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# 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.

## Contents

## 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^{[1]} 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,

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*y*_{i}=mole fraction of component*i*in the gas mixture.

These relations are known as Kay’s rule after W.B. Kay,^{[2]} 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.

### High molecular weight hydrocarbon mixtures

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:

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^{[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:

These equations and **Fig 2** 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

The gas deviation factor is then found by using the well-known correlation chart of **Fig. 1**, originally developed by Standing and Katz.^{[3]} 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.*^{[6]} **Figs. 4 and 5** may be used for low-pressure applications after Brown *et al.*^{[7]}

Dranchuk and Abou-Kassem^{[8]} fitted an equation of state to the data of Standing and Katz,^{[3]} which is more convenient for estimating the gas deviation factor in computer programs and spreadsheets. Hall and Yarborough^{[9]} also have published an alternative equation of state. The Dranchuk and Abou-Kassem^{[8]} equation of state is based on the generalized Starling equation of state and is expressed as follows:

where and where the constants *A*_{1} through *A*_{11} are as follows:

*A*_{1}= 0.3265*A*¬ = –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*A*_{11}= 0.7210

Dranchuk and Abou-Kassem^{[8]} 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^{[8]} 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.^{[8]} The equation also can be solved on a spreadsheet using the nonlinear-equation-solver option, which is discussed in more detail elsewhere.^{[10]} 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^{[4]} are valid only for mixtures of hydrocarbon gases. Wichert and Aziz^{[11]} 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^{[11]} 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^{[11]} correlation first calculates a deviation parameter *ε*:

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:

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. 6** 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^{[11]} 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.*^{[12]} 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:

where,

*y*_{i}are the contaminant compositions and*y*_{j}are the hydrocarbon compositions*α*_{i}and*β*_{i}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.*^{[12]} 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

Using a) the Sutton^{[4]} correlation and the Wichert and Aziz^{[11]} correction, and (b) the method of Piper *et al.*,^{[12]} calculate the *z* factor for a gas with the following properties and conditions:

*γ*_{g}= 0.7- H
_{2}S = 7% - CO
_{2}= 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^{[11]} 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.*,^{[12]}

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.*^{[12]} 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 *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.

## Nomenclature

## References

- ↑ 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.
- ↑
^{3.0}^{3.1}^{3.2}Katz, D.L. 1959. Handbook of Natural Gas Engineering. New York: McGraw-Hill Higher Education. Cite error: Invalid`<ref>`

tag; name "r6" defined multiple times with different content - ↑
^{4.0}^{4.1}^{4.2}^{4.3}^{4.4}^{4.5}^{4.6}^{4.7}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}Brown, G.G. 1948. Natural Gasoline and the Volatile Hydrocarbons. Tulsa, Oklahoma: Natural Gasoline Association of America. - ↑
^{7.0}^{7.1}^{7.2}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 - ↑
^{8.0}^{8.1}^{8.2}^{8.3}^{8.4}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`<ref>`

tag; name "r9" defined multiple times with different content Cite error: Invalid`<ref>`

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tag; name "r9" defined multiple times with different content Cite error: Invalid`<ref>`

tag; name "r9" defined multiple times with different content - ↑ 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.
- ↑
^{11.0}^{11.1}^{11.2}^{11.3}^{11.4}^{11.5}^{11.6}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`<ref>`

tag; name "r12" defined multiple times with different content Cite error: Invalid`<ref>`

tag; name "r12" defined multiple times with different content - ↑
^{12.0}^{12.1}^{12.2}^{12.3}^{12.4}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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## See also

Gas formation volume factor and density