SPEPBM-Ch5

Chapter 5HeptanesĆPlus Characterization 5.1 Introduction Some phase-behavior applications require the use of an equation of state (EOS) to predict properties of petroleum reservoir fluids. The critical properties, acentric factor, molecular weight, and binary-interaction parameters (BIP’s) of components in a mixture are required for EOS calculations. With existing chemical-separation techniques, we usually cannot identify the many hundreds and thousands of components found in reservoir fluids. Even if accurate separation were possible, the critical properties and other EOS parameters of compounds heavier than approximately C20 would not be known accurately. Practically speaking, we resolve this problem by making an approximate characterization of the heavier compounds with experimental and mathematical methods. The characterization of heptanesplus (C7)) fractions can be grouped into three main tasks.1–3 1. Dividing the C7) fraction into a number of fractions with known molar compositions. 2. Defining the molecular weight, specific gravity, and boiling point of each C7) fraction. 3. Estimating the critical properties and acentric factor of each C7) fraction and the key BIP’s for the specific EOS being used. This chapter presents methods for performing these tasks and gives guidelines on when each method can be used. A unique characterization does not exist for a given reservoir fluid. For example, different component properties are required for different EOS’s; therefore, the engineer must determine the quality of a given characterization by testing the predictions of reservoir-fluid behavior against measured pressure/volume/temperature (PVT) data. The amount of C7) typically found in reservoir fluids varies from u50 mol% for heavy oils to t1 mol% for light reservoir fluids.4 Average C7) properties also vary widely. For example, C7) molecular weight can vary from 110 to u300 and specific gravity from 0.7 to 1.0. Because the C7) fraction is a mixture of many hundreds of paraffinic, naphthenic, aromatic, and other organic compounds,5 the C7) fraction cannot be resolved into its individual components with any precision. We must therefore resort to approximate descriptions of the C7) fraction. Sec. 5.2 discusses experimental methods available for quantifying C7) into discrete fractions. True-boiling-point (TBP) distillation provides the necessary data for complete C7) characterization, including mass and molar quantities, and the key inspection data for each fraction (specific gravity, molecular weight, and boiling point). Gas chromatography (GC) is a less-expensive, time-saving alternative to TBP distillation. However, GC analysis quantifies only the mass of C7) fractions; such properties as specific gravity and boiling point are not provided by GC analysis. HEPTANES-PLUS CHARACTERIZATION Typically, the practicing engineer is faced with how to characterize a C7) fraction when only z C7) the mole fraction, ; molecular weight, M C7); and specific gravity, g C7) , are known. Sec. 5.3 reviews methods for splitting C7) into an arbitrary number of subfractions. Most methods assume that mole fraction decreases exponentially as a function of molecular weight or carbon number. A more general model based on the gamma distribution has been successfully applied to many oil and gas-condensate systems. Other splitting schemes can also be found in the literature; we summarize the available methods. Sec. 5.4 discusses how to estimate inspection properties g and Tb for C7) fractions determined by GC analysis or calculated from a mathematical split. Katz and Firoozabadi’s6 generalized single carbon number (SCN) properties are widely used. Other methods for estimating specific gravities of C7) subfractions are based on forcing the calculated g C7) to match the measured value. Many empirical correlations are available for estimating critical properties of pure compounds and C7) fractions. Critical properties can also be estimated by forcing the EOS to match the boiling point and specific gravity of each C7) fraction separately. In Sec. 5.5, we review the most commonly used methods for estimating critical properties. Finally, Sec. 5.6 discusses methods for reducing the number of components describing a reservoir mixture and, in particular, the C7) fraction. Splitting the C7) into pseudocomponents is particularly important for EOS-based compositional reservoir simulation. A large part of the computing time during a compositional reservoir simulation is used to solve the flash calculations; accordingly, minimizing the number of components without jeopardizing the quality of the fluid characterization is necessary. 5.2 Experimental Analyses The most reliable basis for C7) characterization is experimental data obtained from high-temperature distillation or GC. Many experimental procedures are available for performing these analyses; in the following discussion, we review the most commonly used methods. TBP distillation provides the key data for C7) characterization, including mass and molar quantities, specific gravity, molecular weight, and boiling point of each distillation cut. Other such inspection data as kinematic viscosity and refractive index also may be measured on distillation cuts. Simulated distillation by GC requires smaller samples and less time than TBP distillation.7-9 However, GC analysis measures only the mass of carbon-number fractions. Simulated distillation results can be calibrated against TBP data, thus providing physical properties for the individual fractions. For many oils, simulated distillation 1 Cutoff (n-paraffin) boiling point Midvolume (“normal”) boiling point Dp Dp N2 N2 Fig. 5.2—TBP curve for a North Sea gas-condensate sample illustrating the midvolume-point method for calculating average boiling point (after Austad et al.7). Fig. 5.1—Standard apparatus for conducting TBP analysis of crude-oil and condensate samples at atmospheric and subatmospheric pressures (after Ref. 11). provides the necessary information for C7) characterization in far less the time and at far less cost than that required for a complete TBP analysis. We recommend, however, that at least one complete TBP analysis be measured for (1) oil reservoirs that may be candidates for gas injection and (2) most gas-condensate reservoirs. 5.2.1 TBP Distillation. In TBP distillation, a stock-tank liquid (oil or condensate) is separated into fractions or “cuts” by boiling-point range. TBP distillation differs from the Hempel and American Soc. for Testing Materials (ASTM) D-158 distillations10 because TBP analysis requires a high degree of separation, which is usually controlled by the number of theoretical trays in the apparatus and the reflux ratio. TBP fractions are often treated as components having unique boiling points, critical temperatures, critical pressures, and other properties identified for pure compounds. This treatment is obviously more valid for a cut with a narrow boiling-point range. The ASTM D-289211 procedure is a useful standard for TBP analysis of stock-tank liquids. ASTM D-2892 specifies the general procedure for TBP distillation, including equipment specifications (see Fig. 5.1), reflux ratio, sample size, and calculations necessary to arrive at a plot of cumulative volume percent vs. normal boiling point. Normal boiling point implies that boiling point is measured at normal or atmospheric pressure. In practice, to avoid thermal decomposition (cracking), distillation starts at atmospheric pressure and is changed to subatmospheric distillation after reaching a limiting temperature. Subatmospheric boiling-point temperatures are converted to normal boiling-point temperatures by use of a vaporpressure correlation that corrects for the amount of vacuum and the fraction’s chemical composition. The boiling-point range for fractions is not specified in the ASTM standard. Katz and Firoozabadi6 recommend use of paraffin normal boiling points (plus 0.5°C) as boundaries, a practice that has been widely accepted. 2 Fig. 5.27 shows a plot of typical TBP data for a North Sea sample. Normal boiling point is plotted vs. cumulative volume percent. Table 5.1 gives the data, including measured specific gravities and molecular weights. Average boiling point is usually taken as the value found at the midvolume percent of a cut. For example, the third cut in Table 5.1 boils from 258.8 to 303.8°F, with an initial 27.49 vol% and a final 37.56 vol%. The midvolume percent is (27.49)37.56)/2+32.5 vol%; from Fig. 5.2, the boiling point at this volume is [282°F. For normal-paraffin boiling-point intervals, Katz and Firoozabadi’s6 average boiling points of SCN fractions can be used (see Table 5.2). The mass, m i, of each distillation cut is measured directly during a TBP analysis. The cut is quantified in moles n i with molecular weight, M i, and the measured mass m i, where n i + m ińM i. Volume of the fraction is calculated from the mass and the density, ò i (or specific gravity, g i), where V i + m ińò i . M i is measured by a cryoscopic method based on freezing-point depression, and ò i is measured by a pycnometer or electronic densitometer. Table 5.1 gives cumulative weight, mole, and volume percents for the North Sea sample. Average C7) properties are given by ȍm N i MC 7) + i+1 N ȍn i+1 N i+1 N i ȍm and ò C 7) i + ȍV i+1 , i . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (5.1) where ò C7) + g C7)ò w with ò w + pure water density at standard conditions. These calculated averages are compared with measured values of the C7) sample, and discrepancies are reported as “lost” material. Refs. 7 and 15 through 20 give procedures for calculating properties from TBP analyses. Also, the ASTM D-289211 procedure gives details on experimental equipment and the procedure for conducting TBP analysis at atmospheric and subatmospheric conditions. Table 5.3 gives an example TBP analysis from a commercial laboratory. PHASE BEHAVIOR 2 604.60 5. 5.78 64.0 153. This correction factor varies from 1 to 1. the GC response for carbon number Ci starts at the bottom response of n-Ci *1 and extends to the bottom response of n-Ci . The most important application of chromatography to C7) characterization is simulated distillation by GC techniques. Our experience.684 0. may provide detailed information on the compounds separated by GC.3 128.88 11.5 101.98 2. naphthenes.77 77. Consider for example a C7) sample distilled into 10 cuts separated by normal-paraffin boiling points.03 100. 5. liquid chromatography are used to quantify the relative amount of compounds found in oil and gas systems. the baseline should be determined before running the actual chromatogram. this concept is misleading because a unique relation does not exist between molecular weight and mole fraction unless the fractions are separated in a consistent manner.853 0. The main disadvantage of simulated distillation is that inspection data are not determined directly for each fraction and must therefore either be correlated from TBP data or estimated from correlations (see Sec.21 6.57 1.19 15.7830 0. Fig.5 92.47 3.70 25.940 1.8 455.9 287.59 2.97 5. If the same C7) sample is distilled with constant 10-vol% cuts.34 9. A 15% correction to the internal-standard response factor was used to match the two distillation curves.00 SxVi % 4.6 69.9991.8 564. 5.8 58. which are identified up to n-C22. **Water+1.57 35.57 3. Swi +Swi . xVi +100 Vi/2580. 5.4 282.8 347.TABLE 5.8536 0.5 shows the simulated vs.07 12.0 235.8290 0. and distillation at 10 mm Hg+471. to a lesser extent. As an alternative to correcting the internal standard.35 10.6 123.61 6.189 0.847 1. the average boiling points suggested by Katz and Firoozabadi6 (Table 5.8 123. mi /2073.81 4.4 363.89 12.3 427. but the analysis is relatively costly and time-consuming from a practical point of view. As Fig.3 53.624 0.37 100.68 5. the two sets of data will not 3 .2. and aromatics (PNA’s) for carbon-number fractions distilled by TBP analysis.3 199. is that PNA data have limited usefulness for improving EOS fluid characterizations.2 Swi % 4.7459 0.3).19 20.96 11.44 56. However.8047 0.7 294.8466 0.3 Molar Distribution Molar distribution is usually thought of as the relation between mole fraction and molecular weight.18 41.33 12.33 73.97 100.26 71. The mass of carbon number Ci is calculated as the area under the curve from the baseline to the GC response in the n-Ci *1 to n-Ci interval (see the shaded area for fraction C9 in Fig.8708 Mi (g/mol) 96 110 122 137 151 161 181 193 212 230 245 259 266 280 370 Vi (cm3) 123.07 7.4 550.5 ni (mol) 0.1 76.2 45.6 2.8378 0.225 0.92 11.2 to 653°F.02 6.2 to 347°F.03 11.8 591. Maddox and Erbar15 suggest that the reported chromatographic boiling points be adjusted by a correction factor that depends on the reported boiling HEPTANES-PLUS CHARACTERIZATION point and the “paraffinicity” of the composite sample. 5.47 shows schematically.70 2.10 0.313 0.42 2.87 70.8278 0.8236 0. wi +100 *Average taken at midvolume point.1 gi ** 0.0 Average Tbi * (°F) 194.14 12.7 sample. The dominant peaks are for normal paraffins.6 225.99 11.5. checking the accuracy of simulated distillation for SCN fractions greater than approximately C25 is difficult because C25 is usually the upper limit for reliable TBP distillation.3 shows an example gas chromatogram for the North Sea sample considered earlier.69 2.5 173.2 325. For example.98 Reflux ratio+1 : 5.049 wi (%) 4.8 146.70 5.18 3.1 2.6 74.92 76.2) can be used.30 2.80 11.162 1.4 579.97 59.63 100. distillation at atmospheric pressure+201.06 19.1. an internal standard must be used to relate GC area to mass fraction.90 3.78 3. 5. As a good approximation for a paraffinic sample.35 37.4 89.19 79.37 6.97 12.19-23 may improve C7) characterization for modeling phase behavior with EOS’s.1 491.15 and is slightly larger for aromatic than paraffinic samples.8 120.80 16.0 492.5 59.00 53. In fact.7909 0.01 12.95 27.87 9.2 Chromatography.91 80.2 214.35 14. Sophisticated analytical methods.60 1.8034 172 11.41 62.62 78. Recent work has shown that PNA analysis3. and Kw +(Tbi +460)1/3/gi .05 5.09 66.00 xVi % 4.68 4.7711 0. such as mass spectroscopy. TBP distillation curves for the Austad et al. accordingly. xi +100 ni /12.96 5.3.0 381.0 537.16 12.68 68. Several laboratories have calibrated GC analysis to provide simulated-distillation results up to C40. 5. Fig.2°F.08 7.2 401.0 642.96 74.156 12.5 164.850 0.37 43.8 617.479 0.59 100. reflux cycle+18 seconds.74 50. and the factor should be determined on the basis of TBP analysis of at least one sample from a given field. however. distillation at 100 mm Hg+347 to 471.073.8 474. SxVi +SxVi .2 420.15 11.40 48.2 50. Because stock-tank samples cannot be separated completely by standard GC analysis.1—EXPERIMENTAL TBP RESULTS FOR A NORTH SEA CONDENSATE Upper Tbi (°F) 208.4 629.8221 0.82 11.7283 0.80 15. Boiling points are not reported because normal-paraffin boiling-point intervals are used as a standard.40 10. GC and.4 258. ni +mi /Mi . 5.81 2.87 1. mass spectroscopy GC can establish the relative amounts of paraffins.8 303.00 Kw 11.6 653.1 438.8 523.4).00 Fraction C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21) Sum Average mi (g) 90.67 2. This factor will probably be constant for a given oil.00 xi % 7.8 136. Normal hexane was used as an internal standard for the sample in Fig.2 258.322 0.11 12.580.951 1.7658 0. The internal standard’s response factor may need to be adjusted to achieve consistency between simulated and TBP distillation results.8 509.455 0. Detailed PNA information should provide more accurate estimation of the critical properties of petroleum fractions.35 10. Vi +mi /gi /0.049. 076.321 1.168 0.4 10.697 1.915 0.0 858. .004.065 1.9 441. .94 11.9 1.2 674.5 17.120 1.99 12.82 11.4 663.4 602.8 541.7 210.6 6.559 1. .2 10. .749 0.724 pc (psia) 476 457 428 397 367 341 318 301 284 268 253 240 230 221 212 203 195 188 182 175 168 163 157 152 149 145 141 138 135 131 128 126 122 119 116 114 112 109 107 105 ą ąw 0. . . a plot of cumulative mole fraction. .27 11.8 685.523 0.310.418.14 Tc (°R) 914 976 1.2 15.929 0. .1 20.793 0.84 11.545 1.4 915.804 0.661 1.196 0. . .164 0.940 M 84 96 107 121 134 147 161 175 190 206 222 237 251 263 275 291 305 318 331 345 359 374 388 402 416 430 444 458 472 486 500 514 528 542 556 570 584 598 612 626 Defined Kw 12.8 842.92 11.90 11.6 692.3 21.936 0.768 0.1 23.466 1.7 18.3 14.2 456.4 834.9 11. . .1 1.7 1.4 242.0 548.640 1. .195 1.6 935. .222 0.01 12.671 1.379.191 0.2 874.723 0.431.4 606.9 28.03 12.157 1.06 12. .3 26.8 737.596 1.4 25.226 0.2 456. . cumulative average molecular weight.706 1.158 1.5 1.487.6 825.00 12.690 1.876 0.328 Riazi14 Vc (ft3/lbm mol) 5.584 1.0 1.245.754 0.9 7.236 0.1 701. . .8 919.391.2 6.266 0.892 0.843 0.7 1. .5 27.8 produce the same plot of mole fraction vs.226 1.0 304.0 946.816 0.016.7 1.09 12. .5 422.287 1.084 1.96 11.479 0.077 1.194 0.561 0.8 737. .0 304.644 0.6 577.0 406. .1 1.476.206 0. .2 674.0 259.6 9.463.388 1.6 627.3 1.253 0.13 12.6 627.294 1.192 1.6 790. .6 489.07 12.9 1.4 602.902 0.5 24.5 1.650 1.482 1.1 828.249 0.2 804.4 906.8 746.866 0.0 548. . .8 901.5 1.84 11. .5 510.2—SINGLE CARBON NUMBER PROPERTIES FOR HEPTANES-PLUS (after Katz and Firoozabadi6) Katz-Firoozabadi Generalized Properties Tb Interval* Fraction Number 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 *At 1 atmosphere.310 0.618 1.94 11.6 900. .1 1.0 24.0 971.245 0.7 747.6 577.531 1.160 Lower (°F) 97.437 0.366.4 809.4 Upper (°F) 156.4 993.TABLE 5.0 595.294.162 0.885 0.04 12.0 259.601 0.1 1.784 0.498 1. . .027 1.8 901. (5.352.9 1.212 0.6 775.3 385.836 0.263.226.927 0.899 0.0 520.6 755.922 0. .6 692.258 1.87 11.1 27.300 1.912 0.0 640.206. . .4 572.917 0. .826 0.920 0.936 0. **Water+1.896 0.175 1.183 0.7 210.515 1.339.684 0.4 793.044 1.9 19. . .349 0.273 0.727 0.203 0.145.428 1.174 0.630 1.166 0.4 707.10 12.185 0.11 12. .186. .8 651.3 18.7 22.681 1.0 726.ĂĂ . Lee-Kesler12/Kesler-Lee13 Correlations Average Tb (°F) (°R) 147.85 11.6 755.3 Defined Zc 0.89 11.0 475.100.95 11. . .228 1.2 970.931 0.3 346.6 825.6 775.1 1.242 1.04 12.86 11. .447 1. .104 1.7 8.690 0.6 15.87 11.5 24.7 657.5 1.10 12.6 1.96 11.782 0.8 651.316 1.123.849 0.199 0. However.019 1.905 0.262 0.369 1.406.9 25.4 959.7 27.1 1. . .8 851.178 0.86 11.261 1.0 330.4 879.4 809.965 0.6 717.7 1.141 1.7 1. .272 0.14 12.031. j .5 422.207 1.07 12.187 0.1 1.1 1. . .257 0.1 1.608 1.4 617.176 0. .0 14. .2 22.0 858.861 0.5 20.209 0.001.3 385.856 0.0 197.571 1.3 346.4 786.716 1.027.938 0.0 865.6 489. molecular weight.180 0. vs.2) j Q Mi + j+1 ȍz i j+1 . .05 12.7 23. .888 0.166.241 0. .6 717.280.3 1.7 12. .4 369.272 1.04 12.2 1.84 11.879 0.189 0.6 982.934 0. .215 0.452. .122 1.7 21.055.815 0.4 932.7 156.6 866.4 888.881 0.0 520. .925 0.442.4 793.349 1. .1 ăăg*ă 0.1 17.992 1.271 0.933 0.232 0. . .6 766.909 0.9 16.8 892.06 12.8 842.7 1.7 1.3 1.219 0.5 1.172 0.2 874.5 13.269 0. .871 0. ȍz i j Q zi + j+1 N ȍz M i j j ȍz j+1 . (5.325.4 888.04 12.408 1.392 0.1 1.909 0.851 0.13 12.8 26.82 11.5 1. .2 820.1 288.229 0.1 1.170 0.3) 4 PHASE BEHAVIOR . 5 45. . .4—GC simulated distillation chromatograms (a) without any sample (used to determine the baseline). .g. . . and (c) for a crude oil with internal standard (after MacAllister and DeRuiter9). . z Cn +mole fraction of Cn .53 4. .2) can be used when normal paraffin boiling-point intervals are used. In this section.3—STANDARD TBP RESULTS FROM COMMERCIAL PVT LABORATORY Component Heptanes Octanes Nonanes Decanes Undecanes Dodecanes Tridecanes Tetradecanes Pentadecanes plus mol% 1. .10 wt% 2.3. . and A+constant indicating the slope on a plot of ln z i vs. . . .7470 0. . we review methods commonly used to describe molar distribution. .8) for molecular weight of Ci and the assumption that the distribution is infinite. . . .6. More commonly. . A Newton-Raphson algorithm can be used to solve Eqs. .39 31. Strictly speaking. . .10) ȍz + 1 i i+n R . . . . . .7258 0.11) (a) (b) (c) Fig.1 Exponential Distributions.7971 0. . . . .05 2. .10 2.6. 5 . . . .8 42. . . . . . . . . . . . . . . . . . . . .7654 0.18 3. . .08 3. . The Lohrenz-Bray-Clark24 (LBC) viscosity correlation is one of the earliest attempts to use an exponential-type distribution for splitting C7). . .52 3. . . . . . .9) where i+carbon number and z C6 +measured C6 mole fraction. . . . 5. . . .98 0. The constants z Cn and A can be determined explicitly. . .96 2. .7 53. . . .15 2.18 0.2 (after Austad et al. . . (5. .12 1. . . . (b) for a crude oil. . . . . . .57 0. Some methods use a consistent separation of fractions (e. . Constants A1 and A2 are determined by trial and error so that zC 7) + ȍz 40 i+7 i . . . (5. . .62 0. . . . . z i + z Cn exp A[( i * n )]. . .8736 Gravity ągAPI 63. . . (5. .. . . . .30 1. . . . . . . molar distribution is the relation between cumulative molar quantity and some expression for cumulative molecular weight. . .1 50.7808 0. . by SCN) so the molar distribution can be expressed directly as a relationship between mole fraction and molecular weight of individual cuts. .7751 0. constants z Cn and A are given by z Cn + 14 M Cn) * 14( n * 1 ) * h . . . . 5. .61 Density (g/cm3) 0.1 30. Writing the exponential distribution in a general form for any Cn ) fraction (n+7 being a special case). 5. . . . . . (5. . . . .5) and A + lnǒ1 * z CnǓ so that . . (5. should produce a single curve. .9 49. . . . . . . . . The LBC method splits C7) into normal paraffins C7 though C40 with the relation z i + z C exp[A 1(i * 6) ) A 2(i * 6) 2]. . The LBC form of the exponential distribution has not found widespread application. . .4) 6 and z C 7) MC 7) + ȍz M 40 i i+7 i . . .8105 0. . . . . A more general approach uses the continuous three-parameter gamma probability function to describe molar distribution. . . . . . . 5. . . a linear form of the exponential distribution is used to split the C7) fraction. . . . . (5. . . . .8235 0. . (5.7) where i+carbon number. . . . . Note that the LBC model cannot be used when z C7) t z C6 and M C7) u M C40. . 5. . . . . . . .TABLE 5. .3—Simulated distillation by GC of the North Sea gas-condensate sample in Fig. . . . . . . .7).9 40.2 57. .3 Molecular Weight 96 101 114 129 144 163 177 192 330 *At 60°F. . . . (5. Paraffin molecular weights (Mi +14i)2) are used in Eq. Note: Katz and Firoozabadi6 average boiling points (Table 5. . With the general expression M i + 14 i ) h . . therefore. . i.6) are satisfied.5 and 5. . . HEPTANES-PLUS CHARACTERIZATION Fig. . Most methods in this category assume that C7) mole fractions decrease exponentially. . 5.74 0. . . . . .000°F Distillate 1. . oil has a relatively large a of 2. . .628). . and petroleum residues indicates that the upper limit for a is 25 to 30. The three-parameter gamma distribution is a more general model for describing molar distribution. . . . . .15 or determined by fitting measured TBP data. . .250°F Distillate 1. . .700 to 1. . . . . . . .14) The three parameters in the gamma distribution are a. . .9 and 5.13) Fig.5 for reservoir fluids. .27. . They give results for 44 oil and condensate C7) samples that were fit by the gamma distribution with data from complete TBP analyses. .5 amu (molecular weight units) for the 44 samples. and its value usually ranges from 0. . . . . 5. . . . Fig.29 oil and a North Sea oil. . h is fixed). . . (5. . .000 to 1. . bitumen. . 5.10 imply that once a molecular weight relation is chosen (i. . . .200°F Residue Fig.7). .12) are satisfied. .8). then the area of Section i is P0(Mbi )*P0(Mbi *1). . . . . . The Hoffman et al. . (5. . The area of a section is defined as normalized mole fraction z ińz C 7) for the range of molecular weights Mbi *1 to Mbi . . . . Application of the gamma distribution to heavy oils. .2 Gamma-Distribution Model. . where G+gamma function and b is given by b+ MC 7) *h a . .18) 1 . . ..15) 1 * ǒ1 ) 4 ńa 0. . .8. . 5. .19) and P 1 + Q S * a PHASE BEHAVIOR 110 . . .5 to 2. . . . and M C7) The key parameter a defines the form of the distribution. . . . .82. either calculated from Eq. . . . . The absolute average deviation in estimated cut molecular weight was 2. . . 0 bi 0 b i*1 . . h. . all reservoir fluids having a given C7) molecular weight will not have the same molar distribution. .e. a relatively small h of 75. . (5. . The gamma probability density function is p(M) + (M * h) a*1 exp NJ* ƪǒ M * h ǓńbƫNj b a G(a) . Mole fraction zi can be written zi + zC 7) ƪP ǒM Ǔ * P ǒM Ǔƫ . which statistically is approaching a log-normal distribution (see Fig. .3. 5. . . and M C7)+227. .26 and Whitson et al. . . Practically. . . which is one reason why more complicated models have been proposed. . . 5.6—Gamma distributions for petroleum residue (after Brulé et al. . (5. . . . 5. h+93. . . . The continuous distribution p(M ) is applied to petroleum fractions by dividing the area under the p(M ) curve into sections (shown schematically in Fig.17) where P 0 + Q S. . .2. . . Eqs. . . Whitson2. . If the area from h to molecular-weight boundary Mb is defined as P0(Mb ). . a+1 gives an exponential distribution. . Realistically. . (5. . . .7. . . . . . . . .28). An approximate relation between a and h is h[ 6 for reservoir-fluid C7) fractions. . . . . 5.16) Average molecular weight in the same interval is given by Mi + h ) a b P 1ǒM b iǓ * P 1ǒM b i*1Ǔ P 0ǒM b iǓ * P 0ǒM b i*1Ǔ . total area under the p(M ) curve from h to R is unity. .7Ǔ ǒ Ǔ . the North Sea oil is described by a+0. . . . . . 5. . with M C7)+198. .27 discuss the gamma distribution and its application to molar distribution. . . . and i+n ȍz M + M i i R C n) . (5. . . . Parameters for these two oils were determined by fitting experimental TBP data. . The parameter h can be physically interpreted as the minimum molecular weight found in the C7) fraction. h should be considered as a mathematical constant more than as a physical property. . . . . . . . the distribution is uniquely defined by C7) molecular weight. . . . . . . . (5. . . . . .5—Comparison of TBP and GC-simulated distillation for a North Sea gas-condensate sample (after Austad et al. also shown schematically in Fig. 5.7 shows the function p(M ) for the Hoffman et al. By definition. . (5. . . . .25. . . and calculate Mi from Mi + h ) ǒMC 7) * hǓ * Ǔ ǒ Ǔƫ ƪǒQwińQ* M i * Q wi*1ńQ M i*1 Q wi * Q wi*1 .28) h A + z ińz C 7) + P 0ǒM biǓ * P 0ǒM bi*1Ǔ h M bi A + P 0ǒM biǓ h M bi*1 A + P 0ǒM bi*1Ǔ Fig. 7. . . . (5.897056937. . . the cumulative dimensionless molecular-weight variable. . Fig.273 h + 75. . . . furthermore. . . .4 7) a + 0. . . . . 5.10 shows a plot of cumulative weight fraction. . . Mi . . . . Read the value of a from Fig. . 5.3. b Ǔ + where D Mi + N *1 i+1 ȍ (D Mi) 2 . . . . . . The summation in Eq. . . if they are not reported. . The recurrence formula. is used for xu1 and xt1. . . . . 5. . . . . . . . . . . . . . . read Q * M i . . . . . . . Q* Mi + QM i * h . . .482199394. . i i j+1 . . . . . . .. . . . .4 gives three sample outputs from GAMSPL for a+0. . . . A7+ *0.25) Fig. the model average molecular weight. 3. Calculate molecular weights and mole fractions of Fractions i using the best-fit curve in Fig. Calculate experimental weight fractions. . . (5. . . 4. Assume several values of h (e. The sum-of-squares function can be defined as Fǒ a. . . respectively. (5. . . . . . 6. . .27) Note that P0(Mb 0+h)+P1(Mb 0+h)+0. (5. . .g.5. . A+area. The amount and molecular weight of the C26) fraction varies for each value of a. . . Table 5. can be compared with the experimental value as an independent check of the fit. The last fraction is calculated by setting the upper molecular-weight boundary equal to 10. . (5. . . .24) (M i) mod * (M i) exp (M i) exp . .000.22) b . . . .10.23) where A1+*0. For each estimate of h.29 oil (dashed line) and a North Sea volatile oil (solid line). . If the last fraction is not included. including weight fraction and molecular weight for at least five C7) fractions (use of more than 10 fractions is recommended to ensure a unique fit of model parameters). .918206857. The equations for calculating zi and Mi are summarized in a short FORTRAN program GAMSPL found in Appendix A. (5. . 1. . . . . HEPTANES-PLUS CHARACTERIZATION 7 . A4+0. . .8—Schematic showing the graphical interpretation of areas under the gamma density function p(M) that are proportional to normalized mole fraction. . 5.756704078.a + 2. . . Tabulate measured mole fractions zi and molecular weights Mi for each fraction. . 5. . . G(x)1)+xG(x). . . (5. from 75 to 100) and calculate Q* M i for each value of the estimated h.7 M C + 198. . . 5. 5. Enter the curve at measured values of Q wi . . Q wi + ȍ y Ȋ( a ) k ) j j j+0 k+ 0 R ƪ ƫ *1 . . . . . (5. . . . . .988205891. . Also. .10 and choose the curve that fits one of the model curves best. 5. A5+*0. . . MC * h 7) and y + Mb * h . . h . . . . 2.26) vs. . . . .035868343 for 0xxx1. .27 where Q + e *y y a G(a ). . . . After Whitson et al. . . h. . .10 and TBP data.193527818. . 1. .9 plots the results as log zi vs. .21 should be performed until the last term is t1 10*8. . . . . G(1)+1. in accordance with the expected accuracy for measurement of molecular weight. The gamma function can be estimated by30 Gǒ x ) 1Ǔ + 1 ) ȍA x 8 i i+1 i . . . . . . Calculate cumulative molecular weight Q M i from Eq. A3+*0. . . Q wi on a copy of Fig. . . plot Q * M i vs. . . . . .2 M C + 227 7) The gamma distribution can be fit to experimental molar-distribution data by use of a nonlinear least-squares algorithm to determine a. (5. . . . . . Fig. and 2 with h+90 and M C7)+200 held constant. . . . . . . A simple graphical procedure can be used to fit parameters a and h if experimental M C7) is fixed and used to define b.817 h + 93. . . . . . . . the boundary molecular weights are chosen arbitrarily at increments of 14 for the first 19 fractions. . .7—Gamma density function for the Hoffman et al. . . . 5. . .20) S+ Subscripts mod and exp+model and experimental. . . .577191652. the sum-of-squares function does not include the last molecular weight because this molecular weight may be inaccurate or backcalculated to match the measured average C7) molecular weight. . . This sum-of-squares function weights the lower molecular weights more than higher molecular weights. A2+0.21) ȍw . . p(M) Table 5. w i + (z i M i) B (z C 7)M C7)). . . .10. and A8+0. . 5. starting with h as the first lower boundary. . . Normalize weight fractions and calculate cumulative normalized weight fraction Q w i . . 5. A6+0. . . In this simple program. .5 and the following outline describe the procedure for determining model parameters with Fig. . . . . and b. (M C7)) mod + h ) ab. Experimental TBP data are required. 758 152.0336535 0.916 180.424 Weight 96. ăąąąăa+2.0556863 0.9—Three example molar distributions for an oil sample with M C 7)= 200 and h = 90.852 250.0284480 0.0490314 0. 7) ( PHASE BEHAVIOR .525 124. (5. with h+70.0 Cumulative Normalized Mole Fraction.0632969 0.172 139.0225321 0. . .0000000 Molecular Weight 94. calculated with the GAMSPL program (Table A-4) in Table 5.826 236.0369618 0.0252470 0.4—RESULTS OF GAMSPL PROGRAM FOR THREE DATA SETS WITH DIFFERENT GAMMA-DISTRIBUTION PARAMETER a ăąąąăa+0. most reservoir fluids can be characterized with an exponential molar distribution (a+1) without adversely affecting the quality of EOS pre) 1. .23 with little extra effort.6 0.841 222.1052247 0. 5.836 194. .894 539.0 0.0111762 0.852 264. .852 466.0194514 0.852 306.0632444 0.0548597 0.0505020 0.0181808 0. 72.870 250. Although the gamma-distribution model has the flexibility of treating reservoir fluids from light condensates to bitumen.2 0.0890834 1.888 320. .0 } MC h + 90 DM b + 14 ) 0. 5.5 (see Fig.0718284 0.000 Mole Fraction 0.0772607 0.0 f a + 0.5 Fraction Number 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total Average For all three cases h + 90 and M C 7) ăąąąăa+1. .852 138.0160543 0. 5.778 320.852 222.0228471 0.5.0463166 1.0101996 0.0334698 0.852 152.5 and a+2. .0 0.0120910 0.0155959 0.11 shows a Q * M i * Q wi match for the Hoffman et al.10—Cumulative-distribution type curve for fitting experimental TBP data to the gamma-distribution model.588 110.1195065 0. .864 236.852 194.0815774 0.0804707 0.1073842 0.0399302 0. 8 Fig.859 208.852 236.770 348.797 278.4 0. .29) For computer applications.0720157 0. .774 334.0148619 0.0122665 0.16 through 5.038 152.12).886 306.0655834 0.0877762 0. Q wi and Q * M i can be calculated exactly from Eqs. Parameters a and h are determined with M C held constant.0108204 0. .0927292 0.848 208.0322804 0. .0134879 0.0177127 0.892 348.852 348.0000000 200 200 Molecular Weight 99.805 264.784 306.875 264.852 208.852 110.890 334. . and 80 and indicates that a best fit is achieved for h+72.852 320.0281813 0.852 124.0000000 200 + 200.0164152 0.852 166.132 111.0294699 0.2 0.0 Mole Fraction 0.4 0.790 292.5 a + 1. 5.4.0259481 0. . .0925552 0.8 ( 0.0470180 0. Qzi 7) + 200 Fig.852 278.963 166.6 0.819 180.0610991 0.852 334. .0202013 0. 29 oil Fig.2787233 0.814 250.0273900 0.0431719 0.651 Molecular Mole fractions zi are given by z i + z C 7) ǒ Qw i Q* Mi * Q w i*1 Q* M i*1 Ǔ . 5.0 Mole Fraction 0.0380125 0.490 125.883 292.852 292. .0428377 0.0234690 0.8 1.TABLE 5.0926497 0.857 222.0 V a + 2. . 75.0137321 0.796 166.879 278.883 194.0852269 0.852 180.766 420.690 138.0132017 0.0201167 0.1199341 1. 596 3.861 0.5 and h = 72.655 0.351 1.3338 0. HEPTANES-PLUS CHARACTERIZATION Fig.566 0.679 65. their method allows multiple reservoir-fluid samples from a common reservoir to be treated simultaneously with a single fluid characterization.313 0.2799 0.251 0.811 1.9 186.921 0.877 0.733 0.169 0.0266 0.933 0. Calculate a modified b* term.445 0.000 QMi 99. Whitson et al.808 0.526 0.9 178. Specify the heaviest molecular weight of fraction N (recommended value is M N + 2.228 69.8 0.884 0.0034 0. such as g.262 0.266 0. Best-fit model parameters are a = 2.945 1.844 2.322 0. N.936 0.203 ăSzi Mi ă 2.709 0.4 0.905 0.489 0.546 0.697 0.398 0.604 0.0732 0.071 0.433 0.039 0.2 109.434 0.422 0.110 0.907 0.563 0.847 0.079 3.3396 0.0363 0.2 174.0235 0. The following outlines the procedure for applying Gaussian quadrature to the gamma-distribution function.466 3.4 0. Determine the number of C7) fractions.335 0.669 0.555 0.939 0.715 0.857 0.919 0.396 0.815 0.236 0.357 22.754 0.6 0.1 192.9 127.371 0.929 0.7 188.5 151. and obtain the quadrature values Xi and Wi from Table 5.941 0. and w).5 and h = 72.859 0.097 63. molecular weight of fractions for the Hoffman et al.0028 0.0246 0.3483 0.867 0.0130 0.604 2.0326 0.371 5.3444 0. 5.178 8.3072 0.371 0.769 0.719 0.0224 0.203 Qwi 0.749 0.000 h+80 0.29 oil based on the best fit in Fig. 9 .2 142.497 3.8 1.0263 0.5.978 14.848 0.031 66.2 0.008 2. b * + ǒ M N * h ǓńX N .888 0.2035 0.3545 0.995 36.0143 0.8 191.227 67.478 0.036 0.952 1.11—Graphical fit of the Hoffman et al.133 60.0241 0.720 0.194 0.607 58.2398 0.7 zi Mi 2.0956 0.778 0.7 h+70 0.0108 0.940 0.3570 0.5 198.0263 0.29 oil molar distribution by use of the cumulative-distribution type curve.899 0.285 62.6 182.948 1.308 0.27 proposed perhaps the most useful application of the gamma-distribution model.8 120.0058 0.345 0. 5.000 h+75 0.458 45.0 171.811 0.879 0. recommended values are h+90 and a+1.0234 0.832 0.604 5.TABLE 5.0 180.919 0.538 49. 0.000 h+85 0. 5.9 184.905 0.682 0.475 18.3517 0. Each fluid sample can have different C7) properties when the split is made so that each split fraction has the same molecular weight (and other properties.634 0.3180 0.582 1.3593 0.893 0.951 1.067 7.622 0.0039 0.0229 0. 1.0023 0. Qzi Fig. When TBP data are not available to determine these parameters.909 0.872 0.6 (values are given for N+3 and N+5).931 0.841 0.932 0.791 0.928 0.506 0.928 28.526 2.0 Y h + 65 J h + 70 F h + 75 X h + 80 the mole fractions are different for each fluid sample.265 68.152 1.903 0.1443 0.160 0.957 3.0 Cumulative Normalized Mole Fraction.875 0.293 0.627 0.620 0.600 55.676 0.216 0.000 h+72.366 41. pc .760 0.348 0.3 189.671 0.0171 0.0048 0.891 0.11 with a = 2.758 0.574 2.12—Calculated normalized mole fraction vs.6 0.918 0.0072 0.2 134. 3.0087 0.021 10.882 4.0091 0.29 OIL Q* Mi Component i 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 Total zi 0.839 0. Example applications include the characterization of a gas cap and underlying reservoir oil and a reservoir with compositional gradient.396 0.586 0.093 4.640 0.925 0.198 0.0 0.782 0.7 157.570 6.824 0.854 0.744 0.134 52.7 167.497 0.210 0.1197 0.280 0.049 73.5—CALCULATION OF CUMULATIVE WEIGHT FRACTION AND CUMULATIVE MOLECULAR WEIGHT VARIABLE FOR HOFFMAN et al.2941 0.5M C7)).945 1.818 0.799 0. With Gaussian quadrature. while 1.3267 0.384 0.150 0.870 0.0497 0. Tb .787 0.894 0.722 0.316 0.836 0.5 0.683 0.913 0. 2.2627 0.251 0. Tc .123 0.204 0.889 0.950 1.0 104.5.154 73.225 0.457 0.294 0.926 4.3684 Mi 99 110 121 132 145 158 172 186 203 222 238 252 266 279 290 301 315 329 343 357 371 385 399 444 198.963 0.2 0.1709 0.0025 0. Specify h and a.916 0.6 114.0 0.8 162.3684 ąSzi ą 0.000 dictions.196 1.794 0.499 0.825 0.249 0. The volatile oil shown in Fig. 5. specific gravity.2 gives an extended version of the Katz-Firoozabadi property table.30 4. (5.35) This relation is derived from the Riazi-Daubert14 correlation for molecular weight and is generally valid for petroleum fractions with normal boiling points ranging from 560 to 1.986 668 110 83 10*1 7. Because Gaussian quadrature is only approximate. the values of MN .32) Five Quadrature Points (plus fractions) However. The specific gravities. . that Eq. Some methods for estimating specific gravity and boiling point assume that a particular characterization factor is constant for all C7) fractions. . and critical properties. AND WEIGHT FACTORS. g. . Kw +8. . .336 997 238 58 10*5 weights can be used to convert weight fractions. Watson or Universal Oil Products (UOP) Characterization Factor. . . .32. . and for aromatic compounds. . Table 5. . paraffin values). . . Unfortunately.35 is not very accurate for fractions heavier than C20. . . . .294 280 360 279 6. . . . . Check whether the calculated M C7) from Eq.289 945 082 937 ȍ w ńM N j j+7 . The result of this characterization is one set of molecular weights for the C7) fractions.31 Katz and Firoozabadi6 suggest a generalized set of SCN properties for petroleum fractions C6 through C45.5.085 810 005 859 12. either from chromatographic analysis or from a synthetic split. .4. . zi + w i ńM i Three Quadrature Points (plus fractions) 1 2 3 0. .263 560 319 718 1. . . . . (5. M N is given by wN .596 425 771 041 7. inspection data from TBP analysis of a sample from the same field would be the most reliable source of M. For paraffinic compounds. the calculated M C7) may be slightly in error.g. but caution is required to avoid nonphysical adjustments. . and b* must be the same for all samples. However. C N. . . Figs. . and f( X ) + ( X ) a*1 ǒ1 ) ln dǓ a . . 5.. the boiling point of the heaviest fraction must be estimated. . . . the calculated molecular weight M N is often unrealistic because of measurement errors in M C7) or in the chromatographic analysis and because generalized molecular weights are only approximate. the utility of this and other characterization factors is that they give a qualitative measure of the composition of a petroleum fraction. d + exp ǒ a b* *1 MC * h 7) Ǔ . . molecular weights and mole fractions are assumed to be known for C7) fractions. The Watson characterization factor has been found to be useful for approximate characterization and is widely used as a parameter for correlating petroleum-fraction properties.640 800 844 276 3. g. Kw calculated with M C7) and g C7) in Eq. .5 to 11. Experience has shown. . . Molecular 10 . and Tb reflect the chemical makeup of petroleum fractions. . is not known. . . . .90"0. .2. . . . specific gravity for C7) fractions from two North Sea fields. . Inspection properties M. . . Kw +11. is based on normal boiling point. . .99"0. . (5. . . G(a ) dX . .g. . . . . . . of each sample will not be exactly reproduced with this procedure (calculated with Eq. have been proposed. Finally.34) Kw varies roughly from 8. . . . N*1 are taken from Table 5.12 equals the measured value used in Step 4 to define d. vapor pressure. . . 5. the molecular weight of the heaviest fraction. . but the heaviest specific gravity must be backcalculated to match the measured C7) specific gravity. (5. Fc ) must be the same for all mixtures.5579 M 0. 5. . Specific gravities for the C7) fractions can be calculated with one of the correlations given in Sec. Mi + h ) b* Xi . .1 Generalized Properties. For either situation. Generalized properties from a producing region. . is K w [ 4.13A indicate an average K wC7)+ 11. 5.31) where Mi for i+7. The Watson or UOP factor. .. and Tb for each C7) fraction. .4. a complete fluid characterization can be determined with correlations in Sec. from simulated distillation to mole fractions. An approximate relation2 for the Watson factor. The generalized properties for specific gravity and boiling point can be assigned to SCN fractions. (5. The next-best source would be measured TBP data from a field producing similar oil or condensate from the same geological formation. .6—GAUSSIAN QUADRATURE FUNCTION VARIABLES. . TbN can be estimated from a correlation relating boiling point to specific gravity and molecular weight.0. 5. .594 244 968 17 10*2 3.415 774 556 783 2.01 for a range of molecular weights from 220 to PHASE BEHAVIOR . . This situation arises when simulated distillation is used or when no experimental analysis of C7) is available and a synthetic split must be made by use of a molar-distribution model. . Tb . (5.4 InspectionĆProperties Estimation 5. During the remainder of this section.33) MN + N *1 ǒwC ńMC Ǔ * ǒwińMiǓ 7) 7) ȍ i+7 Quadrature function values and weight factors can be found for other quadrature numbers in mathematical handbooks. . and a combination of paraffins and aromatics will obviously “appear” naphthenic. 5.. X. for naphthenic compounds. Calculate the parameter d.611 758 679 92 10*3 2. such as the North Sea.84573 . . . . This can be corrected by (slightly) modifying the value of d. 5.13B has an average K wC7)+11. 5.5 to 13. Calculate the C7) mole fraction zi and Mi for each fraction. h. The molecular weight. however. Both w N and M C7) can be adjusted to give a “reasonable” M N. Eq. . .0 to 12. . . j . g. . . zi + zC 7) ƪ W i f (X i)ƫ.5. . Individual sample values of M C7) and a can. .44). . . W X W 7.5. be different. . 5. These methods are only approximate but are widely used. .…. . and boiling point of C7) fractions must be estimated in the absence of experimental TBP data.785 177 335 69 10*1 1. requiring some adjustment to generalized specific gravities. . . From a mass balance. where the characterization factor (e.g.310°R (C7 through C30). .30) 5. Having defined Mi and gi for the C7) fractions. .01 for a range of molecular weights from 135 to 150. . . .13A and 5. .35 is often constant for a given field. . .038 925 650 16 10*2 5. . 5. and repeating Steps 5 and 6 until a satisfactory match is achieved. Data for the gas condensate in Fig.13B7 plot molecular weight vs. .33 ń3 T1 b Kw 5 g . while each sample has different mole fractions zi (so that their average molecular weights M C7) are honored). such as molecular weight. . . . .110 930 099 29 10*1 2. . . . . in °R and specific gravity.5 to 13.2 Characterization Factors.TABLE 5. . . however.15178 g *0. based on molecular weight and specific gravity. Kw +12.4 (e.37). g C7) .217 556 105 83 10*1 1 2 3 4 5 0. . wi .5. . . Some overlap in Kw exists among these three families of hydrocarbons. 6. Kw. 5. .413 403 059 107 3. When characterizing multiple samples simultaneously. . . The calculated gN also may be unrealistic. The same problem is inherent with backcalculating M N with any set of generalized molecular weights used for SCN Fractions 7 to N*1 (e. .0005). viscosity. but the average characterization factor can be chosen so that the differences are very small ( g"0. . . . Fig.37) The Watson factor satisfying Eq. i .775 would be expected to have a molecular weight of [141 (for K wC7)+ 11. . The behavior of specific gravity as a function of molecular weight is similar for the Jacoby factor and the relation for a constant Kw. . 5. (g C7)) exp.84573 . . 5. . . the spread in K wC7) values will exceed "0. . such as "0. . specific gravity. . . The Jacoby aromaticity factor. . . . . molecular weight for C7) fractions for a North Sea Volatile-Oil Field 3B(after Austad et al. .41) 0.40 overpredict g and Tb at molecular weights greater than [250 (an original limitation of the Riazi-Daubert14 molecular-weight correlation).25 Boiling points. molecular weight for C7) fractions for a North Sea Gas-Condensate Field 2 (after Austad et al. . . . Kw ) Kw Ja Fig. . . . . . .13A—Specific gravity vs. (5. 5. . . Ja ) Present Correlation (Watson Factor. . In this case. . . . . . . . 255. . . T bi + (K wg i) . ǒ g C 7) Ǔ exp + zC N 7) i MC i 7) i ȍǒz M ńg Ǔ i+1 . . . . .82053 . .90. . .7).35 can also be used to calculate specific gravity of C7) fractions determined by simulated distillation or a synthetic split (i. indicates possible error in the measured data.2456 * ǒ1.36) Kw must be chosen so that experimentally measured C7) specific gravity. For the gas condensate in Fig. gi . .18241 Kw . . a C7) sample with specific gravity of 0.0108 M i0.77ńMǓ 11 where A 0 + ȍz M i i+1 0.13A. 5.38) Unfortunately. can be estimated from Eq. .e. . . . . 5.01 when measurements are performed by a commercial laboratory. . . . 3 .36. . . . . . However. Jacoby Aromaticity Factor.14—Specific gravity vs. . 5. (5. . .34 Fig. 5. . . . . . .8ńMǓ . . . .8468 ) ǒ15. If the measured value was 135. When the characterization factor for a field can be determined.142 shows the original Jacoby relation between specific gravity and molecular weight for several values of Ja . Jacoby Correlation (Aromaticity Factor. . . . Assuming a constant Kw for each fraction. . is calculated correctly. . The high degree of correlation for these two fields suggests accurate molecular-weight measurements by the laboratory. . the C7) molecular weight should be redetermined. . Eq. .17947 *1. .36 through 5. flattening at high molecular weights (a more physically consistent behavior). .35 is useful for checking the consistency of C7) molecularweight and specific-gravity measurements. A relation for the Jacoby factor is Ja + g * 0. Significant deviation in K wC7) . . is an alternative characterization factor for describing the relative composition of petroleum fractions. . . Eq.7). . . .40) . . . . . (5. . After Whitson. 5. . .99). . specific gravity calculated with the Jacoby method increases more rapidly at low molecular weights. . .37 is given by Kw + ƪ 0. . . . .. MC 7+ Molecular Weight. . (5. . . . . (5. . can be calculated from g i + 6.Molecular Weight. 5. MC 7+ Fig. an anomalous K wC7) usually indicates an erroneous molecular-weight measurement. In general. Because molecular weight is more prone to error than determination of specific gravity. .39) HEPTANES-PLUS CHARACTERIZATION . Tbi . . .03 for the North Sea fields above. . . . the Watson characterization factor would be 11. . (5.99. which is significantly lower than the field average of 11.13B—Specific gravity vs. . molecular weight for constant values of the Jacoby aromaticity factor (solid lines) and the Watson characterization factor (dashed lines). 5. . Eqs. Ja . . . . when only mole fractions and molecular weights are known).16637 g C zC N 7) A 7) 0 MC 7) ƫ *0. . Whitson26 has fit the seven aromaticity curves originally presented by Yarborough using the equation g i + expƪA 0 ) A 1 i *1 ) A 2 i ) A 3 ln( i )ƫ . . .6 ) 9.97476 6. Boiling point can be estimated from molecular weight and specific gravity with one of several correlations. . . 486.071K w) ln . .5673 2.462 10 5Ǔ M *0. . . Søreide developed an accurate specificgravity correlation based on the analysis of 843 TBP fractions from 68 reservoir C7) samples.3 * ǒ1.4 (a+1.73719 10*1 *7.47) American Petroleum Inst.2855 ) C f (M i * 66) 0. however. . . . when Ja +0) express the relation between specific gravity and molecular weight for normal paraffins.92005 10*2 *1. .7685 g .26 Ya 0.65111 10*1 *9. the Søreide correlation for estimating Tb from M and g. . .72341 *7.3 Boiling-Point Estimation. . 18 0. . . Søreide 35 Correlations. . is related to specific gravity and carbon number.7. . (5.8 gives estimated specific gravities determined with the methods just described for a C7) sample with the exponential split given in Table 5. (5. . .65465 10*2 1.0164 10 *5)T b g . .45) 10 *3Ǔ Mgƫ . . . . 5. 5. .15 shows how the Yarborough aromaticity factor.832. .79T * b 1Nj 10 7ƫ T * b ) NJǒ1 * 0.49) Riazi and Daubert. .922 ) ǒ3. . .15—Specific gravity vs.118 1. .46) Riazi and Daubert. g i + 0. . . .27 ) 0. . A simple relation representing Ya is not available.98T – b 3Nj 10 12ƫ T * . . .02226g 2Ǔ 1Ǔ ƪǒ1. .e.96 T b g expƪǒ5. 272.07gǓ . . .3 0.51274 M + 581. Fig. . Fig.6523 * 3. . . .48) Rao and Bardon. g + 0. . Ya . .02058g 2Ǔ 1Ǔ ƪǒ1. . however.88 10 2Ǔ T b g expǒ0. 5. .43855 10*2 A1 *1.27 Yarborough Aromaticity Factor.34169 10*2 8. .93708 10*4 3.42) M M The first two terms in Eq. . . . b .87273 10*3 2. .7—COEFFICIENTS FOR YARBOROUGH AROMATICITY FACTOR CORRELATION1.07746 10*1 1. .13 . . .00017 10*1 *7.1 0. .08382 10*3 PHASE BEHAVIOR *4. (5. . The following equations also relate molecular weight to boiling point and specific gravity. 14 M + (4.44068 10*1 *1. M C7)+ 200) and g C7)+0.47553 10*1 *4. The Jacoby factor satisfying measured C7) specific gravity (Eq. Søreide also developed a boiling-point correlation based on 843 TBP fractions from 68 reservoir C7) samples. . . 5. .78432 10*1 A2 1.695 exp ƪ * ǒ4. .30947 10*5 1.67141 10*2 1. . . .25800 10*1 *4. . (5.3437 * 720.4 0. (5..11056 10 *3ǓT bgƫ . (API).77982 10*4 4.Cf typically has a value between 0. . 36 M + ǒ2.3287g)T bƫ ) NJ ǒ1 * 0. .8 12 A0 *7. .43) where i+carbon number. . ) 1. . . 5. . . Yarborough1 modified the Jacoby aromaticity factor specifically for estimating specific gravities when mole fractions and molecular weights are known. Table 5. . . carbon number for constant values of the Yarborough aromaticity factor (after Yarborough1).43076 10 *3ǓT b * 9. 5.8828 * 181. .31 and is determined for a specific C7) sample by satisfying Eq. .2 0.43.50) TABLE 5. . (5.39960 *1. . . 37 T ǒ22.1962 *1. . .03522 g 3. .7 gives the constants for Eq. . . . 5. .77 . . .6 0. .44) ǒ Ǔ with Tb in °R. .37.27 and 0. .8 ) J a 0. . Linear interpolation of specific gravity should be used to calculate specific gravity for a Ya value falling between two values of Ya in Table 5. (5. 12 M + ƪ* 12.8 Ǔ.80882g ) 0. .58616 10*3 .311. T b + 1928. . h+90.68K b w ln M + (1.39105 10*1 *2. (5. The aromaticity factor required to satisfy measured C7) specific gravity (Eq.4.17618 10*1 *3. . (5.19267 10*1 5. . . .266 10 *3Ǔ M * 4. Table 5.39412 10*3 *3.0438 0. . 5. We have found that this relation is particularly accurate for gas-condensate systems. . Kesler and Lee. . . . . .53384 g ) ǒ1.38058 10*3 A2 *3.00218T b * 3.4g ) (4. . or. We recommend.42 (i. . in terms of specific gravity. Yarborough tries to improve the original Jacoby relation by reflecting the changing character of fractions up to C13 better and by representing the larger naphthenic content of heavier fractions better. The Jacoby factor can also be used to estimate fraction specific gravities when mole fractions and molecular weights are available from simulated distillation or a synthetic split.0 0. . .77084g * 0.2456 * 1. .97063 *7. any of these correlations can be solved for boiling point in terms of M and g. .80564 10*3 5.37) is determined by trial and error. .37) must be calculated by trial and error. .8468 * 15. . . . 8753 0.078 T b g .0 Kw +12. . Tc . TbF in °F+Tb *459.0121 0. .8966 0. 41 (modified by API36).53) .8 194. w. .8 348.26. . .7951 0.8161 0.7656 0. . critical pressure. (5.8072 0. .0718 0.8554 0.0926 0. .89212579 ) ǒ0.7 ) 811g ) (0.8 180.7660 0.8380 0.7972 0.8 124.8668 0. several reviews22.2985 T c + 19.7789 0.8 264. .1195 0.1052 0.8320 Ja +0. . T c + 645. (5.8426 0.9468 0.1174g)T b ) (0. . .8722 0. .8423 0. .5. 14 T c + 24.11857 0. .07121 ) 1. . .51) ƪǒ 0. Roess.8652 0.8988 0.0566 g * . pc . .TABLE 5. . .0137 0.2395 0.10834003 * ǒ0.8 152.8 166. (5.7486 0.8 292.3596 . .24244 ) 2. .7051 0.2623g) Cavett. .2794 0.9048 0. . .7550 0.8 320.8782 0.0259 0.0156 0.8687 0. .7353 0.and four-constant cubic EOS’s.7684 0.9514 0. specific gravity.8514 0. . vc . .8539 0. .7327 0. .1 Critical Temperature.0557 0.0000 Mi 96. and boiling point. .8898 0.7981 0.55) 5. (5.8 250.0891 1. correlations based on perturbation expansion (a concept borrowed from statistical mechanics) are discussed separately. Nokay.8805 0.8136 0.4669 * 3. . .8466 0. and acentric factor.0295 0.8 334. The following critical-temperature correlations can be used for petroleum fractions. .8 236. . . 43 0.4244 ) 0.7511 0.8442 0.8320 5. Oil gravity is denoted gAPI and is related to specific gravity by gAPI+141.83 ) 1. .0177 0.8318 0.8088 0. (5. . . .8740 0.8 110. .5. .7127 10 *3Ǔƪgǒ T bF ) 100 Ǔƫ .2898 g ) g2 Ǔ 10 *3 T b 13 ƫ HEPTANES-PLUS CHARACTERIZATION .8178 0.7813 0.0816 0. 2 Kesler-Lee. .18) that is intended to highlight differences between the correlations.8604 0. .8028 0.8 222. The following are pc correlations that can be used for petroleum fractions.5.8567 0.8374 0.8335 0. pc correlations are less reliable than Tc correlations.0380 0.8570 0. We present the most commonly used correlations and a graphical comparison (Figs. Now we must consider the problem of assigning critical properties to each pseudocomponent.8314 0.5/g*131. .8253 0. . . Kesler-Lee.8861 0.2 Critical Pressure. .0632 0. .16 through 5.39. .0432 0.8241 0.8926 0. .5309492 ) ǒ0.8075 0.8646 0. . .5 CriticalĆProperties Estimation Thus far.9033 0. .8 278. molecular weight. . . Tc in °R. and vc in ft3/lbm mol. .8 306. 2 10 *2ǓT bF 1 10 5T * b . 5.7286 0. 12 ln p c + 8.8793 0. The units for the remaining equations in this section are Tb in °R. . and critical molar volume is used with the LBC viscosity correlation.8294 0. .7472 0.38890584 ) ǒ0.8598 0.8646 0. we have discussed how to split the C7) fraction into pseudocomponents described by mole fraction. pc in psia.080 0.8189 0.62164 0. .8505 0. .8500 0.8—COMPARISON OF SPECIFIC GRAVITIES WITH CORRELATIONS BY USE OF DIFFERENT CHARACTERIZATION FACTORS gi for Different Correlations With Constant Characterization Factor Chosen To Match g C + 0.8482 0.8320 Ya +0. 42 T c + 768. 12 T c + 341.8623 0.52) 10 *2Ǔ g APIT bF 3 10 *6ǓT bF 2 10 *5Ǔ g APIT bF 2 10 *7Ǔ g 2 APIT bF . Finally. of each component in a mixture are required by most cubic EOS’s.27871T b0. Critical-property estimation of petroleum fractions has a long history beginning as early as the 1930’s. Tc is perhaps the most reliably correlated critical property for petroleum fractions. . 5.7849 0.7913 0.8 138.8 466.8391 0. . .0 200.0490 0.8821 0.67.327116 Riazi-Daubert.40 are available.25. .0201 0.7856 0. . Critical volume.8320 Cf +0.8 208.8238 0. is used instead of critical pressure in the Benedict-Webb-Rubin38 (BWR) EOS. . .0228 0.8697 0. . Critical temperature.7719 0.7133693T bF * ǒ0.8677 0.3634 * 0.8597 0.2864 0.24 Critical compressibility factor has been introduced as a parameter in three.6667ƪgǒ T bF ) 100 Ǔƫ * ǒ0. .0335 0.58848 g 0.9096 0.54) . . .832 7) Fraction 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Total zi 0.8827 0.7177 0. . A4+ *0. Numerically. . . Kesler-Lee.60) where A1+*5.7Ǔ ) A 1 ) A 2 T * 3 br 4 br br 1 ) A ln T ) A T 6 A5 ) A6 T* 7 br 8 br br 5. . . . . . (5. . . . . . . .11047899 * ǒ0. . . . . .2). . w + * 7.8 is based on an accurate vapor-pressure correlation for pure compounds.359T br ) (1. .48271599 ) ǒ0. .7 to Tr +1.500°R assuming a constant Watson characterization factor of 12. .01 for w+1. . . . A5+ 15.17—Comparison of critical-pressure correlations for boiling points from 600 to 1.09648. . . 5. . .169347. 13 (Tbr +Tb /Tc t0. . .6977 g2 Ǔ 10 *7 T 2 b ƫ Ǔ 10 *10 T 3 b.25 for C5. . . .5. . A2+ 6. .1352K w * 0. .7Ǔ w+3 * 1. . . .56) Cavett. .43577. and [0.2087611 ) ǒ0.8) is developed specifically for petroleum fractions. Lee-Kesler. . .30474749 * ǒ0. . . . . . . . 7 ƪǒT cńT bǓ * 1ƫ . .44 defined acentric factor as ǒ Ǔ * 1. .57) Fig.47227 1.8290406 ) ǒ0.16—Comparison of critical-temperature correlations for boiling points from 600 to 1. acentric factor gives a measure of the steepness of the vapor-pressure curve from Tr +0. Pitzer et w 5 * log al.58) 10 9ǓT * g b 2 10 *5ǓT bF 10 *3ǓT bF 10 *4Ǔ g APIT bF 3 10 *8ǓT bF 2 10 *7Ǔ g APIT bF 10 *7Ǔ g 2 APIT bF 2 10 *9Ǔ g 2 APIT bF .648 g ) g2 0.92714.1 for w+0 and p * v /pc +0. . p* v pc .408 * 0. .Fig. .62) PHASE BEHAVIOR . 5.12281 2. A7+*13.500°R assuming a constant Watson characterization factor of 12. 14 p c + ǒ3.8). . and A8+ 0. (5. . . w+ 1 ) A ln T ) A T 6 – lnǒ p cń14. . 14 . . . A6+*15. ƫ . . . The Edmister45 correlation is limited to pure hydrocarbons and should not be used for C7) fractions. . ) * ƪǒ ƪǒ 0.8). . .01063K w)T * br . 42 log p c + 2. .7). . w increases to u1.0 for petroleum fractions heavier than approximately C25 (see Table 5.42019 ) 1.4685 ) 3.3201 . . where p * v /pc +0. . . . . (5. . . . .3 Acentric Factor. . . (5. . . .94120109 * ǒ0.18—Comparison of acentric factor correlations for boiling points from 600 to 1500°R assuming a constant Watson characterization factor of 12.007465K 2 w 1 ) 8. whereas the correlation for Tb /Tc t0. A3+ 1. 5. v+vapor pressure at temperature T+0. 45 logǒ p cń14.904 ) 0. w[0. .01 for methane.6875.2518. .4721. . . 12 (Tbr +Tb /Tc u0. . .59) where p * c (Tr +0. . (5. The three correlations follow.5 for C8 (see Table A. (5. [0. (5.13949619 Riazi-Daubert.28862. . . . . . .3125 2.15184103 ) ǒ0. Fig.61) Edmister. .7T Practically.1 for literature values of acentric factor for pure compounds). The Kesler-Lee12 acentric factor correlation (for Tb /Tc u0. . Critical Volume. . . . . .53262 * 46. . . (5. . .0175691 ) 0. .73) HEPTANES-PLUS CHARACTERIZATION . .1952a ) 104. . . . . .779681 ) and Dg T + exp[5( g P * g )] * 1.5. . . .191017 ) ǒ0.419869 * 0. . . . .500°R. . . . . .284376 ) (0. . .533272 ) ǒ0. .64) Critical compressibility factor. .843593 * 0. . Fig. . then the correlations for petroleum fractions. . . . . (5. .18 compare the various critical-property correlations for a range of boiling points from 600 to 1. . .193168 Dg M . .36159a 3 * 13749. . . . 0. . . Critical Temperature.67) and Dg p + exp[0. . . . The Hall-Yarborough46 critical-volume correlation is given in terms of molecular weight and specific gravity.40 and Pitzer et al.78) and T b + exp(5. 14 v c + ǒ7. . . . . . Z c [ 0. . . . (5. . Kesler et al. .71) Figs. (5. . .7935 . for example. . . . .8888a ) 36. . . .67 through 5. . .00127885T b 0. . . . . . . . . . . . .71419 ) 2.683 . . ) 2f ǒ1 Ǔ.70) x + 0. whereas the Riazi-Daubert14 correlation uses normal boiling point and specific gravity.5 Tb .959468 10 2) ǒ0. p c + p cP ) 2f ǒTT ǓǒVV Ǔǒ1 Ǔ. 46 v c + 0. Zc can be calculated directly from critical pressure.08w . . . . . (5. . .362456 ) 0. .5 Tb ǒ 0.47 first used the perturbation expansion (with n-alkanes as the reference fluid) to develop a suite of critical-property and acentric-factor correlations. . . 5.0398285 * 0. (5. . . . . . . . 1 * 2f 2 c cP c p p cP ƪ 10 *3 ǓT b 3 10 *10ǓT b f p + Dg p ƪǒ 2. .28659q 2 * 39. . . .01721 Dg v .7522q ) 35.14 ) 0.69) f M + Dg M |x| ) ƪ ǒ * 0. . . . . . . . . . . . Twu48 uses the same approach to develop a suite of critical-property correlations. . . . 0. . . . . . . . (5.66 is not particularly accurate (grossly overestimating Zc for heavier compounds) and is used only for approximate calculations. . . . (5. . . . . .948125 Dg T .01T bǓ 13 ƫ ǒ *1 * 11. . critical volume. . .76) Ǔ ƫ 2Ǔƫ and Dg v + expƪ4ǒg 2 * 1. . . 5. . . 5. . .73 must be solved iteratively.4 Critical Volume. . .5 ) 34. . . .5a 12 . . and specific gravity of a paraffin with the same boiling point as the petroleum fraction must be calculated first. . .74) .1928 illustrates the important influence that critical properties have on EOS-calculated properties of pure components. .193a ) . and molecular weight have been developed for petroleum fractions with a perturbation-expansion model with normal paraffins as the reference system. . . . .0434 2. . . . (5.66) f T + Dg T ƪ ƪ * 0. . . . . . . . . .44 * 0. . . 0. . . 2 T T where R+universal gas constant. . . . and Eqs. T where a + 1 * b T cP and q + ln M P . To calculate critical pressure. . . . . . . . . . . .5. 5. The following relations are used to calculate petroleum-fraction properties. (5. . .128624a * 3. Hall-Yarborough. (5. . . . although the properties at higher carbon numbers are only approximate because experimental data for paraffins heavier than approximately C20 do not exist. . . . . . . . . . . . . . . .8544q *1 * 0. . . . . . Vapor pressure is particularly sensitive to critical temperature.77) Molecular Weight. .7a 14)] *8 .4277 ) 252.182421 ) 3. . v c + v cP ) 2f ǒ1 Ǔ. . . . .122488q *2) * 24. .5 Tb Ǔ ƫ . . . . . . .3829 *1. critical volume.328086 . . . . . . .5 Tb Ǔ 2 10 *7ǓT b * ǒ0. Thus. . 5.6 Methods Based on an EOS. . . . .5 Tb ǒ * 0. . . We give his normal-paraffin correlations first. . . Normal Paraffins (Alkanes). . . an initial guess is given by MP [ Tb . T cP + T b 0. Riazi-Daubert. . Reid et al. . . . . (5. . . . .1955 * 0. . Correlations for critical temperature. . . . 2 4 2 . .15 g *0. . 0. .505839a * 1. P*g Critical Pressure. . .5 Correlations Based on Perturbation Expansions. and critical temperature. . .75) Ǔ ƫ Eq. .65) RT c Twu claims that the normal-paraffin correlations are valid for C1 through C100. .025 M 1.56436a * 9481. (5. . . . the Riazi-Daubert19 critical-temperature correlation for toluene overpredicts the experimental value 15 . 5. . .63) . . . . . . . .19629a 0. . . .16 through 5. . 1 * 2f 2 M M p cP + (3. . (5. (5. critical temperature. . . . . . . .3155q 2 . . . . . . . .44 give an approximate relation for Zc . . critical volume. 10 *7Ǔ T b g Paraffin molecular weight. is not explicitly a function of Tb . .00230535T b Dg p .466590 ) 0 .5 Tb . . . For example. .5.72) . . . . . . . . . . . . MP. . . . 0.83354 ) 1. . .5 Tb and Dg M + exp[5(g P * g)]. .68) 3 ln M + ln M P v cP + [ 1 * (0. . . . . . . . T c + T cP ) 2f ǒ1 Ǔ 1 * 2f . . .291 * 0. . 10. .71579q * 0. (5. . Zc .0052T b . . . 1 * 2f 2 v v f v + Dg v 0. is defined as Zc + p cv c . . . .5( g P * g )] * 1. . . . . . . . .012342 * 0. critical pressure. . .5. .5 Tb Ǔ ƫ g P + 0. . . 5. h+90) with Gaussian-quadrature or equal-mass fractions or (2) the exponential distribution (Eq.56 and some three-constant characterizations should provide similar accuracy but are not significantly better).5019 ò ) 208. ò n *1 ( ò6+0.4 .19—Effect of critical temperature on vapor-pressure prediction of toluene with the PR EOS. . . use the Lee-Kesler12/Kesler-Lee13 correlations.7). .7184 10 *6ǓM 2. alternatively.28 5. . saturation pressures ("5 to 15%). the EOS used for mixtures should also predict the behavior of individual components found in the mixture. ln p c + * 0. and both retrograde and near-critical liquid volumes. 2. 5.28).44). Whitson50 suggests fixing the value of w with an empirical correlation and adjusting Tc and pc to match normal boiling point and molar volume (M/g) at standard conditions.1 SRK-Recommended Characterization. . 4. . Improved predictions can be obtained only by tuning EOS parameters to accurate PVT data covering a relatively wide range of pressures. 5. pure component c values are taken from Peneloux et al. 3. 5. Specific gravities should be estimated with the Søreide35 correlation (Eq.a+1. and condensate-liquid dropout ("5 to 10% for maximum dropout.0096707 ò * ǒ3. near-critical saturation pressure and saturation type (bubblepoint or dewpoint). a mathematical split should be made with either (1) the gamma distribution (default 16 Note that the use of acentric factor is circumvented by directly calculating the term m used in the a correction term to EOS Constant A. two of the EOS parameters (Tc . the two characterization procedures give the same results for almost all PVT properties (usually within 1 to 2%). Practically. . Choose the acentric factor to match Tb . . . as discussed later. . . Determine volume-translation coefficients. we recommend the nonhydrocarbon BIP’s given in Chap. Boiling points should be estimated from the Søreide correlation (Eq. volume-shift factor. The recommended EOS methods are less reliable for prediction of minimum miscibility conditions. Calculate critical properties of all C7) fractions (distillation cuts from C7 to Cn *1 and split SCN fractions from Cn through C80) using the correlations T c + 163. .12 ò ) 86. (5. si . . where A0 and A1 are determined by satisfying the experimental-plus density. Split the plus fraction Cn ) (preferably nu10) into SCN fractions up to C80 using Eqs. gas/oil ratios and formation volume factors ("2 to 5%).23 A better (and recommended) approach for cubic EOS’s is to use the volume-shift factor s (see Chap.7 Grouping and Averaging Properties The cost and computer resources required for compositional reservoir simulation increase substantially with the number of compoPHASE BEHAVIOR . the Pedersen et al. This approach proved particularly successful when applied to the BWR EOS for residual-oil supercritical extraction (ROSE).052 ln M ) 0. . pc . . the vapor pressure is accurately predicted because all EOS’s force fit vapor-pressure data.48 ) 1. 5. . 3. .7431 ) 0. w. Some EOS’s are also fit to saturated-liquid densities at subcritical temperatures. Group C7) into 3 to 12 fractions using equal-weight fractions in each group. The measured properties of petroleum fractions. (5.52 6.43475 M * 1877. . For the PR EOS. The effect of critical properties and acentric factor on EOS calculations for reservoir-fluid mixtures is summarized by Whitson. and specific gravity can also be fit by the EOS.13408 ) 2.23 or Jhaveri and Youngren’s53 correlation for the PR EOS. and measured (or assumed) density. with poorer prediction of taillike behavior just below the dewpoint). 5. and compositions (see Sec. ò n). 4 and the modified Chueh-Prausnitz54 equation for C1 through C7) pairs. 5. Other methods for forcing the EOS to match boiling point and specific gravity have also been devised.690 can be used for C7)). Critical properties satisfying these criteria are given for a wide range of petroleum fractions by the PR EOS and the Soave-RedlichKwong (SRK) EOS.574 w * 0.7%. Use the Twu48 (or Lee-Kesler12) critical property correlation for Tc and pc . 5. . use weight-average mixing rules. Alternatively.55 characterization procedure can be used with the SRK EOS. 2. For pure compounds.176 w 2 + 0. . to match specific gravities. . Even with this slight error in Tc . .80) Tc underpredicted← →Tc overpredicted Deviation From Experimental Value. k ij ȱ +A ȧ1 * Ȳ ǒ ń6 1ń6 2v 1 v cj ci ń3 v1 ci ) ń3 v1 cj Ǔȳ ȧ ȴ B . . use Peneloux et al. .2 M M2 and m SRK + 0. % Fig.45). 4.7 through 5. 5.26 In principle. by only 1. . we can expect reasonable predictions of densities and Z factors ("1 to 5%). .6. choosing Cf to match measured C7) specific gravity (Eq. 1. The two recommended C7) characterization procedures outlined previously for the PR EOS and SRK EOS are probably the best currently available (other EOS characterizations.22. Because only two inspection properties are available (Tb and g). Calculate volume-translation parameters for C7) fractions to match specific gravities. SRK BIP’s given in Chap. 4 are used for nonhydrocarbon/hydrocarbon pairs. For each petroleum fraction separately. 1.46 * 3987.37). s.7 and Appendix C).0048122 M ) 0. . M . the average error in vapor pressures predicted by the Peng-Robinson49 (PR) EOS is 16%. 4) to match specific gravity or a saturated liquid density and acentric factor to match normal boiling point. such as the Redlich-Kwong EOS modified by Zudkevitch and Joffe.79) with A+0. boiling point.’s52 correlation for the SRK EOS22. only two of the EOS parameters can be determined. . AAD+absolute average deviation (after Brulé et al. Calculate SCN densities ò i (gi + ò i /0. All hydrocarbon/hydrocarbon BIP’s are set to zero.49 When measured TBP data are not available.6 Recommended C7) Characterizations We recommend the following C7) characterization procedure for cubic EOS’s. . With these EOS-characterization procedures. temperatures.11 and h+*4. alternatively. Brulé and Starling51 proposed a method that uses viscosity as an additional inspection property of the fraction for determining critical properties. .18 and B+6. or multipliers of EOS constants A and B) can be chosen so that the EOS exactly reproduces experimental boiling point and specific gravity.999) using the equation ò i+A0)A1 ln(i). . . . . . .. . Different criteria are used to determine the number of light and heavy pseudocomponents. . (5. . . TcI . . . and Ja plotted vs. . also suggest use of phase diagrams and compositional simulation to verify the grouped fluid description (a practice that we highly recommend). . . . . . . . .7. . . I) making up Pseudocomponent I. . In general. . . . . However. . . .88) 1 8v cI iŮI jŮJ 2 ń3 ń3 z i z j ǒT ci T cjǓ 1ń2 v 1 ) v1 ci cj ǒ Ǔ 3 ƫ B wI + ǒȍ Ǔ ǒȍ Ǔń ǒȍ Ǔ zi iŮI . C8. . .87) where I+1. . . . . . . . . . . . . . . . . or M) and zi +original mole fraction for components (i+1. . . . N) falling within boundaries MI *1 to MI are included in Group I. . v cI + T cI + ƪ ȍȍ ǒ ƪ ȍȍ 1 8 iŮI jŮJ ń3 ń3 zi zj v1 ) v1 ci cj Ǔ ń ȍz 3 iŮI ƫǒ Ǔ i 2 . . . . . and C10)). . Several methods have been proposed for calculating critical properties of pseudocomponents. Li et al. . The number of components used to describe a reservoir fluid depends mainly on the process being simulated. .and weight-average mixing rules and a proportioning factor. . 1. boiling point. . .86. g. .. . . .. f i + F q iz i ) (1 * F) q i w i and w i + zi Mi ȍz M N j j+1 . The following are the main questions regarding component grouping. . . .81) where N+carbon number of the heaviest fraction in the original fluid description. . . .. Whitson et al. . and p cI + Z cI R T cI v cI .85) where qI represents pcI . . With current computer technology. . 5. . . Average specific gravity should always be calculated with the assumption of ideal solution mixing. . . . . and wI . .. . . . C16 through C20. . . the C7) fractions are grouped into two pseudocomponents). . . . . . . . . .91) . .88 through 5. .59 suggest a method for grouping components of an original fluid description that uses K values from a flash at reservoir temperature and the “average” operating pressure. .to eight-component fluid characterizations should be sufficient to simulate practically any reservoir process. . and C21) is substantially better than splitting only the first few carbon-number fractions (e.. . . C9. . j . Whitson2 suggests that the C7) fraction can be grouped into NH pseudocomponents given by N H + 1 ) 3. . . (5.1 How Many and Which Components To Group. . . . . (5. . . . the following rule of thumb reduces the number of components for most systems: group N2 with methane. Li et al. . The simplest and most common mixing rule is ȍz q q + ȍz . . (5. Lee-Kesler13 proposed the mixing rules in Eqs. Lee et al. . . (2) gas cycling above and below the dewpoint of a gas-condensate reservoir. N2. . . .61. . . . . . . . .g. (5.2905 * 0. . . .90) iŮI iŮI Z cI + 0. .e. On the basis of Chueh and Prausnitz’s54 arguments. . . C7. . . and (4) immiscible and miscible gas-injection. . . NH . . . 5.. . . . . .7. from the original fluid description (i+7. . . (5. C11 through C15. . . . .. pc . . . . The groups are separated by molecular weights MI given by MI + MC the method is general and can be applied to any molar-distribution model and for any number of C7) groups. . CO2. . (5.. ..86) with 0xFx1. . . and C1 through C6) and “heavy” components (C7)). .. . . a full-field simulation with fluids exhibiting near-critical phase behavior is not feasible for a 15-component mixture. . (5. . .55 and others suggest use of weight fraction instead of mole fraction. including (1) reservoir depletion of volatile-oil and gas-condensate reservoirs.3 log( N * 7). . . . . Gaussian quadrature is recommended for choosing the pseudocomponents in a C7) fraction. . . . . Molecular weights. F. . . . . . . . . . . i i iŮI . .085w I . . Schlijper’s61 method also treats the problem of retrieving detailed compositional information from pseudoized (grouped) components. Although a simple exponential distribution is used with only two quadrature points (i. . . . . . How should the properties of pseudocomponents be calculated? 5. . . . (5. .e. . . . . iso-butane with n-butane. . How should the components be chosen from the original fluid description? 3. Nonhydrocarbon content should be less than a few percent in both the reservoir fluid and the injection gas if a nonhydrocarbon is to be grouped with a hydrocarbon.nents used to describe the reservoir fluid. . . . . . CO2 with ethane.2 Mixing Rules. Five. . . . . (3) retrograde condensation near the wellbore of a producing well. For example. Coats57 discusses a method for combining a modified black-oil formula with a simplified EOS representation of separator oil and gas streams. . .. .59 approach are valid alternatives. . . . . . . A compromise between accuracy and the number of components must be made according to the process being simulated (i. a detailed fluid description with 12 to 15 components may be needed to simulate developed miscibility in a slim-tube experiment. . . . . . . Behrens and Sandler62 suggest a grouping method for C7) fractions based on application of the Gaussian-quadrature method to continuous thermodynamics. . . . . . . .. most authors have found that broader grouping of C7) as C7 through C10.83) where qi +any property (Tc . . (5. . TcI . . . ȍz M . . .92) 17 . . (5. . The original mixture is divided arbitrarily into “light” components (H2S.82) ȍȍf f k i iŮI jŮJ j ij . . . Wu and Batycky’s63 empirical mixing-rule approach uses both the molar. .89) zi wi zi . . . .. and iso-pentane with n-pentane. g + ȍǒz M ńg Ǔ i i I iŮI i i i iŮI . . w.58 suggest that C7) fractions can be grouped into two pseudocomponents according to a characterization factor determined by averaging the tangents of fraction properties M. . .84) Pedersen et al. according to the expected effect that phase behavior will have on simulated results). . to calculate pcI . A generalized mixing rule for BIP’s can be written k IJ + 7 ǒM N ń MC 7 Ǔ 1ńN H . . . . and wI and fi +average of the molar and weight fractions. . . Mi . (5. . i i I iŮI i iŮI . however.92. . . . . . This method should only be used when C7) fractions are originally separated on a carbon-number basis and for N greater than [20. . 5. . . . . .27 show that HEPTANES-PLUS CHARACTERIZATION where fi is also given by Eq. equal-mass fractions or the Li et al. . . . . . How many components should be used? 2. . . . . qI + ȍf q .. . . Still other pseudoization methods have been proposed60. The “oil” and “gas” pseudocomponents in this model contain all the original fluid components in contrast to the typical method of grouping where each pseudocomponent is made up of only selected original components. . . . . . . . .94) W ai and W bi may include previously determined corrections to the o numerical constants W o a and W b. . 3. 5. This approach to determining pseudocomponent properties. 6. Kay’s64 mixing rule). pseudocritical properties ( pcI . is surprisingly accurate even for VLE calculations. 3. . (5. and wI ) are estimated with any mixing rule (e. and 11 1 1 2 2 1. . Lee et al. . Table 5. (CO2)C2)*C7). such as (N2)C1)*C7). . . Use the simulated PVT data as “real” data for pseudoization based on regression. . and 9 4 4 5 5 1. Simulate a suite of depletion and multicontact gas-injection PVT experiments that cover the expected range of compositions in the particular application. also regress on key BIP’s. Coats requires the pseudoized characterization to reproduce exactly the volumetric behavior of the original reservoir fluid at undersaturated conditions.87 for k I J .g. summarized in the following procedure.87 for k I J . .. 5. and 5 1 1 2 2 1. . w I) and W bI + ǒȍ Ǔń ǒȍ Ǔ zi bi zi iŮI iŮI ǒRT cIńp cIǓb I(T rI. 6. together with Eq. Coats also gives an 18 . saturation pressures.g. . 5. .93) R 2T 2 where a i + W ai p ci a i (T ri. In summary. 4. and 7 3 3 4 4 1. .. . A reduced-component characterization should strive to reproduce the original complete characterization that has been used to match measured PVT data. Use regression to fine tune the W aI and W bI values estimated in Step 4. Use the pseudoization procedure of Coats to obtain WaI and WbI values. TcI . (5. . 4. and the method is easy to implement. . . . 5. .3 Stepwise Regression. . and 4 3 3 4 4 F1 F2 F3 Step 5 N2)C1)CO2)C2 C3)i-C4)n-C4 )i-C5)n-C5)C6 F1 F2)F3* *Indicates the grouped pseudocomponents being regressed in a particular step.7.TABLE 5. 7. w i) . analogous procedure for determining pseudocomponent vcI for the LBC24 viscosity correlation. Coats57 presents a method of pseudoization that basically eliminates the effect of mixing rules on pseudocomponent properties. Repeat Steps 4 and 5 until the quality of the characterization deteriorates beyond an acceptable fluid description.9 shows an example five-step pseudoization procedure. The approach is simple and accurate. This is achieved by ensuring that the mixture EOS constants A and B are identical for the original and the pseudoized characterizations. and use Eq. 2. . and densities) are reasonably close to the predictions for the original (complete) characterization. 5. First. One approach to achieve this goal is stepwise regression. . . . 10. . multicontact gas injection data. . any grouping of a complete EOS characterization into a limited number of pseudocomponents should be checked to ensure that predicted phase behavior (e. Coats’ approach is preferred to all the other proposed methods. . . .58 and Whitson2 consider an alternative method for calculating C7) critical properties based on the specific gravities and boiling points of grouped pseudocomponents. 3. 1. Complete a comprehensive match of all existing PVT data with a characterization containing light and intermediate pure components and at least three to five C7) fractions. . . w I) . Then W aI and W bI are determined to satisfy the following equations. It ensures accurate volumetric calculations that are consistent with the original EOS characterization. . 9. Stepwise regression is the best approach to determine the number and PHASE BEHAVIOR ƪȍ ȍ W aI + iŮI jŮJ z i z j a ia j ǒ1 * k ijǓ ƫńǒȍ Ǔ zi iŮI 2 2 ǒR 2TcI ńp cIǓa I(T rI. .9—EXAMPLE STEPWISE-REGRESSION PROCEDURE FOR PSEUDOIZATION TO FEWER COMPONENTS FOR A GAS CONDENSATE FLUID UNDERGOING DEPLETION Original Component Number 1 2 Original Component N2 CO2 Step 1 N2)C1* CO2)C2* C3 i-C4 n-C4 i-C5 n-C5 C6 F1 F2 F3 Step 2 N2)C1 CO2)C2 C3 i-C4)n-C4* i-C5)n-C5* C6 F1 F2 F3 Step 3 N2)C1 CO2)C2 C3)i-C4)n-C4* i-C5)n-C5)C6* F1 F2 F3 Step 4 N2)C1)CO2)C2* C3)i-C4)n-C4 )i-C5)n-C5)C6* 3 4 5 6 7 8 9 10 11 12 13 kij Wa Wb Wa Wb C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 F1 F2 F3 Regression Parameters 1. . and other nonzero BIP’s involving pseudocomponents from Step 4. w i) ci RT and b i + W bi p ci b i(T ri. ci . 8. Create two new pseudocomponents from the existing set of components. Pennsylvania (1984) 8210. R. M. Katz. 1. M. 1.H. S.59 method.E. New Jersey (1991). I. (1984) 23. 22.R. 20. 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