J Petrol Explor Prod Technol DOI 10.1007/s13202-015-0209-3 ORIGINAL PAPER - PRODUCTION ENGINEERING New explicit correlation for the compressibility factor of natural gas: linearized z-factor isotherms 1 2 3 Lateef A. Kareem • Tajudeen M. Iwalewa • Muhammad Al-Marhoun Received: 14 October 2014 / Accepted: 17 October 2015 Ó The Author(s) 2015. This article is published with open access at Springerlink.com Abstract The compressibility factor (z-factor) of gases is Keywords Z-factor Á Explicit correlation Á Reduced a thermodynamic property used to account for the devia- temperature Á Reduced pressure Á Natural gas tion of real gas behavior from that of an ideal gas. Corre- lations based on the equation of state are often implicit, List of symbols because they require iteration and are computationally P Pressure (psi) expensive. A number of explicit correlations have been P Pseudo-critical pressure derived to enhance simplicity; however, no single explicit pc P Pseudo-reduced pressure correlation has been developed for the full range of pseudo- pr ÀÁ T Temperature (R) reduced temperatures 1:05 Tpr 3 and pseudo-reduced ÀÁ Tpc Pseudo-critical temperature (R) pressures 0:2 Ppr 15 , which represents a significant Tpr Pseudo-reduced temperature research gap. This work presents a new z-factor correlation Ppc Pseudo-critical pressure (psi) that can be expressed in linear form. On the basis of Hall Ppr Pseudo-reduced pressure and Yarborough’s implicit correlation, we developed the v Initial guess for iteration process new correlation from 5346 experimental data points Y Pseudo-reduced density extracted from 5940 data points published in the SPE z-factor Compressibility factor natural gas reservoir engineering textbook and created a linear z-factor chart for a full range of pseudo-reduced temperatures ð1:15 Tpr 3Þ and pseudo-reduced pres- sures ð0:2 Ppr 15Þ. Introduction 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 & Lateef A. Kareem involving natural gases. [email protected] The z-factor is the ratio of the volume occupied by a Tajudeen M. Iwalewa given amount of a gas to the volume occupied by the same [email protected] amount of an ideal gas: Muhammad Al-Marhoun Vactual [email protected] z ¼ ð1Þ Videal 1 Center for Petroleum and Minerals, King Fahd University of Substituting for Videal in the ideal gas equation Petroleum and Minerals, Dhahran, Saudi Arabia PVactual ¼ znRT ð2Þ 2 Department of Earth Sciences, University of Cambridge, Cambridge CB2 3EQ, UK For generalization, the z-factor is expressed as a 3 Reservoir Technologies (ResTec), Dammam, Saudi Arabia function of pseudo-reduced temperature and pressure 123 J Petrol Explor Prod Technol (Trube 1957; Dranchuk et al. 1971; Abou-kassem and Implicit z-factor correlations Dranchuk 1975; Sutton 1985; Heidaryan et al. 2010). Dranchuk et al. (1971) defined pseudo-reduced tem- The three most popular correlations for calculating the perature and pressure as the ratio of temperature and z-factor are implicit. The three correlations, described in pressure to the pseudo-critical temperature and pressure of the following subsections, are well known for their accu- natural gas, respectively: racies, almost unit correlation of regression coefficients and T P low maximum errors. Tpr ¼ ; Ppr ¼ ð3Þ Tpc Ppc Hall and Yarborough’s correlation (Trube 1957) The pseudo-critical properties of gas are the molar abundance (mole fraction weighted) mean of the critical Hall and Yarborough’s correlation is a modification of the properties of the constituents of the natural gas: hard sphere Carnahan–Starling equation of state, with Xn Xn constants developed through regression, and 1500 data Tpc ¼ yiTci; Ppc ¼ yiPci ð4Þ points extracted from Standing and Katz’s original z-factor i¼1 i¼1 chart, as shown in Fig. 2. As a function of specific gravity (air = 1.0), Sutton A P z ¼ 1 pr ; ð7Þ (1985) provides y 2 Tpc ¼ 169:2 þ 349:5cg À 74:0cg ð5Þ where y is the root of the following equation: 2 y þ y2 þ y3 À y4 Ppc ¼ 756:8 À 131:07cg À 3:6cg ð6Þ 2 A4 À A1Ppr þ À A2y þ A3y ¼ 0 ð1 À yÞ3 A plot of 5346 data points covering a full range of À1:2ð1ÀtÞ2 2 3 A1 ¼ 0:06125te ; A2 ¼ 14:76t À 9:76t þ 4:58t ; pseudo-reduced temperatures ð1:15 Tpr 3Þ and pseudo- 2 3 1 reduced pressures ð0:2 Ppr 15Þ is shown in Fig. 1 A3 ¼ 90:7t À 242:2t þ 42:4t ; A4 ¼ 2:18 þ 2:82t; t ¼ Tpr Fig. 1 Plot of experimental measurements of the z-factor 123 J Petrol Explor Prod Technol Fig. 2 Standing and Katz’s compressibility factor chart hi Dranchuk and Abou-Kassem’s correlation (Abou-kassem 2 R2 R y2ð1 þ A y2ÞeðÀA11y Þ þ R y À þ R y2 À R y5 þ 1 ¼ 0 and Dranchuk 1975) 5 11 1 y 3 4 A A A A 0:27P R ¼ A þ 2 þ 3 þ 4 þ 5 ; R ¼ pr 1 1 T T3 T4 T5 2 T This is an eleven-constant modification of the Benedict– pr pr pr pr !pr Webb–Rubin equation of state. The constants were calcu- A7 A8 A7 A8 A10 R3 ¼ A6 þ þ 2 ; R4 ¼ A9 þ 2 ; R5 ¼ 3 lated using a regression method, with 1500 data points Tpr Tpr Tpr Tpr Tpr extracted from standing and Katz’s chart. A1 ¼ 0:3265; A2 ¼À1:0700; A3 ¼À0:5339; A4 ¼ 0:01569; 0:27P z ¼ pr ; ð8Þ A5 ¼À0:05165; A6 ¼ 0:5475; A7 ¼ 0:7361; A8 ¼ 0:1844; yTpr A9 ¼ 0:1056; A10 ¼ 0:6134; A11 ¼ 0:7210 where y is the root of the following equation: 123 J Petrol Explor Prod Technol Dranchuk, Purvis and Robinson’s Correlation (Dranchuk Table 1 Constants of Heidaryan et al.’s correlation et al. 1971) Constants for Ppr 3 Constants for Ppr [ 3 This is a further modification of the earlier obtained DAK A1 2.827793 3.252838 -1 -1 correlation. The DPR has eight constants and requires less A2 -4.688191 9 10 -1.306424 9 10 -1 computational workload to obtain the z-factor. A3 -1.262288 6.449194 9 10 0:27P A4 -1.536524 -1.518028 z ¼ pr ; ð9Þ A5 -4.535045 -5.391019 yTpr -2 -2 A6 6.895104 9 10 -1.379588 9 10 -1 -2 where y is the root of the following equation: A7 1.903869 9 10 6.600633 9 10 hi -1 -1 2 T A8 6.200089 9 10 6.120783 9 10 2 2 ðÀA8y Þ 2 5 5 T4y ð1 þ A8y Þe þ 1 þ T1y þ T2y þ T3y þ ¼ 0 y A9 1.838479 2.317431 -1 -1 A2 A3 A5 A5A6 A10 4.052367 9 10 1.632223 9 10 T1 ¼ A1 þ þ 3 ; T2 ¼ A4 þ ; T3 ¼ ; -1 Tpr Tpr Tpr Tpr A11 1.073574 5.660595 9 10 A7 0:27Ppr T4 ¼ 3 ; T5 ¼ Tpr Tpr A1 ¼ 0:31506237; A2 ¼À1:04670990; Heidaryan, Moghdasi and Rahimi’s Correlation A3 ¼À0:57832720; A4 ¼ 0:53530771; A5 ¼À0:61232032; A6 ¼À0:10488813; Heidaryan et al. (2010) developed a new explicit A7 ¼ 0:68157001; A8 ¼ 0:68446549 piecewise correlation using regression analysis of the z-factor experimental value for reduced pseudo-pres- These correlations are effective; however, they do not sure of fewer and [3(Table1). The correlation has a converge (or converge on wrong pseudo-reduced density total of 22 constants, with a discontinuity at Ppr ¼ 3 values) when the temperature of the systems is close to the (Fig. 2) and correlation regression coefficient of critical temperature. In addition, they are computationally 0.99,963. expensive. It is these limitations that necessitated the 0 ÀÁ 1 A5 2 A9 A11 A1 þ A3 lnðPprÞþ þ A7 lnðPprÞ þ 2 þ lnðPprÞ development of the current explicit correlations. Tpr T Tpr z ¼ ln@ ÀÁpr A A4 2 A8 A10 1 þ A lnðPprÞþ þ A lnðPprÞ þ 2 þ lnðPprÞ 2 Tpr 6 T Tpr Explicit correlations pr ð11Þ Explicit correlations do not require an iterative procedure. For some petroleum engineering applications, it is Therefore, they do not have the problem of convergence as often necessary to compute the derivative of z-factor opposed to implicit correlations. One of the best explicit with respect to pressure or temperature. A function that correlations for evaluation of the z-factor was given by is discontinuous at a certain point is not differentiable at Beggs and Brills (1973). More recent ones are Heidaryan that point (O’Neil 2012). Therefore, the explicit et al. (2010), Azizi et al. (2010) and Sanjari and Lay (2012) correlation developed by Heidaryan et al. (2010) correlations. A short description of some of the explicit cannot be used to evaluate the derivative of the z- correlations is presented in the following subsections. factor with respect to the pseudo-reduced pressure at Ppr ¼ 3(Fig.3). Brill and Beggs’ compressibility factor (1973) 1 À A Azizi, Behbahani and Isazadeh’s Correlation z ¼ A þ þ CpD ; eB pr Azizi et al. (2010) presented an explicit correlation with 20 where constants for a reduced temperature range of 1:1 Tpr 2 0:5 A ¼ 1:39ðTpr À 0:92Þ À 0:36Tpr À 0:10; and reduced pressure range of 0:2 Ppr 11. The corre- 0:066 0:32p2 lation used 3038 data points within the given ranges. B ¼ð0:62 À 0:23T Þp þ À 0:037 p2 þ pr pr pr T À 0:86 pr 10E B þ C pr z A ; 12 F ¼ þ ð Þ C ¼ 0:132 À 0:32 logðTprÞ; D ¼ 10 ; D þ E 2 E ¼ 9ðTpr À 1Þ and F ¼ 0:3106 À 0:49Tpr þ 0:1824Tpr where 123 J Petrol Explor Prod Technol Fig.