Journal of Petroleum and Mining Engineering 20 (1)2018

Journal of Petroleum and Mining Engineering

A Study on Gas Compressibility Factor for Gas-Condensate Systems

Ali M. Wahba *, Hamid M. Khattab, and Ahmed A. Gawish Petroleum Engineering Department, Faculty of Petroleum and Mining Engineering, Suez University- Egypt. *Corresponding author [email protected]

Abstract Gas compressibility factor is the most important gas property. Its value is required in many petroleum engineering calculations. There are many different sources of gas compressibility factor value such as experimental measurements, equations of state, charts, tables, intelligent approaches and empirical correlations methods. In absence of experimental measurements of gas compressibility factor values, it is necessary for the petroleum engineer to find an accurate, quick and reliable method for predicting these values. This study presents a new gas compressibility factor explicit empirical correlation for gas-condensate reservoir systems above dew point pressure. This new correlation is more robust, reliable and efficient than the Keywords previously published explicit empirical correlations. It is also in a simple

Natural Gas,; Compressibility mathematical form. The predicted value using this new correlation can be used as Factor; Empirical Correlation; an initial value of implicit correlations to avoid huge number of iterations. This study Validation. also presents evaluation of the new and previously published explicit correlations.

Introduction important property of natural gas. It accounts for how much the real gas behavior deviates from the ideal gas Naturally occurring gas has many properties such behavior at given condition. According to real gas law, as compressibility factor, density, specific volume, it is expressed as a function of pressure, volume, and specific gravity, isothermal compressibility coefficient, temperature as follows: 푃푉 formation volume factor, expansion factor, and 푍 = (1) 푛푅푇 viscosity. They are required in petroleum engineering calculations such as calculations of gas reserves, gas flow through porous medium, gas pressure gradient in Since 1942, Standing and Katz [1] gas Z-factor production system, gas metering and gas chart has become a standard in petroleum industry compression. Gas compressibility factor is the most which is used to estimate gas compressibility factor. It important gas property as all other gas properties is based on the principle of corresponding states. This depend directly or indirectly on it. Accurate values of principle states that two substances at the same gas compressibility factor are obtained from conditions referenced to critical pressure and critical experimental measurements. These experimental temperature will have similar properties. According to measurements are expensive, time consuming and the principle of corresponding states, gas may be unavailable. They are also not available for all compressibility factor is expressed as a function of reservoir conditions. It is necessary to find an reduced pressure and reduced temperature as accurate, quick and reliable method for predicting follows: these values. So, numerical correlation concept is 푍 = 푓(푃푟 , 푇푟) (2) 푃 introduced in petroleum industry. Several empirical Where: 푃푟 = (3) 푃푐 correlations have been developed to approximately 푇 predict accurate values of gas compressibility factor at 푇푟 = (4) 푇푐 any pressure and temperature conditions. The main objectives of this study are to summarize all available For gas mixture, critical and reduced properties previously published gas compressibility factor are replaced with pseudo-critical and pseudo-reduced empirical correlations, develop a new, simple and properties. The accuracy of pseudo-critical properties accurate gas compressibility factor explicit empirical calculation will affect the accuracy of gas Z-factor correlation for any reservoir conditions and evaluate estimation. the new and other explicit correlations. Several methods for calculation of natural gas Gas compressibility factor which is also called gas pseudo-critical pressure and temperature for gas- deviation factor or simply gas Z-factor is the most condensate reservoir systems have been developed. Page|41 Journal of Petroleum and Mining Engineering 20 (1)2018

These methods are divided into two categories. In this Data Acquisition study, Sutton (1985) [2] modification to Stewart- Huge data points were collected to achieve the Burkhardt-Voo [3] mixing rules (SSBV mixing rules), main objectives of this study. They were divided into Corredor et al. (1992) [4] mixing rules, Piper et al. two sets according to the source of data points: (1993) [5] mixing rules, Al-Sharkawy et al. (2000) [6] general data set and specific data set. General data set mixing rules and Al-Sharkawy (2004) [7] mixing consists of five thousand, nine hundred and forty data parameters are used when natural gas composition is points of gas Z-factor values as a function of pseudo- known while Sutton (1985) [2] empirical correlations, reduced pressure and temperature. They were the Piper et al. (1993) [5] mixing rules, El-Sharkawy-El- result of Standing and Katz chart digitization done by Kamel (2000) [8] empirical correlations and Sutton Poettmann and Carpenter. Statistical distributions (2005) [9] empirical correlations used when natural such as maximum, minimum, mean, median and gas composition is unknown. For some of these range of this data set are shown in Table 2. Specific methods, the critical properties of all components of data set consists of seven hundred and twenty one the gas are required. The critical properties of pure data points of gas Z-factor values as a function of components are well known as given in Table 1. Gas- pseudo-reduced pressure and temperature. These condensate gases contain hydrocarbon-plus fractions data points are measured at pressures above the dew such as heptanes-plus (퐶 +) fraction. So, the critical 7 point pressures of the gas-condensate reservoir properties of the hydrocarbon-plus fractions must be systems. They were prepared from data collected estimated. Several correlations have been developed from unpublished gas-condensate gas PVT reports. to estimate the critical properties of the hydrocarbon- This collected data was reservoir pressure and plus fractions. Kesler and Lee (1976) [10] correlations, temperature, mole fraction of gas chemical Whitson (1987) [11] correlation and Riazi and Daubert composition and gas specific gravity. The statistical (1987) [12] correlations are used in this study. Gas- distributions such as maximum, minimum, mean, condensate gases also contain impurities such as median and range of this collected data are shown in hydrogen sulfide (퐻 푆), carbon dioxide (퐶푂 ), 2 2 Table 3. nitrogen (푁2) and water vapor (퐻2푂) which affect the pseudo-critical pressure and temperature values. Several correlations have been developed to account Research Methodology for the presence of these impurities. Wichert and Aziz To achieve the objectives of this study, MATLAB (1972) [13] correction method, Modified Wichert and Surface Fitting Tool (sftool) was used to develop a new Aziz [9] correction method, Standing (1981) [14] explicit empirical correlation of gas compressibility correction method and Casey [15] correction method factor. EXCEL sheets were used to validate the are used in this study. The corrected pseudo-critical performance of this new correlation. EXCEL sheets pressure and temperature are used to calculate the were also used to evaluate and grade the pseudo-reduced pressure and temperature which are performance of this new and other explicit the parameters of gas compressibility factor empirical correlations. These validation and evaluation were correlations. All of the above mentioned methods are performed using statistical error analysis such as summarized in Appendix A. average absolute percent relative error (AARE%), residual sum of squares (RSS), root mean square error Gas compressibility factor empirical correlations (RMSE), standard deviation (SD) and coefficient of are divided into two categories: implicit and explicit determination (R2) and also with using graphical empirical correlations. Implicit empirical correlations analysis such as cross plot parity line. require iterative solution methods. The most used implicit empirical correlations are Hankinson Thomas Results and Discussion and Phillips (1969) [16], Hall-Yarborough (1973) [17], Dranchuk-Purvis-Robinson (1974) [18], Dranchuk- To develop the new explicit empirical correlation, Abu-Kassem (1975) [19] and Hall and Iglesias-Silva 4000 data points from general data set are entered in (2007) [20] correlations. But, explicit empirical MATLAB Surface Fitting Tool (sftool). Fig. 1 shows the correlations do not require iterative solution surface plot of the new correlation which has the methods. They are used directly to calculate gas following form: compressibility factor. The most used previously 푍 = −0.1284 + 0.3098 푇푝푟 + 0.1427 푃푝푟 + 2 2 published explicit empirical correlations are Papay 0.3222 푇푝푟 − 0.1571 푇푝푟 푃푝푟 + 0.009456 푃푝푟 − 3 2 (1968) [21], Beggs and Brill (1973) [22], Gopal (1977) 0.0963 푇푝푟 + 0.02993 푇푝푟 푃푝푟 − 2 3 [23], Burnett (1979) [24], Kumar (2004) [25], Bahadori 0.00002458 푇푝푟 푃푝푟 − 0.0002861 푃푝푟 (5) et al. (2007) [26], Al-Anazi and Al-Quraishi (2010) [27], Azizi et al. (2010) [28], Heidaryan-Salarabadi- This new empirical correlation is in a simple Moghadasi (2010) [29], Heidaryan-Moghadasi-Rahimi mathematical form. In which, the gas compressibility (2010) [30], Shokir et al. (2012) [31], Sanjari and factor is a function of pseudo-reduced pressure and Nemati Lay (2012) [32], M.A. Mahmoud (2013) [33] temperature. Figure 2 shows the training of this new and Niger Delta (2013) [34] correlations. These explicit empirical correlation using 4000 data points from empirical correlations are summarized in Appendix B. general data set. The statistical parameters values of this training are: RSS = 0.6429, RMSE = 0.0127, AARE% = 0.904, SD = 1.1631 and R2 = 0.9964.

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Figure 1 Surface plot of the new proposed Z-factor explicit empirical correlation.

Figure 2 Accuracy of the new proposed Z-factor explicit empirical correlation for training.

method for Ppc and Tpc calculations are shown in Evaluation and Validation Appendix C from Figure C-1 to Figure C-6. The new and other explicit empirical correlations can be used to predict the Z-factor of gas-condensate Conclusions gases depending on the choice of the correct gas The conclusions emanating from this study are as pseudo-critical pressure and temperature calculation follows: method. There are several methods to calculate gas pseudo-critical pressure and temperature and  A new explicit empirical correlation for Z-factor is accessories methods to account for the presence of obtained in simple mathematical form. heptanes-plus fraction and impurities in gas-  The obtained correlation provides better condensate gases as mentioned above in literature predictions of gas Z-factor values than other explicit review section. From these methods, twelve methods empirical correlations. As it gives the highest are formed when gas composition is known and six accuracy when using any method for calculating gas methods are formed when gas composition is pseudo-critical pressure and temperature either unknown. These methods are evaluated using 721 when gas composition is known or unknown except data points of specific data set. The statistical for using El-Sharkawy empirical correlations for calculating gas pseudo-critical pressure and parameters values of this evaluation are shown in temperature when gas composition is unknown Appendix C from Table C-1 to Table C-18. As shown because of the accuracy of these correlations. from these tables, six explicit empirical correlations have high coefficient of determination (R2) values.  The proposed correlation is recommended for the The cross plots of these correlations when using following pseudo-reduced pressure and Sutton (1985), Standing, Wichert & Aziz and Casey temperature ranges for gas-condensate reservoir systems above dew point pressure:

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1 < 푃푝푟 ≤ 15  The predicted gas Z-factor value using the new 1.05 ≤ 푇푝푟 ≤ 3.0 correlation can be used, out of this new correlation recommended ranges, as an initial value of implicit correlations to avoid huge number of iterations.

Table 1 Physical properties of defined components.

Table 2 Satistical distributions of general data set.

Table 3 Statistical distributions of specific data set.

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yi = Mole fraction of component i in the gas Nomenclature mixture

yN2 = Mole fraction of Nitrogen AARE% = Average absolute percent relative error, Z = Gas deviation factor, dimensionless % o EJ = Sutton SBV parameter, R/psia Greek symbols o 0.5 EK = Sutton SBV parameter, R/psia ϵ = Wichert and Aziz psudo-critical o J = SBV parameter, R/psia temperature adjustment parameter, o 0.5 K = SBV parameter, R/psia oR M = Molecular weight, lb/lb-mole γC7+ = Specific gravity of heptanes-plus Mair = Apparent molecular weight of the air component which has the value 28.964 lb/lb- γg = Gas specific gravity, dimensionless mole γg HC = Specific gravity of hydrocarbon portion M Co2 = Molecular weight of Carbon Dioxide

MC7+ = Molecular weight of heptanes-plus component References MH S = Molecular weight of Hydrogen Sulfide 2 [1] Standing, M.B. and Katz, D.L.: “Density of Natural MN2 = Molecular weight of Nitrogen N = Number of moles of gas, lb-mole Gases,” Trans. AIME, 146, 1942, pp. 140-149. P = Absolute pressure, psia [2] Sutton, R.P.: “Compressibility Factors for High- Molecular-Weight Reservoir Gases,” paper SPE 14265 Pc = Critical pressure, psia th P = Critical pressure of Carbon Dioxide presented at the 60 SPE Annual Technical Conference c Co2 and Exhibition, Las Vegas, NV, September 22-25, 1985, Pc H = Critical pressure of Hydrogen Sulfide 2S P = Critical pressure of Nitrogen pp. 22-25. c N2 [3] Stewart, W.F., Burkhardt, S.F. and Voo, D.: “Prediction Ppc = Pseudo-critical pressure, psia of Pseudo-Critical Parameters for Mixtures,” paper Ppc HC = Pseudo-critical pressure of hydrocarbon portion presented at the AIChE Meeting, Kansas City, MO., May 18, 1959. Ppr = Pseudo-reduced pressure, dimensionless [4] Corredor, J.H., Piper, L.D. and McCain, W.D. Jr.: Pr = Reduced pressure, dimensionless PVT = Pressure, volume and temperature “Compressibility Factors for Naturally Occurring R = The universal gas constant which has the Petroleum Gases,” paper SPE 24864 presented at the th value 10.73 psia.ft3/lb-mole/oR 67 SPE Annual Technical Conference and Exhibition, R2 = Coefficient of determination, fraction Washington, DC, October 4-7, 1992. RMSE = Root mean square error, fraction [5] Piper, L.D., McCain Jr. and Corredor, J.H.: RSS = Residual sum of squares, fraction “Compressibility Factors for Naturally Occurring SD = Standard deviation, % Petroleum Gases,” paper SPE 26668 presented at the th T = Absolute temperature, oR 68 SPE Annual Technical Conference and Exhibition, o Houston, TX, October 3-6, 1993. Tb = Boiling temperature, R o [6] Elsharkawy A.M., Yousef S.Kh., Hashem S. and Alikhan Tc = Critical temperature, R T = Critical temperature of Carbon Dioxide A.A.: “Compressibility Factor for Gas Condensates,” c Co2 paper SPE 59702 presented at the SPE Permian Basin Tc H = Critical temperature of Hydrogen Sulfide 2S T = Critical temperature of Nitrogen Oil and Gas Recovery Conference, Midland, Texas, c N2 o USA, March 21-23, 2000. Tpc = Pseudo-critical temperature, R [7] Elsharkawy, A.M.: “Efficient Methods for Calculations Tpc HC = Pseudo-critical temperature of hydrocarbon portion of Compressibility, Density and Viscosity of Natural Gases,” Fluid Phase Equilib., 218 (1), 2004, pp. 1-13. Tpr = Pseudo-reduced temperature, dimensionless [8] Elsharkawy, A.M. and Elkamel, A.: “Compressibility Factor for Sour Gas Reservoirs,” paper SPE 64284 Tr = Reduced temperature, dimensionless V = Volume, ft3 presented at the 2000 SPE Asia Pacific Oil and Gas Conference and Exhibition, Brisbane, Australia, WHC = Weight fraction of hydrocarbon portion October 16-18, 2000. WNHC = Weight fraction of non-hydrocarbon portion [9] Sutton, R.P.: “Fundamental PVT Calculations for Associated and Gas/Condensate Natural Gas yCo2 = Mole fraction of Carbon Dioxide Systems,” paper SPE 97099 presented at the SPE yC7+ = Mole fraction of heptanes-plus component Annual Technical Conference and Exhibition, Dallas, Texas, October 9-12, 2005, pp. 270-284. yHC = Mole fraction of hydrocarbon portion

yH = Mole fraction of Hydrogen Sulfide 2S Journal of Petroleum and Mining Engineering 20 (1)2018

[10] Kesler, M.G. and Lee, B.I.: “Improve Prediction of [28] Azizi, N., Behbahani, R. and Isazadeh, M.A.: “An Enthalpy of Fraction,” Hyd. Proc., March, 1976, pp. Efficient Correlation for Calculating Compressibility 153-158. Factor of Natural Gases,” J. Nat. Gas Chem., 19 (6), [11] Whitson, C. H.: “Effect of C7+ Properties on Equation- 2010, pp. 642-645. of-State Predictions,” SPEJ, December, 1987, pp. 685- [29] Heidaryan E., Salarabadi A. and Moghadasi J.: “A Novel 696. Correlation Approach for Prediction of Natural Gas [12] Riazi, M.R. and Daubert, T.E.: “Characterization Compressibility Factor,” J. Nat. Gas Chem., 19, 2010, Parameters for Petroleum Fractions,” Ind. Eng. Chem. pp. 189-192. Res., 26 (24), 1987, pp. 755-759. [30] Heidaryan, E., Moghadasi, J. and Rahimi, M.: “New [13] Wichert, E. and Aziz, K.: “Calculation of Z’s for Sour Correlations to Predict Natural Gas Viscosity and Gases,” Hyd. Proc., 51 (5), 1972, pp. 119-122. Compressibility Factor,” J. Petrol. Sci. Eng., 73 (1-2), [14] Standing, M.B.: “Volumetric and Phase Behavior of Oil 2010, pp. 67-72. Field Hydrocarbon Systems,” 9th ed., Society of [31] Shokir, E.M.E.-M., El-Awad, M.N., Al-Quraishi, A.A., Al- Petroleum Engineers of AIME, Dallas, Texas, 1981. Mahdy, O.A.: “Compressibility Factor Model of Sweet, [15] Lee, J. and Wattenbarger, R.A.: “Gas Reservoir Sour, and Condensate Gases Using Genetic Engineering,” Society of Petroleum Engineers, Vol. 5, Programming,” Chem. Eng. Res. Des., 90 (6), 2012, pp. Richardson, Texas, USA, 1996. 785-792. [16] Hankinson, R.W., Thomas, L.K. and Phillips, K.A.: [32] Sanjari, E. and Lay, E.N.: “An Accurate Empirical “Predict Natural Gas Properties,” Hyd. Proc., 48, April, Correlation for Predicting Natural Gas Compressibility 1969, pp. 106-108. Factors,” J. Nat. Gas Chem., 21 (2), 2012, pp. 184-188. [17] Hall, K.R. and Yarborough, L.: “A New Equation of State [33] Mahmoud, M.A.: “Development of a New Correlation for Z-Factor Calculations,” Oil & Gas J., 71 (25), 1973, of Gas Compressibility Factor (Z-Factor) for High pp. 82-92. Pressure Gas Reservoirs,” paper SPE 164587 presented [18] Dranchuk, P.M., Purvis, R.A. and Robinson, D.B.: at the SPE North Africa Technical Conference and “Computer Calculation of Natural Gas Compressibility Exhibition, Cairo, Egypt, April 15-17, 2013. Factors Using the Standing and Katz Correlation,” [34] Obuba, J., Ikiesnkimama, S.S., Ubani, C.E. and Ekeke, Institute of Petroleum Technical Series, No. IP 74-008, I.C.: “Natural Gas Compressibility Factor Correlation 1974, pp. 1-13. Evaluation for Niger Delta Gas Fields,” IOSR Journal of [19] Dranchuk, P.M. and Abou-Kassem, J.H.: “Calculation of Electrical and Electronics Engineering, 6 (4), Jul. - Aug., Z factors for Natural Gases Using Equations of State,” 2013, pp. 1-10. J. Can. Petrol. Technol., 14 (3), 1975, pp. 34-36. [20] Hall, K.R. and Iglesias-Silva, G.A.: “Improved Equations for the Standing-Katz Tables,” Hyd. Proc., 86 (4), 2007, Appendix A. Pseudo-critical pressure and pp. 107-110. temperature calculation methods [21] Papay, J.: “A Termelestechnologiai Parameterek Valtozasa a Gaztelepek Muvelese Soran,” OGIL Musz. Tud Kozl., Budapest, 1968, pp. 267-273. Sutton Modification to SBV Mixing Rules (SSBV [22] Beggs, D.H. and Brill, J.P.: “A Study of Two-Phase Flow Mixing Rules) (1985) in Inclined Pipes,” J. Petrol. Technol., 25 (5), 1973, pp. SBV Mixing rules 1 푛 푇푐 2 푛 푇푐 2 607-617. 퐽 = ( ) [∑푖=1 푦푖 ( )푖] + ( )[∑푖=1 푦푖 (√ )푖] 3 푃푐 3 푃푐 [23] Gopal, V. N.: “Gas Z-Factor Equations Developed for 푛 푇푐 퐾 = ∑푖=1 푦푖 ( )푖 Computer,” Oil & Gas J., August 8, 1977, pp. 58-60. √푃푐 [24] Burnett, R.R.: “Calculator Gives Compressibility Sutton Modification 1 푇 2 푇 Factors,” The Oil and Gas Journal, June 11, 1979, pp. 푐 푐 2 퐹퐽 = ( )[푦 ( )]퐶 + ( )[푦(√ )]퐶 3 푃푐 7+ 3 푃푐 7+ 70-74. 2 퐸퐽 = 0.6081 퐹퐽 + 1.1325 퐹퐽 − 14.004 퐹퐽 푦 + 퐶7+ [25] Kumar, N.: “Compressibility Factors for Natural and 2 64.434 퐹퐽 푦 Sour Reservoir Gases by Correlations and Cubic 퐶7+ 푇 Equations of State,” MS Thesis, Texas Tech University, 푐 2 3 퐸퐾 = (√ )퐶 [0.3129 푦퐶 − 4.8156 푦퐶 27.3751 푦퐶 ] 푃푐 7+ 7+ 7+ 7+ Lubbock, Tex, USA, 2004. ′ 퐽 = 퐽 − 퐸퐽 [26] Bahadori, A., Mokhatab, S. and Towler, B.F.: “Rapidly ′ 퐾 = 퐾 − 퐸퐾 Estimating Natural Gas Compressibility Factor,” J. Nat. ′ 2 푇 = 퐾 ⁄ Gas Chem., 16 (4), 2007, pp. 349-353. 푝푐 퐽′ 푇 [27] Al-Anazi, B.D. and Al-Quraishi, A.A.: “New Correlation 푃 = 푝푐⁄ 푝푐 퐽′ for Z-Factor Using Genetic Programming Technique,” Corredor et al. Mixing Rules (1992) paper SPE 128878 presented at the SPE Oil and Gas 3 푇푐 6 푇푐 퐽 = 훼0 + ∑푖=1 훼푖 푦푖( )푖 + 훼4 ∑푗=1 푦푗( )푗 + India Conference and Exhibition, Mumbai, India, 푃푐 푃푐 6 푇푐 2 2 훼5 [∑푗=1 푦푗( )푗] + 훼6(푦퐶7+푀퐶7+) + 훼7(푦퐶7+푀퐶7+) January 20-22, 2010. 푃푐

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3 푇푐 6 푇푐 퐾 = 훽0 + ∑ 훽푖 푦푖( )푖 + 훽4 ∑ 푦푗( )푗 + Riazi and Daubert Correlations (1987) 푖=1 √푃 푗=1 √푃 푐 푐 푏 푐 6 푇푐 2 2 휃 = 푎 (푀) (훾) 푒푥푝[푑 (푀) + 푒 (훾) + 푓 (푀) (훾)] 훽5 [∑푗=1 푦푗( )푗] + 훽6(푦퐶7+푀퐶7+) + 훽7(푦퐶7+푀퐶7+) √푃푐

Where 푦푖 ∈ {푦퐻 , 푦퐶표 , 푦푁 } and 푦푗 ∈ {푦퐶 , 푦퐶 , … , 푦퐶 } Wichert and Aziz Correction Method (1972) 2푆 2 2 1 2 6 휖 = 120 (퐴 0.9 − 퐴 1.6) + 1.5 (퐵 0.5 − 퐵 4) respectively. ′ 푇푝푐 = 푇푝푐 − 휖 Piper et al. Mixing Rules (1993) 푃 푇′ 푇 푇 ′ 푝푐 푝푐 3 푐 6 푐 푃푝푐 = 퐽 = 훼0 + ∑푖=1 훼푖 푦푖( )푖 + 훼4 ∑푗=1 푦푗( )푗 + [푇푝푐+퐵(1−퐵)휖] 푃푐 푃푐 6 푇푐 2 2 퐴 = 푦퐻 + 푦퐶표 2푆 2 훼5 [∑푗=1 푦푗( )푗] + 훼6(푦퐶7+푀퐶7+) + 훼7(푦퐶7+푀퐶7+) 푃푐 퐵 = 푦퐻 3 푇푐 6 푇푐 2푆 퐾 = 훽0 + ∑푖=1 훽푖 푦푖( )푖 + 훽4 ∑푗=1 푦푗( )푗 + √푃푐 √푃푐 푇 Modified Wichert and Aziz Correction Method 6 푐 2 2 2.2 0.06 0.68 훽5 [∑푗=1 푦푗( )푗] + 훽6(푦퐶7+푀퐶7+) + 훽7(푦퐶7+푀퐶7+) √푃푐 휖 = 107.6 (퐴 − 퐴 ) + 5.9 (퐵 − 퐵 ) Standing Correction Method Where 푦푖 ∈ {푦퐻 , 푦퐶표 , 푦푁 } and 푦푗 ∈ {푦퐶 , 푦퐶 , … , 푦퐶 } 2푆 2 2 1 2 6 푇 = 푦 푇 + 푦 푇 + 푦 푇 + 푦 푇 푝푐 퐻퐶 푝푐 퐻 푆 푐 퐻 퐶표 푐 퐶표 푁 푐 푁 respectively. 퐻퐶 2 2푆 2 2 2 2 푃 = 푦 푃 + 푦 푃 + 푦 푃 + 푦 푃 푝푐 퐻퐶 푝푐 퐻 푆 푐 퐻 퐶표 푐 퐶표 푁 푐 푁 Al-Sharkawy et al. Mixing Rules (2000) 퐻퐶 2 2푆 2 2 2 2 6 푇푐 ⁄ ⁄ 퐽 = ∑푖=1 푦푖( )푖 + {푎0 + [푎1(푦 푇푐 푃푐)]퐶7+ + [푎2(푦 푇푐 푃푐)]푁2 + Casey Correction Method 푃푐 ′ 푇푝푐−(227.2 푦 +1165 푦 ) [푎3(푦 푇푐⁄푃푐)]퐶표 + [푎4(푦 푇푐⁄푃푐)]퐻 } 푁2 퐻 푂 2 2푆 ′′ 2 푇푝푐 = − (246.1 푦푁 − 400 푦퐻 ) 6 푇푐 1−(푦 + 푦 ) 2 2푂 ⁄ 푁2 퐻 푂 퐾 = ∑푖=1 푦푖( )푖 + {푏0 + [푏1(푦 푇푐 √푃푐)]퐶7+ + 2 √푃푐 푃′ −(493.1 푦 +3200 푦 ) 푝푐 푁2 퐻 푂 [푏2(푦 푇푐⁄√푃푐)]푁 + [푏3(푦 푇푐⁄√푃푐)]퐶표 + [푏4(푦 푇푐⁄√푃푐)]퐻 } ′′ 2 2 2 2푆 푃푝푐 = − (162 푦 − 1270 푦 ) 푁2 퐻 푂 1−(푦푁 + 푦퐻 ) 2 Al-Sharkawy Mixing Parameters (2004) 2 2푂 ∑3 푇푐 ∑6 푇푐 퐽푖푛푓 = 훼0 + 푖=1 훼푖 푦푖( )푖 + 훼4 푗=1 푦푗( )푗 + 푃푐 푃푐 The coefficients of corredor et al. mixing rules, 훼5(푦 푀퐶 ) 퐶7+ 7+ Piper et al. mixing rules, Al-Sharkawy mixing rules, Al- ∑3 푇푐 ∑6 푇푐 퐾푖푛푓 = 훽0 + 푖=1 훽푖 푦푖( )푖 + 훽4 푗=1 푦푗( )푗 + √푃푐 √푃푐 Sharkawy mixing parameters and Riazi and Daubert

훽 (푦 푀퐶 ) correlations are shown in Table A-1 through Table A- 5 퐶7+ 7+ 5 respectively. Where 푦푖 ∈ {푦퐻 , 푦퐶표 , 푦푁 } and 푦푗 ∈ {푦퐶 , 푦퐶 , … , 푦퐶 } 2푆 2 2 1 2 6 Papay Correlation (1968) respectively. 3.53 푃 0.274 푃 2 푍 = 1 − 푝푟 + 푝푟 Sutton Correlations (1985) 10 0.9813 푇푝푟 10 0.8157 푇푝푟 2 Beggs and Brill Z-Factor Correlation (1973) 푇푝푐 = 169.2 + 349.5 (훾푔 ) − 74 (훾푔 ) 퐻퐶 퐻퐶 퐻퐶 1−퐴 퐷 2 푍 = 퐴 + + 퐶 푃푝푟 푃푝푐 퐻퐶 = 756.8 − 131 (훾푔 퐻퐶) − 3.6 (훾푔 퐻퐶) 푒푥푝(퐵) Where: 푦퐻퐶 = 1 − 푦퐻 − 푦퐶표 − 푦푁 Recommended range: 1.05 < 푇푝푟 < 2.0 훾푔 −(푦퐻 푀퐻 − 푦2퐶표푆 푀퐶표 −2 푦푁 푀푁2 )/푀푎푖푟 2푆 2푆 2 2 2 2 푃 < 15 훾푔 퐻퐶 = 푝푟 푦퐻퐶 Piper et al. Mixing Rules (1993) Where: 푇 0.5 푐 퐻 푇푐 2푆 퐶표2 퐴 = 1.39 (푇푝푟 − 0.92) − 0.36 푇푝푟 − 0.101 퐽 = 0.11582 − 0.45820 푦퐻 푆 − 0.90348 푦퐶표2 − 6 2 푃푐 푃푐 0.066 2 0.32 푃푝푟 퐻 푆 퐶표2 퐵 = (0.62 − 0.23 푇 )푃 + ( − 0.037) 푃 + 2 푝푟 푝푟 푇 −0.86 푝푟 10 (9푇푝푟−9) 푇푐 푝푟 푁2 2 0.66026 푦푁2 + 0.70729 (훾푔) − 0.099397 (훾푔) 퐶 = 0.132 − 0.32 푙표푔 푇 푃푐 푝푟 푁2 2 푇 (0.3106 − 0.49 푇푝푟+0.1824 푇푝푟) 푐 퐻 푇푐 퐷 = 10 2푆 퐶표2 퐾 = 3.8216 − 0.06534 푦퐻 − 0.42113 푦퐶표 − 2푆 푃 2 Gopal Method (1977) √ 푐 퐻 √푃푐 퐶표 2푆 2 −1.4667 푇푐 푍 = 푃푝푟 (0.711 + 3.66 푇푝푟) − 1.6371 (0.319 푇푝푟 + 0.91249 푦 푁2 + 17.438 (훾 ) − 3.2191(훾 ) 2 푁2 푔 푔 0.522) + 2.071 푃푐 √ 푁2 Recommended range: 1.05 ≤ 푇 ≤ 3.0 El-Sharkawy-El-Kamel Correlations (2000) 푝푟 5.4 ≤ 푃푝푟 ≤ 15 푇 = 195.958 + 206.121 (훾 ) + 25.855 (푊 ) − 푝푐 푔 퐻퐶 Burnett Correlation (1979) 2 2 ′ 푁 6.421(푊푁퐻퐶) + 9.022 (푊퐻퐶) + 163.247 (푊푁퐻퐶) 푍 = 1 + (푍 − 1) (푠푖푛 90 푈) ( ) Recommended range: 1.3 ≤ 푇 ≤ 1.85 푃푝푐 = 193.941 − 131.347 (훾푔) + 217.144 푊퐻퐶 + 푝푟 2 2 푃 ≤ 푃′ 1060.349(푊푁퐻퐶) + 344.573 (푊퐻퐶) − 60.591 (푊푁퐻퐶) 푝푟 푝푟 ′ Sutton Correlations (2005) Where: 푍 = 0.3379 푙푛(푙푛(푇푝푟)) + 1.091 ′ ′ ′ 2 푇 = 164.3 + 357.7 (훾 ) − 67.7 (훾 ) 2 푃푝푟 = 21.46 푍 − 11.9 푧 − 5.9 푝푐 퐻퐶 푔 퐻퐶 푔 퐻퐶 ′ 2 푈 = 푃푝푟/푃푝푟 푃푝푐 퐻퐶 = 744 − 125.4 (훾푔 퐻퐶) + 5.9 (훾푔 퐻퐶) 푒푥푝(푈) 푁 = (1.1 + 0.26 푇푝푟 + (1.04 − 1.42 푇푝푟) 푈)( ) Kesler and Lee Correlations (1976) 푇푝푟

푇푐 = 341.7 + 811 (훾) + (0.4244 + 0.1174 (훾)) 푇푏 + Shell Oil Company (Kumar) Correlation (2004) 푃 (0.4669 − 3.2623 (훾)) (10 5)/푇 푍 = 퐴 + 퐵 푃 + (1 − 퐴) 푒푥푝(−퐶) − 퐷 ( 푝푟) 4 푏 푝푟 10 ⁄ 푃푐 = 푒푥푝[8.3634 − 0.0566 훾 − (0.24244 + Where: 퐴 = −0.101 − 0.36 푇푝푟 + 1.3868 √푇푝푟 − 0.919 ⁄ ⁄ 2 ( −3) ( 0.04275 2.2898 훾 + 0.11857 훾 ) 10 푇푏 + 1.4685 + 퐵 = 0.021 + 3.648⁄훾 + 0.47227⁄훾 2)(10 −7) 푇 2 − (0.42019 + 푇푝푟−0.65 푏 4 2 −10 3 퐶 = 푃푝푟 [퐸 + 퐹 푃푝푟 + 퐺 푃푝푟] 1.6977⁄훾 )(10 ) 푇푏 ] 퐷 = 0.122 푒푥푝[−11.3 (푇 − 1)] Whitson (1987) 푝푟 0.15178 0.15427 3 퐸 = 0.6222 − 0.224 푇푝푟 푇푏 = (4.5579 (푀) (훾) )

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0.0657 퐹 = − 0.037 퐷 = (−2 퐴) (퐵) (퐶) 푇 −0.85 푝푟 퐸 = 퐶+퐷 + 0.9178 퐺 = 0.32 exp[−19.53 (푇푝푟 − 1)] 1.0474 푇푝푟 퐹 = 퐷 − 퐸 Bahadori et al. Correlation (2007) 퐸 2 2 3 푍 = 푎 + 푏 푃푝푟 + 푐 푃푝푟 + 푑 푃푝푟 Azizi et al. Correlation (2010) 푍 = 퐴 + 퐵+퐶 Recommended range: 1.05 ≤ 푇푝푟 ≤ 2.4 퐷+퐸 0.2 ≤ 푃푝푟 ≤ 16 Recommended range: 1.1 ≤ 푇푝푟 ≤ 2.0 2 3 Where: 푎 = 퐴푎 + 퐵푎 푇푝푟 + 퐶푎 푇푝푟 + 퐷푎 푇푝푟 0.2 ≤ 푃푝푟 ≤ 11 2 3 2.16 1.028 1.58 −2.1 푏 = 퐴푏 + 퐵푏 푇푝푟 + 퐶푏 푇푝푟 + 퐷푏 푇푝푟 Where: 퐴 = 푎 푇푝푟 + 푏 푃푝푟 + 푐 푃푝푟 푇푝푟 + 2 3 −0.5 푐 = 퐴푐 + 퐵푐 푇푝푟 + 퐶푐 푇푝푟 + 퐷푐 푇푝푟 푑 푙푛(푇푝푟) 2.4 1.56 0.124 3.033 푑 = 퐴 + 퐵 푇 + 퐶 푇 2 + 퐷 푇 3 퐵 = 푒 + 푓 푇푝푟 + 푔 푃푝푟 + ℎ 푃푝푟 푇푝푟 푑 푑 푝푟 푑 푝푟 푑 푝푟 −1.28 1.37 퐶 = 푖 푙푛(푇푝푟) + 푗 푙푛(푇푝푟) + 푘 푙푛(푃푝푟) + Al-Anazi and Al-Quraishi Correlation (2010) 2 2 퐸 푙 푙푛(푃푝푟) + 푚 푙푛(푃푝푟) 푙푛(푇푝푟) 푍 = + 퐹 5.55 0.68 0.33 1.0482 퐷 = 1 + 푛 푇푝푟 + 표 푃푝푟 푇푝푟 퐴 = −0.06708 푃 + 0.2360 1.18 2.1 2 Where: 푝푟 퐸 = 푝 푙푛(푇푝푟) + 푞 푙푛(푇푝푟) + 푟 푙푛(푃푝푟) + 푠 푙푛(푃푝푟) + (3 퐴) 2−1.427 퐵 = + 0.9178 푡 푙푛(푃푝푟) 푙푛(푇푝푟) 푇푝푟 퐶 = 퐵 − (−2 퐴) (퐵) Heidaryan-Salarabadi-Moghadasi Model (2010)

2 3 퐴5 퐴6 퐴1 + 퐴2 푙푛(푃푝푟) + 퐴3 [푙푛(푃푝푟)] + 퐴4 [푙푛(푃푝푟)] + + 2 푇푝푟 푇푝푟 푍 = 2 퐴9 퐴10 1 + 퐴7 푙푛(푃푝푟) + 퐴8 [푙푛(푃푝푟)] + + 2 푇푝푟 푇푝푟

Recommended range: 1.2 ≤ 푇푝푟 ≤ 3.0 Moghadasi model, Heidaryan-Moghadasi-Rahimi 0.2 ≤ 푃푝푟 ≤ 15 model and Sanjari and Nemati Lay correlation are Heidaryan-Moghadasi-Rahimi Model (2010) shown in Table B-1 through Table B-5 respectively. 퐴 퐴 퐴 퐴 +퐴 ln(푃 )+ 5 + 퐴 [ln(푃 )] 2+ 9 + 11 ln(푃 ) 1 3 푝푟 푇 7 푝푟 푇 2 푇 푝푟 푍 = ln ( 푝푟 푝푟 푝푟 ) 퐴4 2 퐴8 퐴10 1 + 퐴2 ln(푃푝푟)+ + 퐴6 [ln(푃푝푟)] + 2 + ln(푃푝푟) 푇푝푟 푇푝푟 푇푝푟 Recommended range: 1.2 ≤ 푇푝푟 ≤ 3.0 0.2 ≤ 푃푝푟 ≤ 15

For more accuracy, two sets of coefficients

퐴1through 퐴11values were determined by minimizing the residual sum of squares of this proposed correlation. Shokir et al. Correlation (2012) 푍 = 퐴 + 퐵 + 퐶 + 퐷 + 퐸 (2 푇푝푟−푃푝푟−1) Where: 퐴 = 2.679562 2 3 (푃푝푟+푇푝푟) [ ] 푃푝푟 2 푃푝푟 푇푝푟+푃푝푟 퐵 = −7.686825 [ 2 3 ] 푇푝푟 푃푝푟+2 푇푝푟+푇푝푟 2 2 3 퐶 = −0.000624 [푇푝푟 푃푝푟 − 푇푝푟 푃푝푟 + 푇푝푟 푃푝푟 + 2 푇푝푟 푃푝푟 − 2 3 2 푃푝푟 + 2 푃푝푟] (푇푝푟−푃푝푟) 퐷 = 3.067747 2 (푃푝푟+푇푝푟+푃푝푟) 0.068059 2 2 0.041098 푇푝푟 퐸 = + 0.139489 푇푝푟 + 0.081873 푃푝푟 − + 푇푝푟 푃푝푟 푃푝푟 8.152325 푃푝푟 − 1.63028 푃푝푟 + 0.24287 푇푝푟 − 2.64988 푇푝푟 Sanjari and Nemati Lay Correlation (2012) 퐴4 (퐴4+1) (퐴4+2) 2 퐴3 푃푝푟 퐴6 푃푝푟 퐴8 푃푝푟 푍 = 1 + 퐴1 푃푝푟 + 퐴2 푃푝푟 + 퐴5 + 퐴7 + (퐴7+1) 푇푝푟 푇푝푟 푇푝푟 Recommended range: 1.01 ≤ 푇푝푟 ≤ 3.0 0.01 ≤ 푃푝푟 ≤ 15 M.A. Mahmoud Correlation (2013) 2 푍 = 0.702 푒푥푝(−2.5 푇푝푟) 푃푝푟 − 5.524 푒푥푝(−2.5 푇푝푟) 푃푝푟 2 + 0.044 푇푝푟 − 0.164 푇푝푟 + 1.15 Niger Delta Correlation (2013)

푍 = 6.41824 − 0.013363 푃푝푟 − 3.351293 푇푝푟

The coefficients of Bahadori et al. correlation, Azizi et al. correlation, Heidaryan-Salarabadi-

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Table A-1 Coefficients of Corredor et al. mixing rules

Table A-2 Coefficients of Piper et al. mixing rules

Table A-3 Coefficients of Al-Sharkawy et al. mixing rules

Table A-4 Coefficients of Al-Sharkawy et al. mixing parameters.

Table A-5 Coefficients of Riazi and Daubert correlations.

Table B-1 Coefficients of Bahadori et al. empirical correlation.

Table B-2 Coefficients of Azizi et al. empirical correlation.

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Table B-3 Coefficients of Heidaryan-Salarabadi-Moghadasi model.

Table B-4 Coefficients of Heidaryan-Moghadasi-Rahimi model.

Table B-5 Coefficients of Sanjari and Nemati lay empirical correlation.

Table C-1 statistical parameters values for explict emperical correlations when using SSBV, Whitson,Kesler&Lee, Wichert&Aziz and Caseymethod for ppe and tpe.

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Table C-2 statistical parameters values for explict emperical correlations when using SSBV, Whitson,Kesler&Lee, modified Wichert&Aziz and Caseymethod for ppe and tpe.

Table C-3 statistical parameters values for explict emperical correlations when using SSBV, Riazi &Daubert, Wichert&Aziz and Casey method for ppe and tpe.

Table C-4 statistical parameters values for explict emperical correlations when using SSBV, Riazi &Daubert, Modified Wichert&Aziz and Casey method for ppe and tpe.

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Table C-5 statistical parameters values for explict emperical correlations when using SSBV, Riazi &Daubert, Lee, Wichert&Aziz and Casey method for ppe and tpe.

Table C-6 statistical parameters values for explict emperical correlations when using SSBV, Riazi &Daubert, Kesler & Lee, Modified Wichert&Aziz and Casey method for ppe and tpe.

Table C-7 statistical parameters values for explict emperical correlations when using SSBV, Riazi &Daubert, Kesler & Lee, Corredor et al. mixing rules for ppe and tpe.

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Table C-8 statistical parameters values for explict emperical correlations when using piper et al. mixing rules for ppe and tpe.

Table C-9 statistical parameters values for explict emperical correlations when using Al-Sharkawy et al., Whitson and Kesler & Leemethod for ppe and tpe.

Table C-10 statistical parameters values for explict emperical correlations when using Al-Sharkawy et al and riazi & Daubert method for ppe and tpe.

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Table C-11 statistical parameters values for explict emperical correlations when using Al-Sharkawy et al, riazi & Daubert and Daubert and kesler& Lee method for ppe and tpe.

Table C-12 statistical parameters values for explict emperical correlations when using Al-Sharkawy mixing parameters for ppe and tpe.

Table C-13 statistical parameters values for explict emperical correlations when using Sutton(1985), Standing, Wichert& Aziz and Casey method for ppe and tpe.

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Table C-14 statistical parameters values for explict emperical correlations when using Sutton(1985), Standing, Modified Wichert& Aziz and Casey method for ppe and tpe.

Table C-15 statistical parameters values for explict emperical correlations when using piper et al. mixing rules for ppe and tpe.

Table C-16 statistical parameters values for explict emperical correlations when using Al-Sharkawy- El Kamel emprical correlations for ppe and tpe.

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Table C-17 statistical parameters values for explict emperical correlations when using Sutton(1985), Standing, Wichert& Aziz and Casey method for ppe and tpe.

Table C-18 statistical parameters values for explict emperical correlations when using Sutton(1985), Standing, Modified Wichert& Aziz and Casey method for ppe and tpe.

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