Improved K-values correlation for UAE crude oil components at low pressures using PVT laboratory data

Fuel 80 (2001) 117±124

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Improved K-values correlation for UAE crude oil components at low pressures using PVT laboratory data R.A. Almehaideb*, M.A. Abdulkarim, A.S. Al-Khanbashi Department of Chemical and Petroleum Engineering, Faculty of Engineering, UAE University, P.O. Box 17555, Al-Ain, United Arab Emirates Received 25 September 1999; received in revised form 22 March 2000; accepted 27 March 2000

Abstract Several techniques are available in the literature to estimate the K-values. In this paper, results of PVT analysis for 22 crude oil samples from different reservoirs in UAE are used. Sixty-eight single-stage ¯ash laboratory experiments were conducted for these samples. Material balance techniques were used to extract the K-values of crude oil components. These K-values were then correlated and compared with values obtained from published correlations. Comparisons show that current correlations, while they generally give good results for light hydrocarbons in addition to carbon dioxide and hydrogen sul®de, give widely different results for nitrogen and the heptane-plus pseudocomponent. Average absolute deviations in excess of 1000% were observed for nitrogen and in excess of 500% were observed for heptaneplus when current methods were used. The proposed new correlation improves signi®cantly the average absolute deviation for both the heptane-plus fraction and for nitrogen, in addition to improving relatively the average absolute deviation for the C1 ±C6 hydrocarbons, H2S, and CO2. The average absolute deviation for all components was reduced to 28.6% in the new correlation compared to 240% for the Standing correlation and 156.8% for the Wilson correlation. As a test for reliability of the new correlation, bubble point pressures were calculated for 10 samples. The average absolute error for the proposed correlation was 5.2% compared with 6.9% for the Standing correlation, 16.1% for the Wilson correlation, and 7.3% for the Peng±Robinson equation of state. q 2000 Elsevier Science Ltd. All rights reserved. Keywords: K-value correlation; Bubble point pressure; Crude oil

1. Introduction and literature review Methods for accurate prediction of equilibrium ratios or K-values continue to be of large interest to both the chemical and petroleum industries. Design for different separation facilities for crude oil±natural gas mixtures and gas-condensates in addition to pipelines depends to a large part on the availability of accurate K-values to predict the phase behavior of hydrocarbon mixtures under various operating conditions. Several approaches are available to the practicing engineer to obtain K-values. These approaches can be classi®ed as either relying on experimental measurements, or using general predictive tools such as correlations or equations of state (EOS) to predict the phase behavior. K-values are de®ned as the ratio of the mole fraction of each component i in the vapor phase yi over its mole fraction in the liquid phase xi in a multi-component mixture Ki ; yi =xi

…1†

To evaluate K-values for ideal mixtures one can use the * Corresponding author. E-mail address: [email protected] (R.A. Almehaideb).

combination of Raoult's and Dalton's laws to obtain Kvalues as the ratio of vapor pressure p vi of a component, which is a function of temperature, to the total pressure p of the system: yi =xi ˆ pvi =p

…2†

For real mixtures, K-values are dependent on pressure, temperature, and composition of the ¯uid mixture. At low pressures, however, the effect of composition seems to be weak [1], and K-value correlations are typically expressed in terms of pressure and temperature only. Several investigators have proposed empirical methods for obtaining good estimates of K-values for crude oil mixtures. Earlier methods relied on nomographs and other graphical methods to obtain estimates of K-values. More recent methods have expressed these correlations in terms of mathematical expressions to facilitate the computation of K-values, especially with current day computing capability. Several of these correlations to predict the K-values were reviewed in a textbook by Ahmed [2]. In this paper, we will focus on only two widely used correlations that are applicable for low pressure K-values: the Standing [1] correlation, and the Wilson [3] correlation. Also, since our proposed

0016-2361/01/$ - see front matter q 2000 Elsevier Science Ltd. All rights reserved. PII: S 0016-236 1(00)00064-8

118

R.A. Almehaideb et al. / Fuel 80 (2001) 117±124

Nomenclature a,c: correlation parameters oil formation volume factor, bbl/STB Bo: bi : slope of log of vapor pressure curve vs. 1/T, ˆ log(pc/14.7)/(1/TB 2 1/T) for each component i characterization factor for component i Fi : correlated K-value Kcorr: extracted K-value Kexp: equilibrium ratio for component i Ki: Mor: molecular weight of reservoir oil, lbm/lb mol MSTO: molecular weight of stock-tank oil, lbm/lb mol n: average single carbon number mole fraction of gas separated in separator stage j, based on 1 lb mol of feed ng j : ng;ST : mole fraction of gas separated in stock-tank, based on 1 lb mol of feed mole fraction of liquid separated in separator stage j, based on 1 lb mol of feed nLj : nL;ST : mole fraction of liquid separated in stock-tank, based on 1 lb mol of feed bubble point pressure, psia Pb: p: pressure, psia critical pressure for component i, psia pci: separator pressure, psia psep: vapor pressure for component i, psia p vi: %AD: percent average deviation %AAD: percent absolute average deviation gas to oil ratio in stock-tank, SCF/STB RST: gas to oil ratio in separator stage j where j ˆ 1, 2,¼, SCF/STB Rspj: T: Temperature, 8R normal boiling point for component i, 8R TBi: critical temperature for component i, 8R Tci: separator temperature, 8F in Eq. (11) Tsep: xi: mole fraction of component i in liquid phase mole fraction of component i in gas (vapor) phase yi: mole fraction of component i in feed zi : Greek symbols s: standard deviation r STO: density of stock-tank oil, lbm/ft 3 r or: density of reservoir oil, lbm/ft 3 v i: acentric factor for component i Subscripts and superscripts B: boiling c: critical corr: correlated exp: experimental g: gas i: component j: separator stage L: liquid o: oil r: reservoir ST: stock-tank sp: separator v: vapor

R.A. Almehaideb et al. / Fuel 80 (2001) 117±124 Table 1 Parameters for Standing correlation Component

b cycle (8R)

TB (8R)

N2 H2 S CO2 C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6s

470 652 1138 300 1145 1799 2037 2153 2368 2480 2738

109 194 331 94 303 416 471 491 542 557 610

correlation is based on Standing's, we will brie¯y discuss next the main equations for this last correlation. In 1979, Standing [1] presented a simple set of equations to calculate the K-values of crude oil±natural gas mixtures. He correlated the K-value data of Katz and Hachmuth [4] for the Wilcox sand of the Oklahoma City Field. The form of the Standing correlation is a modi®ed version of the form proposed by Hoffman et al. [5] of a linear relationship between the log(Kp) versus the component characterization factor, F, de®ned for each component as: F ˆ b…1=TB 2 1=T†

…3†

where b is the slope of the log of the vapor pressure vs. 1/T; TB, the normal boiling point in 8R; and T, the temperature in 8R. Standing's correlation was expressed as: log…Kp† ˆ a 1 c £ F

…4†

where both a and c are polynomial functions of pressure. Standing extended the correlation to non-hydrocarbon compounds normally present in crude oils, namely, CO2, H2S, and N2 using data of other crude oils. He also provided a correlation for the b and TB values for heptanes and heavier hydrocarbons based on the equivalent carbon number for this fraction, which he correlated with the separation temperature and pressure. Standing adjusted the values of TB and b for C1, C2, N2, Table 2 Fluid properties for sample 30 Data Bo ˆ 2.19 bbl/STB Rsp ˆ 1063 SCF/STB r or ˆ 33.875 lbm/ft 3 Calculations Mor ˆ Sz I £ MWi ˆ 62:75 lbm/lb mol ng ˆ

Rsp Mor ˆ 0:42206 2130:3Bo ror

119

and H2S to better ®t the data. Values for these parameters for all components are reproduced in Table 1. He reported a 2.1% average absolute deviation between the correlation and the experimental data compared to 8.5% if one is to use the original values proposed for TB and b by Hoffman et al. [5] without adjustment. In 1982, Glaso and Whitson [6] tested the general applicability of Standing's K-value correlation using results of experimental PVT tests on 15 samples obtained from parts of the USAÐother than OklahomaÐand other parts of the world, including four samples from the North Sea and one from the Middle East. They evaluated the calculated and experimental values of four PVT properties derived from ¯ash separation tests, namely the total gas to oil ratio (GOR), average gas gravity, stock-tank oil gravity, and formation volume factor. They reported that Standing's correlation gave acceptable predictions for these four PVT properties when compared to experimental values reported for the 15 samples. In this paper, a more thorough investigation of the viability of the Wilson and Standing correlations is tested for UAE crude oil samples by comparing the above correlation results with extracted K-values obtained directly through material balance techniques on separator test results. As a result of the current work, an improved correlation for UAE crudes was formulated using the multi-variable regression techniques.

2. Extracted K-values evaluation Extracted K-values can be obtained from three types of PVT tests, namely differential liberation tests, constant volume depletion tests, and separation tests, provided that measurements of the composition of the gas exiting the PVT cell are performed at each pressure stage. The ®rst two tests are normally carried out at high pressures, approximating reservoir conditions for crude oil and gas-condensate [7,8] Table 3 Separator single-stage data and calculation of K-values for sample 30 at 300 psig and 708F Data

Calculations

Component

z

MW

y

x (Eq. (12))

K (Eq. (1))

H2 S CO2 N2 C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C 6s C71

0.2729 0.0201 0.0008 0.2636 0.0616 0.0477 0.0127 0.0292 0.0188 0.0216 0.0333 0.2177

34.08 44.01 28.01 16.04 30.07 44.1 58.12 58.12 72.15 72.15 86.18 166

0.2877 0.034 0.0016 0.5433 0.0833 0.0328 0.0049 0.007 0.0021 0.0017 0.001 0.0006

0.262092 0.009949 0.000216 0.05934 0.045753 0.058581 0.018396 0.045412 0.030996 0.036133 0.056888 0.376244

1.097707 3.417402 7.415157 9.155701 1.820651 0.559907 0.266359 0.154143 0.067751 0.047049 0.017578 0.001595

120

R.A. Almehaideb et al. / Fuel 80 (2001) 117±124

where n g j is the mole fraction of gas separated in stage j, based on 1 lb mol of feed; n Lj ; the mole fraction of liquid separated in stage j, based on 1 lb mol of feed; r STO, the density of stock-tank oil, lbm/ft 3; MSTO, the molecular weight of stock-tank oil; and the oil formation volume factor, Bo, is expressed as [9]:

Table 4 Range of mole fractions Component

% mole fraction range

N2 CO2 H2 S C1 C2 C3 I-C4 n-C4 I-C5 n-C5 C 6s C71

0.15±0.43 0.39±6.1 0.20±3.68 12.59±53.2 6.05±10.23 4.27±8.83 0.70±1.84 2.59±5.27 0.96±4.98 1.75±4.09 1.14±4.53 21.77±62.18

Mor rSTO ror MSTO n L1 nL2

Bo ˆ

…8†

where r or is the density of reservoir oil, in lbm/ft 3. Dividing Eqs. (5)±(7) by Eq. (8), one can obtain a simple equation to evaluate the gas molar fraction coming from the 1st and 2nd stages and the stock-tank as:

systems, respectively, while separation tests are carried out at pressures approaching surface operating pressures. Data for this study, therefore, were extracted from separator tests using material balance techniques. Relevant equations are reported in a number of textbooks. Here, the equations reported by McCain [9] will be used as a starting point. The equations are normally used to estimate PVT parameters, such as the API of the crude, the GOR, and the oil formation volume factor. These same equations were used in this work in the reverse manner, i.e. to obtain extracted Kvalues using ¯ash separation equations and experimental measurements of the above PVT parameters. For multi-stage separation, the GOR coming from the 1st and 2nd stage separators, and the stock-tank, in standard cubic feet per stock-tank barrel (SCF/STB), can be expressed as [9]: Rsp1 ˆ

2130:3n g1 rSTO nL1 n L2 ¼n L;ST MSTO

…5†

Rsp2 ˆ

2130:3n g2 rSTO nL2 n L3 ¼n L;ST MSTO

…6†

RST ˆ

2130:3n g;ST rSTO nL;ST MSTO

…7†

n g1 ˆ

Rsp1 Mor 2130:3Bo ror

…9†

n g2 ˆ

Rsp2 Mor n L1 2130:3Bo ror

…10†

n gST ˆ

RST Mor nL1 nL2 ¼nLST21 2130:3Bo ror

…11†

Solving Eqs. (9)±(11) using experimental values of Rsp1, Rsp2,¼,RST and Mor, Bob (Bo at the bubble point), and r ob (r o at the bubble point), one can evaluate the gas mole fraction for each separation stage. The liquid mole fractions xi for each stage are then calculated using the component material balance equation given the feed mole fractions zi and the gas mole fractions yi coming out of each stage: xi ˆ

zi 2 ng yi …1 2 n g †

…12†

The ®nal step is the evaluation of Ki s using Eq. (1) given the experimental values of yi s and the calculated values of xi s. The experimental data needed to evaluate K-values are thus: ² compositional analysis of the reservoir/well ¯uid …zi s) and its average molecular weight (Mor); ² the oil density and the oil formation volume factor at the bubble point (r ob, Bob);

Table 5 Parameters for the new proposed correlation for UAE crudes at low pressures Component

b

TB

a0

a1

a2

c0

c1

c2

N2 H2 S CO2 C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C 6s C71

958.217 4537.47 1506.11 1176.018 496.358 773.901 703.898 972.519 2027.732 2127.662 1822.640 Riazi±Daubert correlation

229.144 531.766 1163.80 114.223 378.851 508.584 590.220 623.990 626.456 640.683 715.208

9.43578 2.52312 0.404338 25.0103 1.99508 1.99508 1.99508 1.99508 1.99508 1.99508 1.99508 0.733514

1435.61 2152.256 14698.5 1656.63 423.418 423.418 423.418 423.418 423.418 423.418 423.418 24006.58

253.5503 0.328061 215.4183 27.5942 0.0361687 0.0361687 0.0361687 0.0361687 0.0361687 0.0361687 0.0361687 8.32509

23.18826 0.795361 21.66334 0.996088 1.72688 1.72688 1.72688 1.72688 1.72688 1.72688 1.72688 0.434557

2846.63 25291.21 9186.66 26.1253 2411.283 2411.283 2411.283 2411.283 2411.283 2411.283 2411.283 21091.86

19.0141 6.06910 210.0827 23.60035 0.170883 0.170883 0.170883 0.170883 0.170883 0.170883 0.170883 1.65479

R.A. Almehaideb et al. / Fuel 80 (2001) 117±124

121

Fig. 1. Comparison between K-Standing vs. Kexp.

² the gas/oil ratio for each separation stage (Rsp); and ² the compositional analysis for gas exiting at each stage …yi s). An example of calculations to extract the K-values from laboratory data for one of the samples is shown in Tables 2 and 3. Calculations for the other stages are done in a similar manner, using Eqs. (9)±(11) for each subsequent stage.

4. Results and discussion Extracted K-values were statistically compared in this study with results predicted using the Standing correlation, the Wilson correlation, and the new proposed correlation. The new correlation for UAE crudes was obtained by tuning the Standing correlation parameters to better ®t the UAE data using multi-variable regression. The general form of the Standing correlation is maintained, i.e.: log…Kp† ˆ a 1 c £ F

3. Data sources Data used in this study comprised 804 K-values extracted from single- and multi-stage ¯ash tests on 22 ¯uid samples from different ®elds in UAE with overall ranges of PVT properties as follows: 30:9

,

API

, 48:6

128

,

Rs

, 3871 SCF=STB

516

,

Pb

, 4837 psia

Also, Table 4 shows the range of mole fractions for the components of crude oil used in the study, covering volatile to low-shrinkage oils. Separator conditions from which K-values were extracted were in the range: 70

,

Tsep

, 1208F

40

,

Psep

, 800 psia

…4†

a ˆ a0 1 a1 £ 1026 p 1 a2 £ 1026 p2

…13†

c ˆ c0 1 c1 £ 1026 p 1 c2 £ 1026 p2

…14†

However, the multi-variable regression technique was used to obtain the best values for b, TB, and the coef®cients for a and c (in Eqs. (13) and (14)) as functions of pressure. For better accuracy, separate parameters were obtained for methane C1, considered the main constituent of petroleum systems, for the C2 ±C6 group, for the heptane-plus fraction, and for every non-hydrocarbon component. In addition to minimizing the sum of relative squares of errors, the average percentage error was also minimized to balance out the positive and negative errors and thereby to avoid having a systematic bias in the correlation. For the C71 pseudocomponent, the Riazi±Daubert [10] correlation was used to obtain estimates of TB, Tc, and Pc (and hence b using TB and Pc). The Edmister [11] correlation was also used to obtain the acentric factor, v , for comparison with the

Fig. 2. Comparison between K-Wilson vs. Kexp.

122

R.A. Almehaideb et al. / Fuel 80 (2001) 117±124

Fig. 3. Comparison between Knew correlation vs. Kexp.

Wilson correlation. Both the Riazi±Daubert and the Edmister correlations are widely used for characterizing the plus fraction [2]. The new correlation parameters are listed in Table 5. The following is a statistical comparison of the results obtained by the three published correlations above and the proposed new correlation with extracted values of equilibrium ratios. All components: Figs. 1±3 provide a general comparison between experimental and calculated K-values using the Standing, Wilson, and the new proposed correlation. The smallest K-values on the ®gures represent K-C71 while the highest values represent K-N2. The ®gures show that the Standing correlation signi®cantly overestimated the Kvalues for both C71 and N2, while the Wilson correlation underestimated the extracted K-C71 values and overestimated the N2 values. The proposed correlation, on the other hand, matched the experimental values closely. For the other components, the match was relatively successful for all three correlations. Table 6 is a comparison of the statistical measures for the three correlations. It shows a signi®cant overall improvement in predictions of K-values for the proposed correlation over the Standing and Wilson correlations for UAE data. The following is a group by group comparison of individual crude oil components. C1 ±C6 components: Table 7 lists the statistical comparisons for the three correlations against extracted K-values for the C1 ±C6 hydrocarbons. The comparison shows that the Standing and Wilson correlations both performed relatively well for the C1 ±C6 hydrocarbons. A noticeably high relative error, however, is observed for methane for both correlations. Standing, in his correlation, modi®ed both b and TB for methane. The regressed values for b and TB for methane in the proposed correlation came closer to Hoffman et al. values than to Standing's values. Overall for C1 ±C6s, the

new correlation reduced the %AAD to 20.3% from 27.5% for the Standing correlation and 39.0% for the Wilson correlation. The standard deviation for the proposed correlation also showed a signi®cant improvement over the Standing and Wilson correlations. The parameters for methane were allowed to differ from the C2 ±C6 group in order to obtain the best possible accuracy of estimation for methane, as it is the major constituent of hydrocarbon mixtures. C71 component: Table 8 lists the statistical comparisons of the three correlations against extracted K-values for the C71 pseudo-component. It indicates an order of magnitude improvement in the relative accuracy (%AAD) of the proposed correlation over the Standing correlation and a 1.5 order of magnitude improvement over the Wilson correlation, which underestimated the experimental values by two orders of magnitude. Part of this improvement in the proposed correlation may be attributed to using a general correlation (here Riazi±Daubert) to evaluate b and TB instead of using the correlation indicated by Standing which correlated b and TB with an average carbon number, n, for the plus fraction based on Oklahoma crude oil data. The other part is due to allowing the C71 fraction to have a different set of coef®cients for a and c in the correlations. Further improvement in the accuracy of estimating the C71 fraction was dif®cult due to the unde®ned nature of this fraction. Non-hydrocarbon components: Table 9 lists the statistical comparisons of the three correlations against extracted Kvalues for the non-hydrocarbon components. Results show that the new correlation moderately improves the accuracy of predictions for K-H2S and K-CO2, which were relatively well predicted by the Standing and Wilson correlations. The prediction of these two published correlations of K-N2 was, however, above the extracted K-N2 values, on average, by more than one order of magnitude. The new correlation

Table 6 Statistical comparison of three correlations for all components using UAE data Component

All

Proposed correlation

Standing correlation

Wilson correlation

%AAD

s

%AAD

s

%AAD

s

28.6

42.7

240.0

1519.5

156.8

1046.2

R.A. Almehaideb et al. / Fuel 80 (2001) 117±124

123

Table 7 Statistical comparison of three correlations for C1 ±C6 components using UAE data Component

C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6s All of C1 ±C6s

Proposed correlation

Standing correlation

Wilson correlation

%AAD

s

%AAD

s

%AAD

s

28.7 12.1 10.6 16.1 17.2 25.7 23.3 29.2 20.3

38.77 16.51 13.87 20.61 23.07 35.85 37.15 36.62 29.13

62.7 11.4 13.3 23.4 17.3 26.6 30.8 36.2 27.5

107.6 16.4 17.2 28.3 22.8 48.2 59.6 49.1 51.1

120.4 29.8 14.6 29.5 25.0 32.5 29.3 36.5 39.0

167.0 34.8 18.8 35.1 30.9 44.0 41.7 44.4 66.3

Table 8 Statistical comparison of three correlations for C71 component using UAE data Component

C71

Proposed correlation

Standing correlation

Wilson correlation

%AAD

s

%AAD

s

%AAD

s

64.3

84.7

551.9

797.0

99.0

99.7

Table 9 Statistical comparison of three correlations for non-hydrocarbon components using UAE data Component

H2 S CO2 N2

Proposed correlation

Standing correlation

Wilson correlation

%AAD

s

%AAD

s

%AAD

s

34.5 29.0 68.2

41.3 36.1 81.2

33.0 33.6 2788.2

51.6 51.3 6209.6

37.4 31.5 1937.2

70.0 43.7 4322.2

Table 10 Comparison of bubble point pressures estimated using the three correlations and the PR-EOS Experimental Pb

Standing correlation

%AD

Wilson correlation

%AD

PR-EOS

%AD

New correlation

%AD

652.7 827.7 856.7 664.7 745.7 439.7 790.7 802.7 713.7 939.7

602.7 878.5 734 578.3 756.7 431 758 854 726 1057

7.66 6.14 14.32 12.00 1.48 1.98 4.14 6.39 1.72 12.48

736.2 1008 878.5 690.7 906.7 543 897 984 870 1103

12.79 21.78 2.54 3.91 21.59 23.49 13.44 22.59 21.90 17.38

562 892 761 573 765 409 761 830 733 1003

13.90 7.77 11.17 13.80 2.59 6.98 3.76 3.40 2.70 6.74

656 789 721 612 751 453 749 771 714 841

0.51 4.68 15.84 7.93 0.71 3.02 5.27 3.95 0.04 10.50

%AAD

6.93

improved the accuracy from above 1000% to around 68%. It is worth noting that part of the improvement in accuracy can be attributed to using different sets of coef®cients for the a and c parameters for each non-hydrocarbon component. This is justi®ed based on the fact that chemically, the nonhydrocarbons are different from the mostly paraf®nic series of C2 ±C6s that can be expected to behave in a similar fashion.

16.14

7.28

5.24

As a further check on the accuracy of the new correlation, an independent test is performed by comparing the estimation of the bubble point pressure for ten samples falling within the range of applicability of the proposed correlation. The results of this comparison are listed in Table 10. It shows that the average absolute error for the proposed correlation is 5.2% compared to 6.9% for Standing's, 7.3% for the Peng±Robinson EOS (PR-EOS), and 16.1% for the

124

R.A. Almehaideb et al. / Fuel 80 (2001) 117±124

Wilson correlation. This improvement stems mainly from the better ®t of the hydrocarbons in the proposed correlations, as the non-hydrocarbons and the plus fraction have a small effect on the bubble points evaluated.

5. Conclusions

Acknowledgements The authors would like to acknowledge the management of Abu Dhabi National Oil Co. (ADNOC) and Abu Dhabi Co. for Marine Operations (ADMA-OPCO) for providing the data used in this study and for permission to publish the paper.

As a result of this study, the following conclusions can be made:

References

1. K-values extracted from single- and multi-stage separation experiments provided a direct comparison between experimental and correlated K-values for crude oil components at low pressures. 2. K-values obtained from two widely used correlations, namely the Wilson Correlation and the Standing correlation, compared relatively well with extracted K-values for the C1 ±C6 hydrocarbons and for CO2 and H2S. However, they performed poorly for the C71 fraction and N2. 3. A new correlation is proposed for UAE (and similar) crudes that is based on the same general formula for the Hoffman, Crump, and Hocott and the Standing correlations. However, the parameters for the correlation are modi®ed to ®t experimental data using multi-variable regression. The statistical comparison shows that the new correlation compares favorably well with results from the other two correlations included in the study. 4. This work demonstrates the usefulness of regional correlations developed for different crudes as these allow the engineer to get better estimates for K-values and hence improve the accuracy of phase-behavior related calculations.

[1] Standing MB. A set of equations for computing equilibrium ratios of a crude oil/natural gas system at pressures below 1000 psia. JPT 1979;Sept:1193±5. [2] Ahmed T. Hydrocarbon phase behavior. Gulf Publishing, 1989. p. 244±86. [3] Wilson G. A modi®ed Redlich±Kwong EOS, application to physical data calculations. Paper 15C presented at the Annual AICHE National Meeting, Cleveland, Ohio, May 4±7, 1968. [4] Katz DL, Hachmuth KH. Vaporization equilibrium constants in a crude oil±natural gas system. Ind Eng Chem 1937;29:1072. [5] Hoffmann AE, Crump JS, Hocott RC. Equilibrium constants for a gascondensate system. Trans AIME 1953;198:1±10. [6] Glaso OS, Whitson CH. The accuracy of PVT parameters calculated from computer ¯ash separation at pressures less than 1000 psia. JPT 1982;Aug:1811±3. [7] Bashbush JL. A method to determine K-values from laboratory data and its applications. SPE 10127 presented at the 56th Annual Technical Conference and Exhibition, San Antonio, Texas, 1981. [8] Whitson CH, Torp SB. Evaluating constant volume depletion data. JPT 1983;March:610±20. [9] McCain Jr. WD. The properties of petroleum ¯uids. 2nd ed. PennWell Books, 1990. [10] Riazi MR, Daubert TE. Characterization parameters for petroleum fractions. Ind Eng Chem Res 1987;26(24):755±9. [11] Edmister WC. Applied hydrocarbon thermodynamics, part 4: compressibility factors and equations of state. Pet Ref 1958;37:173±9.