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 2005-
2006 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 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
2 COEFFIECIENT OF ISOTHERMAL OIL COMPRESSIBILITY
2.1 Defining Equations for Isothermal Oil Compressibility 10
2.2 Isothermal Oil Compressibility Correlation and Computation Methods 24
3 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
4 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 Equation of State 66
4.4 Discussion of Results 74
5 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 Definition a Attraction Parameter Term of EOS aP 2 2 A Dimensionless Constant R T 1 Expt − Predicted AAPD Average Absolute Percent Deviation AAPD = N ∑ Exptal API Oil Gravity b Van der Waals co-volume bP 2 2 B Dimensionless Constant R T Bg Formation Gas Volume Factor Bo Formation Oil Volume Factor Bw Formation Water Volume Factor cf Formation Compressibility cg Isothermal Gas Compressibility cL Isothermal Liquid Compressibility co Isothermal Oil Compressibility ct Total System Isothermal Compressibility C Characterization Factor Ki Equilibrium Ratio Mw Molecular Weight p Pressure ppc Pseudo-Critical Pressure ppr Pseudo-Reduced Pressure R Gas Constant Rs Solution Gas-Oil Ratio Sg Gas Saturation So Oil Saturation Sw Water Saturation T Absolute Temperature Tpc Pseudo-Critical Temperature Tpr Pseudo-Reduced Temperature V Molar Volume z Compressibility Factor
Greek Letter
α Parameter of LLS EOS β Parameter of LLS EOS
x
δij Binary Interaction Parameter γ Specific Gravity ρ Density ω Acentric Factor 0.5 ψi Fugacity Parameter ψ i = ∑[x j ()a i a j ] j 0.5 ψ Fugacity Parameter ψ = ∑∑[x i x j ()a i a j ] ij Ω Dimensionless Parameter
Subscript c Critical Property g Gas o Oil r Reduced i Component i
xi 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 pressure-
volume 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.
V1 − V2 co = − (1.1) V1 ()p 2 − p1
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
41 should be V1 or V2, or some value in between . To overcome this limitation, equation
1.1 is usually expressed in partial differential form at constant temperature, as follows
1 ⎛ ∂V ⎞ co = − ⎜ ⎟ (1.2) V ⎝ ∂p ⎠T
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 = coSo + c wSw + cgSg + cf (1.3)
-1 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
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 (1.4) ⎣0.0002637k ⎦
teia = Time to end the infinite-acting period, hours
A = Well drainage area, ft2
5 -1 ct = Total Compressibility, psi
(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 c ()p − p = N B − W + B W oi e i p o e w p (1.5)
S c + S c + c c = o o w w f e B p − p oi ()i (1.6)
N = Initial Oil in Place, STB
Boi = Initial Oil Formation Volume Factor, bbl/STB
-1 ce= Effective Compressibility, psi 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:
()co ()pb −p Bo = Bob e (1.7)
()co ()p−pb ρo = ρobe (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,
1 ⎛ ∂V ⎞ c = − ⎜ ⎟ g V ⎜ ∂p ⎟ ⎝ ⎠T (2.1)
10 1 ⎛ ∂v ⎞ c = − ⎜ ⎟ g v ⎜ ∂p ⎟ ⎝ ⎠T (2.2)
1 ⎛ ∂V ⎞ c = − ⎜ m ⎟ g V ⎜ ∂p ⎟ m ⎝ ⎠T (2.3)
-1 cg = Isothermal Gas Compressibility, psi
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,
nRT V = p (2.5)
Differentiating equation 2.5, with respect to pressure yields,
⎛ ∂V ⎞ nRT ⎜ ⎟ = − ⎜ ∂p ⎟ p 2 ⎝ ⎠T (2.6)
Therefore the isothermal gas compressibility equation for ideal gases can be expressed as;
⎛ 1 ⎞⎛ nRT ⎞ c = ⎜− ⎟⎜− ⎟ g V ⎜ p 2 ⎟ ⎝ ⎠⎝ ⎠ (2.7)
Substituting for V in equation 2.7, yields;
12 ⎛ p ⎞⎛ nRT ⎞ 1 c = ⎜− ⎟⎜− ⎟ = g nRT ⎜ p 2 ⎟ 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;
z V = nRT p (2.10)
Differentiating equation 2.10 in terms of pressure at constant temperature yields;
⎛ ∂V ⎞ ⎡1 ⎛ ∂z ⎞ z ⎤ ⎜ ⎟ = nRT⎢ ⎜ ⎟ − ⎥ ⎜ ∂p ⎟ p ⎜ ∂p ⎟ p 2 ⎝ ⎠T ⎣⎢ ⎝ ⎠T ⎦⎥ (2.11)
Substituting equation 2.11 into equation 2.1 yields;
⎡ p ⎤⎪⎧ ⎡1 ⎛ ∂z ⎞ z ⎤⎪⎫ c = − nRT ⎜ ⎟ − g ⎢ ⎥⎨ ⎢ ⎜ ⎟ 2 ⎥⎬ ⎣ znRT⎦⎪ ⎢p ∂p p ⎥⎪ ⎩ ⎣ ⎝ ⎠T ⎦⎭ (2.12)
Simplifying equation 2.12 yields;
1 1 ⎛ ∂z ⎞ c = − ⎜ ⎟ g 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 ⎞⎛ ∂z ⎞ ⎜ ⎟ = ⎜ ⎟⎜ ⎟ (2.14) ⎝ ∂p ⎠T ⎝ ∂p ⎠⎝ ∂p Pr ⎠
The pseudo-reduced pressure is expressed as;
p = p p pc pr (2.15)
The derivative of the pseudo-reduced pressure with respect to pressure is expressed as;
⎛ ∂p ⎞ 1 ⎜ pr ⎟ = ⎜ ∂p ⎟ p ⎝ ⎠ pc (2.16)
Substituting equation 2.16 into equation 2.14 gives;
16 ⎛ ∂z ⎞ 1 ⎛ ∂z ⎞ ⎜ ⎟ = ⎜ ⎟ ⎜ ∂p ⎟ p ⎜ ∂p ⎟ ⎝ ⎠T pc ⎝ pr ⎠ (2.17)
Substituting equation 2.17 and equation 2.15 into equation 2.13 gives;
1 1 ⎛ ∂z ⎞ c = − ⎜ ⎟ g ⎜ ⎟ p pc p pr zp pc ⎝ ∂p pr ⎠ Tpr (2.18)
This can be better expressed in dimensionless pseudo-reduced compressibility cpr as;
1 1 ⎛ ∂z ⎞ c = c p = − ⎜ ⎟ pr g pc ⎜ ⎟ 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;
1 ⎛ ∂V ⎞ c = − ⎜ ⎟ o V ⎜ ∂p ⎟ ⎝ ⎠T (2.20) where
-1 co = Isothermal Oil Compressibility of crude oil, psi
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;
1 ⎛ ∂B ⎞ c = − ⎜ o ⎟ o B ⎜ ∂p ⎟ o ⎝ ⎠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 c dp = − o o ∫ ∫ Bo pb Bob (2.23)
These results in;
[]co ()pb −p Bo = Bobe (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;
1 v = ρ o (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 ⎛ ∂ρ ⎞ ⎜ ⎟ = ⎜ o ⎟ ⎜ ∂p ⎟ ∂ρ ⎜ ∂p ⎟ ⎝ ⎠T o ⎝ ⎠T (2.27)
Differentiating equation 2.26 with respect to density yields;
∂v 1 = − ∂ρ ρ 2 o o (2.28)
Substituting equations 2.28, 2.27 and 2.26 into equation 2.25 gives;
⎡ 1 ⎤⎡ 1 ⎛ ∂ρo ⎞ ⎤ co = −⎢ ⎥⎢− 2 ⎜ ⎟ ⎥ 1 ρ ρ ⎜ ∂p ⎟ ⎣ o ⎦⎣⎢ o ⎝ ⎠T ⎦⎥ (2.29)
Thus,
1 ⎛ ∂ρ ⎞ c = ⎜ o ⎟ o ρ ⎜ ∂p ⎟ o ⎝ ⎠T (2.30)
21 Assuming constant co at pressures above the bubble point, equation 2.30 can be integrated to give 29 p. 234;
[]co ()p−pb ρo = ρ obe (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:
1 ⎛ ∂V ⎞ 1 ⎛ ∂B ⎞ c = − ⎜ ⎟ = − ⎜ o ⎟ o V ⎜ ∂p ⎟ B ⎜ ∂p ⎟ ⎝ ⎠T o ⎝ ⎠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 ⎛ ∂R ⎞ g ⎜ s ⎟ B ⎜ ∂p ⎟ o ⎝ ⎠T (2.33)
22 Thus the isothermal oil compressibility at pressures below the bubble point pressure is expressed as27.
1 ⎛ ∂B ⎞ B ⎛ ∂R ⎞ c = − ⎜ o ⎟ + g ⎜ s ⎟ o B ⎜ ∂p ⎟ B ⎜ ∂p ⎟ o ⎝ ⎠T o ⎝ ⎠T (2.34)
This is better expressed as;
1 ⎡⎛ ∂Bo ⎞ ⎛ ∂R s ⎞ ⎤ co = − ⎢⎜ ⎟ − Bg ⎜ ⎟ ⎥ B ⎜ ∂p ⎟ ⎜ ∂p ⎟ o ⎣⎢⎝ ⎠T ⎝ ⎠T ⎦⎥ (2.35) 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, oilpsi compressibility as a function of pressure at constant reservoir temperature29 pp 236.
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 = co p pc (2.36) where cr = Pseudo reduced compressibility, dimensionless
-1 co = Isothermal Oil Compressibility, psi 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
o compressibility with the gas solubility Rs, reservoir temperature T, API gravity, gas specific gravity γg and reservoir pressure p. They proposed the following expression:
−1433 + 5R s +17.2(T − 460)−1180γ gs +12.61γ API co = 5 10 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.
⎡ −5 ⎛ psep ⎞⎤ γ gs = γ g ⎢1+ 5.912()10 ()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
o Tsep = actual separator temperature, R
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 = −7.633 −1.497ln p +1.115ln T + 0.533ln γ + 0.184ln R ()o ( ) ( ) ( API )()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()co = −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
5 implemented in the equation through the gas solubility Rs .
The correlation is expressed as:
1 ()a3p co = e (2.41) a1 + a 2 R s 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:
⎡ 0.5 1.175 ⎤ ⎡ ⎛ γ ⎞ ⎤ ⎢a + a ⎢R ⎜ g ⎟ +1.25()T − 460 ⎥ ⎥ ⎢ 1 2 s ⎜ ⎟ ⎥ ⎢ ⎝ γ o ⎠ ⎥ ⎢ ⎣ ⎦ ⎥ ()a3p co = e (2.42) ⎢ a 4 γ O + a 5R s γ g ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ where a1 = 1.026638 a2 = 0.0001553
30 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
− 889.6 + 3.1374R s + 20T − 627.3γ gcorr − 81.4476γ API co = 5 p10 (2.43)
Heavy oils: Modified Vasquez-Beggs correlation
− 2841.8 + 2.9646R s + 25.5439T −1230.5γ gcorr − 41.91γ API co = 5 p10 (2.44)
31 Medium oils: Modified Vasquez-Beggs correlation
− 705.288 + 2.2246R s + 26.0644T − 2080.823γ gcorr − 9.6807γ API co = 5 p10 (2.45)
Light oils: Modified Labedi’s Correlation
−6.1646 1.8789 0.3646 0.1966 ⎛ p b ⎞ −8.98 3.9392 1.349 co = ()10 Bo γ API T − ⎜1− ⎟()10 Bo T ⎝ p ⎠ (2.46)
Where
⎡ ⎛ psp ⎞ −4 ⎤ γ gcorr = γ g psp ⎢1+ 0.5912γ APITsp Log⎜ ⎟10 ⎥ ⎜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 = (4.487462368 + 0.005197040A + 0.000012580A 2 )×10−6 o bp (2.48)
32 where
1.759732076 ⎛ R 0.980922372 γ 0.021003077 ⎞ ⎜ s g + 20.00006358()T − 60 0.300001059 − 0.876813622R ⎟ ⎜ 0.338486128 s ⎟ ⎝ γ 0 ⎠ A = 2 ⎛ 2R −1.713572145 ⎞ ⎜2.749114986 + ()T − 60 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.
2 ln()cofb = 2.434 + 0.475z + 0.048z (2.50)
6 z = ∑ z n n=1 (2.51)
z = C + C x + C x 2 n 0,n 1,n n 2,n n (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.
()p − p c p − (p − p )c p c = b ofb i b ofb i ofi p − p i (2.53)
o Correlation for Coefficient of Isothermal Compressibility Tangent at Some Pressure of Interest.
This gives the tangent or instantaneous compressibility, co.
p − 0.608 + 0.1822ln p b co = cofb + ()(p − p b cofb 0.475 + 0.096z ) p (2.54)
34 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.
RT p = 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.
zRT p = 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
a parameter and b to account for the intermolecular attractive and repulsive forces V 2 respectively.
RT a p = − 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.
3 ⎛ RT ⎞ 2 ⎛ a ⎞ ab V − ⎜b + ⎟V + ⎜ ⎟V − = 0 ⎝ p ⎠ ⎝ p ⎠ p (3.4)
3 2 z − ()1+ B z + Az − AB = 0 (3.5) where the dimensionless parameter A and B are defined as,
aP A = 2 ()RT (3.6)
36 bp B = 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)
⎛ ∂ 2 p ⎞ ⎜ ⎟ = 0 ⎜ 2 ⎟ ⎝ ∂V ⎠ Tc (3.9)
Applying this condition at the critical point, the constant “a” and “b” of the Van der Waal cubic EOS can be determined.
27 ⎛ R 2T 2 ⎞ a = ⎜ c ⎟ 64 ⎜ p ⎟ ⎝ c ⎠ (3.10)
1 ⎛ RT ⎞ b = ⎜ c ⎟ 8 ⎜ p ⎟ ⎝ c ⎠ (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 RT a p = − V − b V()()V + b + b V − b (3.12)
The PR EOS in terms of volume is given as,
⎛ RT ⎞ ⎛ a 2bRT ⎞ ⎛ RTb 2 ab ⎞ V 3 − ⎜ − b⎟V 2 + ⎜ − 3b 2 − ⎟V + ⎜b3 + − ⎟ = 0 ⎜ p ⎟ ⎜ p p ⎟ ⎜ p p ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (3.13)
Or, in terms of the compressibility factor,
z3 − ()1− B z 2 + (A − 3B2 − 2B)z − (AB − B2 − B3 ) = 0 (3.14)
The PR EOS constants are given as,
R 2T 2 a = Ω c α a p c (3.15)
Ω = 0.45724 where a
RT b = Ω c b p c (3.16)
Ω = 0.07780 where b
0.5 2 α = []1+ m()1- Tr (3.17)
2 and m = 0.374964 +1.54226ω − 0.26992ω (3.18)
The value of “m” was later expanded by Robinson et al. for heavier components
(ω > 0.49) 44 p. 50.
2 3 m = 0.3796 +1.485ω − 0.1644ω + 0.01667ω (3.19)
Parameters A and B are as defined in equation 3.6 and 3.7 respectively.
39 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,
RT a P = − 2 2 V − b V + αβV − βb (3.20)
The LLS parameters a, b, α and β are defined as,
()RT 2 a = Ω k c a p c (3.21)
Ω = 1+ Ω −1 z 3 where a ()()w c (3.22)
0.5 2 k = []1+ m()1− Tr (3.23)
2 3 m = 0.14443 +1.06624ω + 0.02756ω − 0.1807ω (3.24)
RT b = Ω c b p and c (3.25)
Ω = z Ω b c w (3.26)
0.361 Ω w = where 1+ 0.274ω (3.27)
1+ z (Ω − 3) α = c w z Ω c w (3.28)
z 2 ()Ω −1 3 + 2Ω 2 z + Ω (1− 3z ) β = c w w c w c i z Ω 2 c w (3.29)
40 The LLS EOS can be written in terms of molar volume as,
⎛ RT ⎞ ⎛ a RTαβ ⎞ ⎛ RTβT2 ab ⎞ V 3 − ⎜ + b − αβ⎟V 2 + ⎜ − βb 2 − bαβ − ⎟V + ⎜b3β + − ⎟ = 0 ⎜ p ⎟ ⎜ p p ⎟ ⎜ p p ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠
(3.30)
or, in terms of the compressibility factor
3 2 2 2 3 z + z []B()α −1 −1 − z[B (α + β)+ αB − A]− [AB − β(B − B )]= 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 multi- component system. The mixture parameters used in the study are defined as,
n n 0.5 a m = ∑∑x i x j ()a i a j δ ij i j (3.32)
3 n 1 ⎡ 3 ⎤ b m = ⎢∑ x i bi ⎥ ⎣ i ⎦ (3.33)
41
n n 0.5 α m = ∑∑x i x j ()α i α j δ ij i j (3.34)
n n 0.5 β m = ∑∑x i x j ()βiβ j δ ij i j (3.35)
δ The binary interaction parameter 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,
0.5 ⎡ ⎤ ⎛Tci ⎞ ⎢⎜ 0.5 ⎟⎥ ⎢⎝ Pci ⎠⎥ ⎢⎛T ⎞⎥ ⎢⎜ cj ⎟⎥ ⎜ P 0.5 ⎟ ⎣⎢⎝ cj ⎠⎦⎥ (3.36)
⎛T ⎞ ⎛T ⎞ ⎜ ci ⎟ < ⎜ cj ⎟ P 0.5 ⎜ P 0.5 ⎟ where, ⎝ ci ⎠ ⎝ cj ⎠
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.
P ()psia = 3.1839 *104 M −0.93426S1.64074T 0.49447C −2.39909 c g b (3.37)
T o R = 66.3775M 0.12286S0.47926T 0.41286C −0.35734 c ( ) g b (3.38)
ω = 4.54949 *10−9 M 0.02445S−2.08511T 2.903798C−1.54424 g b (3.39)
T o R = 108.7017M 0.4225S0.4268 b ( ) g (3.40)
0.293 z c = 1+ 0.375ω (3.41)
3.8501 C = 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 C
Pressure 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,
1 ⎛ ∂V ⎞ c = − ⎜ ⎟ o 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
1 will enable us to determine “ − ” 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 ⎞ The second step is to differentiate the EOS to obtain the “⎜ ⎟ ” term of ⎝ ∂p ⎠T equation 2.20. That is we differentiate the EOS in terms of volume with respect to
1 ⎛ ∂V ⎞ pressure at constant temperature. The two terms “ − ” and “⎜ ⎟ ” are then 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,
RT a p = − 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 3 − ⎜ − b⎟V 2 + ⎜ − 3b 2 − ⎟V + ⎜b3 + − ⎟ = 0 ⎜ 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 2 (V 2 + 2bV − b 2 ) ⎜ ⎟ = ⎜ ∂p ⎟ 2 2 2 2 ⎝ ⎠T 2a()()V + b V − b − RT()V + 2bV − b (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,
2 ⎛ 1 ⎞ ()V − b 2 (V 2 + 2bV − b 2 ) co = ⎜− ⎟ ⎝ V ⎠ 2 2 2 2 2a()()V + b V − b − RT()V + 2bV − b (3.44)
46 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,
1 ⎛ ∂Bo ⎞ Bg ⎛ ∂R s ⎞ co = − ⎜ ⎟ + ⎜ ⎟ (3.45) Bo ⎝ ∂p ⎠T Bo ⎝ ∂p ⎠T
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 Gas Gas Gas
Oil Oil Oil Oil Oil
Oil
P1>Pb P2=Pb P3 Figure 3.4: Schematic Diagram of Flash Liberation Experiment 48 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− ⎟⎥ K= ci e⎣ ⎝ T ⎠⎦ (3.46) i p 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. 1 N = vmin 1− K imax (3.47) 1 N = vmax 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. f ()n N n+1 = N n − v v v f ' n ()v (3.49) z ()K −1 f ()n = i i v ∑ n K −1 +1 i v ()i (3.50) ⎡ z ()K −1 2 ⎤ f ' n = − i i ()v ∑ ⎢ 2 ⎥ i n K −1 +1 ⎣⎢[]v ()i ⎦⎥ (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” z x = i i 1+ n K −1 v ()i (3.52) o Calculate the mole fraction of the component in the vapor phase “yi”. z K y = i i i 1+ n K −1 v ()i (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 2 2 2 3 z − ()1− B z + (A − 3B − 2B)z − (AB − B − B ) = 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 z- factor 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. f B A ⎡2ψ B ⎤ ⎛ z L + 2.414B ⎞ ln Li = i ()()z L −1 − ln z L − B − i − i ln⎜ ⎟ x p B 2.82843B ⎢ ψ B ⎥ ⎜ z L − 0.414B ⎟ i ⎣ ⎦ ⎝ ⎠ (3.54) f B A ⎡2ψ B ⎤ ⎛ z v + 2.414B ⎞ ln vi = i ()()z v −1 − ln z v − B − i − i ln⎜ ⎟ y p B 2.82843B ⎢ ψ B ⎥ ⎜ z v − 0.414B ⎟ i ⎣ ⎦ ⎝ ⎠ (3.55) The parameters of the PR EOS are calculated using the appropriate mole fractions. o Check for convergence of the fugacities 51 2 n ⎛ f ⎞ ⎜ Li −1⎟ < ε ∑⎜ f ⎟ i=1 ⎝ Vi ⎠ (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. f ()n K ()n+1 = K ()n Li i i f ()n 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 2 −4 ∑()ln K i < 10 i=1 (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 L ()1− N v z RT VL = p (3.59) v ()N v z RT Vg = p (3.60) o The total volume VT is calculated using, V = V + V T L g (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, 1 1 ⎛ ∂z ⎞ c = − ⎜ ⎟ g 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 1 zb ()2a − 3pb 2 + (6zpb 2 − 2pb 2 − za)+ ()2 − z ⎛ ∂z ⎞ ()RT 3 ()RT 2 ()RT ⎜ ⎟ = ⎜ ⎟ p 2bp ⎝ ∂p ⎠T 2 2 ()a − 3pb + ()(z −1 + z 3z − 2 ) ()RT ()RT (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. 6zpb2 2zb 2abp z2b za 2pb2 3p2b3 + + − − − − 1 1 ()RT 2 RT ()RT 3 RT ()RT 2 ()RT 2 ()RT 3 c = − (3.63) g p z ⎡ bp ap 3()bp 2 2bp⎤ ⎢3z2 − 2z + 2z + − − ⎥ RT 2 2 RT ⎣⎢ ()RT ()RT ⎦⎥ 53 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, 1 ⎛ ∂V ⎞ c = − ⎜ ⎟ o 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) 2 ⎛ 1 ⎞ ()V − b 2 (V 2 + 2bV − b2 ) co = −⎜ ⎟ ⎝ V ⎠ 2 2 2 2 2a()()V + b V − b − RT()V + 2bV − b (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. Vg VL co = cg + c L (3.66) ()VL + Vg ()VL + Vg 55 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 Volume (ft3/lb mole) Pressure(psia) Exptal PR EOS MPR EOS 5014.7 2.2058 2.2047 2.2672 4514.7 2.2204 2.2201 2.2816 4014.7 2.2360 2.2371 2.2974 3514.7 2.2531 2.2559 2.3148 3014.7 2.2721 2.2770 2.3341 2914.7 2.2760 2.2815 2.3382 2814.7 2.2801 2.2861 2.3424 2714.7 2.2845 2.2909 2.3468 2634.7 2.2884 2.2948 2.3503 2619.7 2.2934 2.2033 2.6633 2605.7 2.2978 2.2083 2.6655 2530.7 2.3236 2.2427 2.6804 2415.7 2.3685 2.2897 2.7081 2267.7 2.4360 2.3597 2.7531 2104.7 2.5264 2.4519 2.8157 1911.7 2.6621 2.5878 2.9152 1712.7 2.8435 2.7688 3.0547 1491.7 3.1163 3.0482 3.2799 1306.7 3.4353 3.3596 3.5454 1054.7 4.0738 3.9944 4.1111 844.7 4.9482 4.8609 4.9103 654.7 6.2960 6.1895 6.1702 486.7 8.5187 8.3257 8.2369 57 Molar Volume Prediction for Oil Well No-4,Good Oil Company, Samson,TX. Pb =2619.7 psia 9.0 8.0 7.0 6.0 5.0 Exptal Model Based on PR EOS 4.0 Molar Volume(ft3/lbmol) 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 7.0 6.0 5.0 Exptal Model Based on MPR EOS 4.0 Molar volume (ft3/lbmol) 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 7.0 6.0 Exptal 5.0 Model Based on PR EOS 4.0 Molar Volume (ft3/lbmol) 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 7.0 6.0 Exptal 5.0 Model Based on MPR EOS 4.0 Molar Volume (ft3/lbmol) 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 7.0 6.0 Exptal 5.0 Model based on PR EOS 4.0 Molar Volume (ft3/lbmol) 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 7.0 6.0 Exptal 5.0 Model Based on MPR EOS 4.0 Molar Volume (ft3/lbmol) 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 Isothermal Compressibility (psia-1) Pressure(psia) Exptal PR EOS MPR EOS 5014.7 1.33E-05 1.21E-05 4514.7 1.32E-05 1.45E-05 1.32E-05 4014.7 1.40E-05 1.59E-05 1.44E-05 3514.7 1.53E-05 1.76E-05 1.58E-05 3014.7 1.68E-05 1.96E-05 1.75E-05 2914.7 1.71E-05 2.01E-05 1.79E-05 2814.7 1.81E-05 2.05E-05 1.83E-05 2714.7 1.91E-05 2.10E-05 1.87E-05 2634.7 2.13E-05 2.14E-05 1.90E-05 2364.7 1.55E-04 2.35E-04 2.06E-04 2114.7 1.83E-04 2.74E-04 2.44E-04 1864.7 2.09E-04 3.26E-04 2.95E-04 1614.7 2.53E-04 3.96E-04 3.65E-04 1364.7 3.15E-04 4.95E-04 4.65E-04 1114.7 4.17E-04 6.44E-04 6.16E-04 864.7 5.76E-04 8.84E-04 8.63E-04 614.7 8.87E-04 1.33E-03 1.32E-03 364.7 1.83E-03 2.42E-03 2.44E-03 173.7 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 0.004 Exptal 0.003 Model Based on PR EOS Co (1/psi) 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 0.004 Exptal MPR 0.003 Model Based on MPR EOS Co (1/psia) 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 3.00E-03 Exptal 2.50E-03 Model Based on PR EOS Co (1/psia) 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 0.003 Exptal 0.003 Model Based on MPR EOS Co (1/psia) 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 0.005 0.004 Exptal Model Based on PR EOS Co (1/psia) 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 0.005 0.004 Exptal Model Based on MPR EOS Co (1/psia) 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. 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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), RT a P = − V − b V(V + b) + b(V − b) (A.1) Multiplying both sides by (V – b), a(V − b) P(V − b) = RT − V(V + b) + b(V − b) (A.2) Simplifying, a(V − b) P(V − b) = RT − 2 2 V + 2Vb − b (A.3) 2 2 2 2 P(V − b)(V + 2Vb − b ) = RT(V + 2Vb − b ) − a(V − b) (A.4) Expanding, 3 2 2 2 2 3 2 2 P(V + 2bV − Vb − bV − 2Vb + b ) = RT(V + 2Vb − b ) − a(V − b) (A.5) Dividing by RT, P a (V3 + bV 2 − 3Vb 2 + b3 ) = (V 2 + 2Vb − b 2 ) − (V − b) RT RT (A.6) 83 ZRT V = But P (Real gas equation), hence (A.6) becomes, 3 3 2 2 2 2 P ⎡Z (RT) Z (RT) 2 Z(RT) 3 ⎤ Z (RT) Z(RT) 2 a ⎛ Z()RT ⎞ ⎢ 3 + b 2 − 3b + b ⎥ = 2 + 2b − b − ⎜ − b⎟ RT ⎣ P P P ⎦ P P RT ⎝ P ⎠ (A.7) Simplifying equation A.7 gives, Z3 (RT) 2 Z2 (RT) P Z2 (RT) 2 Z(RT) Z ba + b − 3b 2 Z + b3 − − 2b + b 2 + a − = 0 P 2 P RT P 2 P P RT (A.8) Multiplying by P2 gives, P3 aPbP Z3 (RT) 2 + bZ2 (RT)P − 3P 2 b 2 Z + b3 − Z2 (RT) 2 − 2bZ(RT)P + P 2 b 2 + aZP − = 0 RT RT (A.9) Dividing by (RT)2 gives, 3 3 2 bP Pb Pb 3 P 2 Pb Pb Pb aP aP bP Z + Z − 3Z + b 3 − Z − 2Z + + Z 2 − 2 = 0 RT RT RT (RT) RT RT RT (RT) (RT) RT (A.10) bP aP B = A = 2 2 But RT and R T , therefore, 3 2 2 3 2 2 Z + Z B − 3ZB + B − Z − 2ZB + B + AZ − AB = 0 (A.13) 3 2 2 2 3 Z − (1− B)Z + (A − 3B − 2B)Z − (AB − B − B ) = 0 (A.14) 84 A.2 Derivation of PR EOS Molar Volume Equation The PR EOS is expressed as, RT a P = − V − b V()V + b + b(v − b) (A.15) Multiplying through by (V – b) gives, a()V − b P()V − b = RT − 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) (A.17) 3 2 2 2 2 3 2 2 P(V + 2V b − Vb − bV − 2Vb + b ) = RT(V + 2Vb − b )− a()v − b (A.18) 3 2 2 3 2 2 P(V + V b − 3Vb + b )− RT(V + 2Vb − b )+ a(v − b) = 0 (A.19) Dividing by p and simplifying gives, ⎛ RT ⎞ ⎛ a RT ⎞ ⎛ RT aP ⎞ V 3 + ⎜b − ⎟V 2 + ⎜ − 3b 2 − 2b ⎟V + ⎜b3 + b 2 − ⎟ = 0 ⎝ P ⎠ ⎝ P P ⎠ ⎝ P b ⎠ (A.20) 85 APPENDIX B DERIVATION OF PR EOS BASED ISOTHERMAL COMPRESSIBILITY EQUATION The coefficient of isothermal compressibility (c) is defined by 1 ⎛ ∂V ⎞ c = − ⎜ ⎟ V ⎜ ∂p ⎟ ⎝ ⎠T (B.1) The Peng-Robinson EOS is expressed as, RT a p = − V − b V()()V + b + b V − b (B.2) Differentiating equation (B.2) gives, ⎛ ∂p ⎞ − RT 2a(V + b) ⎜ ⎟ = + ⎝ ∂V ⎠ V − b 2 2 2 2 T ()()V + 2bV − b (B.3) Simplifying equation B.3 gives, 2 ⎛ ∂p ⎞ 2a()()V + b V − b 2 − RT(V 2 + 2bV − b 2 ) ⎜ ⎟ = ⎝ ∂V ⎠ 2 2 2 2 T ()V − b ()V + 2bV − b (B.4) Inverting equation B.4 gives, 2 ⎛ ∂V ⎞ ()V − b 2 ()V 2 + 2bV − b 2 ⎜ ⎟ = ⎜ ∂p ⎟ 2 2 2 2 ⎝ ⎠T 2a()()V + b V − b − RT()V + 2bV − b (B.5) Therefore the PR EOS based coefficient of isothermal compressibility equation is expressed as, 86 2 1 ⎡ ()V − b 2 ()V 2 + 2bV − b 2 ⎤ co = − ⎢ ⎥ V 2 2 2 2 ⎣⎢2a()()V + b V − b − RT()V + 2bV − b ⎦⎥ (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 3 − ⎜ − b⎟V 2 + ⎜ − 3b 2 − 2b ⎟V + ⎜b3 + − ⎟ = 0 ⎜ p ⎟ ⎜ p p ⎟ ⎜ p b ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ (B.7) 87 APPENDIX C DERIVATION OF THE PR EOS ISOTHERMAL GAS COMPRESSIBILITY EQUATION The compressibility factor equation of the PR EOS is given as, 3 2 2 2 3 Z − (1− B)Z + (A − 3B − 2B)Z − (AB − B − B ) = 0 (C.1) bP aP B = A = 2 2 where RT and R T Replacing “A” and “B” in equation C.1 gives, 2 2 3 ⎛ bp ⎞ ⎡ ap ⎛ bp ⎞ bp ⎤ ⎡⎛ ap ⎞ bp ⎛ bp ⎞ ⎛ bp ⎞ ⎤ Z3 − ⎜1− ⎟Z2 + ⎢ − 3⎜ ⎟ − 2 ⎥Z − ⎢⎜ ⎟ − ⎜ ⎟ − ⎜ ⎟ ⎥ = 0 ⎝ RT ⎠ ()RT 2 ⎝ RT ⎠ RT ⎜ ()RT 2 ⎟ RT ⎝ RT ⎠ ⎝ RT ⎠ ⎣⎢ ⎦⎥ ⎣⎢⎝ ⎠ ⎦⎥ (C.2) The coefficient of isothermal gas compressibility is expressed as, 1 1 ⎛ ∂z ⎞ c = − ⎜ ⎟ g p z ⎜ ∂p ⎟ ⎝ ⎠T (C.3) Differentiating equation C.2 implicitly gives, ⎛ ∂z ⎞ ⎛ ∂z ⎞ ⎡ bp ⎛ ∂z ⎞ z 2 b⎤ ⎡ ap ⎛ ∂z ⎞ za ⎤ 3z 2 ⎜ ⎟ − 2z⎜ ⎟ + 2z ⎜ ⎟ + + ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ ⎢ ⎜ ⎟ ⎥ ⎢ 2 ⎜ ⎟ 2 ⎥ ∂p ∂p RT ∂p RT RT ∂p RT ⎝ ⎠T ⎝ ⎠T ⎣⎢ ⎝ ⎠T ⎦⎥ ⎣⎢()⎝ ⎠T ()⎦⎥ ⎡3()bp 2 ⎛ ∂z ⎞ 6zpb 2 ⎤ ⎡2bp ⎛ ∂z ⎞ 2zb⎤ ⎡ 2abp 2pb 2 3p 2 b3 ⎤ − ⎜ ⎟ + − ⎜ ⎟ + − − − = 0 ⎢ 2 ⎜ ⎟ 2 ⎥ ⎢ ⎜ ⎟ ⎥ ⎢ 3 2 3 ⎥ RT ∂p RT RT ∂p RT RT RT RT ⎣⎢ ()⎝ ⎠T ()⎦⎥ ⎣⎢ ⎝ ⎠T ⎦⎥ ⎣() () ()⎦ (C.4) 88 Simplifying equation C.4 gives, ⎡ 2 ⎤ ⎛ ∂z ⎞ 2 bp ap 3()bp 2bp ⎜ ⎟⎢3z − 2z + 2z + − − ⎥ = ⎝ ∂p ⎠⎢ RT RT 2 RT 2 RT ⎥ ⎣ () () ⎦ 6zpb2 2zb 2abp z2b za 2pb2 3p2b3 + + − − − − 2 RT 3 RT 2 2 3 ()RT ()RT ()RT ()RT ()RT (C.5) 6zpb2 2zb 2abp z2b za 2pb2 3p2b3 + + − − − − ⎛ ∂z ⎞ ()RT 2 RT ()RT 3 RT ()RT 2 ()RT 2 ()RT 3 ⎜ ⎟ = ⎝ ∂p ⎠ ⎡ bp ap 3()bp 2 2bp⎤ ⎢3z2 − 2z + 2z + − − ⎥ ⎢ RT RT 2 RT 2 RT ⎥ ⎣ () () ⎦ (C.6) Therefore the PR EOS based equation for the isothermal gas compressibility is expressed as, 6zpb2 2zb 2abp z2b za 2pb2 3p2b3 + + − − − − 1 1 ()RT 2 RT ()RT 3 RT ()RT 2 ()RT 2 ()RT 3 c = − (C.7) g p z ⎡ bp ap 3()bp 2 2bp⎤ ⎢3z2 − 2z + 2z + − − ⎥ RT 2 2 RT ⎣⎢ ()RT ()RT ⎦⎥ 89 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 = 1 ∑ i (D.3) y = 1 ∑ i (D.4) z = 1 ∑ i (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, y K = i i x i (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, z x = i i n + K n L i V (D.10) Solving for yi, gives, 91 z K y = i i i n + K n L i V (D.11) Combining equation D.10 and D.3 gives, z x = i = 1 ∑∑i n + n K iiL V i (D.12) Combining equation D.11 and D.4 gives, z K y = i i = 1 ∑∑i n + n K iiL V i (D.13) x = 1 y = 1 But ∑ i and ∑ i y − x = 0 Hence, ∑ i ∑ i z K z i i − i = 0 ∑∑n + n K n + n K iiL V i L V i (D.14) Simplifying equation D.14, z ()K −1 i i = 0 ∑ n + n K i L V i (D.15) From equation D.6 nL = 1 - nV (D.16) Combining equations D.15 and D.16 gives, z ()K −1 i i = 0 ∑ n K −1 +1 i V ()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+1 n f ()n ()n = ()n − v V V f' n ()v (D.18) where n and n+1 indicates the iterative steps and the functions are defined as, z ()K −1 f ()n = i i V ∑ n K −1 +1 i V ()i (D.19) ⎡ z ()K −1 2 ⎤ f' n = − i i ()V ∑ ⎢ 2 ⎥ i n K −1 +1 ⎣⎢[]V ()i ⎦⎥ (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. Gas Vg; cg Oil Vt Vt c c o o Oil VL; cL 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, V = V + V t L g (E.1) Differentiating equation E.1 in terms of pressure as constant temperature gives, 94 ⎛ ∂V ⎞ ⎛ ∂V ⎞ ⎛ ∂V ⎞ ⎜ t ⎟ = ⎜ L ⎟ + ⎜ g ⎟ ⎜ ∂p ⎟ ⎜ ∂p ⎟ ⎜ ∂p ⎟ ⎝ ⎠T ⎝ ⎠T ⎝ ⎠T (E.2) The coefficients of isothermal compressibility are defined as, 1 ⎛ ∂V ⎞ c = − ⎜ t ⎟ t V ⎜ ∂p ⎟ t ⎝ ⎠T (E.3) 1 ⎛ ∂V ⎞ c = − ⎜ L ⎟ L V ⎜ ∂p ⎟ L ⎝ ⎠T (E.4) 1 ⎛ ∂V ⎞ c = − ⎜ g ⎟ g V ⎜ ∂p ⎟ g ⎝ ⎠T (E.5) The proposed method for adding the coefficient of isothermal oil compressibility and the coefficient of isothermal gas compressibility is described as, V V c = g c + L c t V g V L t t (E.6) Combining equations E.3, E.4, E.5 and E.6 gives, 1 ⎛ ∂V ⎞ V 1 ⎛ ∂V ⎞ V 1 ⎛ ∂V ⎞ ⎜ t ⎟ = L ⎜ L ⎟ + g ⎜ g ⎟ V ⎜ ∂p ⎟ V V ⎜ ∂p ⎟ V V ⎜ ∂p ⎟ t ⎝ ⎠T t L ⎝ ⎠T t g ⎝ ⎠T (E.7) ⎛ ∂Vt ⎞ Substituting for ⎜ ⎟ in equation E.7, using equation E.2 gives, ⎝ ∂p ⎠T 1 ⎡⎛ ∂V ⎞ ⎛ ∂Vg ⎞ ⎤ V 1 ⎛ ∂V ⎞ Vg 1 ⎛ ∂Vg ⎞ ⎢⎜ L ⎟ + ⎜ ⎟ ⎥ = L ⎜ L ⎟ + ⎜ ⎟ V ⎜ ∂p ⎟ ⎜ ∂p ⎟ V V ⎜ ∂p ⎟ V V ⎜ ∂p ⎟ t ⎣⎢⎝ ⎠T ⎝ ⎠T ⎦⎥ t L ⎝ ⎠T t g ⎝ ⎠T (E.8) Simplifying, 95 1 ⎛ ∂V ⎞ 1 ⎛ ∂V ⎞ 1 ⎛ ∂V ⎞ 1 ⎛ ∂V ⎞ ⎜ L ⎟ + ⎜ g ⎟ = ⎜ L ⎟ + ⎜ g ⎟ V ⎜ ∂p ⎟ V ⎜ ∂p ⎟ V ⎜ ∂p ⎟ V ⎜ ∂p ⎟ t ⎝ ⎠ T t ⎝ ⎠T t ⎝ ⎠T t ⎝ ⎠T (E.9) This shows that equation E.6 is mathematically consistent. 96 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 Pressure- Volume 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 , ⎡⎛ Vt ⎞ ⎤ ⎢⎜ ⎟ ⎥ ⎝ Vb ⎠ ln⎢ 2 ⎥ ⎢⎛ V ⎞ ⎥ ⎢⎜ t ⎟ ⎥ ⎜ V ⎟ ⎣⎢⎝ b ⎠1 ⎦⎥ co = (F.1) p1 − p 2 ⎛ Vt ⎞ where ⎜ ⎟ are Pressure-Volume Relations data at different pressures of interest. ⎝ Vb ⎠ 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 1 ⎛ ∂B ⎞ B ⎛ ∂R ⎞ c = − ⎜ o ⎟ + g ⎜ s ⎟ o B ⎜ ∂p ⎟ B ⎜ ∂p ⎟ o ⎝ ⎠T o ⎝ ⎠T (F.2) 97 Equation F.2 is further simplified by McCain 29 p. 290, to give 1 ⎛ ∂R SD ⎞ ⎡ ⎛ ∂BOD ⎞ ⎤ co = ⎜ ⎟ ⎢Bg − ⎜ ⎟ ⎥ B ⎜ ∂p ⎟ ⎜ ∂p ⎟ oD ⎝ ⎠T ⎣⎢ ⎝ ⎠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 (1/100) x 2 Molecular Weight Component Mol % (lbm/lbm mol) App Mol Weight H2S 0.00 34.08 0.0000 CO2 0.91 44.01 0.4005 N2 0.16 28.02 0.0448 CH4 36.47 16.04 5.8498 C2H6 9.67 30.07 2.9078 C3H8 6.95 44.09 3.0643 i-C4H10 1.44 58.12 0.8369 n-C4 3.93 58.12 2.2841 i-C5 1.44 72.15 1.0390 n-C5 1.41 72.15 1.0173 C6 4.33 86.17 3.7312 C7+ 33.29 218.00 72.5722 Total 100 93.7478 Sp Gr C7+ = 0.8515 Source: McCain29 p. 260 99 Table F.2: Molar Volume Determination from Pressure-Volume Relations for Oil Well No. 4 Vt *ft3/lbmol) Pressure (1) Rel Vol (2) Vsat * Rel = App Mol 3 -1 (psig) (Vt/Vo) Vol ( ft /lb) Wt * (2) co(psi ) 5000 0.9639 0.0235 2.2058 4500 0.9703 0.0237 2.2204 1.32E-05 4000 0.9771 0.0239 2.2360 1.40E-05 3500 0.9846 0.0240 2.2531 1.53E-05 3000 0.9929 0.0242 2.2721 1.68E-05 2900 0.9946 0.0243 2.2760 1.71E-05 2800 0.9964 0.0243 2.2801 1.81E-05 2700 0.9983 0.0244 2.2845 1.91E-05 2620 1.0000 0.0244 2.2884 2.13E-05 2605 1.0022 0.0245 2.2934 2591 1.0041 0.0245 2.2978 2516 1.0154 0.0248 2.3236 2401 1.0350 0.0253 2.3685 2253 1.0645 0.0260 2.4360 2090 1.1040 0.0269 2.5264 1897 1.1633 0.0284 2.6621 1698 1.2426 0.0303 2.8435 1477 1.3618 0.0332 3.1163 1292 1.5012 0.0366 3.4353 1040 1.7802 0.0435 4.0738 830 2.1623 0.0528 4.9482 640 2.7513 0.0672 6.2960 472 3.7226 0.0909 8.5187 100 Table F.3: Differential Vaporization data for Oil Well No. 4 Pressure Bg (res cu Bg(res (psig) Rsd BoD ft/ bbl) dRsd/dp dBod/dRsd bbl/scf) co(p 101 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 / -1 (psig) (Vt/Vo) Vol lb mol) co(psi ) 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 Bg(res (PSIG) Rsd BoD ft/ bbl) dRsd/dp dBod/dRsd bbl/scf) co(1/psia) 2663 928 1.5150 2400 840 1.4750 0.0059 0.3346 0.0005 0.0010 1.34E-04 2150 765 1.4410 0.0065 0.3000 0.0005 0.0012 1.48E-04 1900 690 1.4080 0.0075 0.3000 0.0004 0.0013 1.89E-04 1650 619 1.3760 0.0087 0.2840 0.0005 0.0015 2.27E-04 1400 547 1.3450 0.0105 0.2880 0.0004 0.0019 3.07E-04 1150 476 1.3150 0.0130 0.2840 0.0004 0.0023 4.09E-04 900 407 1.2850 0.0170 0.2760 0.0004 0.0030 5.57E-04 650 336 1.2530 0.0241 0.2840 0.0005 0.0043 8.69E-04 400 262 1.2180 0.0398 0.2960 0.0005 0.0071 1.61E-03 183 183 1.1760 0.0861 0.3641 0.0005 0.0153 4.58E-03 104 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 3 Pressure Rel Vol Vsat * Vt(ft /lb -1 (psig) (Vt/Vo) Rel Vol mole) co(psi ) 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/ Bg(res (psig) Rsd BoD bbl) dRsd/dp dBod/dRsd bbl/scf) co(p 107 APPENDIX G RESERVOIR FLUIDS CRITICAL PROPERTIES TABLE The critical properties used in calculating the constant parameters of the Peng- Robinson EOS are given in Table G.1. 44 pp 162, 13 pp 354 Table G.1: Reservoir Fluids Critical Properties. Critical Critical Temperature Pressure Acentric Critical Molecular Component (oR) (psia) Factor z Weight CO2 547.60 1070.60 0.2301 0.2742 44.01 N2 227.30 493.00 0.0450 0.2916 28.02 C1 343.00 667.80 0.0115 0.2884 16.04 C2 549.80 707.80 0.0908 0.2843 30.07 C3 665.70 616.30 0.1454 0.2804 44.09 i-C4 734.70 529.10 0.1756 0.2824 58.12 n-C4 765.30 550.70 0.1928 0.2736 58.12 i-C5 828.80 490.40 0.2273 0.2701 72.15 n-C5 845.40 488.60 0.2510 0.2623 72.15 C6 913.40 436.90 0.2900 0.2643 86.17 C7 972.36 397.30 0.3495 0.2611 100.20 C8 1023.66 361.05 0.3996 0.2559 114.23 C9 1070.28 332.05 0.4435 0.2520 128.26 C10 1111.86 305.95 0.4923 0.2465 142.29 C11 1150.20 282.61 0.5303 0.2419 156.31 C12 1184.40 263.90 0.5764 0.2382 170.34 C13 1215.00 243.60 0.6174 0.2320 184.37 C14 1247.40 227.65 0.6430 0.2262 198.39 C15 1274.40 214.60 0.6863 0.2235 212.42 C16 1301.40 203.00 0.7174 0.2199 226.45 C17 1324.80 194.30 0.7697 0.2190 240.47 C18 1344.60 184.15 0.8114 0.2168 254.50 C19 1364.40 175.45 0.8522 0.2150 268.53 108 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) ERROR AAPD AAPD PR MPR Pressure(psia) Exptal PR EOS MPR EOS EOS EOS 5014.7 1.33E-05 1.21E-05 4514.7 1.32E-05 1.45E-05 1.32E-05 9.86 0.42 4014.7 1.40E-05 1.59E-05 1.44E-05 14.12 2.95 3514.7 1.53E-05 1.76E-05 1.58E-05 15.09 3.32 3014.7 1.68E-05 1.96E-05 1.75E-05 16.74 4.18 2914.7 1.71E-05 2.01E-05 1.79E-05 17.20 4.46 2814.7 1.81E-05 2.05E-05 1.83E-05 13.49 0.99 2714.7 1.91E-05 2.10E-05 1.87E-05 10.23 2.00 2634.7 2.13E-05 2.14E-05 1.90E-05 0.67 10.62 2364.7 1.55E-04 2.35E-04 2.06E-04 51.36 32.74 2114.7 1.83E-04 2.74E-04 2.44E-04 49.50 33.11 1864.7 2.09E-04 3.26E-04 2.95E-04 56.44 41.59 1614.7 2.53E-04 3.96E-04 3.65E-04 57.00 44.63 1364.7 3.15E-04 4.95E-04 4.65E-04 57.34 47.75 1114.7 4.17E-04 6.44E-04 6.16E-04 54.26 47.63 864.7 5.76E-04 8.84E-04 8.63E-04 53.61 49.84 614.7 8.87E-04 1.33E-03 1.32E-03 50.41 49.26 364.7 1.83E-03 2.42E-03 2.44E-03 32.84 33.52 173.7 5.20E-03 5.42E-03 5.47E-03 4.37 5.20 Avg = 31.36 23.01 110 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 2.2175 2.2330 2.3248 0.70 4.83 4514.7 2.2269 2.2426 2.3345 0.70 4.83 4014.7 2.2368 2.2528 2.3451 0.72 4.84 3514.7 2.2474 2.2640 2.3565 0.74 4.86 3014.7 2.2586 2.2761 2.3689 0.78 4.88 2514.7 2.2707 2.2893 2.3824 0.82 4.92 2014.7 2.2838 2.3039 2.3974 0.88 4.97 1914.7 2.2866 2.3070 2.4006 0.89 4.99 1814.7 2.2893 2.3102 2.4038 0.91 5.00 1714.7 2.2921 2.3134 2.4071 0.93 5.02 1689.7 2.2928 2.3142 2.4079 0.94 5.02 1680.7 2.2969 2.1812 2.4891 5.04 8.37 1670.7 2.3015 2.1860 2.4929 5.02 8.32 1660.7 2.3063 2.1909 2.4968 5.00 8.26 1650.7 2.3111 2.1959 2.5008 4.99 8.21 1627.7 2.3223 2.2076 2.5106 4.94 8.11 1541.7 2.3693 2.2565 2.5506 4.76 7.65 1402.7 2.4636 2.3549 2.6325 4.41 6.86 1244.7 2.6080 2.5060 2.7595 3.91 5.81 1064.7 2.8460 2.7572 2.9765 3.12 4.59 900.7 3.1745 3.0966 3.2800 2.45 3.32 747.7 3.6485 3.5832 3.7261 1.79 2.13 631.7 4.1962 4.1440 4.2506 1.24 1.30 501.7 5.1681 5.1419 5.2026 0.51 0.67 390.7 6.5871 6.6222 6.6328 0.53 0.69 287.7 8.9929 9.1869 9.1431 2.16 1.67 Avg = 2.26 5.00 111 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) ERROR AAPD PR AAPD Pressure (psia) Exptal PR EOS MPR EOS EOS MPR EOS 5014.7 8.24E-06 8.10E-06 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 Avg = 32.44 28.75 112 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 2.6644 2.6147 2.8618 1.86 7.41 1604.7 2.8558 2.8039 3.0107 1.82 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 1.61 3.54 Avg = 2.87 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) ERROR AAPD PR AAPD MPR Pressure (psia) Exptal PR EOS MPR EOS EOS EOS 5014.7 1.09E-05 9.56E-06 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 Avg = 20.37 19.10 114 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 AAPD MPR (psia) Exptal PR EOS MPR EOS AAPD PR EOS EOS 5014.7 9.93E-06 7.33E-06 4514.7 8.46016E-06 1.08E-05 7.84E-06 27.78 7.31 4014.7 8.83457E-06 1.18E-05 8.42E-06 33.79 4.67 3514.7 9.40793E-06 1.30E-05 9.08E-06 38.29 3.49 3014.7 9.97305E-06 1.44E-05 9.83E-06 44.69 1.41 2514.7 1.07315E-05 1.61E-05 1.07E-05 50.40 0.29 2014.7 1.14775E-05 1.82E-05 1.17E-05 58.92 2.03 1914.7 1.20397E-05 1.87E-05 1.19E-05 55.49 0.83 1814.7 1.20253E-05 1.92E-05 1.22E-05 59.83 1.20 1714.7 1.20108E-05 1.97E-05 1.24E-05 64.35 3.24 1689.7 1.20018E-05 1.99E-05 1.25E-05 65.64 3.90 1514.7 2.39E-04 4.05E-04 3.81E-04 69.10 59.23 1364.7 2.72E-04 4.65E-04 4.44E-04 71.12 63.45 1214.7 3.32E-04 5.42E-04 5.25E-04 63.19 58.14 1064.7 3.99E-04 6.43E-04 6.32E-04 61.23 58.50 914.7 4.98E-04 7.80E-04 7.77E-04 56.64 56.11 614.7 9.05E-04 1.27E-03 1.30E-03 40.71 43.51 464.7 1.38E-03 1.78E-03 1.82E-03 28.54 32.04 314.7 2.48E-03 2.78E-03 2.86E-03 12.01 15.34 174.7 5.98E-03 5.31E-03 5.44E-03 11.21 8.98 Avg = 48.05 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 1070.6 0.0049 0.2301 44.01 0.2742 0 227.3 493 0.0053 0.045 28.02 0.2916 0 343 667.8 0.3883 0.0115 16.04 0.2884 0 549.8 707.8 0.0986 0.0908 30.07 0.2843 0 665.7 616.3 0.0953 0.1454 44.09 0.2804 0 734.7 529.1 0.0123 0.1756 58.12 0.2824 0 765.3 550.7 0.0431 0.1928 58.12 0.2736 0 828.8 490.4 0.012 0.2273 72.15 0.2701 0 845.4 488.6 0.0187 0.251 72.15 0.2623 0 913.4 436.9 0.0282 0.29 86.17 0.2643 0 0 0 0.2933 0 252 0 0.8413 622 2677.7 117 The program was originally designed for four different EOS (Soave-Redlich- Kwong (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 !Number of Components Read(5,*) NPSTEPS !Number of Pressure Steps Do i=1,NPSTEPS Read(5,*) PS(i) !Reading Pressures end do 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 !Reading Reservoir Temperature Read(5,*) PB ! 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 November 28, 2006 Student Signature Date Disagree (Permission is not granted.) ______Student Signature Date