Compressibility Factor for Gas Condensates - American Chemical

Energy & Fuels 2001, 15, 807-816

807

Compressibility Factor for Gas Condensates Adel M. Elsharkawy,* Yousef S. Kh. S. Hashem, and Abbas A. Alikhan Petroleum Engineering Department, College of Engineering & Petroleum, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait Received October 4, 2000. Revised Manuscript Received January 19, 2001

The compressibility factor of natural gases is necessary in many petroleum engineering calculations. Some of these calculations are the following: evaluation of a newly discovered formation, pressure drop from flow of gas through a pipe, pressure gradient in gas wells, gas metering, gas compression, and processing. Typically, the gas compressibility factor is measured by laboratory experiments. These experiments are expensive and time-consuming. Occasionally, experimental data became unavailable and the gas compressibility factor is estimated from correlations using gas composition or gas gravity. This paper presents new methods for calculating the gas compressibility factors for gas condensates at any temperature and pressure. The method is based on compositional analysis of 1200 compositions of gas condensates collected worldwide. When the gas composition is known, this study presents a simple mixing rule to calculate the pseudo-critical properties of the gas condensate. The new mixing rule accounts for the presence of the heptane plus fraction and none hydrocarbons. In case the gas composition is unavailable, the study presents a new gas gravity correlation to estimate pseudo-critical properties of the gas condensate. This study also presents an evaluation of eight methods to characterize the plus fraction, three widely used mixing rules, and six methods to calculate the gas compressibility factor. Thus, this study presents an evaluation of one hundred forty-four possible methods of calculating the gas compressibility factor for gas condensates. The accuracy of the new mixing rule and the gas gravity correlation has been compared to other published methods. The comparison indicates that the proposed methods are consistent and provide accurate results.

1. Introduction Knowledge of the gas compressibility factor for gas condensates is necessary in petroleum engineering calculations. Compressibility factors are used in material-balance equations to estimate initial gas in place. It is also used in calculations of gas flow through porous media, gas pressure gradient in tubing and pipelines, gas metering, and gas compression. Typically the gas compressibility factor, Z-factor, is determined experimentally as a part of any standard PVT report. Occasionally, PVT reports are not available and compositional data or gas gravity is used to estimate the Z-factor from correlations. Standing and Katz1 (1942) presented a generalized compressibility factor chart. The chart represents compressibility factors of sweet and dry natural gas as a function of pseudo-reduced pressure (Ppr) and pseudoreduced temperature (Tpr). This chart is generally reliable for sweet natural gases with minor amounts of non-hydrocarbons. It is one of the most widely accepted correlations in the oil and gas industry. The Standing and Katz (SK) chart was developed using data for binary mixtures of methane with propane, ethane, butane, and natural gases having a wide range of composition.2 None of the gas mixtures had a molecular weight in excess of * Corresponding author. Fax (+965) 4849558. E-mail asharkawy@ kuc01.kuniv.edu.kw. (1) Standing, M. B.; Katz, D. L. Density of Natural Gases. Trans. AIME 1942, 146, 140-149.

40. For low molecular weight gases, it was found that the Z-factor estimated from the SK chart has an error in the order of 2 to 3%. However, for gas mixtures whose molecular weight is greater than 40, the SK chart provides inaccurate Z-factors. Methods for estimating gas comprcessibility factors normally are used when a reservoir fluid-depletion study is not available. This practice is acceptable for retrograde gases if the gas condensate is lean; however, if the gas is rich, the reserves may be seriously underestimated if the twophase compressibility factor is not used. The compressibility factor chart (SK) is applicable to most gases encountered in petroleum reservoirs and provides satisfactory prediction for all engineering computations. The calculated volume of gases containing only minor amounts of non-hydrocarbon can be accurate within 97%. Standing and Katz correlation for compressibility factor is valid only for dry-gas systems. Corrective methods are introduced to account for the presence of high molecular weight gases (C7+) and to extend the range of application of the SK chart.3,4 Retrograde gascondensate reservoirs experience liquid fallout during (2) Brown, G. G.; Katz, D. L.; Oberfell, G. G.; Alden, R. C. Natural Gasoline And The Volatile Hydrocarbons; Natural Gas Association of America, Tulsa, OK, 1948. (3) Stewart, W. F.; Burkhardt, S. F.; Voo, D. Prediction of Pseudo Critical Parameters for Mixtures. Paper presented at the AIChE Meeting, Kansas City, MO, 1959. (4) Sutton, R. P. Compressibility Factors for High Molecular Weight Reservoir Gases. Paper SPE 14265, presented at the SPE Annual Technical Meeting and Exhibition, Las Vegas, NV, Sept. 22-25, 1985.

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depletion below the dew point. The two-phase compressibility factor accounts for the formation of a liquid phase. After several decades of existence, the SK chart is still widely used as a practical source of the Z-factor for natural gas.5 Several empirical correlations for calculating the Z-factor have been developed. These correlations are the following: (1) Pappey6 (2) Hall-Yarborough7,8 (3) Dranshuk-Abou-Kassem9 (4) Dranchuk-Purvis-Robinson10 (5) Hankinson-Thomas-Philips11 (6) Brill and Beggs12 TaKacs13 used 180 values of the Z-factor from the SK chart to review the accuracy of the first five correlations. He concluded that the Dranchuk-Abou-Kassem correlation is the best one to fit the SK chart. When gas composition is available, the gas compressibility factor is estimated by one of the six beforementioned methods. These methods use pseudo-reduced pressure and temperature (Ppr,Tpr) to estimate the Z-factor. The pseudo-reduced properties of gas (Ppr,Tpr) are calculated via one of the mixing rules. Three mixing rules are widely used in the petroleum industry to calculate pseudo-critical properties of natural gases. These mixing rules are the following: Key,14 StewartBurkhardt-Voo3 (SBV), and SBV modified by Sutton4 (SSBV). The Key mixing rule is the expressed by the following relationship:

∑ yi Pci Tpc ) ∑ yi Tci Ppc )

∑ yi (Tc/Pc)i] + (2/3) [∑ yi (Tc/Pc)i0.5]2 K ) ∑ [yi (Tc/Pc 0.5)i]

J ) (1/3) [

(5) (6)

Tpc ) K2/J

(7)

Ppc ) Tpc/J

(8)

Using Z-factor data of high molecular weight gases Sutton4 evaluated the accuracy of the SK chart and found that the chart provides satisfactory accuracy for engineering calculations. He observed that a large deviation occurs to gases with high content of C7+. Sutton proposed modifying the SBV mixing rule to minimize this deviation. The Sutton modification of SBV (SSBV):

Fj ) (1/3)[y (Tc/Pc)]c7+ + (2/3)[y (Tc/Pc)0.5]c7+2

(9)

Ej ) 0.6081Fj + 1.1325Fj 2 - 14.004Fjyc7+ + 64.434Fjyc7+2 (10) Ek ) (Tc/Pc0.5)c7+ [0.3129yc7+ - 4.8156yc7+2 + 27.3751yc7+3] (11) J′ ) J - Ej

(12)

K′ ) K - Ek

(13)

Tpc ) K′2/J′

(14)

(1)

Ppc ) Tpc/J′

(15)

(2)

Rayes et al.15 presented a correlation to estimate the two-phase compressibility factor as a function of pseudoreduced pressure and temperature, as follows:

The pseudo-reduced pressure and temperature are expressed by the following relationship:

Ppr ) P/Ppc

(3)

Tpr ) T/Tpc

(4)

The Stewart-Burkhardt-Voo (SBV) mixing rule can be expressed as follows: (5) Ahmed, T. Hydrocarbon Phase Behavior; Gulf Publishing Co.: Houston, TX, 1989. (6) Papay, J. A. Termelestechnologiai Parameterek Valtozasa a Gazlelepk Muvelese Soran, OGIL MUSZ, Tud, Kuzl.: Budapest, 1968; pp 267-273. (7) Hall, K. R.; Yaborough, L. A New Equation of State for Z-Factor Calculations. Oil Gas J. 1973, June 18, 82-85, 90, 92. (8) Yarborough, L.; Hall, K. R. How to Solve Equation-of-State for Z-Factors; Oil Gas J. 1974, Feb. 18, 86-88. (9) Dranchuk, P. M.; Abou-Kasem, J. H. Calculation of Z Factors for Natural Gases Using Equations of State. J. Can. Pet. Technol. 1975, July-Sept., 34-36. (10) Dranchuk, P. M.; Purvis, R. A.; Robinson, D. B. Computer Calculation of Natural Gas Compressibility Factors Using the Standing and Katz correlations; Institute of Petroleum Technical Series, No. IP74-008, 1974; pp 1-13. (11) Hankinson, R. W.; Thomas, L. K.; Philips, K. A. Predict Natural Gas Properties. Hydrocarbon Process. 1969, April, 106-108. (12) Brill, J. P.; Beggs, H. D. Two-phase flow in pipes. INTERCOMP Course, The Huge, 1974. (13) Takacs, G. Comparison Made for Computer Z-Factor Calculation. Oil Gas J. 1976, Dec. 20, 64-66. (14) Kay, W. B. Density of Hydrocarbon Gases and Vapor at High Temperature and Pressure. Ind. Eng. Chem. 1936, Sept., 1014-1019.

Z ) A0 + A1(Pr) + A2(1/Tr) + A3(Pr)2 + A4(1/Tr)2 + A5(Pr/Tr) (16) For 0.7e Pr e 20.0 and 1.1e Tr e 2.1: A0 ) 2.24353, A1 ) -0.0375281, A2 ) -3.56539, A3 ) 0.000829231, A4 ) 1.53428, and A5 ) 0.131987. The pseudo-reduced pressure and temperature in eq 16 are calculated using Sutton’s method. Corredor et al.,16 and Piper et al.17 proposed a mixing rule similar to the SBV rule, eqs 5 and 6. However, they treated the non-hydrocarbons and the C7+ plus fraction differently. Their mixing rule has the following form:

J ) R0 +

∑Ri yi (Tc/Pc)i + R4∑yj (Tc/Pc)j +

∑yi (Tc/Pc)i]2 + R6(yc7+Mc7+) + R7(yc7+Mc7+)2 (17) K ) β0 + ∑βi yi (Tc/Pc0.5)i + β4∑yj (Tc/Pc0.5)j + β5[∑ yj (Tc/Pc0.5)j]2 + β6(yc7+Mc7+) + β7(yc7+Mc7+)2 R5[

(18)

where yi ∈{yH2S,yCO2,yN2} and yj ∈{yc1,yc2, ...,yc6} and R and β are constants. The difference between the Corre(15) Rayes, G. G.; Piper, L. D.; McCain, W. D., Jr.; Poston, S. W. Two Phase Compressibility Factors for Retrograde Gases. SPEFE 1992, Mar., 87-92.

Compressibility Factor for Gas Condensates

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Table 1. Physical Properties of Defined Components

component

molecular weight

critical pressure psia

H2S CO2 N2 C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6

34.08 44.01 28.01 16.04 30.07 44.01 58.12 58.12 72.15 72.15 86.18

1300.00 1071.00 493.00 667.80 707.80 616.30 529.10 550.70 490.40 488.60 436.90

critical temperature °R 672.45 547.45 227.27 343.04 549.76 665.68 734.65 765.32 828.77 845.37 913.37

dor et al. method and the Piper et al. method is that each method has different values for R and β. To calculate the pseudo-critical properties of the gas condensate, Corredor et al. and Piper et al. used the weight fraction of the C7+ rather than the critical properties. Thus, they eliminate the need to characterize the heptane plus fraction. For other mixing rules, to calculate the pseudo-critical properties of natural gas, one needs critical properties of all components of the gas. The critical properties of pure components are well-known. The properties that are used in this paper are given in Table 1. However, the critical properties of the plus fraction must be estimated from correlations. Several correlations are presented to estimate the pseudo-critical properties of the plus fraction. These correlations are the following: (1) Kesler-Lee18 (2) Standing19-Mathew-Ronald-Katz20 (3) Rowe21 (4) Winn22-Sim-Daubert23 (5) Lin-Chao24 (6) Watansiri-Owens-Starling25 (7) Riazi-Daubert26,27 (8) Pedersen28 (16) Corredor, J. H.; Piper. L. D.; McCain, W. D., Jr. Compressibility Factors for Naturally Occurring Petroleum Gases. Paper SPE 24864, presented at the SPE Annual Technical Meeting and Exhibition, Washington, DC, Oct. 4-7, 1992. (17) Piper, L. D.; McCain, W. D., Jr.; Corredor, J. H. Compressibility Factors for Naturally Occurring Petroleum Gases. SPE 26668, Houston, TX, Oct. 3-6, 1993. (18) Kesler, M. G.; Lee, B. I. Improve Prediction of Enthalpy of Fraction. Hyd. Proc. 1976, March, 153-158. (19) Standing, M. B. Volumetric and Phase Behavior of Oil Field Hydrocarbon Systems, 9th printing; Society of Petroleum Engineers of AIME: Dallas, TX, 1981. (20) Matthews, T. A.; Roland, C. H.; Katz, D. L. High-Pressure Gas Measurement. Pet. Refin. 1942, June, 58-70. (21) Rowe, A. M. Internally Consistent Correlation for Predicting Phase Composition of Heptane and Heavier Fractions. Research Report 28, GPA, Tulsa, OK, 1978. (22) Winn, F. W. Simplified monograph presentation, characterization of petroleum fraction. Pet. Refin. 1957, 36 (2), 157. (23) Sim, W. J.; Duabert, T. E. Prediction of Vapor-Liquid Equilibria of Undefined Mixtures. Ind. Eng. Chem. Process Des. Dev. 1980, 19 (3), 380-393. (24) Lin, H. M.; Chao, K. C. Correlation of Critical Properties and Acentric Factor of Hydrocarbon Derivatives. AIChE J. 1984, 30 (6) (Nov.), 153-158. (25) Watansiri, S.; Owens, V. H.; Starling, K. E. Correlation for Estimating Critical Constants, Accentric Factor, and Dipole Moment for Undefined Coal-Fluid Fractions. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 294-296. (26) Riazi, M. R.; Daubert, T. E. Simplify Property Prediction. Hydrocarbon Process. 1980, Mar., 115-116. (27) Riazi, M. R.; Daubert, T. E. Characterization Parameters for Petroleum Fractions. Ind. Eng. Chem. Res. 1987, 26 (24), 755-759. (28) Pedersen, K. S.; Fredensland, Aa.; Thomassen, P. Adv. Thermodyn. 1989, 1, 137.

Thus there are one hundred forty-four methods for estimating the Z-factor. Therefore the first objective of this work is to evaluate the different methods for estimating the Z-factor of gas condensate knowing the composition. The second objective of this work is to introduce a simple mixing rule to account for the plus fraction and the non-hydrocarbons. Natural gases frequently contain hydrogen sulfide, carbon dioxide, and nitrogen that affect the accuracy of the Z-factor calculated from the SK chart. Wichert and Aziz29 presented a method to correct the pseudo-critical properties of natural gases to the presence of these nonhydrocarbon components. The correction factor is

∈ ) 120. (A0.9 - A1.6) + 1.5 (B0.5 - B4)

(19)

where the coefficient A is the sum of the mole fraction of H2S and CO2 and B is the mole fraction of H2S in the gas mixture. The corrected pseudo-critical properties Ppc′ and Tpc′ are

Tpc′ ) Tpc - ∈

(20)

Ppc ′ ) Ppc Tpc′/[Tpc + B(1 - B) ∈]

(21)

When composition is unavailable, the gas compressibility factor is calculated via gas gravity. The gas gravity is used to estimate the pseudo-critical temperature and pressure from either Standing’s correlation or Sutton’s correlation. The former was recommended for sweet and lean gases and the latter for heavy gases. These equations areas follow: Standing19

Ppc ) 706 - 51.7γg - 11.1γg2

(22)

Tpc ) 187 + 330γg - 71.5γg2

(23)

Ppc ) 756.8 - 131.0γg - 3.6γg2

(24)

Tpc ) 169.2 + 349.5γg - 74.0γg2

(25)

Sutton4

Sutton’s gas gravity correlation based on 634 composition from 275 PVT report. Now that many compositions of gas condensates are becoming available in the industry, Sutton’s gas gravity correlation needs to be modified. Therefore, the third objective of this work is to use a larger data bank to develop a new gas gravity correlation. 2. PVT Data Compressibility factors and compositions for 1200 gas condensate samples were available to the current study. The data are obtained from constant volume depletion tests of gas condensate samples. Some of these gas condensate samples have been described in ref 30. (29) Wichert, E.; Aziz, K. Calculation of Z’s for Sour Gases. Hydrocarbon Process. 1972, 51 (5), 119-122. (30) Elsharkawy, A. M.; Foda, S. G. EOS simulation and GRNN modeling of the constant volume depletion behavior of gas condensate reservoirs. Energy Fuels 1988, 12, 353-364.

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Table 2. PVT Data Used in the Study min temperature, °F pressure, psi component mole % H2S CO2 N2 C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 C7+ Mw C7+ γ C7+ Ztp γg

94 200 0 0 0 19.37 1.94 0.61 0.17 0.25 0.12 0.08 0.12 0.22 98.0 0.72 0.41 0.61

max 327 11830 31.82 67.12 12.84 94.73 20.92 12.99 2.60 5.02 1.62 2.09 5.30 17.20 253.0 0.85 1.78 1.89

ave 240 2863 0.92 2.55 1.79 76.28 7.75 3.89 0.82 1.38 0.54 0.55 0.66 2.80 127.0 0.78 0.89 0.85

Others are collected from the literature.31-38 The data bank represents a wide range of gas condensate systems obtained worldwide. For each gas condensate composition, the data include experimentally measured gas gravity, two-phase gas deviation factor at a stage pressure, and temperature reservoir for the produced gas. It also measures the amount of the produced gas, compositional analysis of the produced gas from methane to heptane plus, and molecular weight and specific gravity of the heptane plus fraction. The data cover a wide range of reservoir conditions, temperature ranges from 94 to 327 °F and pressure ranges from 200 to 11 830 psi. A complete description of the data bank is given in Table 2. This table shows that most of the gas condensate samples are sweet gases. Few samples are highly sour, which contains as much as 31.82% hydrogen sulfide, 67.12% carbon dioxide. The data cover a wide range of gas condensates from very lean gases to very rich gases. One of the gas condensate samples collected from the literature37 has an exceptionally high molecular weight of the heptane plus (Mw c7+ ) 253). 3. Results and Discussion The data banks of constant volume depletion of gas condensates were used to check the accuracy of the various methods of calculating gas compressibility factors. These calculations consist of evaluating three (31) Coats, K. H.; Smart, G. T. Application of a regression-based EOS PVT program to laboratory data. SPE Res. Eng. 1986, May, 277299. (32) Drohm, J. K.; Goldthorpe, W. H.; Trengove, R. Enhancing the evaluation of the PVT data. OSEA 88174, presented at the 7th Offshore South Asia Conference, Singapore, 2-5 Feb., 1988. (33) Drohm, J. K.; Trengove, R.; Goldthorpe, W. H. On the quality of data from standard gas condensate PVT experiments. Paper SPE 17768, presented at the SPE Gas Technol. Symp. held in Dallas, TX, June 13-15, 1988. (34) Firoozabadi, A.; Hekim, Y.; Katz, D. L. Reservoir depletion calculation for gas condensate using extended analysis in the Peng Robinson Equation of State. Can. J. Chem. Eng. 1978, 56 (Oct.), 610615. (35) Whitson, C. H. Evaluating constant-volume depletion data. JPT 1983, March, 610-620. (36) McCain, W. D., Jr. The Properties of Petroleum Fluids, 2nd ed.; PennWell Books: Tulsa, 1990. (37) Al-Mahroos, F. M.; Tjoa, G. H. Analysis and phase behavior of Khuff gas condensate system in Bahrain field. SPE 15766, presented at the MEOS, Bahrain, Mar. 7-10, 1987. (38) Keyon, D. E.; Behie, G. A. Third SPE comparative solution project, gas cycling of retrograde condensate reservoirs. JPT 1987, August, 981-997.

Figure 1. Error distribution in gas compressibility factor as a function of heptane content. Key’s mixing rule.

Figure 2. Error distribution in gas compressibility factor as a function of heptane content. SBV mixing rule.

different mixing rules, eight correlations of characterizing the heptane plus fraction, and six correlations for calculating the gas compressibility factor. 3.1. Evaluation of the Existing Methods. Evaluation of the one hundred forty-four possible methods for calculations of gas compressibility factor was accomplished as follows: (1) The critical pressure and critical temperature of the heptane plus content of each gas condensate composition were calculated from one of the above-mentioned eight correlations for characterizing the plus fraction. (2) For each gas condensate composition, the pseudocritical pressure and pseudo-critical temperature of the gas condensate is calculated via one of the three mixing rules discussed in this paper. (3) The pseudo-critical pressure and pseudo-critical temperature of the gas were corrected for the presence of non-hydrocarbons using the Wichert and Aziz method. (4) The corrected values of the pseudo-critical pressure and pseudo-critical temperature of the gas were used to calculate the pseudo-reduced properties. (5) The pseudo-reduced pressure and pseudo-reduced temperature were used to compute the gas compressibility factor for each gas from one of the beforementioned six Z-factor correlations. Tables 3 through 5 show a summary of the calculations for the one hundred forty-four possible methods for calculating the gas compressibility factor of gas condensates. These tables indicate that SSBV mixing rule is more accurate than SBV and Key’s mixing rules, respectively. Figures 1 through 3 show the accuracy of the three mixing rulessKay’s, SBV, and SSBV, respectively. In these plots, the Lin-Chao correlation was used



Figure 3. Error distribution in gas compressibility factor as a function of heptane content. Modified-SBV.

Figure 6. Accuracy of BB method in calculating gas compressibility factor.

Figure 4. Accuracy of HTP method in calculating gas compressibility factor.

Figure 7. Accuracy of DPR method in calculating gas compressibility factor.

Figure 5. Accuracy of Papay’s method in calculating gas compressibility factor.

to characterize the heptane plus fraction and DranchukAbou-Kassem was used to calculate the compressibility factor. As shown in Figure 1, Key’s mixing rule highly underestimates the compressibility factor and has a general trend of increasing the error inthe Z-factor as the heptane plus content is increased. SBV’s mixing rule (Figure 2) also highly underestimates the compressibility factor. However, the SBV mixing rule (Figure 2) and the SSBV mixing rule (Figure 3) show no correlation between the heptane plus content of the gas condensate and error in compressibility factor. Tables 3 through 5 report average relative error (Er), average absolute error (Ea), standard deviation (Sd), and coef-

Figure 8. Accuracy of HY method in calculating gas compressibility factor.

ficient of correlations (R) for the different methods used in calculating the gas compressibility factor. Three of the six methods of calculating Z-factors performed very well. These methods use iterative solution to calculate the gas compressibility factor. They are the following: Dranchuk-Abou-Kassem, Dranchuk-Purvis-Robinson, and Hall-Yarborough, respectively. The other three methodssHankinson-Thomas-Philips, Beggs, and

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Figures 7 through 9, respectively. These figures show that there is no relationship between the errors in the Z-factor and the pressures. The errors in the Z-factor have also been checked against temperature for all six methods. No correlation was found between the errors and the temperature. Accuracy of the eight methods of characterizing the heptane plus fraction is shown in Tables 3 through 5. Tables 3 through 5 illustrate that the methods proposed by Lin-Chao, Kesler-Lee, and Riazi-Daubert, respectively, have the best accuracy. For further investigation of each methods, the accuracy of the before-mentioned methods in characterizing the plus fraction for all gases (C7+ > 1%) as well as rich gas condensates (C7+ > 5%) is considered. The result of the investigation is reported in Table 6. The first part of Table 6 shows that LinChao correlations have the best accuracy for all the gases. However, the second part of Table 6 shows that Kesler-Lee is superior for rich gas condensates. The Riazi-Daubert correlation showed a large deviation for rich gas condensates. In conclusion, out of the 144 possible methods, it was found that Lin-Chao24 correlations, the SBV mixing rule as modified by Sutton,4 and Dranchuk-Abou-Kassem9 resulted in the lowest errors and standard deviation. However, the methods underestimate the gas compressibility factors. Rayes et al. and Corredor et al. discussed the accuracy of Z-factor calculations. They found that the calculated gas compressibility factor using their methods do not match experimental values at low pressures. They attributed the large errors to equilibrium conditions at the low pressures (less than 700 psia). Figure 10 shows relative error distribution at various stages in pressure depletion for gas condensates. For every gas condensate, the pressure at any depletion stage has been divided by the dew point pressure resulting in the dimensionless pressure. This figure shows that the greatest scatter and maximum error occur at the dew-point pressure. Thus,

Figure 9. Accuracy of DAK method in calculating gas compressibility factor.

Papaysgave large errors and standard deviations. The Hankinson-Thomas-Philips correlation has an average relative error and standard deviation in the order of 6%. The method calculates the gas compressibility factor by iterative solution. Hankinson-Thomas-Philips proposed two sets of constants to calculate the gas compressibility factor. The first set is for Pr from 0.4 to 5.0 and the second set is for Pr from 5 to 15. Figure 4 shows that the method is not accurate for calculating the compressibility factor when the reduced pressure (Pr) ranges from 5 to 15. This leads one to conclude that the second set of constants is the reason for the large deviations. Papay’s correlation also shows high errors and standard deviations (51%, and 56%, respectively) regardless of the mixing rules and the method of characterizing the plus fraction. Figure 5 shows that Papay’s correlation has a tendency of larger error at higher pressure. Brill and Beggs correlation (Figure 6) has a tendency of having a larger error in the Z-factor at higher pressure. The errors in the calculated values of the Z-factor for Dranchuk-Purvis-Robinson, Hall-Yarborough, and Dranchuk-Abou-Kassem at various pressures are shown in

Table 3. Accuracy of Calculating the Z-Factor for Gas Condensates Using the Key Mixing Rule method of calculating the Z-factor Papay (1968)

Hall-Yarborough (1973)

characterizing C7+

Er

Ea

Sd

Riazi-Daubert (1980) Riazi-Daubert (1987) Lin-Chao (1984) Kesler-Lee (1976) Winn-Sim-Daubert (1957) Rowe (1978) Standing (1977) Pedersen et al. (1989)

-51.69 -51.69 -51.71 -51.69 -51.15 51.70 -51.76 -50.91

51.74 51.71 51.72 51.71 51.15 51.74 51.77 51.20

56.58 56.53 56.53 56.53 56.10 56.56 56.60 56.75

R

Dranchuk-Abou-Kassem (1975)

Er

Ea

Sd

R

Er

Ea

Sd

R

-1.39 -1.47 -1.41 -1.51 -0.42 -1.42 -1.52 -0.96

1.74 1.79 1.74 1.82 1.83 1.75 1.83 8.77

2.37 2.14 2.36 2.45 3.01 2.37 2.46 12.64

97.26 97.28 97.42 97.23 94.59 97.38 97.16

-1.36 -1.45 -1.37 -1.47 -0.37 -1.39 -1.485 -0.78

1.75 1.79 1.74 1.81 1.89 1.75 1.82 10.54

2.39 2.44 2.38 2.46 3.06 2.39 2.48 14.89

97.26 97.21 97.23 97.15 94.50 97.30 97.08

method of calculating the Z-factor Dranchuk-Purvis-Robinson (1974)

Hankinson-Thomas-Phillips (1969)

characterizing C7+

Er

Ea

Sd

R

Er

Ea

Sd


-1.40 -1.38 -1.28 -1.38 -0.24 -1.29 -1.39 -0.68

1.68 1.76 1.69 1.76 1.91 1.70 1.76 11.23

2.37 2.44 3.36 2.44 3.05 2.38 2.46 15.97

97.30 97.18 97.37 97.18 94.63 96.72 97.12

-6.50 -10.49 -9.01 -6.53 -3.76 -6.32 -6.40 -6.24

6.78 14.96 9.31 6.82 4.90 6.62 6.70 6.55

10.74 26.25 70 11.72 8.67 10.51 10.60 10.40

R

53.29

Brill-Beggs (1974) Er

Ea

Sd

R

-2.11 5.40 -13.14 -2.10 -1.01 -2.03 -2.12 -1.63

2.29 5.5 13.15 2.38 2.03 2.13 2.40 10.61

2.90 6.87 15.13 2.98 3.3 2.91 3.00 15.50

95.89 82.92 95.70 93.53 95.91 95.63



Table 4. Accuracy of Calculating the Z-Factor for Gas Condensates Using the Stewer-Burkhardt-Voo Mixing Rule method of calculating the Z-factor Papay (1968)


characterizing C7+

Er

Ea

Sd


-50.91 -50.94 -50.87 -51.01 -51.18 -51.01 -51.23 -51.91

51.15 51.18 51.15 51.22 51.19 51.22 51.36 52.62

55.96 55.99 55.59 56.03 56.21 56.05 56.16 56.90

R

Dranchuk- Abou-Kassem (1975)

Er

Ea

Sd

R

Er

Ea

Sd

R

-2.21 -2.37 -2.14 -2.48 -0.93 -2.21 -2.53 -2.23

2.42 2.58 2.40 2.67 1.74 2.43 2.70 9.05

2.75 2.93 2.75 3.02 2.83 2.77 3.05 14.77

96.98 96.58 97.03 96.33 95.05 97.00 96.21

-2.17 -2.33 -2.10 -2.43 -0.87 -2.17 -2.48 -2.24

2.38 2.54 2.36 2.67 1.77 2.39 2.66 9.72

2.71 2.88 2.70 2.97 2.86 2.73 3.01 15.53

97.03 96.64 97.11 96.39 94.97 97.06 96.26



characterizing C7+

Er

Ea

Sd

R

Er

Ea

Sd


-2.10 -2.36 -2.04 -2.37 -0.75 -2.10 -2.41 -2.06

2.31 2.56 2.28 2.56 1.77 -2.32 2.60 11.26

2.65 2.91 2.63 2.92 2.83 2.66 2.95 17.23

97.15 96.52 97.25 96.53 95.14 97.20 96.32

-8.04 -12.39 -17.58 -7.91 -4.34 7.91 -8.02 -7.74

8.19 16.11 17.73 8.05 5.01 8.05 8.20 7.90

12.30 27.16 12.58 8.00 12.08 12.28 11.85

R

60.19

Brill- Beggs (1974) Er

Ea

Sd

R

-2.69 4.82 -13.44 -2.93 -1.58 -2.68 -2.90 -2.60

2.89 4.88 13.52 3.11 2.11 2.89 3.14 12.38

3.3 6.11 15.70 3.56 3.29 3.32 3.77 18.47

95.36 86.63 94.64 93.57 95.44 93.47

Table 5. Accuracy of Calculating the Z-Factor for Gas Condensates Using the Modified Stewer-Burkhardt-Voo Mixing Rule method of calculating the Z-factor Papay (1968)


characterizing C7+

Er

Ea

Sd


-50.88 -50.91 -50.85 -50.96 -51.14 -51.01 -51.13 -51.29

51.05 51.07 51.05 51.10 51.15 51.22 51.22 52.30

55.87 55.89 55.85 55.93 56.21 56.00 56.04 57.45

R

Dranchuk- Abou-Kassem (1975)

Er

Ea

Sd

R

Er

Ea

Sd

R

-0.99 -1.13 -0.94 -1.22 -0.571 -0.99 -1.25 1.14

1.46 1.53 1.51 1.56 1.67 1.51 1.57 11.98

1.86 1.91 1.92 1.94 2.60 1.91 1.98 18.87

98.62 98.55 98.51 98.48 95.58 98.55 98.40

-1.43 -1.11 -0.92 -1.19 -0.52 -0.97 -1.22 -1.37

1.67 1.52 1.48 1.54 1.73 1.17 1.55 10.92

2.02 1.89 1.89 1.93 2.70 1.88 1.94 17.98

98.28 98.55 98.55 98.49 95.49 98.58 98.41



Brill- Beggs (1974)

characterizing C7+

Er

Ea

Sd

R

Er

Ea

Sd

R

Er

Ea

Sd

R


-0.88 -1.10 -0.83 -1.11 -0.39 0.88 -1.13 -1.47

1.38 1.47 1.42 1.48 1.75 1.45 1.49 10.26

1.81 1.88 1.86 1.89 2.71 1.85 1.90 17.57

98.18 98.56 98.62 98.56 95.92 98.64 98.42

-6.57 10.50 -6.77 -6.97 -3.90 0.33 6.648 -6.633

6.78 15.04 6.99 6.89 4.76 6.55 6.70 6.45

10.54 26.31 17.56 12.12 7.53 10.16 10.40 10.01

7.57

-1.59 5.53 -12.56 -1.79 -1.22 -1.58 -1.83 -2.24

1.97 5.57 12.77 2.10 1.92 1.99 2.12 10.93

2.37 6.98 15.14 2.52 3.02 2.41 2.54 18.14

97.36 81.64

Table 6. Accuracy of Methods of Characterizing the Plus Fractions method

Er%

Ea%

Sd%

R%

Lin-Choa Kesler-Lee Raizi-Daubert #1

C7+ > 1% -0.92 1.44 -1.24 1.50 -1.43 1.66

1.84 1.88 2.02

98.71 98.64 98.28

Lin-Choa Kesler-Lee Raizi-Daubert #1

C7+ > 5% 0.42 1.53 -0.86 1.43 0.73 15.16

2.22 2.06 22.10

98.97 99.06

gas condensate data do not support the conclusion made by Rayes et al. and Corredor et al. about the accuracy of measurements at low pressure.

18.02 65.84 22.70 17.88 26.00

97.32 95.53 97.61 97.24

3.2. Proposed Mixing Rule. The Standing-Katz chart was developed for sweet and dry gases. Therefore, various attempts were made to extend the validity of the chart to be used for sour gases and gas condensates. To account for the presence of sour gases, Wichert and Aziz proposed corrections for the presence of nonhydrocarbon components in natural gas. Similarly, Sutton introduced a correction factor to the SBV mixing rule to account for the presence of heptane plus. In this paper, a different approach is introduced to account for the presence of C7+ and non-hydrocarbon components, simultaneously. The SBV mixing rule was a starting point for the proposed mixing rule. First, the measured gas compressibility factors and Dranchuk-Abou-Kassem

814


Elsharkawy et al. Table 7. Accuracy of Recent Methods for Calculating the Z-Factor for Gas Condensates method

Er

Ea

Sd

R

Rayes et al. (1992) Corredore et al. (1992) Piper et al. (1993) current study mixing rule #1 mixing rule #2

-8.57 -0.09 -0.06

9.21 1.13 1.17

13.27 1.81 1.84

71.51 98.80 98.76

-0.10 -0.44

1.37 1.33

2.01 2.12

98.28 98.12

analysis resulted in the following equations: Mixing Rule #1.

∑yi (Tc/Pc)i]C1-C6 + [a0+ a1 (yTc/Pc) C7+] K ) [∑yi (Tc/Pc0.5)i]C1-C6 + J)[

(26)

[b0 + b1 (yTc/Pc0.5)C7+] (27)

Figure 10. Error % in calculated gas compressibility factor at various stages of depletion.

where a0 ) -0.0416977, a1 ) 0.8648146, b0 ) -0.8816945, and b1 ) 1.0368581. Equations 26 and 27 are much simpler than the other mixing rules (eqs 5 and 6), and account for the presence of the heptane plus in a much easier way than Sutton’s method (eqs 9 through 13). Equations 26 and 27 are also much simpler than the ones proposed by Corredore et al. and Piper et al. (eqs 17 and 18) as it has fewer terms. For gas condensates that contains some non-hydrocarbon components, we propose the following equations: Mixing Rule #2.

∑

J ) [ yi(Tc/Pc)i]C1-C6 + {[a0 + a1(yTc/Pc)]C7+ + [a2(yTc/Pc)]N2 + [a3(yTc/Pc)]CO2 + [a4(yTc/Pc)]H2S} (28) Figure 11. Relationship between J and properties of C7+.

∑yi(Tc/Pc0.5)i]C1-C6 + {[b0 + b1(yTc/Pc0.5)]C7+ +

K)[

[b2(yTc/Pc0.5)]N2 + [b3(yTc/Pc0.5)]CO2 +

[b4(yTc/Pc0.5)]H2S} (29)

Figure 12. Relationship between K and properties of C7+.

correlation were used to calculate the inferred pseudocritical pressure and inferred pseudo-critical pressure temperature for gas condensates by multiple regression. These inferred values closely fit the Standing-Katz compressibility-factor chart. Second, the inferred values of J and K were calculated using eqs 7 and 8. These inferred values were found to be highly dependent on the amount of C7+ and non-hydrocarbon components and their critical properties. Figures 11 and 12 show that the inferred values {Jinf} and {Kinf} are strongly correlated to the presence of the heptane plus. Most of the gas compositions available in this study represent sweet samples. For these samples, multiple regression

where a0 ) -0.040279933, a1 ) 0.881709332, a2 ) 0.800591625, a3 ) 1.037850321, and a4 ) 1.059063178; and where b0 ) -0.776423332, b1 ) 1.030721752, b2 ) 0.734009058, b3 ) 0.909963446, and b4 ) 0.888959152. Mixing rule #2 (eqs 28 and 29) uses fewer coefficients than the mixing rule proposed by Corredor et al. and Piper et al., eqs 17 and 18. This mixing rule is also consistent as it uses mole fraction and pseudo-critical properties of the heptane plus fraction rather than the weight of the plus fraction. The second mixing rule has the advantage of combining the effect of heptane plus and non-hydrocarbon components. Thus the second mixing rule eliminates the corrections for heptane plus, eqs 9-15, and non-hydrocarbons, eqs 19-21. Table 7 shows the accuracy of the proposed mixing rules as well as the other methods proposed by Rayes et al., Piper et al., and Corredor et al. It is clear from Table 7 that Rayes method has the highest errors and standard deviation. Figure 13 illustrates that the Rayes et al. method has a tendency of giving a large error in the calculated value of gas compressibility at low pressure. The new mixing rules as well as the ones presented by the Corredore and Piper methods have similar accuracy. However, the new ones are simpler and consistent. Figures 14 and 15 show cross plots of the calculated



Figure 13. Error distribution for Rayes et al. method. Figure 15. Cross plot of experimental and calculated Z-factor using mixing rule #2.

Figure 14. Cross plot of experimental and calculated Z-factor using mixing rule #1.

Z-factor using the proposed mixing rules. It is clear from these figures that the values are evenly distributed around unit slope line. In conclusion, it was found that Kay’s mixing rule is suitable for dry and sweet gases of low molecular weight. However, the SSBV mixing rule is suitable for high molecular weight gases with low content of heptane plus. The new mixing rule is much more suitable than the others as it accounts for the presence of much larger amounts of the heptane plus fraction and no hydrocarbons, simultaneously. 3.3. Gas Gravity Correlation. The inferred values of pseudo-critical pressure and pseudo-critical temperature were correlated to gas condensate gravity. Multiple regression resulted in the following equations:

Ppc ) 787.06 - 147.34γg - 7.916γg2

(30)

Tpc ) 149.18 + 358.14γg - 66.976γg2

(31)

Figures 16 and 17 show a comparison between the proposed gas gravity correlation, eqs 30 and 31, and the Standing and Sutton correlations. These figures show that the proposed gas gravity correlation coincides with the Sutton correlation in the gas gravity range from 1.2 to 1.4. To study the accuracy of the proposed gas gravity correlation and compared it to Standing’s correlation and Sutton’s correlation, we have divided the gas gravity into five different ranges from 0.6 to 1.4. Table 8 shows that Sutton’s correlation has the best accuracy in the gas gravity range from 0.6 to 0.8 and from 1.0 to 1.20 followed by the proposed one in this study and Standing, respectively. However, the current gas gravity correla-

Figure 16. Pseudo-critical pressure correlations.

Figure 17. Pseudo-critical temperature correlations.

tion has smaller errors and standard deviation than the Standing and Sutton correlations in the gas gravity range from 0.8 to 1.0 and from 1.2 to 1.4. Overall, the new gas gravity correlations, Sutton correlation, and Standing correlations have an average absolute error (Er) of 1.58%, 1.69%, and 2.07%, respectively. Thus the proposed gas gravity correlation has little improvement over Sutton’s correlation. Figure 18 shows the cumulative error distribution for the proposed gas gravity correlation in this study. The new gas gravity correlation has a frequency of 80% to estimate gas compressibility with less than 2% error and 98% with less than 4% error.

816


Elsharkawy et al.

Table 8. Accuracy of Gas Condensate Gravity Equations current study

Sutton

Standing

gas gravity

Er

Ea

Sd

R

Er

Ea

Sd

R

Er

Ea

Sd

R

0.6-0.8 0.8-1.0 1.0-1.2 1.2-1.4 >1.4 overall

0.49 -0.13 -0.73 -1.11 1.99 0.11

1.62 1.80 2.02 2.27 2.21 1.58

2.34 2.33 2.61 3.39 2.76 2.26

95.97 97.41 99.02 98.77 86.85 97.31

-0.05 -1.08 -0.66 2.14 1.38 -0.53

1.20 2.05 1.77 3.05 2.32 1.69

1.64 3.01 2.55 4.15 3.37 2.52

98.93 97.00 99.82 77.00 84.95 97.60

-1.15 -1.29 -2.18 -5.10 -0.85 -1.39

1.74 1.87 2.97 5.36 6.24 2.07

2.19 2.45 3.46 6.06 7.16 2.74

96.04 96.97 97.93 95.30 81.68 96.69

over-all accuracy of at least 98%. This study also presents a new gas gravity correlation. The new correlation is used to estimate pseudo-critical properties of gas condensate when the composition is unknown. 5. Nomenclature

Figure 18. Error distribution for the new gas gravity correlation.

4. Conclusions A large data bank of gas condensate reservoirs has been used to check the accuracy of various methods for calculating gas compressibility factors. These calculations consist of evaluating three different mixing rules, eight correlations of characterizing the heptane plus fraction, and six correlations for calculating the gas compressibility factor. Evaluation of the 144 possible methods indicates that Lin-Chao24 correlations, the SBV mixing rule as modified by Sutton,4 and the DranchukAbou-Kassem9 method resulted in the lowest errors and standard deviation. However, the methods underestimate the gas compressibility factors. Using the large data bank of gas composition, this study presents new mixing rules to calculate pseudocritical properties of the gas condensate. The new mixing rules are much simpler and consistent than the previously published mixing rules. The new mixing rules account for the presence of the heptane plus in a much easier way than the Sutton, Piper et al., and Rayes et al. methods. The proposed mixing rules, when used with Dranchuk-Abou-Kassem correlation, have

R ) Constant for Corredore et al. β ) Constant for Corredore et al. ∈) Wichert and Aziz pseudo-critical temperature adjustment parameter, °R γg ) Gas specific gravity, (air ) 1) A ) Mole fraction (CO2 + H2S) B ) Mole fraction H2S Ea ) Average absolute error EJ ) Sutton SBV parameter, °R/psia EK ) Sutton SBV parameter, °R/psia0.5 Er ) Average relative error FJ ) Sutton adjustment parameter J ) SBV parameter, °R/psia J′ ) Sutton parameter, °R/psia Jinf ) Inferred value of J parameter, °R/psia K ) SBV parameter, °R/psia0.5 K′ ) Sutton parameter, °R/psia0.5 Kinf ) Inferred value of K parameter, °R/psia0.5 M ) Molecular weight, lb-mol MC7+ ) Molar mass of heptane plus fraction, lb-mol P ) Pressure, psia Pc ) Critical pressure, psia Ppc ) Pseudo-critical pressure, psia Ppr ) Pseudo-reduced pressure R ) Correlation coefficient Sd ) Standard deviation T ) Temperature, °R Tb ) Normal boiling point temperature, °R Tc ) Critical temperature, °R Tpc ) Pseudo-critical temperature, °R Tpr ) Pseudo-reduced temperature yC7+ ) Mole fraction of heptane plus fraction yi ) Mole fraction of component, “i” yi ) Mole fraction of the ith component Z ) Gas compressibility factor EF000216M

Compressibility Factor for Gas Condensates - American Chemical

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