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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 super compressible. 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:
Note that the numerator and denominator of Eq. 1 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:
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.
Compressibility factors for mixtures (or unknown pure compounds) are measured easily in a Burnett 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,
- ppc = pseudocritical pressure of the gas mixture
- Tpc = pseudocritical temperature of the gas mixture
- pci = critical pressure of component i in the gas mixture
- Tci =critical temperature of component i in the gas mixture
- yi =mole fraction of component i in the gas mixture.
These relations are known as Kay’s rule after W.B. Kay, 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 1). It has been found to approximate the convergence of the lines of constant volume on a pressure/temperature diagram.
Fig. 1 – Gas deviation-factor chart for natural gases (from Standing and Katz).
High molecular weight hydrocarbon mixtures
Sutton 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 proposed a modification of a method first proposed by Stewart et al. Sutton’s method is to first define and determine the pseudocritical properties of the C7+ fraction, then calculate the pseudocritical properties of the mixture as follows:
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. 2 was developed by Sutton on the basis of 264 different gas samples.
Fig. 2 – Pseudocritical properties of methane-based natural gases (from Sutton).
Sutton also used regression analysis on the raw data to obtain the following second-order fit for the pseudocritical properties of hydrocarbon mixtures:
These equations and Fig 2 are valid over the range of specific gas gravities with which Sutton worked: 0.57 < γg < 1.68. Using the obtained pseudocritical values, the pseudoreduced pressure and temperature are calculated using
The gas deviation factor is then found by using the well-known correlation chart of Fig. 1, originally developed by Standing and Katz. Compressibility factors of high-pressure natural gases (10,000 to 20,000 psia) may be obtained from Fig. 3, which was developed by Katz et al. Figs. 4 and 5 may be used for low-pressure applications after Brown et al.
Fig. 3 – Gas deviation factors for natural gases at pressures of 10,000 to 20,000 psia.
Fig. 4 – Gas deviation-factor chart for natural gases near atmospheric pressure.
Fig. 5 – Gas deviation-factor chart for natural gases at low reduced pressure.
Dranchuk and Abou-Kassem fitted an equation of state to the data of Standing and Katz, which is more convenient for estimating the gas deviation factor in computer programs and spreadsheets. Hall and Yarborough also have published an alternative equation of state. The Dranchuk and Abou-Kassem equation of state is based on the generalized Starling equation of state and is expressed as follows:
- A1 = 0.3265
- A¬ = –1.0700
- A3 = –0.5339
- A4 = 0.01569
- A5 = –0.05165
- A6 = 0.5475
- A7 = –0.7361
- A8 = 0.1844
- A9 = 0.1056
- A10 = 0.6134
- A11 = 0.7210
Dranchuk and Abou-Kassem 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 < ppr < 30; 1.0 < Tpr < 3.0; and for ppr < 1.0 with 0.7 < Tpr < 1.0.
Dranchuk and Abou-Kassem also found that this equation and other equations of state give unacceptable results near the critical temperature for Tpr = 1.0 and ppr >1.0, so these equations are not recommended in this range.
Because the parameter z is embedded in ρr, 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. The equation also can be solved on a spreadsheet using the nonlinear-equation-solver option, which is discussed in more detail elsewhere. Nonlinear equation solvers are also set up specifically to solve these equations easily.
Significant acid gas fractions
The z-factor chart of Standing and Katz (Fig 1) and the pseudocritical property-calculation methods of Sutton are valid only for mixtures of hydrocarbon gases. Wichert and Aziz 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 (CO2) and hydrogen sulfide (H2S). The Wichert and Aziz 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. 1 or Eq. 7. The Wichert and Aziz correlation first calculates a deviation parameter ε:
where A = the sum of the mole fractions of CO2 and H2S in the gas mixture and B = the mole fraction of H2S in the gas mixture. Then, the value of ε is used to determine the modified pseudocritical properties as follows:
The correlation is valid only in units of T in R and p in psia. It is applicable to concentrations of CO2 < 54.4 mol% and H2S < 73.8 mol%. Note that ε also has units of R. The correction factor, ε, has been plotted against H2S and CO2 concentrations in Fig. 6 for convenience. Note that maximum correction occurs around A = B = 47% or 47% H2S concentration and 0% CO2 concentration. Wichert and Aziz 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.
Fig. 6 - Pseudocritical-temperature-adjustment factor, ε, °F.
Piper et al. have also adapted the Stewart et al. method to develop equations that can be used to calculate the pseudocritical properties of natural gas mixtures that contain nitrogen (N2), CO2, and H2S 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:
- yi are the contaminant compositions and yj are the hydrocarbon compositions
- αi and βi are as given in Table 1.
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. on the basis of 1,482 data points:
Then, the pseudocritical properties can be calculated from J and K in Eqs. 8 and 9.
Example: Calculation of the z factor for sour gas
- γg = 0.7
- H2S = 7%
- CO2 = 10%
- p = 2,010 psia
- T = 75°F.
Solution. (a) First, calculate the pseudocritical properties.
Next, calculate the adjustments to the pseudocritical properties using the Wichert and Aziz parameters.
Next, calculate the pseudoreduced properties:
Finally, looking up the z-factor chart (Fig 1) gives z = 0.772.
(b) Using the method of Piper et al.,
The pseudoreduced properties are then:
Finally, looking up the z-factor chart (Fig 2) 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. is based on a larger data set and has integrated the non hydrocarbon compositions into the method, it is likely to be more accurate.
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).
where po 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.
|a||=||constant characteristic of the fluid|
|ai||=||empirical constant for substance i|
|am||=||parameter a characteristic|
|A||=||sum of the mole fractions of CO2 and H2S in the gas mixture|
|b||=||constant characteristic of the fluid|
|bi||=||empirical constant for substance i|
|bm||=||parameter b for mixture|
|B||=||mole fraction of H2S in the gas mixture|
|Bg||=||gas formation volume factor (RB/scf or Rm3/Sm3)|
|cg||=||coefficient of isothermal compressibility|
|cr||=||dimensionless pseudoreduced gas compressibility|
|C||=||constant with a value of 43 when the temperature is in K, and a value of 77.4 when the temperature is in °R|
|di||=||empirical constant for substance i|
|Ek||=||kinetic energy, J|
|f0, f1||=||functions of reduced temperature|
|Fj||=||parameter in the Stewart et al. equations (Eqs. 6 and 7), K•Pa–1/2|
|J||=||parameter in the Stewart et al. equations (Eqs. 6 and 7), K•Pa–1|
|K||=||parameter in the Stewart et al. equations (Eqs. 6 and 7), K•Pa–1/2|
|Kij||=||constant for each binary pair when used for mixtures|
|Lv||=||molal latent heat of vaporization, J|
|mg||=||mass of gas, kg|
|Ma||=||molecular weight of air|
|=||molecular weight of C7+ fraction|
|Mg||=||average molecular weight of gas mixture|
|n||=||number of moles|
|N||=||number of components in the gas mixture|
|p||=||absolute pressure, Pa|
|pc||=||critical pressure, Pa|
|pci||=||critical pressure of component i in a gas mixture, Pa|
|po||=||base pressure for real-gas pseudopotential, typically atmospheric pressure, Pa|
|ppc||=||pseudocritical pressure of a gas mixture, Pa|
|prc||=||pressure at reservoir conditions, Pa|
|psc||=||pressure at standard conditions, Pa|
|pv||=||vapor pressure, Pa|
|pvr||=||reduced vapor pressure (vapor pressure/critical pressure)|
|R||=||gas-law constant, J/(g mol-K)|
|t||=||ratio of critical to absolute temperature|
|T||=||absolute temperature, K|
|Tc||=||critical temperature, K|
|Tci||=||critical temperature of component i in a gas mixture, K|
|Tpc||=||corrected pseudocritical temperature, K|
|Trc||=||temperature at reservoir conditions, K|
|Tsc||=||temperature at standard conditions, K|
|Vc||=||critical volume, m3|
|=||critical volume of C7+ fraction, m3|
|Vg||=||volume of gas, m3|
|VM||=||molar volume, m3|
|Vrc||=||volume at reservoir conditions, m3|
|VR||=||volume of gas at reservoir temperature and pressure, m3|
|Vsc||=||volume at standard conditions, m3|
|xi||=||mole fraction of component i in a liquid|
|yi||=||mole fraction of component i in a gas mixture|
|z||=||compressibility factor (gas-deviation factor)|
|zrc||=||compressibility factor at reservoir conditions|
|zsc||=||compressibility factor at standard conditions|
|ρpc||=||relative density of C7+ fraction|
|ε||=||temperature-correction factor for acid gases, K|
|γg||=||specific gravity for gas|
|μga||=||viscosity of gas mixture at desired temperature and atmospheric pressure, Pa•s|
|ρg||=||density of gas, kg/m3|
|ρr||=||dimensionless density of gas in Eq. 13 = 0.27 pr/(zTr)|
|ψ||=||real-gas pseudopotential defined by Eq. 5.38|
- 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
- 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.
- Katz, D.L. 1959. Handbook of Natural Gas Engineering. New York: McGraw-Hill Higher Education. Cite error: Invalid
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- 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
- 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.
- Brown, G.G. 1948. Natural Gasoline and the Volatile Hydrocarbons. Tulsa, Oklahoma: Natural Gasoline Association of America.
- 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. Cite error: Invalid
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- 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.
- Wichert, E. and Aziz, K. 1972. Calculate Z's for Sour Gases. Hydrocarbon Processing 51 (May): 119–122.
- 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 Cite error: Invalid
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- 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
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