Real Gas Law
The real gas law is a simplification of the generalized equation of state (EOS) Zfactor definition. The simplification comes from the model only describing nonideal gases and not multiphase fluid systems. The real gas law takes a similar form as the ideal gas law and is defined by
where \(v\) is the molar volume (\(v=V/n\)) and \(Z\) is the simplified Zfactor which, in the case of the real gas law, can be described as the gas deviation factor.
Historical Overview
Early attempts to generalize the gas correction factor was made by Standing and Katz in 1944^{1} by measurements of various petroleum gases and defining the now famous StangingKatz charts. By introducing pseudoreduced variables for the pressure and temperature, they were able to generalize the behavior for a wide range of nonideal pressures and temperatures. With increasing computer power becoming available, function based descriptions of the StandingKatz charts became more applicable like the original and modified BenedictWebbRubin (BWR) EOS models^{2}. Other well known descriptions of the gas deviation factor like the HallYarborough model^{3} are also applied to describe real gases. Modern descriptions of petroleum fluids are typically calculated using cubic EOS models like the PengRobinson or SoaveRedlichKwong EOS models.
StandingKatz ZFactor Charts
Figure 1: StandingKatz chart.
The StandingKatz chart is a best fit model for the Zfactor of various petroleum gases as a function of the specific gas reduced properties. The approach for finding the gas deviation factor is described by:
 Estimate the pseudoreduced pressure and temperature by correlation (if no composition is given) or by Kay's mixing rule (if compositions are given).
 Locate the pseudreduced pressure on the xaxis and then move along the yaxis until you cross the correct pseudoreduced temperature isoline.
 Move along the xaxis relative to the point found in step 2 to the left (if the pseudopressure is less than 7) or right (if the pseudopressure is greater than 7) to find the estimate of the Zfactor.
HallYarborough Estimation of Gas Zfactor
The HallYarborough approach to estimating the gas Zfactor is an iterative approach using the pseudo reduced properties of the gas (\(p_{pr}=p/p_{pc}\) and \(T_{pr}=T/T_{pc}\)) and the CarnahanStanding hardsphere EOS given by
where
and
The reduced density parameter (\(\hat{\rho}\)) is defend by solving
where
and the derivative is given by
where
The procedure for the HallYarborough Zfactor is given below and more details about the procedure can be found in the SPE Monograph Phase Behavior^{4}:
 Guess reduceddensity parameter (\(\hat{\rho}\))
 Calculate objective function (\(f(\hat{\rho})\)) and its derivative (\(f'(\hat{\rho})\))
 Check if objective function is less than the tolerance (\(f(\hat{\rho})<\epsilon\))
 If the objective function is less than the threshold then the reduceddensity parameter is correct and the user should move on to step 5. If the objective function is not less than the threshold then the reduced density parameter must be updated by Newton's method and the used must return to step 2.
 With the converged reduceddensity parameter, calculate the Zfactor from the CarnahanStanding hadspere EOS equation \eqref{eq:CS_eos}.
Estimates for PseudoCritical Properties
Note
All the correlation below are using absolute temperatures in Rankine (\(^\circ\)R) and absolute pressures in psia.
There are several methods for estimating the pseudocritical properties of a gas. The first approach is by correlation. This approach is typically used if there is no available composition or component properties (\(p_c\), \(T_c\) or \(MW\)). Below there are two correlations given for estimating the pseudocritical properties. The second approach to estimating the pseudocritical properties is by applying Kay's mixing rule.
Sutton Correlation for PseudoCritical Properties
whitson comment
From Curtis Hays Whitson: We have found that using the Sutton correlation yields better results than Kay's mixing rule.
where \(\gamma_g\) is the gas specific gravity (\(\gamma=\rho/\rho_{ref}\)).^{5}
Standing Correlation for PseudoCritical Properties
For dry gases where \(\gamma_g < 0.75\) the pseudocritical properties can be estimated from
For wet gases where \(\gamma_g >= 0.75\) the pseudocritical properties can be estimated from
where \(\gamma_g\) is the gas specific gravity (\(\gamma=\rho/\rho_{ref}\)).^{6}
Kay's Mixing Rule for PseudoCritical Properties
Note
See article on PVT propertis for more details and example values for the molecular weights and critical properties.
Given a mixture \(y_i\) containing a gas composition where the molecular weight and critical properties are known for each component. With this information the pseudocritical properties can be calculated from

M.B. Standing and D.L. Katz. Vaporliquid equilibria of natural gascrude oil systems. Transactions of the AIME, 155:paper SPE–944232–G, 1944. ↩

M. Benedict, G. B. Webb, and L. C. Rubin. An empirical equation for thermodynamic properties of light hydrocarbons and their mixtures i. methane, ethane, propane and nbutane. The Journal of Chemical Physics, 8:334–345, 1940. doi:https://doi.org/10.1063/1.1750658. ↩

K. R. Hall and L. Yarborough. A new equation of state for zfactor calculations. Oil Gas J, 71:82, 1973. ↩

C. H. Whitson and M. R. Brulé. Phase behavior. Volume 20. Henry L. Doherty Memorial Fund of AIME, Society of Petroleum Engineers …, 2000. ↩

R.P. Sutton. Compressibility factors for highmolecularweight reservoir gases. In SPE Annual Technical Conference and Exhibition, paper SPE–14265–MS. Society of Petroleum Engineers, 1985. doi:https://doi.org/10.2118/14265MS. ↩

M.B. Standing. Volumetric and phase behavior of oil hydrocarbon system. Society of Petroleum Engineers of AIME, Dallas, 1981. ↩