Khaled Ahmed Abd-el Fattah
Cairo University
Giza, Egypt
Eight popular methods for obtaining the z-factor for natural gas are compared to the Standing-Katz compressibility chart. Inaccurate Z-factors can lead to erratic gas reserves estimates and produced volume metering errors.
Error magnitudes associated with the eight correlations are graphed in Figs. 1 (134674 bytes)and 2 (166715 bytes). To guide designers and experimenters in selecting the best methods for their particular applications, a recommended range of applicability is provided for each method to ensure acceptable compressibility factor values.
Method accuracy has been determined based on 5,940 original data points for pseudoreduced temperatures (Tr) and pseudoreduced pressures (Pr) in the ranges of 1.05 _
Z-FACTOR
After four decades, the Standing-Katz Z-factor chart is still widely used as a practical source for obtaining natural gas compressibility factors. The chart is based on experimental data on gas mixtures at pressures up to 10,000 psia.1
With the advent of computers, however, the need arose to find a convenient technique for calculating Z-factors in gas reservoir pro-grams that did not involve feeding in the entire correlation chart from which Z-factors could be retrieved by table look-up. As a result, this required a simple mathematical description of the Standing-Katz Z-factor chart.
The empirical correlations for calculating Z-factors fall into two main categories.
- Those which attempt to analytically curve fit the Standing- Katz isotherms.3-5
- Those which compute Z-factors using an equation of state.3 6-9
Gas properties which can be derived from Z-factor values are:2
- Gas density
- Supercompressibility
- Gas formation volume factor
- Expansion factor.
CURVE FITTING
The correlations that curve fit the Standing-Katz isotherms are obviously simple and fast. Several authors have reported Z- functions using regression analysis methods.3-5
Papay 5 10 proposed the following simplified equation for calculating Z-factors (see nomenclature box)(20300 bytes):
See Formula
Brill and Beggs 6 11 (BB) presented the following equation:
See Formula
Gopal 4 suggested the following straight-line equation to fit the Standing-Katz chart:
See Formula
where: A, B, C, and D are correlation constants (Table 1)(47086 bytes) and
In the Leung method, 12 the Standing-Katz Z-factor chart was approximated through least-squares method and with the following mixed power polynomial:
See Formula
Table 1 (47086 bytes) lists the coefficients B1, B2, etc.
All equation of state methods for obtaining the Z-factor involve a trial and error solution.3 6-9
Dranchuk, Purvis, and Robinson developed two methods.
The first method (D)3 is derived from the Benedict-Webb- Rubin equation of state.
See Formula
Table 1 (47086 bytes) lists the coefficients A1 to A8.
In the second method (DPR),3 13-15 the following equation is suggested:
See Formula
Y is calculated iteratively by means of following relation:
See Formula where: A = 0.06423
B = 0.5353Tr - 0.6123
C = 0.3151Tr - 1.0467-(0.5783/Tr2)
D = Tr
E = 0.6816/Tr2
F = 0.6845
G = 0.27 Pr
Dranchuk and Abu Kassem (AK)7 16 17 derived the following equation based on the Starling equation of state.
See Formula
Table 1 (47086 bytes) lists the constants A1 to All.
Hall and Yarborough 8 9 18 used the Starling equation of state to arrive at the following relation: See Formula
Y is obtained from the following equation:
See Formula
ERROR ANALYSIS
The accuracy of correlations relative to the experimental values can be determined by various statistical means.19 This study used error and average absolute percent relative error. Error equals deviation of the calculated values from the original ones and was defined by:
error = Zexp - Zest (16)
Average absolute percent relative error is defined as:
See Formula
Ei is the relative deviation in percent of an estimated value from an experimental value and is defined by: See Formula
Zest and Zexp represent the estimated and experimental values respectively.
COMPARISONS
Fortran computer programs were developed to compare the correlations.
From the original Standing-Katz chart, 5,940 Z-factor values were determined in the ranges of 1.05 _
Deviation from experimentally determined data, expressed as error, was calculated for each method. Detailed results for all Tr - Pr pairs are shown in Figs. 1 (134674 bytes)and 2 (166715 bytes). These figures allow the selection of pseudoreduced temperature and pressure ranges within which calculation errors are below some specified maximum limit.
From this analysis, it appears that the Gopal method is not suitable for calculating Z-factors at Tr = 1.05 and 1.0 _
The Brill and Beggs (BB) method is suitable for calculating Z-factors at 1.15 < 15.0. It gives, however, large errors for Tr > 2.4.
Leung's method gives large error values for Pr -
Papay's (PAP) method is suitable for calculating Z-factors for 1.6 _
From Table 2 (28915 bytes), one can select the Z-factor calculation methods that provides the required accuracy in the parameter ranges (Tr and Pr) Excluding the original experimental data points for 1.05 _
It appears that for ranges of 1.05 -
For a hypothetical gas composition of 90% methane and 10% ethane, the corresponding extreme values of pressure and temperature for which the discussed Z-factor calculations are applicable are 135-10,150 psia and -77 to 633 F. This range covers the expected conditions encountered in most natural gas reservoir, production, processing, and transmission systems.