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1988 Prediction of Methane Solubility in Geopressured Brine Solutions by Application of Perturbation Theory. Yangtzu Chao Louisiana State University and Agricultural & Mechanical College
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Prediction of methane solubility in geopressured brine solutions by application of perturbation theory
Chao, Yangtzu, Ph.D.
The Louisiana State University and Agricultural and Mechanical Col., 1988
UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106
Prediction of Methane Solubility in Geopressured Brine Solutions by Application of Perturbation Theory
A Dissertation
Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy
in
The Department of Chemical Engineering
by Yangtzu Chao B.S. Tunghai University, Taiwan, 1976 M.S. Illinois Institute of Technology, 1981 M ay 1988 Acknowledgement
I would like to express my sincere gratitude to my major professor, Dr. Adrain E.
Johnson, Jr., who initiated me into this reserach and guided me with patience, encourgement, and understanding throughout this study. Thanks are due to Dr. Armando B. Corripio, Dr.
Arthur M. Sterling, Dr. Frank R. Groves, Dr. Tryfon Charalampopoulos, and Dr. Elvin
Choong for their time and cooperation in serving on my graduate committee.
The financial support from Chemical Engineering Department at LSU during my graduate study here is deeply appreciated. I would also like to thank Dr. Ralph W. Pike and the Mineral Research Institute for providing me with fellowship support during several of the summer periods. Special thanks are due to Mr. Rueyder Jeng for his assistance in word processing and figure plotting in this report. Also, thanks to my parents and my wife for the sacrifices they endured and the aboudant love and confidence they shared with me.
Above all, thanks and praises are due unto the name of Jesus Christ for His comfort, wisdom, love, and many many answering prayers throughout these years.
i i Table of Contents
Page
Acknowledgement ...... ii
Table of Contents ...... iii
List of Tables ...... vii
List of Figures ...... ix
A bstract ...... xii
List of Symbols ...... xiv
C hapter 1. Introduction ...... 1
1-1 The Geopressured Energy Resource in The U.S ...... 1
1-2 DOE-Sponsored Research on Geopressured Energy And Methane Solubility ...... 3
1-3 Goal of This Research And Nature of The Problem ...... 5
1-4 Correlation of Henry’s Law Constant ...... 6
1-5 Setchenow Equation ...... 7
1-6 Electrostatic Theory ...... 8
1-7 Scaled Particle Theory ...... 11
1-8 Perturbation Theory ...... 13
1-9 Summary ...... 19
Chapter 2. Previous Work on Correlation of Methane Solubility Data ...... 21
2-1 Introduction ...... 21
2-2 Experimental Work And Early Correlations ...... 21
2-2-1 Correlation of Haas ...... 24
2-2-2 Correlarion of Blount ...... 25
2-2-3 Correlation of Coco And Johnson ...... 28 Page
Chapter 3. Development of Govering Equations of Phase Equalibria for A
Complicated System of Gases, Electrolytic Salts, And Water ...... 35
3-1 System Description ...... 36
3-2 Solubility of Gases in Liquids ...... 39
3-3 Perturbation Approach ...... 48
3-3-1 Introduction to Statistical Mechanics ...... 48
3-3-2 Perturbation Theory to Obtain Chemical Potential ...... 50
3-3-3 Relation between Apparent Ideal Solution Fugacity And Chemical Potential .... 57
3-3-4 Expression of Apparent Ideal Solution Fugacity for Geopressured System ...... CO
3-4 Development of The Partial Molar Volume of A Gas Solute ...... 63
3-5 Development of The Isothermal Compressibility of The Partial Molar Volume ...... 6 6
3-6 Gas Solubility at High Pressure ...... 67
3-7 Dissociation of Weak Electrolyte ...... 69
Chapter 4. Method of Solution ...... 72
4-1 Vapor Phase ...... 74
4-2 Liquid Phase ...... 77
4-2-1 Solution Density ...... 77
4-2-2 Saturation Vapor Pressure ...... 83
4-2-3 Poynting Factor ...... 83
4-3 Physical Parameters ...... 85
4-4 Computer Algorithm ...... 8 8
iv Page
4-5 Analysis of Initial Results And Decisions Made to Improve Predictions ...... 91
4-5-1 Possible Effect of W ater Density on Methane Solubility ...... 91
4-5-2 Possible Effect of Temperature on Methane Solubility
via the Lennard-Jones Parameter a ...... 95
4-5-3 Possible Improvement of The Model by Adding Charge-Dipole
And Dipole-Dipole Interactions ...... 98
4-6 Parameters Fitting ...... 101
4-6-1 C H 4 /H 20 System ...... 106
4-6-2 C II4 /H 20 /N a C l System ...... 109
4-6-3 C 2 II6 /II20 System ...... 112
4-6-4 C 0 2 /HoO System ...... 113
4-7 Summary ...... 113
Chapter 5. Results And Comparisons ...... 114
5-1 The Effect of Temperature on Methane Solubility for Methane-Water Mixtures ..... 114
5-2 The Effect of Pressure on Methane Solubility for Methane-Water Mixtures ...... 121
5-3 The Effect of NaCl Salt on Methane Solubility ...... 129
5-4 The Effect of Other Gases on Methane Solubility in Water And in Brine Solutions 140
5-4-1 C H 4 /C 2H 6 /H 20/NaCl System ...... 140
5-4-2 C H 4 / C 0 2/H 20/NaCl System ...... 145
5-5 Summary ...... 154
Chapter 6. Conslusions And Recommendations ...... 155
V Page
References 160
Appendices ...... 167
A Equation of State Developed by Nakamura et al ...... 167
B Orientation-Averaged Molecular Interaction ...... 171
C Potential Energy for Water Molecule ...... 173
D Computer Program ...... 174
V ita ...... 1 9 8
vi List of Tables
Page
Table 2-1 Available published data on methane solubility in water and water-NaCl solution..22
Table 2-2 A comparison of solubilities of methane in water calculated by Haas, Blount,
and Coco-Johnson’s correlations with published data ...... 27
Table 3-1 Classification of subscript number used in this study and computer program .... 38
Table 3-2 Intermolecular potential energy used in this study ...... 61
Table 4-1 Comparison of various correlations of liquid water density with steam tables .... 79
Table 4-2 Comparison of A p predicted by equation (4-21) with Rogers and Pitzer’s
experimental density data for NaCl-H20 solution ...... 82
Table 4-3 Sensitivity study to show the effect of various physical parameters on
predicted methane solubilities and partial molar volumes ...... 8 6
Table 4-4 Numerical values for physical parameters as reported in the literature ...... 87
Table 4-5 Individual Fitted <7 h 2o and for various data sets along with its
corresponding partial molar volume of methane ...... 104
Table 4-6 Partial F test of parameter fitting of CH 4 /H 20 system ...... 107
Table 4-7 Comparison of RMSD of present model with Blount’s polynomial in NaCl
brine solution ...... I l l
Table 5-1 Comparison of the magnitude of each terms in equation 3-56 for
methane-water system ...... 118
Table 5-2 Contribution of various terms in equation 3-86 ...... 122
Table 5-3 Comparison of V q jj and /?q jj predicted by this study with reported values
in C H 4 /H 20 system ...... 123
Table 5-4 Calculated vapor and liquid phase methane fractions in CH 4 /H 20 system ... 127 Table 5-5 Calculated salting-out coefficient based on our model ...... 136
Table 5-6 Reported values for salting-out coefficient of CH4-NaCl pair ...... 138
Table 5-7 Experimental gas solubility data of C 02, C 2 H 6 , and C H 4 in w ater ...... 141
Table 5-8 Calculated vapor and liquid phase thermodynamic properties for
C 2H 6 /CH 4 /H 20/NaCl system at 423 K, 1530 atm, and 1.83 m ...... 144
Table 5-9 Ratio of undissociated C 0 2 to the total amount of C 0 2 in water at 423 K
and at different ^ 7 a /llj. ^ values ...... 148
Table 5-10 Calculated vapor and liquid phase thermodynamic properties for
C 0 2 /CH 4 /II 2 0/NaCl system at 423 K, 874 atm, and 0.89 m ...... 152
Table A-l Comparison of predictions of fugacity coefficients of SRK and Nakamura’s
equation of state in CH 4 /H 20 system ...... 170
v iii List of Figures
Page
Figure 1-1 Lennard-Jones potential ...... 14
Figure 1-2 Hard sphere potential ...... 14
Figure 1-3 Square well potential ...... 14
Figure 1-4 The split of potential according to Barker-Henderson ...... 16
Figure 1-5 The split of potential according to Weeks-Chandler-Andersen ...... 16
Figure 3-1 Schematic representation of the system under consideration ...... 37
Figure 3-2 Extrapolation of liquid saturation pressure into hypothetical liquid region ...... 42
Figure 3-3 Variation of Henry's law constant with composition in a ternary system ...... 44
Figure 4-1 Comparison of A p of our correlation (equation 4-20) with experimental data
of Rogers and Pitzer ...... 80
Figure 4-2 Flow chart of program to compute methane solubility given system specifications
and physical parameters of components ...... 89
Figure 4-3 Initial comparison of the discrepancies between predicted methane
solubility and experimental data of Culberson and McKetta ...... 92
Figure 4-4 The difference of predicted methane solubility in pure water using
various correlation of water density ...... 93
Figure 4-5 Minimum methane solubility in pure water predicted by our model without
adjusting the a parameter of water from the initial values chosen from the
literature ...... 96
Figure 4-6 The effect of a linear variation of
methane solubility in pure water ...... 97
Figure 4-7 Fitting of ir H i) 0 and *n potential with experimental solubility data....102
Figure 4-8 Fitting of in L-J potential with experimental solubility data using
constant value ...... 1 1 0
ix Figure 5-1 Minimum methane solubility in pure water around 344 K predicted by our model
and the experimental data of Culberson and M cKetta ...... 115
Figure 5-2 Maximum apparent ideal solution fugacity of methane predicted
by our model around 360 K at various pressure ...... 117
Figure 5-3 Comparison of methane solubility in pure water predicted by different correlations
with experimental data of Price and Sultanov ...... 119
Figure 5-4 Comparison of methane solubility in pure water predicted by Blount, Coco,
and our model with experimental data of Price and O’Sullivan ...... 124
Figure 5-5 Comparison of methane solubility in water predicted by Blount, Coco,
and this study with experimental data of Sultanov and McKetta ...... 125
Figure 5-6 Fugacity coefficient of pure methane from Choi’s study and that in
methane-water mixture from SRIv equation of state in this study over
moderate presures and temperatures ...... 128
Figure 5-7 Salting effect on methane solubility at 600 atm and moderate temperature ..... 130
Figure 5-8 Salting effect on methane solubility predicted by Blount, Coco, and this study... 131
Figure 5-9 Comparison of methane solubility in low and high salt content brine
solutions predicted by various correlations with Blount’s data ...... 132
Figure 5-10 Comparison of methane solubility in 0.89m brine solution at low temperature
and low pressure conditions ...... 133
Figure 5-11 Comparison of methane solubility at low temperatures and pressures
in 0.89m brine solution ...... 134
Figure 5-12 The existance of a minimum salting out coefficient over the temperature
range and the variation of Ks with pressure at m =l based on our model .... 137
Figure 5-13 Predicted partial molar volume of methane at various salt concentration ...... 139
Figure 5-14 The effect of C 2 H 6 011 the methane solubility in brine solution at
different dissolved gas ratio ...... 142 Figure 5-15 The ratio of methane to ethane in the vapor phase predicted by our model at
Blount’s experimental condition of 1530 atm, 423 K, and molality of 1.83 .... 146
Figure 5-16 Predicted methane solubility and total gas dissolved in 1.82m NaCl solution at
two different pressure and various C 0 2 percentage in the dissolved gas ...... 149
Figure 5-17 Comparison of predicted methane and total gas dissolved in 0.8898m
NaCl soluition with Blount’s experimental data at various C 0 2
percentage in the dissolved gas ...... 150
Figure 5-18 The ratio of CH 4 to C 0 2 in the vapor and phase predicted by our
model at Blount’s experimental condition of 874 atm, 423 K, and a
salt content of 0.89m ...... 153
xi Abstract
The ability to predict methane solubility in underground reservoirs is essential to eventual
exploitation of geothermal-geopressured reservoirs as an alternate energy source. In this study,
second-order Leonard-Barker-IIenderson perturbation theory was applied to brine solutions
containing II 2O, C H 4, C 2 II6, C02, and NaCl to develop a fundamentally-based
thermodynamic approach to handle more effectively the broad temperature and pressure ranges
encountered in underground reservoirs, and to include the effect of the presence of various salts
and other gases on methane solubility.
The apparent ideal solution fugacity, f°', for each solute gas is proposed and developed
through perturbation theory as a direct measure of the net forces acting upon the solute gas
from all the species in the system. Since f °l depends on the brine composition, there is no need
to use a Henry’s law constant evaluated at infinite dilution along with an activity coefficient
that varies with the solute gas mole fraction. Inst' id, the product, (x^•f°,), gives the fugacity of each solute gas in the equations representing phase equilibrium. Both the partial molar volume of solute gas and the isothermal compressibility of its partial molar volume were self- generated from the perturbation theory approach to predict the gas solubility at high pressure, and they both agreed well with reported values.
Essentially all published experimental data on methane solubility were utilized to test the relationships developed in this work. The ranges of conditions covered by this study are: temperatures from 298 to 589 K, pressures from 10 to 2000 atm, and salinities from 0 to 5 m.
The parameters used in this study were all based upon reported literature values. Because of the extreme sensitivity of calculated methane solubilities to the values of a parameter in the
Lennard-Jones potential, it was found necessary to determine the best values for these
xii parameters for each component to insure a minimum least-squares global fit of the solubility data. In addition, it was found that allowing the a parameter for water to vary with temperature significantly improved the overall global fit to the data.
x iii List of Symbols
A Helmholtz free energy
Ac configurational Helmholtz free energy
A° configurational Helmholtz free energy of reference system aa molecular activity aj ionic diameter a_|_ activity of cation a— activity of anion
D Dielectric constant of salt solution
D0 Dielectric constant of water d diam eter ej ionic charge fj fugacity of component i f? fugacity of component i at reference state
apparent ideal solution fugacity of solute gas i gc partial molar Gibbs energy for cavity formation gj partial molar Gibbs energy for interaction g° radial distribution function of reference system h Planck’s constant
H? Henry’s constant of component i in pure water
Ho i Henry’s constant of solute 2 in pure solvent 1
H, Henry’s law constant of component i in the solution
K Boltzmann constant
Kj salting coefficient
xiv m^ Stoichiometric concentration ma molecular concentration m 5 salt molality m^_ cation molality m — anion molality
Nj number of molecules of species i in mixture
N0 Avogadro’s constant
P pressure
Pj partial pressure of component i
PJ saturation pressure of component i at temperature T
Pp perturbation part of pressure equation
Its P ' hard sphere equation of state
Q partition function
Qc configurational partition function
Qj quadrupole mement
R gas constant r intermolecular distance
S salinity in grams per liter
T temperature t temperature in degree Fahrenheit
U(r) total potential energy
U°(r) reference system potential energy
U^(r) perturbed potential energy
V volum e
Vj liquid molar volume of component i
Vs liquid molar volume of salt
XV vpr hs moe rcin o oet i ponent com f o fraction ole m phase vapor f Y f pata moa vlme fcmpnn i ifnt dilution infinite t a i energy ’°j ponent \ com of e potential volum pair olar m V,-• artial p Vf° , lqi pae l fato o o oet i ponent com of fraction ole m phase X liquid X,- Vfj cmpesblt factor pressibility com z c ofgrtoa integral configurational Zc 0
OO ater w pure in fraction ole m solute ar oeta eeg o eeec system reference of energy potential pair etre pi ptnil energy potential pair perturbed ofgrtoa itga o eeec system reference of integral configurational xvi Greek Symbol
a perturbation parameter, measuring inverse steepness of repulsive potential
a,- polarizabilitv of component i
0 Boltzmann factor (1/KT)
/?,: isothermal compressibility of partial molar volume of solute gas i
0 O compressibility of pure water
activity coefficient of component i
7 _j_ mean ionic activity coefficient
e energy parameter in Lennard-Jones potential
9 polar angle
A perturbation parameter, measuring strength of attractive potential
p t chemical potential of component i
hs //,' chemical potential of a hard sphere fluid lis r Pi ’ reduced chemical potential for a hard sphere fluid
chemical potential of an ideal gas
Pi dipole moment of component i
£n reduced density
p solution density
Pi number of i molecule per unit volume
p* density of pure water
A p density increase due to the presence of salt
cr distance parameter in the Lennard-Jones potential
xvii Chapter 1 Introduction 1-1 The Gcopressured Energy Resource in The U.S. The shock of the energy crisis in the 70’s focussed considerable attention in the U.S. on the development of alternative energy resources for the future. It was recognized by the U.S. Department of Energy (DOE) that the various alternate energy resources in the U.S. need to be identified and characterized prior to the time, which surely must come, when the oil and gas resources of the world are depleted. One energy source that was selected for study by DOE is the group of geopressured-geothermal reservoirs that are found in the United States in the northern Gulf of Mexico basin, mainly along the Texas and Louisiana Gulf Coasts, at depths of G000 ft to 15000 ft. As a result of DOE-sponsored effort, the locations of these reservoirs are now generally known, and some information about their temperatures, pressures, and salinities has been obtained from logs of oil and gas wells previously drilled in the immediate vicinity of each geopressured reservoir. In addition, some special test wells were drilled at DOE expense during the late 70‘s and early 80’s to gain more information ( including methane content ) about a few of the high-potential reservoirs, but well logs of previously drilled wells remain the primary source of information for estimating the nature and amount of this energy resource. The energy contained in the geopressured reservoirs is classified as : (A) thermal energy from the hot brine. (B) mechanical energy from the high fluid pressure of the reservoirs, and (C) combustive fuel energy from the natural gas, mostly methane, dissolved in the underground brines (Gregory, 1981). Early estimates for the total in-place resource were up to 1 100.000 quads ( 1 quad = 1.0 xlO 15 Btu ) of low level (200 - 250 F) thermal energy and 60,000 quads of dissolved methane. ( The estimated available mechanical energy was several orders of magnitude less than these two.) The economically recoverable resources were estimated to be as much as 300 quads of thermal energy and 800 quads of methane (Westhusing, 1981). Because of previous success in the exploiation of high temperature geothermal wells in the far western U.S., initial interest focused primarily on the heat that could be extracted from the hot brine and converted to electrical energy through turbines at the surface. In 1974, 30.4% of the energy consumed in the United States came from natural gas (Campbell, 1977). As a result of both increasing demand and the de-regulating Natural Gas Policy Act of 1978 by Congress, the price of deregulated natural gas soon rose by ten fold above pre-1973 levels to the $7-8/1000-SCF range. As a result, economic interest in geopressured resources shifted from the low-level thermal energy of the reservoirs to the fuel energy of the dissolved methane in the brine. At that time, techno-economic studies of specific high-potential reservoirs, such as those made by Johnson et al. (1980), indicated that natural gas prices must equal or exceed the 8S/1000-SCF level for the geopressured energy resource to become economically viable. Since 1983, conservation efforts initiated in the 70’s, coupled with a world-wide energy ‘ glut ’ has forced the price of both oil and natural gas back down to more moderate levels ( 815-20/barrel and SI.5 - 2.0/1000 SCF ), although these prices are still about twice the price levels that existed prior to 1973. Hence, tentative projects for exploitation of the geopressured methane have been shelved by industry until such time that gas prices again reach economically attractive levels. The methane entrained in geopressured reservoirs remains a vast but dilute resource that continues to be unexploited because drilling and production costs are high and because many uncertainties are associated with its exploitation and commercialization. 1-2 DOE-Sponsored Research on Geopressured Energy and Methane Solubility During the late 70’s and early 80’s, an extensive research effort was sponsored by DOE to obtain more reliable assessments of the geological, engineering, enviromental, legal, and social aspects of developing the geopressured resource in order to provide a data base for that future time when this resource becomes economically viable. Among the many related sub-projects carried out under DOE sponsorship was one to study the solubility of methane in brine at conditions equivalent to the underground pressure, temperature, and salinity ranges of known reservoirs. A quantitative description of vapor-liquid equilibria pertinent to reservoir conditions is very important for estimating the methane content of a prospective reservoir. A. E. Johnson. Jr.. who served as the director of a multi-disciplinary, DOE-sponsored project at that time, undertook as one of the sub-tasks of the DOE project a study of the then- available methane solubility data. Coco and Johnson (1981) developed a computer subroutine called SOLUTE to correlate the available solubility data based on a fundamentally sound thermodynamic approach. SOLUTE can be used to predict the solubility of methane in a brine at given salinity, temperature, and pressure conditions. It therefore provided the LSU teclmo-economic computer model of a geopressured, geothermal reservoir with the capability of calculating the maximum (saturated) amounts of methane which could be dissolved in the brine solution of the reservoir, given the salinity of the brine and its temperature and pressure. This work was reasonably successful in correlating the then-available data, but it revealed that there remained a strong need to improve the model used to correlate the data by extending the model’s fundamental basis. The correlation which was developed does not consider the presence of C 0 2 in the dissolved gas, which is the major secondary gas component in geopressured brines, ranging from two to twelve mole percent. Some preliminary data from ‘recombination’ experiments run by IGT (Institute of Gas Technology) on gas from geopressured test wells had suggested that the effect of C 0 2 on total gas dissolved is both substantial and highly nonlinear. Some later DOE-sponsored experimental results by Blount et. al. (1982) indicated that the solubility of methane is enhanced by an extremely low concentration of C 0 2 but is reduced at higher concentrations of C 02. It also revealed the amount of C 0 2 dissolved in geopressured brines can be much greater than originally suspected. The presence of higher-molecular-weight hydrocarbons, mainly ethane, may also play an important role in a comprehensive study of methane solubility. Blount et al. (1982) reported that at low concentrations, ethane salted methane into solution, while above 6 to 8 mole percent ethane of the dissolved gas in solution, methane was strongly salted out by the ethane. This is a significant departure from that observed by Amirajatari and Campbell (1972) who found that solubility of the binary methane-ethane mixture is greater than the solubility of the pure component at. the same temperature and pressure. In addition, geopressured brines contain various salts other than sodium chloride, primarily calcium salts. The previous treatment of Coco and Johnson (1981) correlated data of brine containing only sodium chloride salt and dissolved gases containing only methane. 1-3 Goal of This Research and Nature of The Problem In this research, our goal was to (1) expand the methane solubility correlation to include additional gases and additional salts, ( 2 ) develop an even more fundamental thermodynamic approach, compared to the ‘classical’ solution thermodynamics applied by Coco and Johnson, and (3) extend the ‘global’ nature of the correlation over a wider range of experimental conditions, while at the same time improving the accuracy (‘sum of squares’ error) between the predicted values and experimental data. The coexistence in aqueous solution of strong electrolytes such as N ad together with dissolved gases such as CII4, C 02, and C 2 H 0 creates a complex intermolecular force field that affects the solubility of methane, which for extremely dilute gas concentrations can be expressed in terms of its Henry’s law constant in the liquid phase and its fugacity coefficient in the equilibrium gas phase. Therefore, the success of any methane solubility prediction method is highly dependent on developing an effective and fundamentally correct model to account for the variation of the Henry's law constant of methane with temperature, pressure, and both the electrolyte and dissolved gas composition of the brine. The solubility prediction is also, of course, dependent upon the gas phase fugacity coefficient of methane, which in this work was calculated from the SRK equation of state. 6 1-4 Correlation of Henry’s Law Constant Extensive research has already been done to correlate Henry’s law constants of various gases in binary non-electrolyte solutions. Hayduk and Laudie (1973) plotted straight lines for lull vs. 1/T for a number of gases in a single solvent and found that they all intersected at a single value of 1 nHc at the solvent critical temperature Tc. Therefore, a single measurement of InH can be combined with lnHc t.o obtain values at other temperature. Extensions of this concept have been made to more complex solvents, but it is not clear how it can apply to a system in which II goes through a maximum with temperature. Yorizone and Mivano (1978) developed from a three-parameter corresponding state theory a generalized correlation for predicting Henry’s law constant. But this work was limited to non-polar mixtures. Benson and Krause (1976) studied thoroughly the solubilities of simple gases in water and developed an empirical equation valid from 0 to 50 C. Cysewski and Prausnitz (1976), based on Alder’s perturbed hard sphere equation of state, developed a semi-empirical correlation of Henry’s law constant over a wide temperature range, yet the accuracy of the prediction was not very high. Other attempts include the work of Nakahara and Ilirata (1969) for hydrogen-hydrocarbon mixtures and those of Brelvi (1980) and Sagara and Saito (1975) for variations on the formulae of solubility parameters in regular solution for non-polar systems. In contrast to the relatively extensive studies that have been conducted on Henry’s law constants of gases in binary non-electrolyte solutions, little attention has been given to that of gases in electrolyte solutions. Early approaches correlated the ‘salting ’ effect through the semi-empirical Setchenow equation (1889). With recent improvements in our understanding of solution theories, typified by scaled particle theory and statistical mechanical perturbation theory, a more fundamental approach to this subject can be attempted. In the following section, this topic is developed more fully. 1-5 Sctchenow Equation A well known phenomenon is that when salt is added to a carbonated beverage, the dissolved C 0 2 gas bubbles out from the liquid solution. This phenomenon has long been recognized and is termed the ‘salting-out’ effect. Mathematically, the solubility of a non- electrolvte gas in a salt solution was first related by the semi-empirical Setchenow equation (1889) : log “X " = K S 111 s (1-1) where, for a specified temperature and pressure of the system, X is the solubility of the dissolved gas in an aqueouus salt solution- of concentration ms, XD is its solubility in pure water, and Ks is the salting coefficient. In its original application, the Setchnow equation was applied to aqueous salt solutions in equilibrium with a gas phase at moderate temperatures and pressures, so that the water content of the gas phase at equilibrium is essentially zero. The Setchnow equation is valid primarily at low salt concentrations, and Ks has a specific value for each pair of salt and dissolved non-electrolyte gas. The effect of salt on gas solubility is actually a complex phenomenon. In some cases, adding salt enhances the gas solubility (salting-in), while in other cases the effect is the opposite (salting-out). Clearly, a positive value of Ks in equation (1-1) corresponds to a salting-out effect, whereas, a negative Ks value indicates salting-in. It is easily recognized that the ratio of solubilities expressed in the Setchenow equation actually reflects the effect of the salt on the fugacity, f,, of the dissolved gas, which at equilibrium with a given gas phase must be the same as for the pure water case. Since ff = II{ X; for the gas in a dilute binary 8 solution, equation ( 1- 1) can be rewritten as log = Ks m s ( 1-2 ) Where II® is Henry’s constant of the solute gas in pure water, and H£ m is Henry’s constant in the electrolyte solution. Various attempts were made to predict the salting coefficient Ks. Section 1-6 discusses more on this. 1-6 Electrostatic Theory The electrostatic theory was proposed originally by Debye and McAulay (1925). It treats the solvent as a continuous dielectric. The amount of work necessary to discharge the ions in pure solvent of dielectric constant D 0 and recharging them in a solution of dielectric constant D containing the non-electrolyte is related to the salting coefficient I\s through equations (1-3) and (1-4) : tv ______&_Np ______„ ,■ s “ 2.303 xlOOO K T D 0 A- a £ D = D0 ( 1 - 0 n) (1-4) where N 0 is the Avogadro’s number D is the dielectric constant of the solution D 0 is the dielectric constant of water n is the number of molecules of non-electrolyte solute per cubic centimeter ci is ionic charge a,- is ionic diameter i/j is the number of ions of type i per mole of electrolyte Species which lower the dielectric constant should be salted out by all electrolytes. This theory predicts the right order of magnitude of K, values. However, the predicted Ks values vary very little with the nature of the electrolyte (Masterton, 1970), and the theory fails entirely to predict a shift from salting-out to salting-in of a nonelectrolyte gas in different electrolyte solutions, such as 7 -Butyrolactone in NaBr and Nal solutions (Long, 1952). This theory was later refined by Long and McDevit (1952) to relate the sign of Ivs to the influence of salt on the solvent structure by equation (1-5): CO ______CO K _ V,- (Vs ~ Vs ) , . Ks “ 2.3RT /30 where Vs is the liquid molar volume of salt CO V,- is the partial molar volume of the non-electrolyte at infinite dilution CO Vs is the partial molar volume of the electrolyte at infinite dilution f30 is the compressibility of pure water In this theory, the Vs value is difficult to evalute, and predictions are usually of incorrect order of magnitude (Masterton, 1969). This theory gives relative values of K 4 for different salts with the same solute which fall in the correct order. However, the absolute value of Iv 5 calculated are in poor agreement with experiment (Masterton, 1970). Conway, Desbiyers, and Smith (1964) took into account the dielectric saturation effect. Each ion is assumed to have a primary hydration shell which contains n water molecules. These water molecules are so tied-up with the ion, therefore, that they lose their availability to further dissolve other solute molecules. Outside of this shell, water molecules remain effective and the dielectric constant is assumed to be that of the pure water. Good agreement was usually found between predictions and actual experiments for dilute electrolyte solutions, but not at all at higher concentrations of electrolyte, since negative solubilities are predicted (Tiepel, 1973). For solutions containing two salts, the salting effect can’t be explained by a single salting coefficient nor by a simple product of two coefficients. Though there were some mixing rules proposed by Gordon, and Thorne (1967), no theoretical basis to support them was given. In the sense that salting-out is an effect which increases the activity of the dissolved gas in solution, while salting-in decreases it, the salt effect results from the combined effect of all intermolecular forces upon the gas molecules. Presumably, the more detailed and accurate the consideration of these forces becomes, the more comprehensive the model can be. But the cost of additional detail is that there are more electrostatic parameters for which values are needed. This is the primary weakness of all the electrostatic dependent theories. 11 1-7 Scaled Particle Theory The scaled particle theory of Reiss, Frisch, Helpfand, and Lebowitz (1959, 1960) yields an approximation of the reversible work required to introduce a spherical particle of solute i into a solvent. Pierotti (1963,1965) considered the solution process to consist of two steps : (A) the creation of a cavity in the solvent of a suitable size to accommodate the solute molecule, and (B) the introduction into the cavity of a solute molecule which interacts with the solvent. Henry’s constant is given by : i„ + a ( i - 6) RT RT + RT 1 0j where Yx is the molar volume of the solvent, gc is the partial molar Gibbs energy for cavity $ formation, gj is the partial molar Gibbs energy for interaction, and H , 1 ^ is the H enry’s constant of solute 2 in pure solvent 1 at saturation pressure of component 1 at temperature T. The Gibbs energy, g c , for cavity formation is a known function of the temperature, the molar volume of the solvent, and the hard sphere diameters of the solvent and solute molecules respectively. This theory was later extended by Shoor and Gubbins (1969) to obtain an equation for the solubility of a non-electrolyte in an aqueous salt solution. The salting out effect was explained as due to changes in the cavity work term. Such changes arise primarily from nonpolar solute-ion interactions, and not from the ionic charges themselves. Masterton and Lee (1970) applied Shoor and Gubbins’ model to get a general expression for the salting- out coefficient in terms of the apparent molar volume of the salt, and the diameters and polarizabilities of cation, anion, and non-electrolyte gas applicable to any salt-nonelectrolyte gas pair. Scaled particle theory suffers from the fact that it is only tractable for a hard sphere fluid. The basic assumption that molecules possess hard cores sometimes leads to predicted heats of solution that are too high (Shoor, 1969), and for large gas solute molecules, salting-out coefficients calculated based on this theory are in poor agreement with experiment (Masterton, 1970). Schulze and Prausnitz (1981) applied scaled particle theory to correlate phase equilibrium data of aqueous systems over a wide range of temperature (0 - 300 C). Agreement with experimental data was achieved by allowing a slight temperature dependence of the mixing rule on the diameter of individual molecules. However, this work is limited to low pressure systems where Henry's law holds. Therefore, as we shall discuss in our parameter fitting procedure in chap 4, the parameters they obtained to fit their solubility data could predict unreasonable results for other thermodynamic properties. This may explain why the size parameter they obtained for carbon monoxide (0.395 nm) is much bigger than that for carbon dioxide (0.332 nm). Choi (1982) compared the correlation of Schulze and Prausnitz with his experimental data for both methane and nitrogen in pure water at high pressure and found very poor agreement. Scaled particle theory is an improvement over electrostatic theory in the sense that it can include the molecular forces into consideration. However, the treatment is not general enough and an even more detailed approach is needed. 1-8 Perturbation Theory It is a well known principle that when a physical problem can not be exactly represented mathematically, one can often make progress by a series of successive approximations. Perturbation theory is one way of doing this within the context of solution theory. In the last three decades, numerous efforts have been made to apply perturbation theory to the understanding of the liquid state. Perturbation theory for the Helmholtz free energy provides us a method of relating the thermodynamic properties of a real system to those of a somewhat ideal reference system through Taylor’s expansion of the partition function. Longuet-IIiggins (1951) was the first one to expand the free energy of a liquid mixture about that of an ideal solution. His method can only be applied to near-ideal solution. Later Zwanzig (1954) showed how to relate the properties of a high temperature non-polar gas with a Lennard-Jones potential (Figure 1-1) to those of a rigid sphere fluid (Figure 1-2). None of the above developments got much attention because it was assumed that the perturbed potential has to be very small to obtain meaningful results. The art of applying perturbation theory consists of choosing a reference system with two primary concerns : (a) it is simple enough to handle and (b) it contains as much as possible of the physically important parts of the real system. Unfortunately, desiderate (a) and (b) are generally incompatible. The first generally successful approach was that due to Barker and Henderson (1967 a). They recognized that the failures of previous attempts at low temperature of Smith and Alder (1959), Frisch et al. (1966), and Mcquarrie and Katz (1966) were due either to the lack of a satisfactory treatment of the softness of the repulsive forces, with consequent extreme sensitivity to the choice of hard sphere diameter, or to an unfortunate choice of separation into unperturbed and perturbing potentials. Barker and Henderson used a hard sphere system and chose the hard sphere diameter of the reference fluids in order to minimize the effect of the 14 Figure 1-1 Lennard- Jones potential U(r) Figure 1-2 Hard sphere potential U(r) <7 — 0 I r € \ — R cr-— Figure 1-3 Square well potential 15 repulsion. That is, they used the freedom of choosing the diameter to throw as much as possible the effects of the repulsion into the zeroth order expansion term. Barker and Henderson were able to obtain excellent, results even for pure liquids by changing the expansion procedures for both square well potential (Figure 1-3) ( Barker, et al. 1967 a) and Lennard- Jones potential (Barker, et al. 1967 b). Successful extension of their work to liquid mixture was made by Leonard, Henderson, and Barker (1970). Realizing the effect of the repulsive forces in determining the structure and thermodynamics of simple fluids. Weeks, Chandler, and Andersen (W-C-A) proposed a new first order perturbation approach in 1971. The primary difference between their approach and the Barker Henderson approach is the way in which the intermolecular potential is divided into an unperturbed and a perturbed part, as shown in Figure 1-4. Instead of separating the potential into positive and negative parts as in Barker-Henderson theory, Figure 1-5, they separated the potential into a part containing all the repulsive forces where < 0 , and a part containing all the attractive forces where > 0. The separation can be described as follow : U(r) = U°(r) + Up(r) (1-7) U°(r) = U(r) + c r < i'o ( 1-8 ) 0 r > ro (1-9) r < r0 ( 1- 10) U(r) r > r0 (1- 11) Where U(r) is the real potential; U° is the reference potential; UP is the perturbed potential; u(0 l € Figure 1-5 Figute 1-4 The split of potential according to Barker-Ilenderson The split of potential according to Weeks-Chandler-Andersen CTi and rQ is the location of the lowest potential energy. Use of the WCA reference system is considered more realistic than a hard sphere system (McQuarrie, 1973), and the perturbation series was found to converge more rapidly at high density (Verlet, et al. 1972). However, since there are no purely repulsive fluids in nature, the properties of the reference fluid must be determined from a theoretical calculation. This somewhat offsets the advantage it has over the B-H perturbation theory. Lee and Levesque (1973), has extended W-C-A theory to mixtures. Since the development of the B-H and W-C-A methods, much work has been done on perturbation theory of spherical and nonspherical molecules through the use of models based on a more sophiscated potential or a more suitable reference system. Gubbins (1973), Gubbins, Shing, and Street (1983), Gubbins and Twu (1978 a,b), and Mansoori and Haile (1983) have written general reviews on these recent developments. Most of the research on this subject has been at too microscopic a level to have much engineering application to our system, as we shall see in chapter 3 that the evaluation of Helmholtz free energy requires the knowledge of radial distribution function and reference system which are not usually available except for very simple system. The evaluation of chemical potential and hence the Henry’s law constant need even more work. As a result, use of perturbation theory to predict Henry’s constant have been limited to very simple systems which allow some simplifications to be made. Neff and McQuarrie (1973) derived a equation for Henry’s constant using the result of Leonard, Henderson, and Barker’s (1970) first, order perturbation theory. Good agreement was obtained for Neon in Argon. Uno et al. (1975) included the second order perturbation term and applied it to 22 systems of binary mixtures. They found that a set of ‘effective’ Kihara potential parameters must be found to perform the calculation with reasonable accuracy. Goldman (1977) applied the WCA perturbation theory to simple binary mixtures with success. All these works were concerned with simple binary fluids. For our system, containing polar water molecules and electrolytes as well as gaseous components, it seemed infeasible to consider including the orientation effect and other latest advancements in perturbation theory at this stage of perturbation theory development. It was hoped that these minute details can be included into a macroscopic adjustment. The work of Tiepel and Gubbins (1972 a,b) on the application of perturbation theory to the prediction of Henry’s law constants in electrolyte solutions appeared to be very promising for the development of a global macroscopic model for our specific system of interest. Tiepel and Gubbins (1973) also proposed a model based on the second order perturbation theory of Leonard, Henderson, and Barker (1970) for the thermodynamic properties of gases dissolved in an electrolyte solution. The final equations closely resemble those of scaled particle theory, but the hard sphere diameters are temperature dependent. In this approach, the Helmholtz free energy of a fluid is expressed as a Taylor series expansion about that of a fluid mixture of hard spheres of different diameters with respect to a and A, which measure the inverse steepness of the repulsive potential and the depth of the attraction part of the potential, respectively. The equations of perturbation theory can be made to yield those of scaled particle theory as a special case. A distinct advantage is that numerical values for most of the fundamental parameters needed are readily available from the literature or can be correlated from solubility experiments with the system. Although this model showed the most promise over the others, it had not yet been tested for complex systems of the type we were going to deal with. 1-9 Summary This research is aimed at the problem originally sponsored by DOE : predicting how much methane (and other gases) is dissolved at equilibrium, knowing the temperature and pressure of the system, the electrolyte content (salinity) of the gas-free aqueous phase, and the composition (relative mole ratios) of the dissolved gases in the geopressured reservoir. In approaching the solution of this problem, we outlined the original objectives of this research as follows : (a) to develop a model for predicting Henry’s law constant and its associated activity coefficient for each dissolved gas specie using perturbation theory to account for the various intermolecualr interactions in the liquid phase. (b) to incorporate all available experimental data at various conditions and use parameter- fitting techniques as needed to obtain a global model for predicting methane solubility. (c) to gain insight into the various effects of temperature, pressure, salinity, and other gas components on methane solubility. In chapter 1, we have introduced the purpose and relevant background of this research. Various aspects and approaches to this general type of problem were reviewed, including the Setchnow equation, electrostatic theory, scaled particle theory, and perturbation theory. Chapter 2 reviews specifically the sub-system consisting of pure methane gas dissolved in pure water or in brine solutions. Experimental data on this system are presented. Various previous correlations are discussed. The work of Coco and Johnson (1981), which was the precursor to this research, is discussed at length. In chapter 3, we address the problem often encountered in multi-component phase equilibrium, that is, the variation of Henry’s law constant and its associated activity coefficient with composition of the liquid phase. A new approach through a newly defined quantity, the apparent ideal solution fugacity, f^, is presented which enables us to bypass the difficulties of using Henry’s law- constant in multicomponent systems. An expression for {°' is developed through an application of second-order Barker-IIenderson perturbation theory. Expressions for partial molar volume of the solute and the isothermal compressibility of the partial molar volume are also developed to account for the variation of the apparent ideal solution fugacity with total pressure through the Krichevsky-Kasanovsky equation (1935). In chapter 4, we discuss in detail all the calculations involved in our model together with the computer algorithm used to solve simultaneously the thermodynamic equations. We discuss our reasoning for adjusting the Chapter 5 compares solubilities calculated from this study with experimental data and with correlations of previous investigations. The effect of temperature, pressure, salt, and other gases on methane solubility is discussed. Finally, in chapter 6 , we present some conclusions and recommendations that were developed from this research. Chapter 2 Previous Work on Correlation of Methane Solubility Data 2-1 Introduction The aqueous solubility of methane at high temperature and pressure is of interest for several reasons, the most important of which is that relatively large quantities of methane can be dissolved in the vast geopressured water reservoirs located in Louisiana and Texas (even though the methane content of the water, expressed as mole fraction or as SCF/Barrel of water, is quite small). A quantitative understanding of methane solubility is required as a first step for estimating both the in-place and the recoverable quantities of methane gas in geopressured reservoirs. 2-2 Experimental Work and Early Correlations A number of experimental investigations have been reported on methane solubility , as shown in Table 2-1. Until DOE became interested in geopressured reservoirs, most of the experimental work dealt only with methane in pure water. Little attention was devoted by most investigators to brine solutions. An extensive experimental study was performed recently by Blount et. al. (1982) under the sponsorship of DOE. It covered broad ranges of temperature, pressure, and salinity, and also included some experiments with gas mixtures containing ethane and C0-, . It is by far the most complete body of data collected on this system . 21 22 Table 2-1 Available published data on Methane solubility in water and water-NaCl solution A uthors No. of T P NaCl O ther (data points (C) (atm ) w t % gases C ulberson et al.(1951) 72 25-71 20-690 0 none O'Sullivan et al.(1970) 50 51-125 100-600 0-19 none Sultanov et al.( 1972) 71 150-360 50-1080 0 none N am iot et al.(1979) 14 50-350 295 0; 5.5 none Price et al.(1979) 71 154-354 35-1950 0 none B lount et al.(1982) 670 100-240 136-1530 0-26 none 129 25-71 7-136 5-15 none 27 149 340-1530 5-15 C0 2 26 150 678-1530 10 c 2h 6 Several correlations of methane solubility data existed before the research program initiated by Coco and Johnson in 1980 was undertaken. This research is the most recent phase of that program. The earliest correlation consisted of some experimentally-based curves of Culberson-Mcketta (1951), which were in general use for many years prior to 1976 for predicting the solubility of methane in pure water. During that time period, the effect of salt content on methane solubility was estimated from curves of Isokrari (1976), who used a salting-out type of correction factor proposed by Brill and Beggs (1975). An empirical polynomial fit of the original Culberson-McKetta. data was developed by Garg, et al. (1977), and an analytical expression for the salting-out correction factor as a function of salinity was given by Prichett. et al. (1979), which was based solely on the very limited data for salt solutions of O’Sullivan and Smith (1970). The salting-out coefficient was assumed to be invariant with temperature and pressure. 2-2-1 Correlation of Haas The first correlation based on more extensive data than the original Culberson- McKetta sources was a semi-empirical correlation for methane solubility in pure water proposed by Haas (1979). It was based on the pure water data of Culberson-Mcketta, Sultanov, et al. (1972), and Duffy, et al. (1961). The Haas correlation procedure involved subtracting the vapor pressure of pure water from the total pressure to estimate the partial pressure of the methane, Pqjj , in the vapor phase and then plotting the methane content of the water vs. Iii^Xqjj j Pqjj j to obtain straight lines at constant temperature. These plots will be straight lines only over the range in which (1) Henry’s law holds for the liquid phase. ( 2 ) in the vapor phase the fugacity coefficient of the methane is constant over the pressure range, and (3) the activity coefficient of the water is 1.0. The slopes and intercepts of these lines were fitted to polynomials in T (C). Haas proposed that, for water-NaCl solutions, a constant salting-out coefficient of 0.11 be used, based on the data of O’Sullivan and Smith. Haas’s correlation was a definite contribution for predicting solubility of methane in brine solution at high pressure. But the approximations and simplifying assumptions caused it not to be as globally accurate as one might desire, particularly at high pressure, moderate-to-high methane solubilities, and high electrolyte content. 2-2-2 Correlation of Blount Blount, et al. (1979) developed by linear regression a completely empirical polynomial equation to fit his own solubility data for methane dissolved in NaCl solutions. His equation involved many empirical coefficients for various polynomial-type terms consisting of products of the parameters T, P, and salinity raised to various powers. Unfortunately, methane solubilities predicted by this equation for pure water proved to be as much as 25% higher than the Culberson-Mcketta curves or those calculated from the correlation of Haas. After modifying his calculated solubilities to correct for an error in experimental liquid volume, Blount (1982) later published the following revised polynomials : A. In C1I 4 = - 1.4053 - 0.002332 t + 6.30 xlO "6 t 2 - 0.004038 S - 7 .5 7 9 xlO ' 6 P + 0.5013 InP + 3.235 xlO ' 4 t InP (2-1) Standard error of regression = 0.0706 Multiple R = 0.9944 B. In C II 4 = - 3.3544 - 0.002277 t + 6.278x10'® t 2 - 0.004042 S + 0.9904 InP - 0.0311 (lnP)2+ 3.204xl0'4 t InP (2-2) Standard error of regressions = 0.0709 Multiple R = 0.9943 W here C H 4 is in standard cubic feet (scf) of dissolved methane per petroleum barrel (42 gallons) at 25 C and one atmosphere; t is temperatures in degree Fahrenheit; S is salinity in grams per liter; and P is pressure in psi. The two different correlation equations reported by Blount et al. contain some terms that differ, but the calculated values of methane solubility are virtually identical over the recommended range of applicability, as suggested by the standard 26 error and multiple regression coefficient of each equation. Blount's revised correlations are claimed to be valid only for the temperature range from 160 to 464 F and the pressure range from 3500 to 22500 psi. They should not be used for pressures and temperatures outside these ranges, as the polynomials behave erratically when extrapolated. Also, his correlations are based mainly on the experimental data he obtained for brine solutions. He did not include any methane/pure-water experimental data in his analyses so that prediction of zero molality conditions is actually an extrapolation of his brine data outside of the range in which the parameters were fit. Therefore, these correlations show very large root mean square deviations ( about 0.4 ) when compared with the experimental data of Sultanov (1972). Culberson-McKetta (1951), Price (1979), and O’Sullivan and Smith (1970) for methane in pure water, as shown in Table 2-2. All of the previous attempts at correlating methane solubility data described above were either graphical, empirical or based on only the simplest of thermodynamic relationships for phase equilibria. One exception is the work of Larsen and Prausnitz (1984) which also was sponsored by DOE. They developed an equation of state for the methane-water system over a wide range of temperature and pressure based on experimental residual thermodynamic properties using an extended form of corresponding state theory with a molecular shape factor. Their procedure included adjusting a set of binary parameters to fit the experimental water/methane data at each temperature but without much physical meaning attached to them. Although good agreement with experimental data were reported for methane in pure water, no numerical figures were given for us to compare with other correlations. Also, their work were limited to methane in pure water. The effect of salt and other gases on methane solubility were not included in their study. Table 2-2 A comparison of Solubilities of Methane in Water Calculated by Haas, Blount, and Coco-Johnson’s Correlations with Published Data Data Set No. of T Pressure R. M. S. D*. points (F) (psia) Haas Blount Coco S ultanov 10 302 715-15650 .0325 0 .1 0 0 0 0.0628 (1972) 11 392 711-15650 .0777 0.0826 0.0591 11 482 1422-15650 .0463 0.2050 0.0663 10 572 2134-15650 .0540 0.4160 0.0306 9 662 2845-15650 .0364 1.3400 0.0715 9 626 2845-25650 .0717 0.6470 0.0600 6 680 3556-11379 .0714 0.7790 0.1610 to tal 66 .0560 0.6270 0.0748 Culberson 12 77 341-9300 .0843 0.0 0 0 0 0.1930 et al 12 100 330-9895 .2100 0.0000 0.0570 (1951) 12 160 331-9865 .0227 0 .0 0 0 0 0.0249 12 2 2 0 333-8190 .0257 0.1240 0.0259 12 280 336-9835 .0329 0.1300 0.0436 12 340 323-9995 .0276 0.1430 0.0413 to tal 72 .0419 0.1330 0.0870 Price 8 309 2204-23778 .1070 0.0958 0.0884 (1979) 7 403 2323-27908 .0525 0.0792 0.0530 6 430 5332-20530 .0548 0.0620 0.0554 12 453 2160-23837 .0814 0.1190 0.0693 9 536 2866-27393 .2080 0.1810 0.1070 7 558 1567-24498 .2180 0.2650 0.1040 7 601 3631-27746 .2750 0.1520 0.0970 to tal 56 .1610 0.1470 0.0848 O ’sullivan 6 125 1470-8818 .0251 0.1300 0.0151 and Sm ith 6 217 1484-8876 .0362 0 .1 1 0 0 0.0164 (1970) 6 257 1514-8935 .1140 0.2370 0.0953 to tal 18 .0706 0.1680 0.0565 N am iot (1979) 7 122-662 4595 .0465 0.2860 0.0310 T O T A L FO R ALL DATA 219 77-6S0 711-27908 .0930 0.4140 0.0794 * RMSD is defined in equation 4-32 28 2-2-3 Correlation of Coco and Johnson In recent years, computers have provided us with the capability to treat multicomponent vapor-liquid equilibria of systems with more fundamental, albeit more complicated, equations. (The UNIFAC and UNIQUAC approaches to vapor-liquid equilibria calculations for non-electrolytic systems can be applied successfully only as computer programs.) It would appear that the available data for this system should also be amenable to an improved analysis based on fundamental thermodynamic relationships. This was the starting thesis for Coco and Johnson's (1981) work, and the correlation that was developed by them for the methane-water-NaCl system proved to be superior to the other correlations that existed at that time. Coco and Johnson (1981) undertook a sub-task to develop a computer subprogram to predict the equilibrium methane content of an underground geopressured reservoir, given an estimate of its temperature, pressure and salinity. A brief summary of the development of the equations used by Coco and Johnson for predicting methane solubility in brine solutions follows. The fundamental relationships defining vqpor-liquid equilibrium conditions are well known. For the binary system, water (1) and methane (2), they are : rp/y (2-3) (2-4) (2-5) 29 f£ = 4 (2 -6 ) For methane dissolved in pure water, the fugacities in equations (2-5), and (2-6) may be replaced by their equivalent thermodynamic expressions to give: ^ yiP = (2-7) t>2 y 2 P = x 2 H 2 x (2-8) Where y and x are the mole fraction in vapor and liquid phases respectively. In the Coco and Johnson’s subprogram SOLUTE, the value of ^f, the fugacity coefficient of water vapor at saturation pressure was calculated from an equation of state for pure water given by Keenan, et al. (1969). Values for (1976) for hydrocarbon systems containing water vapor. The Poynting correction factor, i was calculated from a fundamental thermodynamic equation (Prausnitz, 1969), using an equation for the liquid molar volume for water as a function of temperature by Yaws (1977). Of course, Henry’s law constant, Ho i , in equation (2-8) was not a priori calculable for methane in water. In the Coco and Johnson’s approach, all the available published experimental data were used to obtain a correlation of Henry’s law constant with temperature, pressure, and methane content of the liquid phase. The effect of pressure on Henry’s law constant at infinite dilution has been shown to be (Prausnitz, 1969) : d ln( H 2 1)t _ V 2_ (e) 8V ~ R T 1 ' 30 where V 2 is the partial molar volume of the gas (methane) in the solvent at infinite dilution. Integration of equation (2-9) between the limits of zero and P gives: P C O ( ® 2,1 )t , P = ( ^2,1 )t , o exP ^ (2-10) 0 where ( H, j )-p G is the value of Henry’s law constant at zero pressure. It is reasonable to C O consider that V 2 itself varies linearly with pressure over the range of interest. CO V 2 = b 2 ( 1 — 2C,P ) (2-11) where C, is recognized as the compressibility of the partial molar volume, which is a positive quantity. Substitution (2-11) into (2-10) and integrating gives: ^n( ^ 2,1 )t, P ~ ( 3 2)t + p^( ^ — C2P ) P (2-12) The term (a 2)q- is the Henry’s constant for methane at zero pressure and T. The effect of temperature on Henry’s law constant is given by (Prausnitz, 1969) : <91n ( H2 x )t p _ Aho (9 T -1 ~ R } where A h 2 is the partial molar enthalpy change of solution of the methane in water at infinite dilution. This relationships is not quite as useful as equation (2-9) for the effect of pressure, because less can be said in general about the behavior of A h, . ( It may pass through zero, and may have a positive or a negative slope when plotted vs temperature.) If a second order relationship with temperature for Ah 2 is assumed over the range of interest, the following equation results from correlating the data for a binary mixture with both 31 temperature and pressure. (In this equation, the effect of pressure on parameters ax, a 2 and a 3 or of temperature on parameters b 2 and c 2 is considered negligible.) (ln( II °2A )Tt p = aQ + I - i ( T - T0) + (1 - C 2P ) P (2-14) 6 To account for the effect on Henry’s law constant of the methane content of the liquid , the two suffix Margules equation should approximates the change in the activity coefficient of the water quite well for dilute solutions. ln 7 l = lXT X2 (2-15) The activity coefficient of the methane is then found from the Gibbs-Duhem equation as (Prausnitz, 1969) : ln *>2 = W f { X2 1 ) (2-16) It is important to realize that equation (2-14) is valid only for infinite dilute conditions of methane in pure water. Equation (2-16) was added to (2-14) to account for the methane content effect for the range of solubility encountered in actual solutions of methane and water. Based on these relationships, the equation Coco and Johnson developed to correlate all the available data of Henry’s law constant for methane in pure water was: In( H )Ti p = aQ + |I( I - J- ) + jg ( T - T0) +ITT (1 + cP)p + WT{ x2 “ 1 ) <2-17) 32 where aQ = 10.407 ax = — 6814.8 cal/ gmole a 2 = — 0.0533 cal/ gmole-K 2 b = 62.33 + 0.007338 (T-T 0 ) c c / gmole c = — 9.149 xlO ' 5 a tm ' 1 d = 22.73 - 549.8 ln(T/T 0 ) cal/ gmole-K Rc = 1.987 cal/ gmole-K Rg = 82.05 cc-atm/ gmole-K When salt is added to the binary CII 4-H20 solution, the Henry’s law constant of CH 4 will change. Had a thermodynamic relationship for —— existed, it would have been easv to salt extend equation (2-17) to brine solutions. Unfortunately, no such fundamental relation has been found. The only available one is the familiar but semi-empirical Setschenow equation (1-1). This equation can be written in another form as: H2, m = HS l e ksmi (2-18 ) The e ^,nis term is a correction factor for the effect of salt content, with Ka, the salting-out coefficient, the quantitative measure of this effect. This equation can satisfy the boundary conditions as : ms > 0 H2) m > H£ x (2-19a) m s ------> oo x 2 > 0 (2-19b) Coco and Johnson used equation (2-18) in their analysis. An empirical relationship for the effect of temperature on K, based on available experimentally determined values of Ks was used for the temperature range between 323 K and 623 K Ks = K 0 + Kj (T - T 0 ) + K, ( T - T 0 ) 2 (2-20) The parameters and their 95% confidence limits were determined K0 = 0.08 ± 0.00973 cc/gmole Kj = 0.0002751 ± 0.0000653 cc/gmole-K Iv2 = 4.39 xlO "6 ± 1.59 xlO "6 cc/gmole-K 2 T 0 = 455.65 K (arbitrary reference temperature) The final equation for correlating Henry’s law constant in brine solution was then obtained by combining equation (2-17), (2-18), and (2-20) : ln II2) m = aG + |T( 1 _ _L ) q. _ 2 (x _ x o) + j^ 7p (1 + cP)P + j^ ( x| — 1 ) c o c g + 2.303ms (kg + Kx (T - T 0 ) + K 2 ( T- T0)2) (2-21) The work of Coco and Johnson marks the first attempt to correlate the available data through the application of known fundamental thermodynamic relationships without making simplifying assumptions. It resulted in a lower RMSD value than either Haas’ correlation and Blount’s polynomial when applied over all the available data on methane solubility in pure water, as shown in Table 2-2. As shown in equation (2-21), the dependence of Henry’s law constant upon temperature, pressure, and salt content of the liquid phase is expressed in terms of a number of physically meaningful parameters. For example, in equaton (2-21): 34 aj is the partial molar enthalpy of methane in water at T b is the partial molar volume of methane c is the com pressibility d is the Margule’s coefficient, for the activity coefficient of methane in water Although equation (2-21) was by far the most effective correlation yet available, it still had its limitations. First of all, it does not account for a system containing other dissolved gases such as C 0 2 and C 2H 6 , both of which are present in geopressured brine. To do so would involve determination of additional similar parameters for each gas from experimental data, and then allowing somehow for any interactive effects of the gases in solution upon each other. Secondly, some of the parameter values which resulted from correlating Henry’s Law constant are subject to criticism. For example, the partial molar volume of methane in water was correlated to be 61.18 cm 3/g-mole at 298 Iv, while the independently obtained experimental value is about 37.0 (Ivrichevsky, 1945). Thirdly, the validity of the Setchenow equation had been assumed, which is not necessarily true for concentrated salt solutions nor is there any fundamental basis for extending it to a solution of mixed salts. One possible extension of equation (2-18) to include correction factors for other components would give: H 2, m - II2 j e klsmis e k2sl" 2s ...... e kn*mnj (2-22 ) The primary assumption behind this is the additivity of the effects of various salts and other- dissolved gases on the activity coefficient of the dissolved methane. In other words, no interactions exist among the various component species themselves that would affect their individual component interactions with methane. This clearly may not be the case for our system of interest. An even more fundamental approach was needed to improve the theoretical basis of the correlation before it could be extended to include other salts and other gases. Chapter 3 Development of Governing Equations of Phase Equilibria for A Complicated System of Gases, Electrolytic Salts, And W ater In this chapter, we develop the working equations we used for calculating phase equilibria for our system. Starting with a general definition of the system under consideration in section 3-1, we next discuss equilibrium relationships in section 3-2 using traditional liquid- phase activity coefficients and reference states. Because of the difficulties that exist in applying Henry's constant and its associated activity coefficient for multicomponent systems, we then propose a new approach, involving a new thermodynamic property, f°', that we have chosen to call the apparent ideal solution fugacity of pure component i. In section 3-3, after a brief review of statistical mechanics, we show how the chemical potential of a component in a real liquid solution can be calculated through perturbation theory. We then proceed to relate this chemical potential to our newly-defined property, { f , where the fugacity f^ of a gas component in the liquid phase is equal to f^x,-. After defining the potential energy relationship for the components in our system, an expression for the apparent ideal solution fugacity, f?*, of a real gas component i at. the saturation pressure of the solvent is obtained for our specific system. Sections 3-4 and 3-5 present analytical results we obtained from perturbation theory for determining the partial molar volume of a gas component and the isothermal compressibility of the partial molar volume, respectively. These two formulae are then used to adjust the expression for apparent ideal solution fugacity, i f , from the solvent saturation pressure up to the high pressure of the system in section 3-6. Finally, in section 3-7, we discuss the method that was used to evaluate whether partial dissociation of the C 0 2 molecule in the liquid phase is important in our model. 3 5 36 3-1 System Description This research was aimed at developing a more fundamental, thermodynamic-oriented global model for predicting the methane content of geopressured brine reservoirs than that of any previous research. After existing millions of years buried underground far beneath the surface of the earth, the geopressured brine is considered to contain dissolved methane and other gases in equilibrium with its corresponding gas phase at the moderate temperature and very high pressure conditions of the reservoir. This equilibrium state is considered to be true for all volatile components in the system, such as I1 20 , C 0 2, and C 2H6. A schematic representation of the equilibrium system is shown in Figure 3-1. The non-volatile strong electrolytes present in the brine, such as NaCl and CaCl2, are assumed to dissociate completely into their ionic forms. The possibility of chemical reaction between water and C 0 2 to produce IICO 3 and ions is also considered. To fix the system description, we identify all the possible species in the system by subscripts 1 through 18. This notation is used throughout this study and also in the computer program. As shown in Table 3-1, some of the subscript numbers are reserved for future consideration of species other than those studied so far in this research. To define a phase equilibrium problem that can be solved, several specifications besides system temperature and pressure must be made. The specification set used in this research is : The molality for each strong electrolyte is specified in the liquid phase on a gas-free basis, i.e., before any gases are dissolved, and the ratio of moles of each dissolved gas other than methane, such as C 2I i 6 or C 02, to the moles of CII 4 dissolved is specified. Other specification sets could be chosen, but these proposed specifications are equivalent to the problem originally sponsored by DOE, i.e., for known (or expected ) ratios of CH 4 to C 0 2 a n d /o r C 2H 6 in the dissolved gas phase, and known (or expected ) gas-free brine composition, how much dissolved -- T Vapor Phase C,H« G02 4- H20 ?s H + -f HC03” O r Liquid Phase Figure 3-1 Schematic representation of the system under consideration 38 .Table 3-1 Classification of subscript number used in this study and computer program. Subscript Component Classification 1 H 20 Solvent 2 Solvent H 20 ) 3 c h 4 Non dissociate 4 CoH 6 gases 5 other gas •• 6 CO, Dissociate gases 7 other gas 8 N a+ 9 C a + 2 Strong 10 other cation 11 c i - Electrolyte 12 I- 13 otlier anion 14 H+ 15 IICO t W eak 16 other weak 17 electrolyte Electrolyte 18 CII4 can be expected per barrel of brine at the known or expected temperature and pressure of the reservoir. 3-2 Solubility of Gases in Liquids In this section, we give a brief review of the equations needed in dealing with solubility of gases in an aqueous solution and then propose that the expression for fugacity of a solute gas i in the liquid phase can be improved by defining a new thermodynamic property, {°'. T he term ‘solubility of a gas’ generally refers to the liquid phase concentration of a component that would not exist as a pure liquid at the system temperature T and pressure P, either because its vapor pressure is much higher than P or its critical temperature is less than T. The thermodynamic condition necessary for each component i in a system containing two phases, vapor and liquid, at equilibrium can be stated as : the escaping tendency of component i from both phases must be equal. This condition, by definition, is expressed mathematically as: the chemical potentials and, therefore, the fugacities of component i in both phases are equal at equilibrium: ff( T, P , Y ) = ff ( T , P, X ) (3-1) where Y is the vector set of vapor mole fractions and X is that of liquid mole fractions. The fugacity of component i in the gaseous phase can be expressed by : (3-2) where traditional approach is to define this fugacity in terms of a measurable quantity as reference state and adjust it to system conditions through the activity coefficient as in equation (3-3): ff = 7i xf 1? (3-3) where f? is the reference state fugacity chosen for component i. It is important to realize that the activity coefficient is meaningless until the reference state is specified. The conventional reference state for a solvent component is its pure liquid state at the temperature and pressure of the system. Therefore, 7t ------> 1 as x ,------> 1 (3-4) This is known as the normalization of the activity coefficient according to the Lewis-Randall (Prausnitz, 1969) rule. The fugacity of the pure liquid reference state can be expressed as : p V f? = # P* exp L d P (3-5) RT where P* is the saturation vapor pressure of pure liquid i at temperature T; For a solute gas component, whose critical temperature is usually lower than the system temperature, equation 3-5 bears no physical meaning because the solute gas does not exist as pure liquid at the temperature of the system. Thus one cannot know the vapor pressure upon which the pure liquid is based as in equation (3-5). A linear extrapolation on a semi-logarithmic plot of saturation pressure versus reciprocal absolute temperature as shown in Figure 3-2 (Prausnitz, 1969) has been used to define a hypothetical pure liquid vapor pressure but this is not generally accepted, especially when system temperature is well above the critical temperature of the solute gas. An alternative way to deal with this problem of the reference state of a solute gas is through the concept of Henry’s law. Originated through experimental observations at low pressure, Henry’s law states : ff = H° x,- when x, ------> 0 (3-6) Where 11° is the Henry’s constant for component i in this system. Since H° is an experimentally accessible quantity, it is suitable to define it as a reference state upon which the activity coefficient is based to adjust to the condition of a real system. But we need to realize that Henry’s law and a reference state based on Henry’s constant are two separate and distinct concepts (Van Ness and Abbott, 1979). For a binary system, a reference state based on the Henry’s constant gives : f? = li H° (3-7 a) where > 1 as x ;------> 0 (3-7 b) Critical Point iA Figure 3-2 Extrapolation of liquid saturation pressure into hypothetical liquid region Since the solvent (equation 3-4) and solute (equation 3-7) are not normalized in the same way, this approach is usually referred to as the unsymmetric convention for normalization. However, for a multicomponent system, care must be taken to specify exactly the condition that gives */,■ —> 1. This is shown in Figure 3-3 (Van Ness and Abbott, 1979) for the case of a ternary system. Let us assume that component 3 is supercritical and, therefore, Henry’s constant must be used for reference state. In plane 3-1, H 3>1 is the Henry’s constant for binary system 3-1. So is II 3 2 for the binary system 3-2. However, for an intermediate solvent composition, the vertical plane representing a constant ratio of X 1/X 2 intersects the X3=l axis at lnH3m. It is easy to see that different ratios of X j/X 2 will result in different values of lnH 3 m in general. Hence, there is no single value of H 3 m to characterize a unique reference state for component 3 in a ternary system. Instead, there are an infinite number of reference states for component 3, depending upon the composition of the component 3-free mixture in which component 3 is dissolved. Thus, both the reference state fugacity H3)m, and the activity coefficient associated with it are dependent upon the composition of the solution. A common practice for treating multicomponent mixtures of solute gases dissolved in one pure solvent is to use for each solute gas the individual binary Henry’s constant at infinite dilution conditions and to correct the variation of the activity coefficient of each gas component with the solution composition through Two-Suffix Margules or Wilson’s equations, as in the work of Williams (1987). However, as pointed out by Williams, the number of independent parameters to be determined in Wilson’s equation could reach as many as eight for a simple ternary system. This treatment would become even more complex as the number of gas components increases and it becomes essentially infeasible to use this idea of infinite dilution reference states for the multicomponent mixture of gases and salts of this study. 44 In Hi* 3,2 In fi 1 Figure 3-3 Variation of Henry’s law constant with composition in a ternary system 45 In tliis research, we applied a new approach to treat gas solubility in a liquid. Our treatment is based on an idea proposed by Reed and Gubbins (1973), for calculating Henry’s law constant of a solute gas from perturbation theory, but not many people are yet aware of its full significance for handling multicomponent gas-liquid equilibrium problems. Through the derivation presented in section 3-3-3, we shall see that a new quantity, f?,) which is actually the product of a Henry’s constant and its corresponding activity coefficient for a gas component in the system, can be evaluated directly for each solute gas using perturbation theory. We refer to this quantity, f^, as the apparent ideal solution fugacity of pure component i. Our reasoning for introducing this new symbol for a thermodynamic property of a component is as follows. For a multicomponent system, the use of a Henry’s law constant (which is a chosen reference state fugacity of gas component i) multiplied by its associated activity coefficient, which varies with composition, affords no real conceptual advantage. It did provide a conceptual advantage for binary mixtures because the Henry’s law constant was unique for each binary system and composition-independent, with the composition dependence of the fugacity of i clearly accounted for by the activity coefficient alone. This advantage is completely lost for multi-component mixtures simply because the Henry’s law constant for each gas component i must also be a function of the composition of the component i-free liquid phase, so there is no longer an isolation of the composition dependence in only one parameter, the activity coefficient. Furthermore, the activity coefficient of each solute gas depends on the concentration of all the dissolved gases. We propose that it is therefore more useful to replace the combined product of the two parameters with only one composition-dependent symbol (and property), which can be thought of conceptually as the apparent ideal solution fugacity of component i at the system conditions because it needs only to be multiplied by x,-, the mole fraction of component i, to give the fugacity of component i in the mixture. For truly ideal solutions this quantity would simply be f?, the pure component reference state fugacity for each component in the solution. For non-ideal solutions, the proposed new thermodynamic property i f , which is indeed mixture composition dependent, represents the apparent ideal solution fugacity of component i at the temperature and pressure of the system, and, of course, it is a function of the composition of the liquid mixture, including component i. As we will see later, there is an important advantage of replacing the product of Henry’s law constant and its associated activity coefficient with f?1 : it is that f^ can be calculated from perturbation theory by the methods derived in this chapter. It should be noted that since i{ = x f f° , we could have used the notation jA to represent the apparent ideal solution fugacity of pure component i, but we chose to emphasize that this quantity can be considered to be a thermodynamic property of the solute gas in the liquid mixture by referring to it by the single symbol i f . This quantity is a measure of the effect of all the species in the solution on the fugacity of solute gas i. Accordingly, using the new quanty i f , for each gas component, the phase equilibrium relation for a system consisting of H 20 , C II4. C 2H6, and C 0 2 can thus be written as: for w ater, V c — dP (3-8) >’i p = Ti *i 4>i exp R T aJ for C IL : ^3 y3 P — x3 ( 5^) — x3 ( P, (3-9) for C 2H 6 : ^ 6 -v6 ^ = x 6 ( 5 ^ ) “ x 6 ( * 6 ,)t, P, x (3-H) where P x the apparent ideal solution fugacity of a gas component i in the mixture at T and P. The left hand side of these equations can be calculated from a suitable equation of state. But the apparent ideal solution fugacities f | , 1 f ^ and fg* are not predictable via the equation of state approach and have to be evaluated through some solution model. As we have already mentioned in chapter 1, perturbation theory provides such a way in the context of solution theory. 48 3-3 Perturbation Approach 3-3-1 Introduction to Statistical Mechanics Classical thermodynamics supplies the relationships between many macroscopic properties of a system but does not generate information concerning the magnitude of any one. Statistical mechanics, on the other hand, is the science of drawing macroscopic conclusions from microscopic hypotheses about systems. Based on knowledge of the nature of the particles that constitute a system and on quantitative predictions of the interactions among them, statistical mechanics provides a way of computing the macroscopic thermodynamic properties through the bridge of the canonical partition function Q : Q = Qint(T, N) • Qc(N, V,T) (3-12) Where Qjnt is the part of partition function due to vibrational, rotational, and electronic contributions which is independent of density and is therefore the same for a real fluid or solid as for an ideal gas. We will concentrate on the configurational partition function, Qc, which is dependent on density and is affected by intermolecular forces. Qc is defined as : Qc (3-13) 1/2 Where A = h /(27rmKT ) , h is the Planck’s constant, K is the Boltzmann constant, N is the number of molecules, and Zc is the configurational integral defined as : Zc = / ...... / exp [ - U(r) /KT ] dr (3-14) Where U(r) is the total intermolecular potential energy. The key to evaluating 49 thermodynamics properties of a real fluid through statistical mechanics is to relate the configurational partition function, Qc, to the configurational Helmholz free energy AC(N, V, T) which has the same set of independent variables N, V, and T as in the case of canonical ensemble. Among all of the thermodynamic functions, the configurational Helmholz free energy is directly proportional to the logarithrm of the configurational partition function. K T ln Qc (3-15) and all thermodynamic preperties can be derived starting with its total differential form. Therefore, from dA = — SdT - PdV + £ //; dNj (3-16) For example, we can get dA = k t 9 y (3-17) <9V T , N= dv T, N: and 8A K T d InQ Vi (3-18) d Nf T, V, N. 0N,. T , V, N. J O ^ i) J 0 # 1) 50 3-3-2 Perturbation Approach to Obtain Chemical Potential The total potential energy U(r) appearing in the Zc equation is usually assumed to be pairwise additive as a convenient approximation since little is known about three-, four- or higher-body forces (Prausnitz, 1969): U(r) = EE V;. (r) (3-19) i < j J where V— (r) is the intermolecular potential between molecule i and molecule j . The perturbation approach relates the properties of the real system to those of a reference system. In order to do so, Zwanzig (1954) introduced a perturbation paremeter, A, to relate the intermolecular potential of a real fluid to that of a reference fluid as : Vij ( r- A ) = (r) + A Vf, (r) (3-20) Where V° is the reference potential and Vp is the perturbed potential, when A = 1, equation 3-20 gives the potential of the real system; A=0 gives the potential of the reference system. By substituting this new formula into the configurational integral, Zc, we get : Zc = / ...../ exp { £, E ( V°, (r) + A V?,. (r))} d(r) (3-21) From equation 3-13 and 3-15, Ac is directly propotional to lnZc -3N Ac = - KT In ( Zc ) (3-22) 51 A Taylor's series expansion of lnZc about the reference system through A will in turn relate Ac to the Helmholtz free energy of the reference system. Therefore, 8 InZ, 1 2 d2 lnZ c ln Zc = ln Z? + A + (3-23) OX + 2 — Q ^ ~ and correspondingly, A c = A? + A , 1 \2 d Ac + (3-24) 3X + 2 5A2 A = c A — o Where Ac is the configurational Helmholtz free energy of the reference system. This briefly explains how the perturbation is performed through a parameter as in equation 3-20. A common choice for the reference system is a system of hard spheres, with potential defined as: V °. u oo r < d = 0 r > d (3-25) where d is the hard sphere diameter. The hard sphere reference has the advantages of its simplicity and close resemblance to the real system. Many properties of real fluids are similar to those of hard sphere fluids (Reiss, 1965). The properties of hard sphere fluids are now accurately known both from theory (Boublik, 1970) and from computer simulations (Barker and Henderson, 1971). The main disadvantage of using a hard sphere reference is that the resulting calculations are very sensitive to the value chosen for the hard sphere diameter. Also, the temperature dependence of properties is poorly predicted. This can be shown from the fact that the derivative of the configurational integral, Zc, with respect to temperature is zero for a hard sphere fluid. A way of overcoming this difficulty was first suggested by Barker and Henderson (1967 a). By combining the techniques of Zwanzig for treating the attractive potential and of Rowlinson (1964 a,b) for dealing with the softness of the repulsive potential, they defined a modified potential function W( r, cr, d, A, a) by the relations : - V(d + ^ ) r < or ( W ( r, a, d, A, a ) = r > a (cr — d) + d L AV(r) r > a (3-26) Where V is the potential of the actual fluid, cr is customarily taken to be that point at which the potential V(r) passes through zero and d is an arbitrary length parameter to be specified later. There are two perturbation parameters, A and a, in this equation. The parameter a varies the steepness of the modified potential in the repulsive region. The parameter A varies the depth of the potential in the attractive region. This modified potential is written so that when A= a — 0, the potential W becomes that for hard sphere of diameter d. When A= a= 1, W becomes the actual fluid V(r). With these two perturbation parameters, a double series expansion can be performed as in equation 3-23. The configurational integral can be written as: d lnZ , ln Zc = ln Z? + A 9 lnZ " + a d \ Ax — O da a = o + i{ [ d~ lnZ c d 2 lnZc + 2 Q A . ^ i ^ + a" (3-27) dXda d \ 2 a = A = o d a 2 a — o } 53 The corresponding configurational Helmholtz free energy can be written as : d A , , d A c Ac = A? + X + a - ^ + + (3-28) dX da 2 dX 2 X = o A = < The differentiations included in equation 3-28 involve lengthy algebra. The final results obtained by Barker and Henderson (1967) for single component system is as follow : Ac = A ° + 2 a 7rN K T p d 2g°(d) | d - ( 1 - exp( -/?V(r))) dr | COO + 2A7rpN g° (r) Vp(r) r 2 dr + higher order terms (3-29) The value of d is chosen so that the first order term in a expansion vanishes. Therefore, d = ( 1 - exp( —/?V(r))) dr (3-30) Since 3 = 1/KT, d is a temperature-dependent effective hard sphere diameter. Barker and Henderson (1967) argued that with this choice of d, the second order terms in aA and a 2 are considerably smaller than the A 2 term. Leonard et al. (1970) had extended Barker Henderson’s theory of a single component fluid to a liquid mixture using a mixture of hard spheres as the reference fluid. The first order terms they obtained are : oo dAc = 2 Trp-y E E Xj x. v?.(r) g°. r 2 dr (3-31) d X = „ i j J J (T dA, = - 2tt/>2VKT E E Xj X. dfj (r) g ° (d„) ( d „ - 6V} ) (3-32) da *i j i « 54 In the above equations, the hard sphere diameters are given by: dj, = SVl (3-33) d ij = \ ( d ii + djj ) (3-34) and r'j ( 1 — exp( —/3V°j(r))) dr (3-35) Equation 3-32 becomes zero for single component since di8- = 6it-. But the additive assumption of equation 3-34 can not annul all the first order term in a. However, this contribution is usually small since the difference between d and 6 is very small for mixtures of molecules of similar sizes. But this term becomes significant for widely varying sizes. The second order d2A term— has also been evaluated (Tiepel, 1971), and involves mixture distribution functions 0 A of order 2, 3, and 4. The main difficulty in the use of equation 3-31 is the calculation of the reference mixture distribution functions, gfj(r). Tiepel and Gubbin (1972 a, 1973) assumed that g°j(r) = 1 for r > tr; ■ and is zero for r < a {j. A similar approximation is made for the third and fourth order distribution function. The second order term can thus be approximated as : COO d2 A, = 5 £ Xi x. ' ffij ( V fj(r ) ) 2 r 2 dr (3-36) dX A = o By taking the derivative of the Helmholtz free energy with respect to Nj, as in equation 3-18, we can obtain an expression for chemical potential of component i as : 55 ,.hs , _d_ dAi + d dA c i + 0N , dX 3N ,• da \ — 0 a = o 1 8 d 2 A, + (3-37) 2 0N* d A2 A = c w here d dAc = - 4irKT £ Pj d?- (3-38) 3N,- da Ck — o _d_ dAc f ° ° = 4 tt £ P : (3-39) 3N,- 5 A a .. Vij(r) r“ dr A = o and ‘X) d d2 A, cr. - ( V fj(r ) ) 2 r 2 dr (3-40) 0 N,- 3 A2 f t ? 'j >3 The chemical potential of a component in a hard sphere mixture /ijls was obtained by Tiepel (1971) as : hs l _ i P i ln M i l ( i - h r Z3 2 (1 — £ 3) 2 _ ( f|i)3 _ {3, + (3-41) 56 The first term on the right hand side of equation 3-41 is the chemical potential of an ideal gas from statistical mechanics. The remaining terms were obtained from the equation of state for hard spheres by Carnahan and Starling (1969) for pure fluids, and extended to mixtures by Boublik (1970) as: phs 6KT ( £o I I 3^2 ^3 £2 \ /o s - — ( rq; + rrrrF + ~ ) <3-42) where in = | P-} dj1 (3-43) 57 3-3-3 Relation Between Apparent Ideal Solution Fugacity And Chemical Potential The apparent ideal solution fugacity, f°', for component i in a liquid mixture of known composition can be derived from the chemical potential of component i as follows: For a gas component dissolved in a liquid, the equilibrium condition is that the chemical potential of solute gas i in the vapor phase equals that in the liquid phase. = /4 (3-44) The chemical potential of a component in a real gas is related to its fugacity f,- through the definition d ( Hi ) = I\T d ( In fj ) (3-45) integration from a reference low pressure P*of an ideal gas to P, gives: M T , P ) - ( T, P* ) = KT ln £ (3-46) r i where P, is the partial pressure of component i, and is the chemical potential of component i in an ideal gas. The chemical potential of component i in an ideal gas can be calculated from statistical mechanics equation 3-12 and 3-18 with no intermolecular interaction. Therefore, ZC = V N (3-47) 58 and with the ideal gas equation of state PV=NKT and the Stirling’s Approximation ln N! = N InN — N, we can get p|d = - KT ln( A : 3 V qjnt ) = KT l n - ^ + KT ln Pi (3-48) i K iq , Where ( q-nt )N = Qjnt . Substitute equation 3-48 into 3-46 gives: pf = K Tlnr^j^ + KT ln i, (3-49) K Tq'd Equating equations 3-49 and 3-37 at equilibrium, gives : K T ln— + KT ln f = p!is + JL % + dA c ON,- da K T q ” i ON,- dX a = o , 1 _d_ 0 - A e * * n (3-50) 2 ON,- dX2 A — o The expression of pj ' .llS in equation ; 3-41 is substituted into equation 3-50 to get hs,r in £i 1 0 ( dAc , dAc , X ^ Ar- ^ J_ InK T -i- — (3-51) Pi ~ IvT ON, V 0A + da + 2 d\~ 2 ) + lnK T + KKT Where pj1S,r = pjis — pjd is the residual chemical potential for the hard sphere reference system. With p^px,-, the final result after rearrangements is : hs,r f,- in in f o! 1 0 ( dAc . dAc . 1 0 Ac \ , Pi , ln T/’rr „ (3-52) X,. - ln fi - KT ON, { dx + da + 2 OX2 ) K T + lnKT/? The significance of this equation is that the quantity, lnf^, needed to solve multicomponent phase equilibrium problems can be calculated directly by the right hand side of equation (3-52), which measures the interaction among the gas solute and all the other species in the solution. The conventional reference state 11° for the solute gas at infinite dilution condition is no longer needed although it still can be evaluated easily by allowing xf to approach zero as : H t, m = ]c»jn, ir (3-53) It should be noted that since { f is actually a function of the liquid composition as well as T and P of the system, any solution procedure for finding the amount of gases dissolved in a liquid would obviously be iterative, converging to a final value, for liquid and vapor compositions that satisfies all the component fugacity equations simultaneously. 60 3-3-4 Expression of Apparent Ideal Solution Fugacity for Geopressured System The exact solution for the value of apparent ideal solution fugacity depends on the intermolecular potentials. For our specific case of CH4, C 02, and C 2H 6 non-polar solutes dissolved in an electrolyte solution, the intermolecular potentials involved in the system are listed in Table 3-2. The non-polar part of each potential (including the salt ions) is approximated by the Lennard-Jones potential, and the angle-averaged expression is used for the dipole-induced dipole interaction between water molecules and the solutes. The inclusion of Charge-Dipole and Dipole-Dipole interactions will be discussed in section 4-5-3. For solute C 02, the dipole-quadrupole interaction between water and C 0 2 molecules is also considered. In Table 3-2: jj-y is the dipole moment of water a,- is the polarizability Qj is the quadrupole moment q; is the charge The subscripts are as defined in Table 3-1. As usual, the geometrical mixing rule is used for the energy parameter in Lennard-Jones potential 1 = ( <,•< £jj ) 2 (3-54) and the arithmetic mean is used for the distance parameter After substituting these potentials into equation 3-52 and performing the integrations, we obtain the following final expression for apparent ideal solution fugacity of component i. This Table 3-2 Intermolecular potential energy used in this study v„ = 4c„ ( y - (^)«) - V33 = 4c33 ( ) " - - ( ^F)6 ) - 7-0 V„ = 4c* ( (2il )” - ( ^ v«» = ( c-p ),! - ('-py) - ^ V33 = 4 .,, ( (“¥ )12 - ( ^P)6 ) ' v,,„ = 4,1U1 ( (^ r y - ( ^ ) 6) V31 = 4 ',i ((¥),! ~ (^)6) - ^ V34 = 4(3< ( (2 p )■■ - ( °-py) - v 3, = 4£3! ( ( ¥ ),! - (^y ) - V,« = 4f38 ( p )'2 - ( '+Y ) - 62 can be evaluated for a given liquid mixture, T, P, and solution density for each component by a rather lengthy computer subroutine. hs,r dp-. ln f?' = 1 + K T ^ K T dX K T da X — o a = o 1 + ln K T /7 (3-56) + 2K T o X — o where OfX: -11.17 3 4.188 Piflia i 2.5133 PiPQi K T dX K T ^ pi d 'j KTKT X — O 1.2566 PiPF'iQi (3-57) KT d 2 a*. £u = ( X. ^ 4 ~ 0.4274 2K T dX 2 P,-/?2Q f , p ip Q i Pi , PiPi ai + 0.7244 ^•17 "+■ _ _ i -a T a i 1 52 a}* , 2Ap1eil0 p ‘iQi 16 tgPiP~ia i , PiPifiQlotj \ (3-58) 187a?, 45 afx + llo-“ ) and (3-59) K T dot a - 0 ~ 4n ^ P'i d^' ^ d'J ^ 63 3-4 Development of The Partial Molar Volume of A Gas Solute The partial molar volume of solute i can be obtained by differentiating the chemical potential of i with respect to pressure, holding temperature and composition constant. The relation between chemical potential and pressure P is not explicit; therefore, the chain rule is used to relate the differentiation to volume. Then, volume is related to the parameter £ through its definition (3-61) dfi\ dfi j av av dtn (3-62) dP ~ T , x ~ av T , > op T , > d£n av ap T , x and hence ^ s o £o av v (3-63) (3-64) 0V (3-65) dV ^ £ 3 (3-66) dV 64 0\7 In order to evaluate in equation 3-62, we need an expression for pressure. This is done through equation 3-17 using the same approximation as earlier. The result is _ pllS _ 167T 'p ^ o o e cr3 — — o o' V — 9 ^ ^ Pi «'J !J 3 ^1 ^1 ^3 2T Z P i Q ? ( (3-67) i * ’ V 10 asa 5(7/ <9P The . term can be evaluated as follow o\r j9P_ _ 8 P hs , c>PP av - av + av the first term on the right hand side results from the pressure equation for hard sphere and the second term is the perturbation result. Substituting equation 3-42 into 3-68 and using the result of 3-63 through 3-66, we can obtain aPhs _ 6KT / _ _Jq ______9 dw *v v I (i- -3 ,-3 c2 + 4 ** ) (3-69) and a P P 327T V' « n c /r-3 877 n -2 av _ 9v £1 £ J P i P i e,j < T i j 3V Pl P l ? J 7 ^ ) (3-70) V I V 10 <7^ 5 (j/! ) 6 5 The isothermal compressibility is defined as i av _ (3-71) V dP ~ hs dP + dP1 av 5V dfl: To evaluate • vjy hi eqution 3-62, again, we separate it into two parts as in equation 3-72 : dfi j 9Hi d£n av + (3-72) T , x a ^ n dV T , x where i + + — 2- d;o + 3 ^ df a*n av T , x l - £ a (1 - ^ 3 ) 2 ( l - e 3 ) 5 d 3 R o (1 -43)~ + + 9*2 ~ 4*3$ + * 1 * 2 (3-73) 1 I (1 - * 3 ) 4 and a^t agn i a ^ i , j dh _ i aa2 - n/*! (3-74) a « n a v KT <9a ^ KT 5A ^ 2KT d \ 2 T , x The right hand side terms of equation 3-74 have been obtained as in equation 3-57 through 3- 59. The final expression for partial molar volume of a solute gas component is : V, RT ( a/K1S,r a^n (3-75) <91■,hs dP1 \ a$n av ^ a^n av ; av + av 6 6 3-5 Development of Isothermal Compressibility of Partial Molar Volume The isothermal compressibility of partial molar volume /?,■ is defined as: i d V = -Jr-w' i - (3-7C) Following the same principles used in developing the partial molar volume, we have developed the expression for /?,•. The procedure is rather tedious, we only present the final result here: IB ( d V\ V i V Q \ ) (»-*#) ; (3-77) 0 hs hs dV d P 1 ap app \2 av + av ■) ( av + av where m _ -£3 _ 3 ^ (1 + £3) , _ 18^1 + 3gt -Hrtl ,2 d f ( l - ^ 3 ) 2 ( 1 - ^ a ) 3 ( l - ^ 3) 4 ( i - s 3f ^ £ 0 ~So£ 3 — £oS3 + £0 ^ 3 ~ 6^1^2^3 — + 24^ 9/tj a ^ n (3-78) $ = -5 J — £ 0 + ^0^3 +^0^3 ~ £o£| +6^i^2^3 *•(1 - £ 3)~ - 12e1ea - 27£| +7t#a -5^|eI + £2^3 (3-79) 67 3-6 Gas Solubility at, High Pressure The equations developed in section 3-4 are very general in the sense that they can be used to evaluate apparent ideal solution fugacity of a gas component at any pressure should the solution density data be available at that pressure. But for our high pressure system, all reported experimental density and correlations for brine solutions were obtained only at rather low pressures. In order to utilize the low pressure density data, the apparent ideal solution fugacity must first be calculated at a low pressure, then converted to the desired high pressure. The effect of pressure on apparent ideal solution fugacity can be obtained through its definition: ^ ' (3-80) therefore, the pressure effect on f^ can be sought by the pressure effect on the fugacity by the exact equation: a 44 Vi « ( f"’) —— ------*----- 1 (3-81) OP ~ R T ~ <9P where V, is the partial molar volume of solute i in the liquid phase. Integrating equation 3-81 from a reference pressure Pr to the system pressure, P, gives, P V ln = In i f + dP (3-82) \ /P \ /pr Pr For sparingly soluble gases, it is convenient to set the reference pressure as the saturation 68 pressure of the solvent. Therefore, equation 3-82 can be combined with equations 3-2 and 3-80 to give: .P ,-T In + ^ dP (3-83) ? ! p i where ln I i f ) can be calculated from equation 3-52 using the density data at temperature V / P i T and pressure Pi. But in order to evaluate the integration term, we have to establish some kind of relationship between V* and P. If a constant Vj over the entire pressure range is assumed, this lead to the Krichevsk-Kasarnovsky type of equation V P - ’ °i — i„ I r®1 . . ,■ (r -___ n) 1” -TfT^ = 111 I fr | + — S?T ---- “ (3-84) p RT An improvment of equation 3-84 is based on a proposal by Namiot (1960) who suggested that a linear relationship of V, with P would be more accurate. V f = vfi ( 1 - Pi ( P - Pf ) ) (3-85) Where /?f is the coefficient of isothermal compressibility of the partial molar volume. Expressions for both V,- and /?,• were developed through perturbation theory in equation 3-75 and 3-77. Namiot’s improvement was employed by Coco and Johnson in their study of methane solubility. Choi (1982) also adopted this idea and developed an analytical expression upon substituting equation 3-85 into the integration term. Our final working equation for solute i is : This pressure correction was applied in this research, using equations 3-75 and 3-77 to generate the required values for Vj and /?; . 3-7 Dissociation of weak electrolyte In the expressions for apparent ideal solution fugacity (3-56), and the partial molar volume (3-75), we need to know the molecular concentration of each possible species in the system. This includes the ionic concentration of HCO 3 and H"*" which result from the reaction between C 0 2 and II 20. Our derivation so far has been assuming that this reaction is not important, i.e., all C 0 2 exists in its molecular form. How do we evaluate this assumption ? In another words, how do we relate the molecular concentration of C0 2 to its bulk concentration which is usually reported by satandard quantitative analysis techniques. This can be done as proposed by Edward et al. (1975, 1978) through four principles. The first one is the overall mass balance in the liquid phase: m A = m a + \ ( m_|_ + rn_ ) (3-87) where is the bulk molar concentration of weak electrolyte C 02; ma is the concentration of undissociated weak electrolyte, and m_^_ and m _ are the concentration of cation and anion, respectively. The second principle is a charge balance in the liquid phase: 70 m_j_ = m_ (3-88) The third principle is the chemical equilibrium between the undissociated and dissociated forms of weak electrolyte. a i a _ K (3-89) where Iv is the dissociation equilibrium constant and a_j_, a_, and aa are the activities of cation, anion, and hydrated molecular form of the weak electrolyte. The fourth principle is the equilibrium between vapor and liquid phase. /*a = /'a (3-90) For a 1:1 type of electrolyte, equation 3-89 can be written as m , m _ 7 , K = ma 7a (3' 91) where 7 ^ is the mean ionic activity coefficient defined by 7± = (7+ 7— ) l / 2 (3-92) Activity coefficients are normalized in the manner customary for dilute solution > 1 as m. - > 0 (3-93) 7a -> 1 where i stands for all solute species. To use equation 3-91 to relate activity coefficients to concentration, we need a correlation for the equilibrium constant. Edwards, et al. (1975) developed the following relationship by using Van’t Hoff equation and a second order Taylor expansion for heat of reaction in terms of temperature: ln K = C j + C 2/ T + C 3 ln T + C 4 T (3-94) where Cj -- C 4 are parameters which must be determined from experimental data. From equations 3-87, 3-88 and 3-91 we can get : 2 itia m „ = ...... -A (3-95) K - if - J + 2 m A + + 4mAi K t , This equation allows us to evaluate the amount of molecular form of C 02, which exists in the solution, knowing its bulk concentration m ^. Chapter 4 Method of Solution With the development of a means to calculate the apparent ideal solution fugacity of a solute gas in chapter 3, we are ready to solve the phase equilibrium problem. Equations 3-8 and 3-86 are the two basic relationships to work with for solvent water and solute gas component i, respectively. (Since the non-volatile salts can be considered absent from the vapor phase, no phase equilibrium equation is needed for them. ) p V f ^ 1 >'l P = x i 01 P 1 exP ppji d P (3-8) to ^ = to (V ) + ( F - r? ) - & ?* (3-86) P i1 Two additional equations are needed for the complete description of the system. They are : £ Y,- = 1 (4-1) E X,. = 1 (4-2) In the case of methane dissolved in pure water, according to Gibb’s phase rule F = C - P + 2 (4-3) 72 where F stands for the degrees of freedom, C for the number of components and P for the number of phases, giving two degrees of freedom. Therefore, by specifying the system temperature T and pressure P, a binary system is fixed at a unique set of vapor and liquid compositions which can be solved for from the above equations. In the case of methane and ethane both dissolved in brine solution, we have C=4, P=2, and hence four degrees of freedom. Besides specifying T and P, two more relationships must be stated to fix the system. The dissolved gas ratio ^ R = XC^H / ^ and the molality of the salt m in the gas-free liquid were chosen in our study. In equation 3-86, the apparent ideal solution fugacity of component i, f^, can be calculated from equation 3-56, given the solution density (molecule/cm3) at system temperature T and pressure P. The partial molar volume Vt- and its slope with pressure 3i can be found from equation 3-75, and 3-77 respectively. We shall now take a close look at how those terms involved in equations 3-8, and 3-86 are evaluated in this research. Section 4-1 deals with the SRK equation of state used in the vapor phase fugacity calculation. Section 4-2 describes the liquid phase calculations, which include a density correlation of the brine solution, saturation pressure and Poynting correction. In section 4-3, we look at the physical parameters used in this study and the computer algorithm used for solving the vapor-liquid equilibrium equations simultaneously. Some calculation results are also presented from which we conclude the need for some parameter adjustments to fit the solubility data better. Section 4-4 completes this chapter with the results of the parameter fitting procedure. 4-1 Vapor Phase The fugacity of any volatile component i in the vapor phase can be expressed by equation 3-2 : f f = <}>. y< P (3-2) with pressure already specified for the system, we need to know the mole fraction and fugacity coefficient ^ to calculate ff. However, as we shall see in equation 4-14, The Soave-Redlich-Ivwong (SRK) equation of state (Soave, 1972) has proved to be a very versatile tool for most hydrocarbon applications, and is rapidly gaining acceptance by the hydrocarbon processing industry. Although this equation of state is not very suitable for polar molecules, we believe that at the relatively low water concentrations in the vapor phase of our study, it still should provide acceptable results. To check this belief, a comparison of SRK with a newly developed equation of state from Nakamura and Prausnitz (1976) for the gas mixtures of our system was made, which showed no accuracy advantage for the latter, even though it was especially developed for both polar and non-polar systems. Nakamura’s equation was reported to be valid up to 400 atm, which is too low for our system of high pressure up to 2000 atm. Besides, the computer time spent does favor the use of SRK. Nakamura’s equation is presented in appendix A with its calculated results compared to those of SRK. The SRK equation of state and the associated mixing rule used in this research are reviewed here : 75 P = . RT , fl(T) (v - b J V (v + b) where b = 0.0866R Tc/ Pc and A depends on temperature according to : ft(T) = A(Tc)a(T) (4-5 a) ffl(Tc ) = 0.42747 R 2 Tc / Pc (4-5 b) and a(T) = ^ i + (o.480 + 1.574 u> - 0.176 w2 ) ( 1 - ( ^ ) 1 /2 ) (4-6) where w is the acentric factor. By defining the following : V = z R T /P (4-7) A = A (T ) P / R 2 T 2 (4-8) B = b P /R T (4-9) The compressibility factor z can be obtained by solving the cubic equation : z3 - z2 - z (A -B -B 2 ) -AB =0 (4-10) For mixtures : A = E E y.- y,- (4-11) » j B = E y ,-b j (4-12) 76 where, * * j (1 - K {j ) ft4 Hj (4-13) Now, the fugacity coefficient of a component in a mixture can be computed by (4-14) according to Seader (1982) : 4>i = exp( (z-1) Bj / B — ln(z-B) - A/B (2A -/ 2 / A ^ 2- B 4/B) G ) (4-14 a) where G = In (l+ B/z ) (4-14 b) 77 4-2 Liquid Phase The activity coefficient of water, 7 X, in equation 3-8 was assumed to be 1. We made this assumption because in most of our system conditions, the mole fraction of water in the liquid phase was well above 98%, and since very little water is present in the equilibrium vapor phase anyway, there seemed no reason to try to improve on this assumption. 4-2-1 Solution Density The density of a NaCl salt solution (g/cm3) can be represented as: p = p* + Ap (4-15) where p* is the density of pure water at T and A p is the density difference due to the presence of salt at temperature T. There are several correlations available for the density of pure water. Here we compare three of them : Yaw’s correlation : (1972) p* = 0.3471 x 0.274 * 1 T//T<^ ' (4-16) Keenan’s correlation : (1969) ______1+0.134289X (T c - T ) l / 3 -3 .9 4 6 2 6 3 xlO ' 3 (T c - T ) P* = ^3.1975 —0.3151548(TC—T )l/3 —1.203374 x10~3(Tc—T )+7.48908 x10’ 1 3 (T c —T ) 4^ (4-17) Chen’s correlation : (1978) (valid from 273 to 328 K ) p* = 0.9998395 + 6.7914 x 10'5T -9.0894 x 10 ' 6 T 2 +1.0171 x 10 "7 T 3 - 1.2846 x 10 "9 T 4 + 1.1592 x 10 ' 11 T 5 - 5.0125 x 10 ' 14 T 6 (4-18) Table 4-1 compares these correlation with steam tables and shows that Keenan’s correlation is the most accurate one. The following correlation for A p as a function of temperature and molality was made available by Lo Surdo and Millero (1982) from 273.15 Iv to 323.15 K and from m=0.1 moles/Kg-HoO to saturation: A p = (45.5655 m - 0.2341 m T + 3.4128xl0"3 m T 2 -2.703xl0'5 m T 3 + 1.4037 xlO"7 m T 4 - 1.8527 m 3/2 + 5.3956 xlO'2 m 3/2T - 6.2635 xlO"4 m 3/2T 2 - 1.6368 m 2 - 9.5653 xlO*4 n rT + 5.2829 xlO"5 m 2T 2+0.2274 m 5/2) xlO '3 (4-19) Unfortunately, this correlation is valid only at low temperatures (from 273 to 323 K) and does not cover much of the temperature range of our system. We first thought that perhaps the increased amount of density due to the presence of salt is not a function of temperature or at most is a linear variation with temperature. Therefore, we used a linear extrapolation of Lo Surdo and Millero’s correlation to a higher temperature. This turned out to be a serious mistake, as the increased density due to the present of salt is in fact a strong nonlinear function of temperature. As shown in Figure 4-1, it goes through a minimum at around 340 to 380 K (Rogers and Pitzer, 1982). Therefore, when the linear extrapolation was used, the a Table 4-1 Comparison of various correlations of liquid water density with steam tables. T(K) Millero Yaw Keenan Steam Table 273.15 0.999839 0.999761 0.99980 293.15 0.998204 0.997927 0.99820 313.15 0.992216 0.992090 0.992160 328.15 0.985693 0.985661 0.985710 348.15 0.966709 0.974848 0.974850 368.15 0.947256 0.961864 0.961910 388.15 0.927203 0.946984 0.947150 408.15 0.906474 0.930365 0.930490 428.15 0.884973 0.912062 0.912240 458.15 0.851010 0.881386 0.881520 478.15 0.826994 0.858608 0.858810 498.15 0.801624 0.833701 0.833890 513.15 0.781527 * 0.813405 0.813600 533.15 0.752970 0.783764 0.784010 558.15 0.713552 0.741401 0.741670 573.15 0.687168 0.712158 0.712450 598.15 0.636069 0.653646 0.654020 628.15 0.549480 0.549694 0.553400 638.15 0.502150 0.491077 0.495050 80 0.200 — Our Correlation ■ Rogers'Data 0.150- o u Molality= 3.061 o > 0.100 0.050- Molality= 0.978 Temperature ( K) Figure 4-1 Comparison of Ap of our correlation (equation 4-20) with experimental data of Rogers and Pitzer parameters of methane and ions that were required to fit the solubility data for the salt solution increased with temperature, which is not reasonable. In order to extend this correlation more accurately to higher temperatures, we used the experimental density data of Rogers and Pitzers (1982) for NaCl-water system and fitted these data with the same formula as proposed by Lo Surdo and Millero using a GRG2 search method. The resulting formula is as follows. A p = (40.2204 m - 0.013548 mT + 1.25326 xlO ' 3 m T 2 -0.869929 xlO ' 5 m T 3 + 0.352779 xlO ' 7 m T 4 - 2.33803 m 3 / 2 + 0.24002 xlO ' 2 m 3/ 2 T - 1.8341 xlO ' 4 m 3 /2 T 2 -0.570926 n r + 1.04261 xlO ' 4 nrT + 1.13663 xl0" 5 m 2T 2 + 6.06576m5/2)x l0 '3 (4-20) with RMSD = 0.00488 and AD = —0.794x 10 "4 ( RMSD and AD are defined in equations 4-32 and 4-31 ). Table 4-2 compares the prediction with the experimental data of Rogers and P itzer. With the gas-free solution density correlated from the above equations, the solution density with the dissolved gases was then corrected according to the relation: Pgas-free solution ^solution = 1 v ------(4-21) i (solute gas) This equation assumes that the volume of the solution will not be changed upon adding solute gas to it. It presents no problem to our system since the solubilities of gases under consideration are very low. For high soluable gases, the solution volume can be adjusted from the partial molar volume of the dissolved gas. Our current model does not include this adjustment. 82 Table 4-2 Comparison of A p predicted by equation (4-21) with Rogers and Pitzer's experimental density data for NaCl-H20 solution M olality T = 348 Iv T = 373 K T = 423 K of N aCl exp. calc. exp. calc. exp. calc. 0.0520 0 .0 0 2 1 0 0 .0 0 2 1 0 0.00210 0.00212 0.00230 0.00227 0.2719 0.01061 0.01056 0.01065 0.01063 0.01126 0.01134 0.5571 0.02123 0.02127 0.02138 0.02138 0.02261 0.02273 1.0360 0.03867 0.03866 0.03891 0.03882 0.04100 0.04106 3.0610 0.10630 0.10624 0.10670 0.10638 0 .1 1 1 2 0 0.11107 3.2428 0.11192 0.11192 0.11226 0.11205 0.11652 0.11688 4.3933 0.14618 0.14660 0.14664 0.14660 0.15192 0.15212 M olality T = 448 K T = 473 K of NaCl exp. calc. exp. calc. 0.0520 0.00240 0.00239 0.00260 0.00257 0.2719 0.01186 0.01191 0.01254 0.01272 0.5571 0.02368 0.02379 0.02514 0.02531 1.0360 0.04284 0.04282 0.04532 0.04538 3.0610 0.11468 0.11510 0.12050 0.12018 3.2428 0.12040 0.12059 0.12571 0.12628 4.3933 0.16321 0.16292 4-2-2 Saturation Vapor Pressure There are many correlations available for the saturation vapor pressure of water. The Antoine equation is widely used at temperatures between 273 to 443 K, but it fails to produce accurate results at higher temperatures. We chose to use the correlation by Keenan and Keyes (1969). This correlation is extremely accurate and introduces little error: £J5 exp ( r x 10 ' 5 (Tc -T) £ F,- (0.65 -0.01 T)!' _1 ) (4-22) Pc where Ps = saturation vapor pressure (BAR) Pc = critical pressure (220.88 BARS) T = saturation temperature (C ) Tc = critical temperature ( C ) T = 1000/ (T+273.) F, = parameters Fj = -741.9242 F 2 = -29.7210 F 3 = -11.55286 -0.8685635 F 5 = 0.1094098 F 6 = 0.439993 F 7 = 0.2520658 Fs = 0.05218684 4-2-3 Poynting Factor The Poynting correction is an exact thermodynamic relationship that accounts for the effect on the fugacity of pure water of raising the pressure on the pure liquid water to a pressure P greater than its saturation vapor pressure P|. 8 4 f V c = exp ^ dP (4-23) P i The Poynting factor behaves essentially exponentially with pressure. It is small at low pressure but may become large at extremely high pressure. Since our system pressure is very high, the usual simple incompressible assumption made for VJ may not be valid, and we need a relation between the liquid molar volume and pressure over the entire pressure range. For this purpose, we used the global correlation proposed by Cheuch and Prausnitz (1969) : / 9 Zc N ( P — Pvp ) \ P = p A 1 + ------^^ j (4-24) where Zc is the critical compressibility factor. N is a function of the acentric factor w and the reduced temperature. N =(1.0 - 0.89 w) ( exp (6.9547 - 76.2853 Tr + 191.306 Tj? - 203.5472 T? + 82.7631Tf )) (4-25) The saturation liquid density, p s, can be obtained from Keenan’s correlation for water density in our system. 85 4-3 Physical Parameters In evaluating the apparent ideal solution fugacity, as in equation 3-56, we need numerical values for those physical parameters involved in the potential energy. Those are the distance (cr) and energy (e) parameters in the Lennard-Jones potential, the polarizability (a), the dipole moment (//) and the quadrapole moment (Q). It has been established (Tiepel, 1973; Schulze, 1981) that the distance paremeter, methane gas solubility. We also confirmed this from a sensitivity study with our model that variations of e and a within their reported ranges have little effect on the predicted methane solubility. Table 4-3 shows the results of the sensitivity studies. In this table, the first row served as a base to be compared with. In each consecutive row, we changed the value of only one physical parameter from the base value at a time. The predicted methane solubilities in pure water were then compared with the first row value and the results were listed in the last row. From this comparison, we can see that the change of the predicted methane solubility per unit change of the studied physical parameter (AX/A) is quite different. For the polarizability (a), it has no obvious influence on the predicted methane solubility when we switched from one reported value to another. For the energy parameter (e), its effect is also very small. But for the distance parameter (cr), a slight change in either or < 7'cH 4resu*te<^ *n substantial change in the predicted methane solubility. Therefore, we decided to find the best value for the cr parameter of each component and chose to use one of the reported values for the other parameters from literature. Table 4-4 lists the values we found in the literature. The * indicates the value used in this study. Table 4-3 Sensitivity study to show the effect of various physical parameters on predicted methane solubilities and partial molar volumes x c h 4 VCII4 f n / k *11 « 3 sA *33 « n a 33 0.00180 36.64 96.3 2.725 148.6 3.877 1.59 2.60 0.00160 36.45 85.3 2.725 148.6 3.877 1.59 2.60 0.00164 36.54 96.3 2.735 148.6 3.877 1.59 2.60 0.00156 36.91 96.3 2.725 137.0 3.877 1.59 2.60 0.00178 36.71 96.3 2.725 148.6 3.880 1.59 2.60 0.00180 36.64 96.3 2.725 148.6 3.877 1.69 2.60 0.00180 36.64 96.3 2.725 148.6 3.877 1.59 2.70 AX 0.0000182 0.016 0.0000207 0.00667 0 0 A en is the energy parameter for water in L-J potential e33 is the energy parameter for methane in L-J potential & 11 is the distance parameter for water in L-J potential cr33 is the distance parameter for methane in L-J potential a n is the polarizability for water Table 4-4 Numerical values for physical paremeters as reported in the literature (A) (K) 10~24cm 3 e.s.u. cm e.s.u.cm 2 C om ponent a e/K a Jii Q, h 2o 2.52 775 1.59 * 1.84 * 2.641 809.1 3.329 209.1 2.750 85.3 * 2.860 96.3 2.980 96.3 3.023 96.3 CI14 3.958 141.6 2.60 * 3.820 137 2.70 3.758 148.6 * 3.882 148.2 3.817 c , h 6 4.443 215.7 * 4.47 * -0.65 * 4.847 2 1 0 .6 4.33 4.341 247.2 4.420 230.0 CO 2 3.732 262.4 * 2.594 * -4.3 3.996 190 4.59 4.486 189 1.8 3.941 195.2 2.7 N a+ 1.976 91.7 * 0.179 * 1.9 0.21 ci- 3.744 245.8 * 3.66 * 3.75 243.1 3.02 3.62 * indicates the values chosen for use in this study The values of cr used in this study are discussed in section 4-6 The above values are found from the following references 1. Tiepel and Gubbins, CIChE, 361 (1972) 2. Tiepel and Gubbins, I&EC, Fund.,12 ,18 (1973) 3. Tiepel and Gubbins, J. Phy. Chem., 76, 21, 3044 (1972) 4. Shoor and Gubbins, J. Phy. Chem., 73,3, 498 (1969) 5. Masterton and Lee , J. phy. Cliem., 74, 8 , 1776 (1970) 6 . Reid, Prausnitz and Sherwood, The properties of gasesand liquids (1972) 7. Moore, Physical Chemistry (1972 ) 8 . Orcutt, J. Phy. Chem.,39,3, 605 (1963) 4-4 Computer Algorithm As shown in Table 3-1, the computer code was developed not with only our specific system components in mind but also with a built-in flexibility for handling other gas components, salts, and solvents in the future. The program can be divided into two parts. The main program, which does not change for different system conditions and a user-supplied subroutine, which depends on the system conditions. The user subroutine provides information about the system conditions of temperature, pressure, and salinity. It tells what components are present in the system through logical statements. The gas-free liquid solution density is also evaluated in this subroutine together with an initial guess of the solubility of component 3 (methane). The main program consists of several subroutines to evaluate and update those terms in equation 3-8 and 3-86 based on the conditions provided in the user subroutine. Equation 4-1 is used for a stopping criterion. The calculation sequence for determining the gas solubility in a specific brine solution at fixed temperature and pressure is listed below along with a flow chart in Figure 4-2 . The complete computer program is listed in appendix D. (1) Specify temperature, pressure, molality of salts, the system components, and the ratios of other gas solutes to dissolved methane in the liquid phase. (2) Calculate solvent water density, saturation vapor pressure, and gas solute-free solution density. (3) Assume a methane solubility x 3 . Calculate solution density including dissolved gases and calculate the corresponding liquid phase composition based on specified ratios of dissolved gas compositions. (4) Calculate f?*, which is the right hand side of equation 3-56 for each solute at T, P, and at the current estimate of liquid phase composition. 89 S ta rt User provides T,P m, components, ratios of dissolved gases, and physical parameters A ssum e X q jj Calculate f^ for each solute gas, also calculate x t for all other solute gases based on specified ratios guess 4>i f° r each vapor component Solve for Y, Compute using SRK Adjust d>; by Secant method no new old yes no Adjust X ^j| by Secant m ethod yes no old new old yes o u tp u t Stop Figure 4-2 Flow chart of program to compute methane solubility given system specifications and physical parameters of components Since the fugacity coefficient is a non linear function of the vapor phase mole fraction according to equation 4-14, Secant method is used to converge on (5) Assume < for each vapor component. (6 ) Solve for yt- from equation 3-8 and 3-86. (7) Update (8 ) Iterate on steps 6 , and 7 by Secant method until convergence on ,- is achieved. (9) Iterate on steps 4 to 8 by secant method until both Yi ~ l ) a n ^ X few)/ X °^^ are within desired tolerances. 91 4-5 Analysis of Initial Results and Decision Made to Improve Predictions In this section, a review of the results that were achieved from using the methane solubility prediction program is presented. We believe it is important to explain what results we obtained and in what system, so that the subsequent modifications that were made to the physical parameters and to the program can be justified to the reader. First, to evaluate our model to see qualitatively whether it predicted reasonable results, constant values for the a parameters of methane and water were chosen within the reported literature ranges and solubilities were calculated to compare with the experimental data of Culberson and Mcketta (1951) at 377 K and 240 atm. Figure 4-3 plots the predicted Methane solubility vs. pressure at zero salinity and two different temperatures. At relatively low temperature and pressure, the prediction has less than 1% deviation from the experimental data, but the deviations increased as temperature and pressure increased. The same conclusion was drawn when calculated results were compared with other experimental data sets for methane in pure water. Obviously, there were some effects of temperature and pressure which our model at this point did not handle adequately, especially for the temperature effect. 4-5-1 Possible Effect of W ater Density on Methane Solubility A plot of calculated methane solubility in pure water vs. temperature is shown in Figure 4-4 for comparison with the McKetta and Culberson’s data. Our model does predict a rise in solubility with increasing temperature, yet the curvature is not large enough to accomodate the experimental data. One possible explaination of this discrepancy was offered by Potter and Clyrne (1978). They studied solubilities of six simple gases in water and correlated the solubility data using the density of saturated water as the independent variable Methane Solubility X 0.0010 0.0060 0.0070 0.0020 0.0030 0.0040 0.0050- 0.0080 iue - Iiil o aio o h dsrpnis ewen rdce methane m predicted een betw discrepancies the of parison com Initial 4-3 Figure □ - solubility and experim ental d a ta of C ulberson and M cK etta. Physical Physical etta. cK M and ulberson C of ta a d ental experim and solubility aa tr ue ae cC4 .5 , 'o = 2.786 = o'HoO , 3.758 crCH4= : are used eters param cet' Dt a 7 K Cl.Rsls t37 K 377 at Results Calc. K— K 377 at 444 at Data Results McKetta's Calc. K-- ■ 444 at Data McKetta's □ Pressure (atm) 92 Methane Solubility X 0 0 0 0.0030- 0.0040- 0.0050- 0.0060- 0.0070- 0.0080r- . . . 000 0010 0020 ^ - - Figure 4-4 T he difference of predicted m ethane solubility in pure w ater using using ater w pure in solubility ethane m predicted of difference he T 4-4 Figure 8 30 0 40 2 40 4 40 6 40 8 40 0 5 500 490 480 470 460 450 440 430 420 410 400 390 380 Pressure 240 atm = ac eut Uig ac eut Uig cet' Data McKetta's ■ Using Results Calc. Using— Results Calc. aiu creain fwae density ater w of correlation various = 0 = m a' Creain enns Correlation Keenan's Correlation Yaw's for Water Density for Water Density Water for DensityWater for Temperature K) ( 0 93 instead of temperature. Their study indicated that the thermodynamic properties of dense fluid mixtures are more sensitive to the water density than to the temperature. Originally we were using Yaw’s correlation for liquid density. From Table 4-1 we do see the deviation of Yaw’s predicted densities of water from those of the steam tables increases as temperature increases. We wanted to find out if this deviation is a possible reason for the methane solubility discrepancy at high temperatures. So we switched to Keenan’s correlation for water density which agrees almost perfectly with the steam tables. The results are changed only slightly and in the wrong direction. Therefore, we concluded that improving the density correlation did not improve the methane solubility discrepancies. 95 4-5-2 Possible Effect of Temperature on Methane Solubility via The L-J Parameter a The reported methane solubilities of Culberson and McKetta goes through a minimum at about 344 K, as seen in Figure 4-5. Is our model capable of predicting this type of behavior? If not, we probably need to reconsider the validity of our model; if it does predict a minimum in solubility vs temperature, then we want to see what can be done to improve our prediction. Figure 4-5 shows clearly that a minimum is indeed predicted by our model. It also suggests that if we can shift our prediction curve a little to the left, then we should have a better overall prediction. This indicates that some parameters in the model should have been set at values different from the current ones, or, perhaps even varying with temperature. In the light of the above discussion, when we searched for a temperature effect in our model, we concluded that adjustment of the hard sphere diameters for methane and/or water in equation 3-30 offered the only source of possible improvement due to temperature in the current model. Real particles do not possess hard cores anyway, and the correspondence between the hard core diameter and the cr parameter in the L-J potential, as shown in equation 3-30 suggests that if we were to allow a to change with temperature, this should result in a temperature effect, which might reduce the discrepancy. The case for a temperature dependence of cr parameters had been discussed earlier by Pierotti (1967) and Mayer (1963) in connection with applications of scale particle theory to gas solubility and surface tension. Schulze and Prausnitz (1981) also studied gas solubility at low pressure using scale particle theory. They found out that the experimental data could often be fit well only if the key parameter tr were allowed to be slightly dependent on temperature. Figure 4-6 shows the tremendous change in solubility prediction from Figure 4-5 that results from letting the 96 Methane Solubility X 0 0.0030- 0.0040 0.0050 0.0060 0.0070 0.0080 . 0020 iue - Te fet fa ier aito tmeaue n predicted on temperature f o variation linear a of effect The 4-6 Figure 7 30 9 40 1 40 3 40 5 40 7 40 9 500 490 480 470 460 450 440 430 420 410 400 390 380 370 - cet' Data McKetta's ■ h oi ln i peitd ih .8 — .0l* T , T 0.60xlO*3 — 2.786 = with predicted is line solid The h ds ln i peitd ih .8 — 85xl 3 . T '3 l0 x 5 .8 0 — 2.786 = with predicted is line dash the water. pure in solubility ethane m Temperature K) ( esr = 20 atm 240 = ressure P 97 98 4-5-3 Possible Improvement of The Model by Adding Dipole-Dipole and Charge-Dipole Interactions In our model, the effect of pressure on methane solubility is manifested through the partial molar volume of solute gas as in equation 3-60. Recall that in deriving the equation for apparent ideal solution fugacity, only the solute-solute and solute-solvent interactions were required. The solvent-solvent and ion-solvent interactions were not needed as they do not affect the fugacity of the dissolved methane. But to develop the partial molar volume equation, we needed to evaluate the total pressure equation 3-67, which contains all the possible interactions in the solution. When we add the angle-averaged Charge-Dipole and Dipole-Dipole interactions, as in appendix B, to the pressure equation, the resulting magnitude of every contribution for a system at T= 324.5 Iv, P= 300 atm and 4m NaCl brine solution is as follows : (unit= erg/cm 3 ) non-polar dipole-induced dipole-dipole charge-dipole dipole - 0 .4 3 7 x l0 10 — 0.226xl010 -0 .1 6 3 x lO 11 - 0 .3 6 x lO 12 As a result, in equation 3-69 for the hard sphere : V = 0.212 x 10 11 oV and in equation 3-70, substitution of the above contributions gives : V = sum of all interactions = —0.38 x 10 12 oV 99 The resulting value of the isothermal compressibility of the solution calculated in equation 3-71 becomes negative, —0.276 xlO"11, which is obviously physically incorrect. Therefore, the attempt to include the Dipole-Dipole and Dipole-Charge interactions had to be abandoned. This was the case in Tiepel and Gubbins’ work (1972 a), although not stated explicitly. The reason that we could not include them, we believe, is that most of the reported values for Lennard-Jones potential parameters of water have already incorporated the effect of Dipole- Dipole interactions as shown in appendix C. Thus, in order for us to include these interactions, the Lennard-Jones parameters would have to take on substantially different values from the reported values. We decided not to pursue this any further, leaving for a later project the possibility of modifying the L-J parameters enough to permit including the Dipole-Dipole and Dipole-Charge interactions that we have omitted. Upon reaching the above conclusion we decided that we needed to be able to ascertain whether our predictons of partial molar volume ( without the Dipole-Dipole and Charge-Dipole interactions ) were reasonably correct. Experimental values of partial molar volume of methane in water from Krichevsky et al. (1945), together with the correlation of Brelvi and O'Connell (1972), were chosen for comparison purposes. From a given specific volume of a solvent at a system temperature T, O'Connell’s correlation permits the evaluation of the partial molar volume of a solute gas at infinite dilution through the following equations: _coP? V 2 = KJ RT ( 1 - C°12 ) (4-26) where In ( l + p K } RT) = - 0.42704 ( p - 1 )+ 2.089 ( p — l ) 2 -0.42376( p - l ) 3 (4-27) 100 ln ( - C 12 -^ -jr ) 0,52 = -2.4467 + 2.12074 p for 2 < p < 2.785 = 3.02214 - 1.87085 p + 0.71955 p 2 for 2.785< ~p <3.8 (4-28) Here p =pV*, is the density of the pure solvent reduced by its characteristic volume and V*, is a characteristic volume for substance i in unit of cm 3 /mole. The values of V* for methane and water are 99.5 and 46.4, respectively, as reported in the original paper. The comparison between our model predictions of the partial molar volume of methane in water and O’Connell’s correlations are presented later in section 5-2. 4-6 Parameter Fitting From the previous discussion, we have found that a slight change of c ^ with temperature can result in sustantially different predictions for methane solubility. (This is also true for changing Ideally, we would like to use all the data points together and search for the cr parameters that simultaneously result in the best global fit of the data in one run. But there are some difficulties preventing us from doing so. The first concern is the computer time that would be required. While the model itself requires many iterations for each solubility calculation, the program, when imbedded in a GRG2 search method, requires many more. Secondly, most search methods, including GRG2 become almost ineffective when too many parameters are adjusted simultaneously. Thirdly, we really didn’t know what range to permit the parameters to vary within. Since our prediction is very sensitive to In order to handle these massive data sets effectively, we chose first to fit individual data sub-sets at constant temperature to see if we could obtain some clues to the fitting of the a parameters. Figure 4-7 shows some results from using GRG2 to determine a parameters for water and methane with experimental data at constant temperature conditions. We can see Distance Parameter of L-J Potential 2.60 2.80 3.00 3.20 3.40 3.60 3.80 4.00 2 30 6 30 0 40 4 40 8 50 2 540 520 500 480 460 440 420 400 380 360 340 320 L . . L 1 1 ^ H 7^ iue - Ftig f _L_ ■ IB ... 1 ..... + □ + □ _1_ CH4 H20 + + i + + i □ □ 102 103 that there are certain trends evident for a changing with temperature for both water and methane. For some temperatures, however, the best value for the a parameter seemed to be put of the general trend, but it also happens that the corresponding calculated partial molar volume was in error for these data as shown in Table 4-5. This is the case, for example, with Sultanov’s data at 523 K, where the predicted partial molar volume of 39.0 is even less than most of the predicted values at lower temperatures. Throughout our calculations, we have found out that gas solubility increases with a decreasing value of 0"h2o or <7CH4 ’ a'so a change in 0"h2o was f°und to have a bigger effect on solubility than that one in c r ^ ^ . However, when comparing the partial molar volumes, the effect of the cr values is different. V Ch 4 increases rapidly with increasing while it decreases slowly with increasing 0 ^ 0 ' With this in mind, it was possible to adjust both 0h2O and To simplify further the fitting procedure, we assumed that the cr parameters for each component would not be affected by the appearance of other species in the mixture. This enabled us to break the data into small groups, determine and 0 h 2O ^or ^ie meHiane_ water group, and then using these values to determine values for the other cr parameters from other data groups separately and sequentially. Also, instead of using partial molar volume as constraint equations, we simply used our own understanding of the system and judgements to obtain meaningful results. The procedure followed for fitting a parameters is listed below: (1). Search for (2). Find cr , , ctq- from CH 4 /H 20 / NaCl system. The massive 670 Na data points of Blount was averaged to reduce to 215 points. (3). Find 0c.,H 6 fr° m C 2H 6/II20 system. (Culberson & McKetta’s data) Table 4-5 Individual fitted with its corresponding partial molar volume of methane CH4 T (K) A uthor £Th 2o 344 M cK etta 2.77502 3.75936 37.7 377 .. 2.74257 3.82572 40.9 411 2.72034 3.83369 42.6 444 2.73162 3.74893 40.9 324.5 O ’Sullivan 2.76616 3.78532 38.5 375.5 • • 2.73305 3.85415 42.0 398 2.68448 3.98415 48.4 423 Sultanov 2.73064 3.77348 40.8 473 • • 2.68205 3.75677 43.4 523 •• 2.70835 * 3.57205 39.0 373 Blount 2.73560 3.83062 41.2 427 Price 2.67047 3.90958 46.9 479 2.66976 3.79441 44.5 494 • • 2.61816 3.87702 49.5 507 •• 2.62042 3.76725 46.6 105 (4). Find (TCOr) from C 02/ H20 system. (Tdedheide k Frank’s data) (5). The final set of parameters were then used to predict methane solubility in CH4/ C02/ II20 / NaCl and CH4/ C 2H 6/ H 20 / N ad systems and compared with reported data of Blount, which were not used in the parameter fitting steps. The GRG2 search was carried out on the IBM VM facility at LSU Chemical Engineering Department. The objective was to minimize the function: PVD. /> r, ] /> fy OBJ (4-29) where x JXP‘ is the reported experimental data j and Xj a^0- is the calculated solubility at data j condition, which is a function both of the parameters being searched and the conditions of temperature, pressure, and salt molality. (4-30) 106 4-6-1 CII4 / H20 System In the C II 4/H 20 system, 147 data points from various authors were used to search for exp. calc. n j ~ Xj £ exp. j = i ______AD = j (4-31) N vexp. vcalc. \ 2 ( J x“ ') J ) y * RM SD = ^ ------I (4-32) The objective terms (OBJ) is defined as in equation 4-30. Constant values for 2.8769 and Cqjj = 3.3770 were found in run number 1 but at a very high RMSD. The partial molar volume of methane thus obtained varied from 26.2 at 298 K to 33.9 at 565 K, which is too low compared to the experimental values reported in Table 5-3. When we allowed c t^ q to change linearly with temperature in run 2 , we found that a H20 = 2.92191 - 4.88508 x 10 ' 4 T crCH^ = 3.7904 1 0 7 Table 4-6 Partial F test of parameter fitting of CH4/H20 system. run P aram eter AD RMSDOBJ P a rtia l F no fittin g T est 1 ° H 20 = 2.8769 -0.0689 0.214 6.7069 a c h 4 = 3.3 7 7 0 2 4.88508xl0"4 T -0.0322 0.147 3.16725 162 3 ^H 20 = 2.79254- 1 .4 7 7 3 9 x l0 ~4 T -0.0253 0.121 2.16072 6 6 .8 8 crCH =4.19652- 1.04113xl0"3 T 4 = 2 .4 7 3 0 7 + 0.179337x10"2 T -0.0079 0.0692 0.704 503.8 -0.29952x 10 ’ 5 T 2 5 37437+ 0.248406x10”2 T -0.45672xl0"5 T 2 -0.0136 0.0682 0.684 2.69 +0.119xl0"8 T 3 crCH 4 = 3.8 7 0 2 108 But, the RMSD was still too high. The VCH^ values now varied from 38.33 at 298 K to 51.04 at 565 K, which is too low at the high temperature. Would letting cr (-^not be constant improve the fitting further? We found that when we let both cr^^and f (-|20 vaiT linea.rly with temperature in run 3, we obtained : °H„o = --79254 - 1.47739 xlO ' 4 T with RMSD decreasing only a small amount. But the problem now was that the VCH^ value remained almost constant from 42.22 at 298 K to 43.36 at 565 K. And in the vicinity of 323 K, V Ch 4 actually decreased with increasing temperature. It was previously mentioned that the VCh .j value is affected very largely by the cr^^value used. As a CH4 = 3.87711 with RMSD = 0.0692 which was a very substantial improvement, and the V C(_|4 thus calculated varied from 42.0 at 298 K to 62.2 at 565 K which, is quite close to the reported values in Table 5-3. In run 5, a cubic relation of w^h temperature was tried. The last column in Table 4-6 lists the statistical partial F test (Draper and Smith, 1967) value so that it measures the significance of the reduction in RMSD achieved compared to the previous run. The F value for including the T 3 term was too low compared to F required for 99% confidence (6 .8 ) and therefore the term was rejected. Figure 4-8 plots the individual fit of from the various experimental data sets vs. temperature. Compared with Figure 4-7, we see that the overall data can now be represented very well . 4-6-2 C n 4 / I I 20 / NaCl System : The Lennard-Jones parameters for ions do not seem to be reported very widely in the literature. Although crystal radii can be used to provide approximate values for the works of Shoor and Gubbins (1967) and Masterton and Lee (1970). Other than convenience and availability, there is no reason to assume that this is the exact value of the size parameter to use for ions in solution. Benson and Copeland (1963) estimated that the ionic radii in solution should exceed crystal radii by at most 0.02 A. Tiepel and Gubbins (1973) reported a 1.04 times the crystal diameter for the < 7jon parameter. In view of the sensitivity of the calculated solubilities to the a values, we decided to fit and Cl" based on the cr parameters correlated for water and methane in the CH 4 /H 20 system as follows: cr. = C x cr ion pauhng W here °rp a u jin g were 1.9 A for Na~*~ and 3.6 A for Cl- , as determined by Pauling (1960). The best value for C which best fit Blount’s data for methane-brine was 1.012. Table 4-7 compares solubilities calculated from this model with those of Blount’s correlation. For Blount’s data above 373 Iv, the Blount’s polynomial does fit his own data somewhat better than does our model. Blount’s polynomial correlation, however, can not be extended to fit his Distance Parameter of L-J Potential 2.30 2.40 2.50 2.60 2.70 2.80 iue - Fitng o oH) i LJ oeta wih xei na slblt dat ta a d solubility ental experim ith w potential L-J in orHi)Q of g ittin F 4-8 Figure utnvs aa Blount's data ♦ Sultanov’s data □ rc' data Price's f - c e t' dat O'sulfivan's data ^ ta McKetta's a d sn cntn value constant using 0 450300400 Temperature K) ( 500 □ + H20 550 0 650 600 o n Ill Table 4-7 Comparison of RMSD of present model with Blount’s Polynomial in NaCl brine solution. This Study B lount A uthor T ( K) D a ta pts RMSD RMSD B lount 373 49 0.0947 0.0623 et al. 408 54 0.0619 0.0560 443 49 0.0691 0.0529 478 42 0.0967 0.0763 512 19 0.0849 0 .0 2 1 0 O ’Sullivan 324.5 11 0.0507 0.0811 et al. 375.5 10 0.0844 0.1454 398.0 11 0.2520 0.3367 B lount 298-373 34 0.1750 1.094 et al. own low pressure and low temperature data however. Also, there were very large deviations of Blount’s polynomial from O’Sullivan and Smith’s experimental data on methane-brine, which were at moderate temperatures. Our model, on the other hand, can be seen to be valid over the entire temperature and pressure range of Blount’s data as well as O’Sullivan and Smith’s data for the methane-brine system. For the pure water system, as we discussed in the previous section, our prediction is far better than any of the previous work on this subject, including Blount’s correlation. We shall discuss this further in chapter 5. 4-6-3 C 2H 6/ H 2O system : The parameter equation we obtained from the CH 4 /H 20 system was used in this step to find the best values- The experimental data we used for the C 2H 6/H 20 system was from Culberson and McKetta (1950), which includes 45 data points and covers pressures from 130 to 670 atm and temperatures from 310.93 to 444.26 K. The constant value of crc^Hg "’hicli best fit the data was : with OBJ = 0.00731373, AD = -0.00215 and RMSD = 0.0403. This value was then used with the previous findings on ^ch 4> <7H20> a + > an ^ a C\- *° predict the effect of C 2H 6 on methane solubility in NaCl brine solutions. This will be discussed further in section 5-4. 4-6-4 C 0 2/ H20 system : Again, the <^h 20 parameter equation we obtained from the CH 4 / H 20 system was used in this step to find the best values. The experimental data set we used for the C 0 2/H 20 system were from Tdedheide and Franck (1963), which contains 45 data points from 383 to 443 K and 99 to 1480 atm. Because of the widely reported quadrupole moment for the C 0 2 molecule, we found it necessary to include it as another parameter to be determined in addition to the parameter. Our results were : QC0 2 = 0.9315 with AD = -0.0118, and RMSD = 0.0791 . 4-7 Summary We have presented in this chapter all the details of the calculations involved in our model together with a computer algorithm to solve simultaneously the thermodynamic equations for vapor-liquid equilibria in our system. The most significant parameters that affect our predictions were found to be the a parameters of each component in the Lennard- Jones potential. We have thus discussed at length our data fitting and parameter determination procedure. We will next make extensive comparisons of our model predictions of methane solubility with experimental data and other correlations in chapter 5. Chapter 5 Results and Comparisons In chapter 4, we reported on the method whereby the a parameters for our system were determined using the available methane solubility data. Based on these parameters, we now address the questions : what is the effect of temperature, pressure, salinity, and other gas components on methane solubility ? We also want to compare the results of this study with those of previous studies. 5-1 The Effect of Temperature on Methane Solubility for Methane-Water Mixtures Only As was discussed in section 4-3, experimental results show that methane solubility goes through a minimum at about 344 Iv. Figure 5-1 compares the effect of temperature predicted by this model with McKetta’s data for methane-water at various pressures. The correspondence with the data is exceptionally good over the entire range, although there is a slight tendency at high pressure for the predictions to be low in the low temperature range and high in the high temperature range. For both the data and the prediction, at P= 22.5 atm, the existence of a minimum solubility is not clear, but as pressure increases further, it becomes evident. The partial molar heat of solution Ah 3 provides a quantitative measure of the temperature dependence of solubility, as given by the thermodynamiclly exact equation 2-13 : d in ( 7 3 Hg) _A h 3 5 111 ( f3 0 <9T _1 R d T 1 (2-13) 114 115 0.0080 0.0070- - 0.0060- X iS'0.0050- _ o i 0.0040- Q» J«u 0.0030!— 0.0020- 89 0.0010- P = 22.5 atm 0.0000 340 360 380 400 Temperature ( K) Figure 5-1 Minimum methane solubility in pure water around 344 K predicted by our model and the experimental data of Culberson and McKetta 116 In order to duplicate the minimum solubility behavior, Coco and Johnson had to assume a linear relationship of Ah 3 with temperature, then use equation 2-13 to adjust H 3 w ith temperature, and find values for the slope and intercept of the Ah 3 vs temperature line that caused a minimum in solubility to occur at the desired temperature. Figure 5-2 plots the logarithm of the apparent ideal solution fugacity, Inf?*, vs. temperature at various pressures. Clearly, it is not a monotonic function of temperature. Table 5-1 shows the magnitude that each term contributes to lnf^ as in equation 3-56, the perburbation expansion equation, as calculated by our model. hs,r n In f?' = fh_____ (. 1 111 l l — T.''T'K T K K T OX , K T d a A — 0 Ql — o + InKTp (3-56) 2K T \ — 0 T he ln K T /9 term in Table 5-1, which results from the sum of the invidual densities, changes little with temperature, but it does increase gradually as temperature increases. The first order term in a is only a small contribution to the total perturbation while the first and second order terms in A provide most of the adjustment from the hard sphere reference state, and they both decrease with increasing temperature as expected, since real fluids behave as hard sphere fluids in the high temperature limit (Tiepel 1973). The ^/i^S,r/KT^ term is from the hard sphere equation of state which also decreases with increasing temperature. The table shows that the sum of these terms results in a maximum lnf^at about 344 K at a pressure of 170 atm. This, of course, causes a minimum in methane solubility at this point. This minimum was predicted from the perturbation model by equation 3-56. Figure 5-3 compares our calculated results with Blount’s correlation and Coco and Johnson’s predictions at higher temperature and pressure conditions. It is very difficult to In f! ( Apparent Ideal Solution Fugacity) 11.40- 11.60- 12.00 10.60- 10.80- 11.20 300 Figure 5-2 M axim um app aren t ideal solution fugacity of m ethane predicted predicted ethane m of fugacity solution ideal t aren app um axim M 5-2 Figure y u moe aon 30 v vros pressure various t a Iv 360 around odel m our by = 558 atmP = = 240 atmP = Temperature K) ( 8 40 2 440 420 400 380 117 Table 5-1 Comparison of the magnitude of each terms in equation 3-56 for methane-water system hs.r 1 9 V\ 9 2 1 9 V; T P Vi ' l V\ InK T /i In f?' K T K T dX 2K T 5 A2 K T d a 298 170 9.988 -5.885 -0.631 -0.058 7.213 10.627 311 175 9.808 -5.618 -0.578 -0.056 7.251 10.807 34-1 174 9.234 -5.009 -0.468 -0.051 7.336 11.042 377 173 8.567 -4.503 -0.389 -0.046 7.405 11.034 411 169 7.825 -4.064 -0.329 -0.039 7.462 10.855 T in degree Iv and P in atm Methane Solubility X 0.0200 0.0100 0.0150 0.0250 iue - Co aio ofmehn ouiiy n ue tr rdce by predicted ater w pure in solubility ethane m f o parison om C 5-3 Figure hs td utnvs Data Sultanov's Correlation Coco's ■ Study This — Correlation Blount's - - rc' Data Price's □ = 00 atm 1070 = P ifrn creain wih xei na dat o rc n Sultanov and Price of ta a d ental experim ith w correlations different Temperature K) ( = 50 atm 580 = P 119 120 make many X vs T plots like this one from the available experimental data because most of the experiments were run by keeping temperature constant while varying pressure. But we do find these data points from Price (1979) and Sultanov (1972) which are at pressure conditions very close to each other to present on the same figure. From this comparison, we see that this study and Coco and Johnson’s correlation have about equally accurate predictions at the lower pressure of 580 atm and are better than Blount’s correlation, which is accurate only at lower temperature. At the higher pressure of 1070 atm, the three all give roughly equivalent results, with Coco and Johnson’s curve deviating the most at low temperatures, and this correlation and Blount’s correlation deviating the most at the higher temperatures. In summary, our model, based on perturbation theory, accurately duplicates the minimum methane solubility that occurs in the low temperature range without having to use equation 2-13 as the basis for the temperature dependence. In the higher temperature range, our model performs equal to or better than previous correlations in respresenting the temperature effect on methane solubility. 121 5-2 The Effect of Pressure on Methane Solubility for Methane-Watcr Mixtures Only As we have discussed in Chapter 4, the pressure effect on gas solubility enters into our calculation through the partial molar volume of the solute gas. Recall that the apparent ideal solution fugacity at system pressure can be corrected from that at saturation pressure using the Krichevsky-Kasarnovsky equation. Table 5-2 lists the contribution from various terms shown in equation 3-86 : p s V,1 v f i p 2 In (3-86) = m + TuMp-ps0- 2R T _ps /TV 1 P Generally speaking, Namiot’s correction, —— , which is the fifth term in Table 5-2, accounts for about 3% in the corresponding f°' due to the pressure increase at the lower pressure ( P = 671 atm ). But at higher pressure (1900 atm ), it accounts for about 20% change in { f . Therefore, it is important to include this correction at high pressure. Table 5-3 compares our predictions of V; and j3i for methane in pure water with reported values from the literature. Excellent agreement with experimental values was found. Our prediction is clearly more acceptable than Coco’s correlation of partial molar volume from his curve fitting results. The predicted j3i values show a minimum around 373 Iv, which was not known before, but the magnitude of these /?,• values are compatible with Choi’s results that were based on his experiments with the methane-water system. Figures 5-4 and 5-5 show that methane solubility in water always increases with increasing pressure. This study shows that our correlation gives the best global fit to the experimental data of various authors, but Coco and Johnson’s correlation is also very accurate for representing the whole data set. Blount’s correlation gives the biggest deviation, sometimes trending substantially away from the data. 122 Table 5-2 Contribution of various terms in equation 3-86 — P s ps V f 1 / p p s \ Pi v f i p 2 T (K ) P (atm ) In ( V ) R T V 1 ) 2R T M,,, 344 22.5 10.9570 0.0334 0 .0 0 0 0 10.9904 344 671.3 11.0636 1.0047 0.0290 12.0393 479 539.0 10.2807 0.6475 0.0183 10.9100 479 1900.0 10.4214 2.2965 0.2225 12.4954 565 188.5 9.0752 0.1504 0 .0 0 0 2 9.2254 565 1500.4 9.4473 1.7530 0.1847 11.0156 Table 5-3 Comparison of Vqjj and /?qjj predicted by this study with reported values in C H 4 /H 20 system — P s V , 1 ( cm 3 /g-m ole) Pi ( M P a ) ' 1 T (K ) This Coco Brevil’s This study (calculated) correlation reported study reported 298 42.1 61.2 37.1 37.0 * 0.000975 323 42.3 61.4 38.9 38.0 * 0.000880 0.00075 ** 373 43.2 61.7 43.8 0.000858 0.00095 ** 423 45.4 62.1 52.0 0.000932 * from experimental data of Krischevsky et al. (1945) ** from experimental data of Choi (1982) Methane Solubility X 0 0.020 0.030 0.040 . 010 - 0 0 30 0 50 0 70 0 90 00 10 20 1300 1200 1000 1100 900 800 700 600 500 400 300 200 00 Figure 5-4 C om parison of m ethane solubility in pure w ater predicted by B lount, lount, B by predicted ater w pure in solubility ethane m of parison om C 5-4 Figure oo ad u moe wih xei na dat o rc n O’Sullivan O and Price of ta a d ental experim ith w odel m our and , Coco = 355 K 375.5 = T = 53 K 553 = T Pressure (atm) — This Study Study This — - luts Correlation Blount's - - 'ulvns Data O'Sullivan's ♦ rc' Data Price's □ oos Correlation Coco's 124 125 0.0140F 0.0120- T = 473 K X 0.0100 Sultanov's Data 1o 0.0080 GO - - Blount's Correlation J 0.0060 o> Coco's Correlation 0.0040 — This Study 0.0020 T = 298 K □ McKetta's Data 0 100 200 300 400 500 600 700 900 1000 1100 Pressure (atm) Figure 5-5 Comparison of methane solubility in water predicted by Blount, Coco, and this study with experimental data of Sultanov and McKetta 126 The equilibrium gas phase has very little water in it until the temperature exceeds 373 K according to our calculations as shown in Table 5-4. At low temperature (< 373 K), therefore, the fugacity of the methane in the vapor mixture is presumably equal to that of pure methane times its mole fraction and hence its fugacity coefficient equal to that of the pure methane at the same temperature and pressure according to Lewis’s rule. The fugacity coefficient of pure methane can be written as (Prausnitz, 1969): P ( z - 1) In f t = d P (5-1) 0 Since the compressibility factor, Z, of pure methane goes through a minimum at low temperatures, we can expect the fugacity coefficient of pure methane and hence for methane in the gas mixture of our system to exhibit the same behavior. This is shown in Figure 5-6, where the solid lines are based on our calculation of the fugacity coefficient of methane in the equilibrium gas mixture using the SRK equation of state. The squares were read from the figure by Choi (1982), which was based on the volumetric data of Olds et al. (1943) for pure methane. The excellent agreement at 323 K shows that our calculations based on SRK equation of state for the gas phase was adequate. At 373 K, however, the disagreement becomes evident as we should expect, since more water is present in the vapor phase and it causes the fugacity coefficient of methane in the mixture deviates from that of pure methane at the same temperature and pressure. Table 5-4 Calculated vapor and liquid phase methane fractions in CH4/H 20 system T (Iv ) P (atm ) x c h 4 y c h 4 324.5 200 0.00225 0.99654 324.5 600 0.00374 0.99460 373.0 696 0.00448 0.98280 373.0 1520 0.00607 0.98430 473.0 387 0.00792 0.89801 523.0 1064 0.00227 0.83982 Fugacity Coefficient of Methane iue - Fgct cefcet fpr mtaefo Co’ td ad ht in that and study Choi’s from methane pure of coefficient Fugacity 5-6 Figure — This Study , For Equilibrium Gas Mixtures of Methane & Water Water & Methane of Mixtures Gas Equilibrium For , Study This — ■ Choi's Study , For Pure For , Study Choi's ■ ehn-ae mxuefo SK qain fsae n hs td over study this in temperatures state and of presures equation SRK moderate from mixture methane-water 0 400 300 Pressure (atm) = 28 K 298 = T 128 5-3 The Effect of Salt on Methane Solubility As salt is added to the CH 4 /H 20 system, the methane solubility decreases. This salting-out behavior is shown in Figure 5-7 at a constant pressure of 600 atm. Again, the calculated solubility goes through a minimum at about 344 K, according to our prediction. Also shown in Figure 5-7 are some experimental data from O’Sullivan and Smith (1970) which show little variation with temperature. This behavior is contrary to Culberson and McKetta’s results for pure water (m=0) and also to Blount’s finding for brine solutions at low temperature and pressure. The correlation results of Blount et al. are close to our prediction above 344 K, but disagreed below that temperature, as their correlation was acknowledged to become generally invalid at low temperatures and pressures. Figures 5-8 plots methane solubility against molality. The salting-out effect becomes more evident as the salt content increases, since more methane will be salted out. The slopes of the two curves in Figure 5-8 show that, the salting effect is affected by both temperature and pressure. We will discuss this further as we present the results of salting-out coefficients in Figure 5-11. The correlation of Coco and Johnson is also plotted in Figure 5-8 and our study shows better results than previous studies. In Figure 5-9 is plotted methane solubility vs pressure at different salt molalities. The relative improvement of this study over Coco and Johnson’s work is further exhibited here. While Blount’s correlation is completely invalid for these low temperature and pressure conditions, our model, on the other hand, predicts very accurate results even though the low temperature data of Blount were not included in our parameter fitting. This is shown in Figures 5-10 and 5-11 at different low temperature and low pressure conditions. Compared to Blount’s data, our model is again significantly better than the previous work of Coco and Johnson. Also notice in Figure 5-11 that the solubility of methane at 297 K is greater than that at 324 K. This indicates the existence of a minimum solubility, contrary to O’Sullivan and Smith’s data. It appears that the experimental error in O’Sullivan and Smith’s data must Methane Solubility X 0.0040- 0.0060 0.0010b 0.0020b 0.0030 0.0050 hs Study This — Correlation Blounfs - - 'ulvns Data O'sullivan's iue - Slig fet n taeslblt a 60 t n moderate and atm 600 at solubility ethane m on effect Salting 5-7 Figure temperature Temperature K) ( = 0 atm 600 = p 130 Methane Solubility X 0.0020 0.0120 0.0140 .00478 K P=890 atm 0 9 8 = P K, 8 7 4 0.0060- 0.0080 0.0100 - iue - Slig fet n taeslblt peitd y lut Coco, Blount, by predicted solubility ethane m on effect Salting 5-8 Figure 5 15 25 35 4.50 3.50 2.50 1.50 .50 a I oaiy mlsK water) NaCI Molality (moles/Kg n ti study this and Bon' Data Blount's ■ luts Correlation Blount's - hs Study This oos Correlation Coco's 131 Methane Solubility X 0.0000 0 0.0020 0.0030 0.0040 0.0050 0.0060 . 0 1 0 0 - Figure 5-9 C om parison of m ethane solubility in low and high salt content brine brine content salt high and low in solubility ethane m of parison om C 5-9 Figure 0 40 0 80 00 20 40 1600 1400 1200 1000 800 600 400 200 Bon' Data Blount's ■ hs Study This oos correlation Coco's luts Correlation Blount's ouin peitd y aiu orltos t Bluts ta a d lount’s B ith w correlations various by predicted solutions es e at ) tm (a re ressu P m = 7 K 373 = T m 5.03 0.8744 132 x Methane Solubility 0.0000(jL 0.00040 0.00020 0.00060 0.00080 0.00100 0.00120 0.00 0.00160 iue -0 oprsn fmtaeslblt i 08m rn t o temperature low at brine 0.89m in solubility methane of Comparison 5-10 Figure - — 0 0 0 0 100 80 60 40 20 ✓/ i i i i ^ / and low pressure conditions pressure low and a ^ X ,.** .♦ BIB ✓ / • —— /* ••X x X rsue (atm) Pressure = 297 K^x ^ K 7 9 2 = T = 0. 9 .8 0 = m ..... i __ hs Study This oos Correlation Coco's luts Correlation Blount's Data Blount's i i i i * * " 2 140 120 p ^ x ^ 133 Methane Solubility X 0.00020 0.00140 0.00160 0.00040 0.00060 0.00080 0.00100 0.00120 iue -1 oprsn fmtaeslblt a lw eprtrs n pressures and temperatures low at solubility methane of Comparison 5-11 Figure 20 in 0.89m brine solution brine 0.89m in 060 40 rsue (atm) Pressure m 9 K 297 0 0 10 140 120 100 80 0.89 2 K 324 hs Study This oos Correlation Coco's luts Data Blount's 134 1 3 5 have hidden the minimum predicted by the correlation in Figure 5-7. The salting-out effect defined by equations 1-1 and 1-2 can be calculated from our model. Table 5-5 lists some of the results of these calculations. The salting-out coefficient is only a weak function of the salt content, as shown by the case of T= 298 K, P= 200 atm, and a molality change from 0.4 to 4.0. The I values show a moderate dependence on pressure. For instance, at T= 373 K, m= 2.0, Ka change from 0.1107 to 0.09445 when pressure increases from 200 to 1500 atm. This is a change of 15% over the pressure range. Previously, Coco and Johnson (1981) treated their Ka values by averaging over pressure. Also, of course, the Ka value changes dramatically with temperature. It even goes through a minimum at about 380K. This was also observed in Coco and Johnson’s work. They found the Ks to reach a minimum value of 0.076 at around 424 K. Since there was error in Blount’s experimental data (1979), their conclusion may be in doubt. However, the qualitative behavior of Ivs vs temperature is confirmed in this study and also in the study of C 0 2 in NaCl solution by Ellis and Golding (1963). Figure 5-12 plots Ka vs temperature at various pressures. Table 5-6 is a summary of the reported Ka values. Our prediction of Ks falls in the midst of the reported values quite well. Tiepel and Gubbins (1972a) experimentally studied the variation of partial molar volume of solute gases with salt concentration and conclude that for a salting-out system, adding electrolyte causes the partial molar volume of the gas to decrease. This study also predicts this phenomenon. Figure 5-13 presents the results. Table 5-5 Calculated salting-out coefficient based on our model T (K) P (atm) m In (ff) * In ( f f ) K s 298 200 0.4 10.9375 11.0518 0.0497 0.12416 298 200 1.0 10.9375 11.2196 0.1225 0.12251 298 200 2.0 10.9375 11.4944 0.2419 0.12094 298 200 4.0 10.9375 12.0355 0.4769 0.11922 298 600 0.4 11.6453 11.7513 0.0460 0.11507 298 600 1.0 11.6453 11.9073 0.1138 0.11377 298 600 2.0 11.6453 12.1642 0.2254 0.11269 298 600 4.0 11.6453 12.6751 0.4472 0.11181 298 1000 0.4 12.3011 12.4019 0.0438 0.10948 298 1000 1.0 12.3011 12.5505 0.1083 0.10832 298 1000 2.0 12.3011 12.7959 0.2149 0.10746 298 1000 4.0 12.3011 13.2863 0.4279 0.10697 324 200 2.0 11.2198 11.7710 0.2394 0.11968 324 600 2.0 11.8748 12.3901 0.2238 0.11190 324 1000 2.0 12.4832 12.9753 0.2137 0.10685 324 1500 2.0 13.2025 13.6747 0.2051 0.10254 343 200 2.0 11.3053 11.8275 0.2268 0.11338 343 600 2.0 11.9323 12.4205 0 .2 1 2 0 0.10600 343 1000 2.0 12.5144 12.9803 0.2023 0.10117 343 1500 2.0 13.2025 13.6492 0.1940 0.09701 373 200 2.0 11.2941 11.8040 0.2215 0.11073 373 600 2.0 11.8884 12.3645 0.2067 0.10337 373 1000 2.0 12.4374 12.8915 0.1972 0.09859 373 1500 2.0 13.0848 13.5197 0.1889 0.09445 398 200 2.0 11.1787 11.6904 0 .2 2 2 2 0 .1 1 1 1 0 408 200 2.0 11.1121 11.6247 0.2227 0.11133 478 200 2.0 10.4282 10.9568 0.2296 0.11479 523 200 2.0 9.858 10.4582 0.2607 0.13033 * indicates in pure water Salting-out Coefficient Ks 0.090 0.100 0.110 0.120 0.130 0.140 iue -2 h eitne famii m slig u cefcet vr h temperature the over coefficient out salting um inim m a of existance The 5-12 Figure 300 ag ad h vrain fK wt pesr a m=l ae o or model our on based l = m at pressure with Ks of variation the and range acltd Results Calculated 340 ae o Or Model Our on Based eprtr ( K) ( Temperature 380 0 atm 200 1000 ^ 0 atm 600 = 50 atm 1500 420 n r t a 6 500 460 137 1 3 8 Table 5-6 Reported values for salting-out coefficient of CH4-NaCl pair A uthor Ks Method Used Blount et al. (1981) 0.1025 exp. determined Susak et al. (1980) 0.1290 assum ed Haas (1979) 0.1 1 0 O’Sullivan’s data Morrison et al. (1952) 0.1270 exp. d a ta Long et al. (1952) 0.167 M-L theory Masterton (1970) 0.131 scaled particle theory Tiepel (1973) 0.133 perturbation theory Coco et al. (1981) 0.076 - 0.175 thermodynamic correlation T his Study 0.095 - 0.14 perturbation theory correlation K is in (Ljnok_V i \lvg-water/ Partial Molar Volume of Methane (cc/mole) 38 40 42 44 46- 48 iue -3 rdce pril oa ou o tae t aiu sl concentration salt various at ethane m of e volum molar partial Predicted 5-13 Figure 0 5 1.00 .50 00 Molality water) (moles/Kg 9 K 298 = 408 K 8 0 4 = T 2.00 3.00 = 478 K 8 7 4 = T ac Results Calc. K 4.00 139 140 5-4 The Effect of Other Gases on Methane Solubility in Water And in Brine Solutions The solubility of C 0 2 in water is about 6 times greater than that of methane, and methane solubility in water is about twice that of ethane as indicated in Table 5-7, which gives experimental solubilities for the pure gases in water. How will the presence of C 0 2 or C 2H 6 (which are always present to some extent) affect the solubility of methane in underground brine solutions ? In this section, we address this question. We will examine the change of the thermodynamic properties that take place both in the gas phase and liquid phase upon adding another gas component to the system. 5-4-1 CH4/ C 2H 6 / II20 / NaCl System Based on our calculated results, the presence of ethane in a brine solution always salts-out methane. The curves in Figure 5-14 show the results of our model predictions for total gas dissolved, methane solubility, and ethane solubility as a function of ethane content of the dissolved gas. At low ethane mole fraction, the total gas solubility approaches that of pure methane solubility in brine, while at the other extreme, it reaches the pure ethane solubility in brine. This is contrary to the results of Blount’s experiments. He found that as the ethane content of the solute is increased from zero, ethane at first salts methane into solution, while above 6 to 8 mole percent ethane in the dissolved gas in solution, methane is salted out by ethane. Blount’s data are also shown in Figure 5-14. His data for ethane agree well with our calculated ethane curve throughout his experimental range. For methane, there is good agreement only at very low ethane content and also when ethane content exceeds 15 mole percent. No experimental data are available for comparison at higher ethane concentrations for this time. Since all the parameters we used to make our calculation in this system are based on the results we obtained from parameter fitting at other system conditions from Table 5-7 Experimental gas solubility data of C 02, C2H6 , and CH4 in water T ( K) P (atm ) X^ X^ XC 2H 6 411 242 0.00301 (a) 0.0015 (b) 423 197 0.0215 (c) 0.00300 (d) 473 493 0.047 (c) 0.00896 (d) 0.005 (e) (a) Culberson and McKetta (1950) (b) Culberson and McKetta (1949) (c) Takenouchi and Kennedy (1964) (d) Sultanov et al. (1972) (e) Hayduk (1982) Gas Solubility X 0 0.0030 0.0040 0.0050 0.0060 0070- 0 7 0 .0 0 0.0080 . 0010 iue -4 h efc o H o te tae ouiiy n rn slto at solution brine in solubility ethane m the on 2H6 C of effect The 5-14 Figure - 0% C2H6 100% ifrn dsovd a ratio gas dissolved different 26 I Te sovd Gas issolved D The In % C2H6 0% CH4 100% C2H6 oa a Dissolved Gas Total Bon' Mtae Data Methane Blount's ■ Bon' Ehn Data Ethane Blount's ♦ CH4 1.8311530 T= atm 142 experimental data of various authors, we consider the results of the calculated predictions made here to be successful. They may certainly be improved should more experimental data on ethane-methane mixtures become available. The discrepancy between predicted and experimental methane solubility where ethane is 6 - 8 mole percent of the dissolved gas is a fundamental one which can not be resolved by better parameter fitting of our model. In order to salt in methane while adding ethane to the system, there either must be some other molecular attractive forces between methane and ethane not accounted for in our model or else the presence of the ethane must somehow make the enviroment (solvent) more suitable to dissolve methane. The first possibility seems very unlikely because no such attractive force can be identified and even if there is one, the attractive force would be proportional to the ethane mole fraction in the dissolved gas and thus make the methane become salted-in even more as the ethane content increased. Therefore, the second possibility would seem to be the more likely one. We really can not say much about the second possibility except that we know of a somewhat similar unusual behavior of aqueous solutions that has been observed and is generally referred to as ‘hydrophobic interaction’, which gives a favorable free energy for increased solubility (O’Connell 1987) only at low concentrations. According to this theory, it is possible that two gases dissolved in water will cause a ‘salting-in’ behavior for each other at low concentrations. The hydrophobic interaction itself is a broad area of research and was not originally intended to be included in this study. Blount’s experimental data alone are not sufficient to draw a definite conclusion that this behavior actually exists in this system. Table 5-8 lists the calculated results of importance for this system. The apparent ideal solution fugacity of methane decreases less than 8 % from no ethane present to 99% ethane in the dissolved gas in liquid, but the ratio of the apparent ideal solution fugacities of Table 5-8 Calculated vapor and liquid phase thermodynamic properties for C 2H 6/CH 4/H 20/NaCl system at 423 K, 1530 atm and 1.83m. a c , h 6 v fO t f° 1 ^ C H 4 x c h 4 X C 2 H 6 y c h 4 X c 2 H 6 * C 2 H 6 c h 4 f C 2 H 6 x c h 4 0 0 . 0 0 6 2 0 0 . 9 6 5 5 0.0000 2.068 494347 0.00901 0.0060 5.41E-5 0.9402 0 . 0 2 4 3 2 . 0 7 0 1 . 7 5 2 4 9 3 4 6 5 1 1 9 9 0 0 1 0 . 0 1 1 2 1 0.0060 6.73E-5 0 . 9 3 4 1 0 . 0 3 0 1 2 . 0 7 1 1.749 493253 1 1 9 7 8 0 3 0 . 0 3 9 9 2 0 . 0 0 5 6 2 . 2 4 E - 4 0.8605 0.1008 2 . 0 8 0 1.719 490676 1 1 8 9 4 4 8 0.04713 0.0055 2.59E-4 0.8434 0.1171 2.083 1 . 7 1 3 4 9 0 0 7 6 1 1 8 7 0 7 1 0 . 0 8 5 4 7 0 . 0 0 5 0 4 . 2 7 E - 4 0 . 7 6 1 6 0 . 1 9 5 6 2 . 0 9 8 1 . 6 8 8 487169 1177613 0 . 0 9 0 9 1 0 . 0 0 5 0 4 . 4 8 E - 4 0.7510 0.2057 2 . 1 0 0 1.685 486793 1176436 0 . 1 4 5 8 3 0 . 0 0 4 4 6 . 4 2 E - 4 0 . 6 5 7 9 0 . 2 9 4 9 2 . 1 2 1 1.664 483411 1 1 6 5 8 9 6 0 . 1 5 8 4 2 0 . 0 0 4 3 6 . 7 3 E - 4 0 . 6 3 9 5 0 . 3 1 2 6 2 . 1 2 6 1 . 6 6 1 4 8 2 7 3 2 1 1 6 3 5 6 6 0 . 1 7 8 0 8 0.0041 7.31E-4 0.6130 0.3379 2 . 1 3 3 1 . 6 5 6 4 7 8 2 6 4 1 1 4 9 6 8 7 0 . 2 0 8 3 3 0 . 0 0 3 9 8 . 1 3 E - 4 0 . 5 7 5 6 0 . 3 7 3 8 2 . 1 4 3 1 . 6 5 1 4 7 6 8 6 4 1 1 4 5 0 9 7 0.24894 0.0036 8 . 9 6 E - 4 0 . 5 3 1 7 0 . 4 1 5 8 2 . 1 5 5 1 . 6 4 5 4 7 5 2 0 0 1 1 3 9 3 8 6 0.50000 0.0026 0.00129 0 . 3 5 9 4 0 . 5 8 1 0 2 . 2 0 8 1 . 6 3 3 4 7 1 8 3 1 1 1 2 8 0 4 9 2 . 0 0 0 0 0 .00092 0.00185 0.1217 0.8090 2 . 2 9 1 1 . 6 3 5 4 6 1 5 7 2 1 0 9 4 7 1 0 10.0000 .00021 0.00209 0.0268 0 . 9 0 0 1 2 . 3 2 6 1 . 6 4 0 4 5 7 1 8 7 1 0 8 0 5 7 1 5 0 . 0 0 0 0 .00004 0.00215 0.0055 0.9206 2 . 3 3 4 1.641 456175 1077334 1 0 0 . 0 0 0 .00002 0.00215 0.0027 0.9232 2 . 3 3 5 1 . 6 4 2 4 5 6 0 4 5 1 0 7 7 3 3 4 200.000 .00001 0.00216 0.0014 0 . 9 2 4 5 2 . 3 3 5 1.642 455979 1077334 145 methane to ethane remains virtually constant. The fugacity coefficients in the gas phase do not vary too much either and the ratio of the two remain almost fixed. If we rearrange the phase equilibrium equations for the two component, we can get : CH. K Y, CH. C,H2 n 6 / f°3 ' 4>4 a CH. c 2h 6 - = constant K, X CH4 "C9H2 n 6 y (5-2) This expression tells us that the relative volatility of CH 4 to C 2H 6 in w ater a t 423 K is approximately constant over the entire composition range and is approximately 0.35 as shown in Figure 5-15. This is, of course, a reversal of the relative volatilities that methane and ethane exhibit in hydrocarbons. Presumably, this reflects the different net attractive forces between water and these two components, and it is predicted very nicely by our model. 5-4-2 C 0 2 / C H 4 / II20 / NaCl System Before we proceeded to use our model to calculate methane solubilities for this system, we needed to decide whether the ionic reaction between C 0 2 and H20 is important. Recall the equation in 3-95 : 2 m ’ m „ = (3-95) K I f j + 2 m A + \ K I + 4mA K I f YCH4 / YC2H6 40 30 10 iue -5 h rto fmehn t ehn i te ao pae rdce by predicted phase vapor the in ethane to ethane m of ratio The 5-15 Figure - acltd eut a h Conditions The at Results Calculated f luts iud hs Data Phase Liquid Blount's of and salt m olality of 1.83 of olality m salt and u mdl t luts xeietl odto o 50 t 43 K, 423 , atm 1530 of condition experimental Blount’s at model our C 4 XC2H6 XCH4 / 146 147 The equilibrium constant, was found to be 0.179 x 10 "6 mole/ Kg-water at 423 K based on a correlation of equation 3-94 by Edward, et al. (1978). This value was used to calculate the dissociation ratio at different Iv values as shown in Table 5-9. From this table, we conclude that it is not important to consider the association effect of C 0 2 in this study. Burmakibn, et al. (1982) found that at room temperature, only 0.0005 of the C 0 2 in NaCl brine solution is present as H 2CC>5". Figure 5-16 compares our predicted curves of gas solubilities vs C 0 2 content of the dissolved gas with Blount’s experimental data for the C 0 2/CH 4 /H 20/NaCl system. The agreement is very good up to 60 percent of C 0 2 in total dissolved gas at the two experimental pressures of 1530 and 8 8 6 atm, a salt molality of 1.82m, and at temperature of 423 K. The deviations of the experimental data from the predicted curves at C 0 2 contents between 10% to 30% was explained by Blount et al. (as for the ethane-methane system in the previous section) as an initial increase in methane solubility due to the presence of the other gas followed by a drop in methane solubility as the percentage of the other gas increased further. Our model again does not predict this phenomenon. Since the phenomena is quite unusual, and the few data points of Blount et al. are not yet confirmed by other investigators, we believe it is possible that the deviation of the data points from our predicted curve around 10 to 30 C 0 2 percentage are due more to experimental error than a salting-in effect. A more significant disagreement between Blount’s data and our predictions is shown in Figure 5-17. Here we see that for P=1530 atm, T=423 Iv, and salt molality =0.89 at low C0 2 concentrations, our predictions agree well with his data, despite the fact that some of the data show a slight salting-in effect of C 0 2 on methane, but at high C 0 2 concentrations (80%), our predictions of total gas dissolved and of methane solubility are almost twice those of Blount’s data. Interpolation suggests that we would expect that at 60% C 02, our prediction would be about 1.9 times higher than his data. But we found in Figure 5-16, our predictions for this C 0 2 148 Table 5-9 Ratio of undissociated C 0 2 to the total amount of C 0 2 in water at 423 K and at different f 7 a / 7 J values 7 a h 2± K ( 7 a h \ ) m a /m A 0.0001 0.179x 10‘10 1 1 0.179x 10'6 1 50 0.877 x 10'5 0.997 500 0.893x 10'4 0.991 2000 0.358x 10*3 0.981 10000 0.179x 10'2 0.959 Gas Solubility X 0 0.0050 0.0060 0.0070 0.0080 0.0090 0.0100 0.0110 0.0120 . 0040 iue -6 rdce mehn ouiiy n ttl a dsovd n .2 NaCl 1.82m in dissolved gas total and solubility ethane m Predicted 5-16 Figure 0 - — ehn (hs Study) (This methane ■— Study) (This gas total 1 - Bon' Mtae Data Methane Blount's ■ Bon' TtlGs Data TotalGas Blount's □ 0 0 0 0 50 40 30 20 10 ouin t w dfeet rsuead aiu C02 percentage 2 0 C various and pressure different two at solution n h isle gas dissolved the in = 86 atm 886 = p ' - a 0 % n h Dsovd Gas Dissolved The % In C02 a ✓ = 1.82 = m =1530 atm 0 3 5 1 = p 60 ,'u 149 Gas Solubility X 0.030 0.040 0.020 0 0.010 , 0 0 0 . iue -7 oprsn fpeitd ehn ad oa gs isle i 0.8898m in dissolved gas total and methane predicted of Comparison 5-17 Figure Bon' TtlGs Data TotalGas Blount's □ oa a (hs Study) (This Total Gas luts ehn Data Methane Blount's ehn (hs Study) (This Methane ecnaei te isle gas dissolved the in percentage al outo wt Bon’ eprmna dt t aiu C02 0 C various at data experimental Blount’s with soluition NaCl 0 % n h Dsovd Gas Dissolved % The In C02 7 am T= atm 874 CH4 D 0X CH4 100X D D □ 100% C02 8898 9 8 .8 0 150 151 content agreed well with his experimental data for 1.82 salt molality. Although not conclusive, this suggests to us that the experimental data, are not consistent. Furthermore, as the total dissolved gas concentration approaches the saturation concentration for C 02, which is over 6 times greater than the solubility of methane (Table 5-7), our prediction shows a gradual smooth approach to the final solubility while Blount et al., based on his data at 80% C 0 2 content, argued that there must be a sharp upward inflection in the total gas curve in the region of 90 to 95 mole percent C 02. Unfortunately, Blount et al. had no data in this region to support this statement. Table 5-10 lists calculated results for T= 423 K, P = 874 atm and a salt molality = 0.8898. The same conclusions concerning the constancy of the ratios of apparent ideal solution fugacities, the fugacity coefficients, and the relative volatilities of the two components can be drawn as in the previous section for the methane-ethane system. Figure 5-18 shows the slope of ( y Ch 4 /y Cq,) vs- (^CH 4 /^co ,) *s 8*8 which, of course, reflects the higher solubility of C 0 2 compared to that of methane. An important conclusion concerning the effect of a second gas on methane solubility can be drawn based on the model-predicted constancy of the ratios of the fugacity coefficients and the apparent ideal solution fugacities. First, simplified molecular considerations suggest that if the concentration of a second gas in the liquid phase is low enough, it should not affect appreciably the escaping tendency of the methane as measured by its apparent ideal solution fugacity. Also, in the vapor phase, the fugacity coefficient of the methane may be only slightly affected by a second gas at all concentrations. When this is true, the effect of a second gas on methane solubility could only be from its diluent effect in the vapor, lowering the concentration of the methane, and hence the solubility of the methane proportionally. The results of Tables 5-8 and 5-10 have shown that the ratios of the fugacity coefficient, the Calculated vapor and liquid phase thermodynamic properties for C02/CH4/II20/NaCl system at 423 Iv, 874 atm and 0.89 m v f o i fO t x c h 4 X C 0 2 1 c h 4 y Co 2 ^CH4 4> c o 2 c h 4 fc o 2 0.00600 0.00010 0.9575 0.0012 1.327 0.831 188464 12382 0.00580 0.00087 0.9433 0.0149 1.317 0.825 187170 12333 0.00571 0.00143 0.9334 0.0245 1.318 0.824 188401 12382 0.00557 0.00223 0.9189 0.0386 1.318 0.822 190195 12457 0.0049.3 0.00592 0.8486 0.1068 1.322 0.813 198822 12823 0.00455 0.00819 0.8024 0.1515 1.326 0.807 204420 13056 0.00420 0.01050 0.7541 0.1982 1.338 0.804 212798 13400 0.00400 0.01190 0.7236 0.2275 1.343 0.801 217203 13575 0.00370 0.01330 0.6923 0.2576 1.347 0.798 221066 13739 0.00340 0.01500 0.6469 * 0.3010 1.344 0.789 222969 13822 0.00296 0.01776 0.5767 0.3679 1.356 0.781 230884 14143 0.00237 0.02137 0.4770 0.4621 1.379 0.771 242088 14589 0.00188 0.02443 0.3856 0.5476 1.406 0.764 252248 14973 0.00162 0.02600 0.3363 0.5933 1.425 0.762 257694 15184 0.00138 0.02753 0.2867 0.6389 1.446 0.759 263176 15398 0.00030 0.03390 0.0706 0.8330 1.581 0.761 287740 16318 YCH4 / YC02 600 500 700 800 300 200 0 0 1 iue -8 h rto fC4 o i te ao ad hs peitd y our by predicted phase and vapor the in 2 0 C to CH4 of ratio The 5-18 Figure acltd sls tTe Conditions The at esults R Calculated salt content of 0.89m of content salt oe a Bon’ xeietl odto o 7 am, 2 K ad a and K, 423 , atm 874 of condition experimental Blount’s at model f luts iud e Data se a h P Liquid Blount's of C 4 XC02 / XCH4 153 apparent ideal solution fugacity, and the relative volatility of the two gases were indeed essentially constant over the range of realizable vapor and liquid compositions. This suggests strongly that the simplifying assumptions stated above do approximately hold true when the second gas is C 2 H 6 or C 02. These simplifications also make it difficult to justify the salting-in effect reported by the experimental results of Blount et al.. 5-5 Summary We have presented our predictions of methane solubility in this chapter. From the comparisons both with other correlations and available experimental data, we considered our model to be successful both in its global nature to represent gas solubilities at geopressured conditions over a broad range of temperature, pressure, salt content, and in its ability to predict methane solubility in the presence of other gases using model parameters determined from individual gas solubility data. In chapter 6 , we will present our conclusions and recommendations based on this research. Chapter 6 Conclusions and Recommendations In this research, we set out to develop a global model to predict methane solubility in geopressured reservoirs knowing the temperature, pressure, and salinity of the system ( when other gases are present, the expected ratios of each solute gas to the solute methane gas must also be specified ). A simplified version of the second order Leonard-Henderson-Barker’s perturbation theory was used to calculate a newly defined thermodynamic property, f?\ the apparent ideal solution fugacity of a solute gas in the solution. Using this quantity for each solute gas, the vapor-liquid equilibrium equations of the system can be solved by an iteration solution procedure to give the desired result, a prediction of methane solubility, as well as the composition of the equilibrium vapor phase. We have demonstrated why and how the simplified model was modified by introducing a temperature dependence for the L-J cr parameter for water and the modified model was then fit to all the experimental solubility data available at various system conditions. By choosing values for the cr parameters of each component within the ranges reported in the literature, the resulting model gives excellent agreement with other thermodynamics properties, such as the partial molar volume and the isothermal compressibility of the partial molar volume , V,- and /?,■ of solute gases. The development of the concept of an apparent ideal solution fugacity of pure component i, f°', which is equivalent to the product of Henry’s constant and its corresponding activity coefficient ( 11° ), enabled us to treat gas solubility in a multicomponent system in a straight-forward manner. There was no advantage in retaining the concept of Henry’s constant at infinite dilution ( developed for binary systems ) corrected for a composition effect through a Margules or other type of activity coefficient model, because the Henry’s constant itself is 155 156 composition dependent in multicomponent systems. The apparent ideal solution fugacity of pure component i is itself the single measure of the effect of intermolecular forces between the solute i molecules and all the other components and hence is a function also of the concentration (molecular density) of each individual component in the liquid phase. Experimental data on total gas-free solution density were needed and were correlated into a function of temperature and salinity for this system. The apparent ideal solution fugacity at a total pressure equal to the vapor pressure of the solvent, using a low pressure solution density, was then corrected to high pressure values through a Krichevsky-Illinskaya type of equation. The required partial molar volume of the solute gas and its coefficient of isothermal compressibility were self-generated through the perturbation approach. Although it was not done in this work, a final correction to the total solution density based on the partial molar v olumes of each solute gas could be included in the calculation procedure should solute gas concentrations be high enough to warrant it. The effect of pressure on methane solubility was checked both by comparison with experimental solubility data and also by comparison of the corresponding predicted partial molar volume of the solute gas with experimental data. Agreement in both cases indicates that our model is quite valid from 10 to 2000 atm. The predicted coefficient of isothermal compressibility of the partial molar volume of the solute gas is of the right order of magnitude compared with other reported values. Our calculations also reveal that the effect of variation of partial molar volume with pressure on the methane solubility can not be neglected when the pressure exceeds about 1500 atm. A very interesting feature of our model is that it successfully predicts a minimum methane solubility around 344 Iv, in good agreement with available data. The corresponding apparent ideal solution fugacity also shows a maximum at that temperature. Experimental 157 solubility data agree well with our predictions all the way from 298 to 573 K. The salting-out effect is accurately predicted by our model. Our model fits Blount’s experimental data better than the previous correlation of Coco and Johnson and other authors, all of which used a salting-out coefficient that had to be empirically correlated for T, P, and salt content. Salting-out coefficients back-calculated from our model-predicted solubilities varied both with temperature and pressure but remained relatively fixed for different salt concentrations. These salting-out coefficient also show a minimum at around 380 K, in good agreement with other reported values. Our model produced the lowest deviation in terms of RMSD among all the correlations developed to date for the system of methane in pure water as well as for methane in NaCl brine solutions (Larsen and Prausnitz’s work on pure water not included in the comparison because of lack of available results). It requires by far the fewest parameters to be determined from a fit of the data. The model has proved to be extremely successful when extended to conditions outside those for which the parameters were fitted. It also has the capability of readily handling components other than those we have studied in this research with a minimum of new data. The effect of other salts on gas solubility can be predicted once the solution density of mixed salt solutions is available, and the effect of other gases that do not form new components in solution require merely estimates of Lennard-Jones parameters and approximate electrostatic parameters to produce resonably accurate predictions. We have also fit the solubility data of ethane and carbon dioxide in water and used these results to predict the effect of other gas components on methane solubility. Comparison of these predictions with Blount’s sparse solubility data shows generally acceptable agreement for moderate dissolved gas ratios in both cases. But at very low and high dissolved gas ratios, 158 we found large discrepancies between the model predictions and Blount’s data. At present, no definite conclusion can be drawn due to lack of sufficient experimental evidence, but suffice it to say that we suspect the validity of the unusual behavior of Blount’s data until further data are made available. As far as our initial goals are concerned, we have successfully achieved them. This certainly does not mean that our model is complete. We realize that there are still many simplifications that were made in the model which approximate reality. However, we also realize that any further removal of these simplifications would require a tremendous amount of additional effort. Since the RMSD of our model is already quite small, it seems unlikely that an attempt to remove some of the simplifications could be properly confirmed or rejected by statistical tests for substantial further improvement in the global fit of the correlation. But, we do feel that the following areas might bg considered to extend the model’s applicability and increase its robustness : (1) The intermolecular forces included in the pressure equation (3-67) are not explicitly complete because only the non-polar part of intermolecular forces are separately accounted for. The dipole-dipole and dipole-charge interactions are assumed to be already included in the equivalent Lennard-Jones potential that is being used. If these missing interactions were included explicitly, then the Lennard-Jones potential parameters would have to be modified accordingly. If this modification were attempted, it would not necessary have a direct effect on the calculation of the apparent ideal solution fugacity, but it would certainly affect the calculated partial molar volume of the solute gas. In addition, the current approximation may introduce a bigger error as the salt concentrations increase. A revision of the pressure equation to include all the possible interactions more explicitly is desirable, although it may not improve the global fit of the model to the available data very much. 159 (2) The solvent concentration was assumed to be high enough to make its activity coefficient equal to 1. This assumption would need to be modified at higher salt concentrations. (3) More experimental data are needed to evaluate more conclusively the effect of other gases on methane solubility. The possible hydrophobic interaction of Blount’s data might lead to improvements in the model if it proves to be real. (4) Solution density data on salt solutions are needed before we can extend our model predictions to include the effect of mixed salts on methane solubility. Also, at present, the cr parameters for both the anion and the cation are expressed by the same factor times their Pauling diameters. This could also be changed should solubility data with other salts be made available. 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(1976) developed a perturbed hard sphere equation of state for a gaseous mixture containing highly polar molecules such as water and ammonia. This equation of state can be used to calculate fugacity coefficients for components in gaseous mixtures often encountered in chemical technology over a wide temperature range and for pressures up to about 5000 psia. This equation is expressed by: R T | + £ 2 —£ 3 ______p = ir — I - ——JN (A-!) (1 - £)3 ) v( V + c) with the reduced density defined by : £ = (A -2 ) parameter b reflects the hard core size of the molecule and a reflects the strength of attractive forces. They are both temperature dependent. a = a + (3/ T (A-3) log b = - 7 - 6 T (A-4) where a , (3, j, and 6 are empirical constants. C is a small positive constant independent of temperature. For a mixture, upon assuming a one-fluid theory, the above constants become composition dependent and are given by the following relations : 167 168 m 3M = E y,y«'i bh £ i ( A - 5 ) = i m M = E y m m m = EEE E y < y j a a ( A - 7 ) .■=1 3=1 where M stands for the mixture and m is the number of components. aij — a ij + Pij / T (A -8 ) and a ij = Q l j ) + Q\j ( A - 9 ) Pi j — (A -1 0 ) C onstants a - and [3,■ j reflect intermolecular forces of attraction between molecules i and j. Nakamura et al. proposed to retain as adjustable parameters to fit binary data while fixing the other parameters according to the following arbitrary rule : (l) / (i) (l) \i/2 aH = { Qi a3 ) (A-ll) P?? = ( P S1} P f )1/2 (A-1 2 ) Pi? = \ ( Pi0) + p T ) (A-13) 169 The expression they obtained for fugacity coefficient of component k in a mixture is : — 2 £ 2 \ m In CO RTV £ yj aks 1 + ( £ ) (> ST* )= i a M C k ( ~ l) n (n + l) + n + 2 ( , 1 , )" + 1 ) RTV 2 (V n^ = l M In | 1 (A-14) X ( ^ L ) » 2 In Z Equation A-14 was tested against equation 4-14 from SRK equation of state for the system CH 4 /II20 at various temperatures and pressures. Table A-l shows the results. 170 Table A-l Comparison of predictions of fugacity coefficients of SRK and Nakamura’s equation of state in CH 4 /H 20 system T P calculation using SRK calculation using Nakamura (K) (atm) Y C H 4 344 22.5 .9832 .98 .857 .0004 .9845 .975 .925 .0004 344 64.2 .992 .947 .644 .0009 .9935 .933 .798 .0009 344 444 .9918 .897 .116 .0032 .9966 .827 .280 .0031 377 243 .9831 .935 .320 .0026 .9874 .882 .507 .0025 377 672 .9813 1.085 .135 .0045 .9911 .993 .284 .0044 411 670 .9639 1.15 .198 .0057 .9791 1.052 .343 .0055 444 176 .9155 1.003 .565 .0035 .931 .962 .693 .0034 444 680 .9367 1.208 .256 .0079 .9581 1.103 .380 .0076 479 51 .6138 1.018 .832 .0013 .6361 1 .0 0 1 .883 .0013 479 290 .8676 1.061 .482 .0070 .8939 .997 .602 .0068 479 1286 .9106 1.783 .270 .0145 .9333 1.311 .383 .0139 494 363 .8441 1.104 .447 .0094 .8755 1.026 .560 .0092 494 620 .8645 1.235 .344 .0127 .8965 1.127 .450 .0 1 2 2 Appendix B: Orientation-averaged Molecular Interaction The potential energy of two permanent dipoles i and j can be expressed as (Prausnitz, 1969) : y ij = ~ ^ r g (A-15) (A-15) r with g = sin 9{ sin 0 • cos 4’ij ~ 2 cos 9{ cos 9j (A-16) where the angle 9 and ip giye the orientations of the dipole axes as shown below: + e + e -e The average potential energy V,- ■ between two dipoles is found by averaging over all orientation with each orientation weighted according to the following equation by Rowlinson (1958): T_ , / exp { — /K T ) d fi v »j = ~ K T 111------J - J q ------(A-17) Where df2 is the element of the solid angle : d fi = sin 9j sin 9j d 9i <19j d 4’ij (A-18) 171 w ith 0i and 0j varying from 0 to 7r rad. and i/>, j from 0 to 2 tt rad., the exponential function in A-17 is expanded into infinite series. When the temperature is not too low, this series converges rapidly. Therefore , ...... =_ — K T In /' d fi = - K T In ( 1 + -lj f t ^ + V 3r6 (Ij s ' (A-19) 3K Tr Appendix C : Potential Energy for Water Molecule The potential energy for water molecule can be expressed as the sum of non-polar and polar potential. The non-polar part is expressed by Lennard-Jones potential and the polar part by the Dipole-Dipole interaction as in equation A-19 and the Dipole-Induced Dipole interaction. V , = *?. ( ( 4 * ) 12 - ( ^ ) 6 ) - - A - (A-20) We can combine the last two terms with the r " 6 term in the Lennard-Jones potential to obtain: . = “'n ( ( 4 11 ) 12 - ( ^ ) 6 ) (A-21) w here a ll and e 11 are related to ajj and by: /_0 \6 ( * n ) 6 = (A-22) *11 = «n ( * ) 2 (A-23) w hen 5 = 1 + ------4 o 6 + Q0n 6 (A-24) 12 K T €°n {cr°nf 4 e°n (cr°n f 173 Appendix D : Computer Program 174 * * * * * * * PURPOSE OF EACH SUBROUTINE AND ENTRY * * * * * * * * * * * * * MAIN MAIN PROGRAM TO CALCULATE GAS SOLUBILITY USER SUBROUTINE TO PROVIDE SYSTEM INFORMATION ANION INFORMATION ABOUT ANION CATION INFORMATION ABOUT CATION DENSE PROVIDE SOLUTE GAS FREE SOLUTION DENSITY DGAS INFORMATION ABOUT OTHER DISSOCIATED GASES PHIINT IN IT IA L GUESS OF FUGACITY COEFFICIENT RATIO PROVIDE INFORMATION ABOUT DISOLVED GAS RATIO SALINE PROVIDE INFORMATION ABOUT SALT MOLALITY SOLVEN PROVIDE INFORMATION ABOUT SOLVENT OTHER THAN WATER VNGAS INFORMATION ABOUT OTHER NON-DISSOCIATED GAS SOLUTE TIEPEL SUBROUTINE TO CALC. APPARENT IDEAL SOLUTION FUGACITY AISF CALCULATE APPARENT IDEAL SOLUTION FUGACITY DIAT CALCULATE DIAMETER OF EACH PARTICLE TSAI CALCULATE REDUCED DENSITY AS IN EQN. (3 -4 3 ) SOFT CALCULATE SOFT SPHERE CHEMICAL POTENTIAL UHARD CALCULATE HARD SPHERE CHEMICAL POTENTIAL SRK SUBROUTINE TO CALCULATE VAPOR PHASE PROPERTIES POYN CALC. POYNTING FACTOR OF SOLVENT SATP CALC. SATURATION PRESSURE OF PURE SOLVENT SRKPS CALC. FUGACITY COEFF. OF PURE SOLVENT AT SAT. PRES. SRKM CALC. FUGACITY COEFF. OF I IN MIXTURE BY SRK E.O.S. ITER SUBROUTINE TO PERFORM ITERATION NORMI NORMALIZE VAPOR PHASE MOLE FRACTION NORM2 ADJUST GAS SOLUBILITY BY SECANT METHOD NORM3 ADJUST FUGACITY COEFFICIENT BY SECENT METHOD SOLVE SOLVE FOR VAPOR AND LIQUID PHASE MOLE FRACTION START1 CALC. IN ITIA L MOLECULAR DENSITY OF EACH COMPONENT POTEN EVALUATE POTENTIAL ENERGY OUTPUT USER PROVIDE FORMAT TO OUTPUT RESULTS *************** KEY VARIABLE DEFINITION **************** AM MOLALITY BE ISOTHERMAL COMPRESSIBILITY OF PARTIAL MOLAR VOLUME CA INTERACTION PARAMETER IN SRK E.O.S. COMPY COMPRESSIBILITY DEFINED IN EQN. (3 -7 1 ) D HARD SPHERE DIAMETER DELTAD DENSITY INCREASE DUE TO THE PRESENCE OF SALT (4 -1 9 ) DIPOLM DIPOLE MEMENT DPMVDV DEFINED AS EQN. (3 -7 8 ) DXHSDV DEFINED AS EQN. (3 -7 9 ) DSOL DENSITY OF PURE SOLVENT (4 -1 7 ) DUDA AS DEFINED IN EQN. (3 -5 9 ) EPSI ENERGY PARAMETER IN L-J POTENTIAL FIRST AS DEFINED IN EQN. (3-57) H APPARENT IDEAL SOLUTION FUGACITY HK BOLTZMANN CONSTANT HP2 NAMIOT'S CORRECTION PHIN NEWLY CALCULATED VALUE OF FUGACITY COEFFICIENT PHIO INITIAL VALUE OF FUGACITY COEFFICIENT PHO MOLECULES PER CUBIC CENTIMETER PHP POYNTING FACTOR PHS FUGACITY COEFFICIENT AT SATURATION PRESSURE PMV PARTIAL MOLAR VOLUME (3-75) POLAR POLARIZABILITY PS SATURATION PRESSURE OF SOLVENT Q QUADRUPOLE MOMENT R DISSOLVED GAS RATIO R1 ACTIVITY COEFFICIENT OF WATER SEC AS DEFINED IN EQN. (3-58) SIGMA DISTANCE PARAMETER OF L-J POTENTIAL UHKT DEFINED AS EQN. (3-41) VHS DEFINED AS EQN. (3-73) WEAG REDUCED DENSITY AS IN EQN. 3-43 XF LIQUID PHASE MOLE FRACTION XHS DEFINED AS EQN. (3-69) XP DEFINED AS EQN. (3-70) Y VAPOR PHASE MOLE FRACTION ooo oo oo oo ooo o o o COMPONENT SHOWN SAMPLE CLASSIFICATION IS TABLE SHOWN 3-1 CC USER IN IS INPUT FOLLOW AS CC * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * if * * rtrt * rtrt * * * rt * * * * * * rtrt * * * * * * rt* * ** * rt * * rtIlf* * * * rtrtrt * * * rt * yertrt * * rtrt * * * * * * * * rtrt * * * rt * * * * rt * * * * rtrt * * * * * * * * * * * * * * * * * * * * * P0LAR(18),DIP0LM(2) 1 BASE(7) PH0(18), PR(7), TR(7), 1 + .48908D-13*DT**4) 7 + (3.1975D0-0.3151548D0*DT**(l./3.)-1.203374D-3*DT / 1 1 *KT +.24A*25* 1.0D-3 +5.2829D-5*AM**2 +0.2274*AM**2.5)* -9.5653D-4*AM**2*TK *TK*TK *AM**2 -1.6368D0 -6.2635D-4*AM**1.5 *TK*TK +5.3956D-2*AM**1.5*TK 1 +1.4037D-7*AM*TK**4 -1.8527D0*AM**1.5 *AM*TK**3 1 -2.703D-5 1 1 =5. K T=350. COMMON ), (7 Y /VAR1/ 0MEGA(7),T0L4, EPSK(18,18), COMMON SIGMA(18,18), /PARA1/ TK=T-273.15D0 Nl=3 BASE(3)=.TRUE. COMMON PS(7),PHP(2),PHS(2),ZC(2),ADEN(2),BDEN(2),CA(7,7) /VAR3/ SL( D+.14490 D*( /3.)-3.946263D-3*DT) . 3 - ) . 3 ./ DT**(1 1342489D0* .D0+0. DS0L=(1 DT=647.11D0-T X(N1 C0MP(11)=.TRUE. C0MP(8)=.TRUE. C0MP(4)=.TRUE. C0MP(3)=.TRUE. C0MP(1)=.TRUE. ENTRY SYSTEM ,NO ) ,DIDP(7 ) PNP(7 , ) ,QC(7 COMMON ) Q(7),QD(7),QQ(7 /VAR9/' O M N VR/ 01 TL, 03 H,LWIEA, TR2 ITERA3 ITERA2, HK,DL0W,ITERA1, T0L3, COMMON T0L2, T0L1, /VAR2/ C0MM0N/SYS1/S(18),D(18),DS0LN,T,P,AM,R1,I1,I2,I3,N1 RELAX(7), PC(7), TC(7), COMMON R(7,7),C0MP(18), /SYS/ ETD(555D*M O24D* M T +.18-* M TK*TK AM* +3.4128D-3* TK AM* DELTAD=(45.5655D0*AM -O.2341D0* SOME CORRELATION FORMULA FOR THE SOLUTION ENTRY DENSITY DENSE RETURN ATM P=1000. SUBROUTINE USER MLCT EL 8AH 0-Z) LOGICAL COMP, BASE REAL *8(A-H, IMPLICIT F T G. 4.) GO 71 TO 348.0) .GE. (T IF m # = ) XF( ...... ,HN7,PI() A H7, TOTY,ITMAX H(7), A, PHI0(7), ),PHIN(7), 8 1 WATER SOLVENT AS SYS PRESSURE ATM IN SYS TEMP IN NTA GUESS CH4 OFSOLUBILITY INITIAL DISSOLVED BASED ON GAS IS RATIO NACL SALT AS CH4 AND C2H6 DISSOLVED IN SOLUTION CH4 K 177 178 DSOLN=DSOL+ DELTAD GO TO 711 71 TK=T-348. DELTAD=(40.2204*AM-0.013548*AM* TK +1.25326*1.0-3* AM* TK*TK 1 -0.869929*1.D-5 *AM*TK**3 +0.352779*1.D-7*AM*TK**4 1 -2 .33803*AM**1.5 -0.240002*1.D-2*AM**1.5*TK-1.8341*1.D-4*AM**1.5 1 *TK*TK-0. 570926*AM**2+1.04261*1.D-4*AM**2*TK+1.13663*1.D-5 1 *AM**2*TK*TK +0.0606576*AM**2.5)* 1.0D-3 DSOLN=DSOL+ DELTAD C CONVERT TO (M0LECULE/CM3) 711 TOTM1=1000.DO/18.015D0 + AM*2.D0 TOTS=1000.DO+58.4428D0 *AM DSOLN=DSOLN/TOTS*TOTM1* 6.022D23 RETURN C C ENTRY PHIINT C C PHIO(1)=0.2 PHI0(3)=1.2 PHI0(4)=1.3 RETURN C C C ENTRY SOLVEN C C TC(2)=#### K ...... PC(2)=### ATM ZC(2)=### ADEN(2)= ! SKIP THIS SECTION IF WATER IS BDEN(2)= . THE SOLVENT OMEGA(2)= SIGMA(2,2)= EPSK(2,2)= DIPOLM(2)= POLAR(2)= ...... RETURN C C ENTRY VNGAS C C TC(5)=### K ...... PC(5)=### ATM OMEGA(5)=### ! SKIP THIS SECTION IF NO OTHER SIGMA(5,5)=### . NON-DISSOCIATE GAS PRESENT EPSK(5,5)=### POLAR(5)=#### Q(5)=### ...... RETURN C ENTRY DGAS PROVIDE INFORMATION AS IN VNGAS . FOR COMPONENT 7 RETURN ENTRY RATIO R(4,N1)= .... RATIO OF C2H6 TO CH4 RETURN ENTRY CATION SIGMA(10,10)= SKIP IF NO COMPONENT 10 PRESENT EPSK(10,10)= RETURN ENTRY ANION SIGMA(13,13)= SKIP IF NO COMPONENT 13 PRESENT EPSK(13,13)= RETURN ENTRY SALINE S(8)= MOLALITY OF NA ION S ( ll)= . MOLALITY OF CL ION RETURN END USER PROVIDE VARIABLE AND FORMAT C o o o o o o SST,DXPDV,DXHSDV,ALKTP,SUMY,DELTAD,DS0L,C0RR2,PPS,PP,DD,DC1 DUDA(7),DGDN(18,18,7) 1 SQD(7),SPQD(7),SPQID(7),SQCP(7),SQQC(7) 1 PMVV(300) HP2(7),HPS(7),HLN(7),FIRST(7),SEC(7),SP(18), 1 ),VAL(18),DUPDV(7),PMV(7),DPMVDV(7),BE(7),HPSLN(7),HP1(7), 8 PN(1 1 POLAR(18),DIP0LM(2) 1 BASE(7) PHO(18), PR(7), TR(7), 1 CO M M ON /VAR9/ Q(7),QD(7),QQ(7),QC(7),PNP(7),DIDP(7),N0 COMMON Q(7),QD(7),QQ(7),QC(7),PNP(7),DIDP(7),N0 /VAR9/ COMMON WEAG(4),UHKT(18),PIDIL(18),EPSI(18,18),VHS(7), TOTY, ITMAX /VAR5/ COMMON H(7), XINT,TOTPHO PHS(2),ZC(2),ADEN(2),BDEN(2),CA(7,7) A, /VAR4/ ITERA3COMMON PHP(2), ITERA2, PS(7), HK,DL0W,ITERA1, PHI0(7), /VAR3/ COMMON T0L2.T0L3, PHIN(7), T0L1, /VAR2/ COMMON Y(7),XF(18), /VAR1/ * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * it * ** * * * ** * *** * * * * * * * SUBROUTINE * * * OUTPUT * * * * * * * * O M N VR3PIT7) PHIN1(7),PHIP(7) COMMON ), /VAR13/PHI0T(7 COMMON F0(7),YN(7),Y0(7) FN(7), /VAR12/ COMMON IJ(18,18),DEL(18,18),GHS(18,18),DUDA1(7),DUDA2(7), D /VAR11/ COMMON QDP,QQP,QCP,SPP(7),SDID(7),SPD(7),SPQ(7),SQQ(7), /VAR10/ COMMON EPNT, SDIDXP, XHS, EDID, PNT, S22XP,SXP, XP, COMPY, DID, /VAR7/DE, 0MEGA(7),T0L4, EPSK(18,18), COMMON SIGMA(18,18), /PARA1/ COMMON/SYS1/S(18),D(18),DS0LN,T,P,AM,R1,I1,I2,I3,N1 RELAX(7), PC(7), TC(7), COMMON R(7,7),C0MP(18), /SYS/ MLCT EL 8 AH 0-Z) LOGICAL COMP, (A-H, BASE REAL *8 IMPLICIT * * * * * ****** y * * * ye * * * * * * * * * * * * * * * * * * * * * * * * * * * 180 181 C Q ***************************************************** c C MAIN PROGRAM C Q ** ****** *************** ********* ***** ******* ********** C IMPLICIT REAL *8 (A-H, O-Z) LOGICAL COMP, BASE COMMON /SYS/ R(7 ,7 ),COMP(18), TC(7), PC(7), RELAX(7), 1 TR(7), PR(7),PH0(18), BASE(7) COMMON/SYSl/S(18),D(18),DS0LN,T,P,AM,R1,I1,I2,I3,NI COMMON /PARA1/ SIGMA(18,18), EPSK(18,18), 0MEGA(7),T0L4, 1 POLAR(18),DIPOLM(2) COMMON /VAR1/ Y(7),XF(18), PHIN(7), PHI0(7), A, H(7), TOTY,UMAX COMMON /VAR2/ TOL1, TOL2,TOL3, HK,DLOW,ITERA1, ITERA2, ITERA3 COMMON /VAR3/ PS(7),PHP(2),PHS(2),ZC(2),ADEN(2),BDEN(2),CA(7,7) COMMON /VAR4/ XINT,TOTPHO COMMON /VAR12/ FN(7), FO(7),YN(7),YO(7) COMMON /VAR13/PHIOT(7 ), PHIN1(7),PHIP(7) R l= l.DO CALL SYSTEM CALL PHIINT 10 CALL DENSE IF(.NOT. COMP(l)) CALL SOLVEN IF(.NOT. C0MP(5)) GO TO 100 CALL VNGAS 100 IF(.NOT. COMP(7)) GO TO 200 CALL DGAS 200 CALL RATIO IF(.NOT. COMP(IO)) GO TO 210 CALL CATION 210 IF(.NOT. COMP(13)) GO TO 220 CALL ANION 220 CALL SALINE IF(.NOT. (COMP(16).OR.COMP(17).OR.COMP(18))) GO TO 800 800 CALL DIAT CALL SATP CALL SRKP CALL SRKPS CALL POYN C 14=1 13=1 914 12=1 950 CALL START1 11=1 CALL TSAI 912 CALL UHARD CALL SOFT CALL AISF CALL SOLVE 911 CALL N0RM1 13=13+1 CALL SRKM DO 900 1=1,7 IF (.NOT. COMP(I)) GO TO 900 E=(PHIN(I)—PHIO(I) )/PHIO(I) E=ABS(E) IF(E-TOLl) 900,900,910 900 CONTINUE CALL SOLVE GO TO 999 910 DO 920 1=1,7 IF ( I I .LE. 2) GO TO 919 IF (.NOT. COMP(I)) GO TO 920 EE=(PHIOT(I)—PHINl(I))/PHIOT( I ) IF (ABS(EE) .GT. TOL2 ) GO TO 919 920 CONTINUE GO TO 940 919 CALL NORM3 11= 11+1 CALL SOLVE GO TO 911 999 TOTY=O.DO DO 998 1=1,7 TOTY=TOTY+Y(I) 998 CONTINUE A=T0TY-1.0D0 IF (ABS(A)-T0L2) 930, 930, 940 940 CALL NORM1 CALL NORM2 844 12=12+1 GO TO 950 930 IF ( ABS((XF(N1)-XINT)/XINT) .LE. TOL2) GO TO 830 GO TO 940 830 CALL OUTPUT STOP END 183 C Q ***************************************************** c SUBROUTINE TIEPEL C Q ****************************************************** c IMPLICIT REAL *8 (A-H, O-Z) LOGICAL COMP, BASE COMMON /SYS/ R(7,7),C0MP(18), TC(7), PC(7), RELAX(7), 1 TR(7), PR(7),PHO(18), BASE(7) C0MM0N/SYS1/S(18),D(18),DS0LN,T,P,AM,R1,I1,I2,I3,N1 COMMON /PARA1/ SIGMA(18,18), EPSK(18,18), 0MEGA(7),T0L4, 1 POLAR(18),DIP0LM(2) COMMON /VAR2/ TOL1, TOL2, TOL3, HK,DLOW, ITERA1, ITERA2, ITERA3 COMMON /VAR1/ Y(7),XF(18), PHIN(7), PHIO(7), A, H(7), TOTY,ITMAX COMMON /VAR5/ WEAG(4 ) ,UHKT(1 8 ),PIDIL(1 8 ),EPSI(1 8 ,1 8 ),VHS(7), 1 PN(1 8 ),VAL(18),DUPDV(7), PMV(7),DPMVDV(7),BE(7),HPSLN(7),HP1(7), 1 HP2(7),HPS(7),HLN(7), FIRST(7),SEC(7),SP(18), PMVV(300) COMMON /VAR7/DE,PNT,DID,EPNT,EDID,SDIDXP,S22XP,SXP,XP,XHS,COMPY, 1 SST,DXPDV,DXHSDV,ALKTP,SUMY,DELTAD,DSOL,CORR2,PPS,PP,DD,DC COMMON /VAR9/ Q(7),QD(7),QQ(7),QC(7),PNP(7),DIDP(7),NO COMMON /VAR10/ QDP,QQP,QCP,SPP(7),SDID(7),SPD(7),SPQ(7),SQQ(7), 1 SQD(7),SPQD(7),SPQID(7),SQCP(7),SQQC(7) COMMON /VAR11/ DIJ(18,18),DEL(18,18),GHS(18,18),DUDA1(7),DUDA2(7), 1 DUDA(7), DGDN(18,18,7) C C ENTRY DIAT C C SIGMA(1,1)=2.473070+0.179337D-2*TDATA(LL)-0.299520D-5*TDATA(LL)**2 EPSK(1,1)=36000./SIGMA(1,1)**6 DO 209 J=1,18 IF (.NOT. COMP(J)) GO TO 209 EPSI(J,J)=EPSK(J,J)*HK 209 CONTINUE DO 208 J=1,18 IF (.NOT. COMP(J)) GO TO 208 DO 308 K=1,18 IF(.NOT. COMP(K)) GO TO 308 EPSI(J,K)=(EPSI(J,J)*EPSI(K,K))**0.5 SIGMA(J,K)=0.5D0*(SIGMA(J,J)+SIGMA(K,K)) 308 CONTINUE 208 CONTINUE DO 200 1=1,18 IF (.NOT. COMP(I)) GO TO 200 DO 201 K=1,18 IF (.NOT. COMP(K)) GO TO 201 A1=DL0W B1=SIGMA(I,K) H1=B1-A1 SUM1=P0TEN(I,K,A1)+P0TEN(I,K,B1) TSUMl=SUMl*Hl/3.DO M=1 1 H1=H1/2.D0 M=2*M SUM2=0.DO 4 SUM2=SUM2+2.*P0TEN(I,K,A1+J*H1) IF (J+l-M) 2,3,3 2 J=J+2 GO TO 4 3 SUM21=SUM1+SUM2*2.DO SUM1=SUM1+SUM2 TSUM=SUM21*H1/3.D0 IF (DABS(TSUM-TSUMl)-TOL3) 5,6,6 6 TSUM1=TSUM GO TO 1 5 DEL(I,K)=(SIGMA(I,K)-TSUM) * 1.0D-8 201 CONTINUE 200 CONTINUE DO 2021 1=1,18 IF (.NOT. COMP(I)) GO TO 2021 D(I)=DEL(1,1) 2021 CONTINUE DO 2022 1=1,18 IF ( .NOT. COMP(I)) GO TO 2022 DO 2023 J=1,18 IF (.NOT. COMP(J)) GO TO 2023 DIJ(I,J)=0.5D0*(D(I)+D(J)) 2023 CONTINUE 2022 CONTINUE RETURN C C ENTRY TSAI C C DO 501 L=1,4 PIDI=0.DO DO 500 1=1,18 IF (.NOT. COMP(I)) GO TO 500 PIDIL(I)=PHO(I)*(D(I)**(L-1)) PIDI=PIDI+PIDIL(I) 500 CONTINUE WEAG(L)=PIDI* 5.235987D-1 501 CONTINUE RETURN C C ENTRY UHARD C C DE=1.0D0-WEAG(4) DE2=DE*DE DE3=DE2*DE DE4=DE3*DE DE5=DE4*DE P01=WEAG(1)/DE P02=3. 0D0*WEAG(2) *WEAG(3)/DE2 P03=3.0DO*WEAG(3)**3./DE3 P04=(P03*WEAG(4)/3.ODO) P0=P01+P02+P03-P04 PPS=(6.D0*HK*T*P0/3. 1416D0)*9 .86895D-7 185 DO 600 1=3,7 IF (.NOT. COMP(I)) GO TO 600 UHKT( I)=-DLOG(DE) + PO *D (I)*D (I)*D (I) + 3.0D0*WEAG(3)*D(I)/DE 1 +3.ODO*WEAG(2)*D(I )*D(I)/DE + 4.5D0 *WEAG(3)*WEAG(3)*D(I)*D(I) 1 /DE2 +3.0D0*(D(I)*WEAG(3)/WEAG(4 )) * * 2 .*( DLOG(DE) 1 +WEAG(4)/DE-WEAG(4)*WEAG(4)/2.0D0 /DE2 ) 1 - ( D(I)*WEAG(3)/WEAG(4 )) * * 3 . *( 2.ODO *DL0G(DE)+WEAG(4) 1 * ( 2 .ODO -WEAG(4))/DE) VHS(I)=1.D0+ WEAG(4)/DE +3.DO* WEAG(3)* D(I)/DE2 1 +3.D0*WEAG(2)*D(I)*D(I)/DE2 +9.DO*WEAG(3)*WEAG(3)*D(I)*D(I)/DE3 1 -( WEAG(3)*D(I)/DE)**2. * (3.D0*WEAG(4)/DE+WEAG(3)*D(I)) 1 + D (I)**3. * (WEAG(I)*DE2+ 6.D0*WEAG(2)*WEAG(3)*DE +9.D0*WEAG(3) I * * 3 .- 4 .D0*WEAG(4)*WEAG(3)**3+WEAG(4)**2.*WEAG(3)**3. )/DE4 600 CONTINUE RETURN C C ENTRY SOFT C C XD2=0.D0 XD3=0.DO DO 502 1=1,18 IF (.NOT. COMP( I ) ) GO TO 502 XD2=XD2+XF(I)*D(I)*D(I) XD3=XD3+XF(I)*D(I)*D(I)*D(I) 502 CONTINUE DO 503 1=3,7 IF (.NOT. COMP(I)) GO TO 503 DUDA1(I)=O.DO DO 504 J=1,18 IF ( .NOT. COMP(J)) GO TO 504 GHS(I,J)=1.DO/DE +(3.D0*D(I)*D(J)/(D(I)+D(J))) *XD2*WEAG(4) 1 /(XD3*DE2) + 2.D0*(D(I)*D(J)/(D(I)+D(J)))**2 1 *(XD2/XD3)**2 * WEAG(4)**2/DE3 DUDA1(I)=DUDA1(I) +PHO (J)*DIJ(I,J)**2 *(D IJ (I,J )-D E L (I,J )) 1 *GHS(I,J) 504 CONTINUE 503 CONTINUE DO 507 K=3,7 IF (.NOT. COMP(K)) GO TO 507 DUDA2(K)=0.D0 DO 505 1=1,18 IF (.NOT. COMP(I)) GO TO 505 DO 506 J=l,18 IF (.NOT. COMP(J)) GO TO 506 DGDN(I,J,K)= 3.DO*D(I)*D(J)*(WEAG(4)*(D(K)**2-XD2) 1 /XD3/DE2/T0TPH0 + XD2*3.14159D0*D(K)**3/XD3/DE2/6.D0 1 + XD2*WEAG(4)*DE3*3. 14159*D(K)**3/XD3/6. DO 1 - XD2*WEAG(4)*(D(K)**3-XD3)/DE2/XD3/XD3/T0TPH0)/(D(I)+D(J)) DUDA2(K)= DUDA2(K)+ PH 0(I)*P H 0(J)*D IJ(I,J)**2 1 *(DIJ(I,J)-DEL(I,J))*DGDN(I,J,K) 506 CONTINUE 505 CONTINUE 507 CONTINUE DO 508 1=3,7 IF (.NOT. COMP(I)) GO TO 508 DUDA(I)=4.D0*3. 14159*DUDA1(I) +2.D0*3. 14159*DUDA2(I) 186 508 CONTINUE RETURN C C ENTRY AISF C C PNT=O.DO DC=O.DO DD=0.DO DID=O.DO EDID=O.DO EPNT=O.DO SDIDXP=O.DO DO 802 J=1,18 IF (.NOT. COMP(J)) GO TO 802 PN(J)=O.DO SP(J)=0.DO DO 803 K=I,18 IF(.NOT. COMP(K)) GO TO 803 PN(J)=PN(J)+ PHO(J)*PHO(K)*EPSI(J,K)*(SIGMA(J,K)*1.D-8)**3. SP(J)=SP(J)+ (PHO(K)*EPSI(J,K)**2.)*(PHO(J)*(SIGMA(J,K)*1.D-8) 1 **3 ) 803 CONTINUE PNT=(PNT+PN(J)) EPNT=EPNT+SP(J) 802 CONTINUE PNT=-11.17D0*PNT EPNT=-2.553D0* EPNT/HK/T XHS=1. 909859D0*HK*T*( WEAG(1)*DE2+ 6.D0*WEAG(2)*WEAG(3)*DE 1 +9.D0*WEAG(3)**3. + WEAG(4)**2.*WEAG(3)**3. 1 - 4 .DO*WEAG(3 )* * 3 . *WEAG(4))/DE4 C DO 804 1=1,2 IF (.NOT. COMP(I)) GO TO 804 DO 805 J=1,18 IF (.NOT. COMP(J)) GO TO 805 DID=DID+(PHO(J)*POLAR(J)/(SIGMA(I,J)*l.D-8)**3.)*PHO(I)*DIPOLM(I) 1 *DIPOLM(I) EDID=EDID+(PHO(J)*POLAR(J)/(SIGMA(I,J)*1.D“ 8)**3.)*PHO(I) 1 *DIP0LM(I)**2*EPSI(I,J) SDIDXP=SDIDXP+PH0(I)*PH0(J)yrP0LAR(J)*^t2 1 *(DIP0LM(I)**4/(SIGMA(I,J)*l.D-8)**9) 805 CONTINUE EDID=-4. 468D0*EDID/HK/T SDIDXP=-1.3963D0*SDIDXP/HK/T DID=-8.37758D0*DID QDP=O.DO QQP=0.DO QCP=O.DO DO 801 J=3,7 IF (.NOT. COMP(J)) GO TO 801 QDP=QDP-1.25664DO*PHO(J)*Q(J)**2*PH0(I)*DIP0LM(I)**2/(SIGMA(I,J) 1 *l.D-8)**5/HK/T QQP=QQP-2.51327DO*PHO(J)**2*Q(J)**2*(Q(J)/(SIGMA(J,J)*l.D-8)yt*3)** 1 2/(SIGMA(J,J)*l,D-8)/HK/T DO 806 K=8,18 IF (.NOT. COMP(K)) GO TO 806 187 QCP=QCP—0.837758DO*PHO(K)*PHO(J)*Q(J)**2*23.0686D-20/HK/T 1 /(SIGMA(J,K)*l.D-8)**3 806 CONTINUE 801 CONTINUE S22XP=-1.3963DO*PHO(I)*PHO(I)*P0LAR(I)**2*DIP0LM(I)**4 1 /( SIGMA(I,I)*l.D-8)**9/HK/T DC=0.DO DD=0. 804 CONTINUE SXP=O.DO XP=(PNT+DID+DD+DC+(SXP+QDP+QCP+QQP))*(1.+0.*0.109993) C0MPY=1.D0/(XP+XHS) C DO 807 1=1,2 IF (.NOT. COMP(I)) GO TO 807 M=I 807 CONTINUE DO 808 1=3,7 IF (.NOT. COMP(I)) GO TO 808 QQ(I)=O.DO QD(I)=O.DO QC(I)=O.DO PNP(I)=0.DO DIDP(I)=0.DO SPP(I)=0.DO SPQ(I)=0.DO SQQ(I)=0.DO SQD(I)=0.DO SPQD(I)=0.DO SPQID(I)=O.DO SQCP(I)=O.DO SQQC(I)=O.DO SDID(I)=O.DO SPD(I)=0.DO SEC(I)=0-DO DO 812 J=1,18 IF (.NOT. COMP(J)) GO TO 812 SPP(I)=SPP(I)+2.553DO*PHO(J)*EPSI(I,J)**2*(SIGMA(I,J)*1.D-8)**3 1 /HK/T 812 CONTINUE SPQ(I)=-0.427358D0*PHO(I)*Q(I)*(Q(I)/HK)**2*(Q(I)/(SIGMA(I,I) 1 *1.D-8)**7)*EPSI(I,I)/T/T SQQ(I)=0.7244D0*PHO(I)*Q(I)*(Q(I)/HK)**3*(Q(I)/(SIGMA(I,1) 1 *1.D-8)**4)**4/(SIGMA( I,I)*1.D-8)/T/T/T SQD(I)=0.1208DO*PHO(M)*Q(I)*(DIPOLM(M)/HK)**3*(Q(I)/ 1 (SIGMA(I,M)*l.D-8)**4)**3*DIP0LM(M)/(SIGMA(I,M)*l.D-8)/T/T/T SPQD(I)=0.8064D0*PHO(M)*Q(I)*(DIP0LM(M)/HK)**2*(Q(I)/ 1 (SIGMA(I,M)*1.D-8)**4)*(EPSI(I,M)/(SIGMA(I,M)*l.D-8))/T/T SPQID(I)=0.5712DO*PHO(M)*(DIPOLM(M)/HK)**2*(Q(I)/(SIGMA(I,M) 1 *1.D-8)**4)**2*(DIP01M(M)**2/(SIGMA(I,M)*l.D-8)**3) 1 *POLAR(I)/T/T SPD(I)= 2 .234DO*PHO(M)*POLAR(I)*DIPOLM(M)*DIPOLM(M)AEPSI(I,M) 1 /(SIGMA(I,M)*1.D-8)**3/HK/T SDID(I)=0.698D0*PH0(M)/HK/T 1 AP0LAR(I)**2*(DIP0LM(M)**4/(SIGMA(I,M)*l.D-8)**9) SEC(I)=SPP(I)+SPD(I)+SDID(I)+SPQ(I)+SQQ(I)+SQD(I)+SPQD(I)+SPQID(I) 1 +SQCP(I)+SQQC(I) DIDP(I)= 4 .18879DO*DIPOLM(M) 1 *DIP0LM(M)*PH0(M)*P0LAR(I)/(SIGMA(I,M)*l.D-8)**3. 188 PNP(I)=11.170000D0*PN(I)/PHO(I) QQ(I)=2.513274D0*(Q(I)/(SIGMA(I,I)*1.D-8))**4*(PH0(I)/(SIGMA(I,I) 1*1.D-8)**3)/HK/T QD(I)=(1.256637DO*DIPOLM(M)**2*PHO(M)*(Q(I)**2/(SIGMA(I,M)*l.D-8) 1**5)/HK/T)*2. QD(I)=2.*QD(I) FIRST(I)=DIDP(I)+PNP(I)+QQ(I)+QD(I)+QC(I) FIRST(I)=FIRST(I)/HK/T SEC(I)= SEC(I)/HK/T DUPDV(I)= - ( FIRST( I)+SEC(I)+DUDA(I)) PMV(I)=6.022D23*HK*T* COMPY*(VHS(I) + DUPDV(I)) 808 CONTINUE DXPDV=-2.D0*XP DXHSDV=1.909859D0*HK*T*( -WEAG(1 )* (1 .D0-WEAG(4)-WEAG(4)**2. 1 +WEAG(4)**3.) +6.D0*WEAG(2 )*WEAG(3 )*WEAG(4 )* (1 .DO+WEAG(4)) 1 -12.D0*WEAG(2)*WEAG(3)-WEAG(3)**3.*(27.D0+WEAG(4)*(-7.D0) 1 +5.D0*WEAG(4)*WEAG(4)-WEAG(4)**3.))/DE5 C SUMPH=0.D0 DO 810 1=1.18 IF ( .NOT. COMP(I)) GO TO 810 SUMPH=SUMPH+PHO(I) 810 CONTINUE ALKTP=DL0G(HK*T*SUMPH*9.86895D-7) C DO 809 1=3,7 IF (.NOT. COMP(I)) GO TO 809- DPMVDV(I)=-WEAG(4)/DE2 - 3.D0*WEAG(3)*(1.D0+WEAG(4))*D(I)/DE3 1 - ( 3 . D0*WEAG(2)+18. DO*WEAG(3)*WEAG(3)-3.DO*WEAG(2)*WEAG(4)**2. 1 *D(I)*D(I)/DE4 - D(I)*D(I)*D(I)*(WEAG(1)*(1.D0-WEAG(4) 1 -WEAG(4)*WEAG(4)+ WEAG(4)**3.)-6.D0*WEAG(2)*WEAG(3)*WEAG(4) 1 *(1.D0+WEAG(4)) + 12.DO* WEAG(3)*(WEAG(2)+ 2.D0*WEAG(3)*WEAG(3)) 1 )/DE5 - DUPDV(I) C BE(I)= (COMPY* DPMVDV(I) / ( VHS(I)+DUPDV(I)) 1 -( DXPDV+DXHSDV)*C0MPY*C0MPY)*1. 013279D6 C HPSLN(I)=(UHKT(I)+ DUPDV(I)+ ALKTP) C HP1(I)=1.00*(P-PS(M) )*PMV(I)/T/82.05D0 C HP2(I)=BE(I)*PMV(I)/T/82.05D0*(P*P/2.D0-P*PS(M)~PS(M)*PS(M)/2.D0) C HPS(I)= DEXP(HPSLN(I)) H(I)=HPS(I)*DEXP(HP1(I) )/DEXP(HP2(I)) HLN(I)=DLOG(H(I)) 809 CONTINUE RETURN END c Q ********************************************************* C FUNCTION POTEN(I,J,SR) C q ********************************************************* C IMPLICIT REAL *8 (A-H, O-Z) LOGICAL COMP, BASE COMMON /SYS/ R(7,7),C0MP(18), TC(7), PC(7), RELAX(7), 1 TR(7), PR(7),PHO(18), BASE(7) COMMON/SYSl/S(18),D(18),DS0LN,T,P,AM,R1,I1,I2,I3,N1 COMMON /PARA1/ SIGMA(18,18), EPSK(18,18), 0MEGA(7),T0L4, 1 POLAR(18),DIP0LM(2) COMMON /VAR2/ TOL1, TOL2,TOL3, HK.DLOW,ITERA1, ITERA2, ITERA3 COMMON /VAR9/ Q(7),QD(7),QQ(7 ),QC(7 ) ,PNP(7 ),DIDP(7),NO C IF (I .EQ. J) GO TO 1191 GO TO 1192 1191 EI=EPSI(I,1) SI=SIGMA(1,1) QI=Q(I) IF( I .GT. 2) GO TO 91 P0T=(4.0D0*EI*((SI/SR)**12. 1 ~(SI/SR)**6.))/T/HK GO TO 93 91 IF( I .GT. 7) GO TO 92 POT=(4.0D0*EI*((SI/SR)**12. 1 -(SI/SR)**6.))/T/HK GO TO 93 92 P0T=(4.0D0*EI*((SI/SR)**12. 1 -(SI/SR)**6.))/T/HK 93 IF ( POT .GE. 174.DO ) THEN POTEN=0. ELSE POTEN=DEXP(-POT) END IF GO TO 1193 1192 POT=(4.0D0*EPSI(I,J)*((SIGMA(I,J)/SR)**12. 1-(SIGMA(I,J)/SR)**6.))/T/HK IF ( POT .GE. 174.DO ) THEN P0TEN=0. ELSE POTEN=DEXP(-POT) END IF 1193 RETURN END 190 C 0 ******************************************************** C SUBROUTINE SRK C 0 ******** ************************* ** ****** AVr Aye ****** ***** c IMPLICIT REAL*8(A-H , 0-Z) LOGICAL COMP, BASE COMMON /SYS/ R(7,7),COMP(18), TC(7), PC(7), RELAX(7), 1 TR(7), PR(7),PHO(18), BASE(7) C0MM0N/SYS1/S(18),D(18),DS0LN,T,P,AM,R1,I1,I2,I3,N1 COMMON /VARl/ Y(7), XF(18),PHIN(7), PHI0(7), A, H(7), TOTY,ITMAX COMMON /PARA1/ SIGMA(18,18), EPSK(18,18), 0MEGA(7),T0L4, 1 P0LAR(18),DIP0LM(2) COMMON /VAR3/ PS(7), PHP(2), PHS(2),ZC(2), ADEN(2),BDEN(2),CA(7,7) COMMON /VAR13/PHI0T(7 ), PHIN1(7),PHIP(7) DIMENSION SM(7), ALPH(7), SA(7), SB(7), TF1(7),TF2(7) DIMENSION PRS(2), SAS(2), SBS(2) DIMENSION F(8) C C ENTRY SATP C C DATA TCRIT,PCRIT,F/374.136,220.88,-741.9242, -29.7210, -11.55286, 1 -0.8685635, 0.1094098,0.439993,0.2520658, 1 0.05218684 / CT=T-273.15 RT=0.01/T SUM=0.0 DO 300 1=1,8 300 SUM=SUM+(F(I)*((0.65-0.01*CT)**(I-1))) VPH20=PCRIT* EXP(RT*(TCRIT-CT)*SUM) VPH20=VPH20/1.01325 PS(1)=VPH20 RETURN C C ENTRY SRKPS C C DO 1501 1=1,2 IF (.NOT. COMP(I)) GO TO 1501 PRS(I)=PS(I)/PC(I) TR(I)=T/TC(I) SM(I)=0.48508D0+(1 .5517DO*OMEGA(I) ) - ( . 15613DO*OMEGA(I)**2) ALPH(I)=(1.D0+(SM(I)*(1.D0-TR(I)**0.5)))**2 SAS( I )=0. 42747D0*ALPH( I)*PRS( I )/TR( I )**2 SBS(I)=0.08664D0*PRS(I)/TR(I) CAPA=SAS(I) CAPB=SBS(I) C C FIND ROOT OF Z BY NEWTON'S METHOD C C USE IDEAL GAS LAW FOR FIRST ESTIMATION C Z=1.1D0 ABB2=CAPA-CAPB*(1 .DO+CAPB) AB=CAPA*CAPB C C BEGIN NEWTON’S ITERATION C DO 56 NITER=1,ITMAX 13=11*111= 1*1 DELTAZ=-((Z3-Z2+Z*ABB2)-AB) / (3.D0*Z2- 2.D0*Z +ABB2) 1=1+ DELTAZ C C CHECK FOR CONVERGENCE C IF (DABS(DELTAZ/Z) .LT. T0L4) GO TO 992 56 CONTINUE 992 CZ=Z NO=NITER TF1(I)=(CZ-1. ODO)*SBS(I)/CAPB-DLOG(CZ-CAPB) T F2(I)= (CAPA/CAPB)*(2. 0D0*SAS(I)**0.5D0/CAPA**0. 5-(SBS(I)/CAPB))* SDLOG(1 .ODO+CAPB/CZ) PHS(I)=DEXP(TF1(I)—TF2(I)) 1501 CONTINUE RETURN C C ENTRY SRKM C C CAPA=0.D0 CAPB=0.D0 DO 1502 1=1,7 IF (.NOT. COMP(I)) GO TO 1502 CAPB=CAPB+(SB(I)*Y(I)) DO 1503 J = l,7 IF (.NOT. COMP(J)) GO TO 1503 CAPA=CAPA+(Y(I)*Y(J)*(1.-CA(I,J))*((SA(I)*SA(J))**0.5)) 1503 CONTINUE 1502 CONTINUE Z=2.0D0 ABB2=CAPA-CAPB*(1.DO+CAPB) AB=CAPA*CAPB C C BEGIN NEWTON'S ITERATION C DO 61 NITER=1,ITMAX 1113=11*1= 1*1 DELTAZ=-((Z3-Z2+Z*ABB2)-AB) / (3.D0*Z2- 2.D0*Z +ABB2) 1=1+ DELTAZ C C CHECK FOR CONVERGENCE C IF (DABS(DELTAZ/Z) .LT. T0L4) GO TO 993 61 CONTINUE 993 CZ=Z NO=NITER (N Ot /—N + 4—4 * * w * ✓—s O f /—N /*N Q 1— 4-1 00 * * w a . r o O Of < l Q h - <_> a CO * \ i n o /-N 0 0 o 4—1 1— CO CM 1—4 W Q CO • CO 0 0 * O vo vo CO CO CM r*** r^. 1 • 1 Q f-l 1 CM i n CO I O 00 • VO /—N Q -*• s o CM • r - c o o VO CO * O 0 0 * ^ N m * • CL < c n • O l 00 < CL • p H • 1—1 * O < CO VO >wS _1 O 1 o 1 * Of 00 O h g \ ✓~N* w h LU o i n • I— * O CO Q CL O Of CD O N * X Q * o * • O H LU CNJ H —I * H O w l ^ . * a /—> w O O * * O O 4—4 o * M - * in Q a ^ 1 W o * m h \ • in O f f l < Ot I— t—I Q ^NCO o i n c l oo ✓-N □ in * HH O t-4 c * />—N <—I CO w CM cm o CJ o 4—4 O co •—I < I - >co o O \ Q V-/ * • i CD •''-•'i 1— ✓ -NO CM Q O Q LU i—« • >00 4—4 U_ O O I CO Z W o O W C M H- CD a t o o o c c Q _ . CD 00 w 1 co o a t o w * i n * ^ *a- m co Q (/) K .•'■n M t—t M N ^ l- O Q -C O ^ s #^N^\CO U V / «t at O 1—4 • I C_> h- I—t O C l \ H CO *—I f"L Q w cn a. z — o e m u . iH • C Q *********************************************************** c SUBROUTINE ITER C Q *********************************************************** c IMPLICIT REAL*8(A-H,0-Z) LOGICAL COMP, BASE COMMON /SYS/ R(7 ,7 ),COMP(18), TC(7), PC(7), RELAX(7), 1 TR(7), PR(7), PHO(18), BASE(7) C0MM0N/SYS1/S(18),D(18),DS0LN,T,P,AM,R1,I1,I2,I3,N1 COMMON /VAR1/ Y(7),XF(18), PHIN(7), PHI0(7), A, H(7), TOTY, ITMAX COMMON /VAR2/ TOL1, TOL2.TOL3, HK,DLOW,ITERAl, ITERA2, ITERA3 COMMON /VAR3/ PS(7),PHP(2),PHS(2),ZC(2 ),ADEN(2),BDEN(2),CA(7,7) COMMON /VAR4/ XINT.TOTPHO COMMON /VAR12/ FN(7), FO(7),YN(7),Y0(7) COMMON /VAR13/PHIOT(7), PHIN1(7),PHIP(7) C C ENTRY START1 C C TOTR=O.DO DO 116 1=3,7 IF (.NOT. COMP(I)) GO TO 116 TOTR=TOTR+XF(Nl)*R(I,N1) 116 CONTINUE TOTM=55. 5093D0 DO 12 1=8,18 TOTM=TOTM+S(I) 12 CONTINUE TOTM=TOTM/(1 .DO-TOTR) DSOLNl=DSOLN/(1.DO-TOTR) DSOLN2=DSOLNl/TOTM DO 13 1=8,18 IF (.NOT. COMP(I)) GO TO 13 PHO(I)=DSOLN2*S(I) 13 CONTINUE PHO(l)=DSOLN2*55. 5093D0 PH0(N1)=XF(N1)* DSOLN1 DO 14 1=3,7 IF (.NOT. COMP(I)) GO TO 14 PHO(I)=PHO(Nl)*R(I,Nl) 14 CONTINUE TOTPHO=O.DO DO 15 1=1,18 IF (.NOT. COMP(I)) GO TO 15 TOTPHO=PHO(I)+ TOTPHO 15 CONTINUE DO 16 1=1,18 IF (.NOT. COMP(I)) GO TO 16 X F(I)=PHO(I)/TOTPHO 16 CONTINUE RETURN ENTRY SOLVE C C TOTPHO=O.DO DO 1199 1=1,18 IF (.NOT. COMP(I)) GO TO 1199 TOTPHO=PHO(I)+ TOTPHO 1199 CONTINUE DO 1198 1=1,18 IF (.NOT. COMP(I)) GO TO 1198 XF(I)=PHO(I)/TOTPHO 1198 CONTINUE DO 1200 1=1,2 IF(.NOT. COMP(I)) GO TO 1200 Y(I)=R1*XF(I)*PS(I)*PHP(I)*PHS(I)/P/PHI0(I) 1200 CONTINUE DO 1210 1=3,7 IF(.NOT. COMP(I)) GO TO 1210 Y(I)=H(I)*XF(I)/P/PHIO(I) 1210 CONTINUE DO 1235 1=1,7 IF ( .NOT. COMP(I)) GO TO 1235 IF ( I I .NE. 1) GO TO 1236 Y0(I)=Y( I ) GO TO 1235 1236 YN(I)=Y(I ) 1235 CONTINUE RETURN C C ENTRY N0RM1 C C SUMY=0. DO 1300 1=1,7 SUMY=SUMY+Y(I) 1300 CONTINUE DO 1310 1=1,7 IF(.NOT. COMP(I)) GO TO 1310 Y(I)=Y( I)/SUMY 1310 CONTINUE 1311 RETURN C C ENTRY N0RM2 C C IF (12 .GT. 1) GO TO 21 XINT=XF(N1) YINT=Y(N1) PHINT=PHI0(N1) HINT=H(N1) XF(N1)=P* YINT* PHINT / HINT GO TO 22 21 XK1=XF(N1) YK1=Y(N1) PHI0K1=PHI0(N1) HK1=H(N1) 195 FK1=YK1*P*PHI0K1/HK1 FKO=YINT*P*PHINT/HINT IF (ABS(XKl-XINT) .LE. l.D -3 ) GO TO 25 SLOP=(FK1—FKO)/( XK1—XINT) GO TO 27 25 SLOP=0. DO 27 T l= l.DO/(1 .DO-SLOP) TMAX= 4 .DO IF (T1 .GE. TMAX ) GO TO 23 IF (T1 .LE. -TMAX ) GO TO 24 GO TO 26 23 T1=TMAX GO TO 26 24 T1=-TMAX 26 XF(N1)=T1*FK1+ (l.Q O -Tl)*X K l XINT=XK1 YINT=YK1 PHINT=PHIOKl HINT=HK1 22 TOTR=O.DO DO 74 1=3,7 IF (.NOT. COMP(I)) GO TO 74 TOTR=TOTR+XF(Nl)*R(I,N1) 74 CONTINUE PH0(N1)=XF(N1)* DSOLN/(l.DO-TOTR) DO 10 1=3,7 IF (.NOT. COMP(I)) GO TO 10 PH0(I)=R(I,N1)*PH0(N1) 10 CONTINUE RETURN C C ENTRY N0RM3 C C IF ( I I .NE. 1) GO TO 621 DO 622 1=1,7 IF (.NOT. COMP(I)) GO TO 622 PHIOT( I)=PH IO (I) PHIN1(I)=PHIN(I) PHIO(I)=PHIN(I) 622 CONTINUE GO TO 618 621 TMAX= 4 .DO DO 601 1=1,2 IF(.NOT. COMP(I)) GO TO 601 FN(I)=XF(I)*PS(I)*PHP(I)*PHS(I)/P/YN(I) FO(I)=XF(I)*PS(I)*PHP(I)*PHS(I)/P/YO(I) IF (ABS(PHIOT(I)-PHINl(I)) .LE. T0L2 ) GO TO 602 SLOP=(FN(I) —F0 (I) ) / ( PHIN1(I) —PHIOT(I)) EEE=1.DO-SLOP IF (ABS( EEE) .GT. l.D -8 ) GO TO 603 IF( EEE .GT. O.DO) GO TO 604 GO TO 605 602 SLOP=O.DO 603 T1=1.D0/(1.DO-SLOP) IF (T1 .GE. TMAX ) GO TO 604 IF (T1 .LE. -TMAX ) GO TO 605 196 GO TO 606 604 T1=TMAX GO TO 606 605 T1=-TMAX 606 PHIO(I)=T1*FN(I)+(1.DO-T1)*PHIN1(I) 601 CONTINUE DO 607 1=3,7 I F ( .NOT. COMP(I)) GO TO 607 FN(I)=XF(I)*H(I)/P/YN(I) F0(I)=XF(I)*H(I)/P/YO(I) IF (ABS(PHIOT( I ) —PHIN1(I)) .LE. T0L2 ) GO TO 609 S LOP=(FN(I)—F0(I))/(PH IN 1(I)—PHIOT( I )) EEE=1.DO-SLOP IF (ABS( EEE) .GT. l.D -8 ) GO TO 611 IF( EEE .GT. O.DO) GO TO 612 GO TO 613 609 SLOP=O.DO 611 T1=1.D0/(1.DO-SLOP) IF (T1 .GE. TMAX ) GO TO 612 IF (T1 .LE. -TMAX ) GO TO 613 GO TO 614 612 T1=TMAX GO TO 614 613 T1=-TMAX 614 PHIO( I )=T lyc FN( I )+(1. DO—T1 )*PHIN1(I) 607 CONTINUE DO 616 1=1,7 IF( .NOT. COMP(I)) GO TO 616 YO(I)=YN(I) PHIOT(I)=PHIN1(I) PHIN1(I)=PHIO(I) 616 CONTINUE 618 RETURN END o o o o o o o POLAR(18),DIPOLM(2) 1 BASE(7) PHO(18), PR(7), TR(7), 1 A A IOM H / 18D1 , .O 1308-6 / 1.38048D-16 l.DO, , 1.85D-18 ID-24/ . HK / DATA DIPOLM , .66D-24,7 / 179D-24,3 POLAR(1DATA ),POLAR(12)/0. POLAR(8 , 1 ) / DATA /2.594D-24/ POLAR(6) l.DO / DATA POLAR(l),POLAR(3),POLAR(4)/1.59D-24,2.600D-24,4.47D-24 /0.65D-26,0.9315D-26/ l.DO 0.225D0, 72.8D0, DATA l.DO, Q(4),Q(6) l.DO, 9.8D-2, *0.D0/ DATA / POLAR,Q 48.2D0, /25 -3, 1.D0 45.4D0, 0,8.D DATA 304.2D0, 0,l.D OMEGA l.DO, 1.D0, /0.344D DATA 305.4D0, PC /217.6D0, 190.6D0, l.DO, DATA /647.3D0, TC O.DO/ O.DO/ DATA * FIRSTU, SECU,UH,U, UHKT * /90 SEC,SP /32 DATA FIRST, DATA IJ,DEL,GHS,DUDA1,DUDA2,DUDA,DGDN D O.DO/ R,PS,H,PHIN,PHI0,PN,VAL/822*0.D0/ SIGMA,EPSK, S,PHO, * DATA Y.XF, /3261 .FALSE./ DATA COMP, BASE /25* A A A16, A6, / .0 0.10/ 0.10, )/ ,1 CA(6 DATA CA(1,6), 91.700.407.7D0/ / /262.4D0, EPSK(9,9) 85.3D0,148.60,215.7D0 ), ,8 ,EPSK(8 ) EPSK(4,4)/ DATA EPSK(6 ,6 ), 4.493D0/ ,3 /2.4D0 DATA EPSK(1,1),EPSK(3 SIGMA(15,15) 3.6432, / 14), DATA SIGMA(14, SIGMA(12,12) .2936,1.9228,2.06D0/ 11), 3 / DATA / SIGMA(11, ,SIGMA(9,9) ) ,8 ,SIGMA(8 ) DATA SIGMA(6 ,6 SIGMA(4,4)/3.87711,4.42198 ), DATA SIGMA(3 ,3 ,NO ) ,DIDP(7 ) PNP(7 TOTY,ITMAX ,QQ(7),QC(7), COMMON ) H(7), Q(7),QD(7 A, /VAR9/ COMMON PS(7),PHP(2),PHS(2),ZC(2),ADEN(2),BDEN(2),CA(7,7) /VAR3/ ,PHIN(7),PHI0(7), COMMON F(18) ITERA3 (7),X Y ITERA2, HK,DLOW,ITERAl, /VAR1/ COMMON TOL3, TOL2, TOL1, /VAR2/ BLOCK DATA A A TR1IEA, TR3IMX 20,91,0 / ,99,10,200 0 /2 / l.ODO/ 5* -4 ,l.D ITERA3.ITMAX / DATA ITERA1,ITERA2, 1.0D-4 ,R(7,7) ) R(6,6 DATA T0L1,T0L2,T0L3,TQL4,T0L5,DL0W/5* ), ,5 0.5D0/ (5 * ),R DATA RELAX ,4 /7 (4 ),R ,3 DATA R(3 CA(4,3)/-0.0078DO,-0.0078DO/ 0.0933D0.0.0933DO/ DATA CA(3,4), CA(6,3)/ ), DATA CA(3,6 / / O.DO 49* 0.274D0,1.D0,0.229D0,1.D0 DATA CA / l.DO, /190.D0.203.D0/ 385.1D0/ DATA 0.3471D0, ADEN,BDEN,ZC EPSK(15,15) / /245.8, 14), DATA EPSK(14, EPSK(12,12) 11), DATA EPSK(11, OMEGA(7),T0L4, EPSK(18,18), COMMON SIGMA(18,18), /PARA1/ COMMON/SYS1/S(18),D(18),DS0LN,T,P,AM,R1,I1,I2,I3,N1 RELAX(7), PC(7), TC(7), COMMON * * * R(7,7),COMP(?.8). rt * t* r * t* tr /SYS/ rtr * trtrt* r * * * * * * * * * * * rtrtrt * * * * rtrtrt rt * * rtrt * * rt * rt * rtrtrt * * * * ** * * rt ** * * rt rt * rt * * * rtrtrt rt rt * * * rtrtrt * rt* * * * rtrt Hr* * * rtrt * * rt * * rt * * * END LOGICAL COMP, BASE MLCT EL8AH O-Z) REAL*8(A-H, IMPLICIT ,3.400 / 197 VITA Yangtzu Chao was born on May 8 , 1954 in Taiwan. He graduated from Tunghai University with a Bachelor of Science degree in Chemical Engineering in 1976. After two years of military service as an army second Lieutenant, he rejoined the ChE department at Tunghai University and served as a full time teaching assistant. He came to the U.S. in August, 1979 and obtained his Master of Science degree in Gas Engineering from Illinois Institute of Technology in December of 1981. He then enrolled in the Chemical Engineering Department at LSU in August, 1982 and is currently a candidate for the degree of Doctor of Philosophy in Chemical Engineering. Yangtzu Chao married the former Yaling Chien in May of 1980. Their first born was a lovely daughter, Stephanie, in 1984. 198 DOCTORAL EXAMINATION AND DISSERTATION REPORT Candidate: Yangtzu Chao Major Field: Chemical Engineering Title of Dissertation: Prediction of Methane Solubility in Geopressured Brine Solutions by Application of Perturbation Theory Approved: Major/Pj^ofessor and Chairman lean of the Graduate ISchool EXAMINING COMMITTEE: f c - 7 - Date of Examination: November 30, 1987