Accurate Calculations of Compressibility Factor for Pure Gases and Gas Mixtures
Authors: Obeida T.A. , Heinemann Z.E. , Kriebernegg M. 
Published: December 2007

Abstract
An accurate method to predict volumetric behavior of gas mixtures, such as in the case of underground gas storage where the in-situ gas is mixed with the injected gas, is presented in this paper. This method accurately calculates the compressibility (Z) factor of pure hydrocarbon, non-hydrocarbon gases and gas mixtures. These calculations are based on correction functions developed from correlation of data (Z-factor) generated by the Peng-Robinson equation of state. The correction functions are function of gas composition, pressure and temperature, so the Z-factor can be calculated explicitly from gas composition under different reservoir conditions.
Several comparative examples are presented to compare the Z-factors calculated by the correction functions with those calculated by the Peng-Robinson equation of state (PR-EOS) and with measured data published in the literature. The comparison results indicated that the average absolute relative deviation (AARD) is 3% for gas mixtures, 2% for pure hydrocarbon, non-hydrocarbon gases and less than 1% for pure components (methane, nitrogen, carbon dioxide). A stable method of calculating Z-factor for gases from their composition is presented. This method is iteration free so the CPU time is minimized. Accurate values of Z-factor can be calculated which are better than those obtained by linear interpolations.
The correction functions can be incorporated in any non-compositional simulator to calculate the Z-factor directly without any iterative procedures which occur in compositional simulators during the calculations of Z-factor using the equation of state. These functions also eliminate the inaccurate linear interpolations of tabulated Z-values, specially during calculations of Z-factor for gas mixtures, in non-compositional simulators.

Introduction
The compressibility factor is an important property for gases to calculate volume (material) of gases under given conditions (pressure, temperature). Also the Z-factor is important parameter to calculate other gas properties such as the formation volume factor and the coefficient of isothermal compressibility.
It is important to calculate the Z-factor more accurately, specially for gas mixtures, in order to predict the volumetric gas behavior more reasonably. In compositional simulators the calculations of the Z-factor are accurate, but for every condition the cubic equation of state is solved for Z-factor. The solution procedure involves iterations such as in Newton Raphson method. These iterations and convergence checking procedure consumes, some times, a considerable part of CPU time for just calculating gas properties (Z-factor). The CPU time should be used more efficiently in and wisely in the simulator. On the other side in the non-compositional simulators the Z-factor values are tabulated for certain gas composition and pressures and a linear interpolation procedure is used to calculated those Z-factor values which are not listed in the table. This procedure leads to erroneous calculations of Z-factor specially for gas mixtures where the linear interpolations are no longer accurate. The calculation procedure of Z-factor using the correlation functions presented in this paper has two advantages: Obtaining an accurate value of Z-factor and saving CPU time for other more important calculations in the simulator.

Background
Some impurities such as nitrogen and carbon dioxide are often existed in appreciable amounts in natural gases. The Z-factor for non-hydrocarbon components of natural gas in certain corresponding states differ markedly from those of hydrocarbons. This makes the non-hydrocarbon and hydrocarbon components not quit additive. Eilerts, Muller and Carlson studied the compressibility of natural gas and nitrogen mixtures. They proposed a method to calculate the Z-factor for the gas mixture by introducing a correction factor into the additive form as shown in Eq. 1.
(1)
where: Zm = actual Z-factor for gas mixture, Zn = Z-factor of the nitrogen in the mixture,
Zn = Z-factor of hydrocarbon gas,
n = mole fraction of nitrogen in the mixture.
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