-C5
2.76
72.15
1.99
C6
3.67
86.17
3.16
C7+
37.55
209.00
78.48
Total
100
101.52
Sp Gr C7+
=
0.8467
105
Table F.8: Molar Volume Determination from Pressure-Volume Relations for Jacques Unit #5603
Pressure
Rel Vol
Vsat *
Vt(ft3/lb
(psig)
(Vt/Vo)
Rel Vol
mole)
co(psi-1)
5000
0.9672
0.0218
2.2175
4500
0.9713
0.0219
2.2269
8.46E-06
4000
0.9756
0.0220
2.2368
8.83E-06
3500
0.9802
0.0221
2.2474
9.41E-06
3000
0.9851
0.0222
2.2586
9.97E-06
2500
0.9904
0.0224
2.2707
1.07E-05
2000
0.9961
0.0225
2.2838
1.15E-05
1900
0.9973
0.0225
2.2866
1.20E-05
1800
0.9985
0.0226
2.2893
1.20E-05
1700
0.9997
0.0226
2.2921
1.20E-05
1675
1.0000
0.0226
2.2928
1.20E-05
1666
1.0018
0.0226
2.2969
1656
1.0038
0.0227
2.3015
1646
1.0059
0.0227
2.3063
1636
1.0080
0.0228
2.3111
1613
1.0129
0.0229
2.3223
1527
1.0334
0.0233
2.3693
1388
1.0745
0.0243
2.4636
1230
1.1375
0.0257
2.6080
1050
1.2413
0.0280
2.8460
886
1.3846
0.0313
3.1745
733
1.5913
0.0359
3.6485
617
1.8302
0.0413
4.1962
487
2.2541
0.0509
5.1681
376
2.8730
0.0649
6.5871
273
3.9223
0.0886
8.9929
106
Table F.9: Differential Vaporization data for Jacques Unit #5603
Bg (res
Pressure
cu
ft/
(psig)
Rsd
BoD
dBod/dRsd
bbl)
dRsd/dp
1675
691
1.3980
1500
646
1.3800
0.0095
0.2571
0.0004
1350
608
1.3640
0.0106
0.2533
0.0004
1200
569
1.3480
0.0120
0.2600
0.0004
1050
530
1.3320
0.0138
0.2600
0.0004
900
490
1.3150
0.0162
0.2667
0.0004
750
448
1.2980
0.0195
0.2800
0.0004
600
404
1.2790
0.0246
0.2933
0.0004
450
356
1.2580
0.0330
0.3200
0.0004
300
301
1.2320
0.0495
0.3667
0.0005
160
237
1.2000
0.0909
0.4571
0.0005
107
Bg(res
bbl/scf)
co(p<pb)
0.0017
0.0019
0.0021
0.0025
0.0029
0.0035
0.0044
0.0059
0.0088
0.0162
2.39E-04
2.72E-04
3.32E-04
3.99E-04
4.98E-04
6.63E-04
9.05E-04
1.38E-03
2.48E-03
5.98E-03
APPENDIX G
RESERVOIR FLUIDS CRITICAL PROPERTIES TABLE
The critical properties used in calculating the constant parameters of the PengRobinson EOS are given in Table G.1. 44 pp 162, 13 pp 354
Table G.1: Reservoir Fluids Critical Properties.
Component
CO2
N2
C1
C2
C3
i-C4
n-C4
i-C5
n-C5
C6
C7
C8
C9
C10
C11
C12
C13
C14
C15
C16
C17
C18
C19
Critical
Temperature
(oR)
547.60
227.30
343.00
549.80
665.70
734.70
765.30
828.80
845.40
913.40
972.36
1023.66
1070.28
1111.86
1150.20
1184.40
1215.00
1247.40
1274.40
1301.40
1324.80
1344.60
1364.40
Critical
Pressure
(psia)
1070.60
493.00
667.80
707.80
616.30
529.10
550.70
490.40
488.60
436.90
397.30
361.05
332.05
305.95
282.61
263.90
243.60
227.65
214.60
203.00
194.30
184.15
175.45
Acentric
Factor
0.2301
0.0450
0.0115
0.0908
0.1454
0.1756
0.1928
0.2273
0.2510
0.2900
0.3495
0.3996
0.4435
0.4923
0.5303
0.5764
0.6174
0.6430
0.6863
0.7174
0.7697
0.8114
0.8522
108
Critical
z
0.2742
0.2916
0.2884
0.2843
0.2804
0.2824
0.2736
0.2701
0.2623
0.2643
0.2611
0.2559
0.2520
0.2465
0.2419
0.2382
0.2320
0.2262
0.2235
0.2199
0.2190
0.2168
0.2150
Molecular
Weight
44.01
28.02
16.04
30.07
44.09
58.12
58.12
72.15
72.15
86.17
100.20
114.23
128.26
142.29
156.31
170.34
184.37
198.39
212.42
226.45
240.47
254.50
268.53
APPENDIX H
AVERAGE ABSOLUTE PERCENT DEVIATION (AAPD)
The relative errors of the predicted molar volumes and the isothermal oil compressibility
are given in the Tables below.
Table H.1: The AAPD for the Predicted Molar Volume for Oil Well No. 4, Good Oil
Company. Samson, Texas.
Volume (ft3/lb mole)
ERROR
AAPD
AAPD
Pressure(psia)
Exptal
PR EOS MPR EOS PR EOS MPR EOS
5014.7
2.2058
2.2047
2.2672
0.05
2.79
4514.7
2.2204
2.2201
2.2816
0.01
2.76
4014.7
2.2360
2.2371
2.2974
0.05
2.75
3514.7
2.2531
2.2559
2.3148
0.12
2.73
3014.7
2.2721
2.2770
2.3341
0.21
2.73
2914.7
2.2760
2.2815
2.3382
0.24
2.73
2814.7
2.2801
2.2861
2.3424
0.26
2.73
2714.7
2.2845
2.2909
2.3468
0.28
2.73
2634.7
2.2884
2.2948
2.3503
0.28
2.71
2619.7
2.2934
2.2033
2.6633
3.93
16.13
2605.7
2.2978
2.2083
2.6655
3.89
16.00
2530.7
2.3236
2.2427
2.6804
3.48
15.35
2415.7
2.3685
2.2897
2.7081
3.33
14.34
2267.7
2.4360
2.3597
2.7531
3.13
13.02
2104.7
2.5264
2.4519
2.8157
2.95
11.45
1911.7
2.6621
2.5878
2.9152
2.79
9.51
1712.7
2.8435
2.7688
3.0547
2.63
7.42
1491.7
3.1163
3.0482
3.2799
2.19
5.25
1306.7
3.4353
3.3596
3.5454
2.21
3.21
1054.7
4.0738
3.9944
4.1111
1.95
0.92
844.7
4.9482
4.8609
4.9103
1.76
0.77
654.7
6.2960
6.1895
6.1702
1.69
2.00
486.7
8.5187
8.3257
8.2369
2.27
3.31
Avg =
1.73
6.23
109
Table H.2: The AAPD for the Predicted Coefficient of Isothermal Oil Compressibility for Oil Well No. 4, Good Oil Company.
Samson, Texas.
Isothermal Compressibility (psia-1)
Pressure(psia)
5014.7
4514.7
4014.7
3514.7
3014.7
2914.7
2814.7
2714.7
2634.7
2364.7
2114.7
1864.7
1614.7
1364.7
1114.7
864.7
614.7
364.7
173.7
Exptal
1.32E-05
1.40E-05
1.53E-05
1.68E-05
1.71E-05
1.81E-05
1.91E-05
2.13E-05
1.55E-04
1.83E-04
2.09E-04
2.53E-04
3.15E-04
4.17E-04
5.76E-04
8.87E-04
1.83E-03
5.20E-03
PR EOS
1.33E-05
1.45E-05
1.59E-05
1.76E-05
1.96E-05
2.01E-05
2.05E-05
2.10E-05
2.14E-05
2.35E-04
2.74E-04
3.26E-04
3.96E-04
4.95E-04
6.44E-04
8.84E-04
1.33E-03
2.42E-03
5.42E-03
MPR EOS
1.21E-05
1.32E-05
1.44E-05
1.58E-05
1.75E-05
1.79E-05
1.83E-05
1.87E-05
1.90E-05
2.06E-04
2.44E-04
2.95E-04
3.65E-04
4.65E-04
6.16E-04
8.63E-04
1.32E-03
2.44E-03
5.47E-03
Avg =
110
ERROR
AAPD
AAPD
PR
MPR
EOS
EOS
9.86
14.12
15.09
16.74
17.20
13.49
10.23
0.67
51.36
49.50
56.44
57.00
57.34
54.26
53.61
50.41
32.84
4.37
31.36
0.42
2.95
3.32
4.18
4.46
0.99
2.00
10.62
32.74
33.11
41.59
44.63
47.75
47.63
49.84
49.26
33.52
5.20
23.01
Table H.3: The AAPD for the Predicted Molar Volume for Jacques Unit #5603, Rough Ride Field, A.C.T. Operating
Company, Fisher County, Texas.
Volume (ft3/lb mole)
ERROR
Pressure (psia)
Exptal
PR EOS
MPR EOS AAPD PR EOS
AAPD MPR EOS
5014.7
4514.7
4014.7
3514.7
3014.7
2514.7
2014.7
1914.7
1814.7
1714.7
1689.7
1680.7
1670.7
1660.7
1650.7
1627.7
1541.7
1402.7
1244.7
1064.7
900.7
747.7
631.7
501.7
390.7
287.7
2.2175
2.2269
2.2368
2.2474
2.2586
2.2707
2.2838
2.2866
2.2893
2.2921
2.2928
2.2969
2.3015
2.3063
2.3111
2.3223
2.3693
2.4636
2.6080
2.8460
3.1745
3.6485
4.1962
5.1681
6.5871
8.9929
2.2330
2.2426
2.2528
2.2640
2.2761
2.2893
2.3039
2.3070
2.3102
2.3134
2.3142
2.1812
2.1860
2.1909
2.1959
2.2076
2.2565
2.3549
2.5060
2.7572
3.0966
3.5832
4.1440
5.1419
6.6222
9.1869
2.3248
2.3345
2.3451
2.3565
2.3689
2.3824
2.3974
2.4006
2.4038
2.4071
2.4079
2.4891
2.4929
2.4968
2.5008
2.5106
2.5506
2.6325
2.7595
2.9765
3.2800
3.7261
4.2506
5.2026
6.6328
9.1431
Avg =
0.70
0.70
0.72
0.74
0.78
0.82
0.88
0.89
0.91
0.93
0.94
5.04
5.02
5.00
4.99
4.94
4.76
4.41
3.91
3.12
2.45
1.79
1.24
0.51
0.53
2.16
2.26
111
4.83
4.83
4.84
4.86
4.88
4.92
4.97
4.99
5.00
5.02
5.02
8.37
8.32
8.26
8.21
8.11
7.65
6.86
5.81
4.59
3.32
2.13
1.30
0.67
0.69
1.67
5.00
Table H.4: The AAPD for the Predicted Coefficient of Isothermal Oil Compressibility for Jacques Unit #5603, Rough Ride
Field, A.C.T. Operating Company, Fisher County, Texas (Using 11 components).
Isothermal Compressibility (psia-1)
Pressure (psia)
Exptal
5014.7
PR EOS
MPR EOS
8.24E-06
8.10E-06
ERROR
AAPD PR
EOS
AAPD
MPR EOS
4514.7
8.46E-06
8.83E-06
8.68E-06
4.36
2.61
4014.7
8.83E-06
9.49E-06
9.34E-06
7.43
5.66
3514.7
9.41E-06
1.03E-05
1.01E-05
8.95
7.14
3014.7
9.97E-06
1.11E-05
1.09E-05
11.40
9.70
2514.7
1.07E-05
1.21E-05
1.19E-05
12.94
11.17
2014.7
1.15E-05
1.33E-05
1.31E-05
15.79
14.05
1914.7
1.20E-05
1.36E-05
1.34E-05
12.54
10.88
1814.7
1.20E-05
1.38E-05
1.36E-05
14.92
13.26
1714.7
1.20E-05
1.41E-05
1.39E-05
17.39
15.73
1689.7
1.20E-05
1.42E-05
1.40E-05
18.15
16.40
1514.7
2.39E-04
4.32E-04
4.00E-04
80.37
67.07
1364.7
2.72E-04
4.98E-04
4.65E-04
83.21
71.19
1214.7
3.32E-04
5.82E-04
5.49E-04
75.18
65.26
1064.7
3.99E-04
6.91E-04
6.58E-04
73.29
65.20
914.7
4.98E-04
8.38E-04
8.08E-04
68.34
62.22
614.7
9.05E-04
1.36E-03
1.34E-03
50.23
47.89
464.7
1.38E-03
1.88E-03
1.87E-03
36.18
35.36
314.7
2.48E-03
2.91E-03
2.91E-03
17.29
17.46
174.7
5.98E-03
5.47E-03
5.50E-03
8.45
7.95
32.44
28.75
Avg =
112
Table H.5: The AAPD for the Predicted Molar Volume for Jehlicka 1A, Wilshire Oil Co. of Texas, Beaver, Oklahoma.
Volume (ft3/lb mole)
ERROR
Pressure (psia)
Exptal
PR EOS
MPR EOS
AAPD PR EOS
AAPD MPR EOS
5014.7
2.2110
2.2956
2.2779
3.83
3.03
4514.7
2.2226
2.3087
2.2892
3.87
3.00
4014.7
2.2351
2.3229
2.3015
3.93
2.97
3514.7
2.2487
2.3386
2.3149
4.00
2.94
3014.7
2.2633
2.3560
2.3296
4.09
2.93
2914.7
2.2662
2.3596
2.3327
4.12
2.93
2814.7
2.2694
2.3634
2.3359
4.14
2.93
2714.7
2.2726
2.3673
2.3391
4.17
2.93
2677.7
2.2737
2.3688
2.3403
4.18
2.93
2653.7
2.2794
2.2131
2.5787
2.91
13.13
2591.7
2.2949
2.2308
2.5899
2.79
12.86
2507.7
2.3181
2.2569
2.6056
2.64
12.40
2365.7
2.3645
2.3070
2.6378
2.43
11.56
2192.7
2.4575
2.3949
2.6874
2.55
9.36
2012.7
2.5282
2.4780
2.7570
1.99
9.05
1811.7
1604.7
2.6644
2.8558
2.6147
2.8039
2.8618
3.0107
1.86
1.82
7.41
5.42
1392.7
3.1289
3.0740
3.2342
1.75
3.37
1193.7
3.4998
3.4411
3.5483
1.67
1.39
949.7
4.2155
4.1399
4.1746
1.79
0.97
760.7
5.1473
5.0465
5.0142
1.96
2.59
591.7
6.5507
6.4199
6.3198
2.00
3.52
435.7
8.8958
8.7522
8.5807
Avg =
1.61
2.87
3.54
5.35
113
Table H.6: The AAPD for the Predicted Coefficient of Isothermal Oil Compressibility for Jehlicka 1A, Wilshire Oil Co. of
Texas, Beaver, Oklahoma.
Isothermal Compressibility (psia-1)
Pressure (psia)
Exptal
5014.7
ERROR
PR EOS
MPR EOS
1.09E-05
9.56E-06
AAPD PR
EOS
AAPD MPR
EOS
4514.7
1.05E-05
1.18E-05
1.03E-05
4.19
1.55
4014.7
1.12E-05
1.29E-05
1.11E-05
5.24
0.73
3514.7
1.22E-05
1.41E-05
1.21E-05
5.67
0.58
3014.7
1.29E-05
1.55E-05
1.32E-05
9.14
2.55
2914.7
1.31E-05
1.59E-05
1.35E-05
18.99
3.28
2814.7
1.40E-05
1.62E-05
1.37E-05
12.92
2.11
2714.7
1.40E-05
1.65E-05
1.40E-05
15.43
0.12
2677.7
1.35E-05
1.67E-05
1.41E-05
22.29
4.31
2414.7
1.34E-04
2.12E-04
1.88E-04
87.57
40.45
1914.7
1.89E-04
2.96E-04
2.69E-04
12.29
42.55
1664.7
2.27E-04
3.60E-04
3.33E-04
30.43
46.68
1414.7
3.07E-04
4.50E-04
4.23E-04
17.17
37.66
1164.7
4.09E-04
5.82E-04
5.58E-04
9.90
36.35
914.7
5.57E-04
7.95E-04
7.76E-04
4.50
39.26
664.7
8.69E-04
1.18E-03
1.17E-03
8.58
34.84
414.7
1.61E-03
2.05E-03
2.07E-03
26.66
28.70
197.7
4.58E-03
4.66E-03
4.72E-03
55.23
2.90
20.37
19.10
Avg =
114
Table H.7: The AAPD for the Predicted Coefficient of Isothermal Oil Compressibility for Jacques Unit #5603, Rough Ride
Field, A.C.T. Operating Company, Fisher County, Texas (Using 24 components).
Pressure
(psia)
5014.7
4514.7
4014.7
3514.7
3014.7
2514.7
2014.7
1914.7
1814.7
1714.7
1689.7
1514.7
1364.7
1214.7
1064.7
914.7
614.7
464.7
314.7
174.7
Exptal
8.46016E-06
8.83457E-06
9.40793E-06
9.97305E-06
1.07315E-05
1.14775E-05
1.20397E-05
1.20253E-05
1.20108E-05
1.20018E-05
2.39E-04
2.72E-04
3.32E-04
3.99E-04
4.98E-04
9.05E-04
1.38E-03
2.48E-03
5.98E-03
PR EOS
9.93E-06
1.08E-05
1.18E-05
1.30E-05
1.44E-05
1.61E-05
1.82E-05
1.87E-05
1.92E-05
1.97E-05
1.99E-05
4.05E-04
4.65E-04
5.42E-04
6.43E-04
7.80E-04
1.27E-03
1.78E-03
2.78E-03
5.31E-03
MPR EOS
7.33E-06
7.84E-06
8.42E-06
9.08E-06
9.83E-06
1.07E-05
1.17E-05
1.19E-05
1.22E-05
1.24E-05
1.25E-05
3.81E-04
4.44E-04
5.25E-04
6.32E-04
7.77E-04
1.30E-03
1.82E-03
2.86E-03
5.44E-03
Avg =
AAPD PR EOS
AAPD MPR
EOS
27.78
33.79
38.29
44.69
50.40
58.92
55.49
59.83
64.35
65.64
69.10
71.12
63.19
61.23
56.64
40.71
28.54
12.01
11.21
48.05
7.31
4.67
3.49
1.41
0.29
2.03
0.83
1.20
3.24
3.90
59.23
63.45
58.14
58.50
56.11
43.51
32.04
15.34
8.98
22.30
115
APPENDIX I
FORTRAN CODE DOCUMENTATION
This is a short documentation to accompany the FORTRAN program developed
to compute the coefficient of isothermal oil compressibility based on the developed
model.
The input data required to run the program are listed below;
1. Number of components in the hydrocarbon fluid.
2. Number of pressure steps required; including pressures below and above the
bubble point.
3. The list of the pressure steps.
4. The list of the component properties in this order; critical temperature (oR),
critical pressure (psia), component mole fraction, acentric factor, molecular
weight, critical compressibility factor and the specific gravity. It should be noted
that the specific gravity values of the component are not required in the
calculations except for the plus fraction where it is used to calculate the critical
parameters. Therefore the specific gravity values inputted into the program are
zeros except for the plus fraction. The critical parameters of the plus fraction are
also inputted as zeros.
5. The reservoir temperature in degree Rankine
6. The bubble point pressure in psia.
A sample of the input data is displayed below;
11
19
5014.7
4514.7
4014.7
3514.7
3014.7
2914.7
2814.7
2714.7
2677.7
2414.7
1914.7
1664.7
1414.7
1164.7
914.7
664.7
414.7
197.7
114.7
547.6
227.3
343
549.8
665.7
734.7
765.3
828.8
845.4
913.4
0
1070.6
493
667.8
707.8
616.3
529.1
550.7
490.4
488.6
436.9
0
0.0049
0.0053
0.3883
0.0986
0.0953
0.0123
0.0431
0.012
0.0187
0.0282
0.2933
0.2301
0.045
0.0115
0.0908
0.1454
0.1756
0.1928
0.2273
0.251
0.29
0
44.01 0.2742
28.02 0.2916
16.04 0.2884
30.07 0.2843
44.09 0.2804
58.12 0.2824
58.12 0.2736
72.15 0.2701
72.15 0.2623
86.17 0.2643
252 0
622
2677.7
117
0
0
0
0
0
0
0
0
0
0
0.8413
The program was originally designed for four different EOS (Soave-RedlichKwong (S.R.K.) EOS, Modified SRK EOS, Peng-Robinson (P.R.) EOS and Modified PR
EOS). Due to the problem of convergence in the fugacities and the observed error in the
SRK EOS and modified SRK EOS, the predicted values of these two EOS were not
reported. It was observed that the SRK EOS and the Modified SRK EOS gave good
predictions at pressures above the bubble point pressure.
The program reads in the pressure and check if the pressure is above or below the
bubble point pressure. If the pressure is above the bubble point pressure, the molar
volume is calculated and the partial derivative of the molar volume with respect to
pressure at constant temperature is calculated. These two answers are combined to
calculate the coefficient of isothermal oil compressibility.
If the pressure is below the bubble point pressure, the two phase flash calculation
algorithm is called and the isothermal compressibility of the liquid and gas phases are
calculated and combined to give the coefficient of isothermal oil compressibility at
pressures below the bubble point pressure.
It was observed that convergence could not be achieved at a reasonable length of
time for some desired pressures. The two pressures at which this lack of convergence was
observed are at the input pressure 764.7 psia for the reservoir fluid sample from A.C.T
Operating Company and at the input pressure of 2164.7 psia.
The output of the program is formatted to give the calculated molar volumes and
the calculated coefficient of isothermal oil compressibility above the bubble point
pressure. At pressures below the bubble point pressure, the molar volume and the
118
coefficient of isothermal oil compressibility of both the liquid and the vapor phases are
reported.
119
APPENDIX J
DEVELOPED FORTRAN CODE
This is the developed FORTRAN Code to compute the Coefficient of Isothermal
Oil Compressibility for Reservoir Fluids based on the developed model.
Program Flash_Iso_Comp
Implicit None
REAL :: M_W(99), S_GC7 ,Tci(99), Ci, Pci(99)
REAL :: OWi(99),A_Fi(99), Zci(99),S_G(99)
REAL :: Api(99), Bti(99), M_F(99), B_PtC7
REAL :: PR, T,PB, TRi(99), PRi(99),R,PS(99)
REAL :: kij
REAL :: ai_LLS(99),ai_SRK(99),ai_PR(99), bi_LLS(99),bi_SRK(99)
REAL
:: ai_MPR(99),bi_MPR(99),ai_MSRK(99),bi_MSRK(99),am_MPR
REAL
:: am_MSRK,bm_MPR,bm_MSRK
REAL :: am_LLS,am_SRK, am_Pr,bm_LLS,bm_SRK,bm_PR ,bi_PR(99)
REAL :: ami_LLS,ami_SRK,ami_PR,bmi_LLS,bmi_SRK,bmi_PR,TF_PR(99)
REAL :: mi_LLS(99), mi_SRK(99), mi_PR(99), TF_LLS(99), TF_SRK(99)
REAL :: A_PR, B_PR, C_PR, D_PR,A_SRK, B_SRK, C_SRK, D_SRK
REAL :: A_MPR, B_MPR, C_MPR, D_MPR,A_MSRK, B_MSRK, C_MSRK, D_MSRK
REAL :: OBi(99), OAi(99),Coeff(4),RT(3)
REAL :: V_PR, V_SRK, V_LLS, DVPT_PR, DVPT_SRK, DVPT_LLS
REAL
:: V_MPR,V_MSRK,CO_MPR,CO_MSRK,V_PRG,V_MPRG,V_SRKG
REAL
:: V_PRL,V_MPRL,V_SRKL,V_MSRKL,V_MSRKG
REAL :: CO_PR, CO_SRK, CO_LLS
REAL ::CO_PRG,CO_PRL,CT_PR,CO_SRKG,CO_SRKL,CT_SRK,CO_MPRG
REAL
:: CO_MPRL,CT_MPR,CO_MSRKG,CO_MSRKL,CT_MSRK
REAL :: NLLS1, NLLS2, DLLS1, DLLS2,NSRK1,NSRK2,DSRK1,DSRK2
REAL :: NPR1,NPR2,DPR1,DPR2,NMPR1,NMPR2,DMPR1,DMPR2
REAL :: NMSRK1,NMSRK2,DMSRK1,DMSRK2,DVPT_MPR, DVPT_MSRK
REAL
:: VTPR,VTSRK,VTMPR,VTMSRK,Mtype
INTEGER :: i,j, NCOMP, NPSTEPS,II
REAL :: g
R=10.7315
Mtype = 0
g =1.0
open(UNIT=5,FILE='INPUT10.txt',STATUS='old')
open(UNIT=6,FILE='OUTPUT10.txt',STATUS='unknown')
Read(5,*) NCOMP
Read(5,*) NPSTEPS
!Number of Components
!Number of Pressure Steps
Do i=1,NPSTEPS
Read(5,*) PS(i)
end do
!Reading Pressures
120
Do i=1,NCOMP
Read(5,*) Tci(i),Pci(i),M_F(i),A_Fi(i),M_W(i),Zci(i),S_G(i)
end do
Read(5,*) T
Read(5,*) PB
!Reading Reservoir Temperature
! Reading Bubble Point
! Computing the Properties of the plus fraction - Pc=0
Do i=1,NCOMP
if (Pci(i).EQ.0) Then
Ci = 3.8501/(1.54057-(0.02494*(M_W(i)**0.5)))
B_PtC7 = 108.7017*(M_W(i)**0.4225)*(S_G(i)**0.4268)
A_Fi(i) = 4.5494E-9*(M_W(i)**0.02445)*(S_G(i)**(-2.08511))
& * (B_PtC7**2.903798)*(Ci**(-1.54424))
Zci(i)=0.293/(1+(0.375*A_Fi(i)))
Tci(i)=66.3775*(M_W(i)**0.12286)*(S_G(i)**0.47926)
& *(B_PtC7**0.41286)*(Ci**(-0.35734))
Pci(i) =31839*(M_W(i)**(-0.93426))*(S_G(i)**(1.64074))
& * (B_PtC7**0.49447) * (Ci**(-2.39909))
end if
end do
Write (6,10)"P", "VSRK","VMSRK","VPR","VMPR","CoSRK","CoMSRK"
& ,"CoPR","CoMPR"
10 Format(A8,2x,A7,2x,A7,2x,A7,2x,A7,A10,2x,A10,2x,A10,2x,A10)
Do II=1,NPSTEPS
If (PS(II).GE.PB) Then
PR = PS(II)
CALL Comp_EOS(PR,V_SRK,V_MSRK,V_PR,V_MPR,CO_SRK,CO_MSRK
& ,CO_PR,CO_MPR)
Write(6,20)PR,V_SRK,V_MSRK,V_PR,V_MPR,CO_SRK,CO_MSRK,CO_PR,CO_MPR
20
Format(1x,F7.2,1x,F10.6,1x,F10.6,1x,F10.6,1x,F10.6,2x,E20.4,2x
& ,E20.4,2x,E20.4,2x,E20.4)
Else
If (g.EQ.1.0) Then
g=2.0
Write(6,40)"PR","V_SRKL","V_SRKG","VTSRK","V_MSRKL","V_MSRKG"
& ,"VTMSRK","V_PRL","V_PRG","VTPR","V_MPRL","V_MPRG","VTMPR"
& ,"CO_SRKL","CO_SRKG","CT_SRK","CO_MSRKL","CO_MSRKG","CT_MSRK"
& ,"CO_PRL","CO_PRG","CT_PR","CO_MPRL","CO_MPRG","CT_MPR"
121
40
Format (A8,2x,A7,2x,A7,2x,A7,2x,A7,2x,A7,2x,A7,2x,A7,2x,A7,2x,A7
&,2x,A7,2x,A7,2x,A7,2x,A10,2x,A10,2x,A10,2x,A10,2x,A10,2x,A10,2x
&,A10,2x,A10,2x,A10,2x,A10,2x,A10,2x,A10)
end if
PR = PS(II)
CALL Flash(PR,V_SRKL,V_SRKG,VTSRK,V_MSRKL,V_MSRKG,VTMSRK
& ,V_PRL,V_PRG,VTPR,V_MPRL,V_MPRG,VTMPR,CO_SRKL,CO_SRKG,CT_SRK
& ,CO_MSRKL,CO_MSRKG,CT_MSRK,CO_PRL,CO_PRG,CT_PR,CO_MPRL,CO_MPRG
& ,CT_MPR)
Write(6,30)PR,V_SRKL,V_SRKG,VTSRK,V_MSRKL,V_MSRKG,VTMSRK
& ,V_PRL,V_PRG,VTPR,V_MPRL,V_MPRG,VTMPR,CO_SRKL,CO_SRKG,CT_SRK
& ,CO_MSRKL,CO_MSRKG,CT_MSRK,CO_PRL,CO_PRG,CT_PR,CO_MPRL,CO_MPRG
& ,CT_MPR
30
Format(F10.2,2x,F10.6,2x,F10.6,2x,F10.6,2x,F10.6,2x,F10.6,2x
& ,F10.6,2x,F10.6,2x,F10.6,2x,F10.6,2x,F10.6,2x,F10.6,2x,F10.6,2x
& ,E20.6,2x,E20.6,2x,E20.6,2x,E20.6,2x,E20.6,2x,E20.6,2x,E20.6,2x
& ,E20.6,2x,E20.6,2x,E20.6,2x,E20.6,2x,E20.6)
end if
end do
!*******Subroutine Comp EOS*****************************************
contains
Subroutine Comp_EOS(PR,V_SRK,V_MSRK,V_PR,V_MPR,CO_SRK,CO_MSRK
& ,CO_PR,CO_MPR)
REAL, INTENT(IN) :: PR
REAL, INTENT(OUT) ::CO_SRK,CO_MSRK,CO_PR,CO_MPR,V_SRK
REAL, INTENT(OUT) ::V_MSRK,V_PR,V_MPR
!****************************************************************!
Do i=1,NCOMP
OWi(i) = 0.361/(1+(0.274*A_Fi(i)))
mi_LLS(i) = 0.14443 + 1.0662*A_Fi(i) + 0.02756*A_Fi(i)
& *A_Fi(i) - 0.18074*(A_Fi(i)**3)
TRi(i)=T/Tci(i)
PRi(i)=PR/Pci(i)
OBi(i)=Zci(i)*OWi(i)
OAi(i)=(1+(OWi(i)-1)*Zci(i))**3
TF_LLS(i)=(1+mi_LLS(i)*(1-TRi(i)**0.5))**2
mi_SRK(i) = 0.480 + 1.574*A_Fi(i) - 0.176*A_Fi(i)**2
mi_PR(i) = 0.37464 + 1.54226*A_Fi(i) - 0.26992*A_Fi(i)**2
122
TF_SRK(i)=(1+(mi_SRK(i))*(1- TRi(i)**0.5))**2
TF_PR(i)=(1+(mi_PR(i))*(1- TRi(i)**0.5))**2
ai_LLS(i) = (TF_LLS(i)*OAi(i)*(R*Tci(i))**2)/Pci(i)
bi_LLS(i) = ((OBi(i)*R*Tci(i))/Pci(i))
ai_SRK(i)= (0.42748*TF_SRK(i)*(R*Tci(i))**2)/pci(i)
bi_SRK(i) = (0.08664*R*Tci(i))/pci(i)
ai_PR(i) = 0.45724*TF_PR(i)*(R*Tci(i))**2/pci(i)
bi_PR(i) = 0.07780*R*Tci(i)/pci(i)
ai_MPR(i) = ai_LLS(i)
bi_MPR(i) = bi_LLS(i)
ai_MSRK(i) = ai_LLS(i)
bi_MSRK(i)= bi_LLS(i)
END DO
! Solving for Mixture Parameters
am_LLS = 0
am_SRK = 0
am_Pr = 0
bmi_LLS = 0
bmi_SRK = 0
bmi_PR = 0
DO i=1,NCOMP
DO j=1,NCOMP
IF ( (Tci(i)/Pci(i)**0.5) < (Tci(j)/Pci(j)**0.5) ) THEN
kij = ((Tci(i)/Pci(i)**0.5) / (Tci(j)/Pci(j)**0.5) )**0.5
ELSE
kij = ((Tci(j)/Pci(j)**0.5) / (Tci(i)/Pci(i)**0.5) )**0.5
END IF
kij =1
ami_LLS=M_F(i)*M_F(j)*(ai_LLS(i)*ai_LLS(j))**0.5*kij
ami_SRK=M_F(i)*M_F(j)*(ai_SRK(i)*ai_SRK(j))**0.5*kij
ami_PR =M_F(i)*M_F(j)*(ai_PR(i)*ai_PR(j))**0.5*kij
am_LLS = am_LLS + ami_LLS
am_SRK = am_SRK + ami_SRK
am_PR = am_PR + ami_PR
am_MSRK=am_LLS
am_MPR = am_LLS
END DO
123
bmi_LLS = bmi_LLS + ( M_F(i)*bi_LLS(i) )
bmi_SRK = bmi_SRK + ( M_F(i)*bi_SRK(i) )
bmi_PR = bmi_PR + ( M_F(i)*bi_PR(i) )
END Do
bm_LLS = bmi_LLS
bm_SRK = bmi_SRK
bm_PR = bmi_PR
bm_MPR = bm_LLS
bm_MSRK=bm_LLS
!****************************************************************!
A_PR= 1
A_MPR=1
A_SRK = 1
A_MSRK=1
A_PR= 1
A_MPR=1
B_PR=bm_PR-R*T/PR
C_PR=(am_PR/PR)-3*bm_PR**2-2*bm_PR*R*T/PR
D_PR=bm_PR**3+(bm_PR**2)*R*T/PR-am_PR*bm_PR/PR
B_MPR=bm_MPR-R*T/PR
C_MPR=(am_MPR/PR)-3*bm_MPR**2-2*bm_MPR*R*T/PR
D_MPR=bm_MPR**3+(bm_MPR**2)*R*T/PR-am_MPR*bm_MPR/PR
B_SRK=-1*R*T/PR
C_SRK=am_SRK/PR-bm_SRK**2-R*T*bm_SRK/PR
D_SRK=-1*am_SRK*bm_SRK/PR
B_MSRK=-1*R*T/PR
C_MSRK=am_MSRK/PR-bm_MSRK**2-R*T*bm_MSRK/PR
D_MSRK=-1*am_MSRK*bm_MSRK/PR
!***Calculating the total molar volume using PR EOS*********
Coeff(1)=A_PR
Coeff(2)=B_PR
Coeff(3)=C_PR
Coeff(4)=D_PR
124
Call Cubic_Solver(Mtype,Coeff,RT)
If (RT(3).EQ.0.0) then
RT(3)=RT(1)
endif
If (RT(2).EQ.0.0) then
RT(2)=RT(1)
endif
If (RT(1).EQ.0.0) then
Write(6,*)"Error All roots are zeros"
endif
If (RT(1).LE.RT(2)) Then
V_PR = RT(1)
else
V_PR=RT(2)
endif
If (V_PR.LE.RT(3)) Then
V_PR = V_PR
else
V_PR=RT(3)
endif
!***Calculating the total molar volume using Modified PR EOS*********
Coeff(1)=A_MPR
Coeff(2)=B_MPR
Coeff(3)=C_MPR
Coeff(4)=D_MPR
Call Cubic_Solver(Mtype,Coeff,RT)
If (RT(3).EQ.0.0) then
RT(3)=RT(1)
end if
If (RT(2).EQ.0.0) then
RT(2)=RT(1)
end if
If (RT(1).EQ.0.0) then
Write(6,*)"Error All roots are zeros"
endif
If (RT(1).LE.RT(2)) Then
V_MPR = RT(1)
else
V_MPR=RT(2)
endif
If (V_MPR.LE.RT(3)) Then
V_MPR = V_MPR
else
V_MPR=RT(3)
endif
!***Calculating the total molar volume using SRK EOS*********
Coeff(1)=A_SRK
Coeff(2)=B_SRK
Coeff(3)=C_SRK
Coeff(4)=D_SRK
Call Cubic_Solver(Mtype,Coeff,RT)
125
If (RT(3).EQ.0.0) then
RT(3)=RT(1)
end if
If (RT(2).EQ.0.0) then
RT(2)=RT(1)
end if
If (RT(1).EQ.0.0) then
Write(6,*)"Error All roots are zeros"
endif
If (RT(1).LE.RT(2)) Then
V_SRK = RT(1)
else
V_SRK=RT(2)
endif
If (V_SRK.LE.RT(3)) Then
V_SRK = V_SRK
else
V_SRK=RT(3)
endif
!***Calculating the total molar volume using Modified PR EOS*********
Coeff(1)=A_MSRK
Coeff(2)=B_MSRK
Coeff(3)=C_MSRK
Coeff(4)=D_MSRK
Call Cubic_Solver(Mtype,Coeff,RT)
If (RT(3).EQ.0.0) then
RT(3)=RT(1)
end if
If (RT(2).EQ.0.0) then
RT(2)=RT(1)
end if
If (RT(1).EQ.0.0) then
Write(6,*)"Error All roots are zeros"
endif
If (RT(1).LE.RT(2)) Then
V_MSRK = RT(1)
else
V_MSRK=RT(2)
endif
If (V_MSRK.LE.RT(3)) Then
V_MSRK = V_MPR
else
V_MSRK=RT(3)
endif
!*********************************************************************
NSRK1=(V_SRK-bm_SRK)**2
NSRK2=(V_SRK**2+V_SRK*bm_SRK)**2
DSRK1=am_SRK*(2*V_SRK+bm_SRK)*(V_SRK-bm_SRK)**2
DSRK2=R*T*(V_SRK**2+bm_SRK*V_SRK)**2
DVPT_SRK=NSRK1*NSRK2/(DSRK1-DSRK2)
NMSRK1=(V_MSRK-bm_MSRK)**2
NMSRK2=(V_MSRK**2+V_MSRK*bm_MSRK)**2
126
DMSRK1=am_MSRK*(2*V_MSRK+bm_MSRK)*(V_MSRK-bm_MSRK)**2
DMSRK2=R*T*(V_MSRK**2+bm_MSRK*V_MSRK)**2
DVPT_MSRK=NMSRK1*NMSRK2/(DMSRK1-DMSRK2)
NPR1=(V_PR-bm_PR)**2
NPR2=(V_PR**2+2*bm_PR*V_PR-bm_PR**2)**2
DPR1=2*am_PR*(V_PR+bm_PR)*(V_PR-bm_PR)**2
DPR2=R*T*(V_PR**2+2*bm_PR*V_PR-bm_PR**2)**2
DVPT_PR=NPR1*NPR2/(DPR1-DPR2)
NMPR1=(V_MPR-bm_MPR)**2
NMPR2=(V_MPR**2+2*bm_MPR*V_MPR-bm_MPR**2)**2
DMPR1=2*am_MPR*(V_MPR+bm_MPR)*(V_MPR-bm_MPR)**2
DMPR2=R*T*(V_MPR**2+2*bm_MPR*V_MPR-bm_MPR**2)**2
DVPT_MPR=NMPR1*NMPR2/(DMPR1-DMPR2)
CO_SRK=-1*DVPT_SRK/V_SRK
CO_PR=-1*DVPT_PR/V_PR
CO_MSRK=-1*DVPT_MSRK/V_MSRK
CO_MPR=-1*DVPT_MPR/V_MPR
END Subroutine Comp_EOS
!*************Subroutine Flash Calculations***********************
Subroutine Flash(PR,V_SRKL,V_SRKG,VTSRK,V_MSRKL,V_MSRKG,VTMSRK
& ,V_PRL,V_PRG,VTPR,V_MPRL,V_MPRG,VTMPR,CO_SRKL,CO_SRKG,CT_SRK
& ,CO_MSRKL,CO_MSRKG,CT_MSRK,CO_PRL,CO_PRG,CT_PR,CO_MPRL,CO_MPRG
& ,CT_MPR)
Implicit None
Real, Intent (IN)
:: PR
Real, Intent (OUT) :: V_PRG,V_MPRG,V_SRKG,V_MSRKG,V_PRL,V_MPRL
Real, Intent (OUT) :: V_SRKL,V_MSRKL,CO_SRKG,CO_SRKL,CT_SRK
Real, Intent (OUT) :: CO_MSRKL,CT_MSRK,CO_PRG,CO_PRL,CT_PR
Real, Intent (OUT) :: CO_MPRG,CO_MPRL,CT_MPR,CO_MSRKG
Real, Intent (OUT) :: VTPR,VTSRK,VTMPR,VTMSRK
Real :: Zi(99),Ki(99),Kimin,Kimax,NVmin,NVmax,NV,FNV,FPNV
Real :: Ai(99),Bi(99),SAi,SBi,TOL,maxx,minn,Liqxi(99),Vapyi(99)
Real :: ZLPR,ZGPR,ZLSRK,ZGSRK,ZLMPR,ZGMPR,ZLMSRK,ZGMSRK
Real :: FLPR(99),FGPR(99),FLMPR(99),FGMPR(99),FLSRK(99),FGSRK(99)
Real ::FLMSRK(99),FGMSRK(99),SFPR,SFSRK,SFMPR,SFMSRK,CHECK,Counter
Real :: LiPR(99),ViPR(99),LiSRK(99),ViSRK(99),LiMPR(99),ViMPR(99)
Real :: LiMSRK(99),ViMSRK(99),ORIGXI(99),ORIGYI(99)
Real :: BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK
Real :: BBMSRK,NV1,NVPR,NVMPR
127
TOL=0.001
Zi = M_F
Kimin=50
Kimax=0
SAi = 0
SBi = 0
FNV = 0
FPNV = 0
SFPR=0
SFSRK=0
SFMSRK=0
SFMPR=0
CHECK=0
Counter = 0
!write(6,*) " Zi = " , Zi , M_F
Do i=1,NCOMP
Ki(i)=Pci(i)*exp(5.371*(1+A_Fi(i))*(1-(Tci(i)/T)))/PR
Ai(i)= Zi(i)*(Ki(i)-1)
Bi(i)=Zi(i)*(Ki(i)-1)/Ki(i)
SAi=SAi + Ai(i)
SBi=SBI + Bi(i)
end do
!write (6,*) " Ki = ", Ki
Do j=2,NCOMP
If (Ki(j-1).LT.Ki(j)) Then
minn=Ki(j-1)
maxx=Ki(j)
else
minn=Ki(j)
maxx=Ki(j-1)
end if
If (Kimin.LT.minn) then
Kimin=Kimin
else
Kimin=minn
endif
If (Kimax.GT.maxx) then
Kimax=Kimax
else
Kimax=maxx
endif
end do
NVmin=1/(1-Kimax)
NVmax=1/(1-Kimin)
If (NVmin.GT.0.0) then
Write(6,*) "NVmin is > 0.0-May converge at trivial Soln"
endif
128
If (NVmax.LT.1.0) then
Write(6,*) "NVmax is < 1.0-May converge at trivial Soln"
endif
!
NV = SAi/(SAi-SBi)
NV=0.5
!Write(6,*) " NVmin = ", NVmin, "NVmax = " , NVmax
!***** Bisection Method *************************************
If ((NV.LT.NVmin) .OR. (NV.GT.NVmax)) then
Call bisect(NVmin,NVmax,Zi,Ki,NV)
endif
!*******Using Newton Raphson*********************************
100
Do i=1,NCOMP
FNV= FNV + Zi(i)*(Ki(i)-1)/(NV*(Ki(i)-1)+1)
FPNV = FPNV + (1)*Zi(i)*(Ki(i)-1)**2/((NV*(Ki(i)-1)+1)**2)
enddo
NV1= NV - (FNV/(-1)*FPNV)
If (ABS(NV1-NV).GT.TOL) then
NV= NV1
Counter = Counter + 1
If (Counter .GT. 80000000) then
Write(6,*) " Newton Raphson Called 80 million times"
Stop
endif
FNV=0
FPNV = 0
Goto 100
endif
!***** Bisection Method *************************************
If ((NV.LT.NVmin) .OR. (NV.GT.NVmax)) then
Call bisect(NVmin,NVmax,Zi,Ki,NV)
endif
Do i=1,NCOMP
Liqxi(i)= Zi(i)/(1+NV*(Ki(i)-1))
Vapyi(i)= Zi(i)*Ki(i)/(1+NV*(Ki(i)-1))
Enddo
ORIGXI = Liqxi
ORIGYI = Vapyi
!write (6,*) "NV = ", NV
!write (6,*) "NVmin = ", NVmin
!write (6,*) "NVmax = ", NVmax
129
!**********Checking Fugacity Constraint for PR EOS**************
Call Zfactor_FugacPR(PR,Liqxi,Vapyi,NV,ZLPR,ZGPR,FLPR,FGPR,BAPR
&,BBPR)
Do j=1,NCOMP
SFPR = SFPR + (((FLPR(j)/FGPR(j))-1)**2)
end do
LiPR = Liqxi
ViPR = Vapyi
!write (6,*) " SFPR = ", SFPR
111
IF(Abs(SFPR).GT. TOL) Then
!write (6,*) " SFPR = ", SFPR
!Write (6,*) " Function Called"
SFPR = 0.0
Do i=1, NCOMP
Ki(i) = Ki(i)*FLPR(i)/FGPR(i)
Ai(i)= Zi(i)*(Ki(i)-1)
Bi(i)=Zi(i)*(Ki(i)-1)/Ki(i)
SAi=SAi + Ai(i)
SBi=SBI + Bi(i)
end do
!write (6,*) " Ki = ", Ki
!*****Check if Ki is Converging to a trivial Solution***********
Do i=1, NCOMP
CHECK = CHECK + (Alog(Ki(i)))**2
end do
If (CHECK.LT.0.0001) Then
Write(6,*) " Trivial Solution is Detected"
end if
CHECK =0.0
Do j = 2, NCOMP
If (Ki(j-1).LT.Ki(j)) Then
minn=Ki(j-1)
maxx=Ki(j)
else
minn=Ki(j)
maxx=Ki(j-1)
end if
If (Kimin.LT.minn) then
Kimin=Kimin
else
Kimin=minn
endif
If (Kimax.GT.maxx) then
Kimax=Kimax
else
Kimax=maxx
130
endif
end do
NVmin=1/(1-Kimax)
NVmax=1/(1-Kimin)
!write (6,*) " NVmin = ", NVmin , "NVmax = ", NVmax
If (NVmin.GT.0.0) then
Write(6,*) "NVmin is > 0.0-May converge at trivial Soln"
endif
If (NVmax.LT.1.0) then
Write(6,*) "NVmax is < 1.0-May converge at trivial Soln"
endif
!
NV = SAi/(SAi-SBi)
!write(6,*) " NV initial = " , NV
!***** Bisection Method *************************************
If ((NV.LT.NVmin) .OR. (NV.GT.NVmax)) then
Call bisect(NVmin,NVmax,Zi,Ki,NV)
endif
!*******Using Newton Raphson*********************************
FNV = 0
FPNV = 0
1000
Do i=1,NCOMP
FNV= FNV + Zi(i)*(Ki(i)-1)/(NV*(Ki(i)-1)+1)
FPNV = FPNV + (1)*Zi(i)*((Ki(i)-1)**2)/((NV*(Ki(i)-1)+1)**2)
enddo
!write (6,*) " FNV =" , FNV , "FPNV = ", FPNV, "NV = " , NV
NV1= NV - (FNV/((-1)*FPNV))
If (((-1)*FPNV) .EQ.0) then
Call bisect(NVmin,NVmax,Zi,Ki,NV)
!write(6,*) "Bisection Called"
NV1 = NV
endif
If (ABS(ABS(NV1)-ABS(NV)).GT.TOL) then
NV = NV1
Counter = Counter + 1
If (Counter .GT. 40000) then
!Write(6,*) " Newton Raphson Called 40,000 times"
endif
FNV=0
131
FPNV = 0
Goto 1000
continue
Endif
!write (6,*) " NV = ", NV , " Ki = ", Ki
Do i=1,NCOMP
Liqxi(i)= Zi(i)/(1+NV*(Ki(i)-1))
Vapyi(i)= Zi(i)*Ki(i)/(1+NV*(Ki(i)-1))
Enddo
!write (6,*) " Got to this point"
Call Zfactor_FugacPR(PR,Liqxi,Vapyi,NV,ZLPR,ZGPR,FLPR,FGPR,BAPR
&,BBPR)
!Write(6,*) " Function Called 22 "
Do i=1,NCOMP
SFPR = SFPR + ((FLPR(i)/FGPR(i))-1)**2
end do
!write (6,*) " SFPR = ", SFPR
LiPR = Liqxi
ViPR = Vapyi
Goto 111
End if
!write (6,*) "Liqxi = " , Liqxi
!write (6,*) "Vapyi = " , Vapyi
!write (6,*) " ki = ", Ki
!write (6,*) " NV = " , NV
!write (6,*) " SFPR = ", SFPR
!write (6,*) " Counter = " , counter
!write (6,*) "ZGPR = ", ZGPR ,"ZLPR = " , ZLPR
!STOP
Liqxi = ORIGXI
Vapyi = ORIGYI
NVPR = NV
!write (6,*) " Got to point close to 222"
!*******MPR EOS Fugacity Constraint***************************
Call Zfactor_FugacMPR(PR,Liqxi,Vapyi,NV,ZLMPR,ZGMPR,FLMPR,FGMPR
& ,BAMPR,BBMPR)
Do i=1,NCOMP
SFPR = SFPR + ((FLMPR(i)/FGMPR(i))-1)**2
end do
132
LiMPR = Liqxi
ViMPR = Vapyi
!**********Checking Fugacity Constraint for PR EOS**************
222
If (SFMPR.GT. TOL) Then
SFMPR = 0.0
Do i=1, NCOMP
Ki(i) = Ki(i)*FLMPR(i)/FGMPR(i)
Ai(i)= Zi(i)*(Ki(i)-1)
Bi(i)=Zi(i)*(Ki(i)-1)/Ki(i)
SAi=SAi + Ai(i)
SBi=SBI + Bi(i)
end do
!*****Check if Ki is Converging to a trivial Solution***********
Do i=1, NCOMP
CHECK = CHECK + (Alog(Ki(i)))**2
end do
If (CHECK.LT.0.0001) Then
Write(6,*) " Trivial Solution is Detected"
end if
CHECK =0.0
Do j = 2, NCOMP
If (Ki(j-1).LT.Ki(j)) Then
minn=Ki(j-1)
maxx=Ki(j)
else
minn=Ki(j)
maxx=Ki(j-1)
end if
If (Kimin.LT.minn) then
Kimin=Kimin
else
Kimin=minn
endif
If (Kimax.GT.maxx) then
Kimax=Kimax
else
Kimax=maxx
endif
end do
NVmin=1/(1-Kimax)
NVmax=1/(1-Kimin)
If (NVmin.GT.0.0) then
Write(6,*) "NVmin is > 0.0-May converge at trivial Soln"
endif
If (NVmax.LT.1.0) then
Write(6,*) "NVmax is < 1.0-May converge at trivial Soln"
endif
!
NV = SAi/(SAi-SBi)
133
!***** Bisection Method *************************************
If ((NV.LT.NVmin) .OR. (NV.GT.NVmax)) then
Call bisect(NVmin,NVmax,Zi,Ki,NV)
endif
!*******Using Newton Raphson*********************************
!write (6,*) " Goto to 2000"
2000
Do i=1,NCOMP
FNV= FNV + Zi(i)*(Ki(i)-1)/(NV*(Ki(i)-1)+1)
FPNV = FPNV + Zi(i)*(Ki(i)-1)**2/((NV*(Ki(i)-1)+1)**2)
end do
If ((-1)*FPNV .EQ.0) then
Call bisect(NVmin,NVmax,Zi,Ki,NV)
endif
NV1= NV - (FNV/((-1)*FPNV))
If (ABS(NV1-NV).GT.TOL) then
NV= NV1
FNV=0
FPNV = 0
Counter = Counter + 1
If (Counter .GT. 800000000) then
Write(6,*) " Newton Raphson Called 80,000 times"
Stop
endif
Goto 2000
Endif
Do i=1,NCOMP
Liqxi(i)= Zi(i)/(1+NV*(Ki(i)-1))
Vapyi(i)= Zi(i)*Ki(i)/(1+NV*(Ki(i)-1))
Enddo
Call Zfactor_FugacMPR(PR,Liqxi,Vapyi,NV,ZLMPR,ZGMPR,FLMPR,FGMPR
& ,BAMPR,BBMPR)
Do i=1,NCOMP
SFMPR = SFMPR + ((FLMPR(i)/FGMPR(i))-1)**2
end do
LiMPR = Liqxi
ViMPR = Vapyi
Goto 222
134
Endif
Liqxi = ORIGXI
Vapyi = ORIGYI
NVMPR = NV
!*******SRK EOS Fugacity Constraint***************************
Call Zfactor_FugacSRK(PR,Liqxi,Vapyi,NV,ZLSRK,ZGSRK,FLSRK,FGSRK
& ,BBSRK,BASRK)
Do i=1,NCOMP
SFSRK = SFSRK + ((FLSRK(i)/FGSRK(i))-1)**2
end do
LiSRK = Liqxi
ViSRK = Vapyi
SFSRK = 0.0
!**********Checking Fugacity Constraint for PR EOS**************
333
If (SFSRK.GT. TOL) Then
SFSRK = 0.0
Do i=1, NCOMP
Ki(i) = Ki(i)*FLSRK(i)/FGSRK(i)
Ai(i)= Zi(i)*(Ki(i)-1)
Bi(i)=Zi(i)*(Ki(i)-1)/Ki(i)
SAi=SAi + Ai(i)
SBi=SBI + Bi(i)
end do
!*****Check if Ki is Converging to a trivial Solution***********
Do i=1, NCOMP
CHECK = CHECK + (Alog(Ki(i)))**2
end do
If (CHECK.LT.0.0001) Then
Write(6,*) " Trivial Solution is Detected"
end if
CHECK =0.0
Do j = 2, NCOMP
If (Ki(j-1).LT.Ki(j)) Then
minn=Ki(j-1)
maxx=Ki(j)
else
minn=Ki(j)
maxx=Ki(j-1)
end if
If (Kimin.LT.minn) then
Kimin=Kimin
else
Kimin=minn
endif
If (Kimax.GT.maxx) then
Kimax=Kimax
else
Kimax=maxx
endif
end do
135
NVmin=1/(1-Kimax)
NVmax=1/(1-Kimin)
If (NVmin.GT.0.0) then
Write(6,*) "NVmin is > 0.0-May converge at trivial Soln"
endif
If (NVmax.LT.1.0) then
Write(6,*) "NVmax is < 1.0-May converge at trivial Soln"
endif
!
NV = SAi/(SAi-SBi)
!***** Bisection Method *************************************
If ((NV.LT.NVmin) .OR. (NV.GT.NVmax)) then
Call bisect(NVmin,NVmax,Zi,Ki,NV)
endif
!*******Using Newton Raphson*********************************
3000
Do i=1,NCOMP
FNV= FNV + Zi(i)*(Ki(i)-1)/((NV*(Ki(i)-1)+1))
FPNV = FPNV + (1)*Zi(i)*(Ki(i)-1)**2/((NV*(Ki(i)-1)+1)**2)
enddo
If ((-1)*FPNV .EQ.0) then
Call bisect(NVmin,NVmax,Zi,Ki,NV)
endif
NV1= NV - (FNV/((-1)*FPNV))
If (ABS(NV1-NV).GT.TOL) then
NV= NV1
FNV=0
FPNV = 0
Counter = Counter + 1
If (Counter .GT. 80000000) then
Write(6,*) " Newton Raphson Called 80,000 times"
Stop
endif
Goto 3000
Counter = Counter + 1
If (Counter .GT. 80000000) then
Write(6,*) " Newton Raphson Called 80,000 times"
Stop
endif
Endif
Do i=1,NCOMP
Liqxi(i)= Zi(i)/(1+NV*(Ki(i)-1))
Vapyi(i)= Zi(i)*Ki(i)/(1+NV*(Ki(i)-1))
Enddo
Call Zfactor_FugacSRK(PR,Liqxi,Vapyi,NV,ZLSRK,ZGSRK,FLSRK,FGSRK
& ,BBSRK,BASRK)
136
Do i=1,NCOMP
SFSRK = SFSRK + ((FLSRK(i)/FGSRK(i))-1)**2
end do
LiSRK = Liqxi
ViSRK = Vapyi
Goto 333
End if
Liqxi = ORIGXI
Vapyi = ORIGYI
!******************MSRK EOS***********************************************
Call Zfactor_FugacMSRK(PR,Liqxi,Vapyi,NV,ZLMSRK,ZGMSRK,FLMSRK
& ,FGMSRK,BAMSRK,BBMSRK)
Do i=1,NCOMP
SFMSRK = SFMSRK + ((FLMSRK(i)/FGMSRK(i))-1)**2
end do
LiMSRK = Liqxi
ViMSRK = Vapyi
SFMSRK = 0.0
!**********Checking Fugacity Constraint for PR EOS**************
444
If (SFMSRK.GT. TOL) Then
SFMSRK = 0.0
Do i=1, NCOMP
Ki(i) = Ki(i)*FLMSRK(i)/FGMSRK(i)
Ai(i)= Zi(i)*(Ki(i)-1)
Bi(i)=Zi(i)*(Ki(i)-1)/Ki(i)
SAi=SAi + Ai(i)
SBi=SBI + Bi(i)
end do
!*****Check if Ki is Converging to a trivial Solution***********
Do i=1, NCOMP
CHECK = CHECK + (Alog(Ki(i)))**2
end do
If (CHECK.LT.0.0001) Then
Write(6,*) " Trivial Solution is Detected"
end if
CHECK =0.0
Do j = 2, NCOMP
If (Ki(j-1).LT.Ki(j)) Then
minn=Ki(j-1)
maxx=Ki(j)
else
minn=Ki(j)
maxx=Ki(j-1)
end if
If (Kimin.LT.minn) then
Kimin=Kimin
else
Kimin=minn
137
endif
If (Kimax.GT.maxx) then
Kimax=Kimax
else
Kimax=maxx
endif
end do
NVmin=1/(1-Kimax)
NVmax=1/(1-Kimin)
If (NVmin.GT.0.0) then
Write(6,*) "NVmin is > 0.0-May converge at trivial Soln"
endif
If (NVmax.LT.1.0) then
Write(6,*) "NVmax is < 1.0-May converge at trivial Soln"
endif
!
NV = SAi/(SAi-SBi)
!***** Bisection Method *************************************
If ((NV.LT.NVmin) .OR. (NV.GT.NVmax)) then
Call bisect(NVmin,NVmax,Zi,Ki,NV)
endif
!*******Using Newton Raphson*********************************
4000
Do i=1,NCOMP
FNV= FNV + Zi(i)*(Ki(i)-1)/((NV*(Ki(i)-1)+1))
FPNV = FPNV + (1)*Zi(i)*(Ki(i)-1)**2/((NV*(Ki(i)-1)+1)**2)
enddo
If ((-1)*FPNV .EQ.0) then
Call bisect(NVmin,NVmax,Zi,Ki,NV)
endif
NV1= NV - (FNV/((-1)*FPNV))
If (ABS(NV1-NV).GT.TOL) then
NV= NV1
FNV=0
FPNV = 0
Counter = Counter + 1
If (Counter .GT. 80000000) then
Write(6,*) " Newton Raphson Called 80000,000 times"
Stop
endif
Goto 4000
Counter = Counter + 1
If (Counter .GT. 80000000) then
Write(6,*) " Newton Raphson Called 80000,000 times"
Stop
endif
Endif
138
Do i=1,NCOMP
Liqxi(i)= Zi(i)/(1+NV*(Ki(i)-1))
Vapyi(i)= Zi(i)*Ki(i)/(1+NV*(Ki(i)-1))
Enddo
Call Zfactor_FugacMSRK(PR,Liqxi,Vapyi,NV,ZLMSRK,ZGMSRK,FLMSRK
& ,FGMSRK,BAMSRK,BBMSRK)
Do i=1,NCOMP
SFMSRK = SFMSRK + ((FLMSRK(i)/FGMSRK(i))-1)**2
end do
LiMSRK = Liqxi
ViMSRK = Vapyi
goto 444
End if
Call CompValue(PR,LiPR,ViPR,NVPR,NVMPR,LiSRK,ViSRK,LiMPR,ViMPR
&,LiMSRK,ViMSRK,ZLPR,ZGPR,ZLSRK,ZGSRK,ZLMPR,ZGMPR,ZLMSRK,ZGMSRK
&,V_PRL,V_PRG,V_SRKL,V_SRKG,V_MPRL,V_MPRG,V_MSRKL,V_MSRKG,CO_PRL
&,CO_PRG,CT_PR,CO_SRKL,CO_SRKG,CT_SRK,CO_MPRL,CO_MPRG,CT_MPR
&,CO_MSRKL,CO_MSRKG,CT_MSRK,VTPR,VTSRK,VTMPR,VTMSRK)
End Subroutine Flash
!**************************Subroutine CompValue******************************
Subroutine CompValue(PR,LiPR,ViPR,NVPR,NVMPR,LiSRK,ViSRK,LiMPR
&,ViMPR,LiMSRK,ViMSRK,ZLPR,ZGPR,ZLSRK,ZGSRK,ZLMPR,ZGMPR,ZLMSRK
&,ZGMSRK,V_PRL,V_PRG,V_SRKL,V_SRKG,V_MPRL,V_MPRG,V_MSRKL,V_MSRKG
&,CO_PRL,CO_PRG,CT_PR,CO_SRKL,CO_SRKG,CT_SRK,CO_MPRL,CO_MPRG,CT_MPR
&,CO_MSRKL,CO_MSRKG,CT_MSRK,VTPR,VTSRK,VTMPR,VTMSRK)
Implicit None
Real, Intent (IN) :: PR,LiPR(99),ViPR(99),LiSRK(99),ViSRK(99)
&,LiMPR(99),ViMPR(99),LiMSRK(99),ViMSRK(99),ZLPR,ZGPR,ZLSRK,ZGSRK
&,ZLMPR,ZGMPR,ZLMSRK,ZGMSRK,NVPR,NVMPR
Real, Intent (OUT) ::V_PRL,VTPR,VTSRK,VTMPR,VTMSRK
&,V_PRG,V_SRKL,V_SRKG,V_MPRL,V_MPRG,V_MSRKL,V_MSRKG,CO_PRL,CO_PRG
&,CT_PR,CO_SRKL,CO_SRKG,CT_SRK,CO_MPRL,CO_MPRG,CT_MPR,CO_MSRKL
&,CO_MSRKG,CT_MSRK
Real :: am_PR,bm_PR,am_SRK,bm_SRK,am_MPR,bm_MPR,am_MSRK,bm_MSRK
REAL :: NMPR1,NMPR2,DMPR1,DMPR2,DVPT_MPR
Real :: NPR1,NPR2,DPR1,DPR2,DVPT_PR,NSRK1,NSRK2,DSRK1,DSRK2
Real :: DVPT_SRK,NMSRK1,NMSRK2,DMSRK1,DMSRK2,DVPT_MSRK
Real :: BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK
Real :: BBMSRK,BiPR(99),AiPR(99),BiMPR(99),AiMPR(99)
Real :: AiMSRK(99),BiMSRK(99),BiSRK(99),AiSRK(99)
Real :: YAIJPR,YAIJSRK,YAIJMPR,YAIJMSRK
Real :: NGG,NGG1,NGG2,NGG3,NGG4,DGG,DGG1,DGG2,DGG3,DZP
139
!****Using PR EOS Liquid***********************************************
Call ParaMix(PR,LiPR,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
V_PRL = ZLPR*R*T/PR
V_PRG = ZGPR*R*T/PR
am_PR = (BAPR*(R*T)**2)/PR
bm_PR = (BBPR*R*T)/PR
NPR1=(V_PRL-bm_PR)**2
NPR2=(V_PRL**2+2*bm_PR*V_PRL-bm_PR**2)**2
DPR1=2*am_PR*(V_PRL+bm_PR)*(V_PRL-bm_PR)**2
DPR2=R*T*(V_PRL**2+2*bm_PR*V_PRL-bm_PR**2)**2
DVPT_PR=NPR1*NPR2/(DPR1-DPR2)
CO_PRL=-1*DVPT_PR/V_PRL
!*****Using PR EOS for Gas Properties************************************
Call ParaMix(PR,ViPR,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
V_PRG = ZGPR*R*T/PR
am_PR = (BAPR*(R*T)**2)/PR
bm_PR = (BBPR*R*T)/PR
!***************************************************************************
NGG1= (6*ZGPR*PR*bm_PR**2)/((R*T)**2) + ZGPR*2*bm_PR/(R*T)
NGG2=(2*am_PR*bm_PR*PR)/((R*T)**3)-(2*PR*bm_PR**2)/((R*T)**2)
NGG3= (bm_PR*ZGPR**2)/(R*T) + (ZGPR*am_PR)/((R*T)**2)
NGG4= 3*(bm_PR**3)*(PR**2)/(R*T)**3
NGG= NGG1 + NGG2-NGG3-NGG4
DGG1= 3*ZGPR - 2*ZGPR + 2*ZGPR*bm_PR*PR/(R*T)
DGG2= am_PR*PR/(R*T)**2 - 2*bm_PR*PR/(R*T)
DGG3= 3*(bm_PR*PR/(R*T))**2
DGG = DGG1 + DGG2 - DGG3
DZP = NGG/DGG
!write(6,*) "NGG =", NGG, "DGG=",DGG
CO_PRG = 1/PR - DZP/ZGPR
!***************************************************************************
140
!
!
NPR1=(V_PRG-bm_PR)**2
NPR2=(V_PRG**2+2*bm_PR*V_PRG-bm_PR**2)**2
!
!
DPR1=2*am_PR*(V_PRG+bm_PR)*(V_PRG-bm_PR)**2
DPR2=R*T*(V_PRG**2+2*bm_PR*V_PRG-bm_PR**2)**2
!
!
DVPT_PR=NPR1*NPR2/(DPR1-DPR2)
CO_PRG=-1*DVPT_PR/V_PRG
VTPR = V_PRL + V_PRG
CT_PR = (V_PRG/VTPR)*(CO_PRG) + (V_PRL/VTPR)*(CO_PRL)
V_PRL=(1-NVPR)*V_PRL
V_PRG= NVPR*V_PRG
VTPR = V_PRL + V_PRG
!*******Using MPR EOS for Liquid Properties*************************
Call ParaMix(PR,LiMPR,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
V_MPRL = ZLMPR*R*T/PR
am_MPR = (BAMPR*(R*T)**2)/PR
bm_MPR = (BBMPR*R*T)/PR
NMPR1=(V_MPRL-bm_MPR)**2
NMPR2=(V_MPRL**2+2*bm_MPR*V_MPRL-bm_MPR**2)**2
DMPR1=2*am_MPR*(V_MPRL+bm_MPR)*(V_MPRL-bm_MPR)**2
DMPR2=R*T*(V_MPRL**2+2*bm_MPR*V_MPRL-bm_MPR**2)**2
DVPT_MPR=NMPR1*NMPR2/(DMPR1-DMPR2)
CO_MPRL=-1*DVPT_MPR/V_MPRL
!*****Using MPR EOS for Gas Properties************************************
Call ParaMix(PR,ViMPR,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
V_MPRG = ZGMPR*R*T/PR
am_MPR = (BAMPR*(R*T)**2)/PR
bm_MPR = (BBMPR*R*T)/PR
!***************************************************************************
NGG1= (6*ZGMPR*PR*bm_MPR**2)/((R*T)**2) + ZGMPR*2*bm_MPR/(R*T)
NGG2=2*am_MPR*bm_MPR*PR/((R*T)**3)-(2*PR*bm_MPR**2)/((R*T)**2)
NGG3= (bm_MPR*ZGMPR**2)/(R*T) + (ZGMPR*am_MPR)/((R*T)**2)
NGG4= 3*(bm_MPR**3)*(PR**2)/(R*T)**3
NGG= NGG1 + NGG2-NGG3-NGG4
DGG1= 3*ZGMPR - 2*ZGMPR + 2*ZGMPR*bm_MPR*PR/(R*T)
DGG2= am_MPR*PR/(R*T)**2 - 2*bm_MPR*PR/(R*T)
DGG3= 3*(bm_MPR*PR/(R*T))**2
DGG = DGG1 + DGG2 - DGG3
141
DZP = NGG/DGG
CO_MPRG = 1/PR - DZP/ZGMPR
!***************************************************************************
!
!
NMPR1=(V_MPRG-bm_MPR)**2
NMPR2=(V_MPRG**2+2*bm_MPR*V_MPRG-bm_MPR**2)**2
!
!
DMPR1=2*am_MPR*(V_MPRG+bm_MPR)*(V_MPRG-bm_MPR)**2
DMPR2=R*T*(V_MPRG**2+2*bm_MPR*V_MPRG-bm_MPR**2)**2
!
!
DVPT_MPR=NMPR1*NMPR2/(DMPR1-DMPR2)
CO_MPRG=-1*DVPT_MPR/V_MPRG
VTMPR = V_MPRL + V_MPRG
CT_MPR = (V_MPRG/VTMPR)*(CO_MPRG) + (V_MPRL/VTMPR)*(CO_MPRL)
V_MPRL=(1-NVMPR)*V_MPRL
V_MPRG= NVMPR*V_MPRG
VTMPR = V_MPRL + V_MPRG
!***********Using SRK for Liquid Properties*******************
Call ParaMix(PR,LiSRK,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
V_SRKL = ZLSRK*R*T/PR
am_SRK = (BASRK*(R*T)**2)/PR
bm_SRK = (BBSRK*R*T)/PR
NSRK1=(V_SRKL-bm_SRK)**2
NSRK2=(V_SRKL**2+V_SRKL*bm_SRK)**2
DSRK1=am_SRK*(2*V_SRKL+bm_SRK)*(V_SRKL-bm_SRK)**2
DSRK2=R*T*(V_SRKL**2+bm_SRK*V_SRKL)**2
DVPT_SRK=NSRK1*NSRK2/(DSRK1-DSRK2)
CO_SRKL=-1*DVPT_SRK/V_SRKL
!*******Using SRK EOS for Gas Properties***********************
Call ParaMix(PR,ViSRK,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
V_SRKG = ZGSRK*R*T/PR
am_SRK = (BASRK*(R*T)**2)/PR
142
bm_SRK = (BBSRK*R*T)/PR
NSRK1=(V_SRKG-bm_SRK)**2
NSRK2=(V_SRKG**2+V_SRKG*bm_SRK)**2
DSRK1=am_SRK*(2*V_SRKG+bm_SRK)*(V_SRKG-bm_SRK)**2
DSRK2=R*T*(V_SRKG**2+bm_SRK*V_SRKG)**2
DVPT_SRK=NSRK1*NSRK2/(DSRK1-DSRK2)
CO_SRKG=-1*DVPT_SRK/V_SRKG
VTSRK = V_SRKL + V_SRKG
CT_SRK = (V_SRKG/VTSRK)*CO_SRKG + (V_SRKL/VTSRK)*CO_SRKL
!***********Using MSRK for Liquid Properties*******************
Call ParaMix(PR,LiMSRK,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
V_MSRKL = ZLMSRK*R*T/PR
am_MSRK = (BAMSRK*(R*T)**2)/PR
bm_MSRK = (BBMSRK*R*T)/PR
NMSRK1=(V_MSRKL-bm_MSRK)**2
NMSRK2=(V_MSRKL**2+V_MSRKL*bm_MSRK)**2
DMSRK1=am_MSRK*(2*V_MSRKL+bm_MSRK)*(V_MSRKL-bm_MSRK)**2
DMSRK2=R*T*(V_MSRKL**2+bm_MSRK*V_MSRKL)**2
DVPT_MSRK=NMSRK1*NMSRK2/(DMSRK1-DMSRK2)
CO_MSRKL=-1*DVPT_MSRK/V_MSRKL
!*******Using MSRK EOS for Gas Properties***********************
Call ParaMix(PR,ViMSRK,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
V_MSRKG = ZGMSRK*R*T/PR
am_MSRK = (BAMSRK*(R*T)**2)/PR
bm_MSRK = (BBMSRK*R*T)/PR
NMSRK1=(V_MSRKG-bm_MSRK)**2
NMSRK2=(V_MSRKG**2+V_MSRKG*bm_MSRK)**2
DMSRK1=am_MSRK*(2*V_MSRKG+bm_MSRK)*(V_MSRKG-bm_MSRK)**2
DMSRK2=R*T*(V_MSRKG**2+bm_MSRK*V_MSRKG)**2
DVPT_MSRK=NMSRK1*NMSRK2/(DMSRK1-DMSRK2)
CO_MSRKG=-1*DVPT_MSRK/V_MSRKG
143
VTMSRK = V_MSRKL + V_MSRKG
CT_MSRK = (V_MSRKG/VTMSRK)*CO_MSRKG + (V_MSRKL/VTMSRK)*CO_MSRKL
End Subroutine CompValue
!***************Bisection Method Subroutine********************
Subroutine bisect(A,B,MF,Ki,C)
Implicit None
Real, Intent(In) :: A,B,MF(99),Ki(99)
Real, Intent(Out) :: C
Real :: DNV,CONV,AA,BB, k,KMAX
DNV=0.0
CONV=0.001
KMAX = 40000
K=1
AA=B
BB=A
C = (AA+BB)/2
Do i=1,NCOMP
DNV=DNV + MF(i)*(Ki(i)-1)/(C*(Ki(i)-1)+1)
end do
2345
If (K .LE. Kmax) Then
K=K + 1
If (ABS(AA-BB).GE.CONV) Then
If (DNV.GT.0.0) Then
AA=AA
BB=C
else
AA=C
BB=BB
endif
!write(6,*) " Counter = " , K , "NV = " , C, "FPV =", DNV
C = (AA+BB)/2
DNV = 0
Do i=1,NCOMP
DNV=DNV + MF(i)*(Ki(i)-1)/(C*(Ki(i)-1)+1)
end do
goto 2345
endif
end if
C=C
If (K .GE. Kmax) Then
C=0
end if
144
End Subroutine bisect
!**********Z Factor Subroutine ***********************************
Subroutine Zfactor_FugacPR(PR,Li,Vi,NV,ZLPR,ZGPR,FLPR,FGPR
&,BAPR,BBPR)
Implicit None
Real, Intent (In) :: PR,Li(99),Vi(99),NV
Real , Intent (Out) :: ZLPR,ZGPR,FLPR(99),FGPR(99)
Real,Intent(OUT) :: BAPR,BBPR
Real :: ZGMSRK,FLMPR(99),FGMPR(99),ZLSRK,ZGSRK,ZLMPR,ZGMPR,ZLMSRK
Real :: FLMSRK(99),FGMSRK(99),FLSRK(99),FGSRK(99)
Real :: BBSRK,BASRK,BAMPR,BBMPR,BAMSRK
Real :: BBMSRK,BiPR(99),AiPR(99),BiMPR(99),AiMPR(99)
Real :: AiMSRK(99),BiMSRK(99),BiSRK(99),AiSRK(99)
Real :: YAIJPR,YAIJSRK,YAIJMPR,YAIJMSRK
Real :: JPR,JMPR,JSRK,JMSRK,JPR1,JPR2,JPR3
Real :: SYAIJPR,SYAIJSRK,SYAIJMSRK,SYAIJMPR,Mtype
Double Precision FGGPR(99)
YAIJPR = 0
YAIJSRK = 0
YAIJMPR = 0
YAIJMSRK = 0
Mtype =0
! Solving for Mixture Parameters
Call ParaMix(PR,Li,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
!***Calculating Coeff for PR Z factor Liquid********
Coeff(1)=1
Coeff(2)= -1*(1-BBPR)
Coeff(3)= BAPR -3*BBPR**2-2*BBPR
Coeff(4)= -1*(BAPR*BBPR-BBPR**2-BBPR**3)
!Write(6,*) " Called ParaMix"
!write (6,*) " Li = " , Li , "Vi = ", Vi , "PR = " , PR
!Write(6,*) "YAIJPR =" , YAIJPR
!write(6,*) " BAPR = " , BAPR
!write (6,*) "BBPR = " , BBPR
!write (6,*) "Coeff = " , Coeff
Call Cubic_Solver(Mtype,Coeff,RT)
145
If (RT(3).EQ.0.0) then
RT(3)=RT(1)
end if
If (RT(2).EQ.0.0) then
RT(2)=RT(1)
end if
If (RT(1).EQ.0.0) then
Write(6,*)"Error All roots are zeros"
endif
If (RT(1).LE.RT(2)) Then
ZLPR = RT(1)
else
ZLPR=RT(2)
endif
If (ZLPR.LE.RT(3)) Then
ZLPR = ZLPR
else
ZLPR =RT(3)
endif
! write (6,*) " ZLPR = " , ZLPR
!******Liquid Fugacity************************************************
Do i = 1,NCOMP
YAIJPR = 0
YAIJSRK = 0
YAIJMPR = 0
YAIJMSRK = 0
Do j = 1, NCOMP
Kij = 1
YAIJPR = YAIJPR + Li(j)*(AiPR(j)*AiPR(i))**0.5*kij
YAIJSRK = YAIJSRK + Li(j)*(AiSRK(j)*AiSRK(i))**0.5*kij
YAIJMPR = YAIJMPR + Li(j)*(AiMPR(j)*AiMPR(i))**0.5*kij
YAIJMSRK = YAIJMSRK + Li(j)*(AiMSRK(j)*AiMSRK(i))**0.5*kij
End do
!********Liquid Fugacity for PR EOS************************************
JPR1=(BiPR(i)*(ZLPR-1)/BBPR)-Alog(ZLPR-BBPR)
JPR2 = (BAPR/(2.828427*BBPR))*((BiPR(i)/BBPR)-2*YAIJPR/BAPR)
JPR3 = Alog((ZLPR+2.414214*BBPR)/(ZLPR-0.414214*BBPR))
JPR = JPR1 + JPR2*JPR3
FLPR(i)= Li(i)*PR*exp(JPR)
!Write(6,*) "FLPR =" , FLPR(i)
!write (6,*) "YAIJPR = " , YAIJPR
end do
146
!********Mol * Z factor*******************
!ZLPR = ZLPR * (1-NV)
!**********Vapor Z Factor********************************************
Call ParaMix(PR,Vi,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
!***Calculating Coeff for PR Z factor Vapor********
Coeff(1)=1
Coeff(2)= -1*(1-BBPR)
Coeff(3)= BAPR -3*BBPR**2-2*BBPR
Coeff(4)= -1*(BAPR*BBPR-BBPR**2-BBPR**3)
!Write (6,*) " Gas Values"
!Write(6,*) "YAIJPR =" , YAIJPR
!write(6,*) " BAPR = " , BAPR
!write (6,*) "BBPR = " , BBPR
!write (6,*) "Coeff = " , Coeff
If (NV.LT.0) Then
Mtype = 1.0
end if
Call Cubic_Solver(Mtype,Coeff,RT)
If (RT(3).GT.0.0) then
RT(3)=RT(1)
end if
If (RT(2).EQ.0.0) then
RT(2)=RT(1)
end if
If (RT(1).EQ.0.0) then
Write(6,*)"Error All roots are zeros"
endif
If (RT(1).GT.RT(2)) Then
ZGPR = RT(1)
else
ZGPR=RT(2)
endif
If (ZGPR.GT.RT(3)) Then
ZGPR = ZGPR
else
ZGPR = RT(3)
endif
!write (6,*) "ZGPR = ", ZGPR, "NV = ", NV
!write(6,*) " BBPR =", BBPR
!******Vapor Fugacity************************************************
Do i = 1,NCOMP
YAIJPR = 0
147
YAIJSRK = 0
YAIJMPR = 0
YAIJMSRK = 0
Do j = 1, NCOMP
Kij = 1
YAIJPR = YAIJPR + Vi(j)*(AiPR(j)*AiPR(i))**0.5*kij
YAIJSRK = YAIJSRK + Vi(j)*(AiSRK(j)*AiSRK(i))**0.5*kij
YAIJMPR = YAIJMPR + Vi(j)*(AiMPR(j)*AiMPR(i))**0.5*kij
YAIJMSRK = YAIJMSRK + Vi(j)*(AiMSRK(j)*AiMSRK(i))**0.5*kij
End do
!********Vapor Fugacity for PR EOS************************************
JPR1=(BiPR(i)*(ZGPR-1)/BBPR)-Alog(ZGPR-BBPR)
JPR2 = (BAPR/(2.828427*BBPR))*((BiPR(i)/BBPR)-2*YAIJPR/BAPR)
JPR3 = Alog((ZGPR+2.414214*BBPR)/(ZGPR-0.414214*BBPR))
JPR = JPR1 + JPR2*JPR3
!Write(6,*) "JPR =" , JPR
FGPR(i)= Vi(i)*PR*exp(JPR)
!write (6,*) "FGPR =" , FGPR(i)
!write (6,*) "YAIJPR = " , YAIJPR
end do
!************Imaginary Root********************
!If (NV .LT.0) then
!If (RT(1).GT.RT(2)) Then
!ZGPR = RT(2)
!else
!ZGPR=RT(1)
! endif
!end if
!***************************************************
!********Mol * Z factor*******************
!ZGPR = ZGPR * NV
End Subroutine Zfactor_FugacPR
!***********************Subroutine Z factor Fugacity MPR EOS*******************
Subroutine Zfactor_FugacMPR(PR,Li,Vi,NV,ZLMPR,ZGMPR,FLMPR,FGMPR
&,BAMPR,BBMPR)
Implicit None
Real, Intent (In) :: PR,Li(99),Vi(99),NV
Real, Intent (Out) :: ZLMPR,ZGMPR,FLMPR(99),FGMPR(99)
Real, Intent(OUT):: BAMPR,BBMPR
Real :: ZGMSRK,FLPR(99),FGPR(99),ZLSRK,ZGSRK,ZLMSRK
Real :: FLMSRK(99),FGMSRK(99),FLSRK(99),FGSRK(99),ZLPR,ZGPR
148
Real :: BAPR,BBPR,BBSRK,BASRK,BAMSRK,BBMSRK
Real :: BiPR(99),AiPR(99),BiMPR(99),AiMPR(99)
Real :: AiMSRK(99),BiMSRK(99),BiSRK(99),AiSRK(99)
Real :: YAIJPR,YAIJSRK,YAIJMPR,YAIJMSRK
Real :: JPR,JMPR,JSRK,JMSRK,Mtype
Mtype =0
Call ParaMix(PR,Li,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
!***Calculating Coeff for MPR Z factor Liquid********
Coeff(1)=1
Coeff(2)= -1*(1-BBMPR)
Coeff(3)= BAMPR -3*BBMPR**2-2*BBMPR
Coeff(4)= -1*(BAMPR*BBMPR-BBMPR**2-BBMPR**3)
Call Cubic_Solver(Mtype,Coeff,RT)
If (RT(3).EQ.0.0) then
RT(3)=RT(1)
end if
If (RT(2).EQ.0.0) then
RT(2)=RT(1)
end if
If (RT(1).EQ.0.0) then
Write(6,*)"Error All roots are zeros"
endif
If (RT(1).LE.RT(2)) Then
ZLMPR = RT(1)
else
ZLMPR=RT(2)
endif
If (ZLMPR.LE.RT(3)) Then
ZLMPR = ZLMPR
else
ZLMPR =RT(3)
endif
!********Liquid Fugacity for MPR EOS************************************
Do i=1, NCOMP
YAIJPR = 0
YAIJSRK = 0
YAIJMPR = 0
YAIJMSRK = 0
Do j = 1, NCOMP
Kij = 1
YAIJPR = YAIJPR + Li(j)*(AiPR(j)*AiPR(i))**0.5*kij
YAIJSRK = YAIJSRK + Li(j)*(AiSRK(j)*AiSRK(i))**0.5*kij
YAIJMPR = YAIJMPR + Li(j)*(AiMPR(j)*AiMPR(i))**0.5*kij
YAIJMSRK = YAIJMSRK + Li(j)*(AiMSRK(j)*AiMSRK(i))**0.5*kij
149
End do
JMPR=(BiMPR(i)*(ZLMPR-1)/BBMPR)-Alog(ZLMPR-BBMPR)+(BAMPR/(2.828427
& *BBMPR))*((BiMPR(i)/BBMPR)-2*YAIJMPR/BAMPR)*Alog((ZLMPR+2.414214
& *BBMPR)/(ZLMPR-0.414214*BBMPR))
FLMPR(i)= Li(i)*PR*exp(JMPR)
end do
!ZLMPR = ZLMPR*(1-NV)
Call ParaMix(PR,Vi,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
!***Calculating Coeff for MPR Z factor Vapor********
Coeff(1)=1
Coeff(2)= -1*(1-BBMPR)
Coeff(3)= BAMPR -3*BBMPR**2-2*BBMPR
Coeff(4)= -1*(BAMPR*BBMPR-BBMPR**2-BBMPR**3)
If (NV.LT.0) Then
Mtype = 1.0
end if
Call Cubic_Solver(Mtype,Coeff,RT)
If (RT(3).EQ.0.0) then
RT(3)=RT(1)
end if
If (RT(2).EQ.0.0) then
RT(2)=RT(1)
end if
If (RT(1).EQ.0.0) then
Write(6,*)"Error All roots are zeros"
endif
If (RT(1).GT.RT(2)) Then
ZGMPR = RT(1)
else
ZGMPR=RT(2)
endif
If (ZGMPR.GT.RT(3)) Then
ZGMPR = ZGMPR
else
ZGMPR =RT(3)
endif
!********Vapor Fugacity for MPR EOS************************************
Do i=1,NCOMP
YAIJPR = 0
YAIJSRK = 0
YAIJMPR = 0
YAIJMSRK = 0
150
Do j = 1, NCOMP
Kij = 1
YAIJPR = YAIJPR + Vi(j)*(AiPR(j)*AiPR(i))**0.5*kij
YAIJSRK = YAIJSRK + Vi(j)*(AiSRK(j)*AiSRK(i))**0.5*kij
YAIJMPR = YAIJMPR + Vi(j)*(AiMPR(j)*AiMPR(i))**0.5*kij
YAIJMSRK = YAIJMSRK + Vi(j)*(AiMSRK(j)*AiMSRK(i))**0.5*kij
End do
JMPR=(BiMPR(i)*(ZGMPR-1)/BBMPR)-Alog(ZGMPR-BBMPR)+(BAMPR/(2.828427
& *BBMPR))*((BiMPR(i)/BBMPR)-2*YAIJMPR/BAMPR)*Alog((ZGMPR+2.414214
& *BBMPR)/(ZGMPR-0.414214*BBMPR))
FGMPR(i)= Vi(i)*PR*exp(JMPR)
end do
!************Imaginary Root********************
!If (NV .LT.0) then
!If (RT(1).GT.RT(2)) Then
! ZGMPR = RT(2)
! else
! ZGMPR=RT(1)
! endif
!end if
!***************************************************
!ZGMPR = ZGMPR*NV
End Subroutine Zfactor_FugacMPR
!***********************Subroutine Z factor Fugacity SRK*******************
Subroutine Zfactor_FugacSRK(PR,Li,Vi,NV,ZLSRK,ZGSRK,FLSRK,FGSRK
&,BBSRK,BASRK)
Implicit None
Real, Intent (In) :: PR,Li(99),Vi(99),NV
Real, Intent (Out) :: ZLSRK,ZGSRK,FLSRK(99),FGSRK(99),BBSRK,BASRK
Real :: ZGMSRK,FLPR(99),FGPR(99),FLMPR(99),FGMPR(99),ZLMPR,ZGMPR
Real :: FLMSRK(99),FGMSRK(99),ZLPR,ZGPR,ZLMSRK
Real :: BAPR,BBPR,BAMPR,BBMPR,BAMSRK,BBMSRK
Real :: BiPR(99),AiPR(99),BiMPR(99),AiMPR(99)
Real :: AiMSRK(99),BiMSRK(99),BiSRK(99),AiSRK(99)
Real :: YAIJPR,YAIJSRK,YAIJMPR,YAIJMSRK
Real :: JPR,JMPR,JSRK,JMSRK,Mtype
Mtype =0
Call ParaMix(PR,Li,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
151
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
!***Calculating Coeff for SRK EOS Z factor Liquid********
Coeff(1)=1
Coeff(2)= -1
Coeff(3)= BASRK - BBSRK - BBSRK**2
Coeff(4)= -1*BASRK*BBSRK
Call Cubic_Solver(Mtype,Coeff,RT)
If (RT(3).LE.0.01) then
RT(3)=RT(1)
end if
If (RT(2).LE.0.01) then
RT(2)=RT(1)
end if
If (RT(1).EQ.0.0) then
Write(6,*)"Error All roots are zeros"
endif
If (RT(1).LE.RT(2)) Then
ZLSRK = RT(1)
else
ZLSRK=RT(2)
endif
If (ZLSRK.LE.RT(3)) Then
ZLSRK = ZLSRK
else
ZLSRK =RT(3)
endif
!********Liquid Fugacity for SRK EOS************************************
Do i=1, NCOMP
YAIJPR = 0
YAIJSRK = 0
YAIJMPR = 0
YAIJMSRK = 0
Do j = 1, NCOMP
Kij = 1
YAIJPR = YAIJPR + Li(j)*(AiPR(j)*AiPR(i))**0.5*kij
YAIJSRK = YAIJSRK + Li(j)*(AiSRK(j)*AiSRK(i))**0.5*kij
YAIJMPR = YAIJMPR + Li(j)*(AiMPR(j)*AiMPR(i))**0.5*kij
YAIJMSRK = YAIJMSRK + Li(j)*(AiMSRK(j)*AiMSRK(i))**0.5*kij
End do
JSRK=(BiSRK(i)*(ZLSRK-1)/BBSRK)-Alog(ZLSRK-BBSRK)+(BASRK/BBSRK)*
& ((BiSRK(i)/BBSRK)-2*YAIJSRK/BASRK)*Alog(1+(BBSRK/ZLSRK))
FLSRK(i) = Li(i)*PR*EXP(JSRK)
end do
ZLSRK = ZLSRK*(1-NV)
Call ParaMix(PR,Vi,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
152
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
!***Calculating Coeff for SRK EOS Z factor Vapor********
Coeff(1)=1
Coeff(2)= -1
Coeff(3)= BASRK - BBSRK - BBSRK**2
Coeff(4)= -1*BASRK*BBSRK
If (NV.LT.0) Then
Mtype = 1.0
end if
Call Cubic_Solver(Mtype,Coeff,RT)
If (RT(3).EQ.0.0) then
RT(3)=RT(1)
end if
If (RT(2).EQ.0.0) then
RT(2)=RT(1)
end if
If (RT(1).EQ.0.0) then
Write(6,*)"Error All roots are zeros"
endif
If (RT(1).GT.RT(2)) Then
ZGSRK = RT(1)
else
ZGSRK=RT(2)
endif
If (ZGSRK.GT.RT(3)) Then
ZGSRK = ZGSRK
else
ZGSRK =RT(3)
endif
!********Vapor Fugacity for SRK EOS************************************
Do i=1,NCOMP
YAIJPR = 0
YAIJSRK = 0
YAIJMPR = 0
YAIJMSRK = 0
Do j = 1, NCOMP
Kij = 1
YAIJPR = YAIJPR + Vi(j)*(AiPR(j)*AiPR(i))**0.5*kij
YAIJSRK = YAIJSRK + Vi(j)*(AiSRK(j)*AiSRK(i))**0.5*kij
YAIJMPR = YAIJMPR + Vi(j)*(AiMPR(j)*AiMPR(i))**0.5*kij
YAIJMSRK = YAIJMSRK + Vi(j)*(AiMSRK(j)*AiMSRK(i))**0.5*kij
End do
JSRK=(BiSRK(i)*(ZGSRK-1)/BBSRK)-Alog(ZGSRK-BBSRK)+(BASRK/BBSRK)*
& ((BiSRK(i)/BBSRK)-2*YAIJSRK/BASRK)*Alog(1+(BBSRK/ZGSRK))
FGSRK(i) = Vi(i)*PR*EXP(JSRK)
153
end do
!************Imaginary Root********************
!If (NV .LT.0) then
!If (RT(1).GT.RT(2)) Then
!ZGSRK = RT(2)
! else
!ZGSRK =RT(1)
!endif
!end if
ZGSRK = ZGSRK*NV
!***************************************************
End Subroutine Zfactor_FugacSRK
!***********Subroutine for MSRK Z factor & Fugacity**************************
Subroutine Zfactor_FugacMSRK(PR,Li,Vi,NV,ZLMSRK,ZGMSRK,FLMSRK
& ,FGMSRK,BAMSRK,BBMSRK)
Implicit None
Real, Intent (In) :: PR,Li(99),Vi(99),NV
Real, Intent (OUT) :: ZLMSRK,ZGMSRK,FLMSRK(99),FGMSRK(99),BAMSRK
Real, Intent (OUT) :: BBMSRK
Real :: ZLPR,ZGPR,ZLSRK,ZGSRK,ZLMPR,ZGMPR,FLSRK(99)
Real :: FLPR(99),FGPR(99),FLMPR(99),FGMPR(99),FGSRK(99)
Real :: BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR
Real :: BiPR(99),AiPR(99),BiMPR(99),AiMPR(99)
Real :: AiMSRK(99),BiMSRK(99),BiSRK(99),AiSRK(99)
Real :: YAIJPR,YAIJSRK,YAIJMPR,YAIJMSRK
Real :: JPR,JMPR,JSRK,JMSRK
Mtype =0
Call ParaMix(PR,Li,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
!***Calculating Coeff for MSRK EOS Z factor Liquid********
Coeff(1)=1
Coeff(2)= -1
Coeff(3)= BAMSRK - BBMSRK - BBMSRK**2
Coeff(4)= -1*BAMSRK*BBMSRK
Call Cubic_Solver(Mtype,Coeff,RT)
If (RT(3).LE.0.01) then
RT(3)=RT(1)
end if
If (RT(2).LE.0.01) then
RT(2)=RT(1)
154
end if
If (RT(1).EQ.0.0) then
Write(6,*)"Error All roots are zeros"
endif
If (RT(1).LE.RT(2)) Then
ZLMSRK = RT(1)
else
ZLMSRK=RT(2)
endif
If (ZLMSRK.LE.RT(3)) Then
ZLMSRK = ZLMSRK
else
ZLMSRK =RT(3)
endif
!********Liquid Fugacity for MSRK EOS************************************
Do i=1,NCOMP
YAIJPR = 0
YAIJSRK = 0
YAIJMPR = 0
YAIJMSRK = 0
Do j = 1, NCOMP
Kij = 1
YAIJPR = YAIJPR + Li(j)*(AiPR(j)*AiPR(i))**0.5*kij
YAIJSRK = YAIJSRK + Li(j)*(AiSRK(j)*AiSRK(i))**0.5*kij
YAIJMPR = YAIJMPR + Li(j)*(AiMPR(j)*AiMPR(i))**0.5*kij
YAIJMSRK = YAIJMSRK + Li(j)*(AiMSRK(j)*AiMSRK(i))**0.5*kij
End do
JMSRK=(BiMSRK(i)*(ZLMSRK-1)/BBMSRK)-Alog(ZLMSRK-BBMSRK)+(BAMSRK
& /BBMSRK)*((BiMSRK(i)/BBMSRK)-2*YAIJMSRK/BAMSRK)*Alog(1+(BBMSRK
& /ZLMSRK))
FLMSRK(i) = Li(i)*PR*EXP(JMSRK)
end do
ZLMSRK = ZLMSRK*(1-NV)
Call ParaMix(PR,Vi,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
!***Calculating Coeff for MSRK EOS Z factor Vapor********
Coeff(1)=1
Coeff(2)= -1
Coeff(3)= BAMSRK - BBMSRK - BBMSRK**2
Coeff(4)= -1*BAMSRK*BBMSRK
If (NV.LT.0) Then
Mtype = 1.0
end if
155
Call Cubic_Solver(Mtype,Coeff,RT)
If (RT(3).EQ.0.0) then
RT(3)=RT(1)
end if
If (RT(2).EQ.0.0) then
RT(2)=RT(1)
end if
If (RT(1).EQ.0.0) then
Write(6,*)"Error All roots are zeros"
endif
If (RT(1).GT.RT(2)) Then
ZLMSRK = RT(1)
else
ZLMSRK=RT(2)
endif
If (ZGMSRK.GT.RT(3)) Then
ZGMSRK = ZGMSRK
else
ZGMSRK =RT(3)
endif
!*****Vapor Fugacity for MSRK EOS************************************
Do i= 1, NCOMP
YAIJPR = 0
YAIJSRK = 0
YAIJMPR = 0
YAIJMSRK = 0
Do j = 1, NCOMP
Kij = 1
YAIJPR = YAIJPR + Vi(j)*(AiPR(j)*AiPR(i))**0.5*kij
YAIJSRK = YAIJSRK + Vi(j)*(AiSRK(j)*AiSRK(i))**0.5*kij
YAIJMPR = YAIJMPR + Vi(j)*(AiMPR(j)*AiMPR(i))**0.5*kij
YAIJMSRK = YAIJMSRK + Vi(j)*(AiMSRK(j)*AiMSRK(i))**0.5*kij
End do
JMSRK=(BiMSRK(i)*(ZGMSRK-1)/BBMSRK)-Alog(ZGMSRK-BBMSRK)+(BAMSRK
& /BBMSRK)*((BiMSRK(i)/BBMSRK)-2*YAIJMSRK/BAMSRK)*Alog(1+(BBMSRK
& /ZGMSRK))
FGMSRK(i) = Vi(i)*PR*EXP(JMSRK)
end do
!************Imaginary Root********************
!If (NV .LT.0) then
!If (RT(1).GT.RT(2)) Then
!ZGMSRK = RT(2)
! else
!ZGMSRK =RT(1)
!endif
156
!end if
ZGMSRK = ZGMSRK*NV
!***************************************************
End Subroutine Zfactor_FugacMSRK
!***********Subroutine Parameter Mixture*******************
Subroutine ParaMix(PR,Ji,BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK,
& BBMSRK,AiPR,BiPR,AiMPR,BiMPR,AiMSRK,BiMSRK,AiSRK,BiSRK,YAIJPR
& ,YAIJSRK,YAIJMPR,YAIJMSRK)
Real, Intent (In) :: PR, Ji(99)
Real, Intent (OUT) :: BAPR,BBPR,BBSRK,BASRK,BAMPR,BBMPR,BAMSRK
Real, Intent (OUT) :: BBMSRK,BiPR(99),AiPR(99),BiMPR(99),AiMPR(99)
Real, Intent (OUT) :: AiMSRK(99),BiMSRK(99),BiSRK(99),AiSRK(99)
Real, Intent (OUT) :: YAIJPR,YAIJSRK,YAIJMPR,YAIJMSRK
Real :: SYAIJPR(99,99),SYAIJSRK(99,99),SYAIJMPR(99,99)
Real :: SYAIJMSRK(99,99),LLi(99)
am_LLS = 0
am_SRK = 0
am_Pr = 0
bmi_LLS = 0
bmi_SRK = 0
bmi_PR = 0
YAIJPR = 0
YAIJSRK = 0
YAIJMPR = 0
YAIJMSRK = 0
LLi = Ji
!****************************************************************************!
Do i=1,NCOMP
OWi(i) = 0.361/(1+(0.274*A_Fi(i)))
mi_LLS(i) = 0.14443 + 1.0662*A_Fi(i) + 0.02756*A_Fi(i)
& *A_Fi(i) - 0.18074*(A_Fi(i)**3)
TRi(i)=T/Tci(i)
PRi(i)=PR/Pci(i)
OBi(i)=Zci(i)*OWi(i)
OAi(i)=(1+(OWi(i)-1)*Zci(i))**3
TF_LLS(i)=(1+mi_LLS(i)*(1-TRi(i)**0.5))**2
157
mi_SRK(i) = 0.480 + 1.574*A_Fi(i) - 0.176*A_Fi(i)**2
mi_PR(i) = 0.37464 + 1.54226*A_Fi(i) - 0.26992*A_Fi(i)**2
TF_SRK(i)=(1+(mi_SRK(i))*(1- TRi(i)**0.5))**2
TF_PR(i)=(1+(mi_PR(i))*(1- TRi(i)**0.5))**2
ai_LLS(i) = (TF_LLS(i)*OAi(i)*(R*Tci(i))**2)/Pci(i)
bi_LLS(i) = ((OBi(i)*R*Tci(i))/Pci(i))
ai_SRK(i)= (0.42748*TF_SRK(i)*(R*Tci(i))**2)/pci(i)
bi_SRK(i) = (0.08664*R*Tci(i))/pci(i)
ai_PR(i) = 0.45724*TF_PR(i)*(R*Tci(i))**2/pci(i)
bi_PR(i) = 0.07780*R*Tci(i)/pci(i)
ai_MPR(i) = ai_LLS(i)
bi_MPR(i) = bi_LLS(i)
ai_MSRK(i) = ai_LLS(i)
bi_MSRK(i)= bi_LLS(i)
END DO
!****************************************************************************!
DO i=1,NCOMP
AiPR(i)= ai_PR(i)*PR/(R*T)**2
BiPR(i)= bi_PR(i)*PR/ (R*T)
AiSRK(i)= ai_SRK(i)*PR/(R*T)**2
BiSRK(i)= bi_SRK(i)*PR/ (R*T)
AiMPR(i)= ai_MPR(i)*PR/(R*T)**2
BiMPR(i)= bi_MPR(i)*PR/ (R*T)
AiMSRK(i)= ai_MSRK(i)*PR/(R*T)**2
BiMSRK(i)= bi_MSRK(i)*PR/ (R*T)
DO j=1,NCOMP
IF ( (Tci(i)/Pci(i)**0.5) < (Tci(j)/Pci(j)**0.5) ) THEN
kij = ((Tci(i)/Pci(i)**0.5)/(Tci(j)/Pci(j)**0.5))**0.5
ELSE
kij = ((Tci(j)/Pci(j)**0.5)/(Tci(i)/Pci(i)**0.5) )**0.5
END IF
kij=1
ami_LLS=LLi(i)*LLi(j)*(ai_LLS(i)*ai_LLS(j))**0.5*kij
ami_SRK=LLi(i)*LLi(j)*(ai_SRK(i)*ai_SRK(j))**0.5*kij
158
ami_PR =LLi(i)*LLi(j)*(ai_PR(i)*ai_PR(j))**0.5*kij
am_LLS = am_LLS + ami_LLS
am_SRK = am_SRK + ami_SRK
am_PR = am_PR + ami_PR
am_MSRK= am_LLS
am_MPR = am_LLS
END DO
bmi_LLS = bmi_LLS + ( LLi(i)*bi_LLS(i) )
bmi_SRK = bmi_SRK + ( LLi(i)*bi_SRK(i) )
bmi_PR = bmi_PR + ( LLi(i)*bi_PR(i) )
END Do
do i= 1, NCOMP
Do j = 1, NCOMP
Kij = 1
SYAIJPR (i,j) = YAIJPR + LLi(j)*(AiPR(j)*AiPR(i))**0.5*kij
SYAIJSRK(i,j) = YAIJSRK + LLi(j)*(AiSRK(j)*AiSRK(i))**0.5*kij
SYAIJMPR(i,j) = YAIJMPR + LLi(j)*(AiMPR(j)*AiMPR(i))**0.5*kij
SYAIJMSRK(i,j) = YAIJMSRK + LLi(j)*(AiMSRK(j)*AiMSRK(i))**0.5*kij
YAIJPR = SYAIJPR (I,J)
YAIJSRK = SYAIJSRK(I,J)
YAIJMPR = SYAIJMPR (I,J)
YAIJMSRK = SYAIJMSRK (I,J)
End do
end do
bm_LLS = bmi_LLS
bm_SRK = bmi_SRK
bm_PR = bmi_PR
bm_MPR = bm_LLS
bm_MSRK=bm_LLS
BAPR=am_PR*PR/(R*T)**2
BBPR=bm_PR*PR/(R*T)
BASRK=am_SRK*PR/(R*T)**2
BBSRK=bm_SRK*PR/(R*T)
BAMPR=am_MPR*PR/(R*T)**2
BBMPR=bm_MPR*PR/(R*T)
BAMSRK=am_MSRK*PR/(R*T)**2
BBMSRK=bm_MSRK*PR/(R*T)
End Subroutine ParaMix
159
!**************Subroutine Cubic Solver******************************
!*****************************Cubic Solver 2******************************
SUBROUTINE Cubic_Solver(Mtype,Coeff,X)
IMPLICIT NONE
!REAL :: X1,X2,X3
REAL :: F,G,H,I,J,K,L,M,N,P,RI
REAL :: K1,K2,K3,K4
Real :: A,B,C,D
REAL :: U_D , T, S
REAL , INTENT (IN) :: Mtype, Coeff(4)
REAL , INTENT (OUT) :: X(3)
X(1)=0.0
X(2)=0.0
X(3)=0.0
A= Coeff(1)
B = Coeff(2)
C = Coeff(3)
D = Coeff(4)
F = (( 3*C/A) - (B*B)/(A*A) ) /3
G = (2*B**3/A**3 - 9*B*C/(A*A) + 27*D/A ) / 27
H = ((G*G)/4) + ((F**3)/27)
K1=F-G
K2=H-G
K3=ABS(K2-K1)
K4=0.0000001
IF (abs(K3).LT.K4) THEN
X(1) = (D/A)**0.3333 * (-1)
END IF
IF (H.LE.0) THEN
I = ((G*G)/4 - H )**0.5
J = I**0.3333
K = ACOS ( (-1 * G)/ (2*I) )
L = -1 * J
160
M = COS (K/3)
N = 3**0.5 * SIN (K/3)
P = B/(3*A) * ( -1)
X(1) = 2*J* COS (K/3) - b/(3*A)
X(2) = L * ( M + N ) + P
X(3) = L * (M -N ) + P
ENDIF
IF (H .GT.0) THEN
RI = ((-1*G)/2) + H**0.5
S = RI**(0.33333)
T = ((-1*G)/2) - H**0.5
If (T.LE.0) Then
T = -1*T
U_D =T**0.33333
U_D = -1 * U_D
Else
U_D =T**0.33333
End If
X(1) = S + U_D - B/(3*A)
!****Looking for imaginary root for gas***********************
If (Mtype .EQ. 1.0) then
X(2) = -1* (S + U_D)/2 - ( B/(3*A))
end if
!**************************************************
! " One Real Root And 2 Imaginary "
! X1 - Real Root
!Imaginary roots
! -1* (S + U_D)/2 - ( B/(3*A)),"+i",(S-U_D)*3**0.5/2
! -1* (S + U_D)/2- ( B/(3*A) ), "- i",(S-U_D)*3**0.5/2
END IF
END subroutine Cubic_solver
!***************End of Program******************************
End Program Flash_Iso_Comp
161
APPENDIX K
VITA
Olaoluwa Adepoju was born in Ibadan, Nigeria. He attended Mayflower
secondary school, Ikenne, Ogun State Nigeria where he graduated with distinctions in
mathematics and chemistry. Ola Adepoju is a Chemical Engineering Graduate from
Ladoke Akintola University of Technology, Ogbomosho, Nigeria, where he graduated
with a first class and was awarded the best graduating student in the faculty of
engineering and technology.
Ola has exemplified himself as a hardworking and dedicated student. He placed
second during the Texas Tech SPE Student paper contest and went ahead to present the
same paper at the 2006 SPE Gulf Coast Student paper contest at Texas A&M, College
Station, Texas. He also presented a Poster at the Graduate School Poster Competition.
Ola worked comfortably in research teams. He worked closely with a Ph.D.
student (Lukeman Adetunji) in his dissertation titled “Thermodynamically Equivalent
Pseudo-Components Validated for Compositional Reservoir Models.”
Ola’s research interest is in Reservoir Engineering and Phase Behavior; he carried
out reservoir simulation and well testing analysis using ECLIPSE Software and Weltest
200.
Ola worked with as a graduate assistant with Dr Lloyd Heinze in the
undergraduate drilling engineering class.
162
PERMISSION TO COPY
In presenting this thesis in partial fulfillment of the requirements for a master’s
degree at Texas Tech University or Texas Tech University Health Sciences Center, I
agree that the Library and my major department shall make it freely available for research
purposes. Permission to copy this thesis for scholarly purposes may be granted by the
Director of the Library or my major professor. It is understood that any copying or
publication of this thesis for financial gain shall not be allowed without my further
written permission and that any user may be liable for copyright infringement.
Agree (Permission is granted.)
Adepoju, Olaoluwa Opeoluwa
Student Signature
November 28, 2006
Date
Disagree (Permission is not granted.)
_______________________________________________
Student Signature
_________________
Date