The compressibility factor (z-factor) of gases is used to correct the volume of gas estimated from the ideal gas equation to the actual value. It is required in all calculations involving natural gases.
The z-factor is the ratio of the volume occupied by a given amount of a gas to the volume occupied by the same amount of an ideal gas:
$$z = \frac{{V_{\text{actual}} }}{{V_{\text{ideal}} }}$$
(1)
Substituting for \(V_{\text{ideal}}\) in the ideal gas equation
$$PV_{\text{actual}} = znRT$$
(2)
For generalization, the z-factor is expressed as a function of pseudo-reduced temperature and pressure (Trube 1957; Dranchuk et al. 1971; Abou-kassem and Dranchuk 1975; Sutton 1985; Heidaryan et al. 2010).
Dranchuk et al. (1971) defined pseudo-reduced temperature and pressure as the ratio of temperature and pressure to the pseudo-critical temperature and pressure of natural gas, respectively:
$$T_{pr} = \frac{T}{{T_{pc} }} , \quad P_{pr} = \frac{P}{{P_{pc} }}$$
(3)
The pseudo-critical properties of gas are the molar abundance (mole fraction weighted) mean of the critical properties of the constituents of the natural gas:
$$T_{pc} = \mathop \sum \limits_{i = 1}^{n} y_{i} T_{ci} ,\quad P_{pc} = \mathop \sum \limits_{i = 1}^{n} y_{i} P_{ci}$$
(4)
As a function of specific gravity (air = 1.0), Sutton (1985) provides
$$T_{pc} = 169.2 + 349.5{{\gamma_{g} }} - 74.0\gamma_{g}^{2}$$
(5)
$$P_{pc} = 756.8 - 131.07\gamma_{g} - 3.6\gamma_{g}^{2}$$
(6)
A plot of 5346 data points covering a full range of pseudo-reduced temperatures \((1.15 \le T_{pr} \le 3)\) and pseudo-reduced pressures \((0.2 \le P_{pr} \le 15)\) is shown in Fig. 1
Implicit z-factor correlations
The three most popular correlations for calculating the z-factor are implicit. The three correlations, described in the following subsections, are well known for their accuracies, almost unit correlation of regression coefficients and low maximum errors.
Hall and Yarborough’s correlation (Trube 1957)
Hall and Yarborough’s correlation is a modification of the hard sphere Carnahan–Starling equation of state, with constants developed through regression, and 1500 data points extracted from Standing and Katz’s original z-factor chart, as shown in Fig. 2.
$$z = \frac{{A_{1} P_{pr} }}{y},$$
(7)
where y is the root of the following equation:
$$\begin{aligned} & - A_{1} P_{pr} + \frac{{y + y^{2} + y^{3} - y^{4} }}{{(1 - y)^{3} }} - A_{2} y^{2} + A_{3} y^{{A_{4} }} = 0 \\ & A_{1} = 0.06125te^{{ - 1.2(1 - t)^{2} }} ,\quad A_{2} = 14.76t - 9.76t^{2} + 4.58t^{3} , \\ & A_{3} = 90.7t - 242.2t^{2} + 42.4t^{3} ,\quad A_{4} = 2.18 + 2.82t, \quad t = \frac{1}{{T_{pr} }} \\ \end{aligned}$$
Dranchuk and Abou-Kassem’s correlation (Abou-kassem and Dranchuk 1975)
This is an eleven-constant modification of the Benedict–Webb–Rubin equation of state. The constants were calculated using a regression method, with 1500 data points extracted from standing and Katz’s chart.
$$z = \frac{{0.27P_{pr} }}{{yT_{pr} }},$$
(8)
where y is the root of the following equation:
$$\begin{aligned} & \left[ {R_{5} y^{2} (1 + A_{11} y^{2} )e^{{( - A_{11} y^{2} )}} } \right] + R_{1} y - \frac{{R_{2} }}{y} + R_{3} y^{2} - R_{4} y^{5} + 1 = 0 \\ & R_{1} = A_{1} + \frac{{A_{2} }}{{T_{pr} }} + \frac{{A_{3} }}{{T_{pr}^{3} }} + \frac{{A_{4} }}{{T_{pr}^{4} }} + \frac{{A_{5} }}{{T_{pr}^{5} }}, \quad R_{2} = \frac{{0.27P_{pr} }}{{T_{pr} }} \\ & R_{3} = A_{6} + \frac{{A_{7} }}{{T_{pr} }} + \frac{{A_{8} }}{{T_{pr}^{2} }},\quad R_{4} = A_{9} \left( {\frac{{A_{7} }}{{T_{pr} }} + \frac{{A_{8} }}{{T_{pr}^{2} }}} \right), \quad R_{5} = \frac{{A_{10} }}{{T_{pr}^{3} }} \\ & A_{1} = 0.3265, \quad A_{2} = - 1.0700, \quad A_{3} = - 0.5339, \quad A_{4} = 0.01569, \\ & A_{5} = - 0.05165, \quad A_{6} = 0.5475, \quad A_{7} = 0.7361, \quad A_{8} = 0.1844, \\ & A_{9} = 0.1056, \quad A_{10} = 0.6134, \quad A_{11} = 0.7210 \\ \end{aligned}$$
Dranchuk, Purvis and Robinson’s Correlation (Dranchuk et al. 1971)
This is a further modification of the earlier obtained DAK correlation. The DPR has eight constants and requires less computational workload to obtain the z-factor.
$$z = \frac{{0.27P_{pr} }}{{yT_{pr} }},$$
(9)
where y is the root of the following equation:
$$\begin{aligned} & \left[ {T_{4} y^{2} (1 + A_{8} y^{2} )e^{{( - A_{8} y^{2} )}} } \right] + 1 + T_{1} y + T_{2} y^{2} + T_{3} y^{5} + \frac{{T_{5} }}{y} = 0 \\ & T_{1} = A_{1} + \frac{{A_{2} }}{{T_{pr} }} + \frac{{A_{3} }}{{T_{pr}^{3} }},\quad T_{2} = A_{4} + \frac{{A_{5} }}{{T_{pr} }}, \quad T_{3} = \frac{{A_{5} A_{6} }}{{T_{pr} }}, \quad T_{4} = \frac{{A_{7} }}{{T_{pr}^{3} }}, \quad T_{5} = \frac{{0.27P_{pr} }}{{T_{pr} }} \\ & A_{1} = 0.31506237, \quad A_{2} = - 1.04670990, \quad A_{3} = - 0.57832720, \quad A_{4} = 0.53530771, \\ & A_{5} = - 0.61232032, \quad A_{6} = - 0.10488813, \quad A_{7} = 0.68157001, \quad A_{8} = 0.68446549 \\ \end{aligned}$$
These correlations are effective; however, they do not converge (or converge on wrong pseudo-reduced density values) when the temperature of the systems is close to the critical temperature. In addition, they are computationally expensive. It is these limitations that necessitated the development of the current explicit correlations.
Explicit correlations
Explicit correlations do not require an iterative procedure. Therefore, they do not have the problem of convergence as opposed to implicit correlations. One of the best explicit correlations for evaluation of the z-factor was given by Beggs and Brills (1973). More recent ones are Heidaryan et al. (2010), Azizi et al. (2010) and Sanjari and Lay (2012) correlations. A short description of some of the explicit correlations is presented in the following subsections.
Brill and Beggs’ compressibility factor (1973)
$$z = A + \frac{1 - A}{{e^{B} }} + Cp_{pr}^{D} ,$$
where
$$\begin{aligned} A & = 1.39(T_{pr} - 0.92)^{0.5} - 0.36T_{pr} - 0.10, \\ B & = (0.62 - 0.23T_{pr} )p_{pr} + \left( {\frac{0.066}{{T_{pr} - 0.86}} - 0.037} \right)p_{pr}^{2} + \frac{{0.32p_{pr}^{2} }}{{10^{E} }} \\ C & = 0.132 - 0.32\log (T_{pr} ), \;D = 10^{F} , \\ E & = 9(T_{pr} - 1)\;{\text{and}}\; F = 0.3106 - 0.49T_{pr} + 0.1824T_{pr}^{2} \\ \end{aligned}$$
Heidaryan, Moghdasi and Rahimi’s Correlation
Heidaryan et al. (2010) developed a new explicit piecewise correlation using regression analysis of the z-factor experimental value for reduced pseudo-pressure of fewer and >3 (Table 1). The correlation has a total of 22 constants, with a discontinuity at \(P_{pr} = 3\) (Fig. 2) and correlation regression coefficient of 0.99,963.
$$z = \ln \left( {\frac{{A_{1} + A_{3} \ln (P_{pr} ) + \frac{{A_{5} }}{{T_{pr} }} + A_{7} \left( {\ln (P_{pr} )} \right)^{2} + \frac{{A_{9} }}{{T_{pr}^{2} }} + \frac{{A_{11} }}{{T_{pr} }}\ln (P_{pr} )}}{{1 + A_{2} \ln (P_{pr} ) + \frac{{A_{4} }}{{T_{pr} }} + A_{6} \left( {\ln (P_{pr} )} \right)^{2} + \frac{{A_{8} }}{{T_{pr}^{2} }} + \frac{{A_{10} }}{{T_{pr} }}\ln (P_{pr} )}}} \right)$$
(11)
Table 1 Constants of Heidaryan et al.’s correlation
For some petroleum engineering applications, it is often necessary to compute the derivative of z-factor with respect to pressure or temperature. A function that is discontinuous at a certain point is not differentiable at that point (O’Neil 2012). Therefore, the explicit correlation developed by Heidaryan et al. (2010) cannot be used to evaluate the derivative of the z-factor with respect to the pseudo-reduced pressure at \(P_{pr} = 3\) (Fig. 3).
Azizi, Behbahani and Isazadeh’s Correlation
Azizi et al. (2010) presented an explicit correlation with 20 constants for a reduced temperature range of \(1.1 \le T_{pr} \le 2\) and reduced pressure range of \(0.2 \le P_{pr} \le 11\). The correlation used 3038 data points within the given ranges.
$$z = A + \frac{B + C}{D + E},$$
(12)
where
$$\begin{aligned} A & = a T_{pr}^{2.16} + b P_{pr}^{1.028} + c P_{pr}^{1.58} T_{pr}^{ - 2.1} + d\ln \left( {T_{pr}^{ - 0.5} } \right) \\ B & = e + f T_{pr}^{2.4} + g P_{pr}^{1.56} + h P_{pr}^{0.124} T_{pr}^{3.033} \\ C & = i\ln \left( {T_{pr}^{ - 1.28} } \right) + j\ln \left( {T_{pr}^{1.37} } \right) + k\ln \left( {P_{pr} } \right) + l\ln \left( {P_{pr}^{2} } \right) + m\ln (P_{pr} )\ln (T_{pr} ) \\ D & = 1 + n T_{pr}^{5.55} + o P_{pr}^{0.68} T_{pr}^{0.33} \\ E & = p\ln \left( {T_{pr}^{1.18} } \right) + q\ln \left( {T_{pr}^{2.1} } \right) + r\ln (P_{pr} ) + s\ln \left( {P_{pr}^{2} } \right) + t\ln (P_{pr} )\ln (T_{pr} ) \\ \end{aligned}$$
$$\begin{aligned} a = 0.0373142485385592; \quad b = - 0.0140807151485369; \quad c = 0.0163263245387186; \quad d = - 0.0307776478819813; \quad e = 13843575480.943800; \quad f = - 16799138540.763700; \quad g = 1624178942.6497600; \quad h = 13702270281.086900; \quad i = - 41645509.896474600; \quad j = 237249967625.01300; \quad k = - 24449114791.1531;\quad l = 19357955749.3274; \quad m = - 126354717916.607;\quad n = 623705678.385784;\quad o = 17997651104.3330; \quad p = 151211393445.064;\quad q = 139474437997.172;\quad r = - 24233012984.0950; \quad s = 18938047327.5205;\quad t = - 141401620722.689; \end{aligned}$$
Sanjari and Nemati’s Correlation
Using 5844 data points, Sanjari and Lay (2012) developed an explicit correlation for the z-factor. This correlation, as with Heidaryan et al. (2010) correlation, has different constants for the values of \(P_{pr}\) below and above 3, but a total of 16 constants (Table 2). The procedure for calculating the z-factor is as follows:
$$z = 1 + A_{1} P_{pr} + A_{2} P_{pr}^{2} + \frac{{A_{3} P_{pr}^{{A_{4} }} }}{{T_{pr}^{{A_{5} }} }} + \frac{{A_{6} P_{pr}^{{(A_{4} + 1)}} }}{{T_{pr}^{{A_{7} }} }} + \frac{{A_{8} P_{pr}^{{(A_{4} + 2)}} }}{{T_{pr}^{{(A_{7} + 1)}} }}$$
(13)
Table 2 Constants of Sanjari and Lay’s correlation
This correlation, however, is less efficient when compared with that of Heidaryan et al. (2010). Its regression correlation coefficient is 0.8757 and its error rate at a certain point can be as high as 90 per cent. For instance, the actual value of z from the experiment for a \(P_{pr}\) of 15 and a \(T_{pr}\) of 1.05 is 1.753, but this correlation gives a value of 3.3024. Therefore, the actual maximum error for this correlation is 104.3206 %.
To resolve the limitations in the application of the existing explicit correlations, a single correlation that is continuous over the entire range of pseudo-reduced pressure is required.