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PEH:Gas Properties
Publication Information
Petroleum Engineering Handbook
Larry W. Lake, Editor-in-Chief
Volume I – General Engineering
John R. Fanchi, Editor
Copyright 2007, Society of Petroleum Engineers
Chapter 5 – Gas Properties
ISBN 978-1-55563-108-6
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Contents
- 1 Molecular Weight
- 2 Ideal Gas
- 3 Critical Temperature and Pressure
- 4 Specific Gravity (Relative Density)
- 5 Mole Fraction and Apparent Molecular Weight of Gas Mixtures
- 6 Specific Gravity of Gas Mixtures
- 7 Dalton’s Law
- 8 Amagat’s Law
- 9 Real Gases
- 10 Real-Gas Law
- 11 Gas Density and Formation Volume Factor
- 12 Isothermal Compressibility of Gases
- 13 Gas Viscosity
- 14 Real-Gas Pseudopotential
- 15 Vapor Pressure
- 16 Further Example Problems
- 17 Nomenclature
- 18 References
- 19 SI Metric Conversion Factors
Molecular Weight
Molecules of a particular chemical species are composed of groups of atoms that always combine according to a specific formula. The chemical formula and the international atomic weight table provide us with a scale for determining the weight ratios of all atoms combined in any molecule. The molecular weight, M, of a molecule is simply the sum of all the atomic weights of its constituent atoms. It follows, then, that the number of molecules in a given mass of material is inversely proportional to its molecular weight. Therefore, when masses of different materials have the same ratio as their molecular weights, the number of molecules present is equal. For instance, 2 lbm hydrogen contains the same number of molecules as 16 lbm methane. For this reason, it is convenient to define the unit "lbm mol" as a mass of the material in pounds equal to its molecular weight. Similarly, a "g mol" is its mass in grams. One lbm mol or one g mol of any compound, therefore, represents a fixed number of molecules. This number for the g mol was determined in 1998 by the U.S. Natl. Inst. of Standards and Technology to be 6.02214199×10^{23}. The number of significant digits shown is the accuracy to which it has been determined experimentally.
Ideal Gas
The kinetic theory of gases postulates that a gas is composed of a large number of very small discrete particles. These particles can be shown to be identified with molecules. For an ideal gas, the volume of these particles is assumed to be so small that it is negligible compared with the total volume occupied by the gas. It is assumed also that these particles or molecules have neither attractive nor repulsive forces between them. The average energy of the particles or molecules can be shown to be a function of temperature only. Thus, the kinetic energy, E_{k}, is independent of molecule type or size. Because kinetic energy is related to mass and velocity by E_{k} = 1/2 mv^{2}, it follows that small molecules (less mass) must travel faster than large molecules (more mass) when both are at the same temperature. Molecules are considered to be moving in all directions in a random manner as a result of frequent collisions with one another and with the walls of the containing vessel. The collisions with the walls create the pressure exerted by the gas. Thus, as the volume occupied by the gas is decreased, the collisions of the particles with the walls are more frequent, and an increase in pressure results. It is a statement of Boyle’s law that this increase in pressure is inversely proportional to the change in volume at constant temperature:
....................(5.1)
where p is the absolute pressure and V is the volume.
Further, if the temperature is increased, the velocity of the molecules and, therefore, the energy with which they strike the walls of the containing vessel will be increased, resulting in a rise in pressure. To maintain the pressure constant while heating a gas, the volume must be increased in proportion to the change in absolute temperature. This is a statement of Charles’ law,
....................(5.2)
where T is the absolute temperature and p is constant.
From a historical viewpoint, the observations of Boyle and Charles in no small degree led to the establishment of the kinetic theory of gases, rather than vice versa. It follows from this discussion that, at zero degrees absolute, the kinetic energy of an ideal gas, as well as its volume and pressure, would be zero. This agrees with the definition of absolute zero, which is the temperature at which all the molecules present have zero kinetic energy.
Because the kinetic energy of a molecule depends only on temperature, and not on size or type of molecule, equal molecular quantities of different gases at the same pressure and temperature would occupy equal volumes. The volume occupied by an ideal gas therefore depends on three things: temperature, pressure, and number of molecules (moles) present. It does not depend on the type of molecule present. The ideal-gas law, which is actually a combination of Boyle’s and Charles’ laws, is a statement of this fact:
....................(5.3)
where p = pressure, V = volume, n = number of moles, R = gas-law constant, and T = absolute temperature.
The gas-law constant, R, is a proportionality constant that depends only on the units of p, V, n, and T. Tables 5.1a through 5.1c present different values of R for the various units of these parameters. The value of the gas constant is experimental, and more-accurate values are reported occasionally. The values in Tables 5.1a through 5.1c are based on the values reported by Moldover et al.^{[1]} Their value was determined from measurements of the speed of sound in argon as a function of pressure at the temperature of the triple point of water. Note that because pV has the units of energy, the value of R is typically given in units of energy per mole per absolute temperature unit [e.g., the appropriate SI value for R is 8.31447 J/(g mol-K), and the appropriate British gravitational (sometimes called the American customary units) value for R is 1,545.35 ft-lbf/(lb-mol°R]. However, sometimes pressure and volume units are more appropriate, such as R = 10.7316 (psia-ft^{3})/ (lb mol-°R).
Critical Temperature and Pressure
Typical pressure/volume/temperature (PVT) relationships for a pure fluid are illustrated in Fig. 5.1. The curve segment B-C-D defines the limits of vapor/liquid coexistence, with B-C being the bubblepoint curve of the liquid and C-D being the dewpoint curve of the vapor. Any combination of temperature, pressure, and volume above that line segment indicates that the fluid exists in a single phase. At low temperatures and pressures, the properties of equilibrium vapors and liquids are extremely different (e.g., the density of a gas is low, while that of a liquid is relatively high). As the pressure and temperature are increased along the coexistence curves, liquid density, viscosity, and other properties generally decrease while vapor density and viscosity generally increase. Thus, the difference in the physical properties of the coexisting phases decreases. These changes continue as the temperature and pressure are raised until a point is reached at which the properties of the equilibrium vapor and liquid become equal. The temperature, pressure, and volume at this point are called the "critical" values for that species. Location C on Fig. 5.1 is the critical point. The critical temperature and pressure are unique values for each species and are useful in correlating physical properties. Critical constants for some of the commonly occurring hydrocarbons and other components of natural gas can be found in Table 5.2.
Specific Gravity (Relative Density)
The specific gravity of a gas, γ, is the ratio of the density of the gas at standard pressure and temperature to the density of air at the same standard pressure and temperature. The standard temperature is usually 60°F, and the standard pressure is usually 14.696 psia. However, slightly different standards are sometimes used in different locations and in different units. The ideal-gas laws can be used to show that the specific gravity (ratio of densities) is also equal to the ratio of the molecular weights. By convention, specific gravities of all gases at all pressures are usually set equal to the ratio of the molecular weight of the gas to that of air (28.967). Although specific gravity is used throughout this chapter, this traditional term is not used under the SI system; it has been replaced by "relative density."
Mole Fraction and Apparent Molecular Weight of Gas Mixtures
The analysis of a gas mixture can be expressed in terms of a mole fraction, y_{i}, of each component, which is the ratio of the number of moles of a given component to the total number of moles present. Analyses also can be expressed in terms of the volume, weight, or pressure fraction of each component present. Under limited conditions, where gaseous mixtures conform reasonably well to the ideal-gas laws, the mole fraction can be shown to be equal to the volume fraction but not to the weight fraction. The apparent molecular weight of a gas mixture is equal to the sum of the mole fraction times the molecular weight of each component.
Specific Gravity of Gas Mixtures
The specific gravity (γ_{g}) of a gas mixture is the ratio of the density of the gas mixture to that of air. It is measured easily at the wellhead in the field and therefore is used as an indication of the composition of the gas. As mentioned earlier, the specific gravity of gas is proportional to its molecular weight (M_{g}) if it is measured at low pressures where gas behavior approaches ideality. Specific gravity also has been used to correlate other physical properties of natural gases. To do this, it is necessary to assume that the analyses of gases vary regularly with their gravities. Because this assumption is only an approximation and is known to do poorly for gases with appreciable nonhydrocarbon content, it should be used only in the absence of a complete analysis or of correlations based on a complete analysis of the gas.
Dalton’s Law
The partial pressure of a gas in a mixture of gases is defined as the pressure that the gas would exert if it alone were present at the same temperature and volume as the mixture. Dalton’s law states that the sum of the partial pressures of the gases in a mixture is equal to the total pressure of the mixture. This law can be shown to be true if the ideal-gas laws apply.
Amagat’s Law
The partial volume of a gas in a mixture of gases is defined as the volume that the gas would occupy if it alone were present at the same temperature and pressure as the mixture of the gases. If the ideal-gas laws hold, then Amagat’s law (that the sum of the partial volumes is equal to the total volume) also must be true.
Real Gases
At low pressures and relatively high temperatures, the volume of most gases is so large that the volume of the molecules themselves may be neglected. Also, the distance between molecules is so great that the presence of even fairly strong attractive or repulsive forces is not sufficient to affect the behavior in the gas state. However, as the pressure is increased, the total volume occupied by the gas becomes small enough that the volume of the molecules themselves is appreciable and must be considered. Also, under these conditions, the distance between the molecules is decreased to the point at which the attractive or repulsive forces between the molecules become important. This behavior negates the assumptions required for ideal-gas behavior, and serious errors are observed when comparing experimental volumes to those calculated with the ideal-gas law. Consequently, a real-gas law was formulated (in terms of a correction to the ideal-gas law) by use of a proportionality term.
Real-Gas Law
The volume of a real gas is usually less than what the volume of an ideal gas would be at the same temperature and pressure; hence, a real gas is said to be supercompressible. The ratio of the real volume to the ideal volume, which is a measure of the amount that the gas deviates from perfect behavior, is called the supercompressibility factor, sometimes shortened to the compressibility factor. It is also called the gas-deviation factor and given the symbol z. The gas-deviation factor is by definition the ratio of the volume actually occupied by a gas at a given pressure and temperature to the volume it would occupy if it behaved ideally, or:
....................(5.4)
Note that the numerator and denominator of Eq. 5.4 refer to the same mass. (This equation for the z factor is also used for liquids.) Thus, the real-gas equation of state is written:
....................(5.5)
The gas-deviation factor, z, is close to 1 at low pressures and high temperatures, which means that the gas behaves as an ideal gas at these conditions. At standard or atmospheric conditions, the gas z factor is always approximately 1. As the pressure increases, the z factor first decreases to a minimum, which is approximately 0.27 for the critical temperature and critical pressure. For temperatures of 1.5 times the critical temperature, the minimum z factor is approximately 0.77, and for temperatures of twice the critical temperature, the minimum z factor is 0.937. At high pressures, the z factor increases above 1, where the gas is no longer supercompressible. At these conditions, the specific volume of the gas is becoming so small, and the distance between molecules is much smaller, so that the density is more strongly affected by the volume occupied by the individual molecules. Hence, the z factor continues to increase above unity as the pressure increases.
Tables of compressibility factors are available for most pure gases as functions of temperature and pressure. Compressibility factors for mixtures (or unknown pure compounds) are measured easily in a Burnett^{[2]} apparatus or a variable-volume PVT equilibrium cell. The gas-deviation factor, z, is determined by measuring the volume of a sample of the natural gas at a specific pressure and temperature, then measuring the volume of the same quantity of gas at atmospheric pressure and at a temperature sufficiently high so that the hydrocarbon mixture is in the vapor phase. Tables of compressibility factors are available for most pure gases as functions of temperature and pressure. Excellent correlations are also available for the calculation of compressibility factors. For this reason, compressibility factors are no longer routinely measured on dry-gas mixtures or on most of the leaner wet gases. Rich-gas/condensate systems require other equilibrium studies, and compressibility factors can be obtained routinely from these data.
If the gas-deviation factor is not measured, it may be estimated from correlations. The correlations depend on the pseudoreduced temperature and pressure, which in turn depend on the pseudocritical temperature and pseudocritical pressure. The pseudocritical temperature and pseudocritical pressure normally can be defined most simply as the molal average critical temperature and pressure of the mixture components. Thus,
....................(5.6)
where p_{pc} = pseudocritical pressure of the gas mixture, T_{pc} = pseudocritical temperature of the gas mixture, p_{ci} = critical pressure of component i in the gas mixture, T_{ci} =critical temperature of component i in the gas mixture, and y_{i} =mole fraction of component i in the gas mixture. These relations are known as Kay’s rule after W.B. Kay,^{[3]} who first suggested their use.
The pseudocritical temperature and pressure are not the actual critical temperature and pressure of the mixture but represent the values that must be used for the purpose of comparing corresponding states of different gases on the z-factor chart (Fig 5.2). It has been found to approximate the convergence of the lines of constant volume on a pressure/temperature diagram.
Sutton^{[4]} found that Kay’s rules for the determination of pseudocritical properties did not give accurate results for higher-molecular-weight mixtures of hydrocarbon gases. He found that they resulted in errors in the z factor as high as 15%. Instead, Sutton^{[4]} proposed a modification of a method first proposed by Stewart et al.^{[5]} Sutton’s^{[4]} method is to first define and determine the pseudocritical properties of the C_{7+} fraction, then calculate the pseudocritical properties of the mixture as follows:
....................(5.7)
....................(5.8)
....................(5.9)
....................(5.10)
....................(5.11)
....................(5.12)
If the composition of the gas is unknown, then a correlation to estimate pseudocritical temperature and pseudocritical pressure values from the specific gravity is used. There are several different correlations available, but Fig. 5.3 was developed by Sutton^{[4]} on the basis of 264 different gas samples. Sutton also used regression analysis on the raw data to obtain the following second-order fit for the pseudocritical properties of hydrocarbon mixtures:
....................(5.13)
....................(5.14)
These equations and Fig 5.3 are valid over the range of specific-gas gravities with which Sutton^{[4]} worked: 0.57 < γ_{g} < 1.68. Using the obtained pseudocritical values, the pseudoreduced pressure and temperature are calculated using
....................(5.15)
The gas-deviation factor is then found by using the well-known correlation chart of Fig. 5.2, originally developed by Standing and Katz.^{[6]} Compressibility factors of high-pressure natural gases (10,000 to 20,000 psia) may be obtained from Fig. 5.4, which was developed by Katz et al.^{[7]} Figs. 5.5 and 5.6 may be used for low-pressure applications after Brown et al.^{[8]}
Dranchuk and Abou-Kassem^{[9]} fitted an equation of state to the data of Standing and Katz,^{[6]} which is more convenient for estimating the gas-deviation factor in computer programs and spreadsheets. Hall and Yarborough^{[10]} also have published an alternative equation of state. The Dranchuk and Abou-Kassem^{[9]} equation of state is based on the generalized Starling equation of state and is expressed as follows:
....................(5.16)
where and where the constants A_{1} through A_{11} are as follows: A_{1} = 0.3265; A_{2} = –1.0700; A_{3} = –0.5339; A_{4} = 0.01569; A_{5} = –0.05165; A_{6} = 0.5475; A_{7} = –0.7361; A_{8} = 0.1844; A_{9} = 0.1056; A_{10} = 0.6134; and A_{11} = 0.7210.
Dranchuk and Abou-Kassem^{[9]} found an average absolute error of 0.486% in their equation, with a standard deviation of 0.00747 over ranges of pseudoreduced pressure and temperature of 0.2 < p_{pr} < 30; 1.0 < T_{pr} < 3.0; and for p_{pr} < 1.0 with 0.7 < T_{pr} < 1.0.
Dranchuk and Abou-Kassem^{[9]} also found that this equation and other equations of state give unacceptable results near the critical temperature for T_{pr} = 1.0 and p_{pr} >1.0, so these equations are not recommended in this range.
Because the parameter z is embedded in ρ_{r}, an iterative solution is necessary to solve the Dranchuk and Abou-Kassem equation of state, but this can be programmed. An example of this is provided by Dranchuk and Abou-Kassem.^{[9]} The equation also can be solved on a spreadsheet using the nonlinear-equation-solver option, which is discussed in more detail elsewhere.^{[11]} Nonlinear equation solvers are also set up specifically to solve these equations easily.
The z-factor chart of Standing and Katz (Fig 5.2) and the pseudocritical property-calculation methods of Sutton^{[4]} are valid only for mixtures of hydrocarbon gases. Wichert and Aziz^{[12]} have developed a correlation to account for inaccuracies in the Standing and Katz chart when the gas contains significant fractions of acid gases, specifically carbon dioxide (CO_{2}) and hydrogen sulfide (H_{2}S). The Wichert and Aziz^{[12]} correlation modifies the values of the pseudocritical temperature and pressure of the gas. Once the modified pseudocritical properties are obtained, they are used to calculate pseudoreduced properties, and the z factor is determined from Fig. 5.2 or Eq. 5.10. The Wichert and Aziz^{[12]} correlation first calculates a deviation parameter ε:
....................(5.17)
where A = the sum of the mole fractions of CO_{2} and H_{2}S in the gas mixture and B = the mole fraction of H_{2}S in the gas mixture. Then, the value of ε is used to determine the modified pseudocritical properties as follows:
....................(5.18)
....................(5.19)
The correlation is valid only in units of T in R and p in psia. It is applicable to concentrations of CO_{2} < 54.4 mol% and H_{2}S < 73.8 mol%. Note that ε also has units of R. The correction factor, ε, has been plotted against H_{2}S and CO_{2} concentrations in Fig. 5.7 for convenience. Note that maximum correction occurs around A = B = 47% or 47% H_{2}S concentration and 0% CO_{2} concentration. Wichert and Aziz^{[12]} found their correlation to have an average absolute error of 0.97% over the following ranges of data: 154 psia < p < 7,026 psia and 40°F < T < 300°F.
Piper et al.^{[13]} have also adapted the Stewart et al.^{[5]} method to develop equations that can be used to calculate the pseudocritical properties of natural gas mixtures that contain nitrogen (N_{2}), CO_{2}, and H_{2}S without making a separate correction. There are two sets of equations, depending on whether the composition or the specific gravity is known. When the gas composition is used, the following equations are developed on the basis of 896 data points:
....................(5.20)
....................(5.21)
where y_{i} are the contaminant compositions and y_{j} are the hydrocarbon compositions , and α_{i} and β_{i} are as given in Table 5.3.
If the composition of the hydrocarbons is unknown but the specific gravity and the nonhydrocarbon compositions are known, the following equations for J and K were developed by Piper et al.^{[13]} on the basis of 1,482 data points:
....................(5.22)
and
....................(5.23)
Then, the pseudocritical properties can be calculated from J and K in Eqs. 5.11 and 5.12.
Example 5.1
Calculation of the z Factor for Sour Gas. Using (a) the Sutton^{[4]} correlation and the Wichert and Aziz^{[12]} correction, and (b) the method of Piper et al.,^{[13]} calculate the z factor for a gas with the following properties and conditions:
γ_{g} = 0.7, H_{2}S = 7%, and CO_{2} = 10%; p = 2,010 psia and T = 75°F.
Solution. (a) First, calculate the pseudocritical properties.
Next, calculate the adjustments to the pseudocritical properties using the Wichert and Aziz^{[12]} parameters.
Next, calculate the pseudoreduced properties:
Finally, looking up the z-factor chart (Fig 5.2) gives z = 0.772.
(b) Using the method of Piper et al.,^{[13]}
The pseudoreduced properties are then:
Finally, looking up the z-factor chart (Fig 5.3) gives z = 0.745.
The two methods give results that differ by 3.6% of the smaller value (z = 0.745), which is within the range of accuracy of either method. Because the method of Piper et al.^{[13]} is based on a larger data set and has integrated the nonhydrocarbon compositions into the method, it is likely to be more accurate.
Gas Density and Formation Volume Factor
The formation volume factor of gas is defined as the ratio of the volume of gas at the reservoir temperature and pressure to the volume at the standard or surface temperature and pressure (p_{s} and T_{s}). It is given the symbol B_{g} and is often expressed in either cubic feet of reservoir volume per standard cubic foot of gas or barrels of reservoir volume per standard cubic foot of gas. The gas-deviation factor is unity at standard conditions; hence, the equation for the gas formation volume factor can be calculated using the real gas equation:
....................(5.24)
The n divides out here because both volumes refer to the same quantity of mass.
When p_{s} is 1 atm (14.696 psia or 101.325 kPa) and T_{s} is 60°F (519.67°R or 288.71°K), this equation can be written in three well-known standard forms:
....................(5.25)
where rcf/scf = reservoir cubic feet per standard cubic feet, RB = reservoir barrels, and Rm^{3}/Sm^{3} = reservoir cubic meters per standard cubic meters. The formation volume factor is always in units of reservoir volumes per standard volumes.
The three forms in Eq. 5.25 are for specific units. In the first two equation forms, the pressure is in psia and the temperature is in °R. In the third form, the pressure is in kPa and the temperature is in K.
The density of a reservoir gas is defined as the mass of the gas divided by its reservoir volume, so it can also be derived and calculated from the real-gas law:
....................(5.26)
Isothermal Compressibility of Gases
The isothermal gas compressibility, c_{g}, is a useful concept that is used extensively in determining the compressible properties of the reservoir. The isothermal compressibility is also the reciprocal of the bulk modulus of elasticity. Gas usually is the most compressible medium in the reservoir; however, care should be taken so that it is not confused with the gas-deviation factor, z, which is sometimes called the compressibility factor.
The isothermal gas compressibility is defined as:
....................(5.27)
An expression in terms of z and p for the compressibility can be derived from the real-gas law (Eq. 5.5):
....................(5.28)
From the real-gas equation of state,
....................(5.29)
hence,
....................(5.30)
For gases at low pressures, the second term is small, and the isothermal compressibility can be approximated by c_{g} ≈ 1/p. Eq. 5.30 is not particularly convenient for determining the gas compressibility because in Fig 5.2 and Eq. 5.16, z is not actually expressed as a function of p but of p_{r}. However, Eq. 5.30 can be made more convenient when written in terms of a dimensionless, pseudoreduced gas compressibility defined as
....................(5.31)
Multiplying Eq. 5.30 through by the pseudocritical pressure gives
....................(5.32)
Charts of the pseudoreduced gas compressibility have been published by Trube^{[14]} and by Mattar et al.,^{[15]} and two of these are shown in Figs 5.8 and 5.9. Mattar et al.^{[15]} also developed an analytical expression for calculating the pseudoreduced compressibility; that expression is
....................(5.33)
Taking the derivative of Eq. 5.10, the following is obtained:
....................(5.34)
Parameters A_{1} through A_{11} are defined after Eq. 5.16. Eq. 5.34 can then be substituted into Eq. 5.33, and the pseudoreduced gas compressibility can be calculated. Then, if the pseudoreduced gas compressibility is divided by the pseudocritical pressure, the gas compressibility is obtained analytically. Either the graphical method or the analytical method can be used, but the analytical method is easier to apply in a spreadsheet, nonlinear solver, or other computer program.
There is also a close relationship between the formation volume factor of gas and the isothermal gas compressibility. It can easily be shown that
....................(5.35)
Gas Viscosity
Just as the compressibility of natural gas is much greater than that of oil, water, or rock, the viscosity of natural gas is usually several orders of magnitude smaller than oil or water. This makes gas much more mobile in the reservoir than either oil or water. Reliable correlation charts are available to estimate gas viscosity. Carr et al.^{[16]} have developed charts (Figs. 5.10 through 5.13) that are the most widely used for estimating the viscosity of natural gas from the pseudoreduced critical temperature and pressure. Fig. 5.10 gives the viscosities for individual components. Fig. 5.11 gives the viscosities for gas at the desired temperature and atmospheric pressure based on the temperature and specific gravity or molecular weight. The viscosity of gas mixtures at one atmosphere and reservoir temperature can either be read from Fig. 5.11 or determined from the gas-mixture composition with Eq. 5.36.
....................(5.36)
where μ_{ga} = viscosity of the gas mixture at the desired temperature and atmospheric pressure; y_{i} = mole fraction of the ith component; μ_{i} = viscosity of the ith component of the gas mixture at the desired temperature and atmospheric pressure (obtained from Fig. 5.10); M_{gi} = molecular weight of the ith component of the gas mixture; and N = number of components in the gas mixture.
This viscosity is then multiplied by the viscosity ratio (from Fig. 5.12 or Fig. 5.13) to obtain the viscosity at reservoir temperature and pressure. Note that Figs. 5.12 and 5.13 (from Carr et al.^{[16]}) are based on pseudocritical properties determined with Kay’s rules. It would not be correct, then, to use the methods of Sutton^{[4]} or Piper et al.^{[13]} to calculate the pseudocritical properties for use with those charts. However, Kay’s rules require a full gas composition. If only specific gravity is known, then the pseudocritical properties would have to be obtained from Fig. 5.3 or Eqs. 5.13 and 5.14. The inserts of Fig. 5.11 are corrections to be added to the atmospheric viscosity when the gas contains N_{2}, CO_{2}, and H_{2}S.
Lee et al.^{[17]} developed a useful analytical method that gives a good estimate of gas viscosity for most natural gases. This method lends itself for use in computer programs and spreadsheets. The method uses the gas temperature, pressure, z factor, and molecular weight, which have to be measured or calculated; the density can be measured or calculated as well. The equations of Lee et al.^{[17]} are for specific units as noted below and are as follows:
....................(5.37)
where , and
Y = 2.4 - 0.2X and where μ_{g} = gas viscosity, cp; ρ =gas density, g/cm^{3}; p = pressure, psia; T = temperature, °R; and M_{g} = gas molecular weight = 28.967 γ_{g}.
For the data from which the correlation was developed, the standard deviation in the calculated gas viscosity was 2.7%, and the maximum deviation was 9%. The ranges of variables used in the correlation were 100 psia < p < 8,000 psia, 100 < T (°F) < 340, and 0.90 < CO_{2} (mol%) < 3.20 and 0.0 < N_{2} (mol%) < 4.80. In using these equations, it is important either to measure the density or to ensure that the z-factor calculation has included the effect of N_{2}, CO_{2}, and H_{2}S using the method of Wichert and Aziz.^{[12]} The equations of Lee et al.^{[17]} were originally written to give the viscosity in micropoise, but the modified form above gives the viscosity in the more commonly used centipoise. This viscosity unit (cp) is also easily converted to the SI unit of Pa•s by dividing by 1,000.
Example 5.2 Properties of Natural Gas. For the gas in Example 5.1, find the (a) density, (b) formation volume factor, (c) viscosity, and (d) isothermal compressibility.
Solution.
(a) The density is calculated from Eq. 5.14:
(b) The formation volume factor is calculated from Eq. 5.13:
(c) The viscosity is determined using the charts of Carr et al.^{[16]} in Figs. 5.10 through 5.13. First, the viscosity for M_{g} = (0.7)(28.967) = 20.3 at p = 1 atm and T = 75°F is read from Fig. 5.11. This gives 0.0102 cp, but corrections are needed for the acid gases. The correction for 10% CO_{2} is 0.0005 cp, and the correction for 7% H_{2}S is 0.0002 cp. Hence, this gives μ_{ga} = 0.0109 cp.
Next, the ratio of μ_{g}/μ_{ga} is read from Fig. 5.13, which gives μ_{g}/μ_{ga} = 1.55. Hence, μ_{g} = (1.55) (0.0109 cp) = 0.0169 cp.
(d) The compressibility is determined by first reading Fig. 5.8 or Fig. 5.9 for the previously calculated values of p_{r} = 3.200 and T_{r} = 1.500 to give c_{r}T_{r} = 0.5. Because T_{r} = 1.500 then c_{r} = 0.5/1.5 = 0.3333. Because c_{r} = c_{g} p_{pc},
Real-Gas Pseudopotential
In the analysis of gas reservoirs, well-test analysis, gas flow in pipes, and other calculations can be made more accurate by the use of the real-gas pseudopotential. This is because the z factor and viscosity that appear in such equations along with pressure terms are dependent on pressure. Consequently, the integral of pressure divided by the z factor and viscosity is defined as a separate parameter called the real-gas pseudopotential and is designated here as ψ(p).
....................(5.38)
where p_{o} is some arbritary low base pressure (typically atmospheric pressure). This integral is usually evaluated numerically using values of z and μ for the particular gas at a particular temperature. Then, the pseudopotential is tabulated as a function of pressure and temperature. Illustrations of the calculation and use of the real-gas pseudopotential are provided elsewhere in this Handbook.
Vapor Pressure
At a given temperature, the vapor pressure of a pure compound is the pressure at which vapor and liquid coexist at equilibrium. The term "vapor pressure" should be used only with pure compounds and is usually considered as a liquid (rather than a gas) property. For a pure compound, there is only one vapor pressure at any temperature. A plot of these pressures for various temperatures is shown in Fig. 5.14 for n-butane. The temperature at which the vapor pressure is equal to 1 atm (14.696 psia or 101.32 kPa) is known as the normal boiling point.
The Clapeyron Equation
The Clapeyron equation gives a rigorous quantitative relationship between vapor pressure and temperature:
....................(5.39)
where p_{v} = vapor pressure, T = absolute temperature, ΔV = increase in volume caused by vaporizing 1 mole, and L_{v} = molal latent heat of vaporization.
Assuming ideal-gas behavior of the vapor and neglecting the liquid volume, the Clapeyron equation can be simplified over a small temperature range to give the approximation
....................(5.40)
which is known as the Clausius-Clapeyron equation. Integrating this equation gives
....................(5.41)
where b is a constant of integration that depends on the particular fluid and the data range. This equation suggests that a plot of logarithm of vapor pressure against the reciprocal of the absolute temperature would approximate a straight line. Such a plot is useful in interpolating and extrapolating data over short ranges. However, the shape of this relationship for a real substance over a significant temperature range is more S-shaped than straight. Therefore, the use of the Clausius-Clapeyron equation is not recommended when other methods are available, except over short temperature ranges in regions where the ideal-gas law is valid.
Cox Chart
Cox^{[18]} further improved the method of estimating vapor pressure by plotting the logarithm of vapor pressure against an arbitrary temperature scale. The vapor-pressure/temperature plot forms a straight line, at least for the reference compound (and usually for most of the materials related to the reference compound). This is especially true for petroleum hydrocarbons. A Cox chart, using water as a reference material, is shown in Fig. 5.15. In addition to forming nearly straight lines, compounds of the same family appear to converge on a single point. Thus, it is necessary to know only vapor pressure at one temperature to estimate the position of the vapor-pressure line. This approach is very useful and can be much better than the previous method. Its accuracy is dependent to a large degree on the readability of the chart.Calingheart and Davis Equation
The Cox chart was fit with a three-parameter function by Calingeart and Davis.^{[19]} Their equation is
....................(5.42)
where A and B are empirical constants and, for compounds boiling between 32 and 212°F, C is a constant with a value of 43 when T is in K and a value of 77.4 when T is in °R. This equation generally is known as the Antoine^{[20]} equation because Antoine proposed one of a very similar nature that used 13 K for the constant C. Knowledge of the vapor pressure at two temperatures will fix A and B and permit approximations of vapor pressures at other temperatures. Generally, the Antoine approach can be expected to have less than 2% error and is the preferred approach if the vapor pressure is expected to be less than 1,500 mm Hg (200 kPa) and if the constants are available.
Lee-Kesler Equation
Vapor pressures also can be calculated by corresponding-states principles. The most common expansions of the Clapeyron equation lead to a two-parameter expression. Pitzer et al.^{[21]} extended the epansion to contain three parameters:
....................(5.43)
where p_{vr} is the reduced vapor pressure (vapor pressure/critical pressure), f^{0} and f^{1} are functions of reduced temperature, and ω is the acentric factor.
Lee and Kesler^{[22]} have expressed f^{0} and f^{1} in analytical forms:
....................(5.44)
and
....................(5.45)
which can be solved easily by computer or spreadsheet. Lee-Kesler^{[22]} is the preferred method of calculation but should be used only for nonpolar liquids.
The advent of computers, calculators, and spreadsheets makes the use of approximations and charts much less advantageous than it was before the 1970s. Values of acentric factors can be found in Poling et al.,^{[23]} who also presented many other available vapor-pressure correlations and calculation techniques, with comments about their advantages and limitations.
Further Example Problems
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Example 5.3
Calculate the relative density (specific gravity) of natural gas with the following composition (all compositions are in mol%):
C_{1} | = | 83.19% |
C_{2} | = | 8.48% |
C_{3} | = | 4.37% |
i-C_{4} | = | 0.76% |
n-C_{4} | = | 1.68% |
i-C_{5} | = | 0.57% |
n-C_{5} | = | 0.32% |
C_{6} | = | 0.63% |
Total | = | 100% |
where the molecular weight of air, M_{a}, is 28.967.
Example 5.4
Calculate the actual density of the same mixture at 1,525 psia and 75°F (a) using Kay’s^{[3]} rules, (b) Sutton’s^{[4]} correlation, and (c) the Piper et al.^{[13]} correlation.
Solution. The density is calculated from
where p = 1,525 psia, M_{g} = 20.424, R = 10.7316 (psia-ft^{3})/(lbm mol°R), and T = 75°F + 459.67 = 534.67°R, and z must be obtained from Fig. 5.3.
(a) Calculate z_{g} from the known composition in Table 5.5.
Using Kay’s^{[3]} rules, we obtain from the known gas composition: T_{pc} =Σy_{i}T_{i} = 393.8°R, T_{pr} = 534.67/393.8 = 1.3577, p_{pc} =Σy_{i}p_{ci} = 662.88 psia, p_{pr} = p/p_{pc} = 1,525/662.88 =2.301,
and from Fig. 5.3, z_{g} = 0.71.
(b) From Sutton’s^{[4]} gas gravity method, γ_{g} = 0.705; then, we obtain from Eqs. 5.7 and 5.8 that
This gives
From Fig. 5.3, we obtain z_{g} = 0.745.
(c) Using the Piper et al.^{[13]} method, we first calculate J and K using
The details of the calculations are found in Table 5.5.
Then,
Finally, looking up the z-factor chart (Fig. 5.3) gives z = 0.745.
Conclusion. Even though the Sutton^{[4]} correlation and the Piper et al.^{[13]} correlation gave slightly different critical properties, the z factors from those two methods are the same. Kay’s^{[3]} rule gives a value that is 4.6% lower, but the result using Sutton’s^{[4]} correlation and the Piper et al.^{[13]} correlation has been shown to be more accurate. The density is then given by
Example 5.5 Calculate the z factor for the reservoir fluid in Table 5.6 at 307°F and 6,098 psia.
The experimental value is z = 0.998.
Solution. Using the Piper et al.^{[13]} method, we first calculate J and K using
The details of the calculation are in Table 5.7.
Then,
Finally, looking up the z-factor chart (Fig. 5.3) gives z = 1.02. This represents a 2% error with the experimental value.
Example 5.6
Calculate the viscosity at 150°F (609.67°R) and 2,012 psia for the gas of the composition shown in Table 5.8.
Solution (by the Carr et al.^{[16]} 'Method). First, calculate the pseudocritical properties using Kay’s^{[3]} rules. The charts of Carr et al.^{[16]} are based on pseudocritical properties determined with Kay’s rules; it would not be correct, then, to use the methods of Sutton^{[4]} or Piper et al.^{[13]} to calculate the pseudocritical properties for use with the viscosity calculation. The details are inTable 5.9.
These parameters are then used to determine the viscosity at 1 atm. First, the viscosity forM_{g}= 20.079 at p = 1 atm and T= 150°F is read fromFig. 5.11. This givesμ_{ga}= 0.0114 cp, but a correction is needed for the nitrogen. The correction for 15.8% N_{2} is 0.0013 cp. Hence, this gives μ_{ga}= 0.0127 cp.
Next, the ratio of μ_{g}/μ_{ga}is read fromFig. 5.13using the pseudoreduced properties calculated above, which givesμ_{g}/μ_{ga}= 1.32. Hence, μ_{g}= (1.32) (0.0127) = 0.0168 cp. This represents a 2.5% error from the experimentally determined value of 0.0172 cp.
Solution (by the Lee et al.^{[17]} 'Method). In this method, the z factor is required; this is most accurately determined with the Piper et al.^{[13]} method, the details of which are in Table 5.10.
Look up the chart of Fig. 5.3, which gives a value of z = 0.91; then,
This method gives a value that is 5.5% less than the experimentally determined value of 0.0172 cp.
Example 5.7 The vapor pressure of pure hexane as a function of temperature is 54.04 kPa at 50°C and 188.76 kPa at 90°C. Estimate the vapor pressure of hexane at 100°C, using all the methods outlined previously.
Solution: Clausius-Clapeyron. The Clausius-Clapeyron equation can be solved graphically by plotting a log of vapor pressure vs. reciprocal absolute temperature and extrapolating. It also can be solved by slopes fitting an equation of the form log(p_{v}) = c/T+b to the two data points. Because the other three methods must be done in American customary units, the Clausius-Clapeyron method also will be converted to those units.
T_{1} = 50°C = 122°F = 581.67°R,
l/1 = 0.0017192°R^{–1},
T_{2} = 90°C = 653.67°R,
1/T_{2} = 0.0015298°R^{–1},
p_{v} at T_{1} = 54.04 kPa = 7.8374 psia,
log p_{v} = 0.89417,
p_{v} at T_{2} = 188.76 kPa = 27.3773 psia,
log p_{v} = 1.43739,
Δlog p_{v} = –0.543195,
1/T_{1} –1/T_{2} = 0.00018936,
and c = slope = –0.543195/0.00018936
= –2868.52°R.
Solving for b, log p_{v} = –2868.52 /T+b yields
b = 5.8257,
T_{3} = 100°C = 212°F = 671.67°R,
and 1/T_{3} = 0.0014888.
Solving for p_{v} at 100°C yields
hence, p_{v} = 35.89 psia = 247.46 kPa.
Alternatively, if the vapor pressure at 70°C is 105.37 kPa and is known, you can use the 70 to 90°C temperature differential to calculate the slope and intercept and ultimately calculate p_{v} = 35.79 psia = 246.79 kPa.
Solution: Cox Chart.^{[18]} From Fig. 5.15, the vapor pressure at 100°C can be approximated between 35 and 36 psia. A larger chart is required for more-precise readings.
Solution: The Calingeart and Davis or Antoine Equation. This can be used by obtaining the Antoine constants from Poling et al.^{[23]} For n-hexane, with temperature in K, these constants are A = 15.8366, B = 2697.55, and C = –48.78. Then,
and p_{v} = 36.68 psia = 252.73 kPa.
Solution: Lee-Kesler. The use of the Lee-Kesler^{[22]} equation requires p_{c}, T_{c}, and ω for n-hexane. These can be obtained from Table 5.2.
p_{c} = 436.9 psia (29.7 atm),
T_{c} = 453.7°F or 913.3°R or 507.4 K,
and ω = 0.3007.
For 100°C,
T_{r} = 0.7351,
(T_{r}) 6 = 0.15782,
ln T_{r} = –0.30775,
and p_{v} = (0.0816)(29.7) = 2.4235 atm = 35.62 psia = 245.6 kPa.
Experimental Value. 35.69 psia = 246.1 kPa.
Conclusions. Lee-Kesler gives the best answer, but the Clausius-Clapeyron method is also very accurate to within 0.17 psi, which is typical if the extrapolation is close to the appropriate range.
---
Nomenclature
References
- ↑ Moldover, M.R., Trusler, J.P.M., Edwards, T.J. et al. 1988. Measurement of the Universal Gas Constant R Using a Spherical Acoustic Resonator. J. Res. Nat. Inst. Stand. Technol. 93 (2): 85.
- ↑ Burnett, E.S. 1936. Compressibility Determinations without Volume Measurements. J. Appl. Mech. 3 (December 1936): A136–A140.
- ↑ ^{3.0} ^{3.1} ^{3.2} ^{3.3} ^{3.4} Kay, W. 1936. Gases and Vapors At High Temperature and Pressure - Density of Hydrocarbon. Ind. Eng. Chem. 28 (9): 1014-1019. http://dx.doi.org/10.1021/ie50321a008
- ↑ ^{4.00} ^{4.01} ^{4.02} ^{4.03} ^{4.04} ^{4.05} ^{4.06} ^{4.07} ^{4.08} ^{4.09} ^{4.10} ^{4.11} ^{4.12} ^{4.13} Sutton, R.P. 1985. Compressibility Factors for High-Molecular-Weight Reservoir Gases. Presented at the SPE Annual Technical Conference and Exhibition, Las Vegas, Nevada, USA, 22-26 September. SPE-14265-MS. http://dx.doi.org/10.2118/14265-MS
- ↑ ^{5.0} ^{5.1} ^{5.2} ^{5.3} ^{5.4} Stewart, W.F., Burkhardt, S.F., and Voo, D. 1959. Prediction of Pseudo-critical Parameters for Mixtures. Presented at the AIChE Meeting, Kansas City, Missouri, USA, 18 May 1959.
- ↑ ^{6.0} ^{6.1} ^{6.2} Standing, M.B. and Katz, D.L. 1942. Density of Natural Gases. In Transactions of the American Institute of Mining and Metallurgical Engineers, No. 142, SPE-942140-G, 140–149. New York: American Institute of Mining and Metallurgical Engineers Inc.
- ↑ ^{7.0} ^{7.1} Katz, D.L. 1959. Handbook of Natural Gas Engineering. New York: McGraw-Hill Higher Education.
- ↑ ^{8.0} ^{8.1} ^{8.2} Brown, G.G. 1948. Natural Gasoline and the Volatile Hydrocarbons. Tulsa, Oklahoma: Natural Gasoline Association of America.
- ↑ ^{9.0} ^{9.1} ^{9.2} ^{9.3} ^{9.4} Dranchuk, P.M. and Abou-Kassem, H. 1975. Calculation of Z Factors For Natural Gases Using Equations of State. J Can Pet Technol 14 (3): 34. PETSOC-75-03-03. http://dx.doi.org/10.2118/75-03-03
- ↑ Hall, K.R. and Yarborough, L. 1973. A New Equation of State for Z-Factor Calculations. Oil Gas J. 71 (25): 82.
- ↑ Towler, B.F. 2002. Fundamental Principles of Reservoir Engineering, Vol. 8. Richardson, Texas: Textbook Series, SPE. Towler, B.F. 2002. Fundamental Principles of Reservoir Engineering. Textbook Series, SPE, Richardson, Texas 8.
- ↑ ^{12.0} ^{12.1} ^{12.2} ^{12.3} ^{12.4} ^{12.5} ^{12.6} ^{12.7} Wichert, E. and Aziz, K. 1972. Calculate Z's for Sour Gases. Hydrocarbon Processing 51 (May): 119–122.
- ↑ ^{13.00} ^{13.01} ^{13.02} ^{13.03} ^{13.04} ^{13.05} ^{13.06} ^{13.07} ^{13.08} ^{13.09} ^{13.10} ^{13.11} ^{13.12} Piper, L.D., McCain Jr., W.D., and Corredor, J.H. 1993. Compressibility Factors for Naturally Occurring Petroleum Gases (1993 version). Presented at the SPE Annual Technical Conference and Exhibition, Houston, 3–6 October. SPE-26668-MS. http://dx.doi.org/10.2118/26668-MS
- ↑ Trube, A.S. 1957. Compressibility of Natural Gases. J Pet Technol 9 (1): 69-71. SPE-697-G. http://dx.doi.org/10.2118/697-G
- ↑ ^{15.0} ^{15.1} ^{15.2} ^{15.3} Mattar, L., Brar, G.S., and Aziz, K. 1975. Compressibility of Natural Gases. J Can Pet Technol 14 (4): 77. PETSOC-75-04-08. http://dx.doi.org/10.2118/75-04-08
- ↑ ^{16.0} ^{16.1} ^{16.2} ^{16.3} ^{16.4} ^{16.5} ^{16.6} ^{16.7} ^{16.8} ^{16.9} Carr, N.L., Kobayashi, R., and Burrows, D.B. 1954. Viscosity of Hydrocarbon Gases Under Pressure. J Pet Technol 6 (10): 47-55. SPE-297-G. http://dx.doi.org/10.2118/297-G
- ↑ ^{17.0} ^{17.1} ^{17.2} ^{17.3} Lee, A.L., Gonzalez, M.H., and Eakin, B.E. 1966. The Viscosity of Natural Gases. J Pet Technol 18 (8): 997–1000. SPE-1340-PA. http://dx.doi.org/10.2118/1340-PA
- ↑ ^{18.0} ^{18.1} ^{18.2} Cox, E.R. 1923. Pressure-Temperature Chart for Hydrocarbon Vapors. Ind. Eng. Chem. 15 (6): 592-593. http://dx.doi.org/10.1021/ie50162a013
- ↑ Calingaert, G. and Davis, D.S. 1925. Pressure-Temperature Charts—Extended Ranges. Ind. Eng. Chem. 17 (12): 1287-1288. http://dx.doi.org/10.1021/ie50192a037
- ↑ Antoine, C. 1888. Tensions des vapeurs; nouvelle relation entre les tensions et les températures. Comptes Rendus des Séances de l'Académie des Sciences 107: 836–850.
- ↑ Pitzer, K.S., Lippman, D.Z., Curl, R.F. Jr. et al. 1955. The Volumetric and Thermodynamic Properties of Fluids. II. Compressibility Factor, Vapor Pressure, and Entropy of Vaporization. J. Am. Chem. Soc. 77: 3433.
- ↑ ^{22.0} ^{22.1} ^{22.2} Lee, B.I. and Kesler, M.G. 1975. A generalized thermodynamic correlation based on three-parameter corresponding states. AIChE J. 21 (3): 510-527. http://dx.doi.org/10.1002/aic.690210313
- ↑ ^{23.0} ^{23.1} Poling, B.E., Prausnitz, J.M., and O’Connell, J.P. 2001. The Properties of Gases and Liquids, fifth edition (paperback). New York: McGraw-Hill Professional.
SI Metric Conversion Factors
°API | 141.5/(131.5+°API) | = | g/cm^{3} | ||
atm | × | 1.013 250* | E + 05 | = | Pa |
bar | × | 1.0* | E + 05 | = | Pa |
bbl | × | 1.589 873 | E − 01 | = | m^{3} |
cp | × | 1.0* | E – 03 | = | Pa•s |
Darcy | × | 9.869 233 | E–01 | = | μm^{2} |
dyne | × | 1.0* | E – 02 | = | mN |
dyne/cm^{2} | × | 1.0* | E – 01 | = | Pa |
ft | × | 3.048* | E – 01 | = | m |
ft^{2} | × | 9.290 304* | E – 02 | = | m^{2} |
ft^{3} | × | 2.831 685 | E – 02 | = | m^{3} |
ft-lbf | × | 1.355 818 | = | J | |
°F | (°F−32)/1.8 | = | °C | ||
°F | (°F+459.67)/1.8 | = | K | ||
hp-hr | × | 2.684 520 | E + 06 | = | J |
in. | × | 2.54* | E + 00 | = | cm |
in.^{2} | × | 6.451 6* | E + 00 | = | cm^{2} |
in.^{3} | × | 1.638 706 | E + 00 | = | cm^{3} |
kW-hr | × | 3.6 | E + 06 | = | J |
lbf | × | 4.448 222 | E + 00 | = | N |
lbf/in.^{2} | × | 6.894 757 | E + 03 | = | Pa |
lbf-s/ft^{2} | × | 4.788 026 | E + 01 | = | Pa•s |
lbm | × | 4.535 924 | E − 01 | = | kg |
mile | × | 1.609 344 | E + 00 | = | km |
N•m | = | J | |||
psi | × | 6.894 757 | E + 00 | = | kPa |
°R/1.8 | = | K |
*