Figure: Standing and Katz compressibility factor chart
Compressibility factor, \(z\):
\(z=\displaystyle\frac{V_{real}}{V_{ideal}}\)
From the ideal gas equation
\( PV_{real}=znRT\)
In general, the compressibility factor \(z\) is expressed as a function of reduced temperature and reduced pressure
(Trube, 1957; Dranchuk et al., 1971; Abou-Kassem and Dranchuk, 1975; Sutton, 1985; Heidaryan et al., 2010).
Dranchuk (1971) defined reduced temperature and reduced pressure as the ratios of temperature and pressure,
respectively, to the pseudo-critical temperature and pseudo-critical pressure of natural gas:
\( T_{pr}=\displaystyle\frac{T}{T_{pc}}\,,\;\; P_{pr}=\displaystyle\frac{P}{P_{pc}}\)
The critical properties of the gas are obtained from the molar-fraction-weighted average of the critical properties of the gas mixture components:
\( T_{pc}=\displaystyle\sum_{i=1}^n{y_iT_{c,i}}\,,\;\; P_{pc}=\displaystyle\sum_{i=1}^n{y_iP_{c,i}}\)
\(P\) : Absolute pressure (bar, kPa, psi)
\(P_{pc}\) : Pseudo-critical pressure (bar, kPa, psi)
\(P_{pr}\) : Reduced pressure
\(P_{c,i}\) : Critical pressure of component \(i\) in the gas mixture (bar, kPa, psi)
\(T\) : Temperature (K, R)
\(T_{pc}\) : Pseudo-critical temperature (K, R)
\(T_{pr}\) : Reduced temperature
\(T_{c,i}\) : Critical temperature of component \(i\) in the gas mixture (K, R)
\(y_{i}\) : Mole fraction of component \(i\) in the gas mixture
Figure: General gas pressure-temperature diagram
\(z\)-Factor Correlations
The most widely used correlations for calculating the \(z\)-factor are given below.
In the correlation formulas below, the pressure unit is psi and the temperature unit is R (Rankine).
Hall and Yarborough correlation (Trube, 1957)
Accuracy range: \(1.05 \leq T_{pr} \leq 3.0\) and \(0.2 \leq P_{pr} \leq 15\).
In the Hall and Yarborough correlation, the value of \(y\) must be obtained iteratively in order to calculate \(z\).
\( z=\displaystyle\frac{A_1 P_{pr}}{y}\)
The value of \(y\) is the root of the following equation and is determined by iteration:
\( -A_1 P_{pr} +\displaystyle\frac{y+y^2+y^3-y^4}{(1-y)^3}-A_2y^2+\displaystyle A_3 \displaystyle y^{\displaystyle A_4}=0\)
The coefficients \(A_1\), \(A_2\), \(A_3\), and \(A_4\) are given below:
\( A_1=0.06125t\displaystyle e^{-1.2 \left(1-t\right)^2} \),
\( A_2=14.76t-9.76t^2+4.58t^3 \),
\( A_3=90.7t-242.2t^2+42.4t^3 \),
\( A_4=2.18+2.82t \),
\( t= \displaystyle\frac{1}{T_{pr}} \)
Dranchuk, Purvis, and Robinson correlation (1971)
Accuracy range: \(1.0 < T_{pr} \leq 3.0\) and \(0.2 \leq P_{pr} \leq 30\), and
\(0.7 < T_{pr} \leq 1.0\) and \(P_{pr} < 1.0 \).
The value of \(z\) is obtained from the following equation:
\(z=\displaystyle\frac{0.27 P_{pr}}{yT_{pr}}\)
The value of \(y\) is the root of the following equation and is determined by iteration:
\( T_4y^2\left(1+A_{8}y^2\right)e^{-A_{8}y^2}+1+T_1y+T_2y^2+T_3y^5+\displaystyle\frac{T_5}{y}=0 \)
\( T_1=A_1+\displaystyle\frac{A_2}{T_{pr}}+\displaystyle\frac{A_3}{T_{pr}^3} \)
\( T_2=A_4+\displaystyle\frac{A_5}{T_{pr}} \)
\( T_3=\displaystyle\frac{A_5A_6}{T_{pr}} \)
\( T_4=\displaystyle\frac{A_7}{T_{pr}^3} \)
\( T_5=\displaystyle\frac{0.27P_{pr}}{T_{pr}} \)
\(A_1=0.31506237\), \(A_2=-1.04670990\), \(A_3=-0.57832720\),
\(A_4=0.53530771\), \(A_5=-0.61232032\), \(A_6=-0.10488813\),
\(A_7=0.68157001\), \(A_8=0.68446549\)
Beggs and Brill compressibility factor correlation (1973)
This is an explicit correlation that gives the value of \(z\) directly without iteration.
\( z=A+\displaystyle\frac{1-A}{e^B}+C P_{pr}^D\)
Where,
\( A = 1.39(T_{pr} - 0.92)^{0.5} - 0.36T_{pr} - 0.10,\)
\( B = (0.62 - 0.23T_{pr} )p_{pr} + \left( {\displaystyle\frac{0.066}{{T_{pr} - 0.86}} - 0.037} \right)p_{pr}^{2} +\displaystyle \frac{{0.32p_{pr}^{2} }}{{10^{E} }},\)
\( C = 0.132 - 0.32\log (T_{pr} ),\)
\( D = 10^{F},\)
\( E = 9(T_{pr} - 1)\;,\)
\( F = 0.3106 - 0.49T_{pr} + 0.1824T_{pr}^{2}\)
Dranchuk and Abou-Kassem correlation (1975)
Accuracy range: \(1.0 < T_{pr} \leq 3.0\) and \(0.2 \leq P_{pr} \leq 30\), and
\(0.7 < T_{pr} \leq 1.0\) and \(P_{pr} < 1.0 \).
The value of \(z\) is obtained from the following equation:
\(z=\displaystyle\frac{0.27 P_{pr}}{yT_{pr}}\)
The value of \(y\) is the root of the following equation and is determined by iteration:
\( R_5y^2\left(1+A_{11}y^2\right)e^{-A_{11}y^2}+R_1y-\displaystyle\frac{R_2}{y}+R_3y^2-R_4y^5+1=0 \)
\( R_1=A_1+\displaystyle\frac{A_2}{T_{pr}}+\displaystyle\frac{A_3}{T_{pr}^3}+\displaystyle\frac{A_4}{T_{pr}^4}+\displaystyle\frac{A_5}{T_{pr}^5} \)
\( R_2=\displaystyle\frac{0.27 P_{pr}}{T_{pr}} \)
\( R_3=A_6+\displaystyle\frac{A_7}{T_{pr}}+\displaystyle\frac{A_8}{T_{pr}^2} \)
\( R_4=A_9 \left( \displaystyle\frac{A_7}{T_{pr}}+\displaystyle\frac{A_8}{T_{pr}^2}\right) \)
\( R_5=\displaystyle\frac{A_{10}}{T_{pr}^3} \)
\(A_1=0.3265\), \(A_2=-1.070\), \(A_3=-0.5339\),
\(A_4=0.01569\), \(A_5=-0.05165\), \(A_6=0.5475\),
\(A_7=0.7361\), \(A_8=0.1844\), \(A_9=0.1056\),
\(A_{10}=0.6134\), \(A_{11}=0.7210\)
Heidaryan, Moghdasi, and Rahimi correlation (2010)
This is an explicit correlation that gives the value of \(z\) directly without iteration.
\(z = \ln \left( {\displaystyle\frac{{A_{1} + A_{3} \ln (P_{pr} ) + \displaystyle\frac{{A_{5} }}{{T_{pr} }} + A_{7} \left( {\ln (P_{pr} )} \right)^{2} + \displaystyle\frac{{A_{9} }}{{T_{pr}^{2} }} + \displaystyle\frac{{A_{11} }}{{T_{pr} }}\ln (P_{pr} )}}{{1 + A_{2} \ln (P_{pr} ) + \displaystyle\frac{{A_{4} }}{{T_{pr} }} + A_{6} \left( {\ln (P_{pr} )} \right)^{2} +\displaystyle \frac{{A_{8} }}{{T_{pr}^{2} }} + \displaystyle\frac{{A_{10} }}{{T_{pr} }}\ln (P_{pr} )}}} \right)\)
\(\begin{array}{ l l l }
\rlap{\text{Table: Constants of the Heidaryan correlation}}\\
\hline
& P_{pr} \leq 3\; & P_{pr}>3\; \\
\hline \hline
A_1 & \;\;\;2.827793 & \;\;\;3.252838 \\
A_2 & -4.688191x10^{-1} & -1.306424x10^{-1} \\
A_3 & -1.262288 & \;\;\;6.449194x10^{-1} \\
A_4 & -1.536524 & -1.518028 \\
A_5 & -4.535045 & -5.391019 \\
A_6 & \;\;\;6.895104 × 10^{-2} & -1.379588 × 10^{-2} \\
A_7 & \;\;\;1.903869 × 10^{-1} & \;\;\;6.600633 × 10^{-2} \\
A_8 & \;\;\;6.200089 × 10^{-1} & \;\;\;6.120783 × 10^{-1} \\
A_9 & \;\;\;1.838479 & \;\;\;2.317431 \\
A_{10} & \;\;\;4.052367 × 10^{-1} & \;\;\;1.632223 × 10^{-1} \\
A_{11} & \;\;\;1.073574 & \;\;\;5.660595 × 10^{-1} \\
\hline
\end{array} \)
Kareem \(z\)-factor correlation (2016)
\(z = \displaystyle\frac{{ DP_{pr} (1 + y + y^{2} - y^{3} )}}{{\left( {DP_{pr} + Ey^{2} - Fy^{G} } \right)(1 - y)^{3} }}\)
\(y = \displaystyle\frac{{ DP_{pr} }}{{\left( {\frac{{1 + A^{2} }}{C} - \displaystyle\frac{{A^{2} B}}{{C^{3} }}} \right)}} ,\)
Where,
\(t = \frac{1}{{T_{pr} }}\),
\(A = a_{1} te^{{a_{2} (1 - t)^{2} }} P_{pr}, \)
\( B = a_{3} t + a_{4} t^{2} + a_{5} t^{6} P_{pr}^{6}, \)
\( C = a_{9} + a_{8} tP_{pr} + a_{7} t^{2} P_{pr}^{2} + a_{6} t^{3} P_{pr}^{3},\)
\(D = a_{10} te^{{a_{11} (1 - t)^{2} }}, \)
\(E = a_{12} t + a_{13} t^{2} + a_{14} t^{3}, \)
\( F = a_{15} t + a_{16} t^{2} + a_{17} t^{3},\)
\(G = a_{18} + a_{19} t \)
\(\begin{array}{ l l }
\rlap{\text{Table: Constants of the Kareem correlation}}\\
\hline \hline
a_1 = 0.317842 & a_{11}= -1.966847 \\
a_2 = 0.382216 & a_{12} = 21.0581 \\
a_3 = -7.768354 & a_{13}= -27.0246 \\
a_4 = 14.290531 & a_{14}= 16.23 \\
a_5 = 0.000002 & a_{15} = 207.783 \\
a_6 = -0.004693 & a_16 = -488.161 \\
a_7 = 0.096254 & a_{17}= 176.29 \\
a_8 = 0.166720 & a_{18}= 1.88453 \\
a_9 = 0.966910 & a_{19}= 3.05921 \\
a_{10}= 0.063069 & \\
\hline
\end{array}
\)
Source:
New explicit correlation for the compressibility factor of natural gas: linearized z-factor isotherms
Lateef A. Kareem, Tajudeen M. Iwalewa & Muhammad Al-Marhoun