Characterizing Hydrocarbon-Plus Fractions

Chapter

2

Characterizing Hydrocarbon-Plus Fractions Reservoir fluids contain a variety of substances of different chemical structure that include hydrocarbon and nonhydrocarbon components. Because of the enormous number components that constitute the hydrocarbon system, a complete chemical composition as well as identification of various components that constitute the hydrocarbon fluid are largely speculative. The constituents of a natural reservoir fluid form hydrocarbon spectrum from the lightest one; eg, nitrogen or methane, through intermediate to very large molecular weight fractions. The relative proportion of these different parts of the fluid can vary over a wide range which results in petroleum fluids showing very different features: eg, dry gas, retrograde, volatile oil, and others.

CRUDE OIL ASSAY Each crude oil has unique molecular and chemical characteristics. These variances translate into crucial differences in crude oil quality. A crude oil assay is essentially the evaluation of a crude oil by physical and chemical testing. It provides data that gives information on the suitability of the crude for its end use and also helps predict the commercial value of the crude. In addition, Information obtained from the petroleum assay is used for detailed equations of state tuning and validation. Crude oil bulk properties are useful for initial screening and tentative identification of genetically related oils. These bulk properties include several laboratory measurements that can be used to quantify the quality of the crude oil. A brief discussion of these properties is given below: n

The American Petroleum Institute (API) gravity: The API gravity is a standard industry property that is designed to quantify the quality of the crude oil. This oil property is defined in a manner, as given below, that suggests a higher value indicates a light oil and a lower value indicates a heavy oil. API ¼

141:5  131:5 γ

where γ is the liquid specific gravity. Equations of State and PVT Analysis. http://dx.doi.org/10.1016/B978-0-12-801570-4.00002-7 Copyright # 2016 Elsevier Inc. All rights reserved.

71

72 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

n

n

n

n

n

n

It should be noted that the API gravity of water is 10 because the water specific gravity is 1, which suggests that the API gravity for very heavy crudes could be less than 10 and varies between 10 and 30 for heavy crudes, to between 30 and 40 for medium crudes, and to above 40 for light crudes. The cloud point: The cloud point is defined as the lowest temperature at which wax crystals begin to form by a gradual cooling under standard conditions. At this temperature the oil becomes cloudy and the first particles of wax crystals are observed. Low cloud point products are desirable under low-temperature conditions. Cloud points are measured for oils that contain paraffins in the form of wax and therefore for light fractions (naphtha or gasoline) no cloud point data are reported. Wax crystals can cause severe flow assurance problems including plugging flow lines and reduction in well flow potential. The Reid vapor pressure (RVP): The RVP is the measure of vapor pressure of lightweight hydrocarbons present above the crude oil surface, the higher the value of RVP, the higher the percentage of light hydrocarbons in the crude oil samples. The pour point: The pour point of a petroleum fraction is defined as the lowest temperature at which the oil can be stored and still be capable of flowing through a pipeline without stirring. The sulfur content: The sulfur content in natural gas is usually found in the form of hydrogen sulfide (H2S). Some natural gas contains H2S as high as 30% by volume. The amount of sulfur in a crude is expressed as a percentage of sulfur by weight, and varies from 0.1% to 6%. Crude oils with less than 1 wt.% sulfur are called low-sulfur or sweet crude oil, and those with more than 1 wt.% sulfur are called high-sulfur or sour crude oil. The flash point: The flash point of a liquid hydrocarbon or any of its derivatives is defined as the lowest temperature at which sufficient vapor is produced above the liquid to form a mixture with air that can produce a spontaneous ignition if a spark is present. Flash point is an important parameter for safety considerations, especially during storage and transportation of volatile petroleum products (ie, LPG, light naphtha, gasoline). The surrounding temperature around a storage tank should always be less than the flash point of the fuel to avoid the possibility of ignition. The refractive index (RI): The RI represents the ratio of the velocity of light in a vacuum to that in the oil. The property is considered as the degree to which light bends (refraction) when passing through a medium. Values of RI can be measured very accurately and are used to correlate density and other properties of hydrocarbons with high reliability. Information obtained from RI measurements can be used to

Crude Oil Assay 73

n

n

n

n

n

n

n

determine the behavior of asphaltenes; specifically, their tendency to precipitate from crude oil. The freeze point: The freeze point is defined as the temperature at which the hydrocarbon liquid solidifies at atmospheric pressure. The smoke point: The smoke points is a property that is designed to describe the tendency of a fuel to burn with a smoky flame. A higher amount of aromatics in a fuel causes a smoky characteristic for the flame and energy loss due to thermal radiation. This property refers to the height of a smokeless flame of fuel as expressed in millimeters beyond which smoking takes place; ie, a high smoke point indicates a fuel with low amount of aromatics. The aniline point: This property represents the minimum temperature for complete miscibility of equal volumes of aniline and petroleum oil. The aniline point is important in characterization of petroleum fractions and analysis of molecular type. The Conradson carbon residue (CCR): This property measures the tendency and ability of the crude oil to form coke. It is determined by destructive distillation of a sample to the remaining coke residue. The CCR property is expressed as the weight of the remaining coke residue as a percentage of the original sample. The acid number: The acid number is a property that determines the organic acidity of a refinery stream. The gross heat of combustion (high heating value): This property is a measure of the amount of heat produced by the complete burning of a unit quantity of fuel. The net heat of combustion (lower heating value): This is obtained by subtracting the latent heat of vaporization of the water vapor formed by the combustion from the gross heat of combustion or higher heating value.

There are different ways to classify the components of the reservoir fluids. Normally, the constituents of a hydrocarbon system are classified into the following two categories: n n

Well-defined petroleum fractions Undefined petroleum fractions

Well-Defined Components Pure fractions are considered well-defined components because their physical properties were measured and compiled over the years. These properties include specific gravity, normal boiling point, molecular weight, critical properties, and acentric factor. Well-defined fractions are grouped in the following three sets:

74 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

n n n

Nonhydrocarbon fractions that include CO2, N2, H2S, etc. Methane C1 through normal pentane n-C5. Hydrocarbon components that include hexanes, heptanes, and other heavier components, (ie, C6, C7, C8, C9, etc.) where the number of isomers rises exponentially. For example, Hexanes “C6” refers to the hexane petroleum hydrocarbon distillation fraction that contains a high proportion of n-hexane in addition to 2-methylpentane, 3-methylpentane, and smaller amounts of other components. Similarly, the Heptanes refer to the heptane petroleum hydrocarbon fraction which contains isomers of the empirical formula C7H16 with the most prevalent components being n-heptane, dimethylcyclopentanes, 3-ethylpentane, methylcyclohexane, and 3-methylhexane. However, the identification and characterizing of each one of these components for us in EOS applications can be a difficult task. Therefore, these components are expressed as a group that consists of a number of components that have boiling points within a certain range. These groups are commonly denoted as single carbon number (SCN) and called pseudofractions because they represent a group of components; eg, C6 group, C7 group, and so forth. The tabulated laboratory data shown in Table 2.1 illustrates both the concept of SCN and pseudofractions.

Katz and Firoozabadi (1978) presented a generalized set of physical properties for the hydrocarbon groups C6 through C45 that are expressed as a SCN, such as the C6-group, C7-group, C8-group, and so on. These generalized properties, as shown in Table 2.2, were generated by analyzing the physical properties of 26 condensates and crude oil systems. The tabulated properties of these groups include the average boiling point, specific gravity, and molecular weight as well as critical properties. Ahmed (1985) conveniently correlated Katz and Firoozabadi’s tabulated physical properties with SCN as represented by the number of carbon atoms “n” by using a regression model. The generalized equation has the following form: θ ¼ a1 + a2 n + a3 n2 + a4 n3 +

a5 n

where θ ¼ any physical property, such as Tc, pc, or Vc n ¼ effective number of carbon atoms of SCN group, eg, 6, 7, …, 45 a1–a5 ¼ coefficients of the equation as tabulated in Table 2.3

Crude Oil Assay 75

Table 2.1 The Concept of Single Carbon Number Wellstream

Single carbon number group “SCN”

C6 group

C7 group

C8 group

C9 group

C10 group

*

Component

Mole %

GPM*

Hydrogen sulfide Nitrogen Carbon dioxide Methane Ethane Propane Iso-butane N-Butane 2,2-Dimethylpropane Iso-pentane N-Pentane 2,2-Dimethylbutane Cyclopentane 2,3-Dimethylbutane 2-Methylpentane 3-Methylpentane Other hexanes n-Hexane Methylcyclopentane Benzene Cyclohexane 2-Methylhexane 3-Methylhexane 2,2,4-Trimethylpentane Other heptanes n-Heptane Methylcyclohexane Toluene Other C-8s n-Octane Ethylbenzene m- and p-Xylene o-Xylene Other C-9s n-Nonane Other C-10s n-Decane Undecanes plus Total

0.002 0.084 1.163 65.410 12.107 5.329 1.280 2.039 0.012 0.899 0.902 0.031 0.006 0.060 0.370 0.221 0.000 0.560 0.098 0.066 0.144 0.235 0.211 0.000 0.213 0.471 0.329 0.286 0.815 0.388 0.088 0.354 0.113 0.547 0.299 0.742 0.247 3.880 100.000

0.000 0.000 0.000 0.000 3.220 1.459 0.416 0.639 0.004 0.327 0.325 0.013 0.002 0.025 0.153 0.090 0.000 0.229 0.035 0.018 0.049 0.109 0.096 0.000 0.092 0.216 0.132 0.095 0.380 0.198 0.034 0.136 0.043 0.285 0.167 0.425 0.151 3.131 12.692

Gallon-per-thousand cubic feet

76 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Table 2.2 Generalized Physical Properties Group SCN

Tb (°R)

γ

Kw

M

Tc (°R)

Pc (psia)

ω

Vc (ft3/lb)

C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20 C21 C22 C23 C24 C25 C26 C27 C28 C29 C30 C31 C30 C33 C34 C35 C36 C37 C38 C39 C40 C41 C42 C43 C44

607 658 702 748 791 829 867 901 936 971 1002 1032 1055 1077 1101 1124 1146 1167 1187 1207 1226 1244 1262 1277 1294 1310 1326 1341 1355 1368 1382 1394 1407 1419 1432 1442 1453 1464 1477

0.69 0.727 0.749 0.768 0.782 0.793 0.804 0.815 0.826 0.836 0.843 0.851 0.856 0.861 0.866 0.871 0.876 0.881 0.885 0.888 0.892 0.896 0.899 0.902 0.905 0.909 0.912 0.915 0.917 0.92 0.922 0.925 0.927 0.929 0.931 0.933 0.934 0.936 0.938

12.27 11.96 11.87 11.82 11.83 11.85 11.86 11.85 11.84 11.84 11.87 11.87 11.89 11.91 11.92 11.94 11.95 11.95 11.96 11.99 12.00 12.00 12.02 12.03 12.04 12.04 12.05 12.05 12.07 12.07 12.08 12.08 12.09 12.10 12.11 12.11 12.13 12.13 12.14

84 96 107 121 134 147 161 175 190 206 222 237 251 263 275 291 300 312 324 337 349 360 372 382 394 404 415 426 437 445 456 464 475 484 495 502 512 521 531

914 976 1027 1077 1120 1158 1195 1228 1261 1294 1321 1349 1369 1388 1408 1428 1447 1466 1482 1498 1515 1531 1545 1559 1571 1584 1596 1608 1618 1630 1640 1650 1661 1671 1681 1690 1697 1706 1716

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

0.271 0.310 0.349 0.392 0.437 0.479 0.523 0.561 0.601 0.644 0.684 0.723 0.754 0.784 0.816 0.849 0.879 0.909 0.936 0.965 0.992 1.019 1.044 1.065 1.084 1.104 1.122 1.141 1.157 1.175 1.192 1.207 1.226 1.242 1.258 1.272 1.287 1.300 1.316

5.6 6.2 6.9 7.7 8.6 9.4 10.2 10.9 11.7 12.5 13.3 14 14.6 15.2 15.9 16.5 17.1 17.7 18.3 18.9 19.5 20.1 20.7 21.3 21.7 22.2 22.7 23.1 23.5 24 24.5 24.9 25.4 25.8 26.3 26.7 27.1 27.5 27.9

Source: Permission to publish from the Society of Petroleum Engineers of the AIME. # SPE-AIME.

Group 1. TBP 77

Table 2.3 Coefficients of the Equation θ

M

Tc (°R)

pc (psia)

Tb (°R)

ω

γ

Vc (ft3/ lbm-mol)

a1 a2 a3 a4 a5

131.11375 24.96156000 0.34079022 2.49411840E-03 468.32575000

926.602244514 39.729362915 0.722461850 0.005519083 1366.431748654

311.2361908 14.6869301 0.3287671 0.0027346 1690.9001135

427.2959078 50.08577848 0.88693418 6.75667E-03 551.2778516

0.31428163 7.80028E-02 1.39205E-03 1.02147E-05 0.991028867

0.86714949 3.4143E-03 2.8396E-05 2.4943E-08 1.1627984

0.232837085 0.974111699 0.009226997 3.63611E-05 0.111351508

Undefined Components The undefined petroleum fractions are those heavy components lumped together and identified as the plus fraction; for example, the C7+ fraction. Nearly all naturally occurring hydrocarbon systems contain a quantity of heavy fractions that are not well defined and not mixtures of discretely identified components. A proper description of the physical properties of the plus fractions and other undefined petroleum fractions in hydrocarbon mixtures is essential in performing reliable phase-behavior calculations and compositional modeling studies. The proper description of these physical properties require the use of thermodynamic property prediction models, such as an equation of state, which in turn requires that one must be able to provide the acentric factor, critical temperature, and critical pressure for both the defined and undefined (heavy) fractions in the mixture. The problem of how to adequately characterize these undefined plus fractions in terms of their critical properties and acentric factors has been long recognized in the petroleum industry. Typically, undefined components are identified by a true boiling point (TBP) distillation analysis and characterized by an average normal boiling point, specific gravity, and molecular weight. In general, the data available that can be used to characterize the plus fractions are divided into the following three distinct groups: n n n

TBP Stimulated distillation by gas chromatography (GC) The plus fraction characterization methods

A brief discussion of the three groups is given next.

GROUP 1. TBP Boiling point of a pure petroleum fraction is the temperature at which it starts boiling and transfers from a liquid state to a vapor state at a pressure of 1 atm. In other words, it is the temperature at which the vapor pressure is

78 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

equal to atmospheric pressure for the petroleum fraction. There are various standard distillation methods through which the boiling point curve can be obtained. In the TBP distillation test, the oil is distilled and the temperature of the condensing vapor as well as the volume of liquid formed are recorded. The TBP test data are used to construct a distillation curve of liquid volume percent distilled versus condensing temperature. In this distillation process, the Heptanes-plus fraction “C7+” is subjected to a standardized analytical distillation, first at atmospheric pressure, then in a vacuum at a pressure of 40 mmHg. Usually the temperature is taken when the first droplet distills over. Ten fractions (cuts) are then distilled off, the first one at 50°C and each successive one with a boiling range of 25°C. For each distillation cut of the C7+, the volume, specific gravity, and molecular weight are determined. Cuts obtained in this manner are identified by the boiling point ranges in which they were collected. The condensing temperature of the vapor at any point in the test will be close to the boiling of the material condensing at that point. A key result from this distillation test is the boiling point curve that is defined as the boiling point of the petroleum fraction versus the percentage volume distilled. The initial boiling point (IBP) is defined as the temperature at which the first drop of liquid leaves the condenser tube of the distillation apparatus with the final boiling point (FBP) defined as the highest temperature recorded in the test. The difference between FBP and IBP is called boiling point range. It should be pointed out that oil fractions tend to crack at a temperature of approximately 650°F and one atmosphere. As a result, when approaching this cracking temperature, the pressure is gradually reduced to as low as 40 mmHG to avoid cracking the sample and distorting TBP measurements. A typical TBP curve is shown in Fig. 2.1. It should be noted that each distilled cut includes a large number of components with close boiling points. Based on the data from the TBP distillation, the C7+ fraction is divided into pseudocomponents or hypothetical components suitable for equations of state modeling application. These fractions are commonly collected within the temperature range of two consecutive distillation volumes, as shown in Fig. 2.1. The distilled cut is referred to as pseudocomponent. Each pseudocomponent, eg, pseudocomponent 1, is represented by a volumetric portion of the TBP curve with an average normal boiling point “Tbi.” The average boiling point of each cut is taken at the mid-volume percent of the cut; that is, between Vi and Vi1. This is given by the integration ð Vi Tb1 ¼

Tb ðV ÞdV

Vi1

Vi  Vi1



ðTb Þi + ðTb Þi1 2

Group 1. TBP 79

Pseudocomponent Tbi from TBP curve

Temperature

Vi

Tb1 =

∫V T dv i−1

Vi − Vi−1

≈

(Tb)i + (Tb)i−1 2

(Tb)i (Tb)i–1 Pseudocomponent 1

Vi–1

Vi

Volume distilled n FIGURE 2.1 Typical distillation curve.

where Vi  Vi1 ¼ volumetric cut of the TBP curve associated with pseudocomponent i Tbt(V) ¼ TBP temperature at liquid volume percent distilled “V” The pseudocomponents from the TBP test are identified primarily by the following three laboratory measured physical properties: n n

n

The boiling point “Tb.” The molecular weight “M,” is measured by the freezing point depression experiment. In this test, the molecular weight of the distillation cut is measured by comparing its freezing point with the freezing point of the pure solvent with a known molecular weight. The fraction specific gravity “γ i” of each distillation cut, is measured by using a pycnometer or electronic densitometer.

The mass (mi) of each distillation cut is measured directly during the TBP analysis. The cut is quantified in terms of number of moles ni, which is given as the ratio of the measured mass (mi) to the molecular weight (Mi); ie, ni ¼

mi Mi

The density (ρi) and specific gravity (γ i) is then calculated from the measured mass and volume of the distillation cut as ρi ¼

mi Vi

80 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Or in terms of the specific gravity as γi ¼

ρi 62:4

The average molecular weight and density of the C7+ from the TBP experiment with N distillation cuts are given by XN MC7 + ¼ Xi¼1 N

mi

n i¼1 i

and XN ρC7 + ¼ Xi¼1 N

mi

V i¼1 i

The calculated molecular weight and density from the distillation test should be compared with those of the measured C7+ values, any discrepancies are attributed and reported as lost materials resulting from the distillation test. Fig. 2.2 shows a typical graphical presentation of the molecular weight, specific gravity, and the TBP as a function of the weight percent of liquid vaporized. The molecular weight, M, specific gravity, γ, and boiling point temperature, Tb, are considered the key properties that reflect the chemical makeup

Tb,

M

Sp.gr 1

1600

0.9

1400

0.8 1200

Tb

800

M

600

0.6 0.5 0.4 0.3

400 0.2 200 0

0.1 0

20

40

60

80

Mass %

n FIGURE 2.2 TBP, specific gravity, and molecular weight of a stock-tank oil sample.

0 100

Specific gravity

0.7 1000

Group 1. TBP 81

of petroleum fractions. Watson et al. (1935) introduced a widely used characterization factor, commonly known as the universal oil products or Watson characterization factor, based on the normal boiling point and specific gravity. This characterization parameter is given by the following expression: 1=3

Kw ¼

Tb γ

(2.1)

where Kw ¼ Watson characterization factor Tb ¼ normal boiling point temperature, °R γ ¼ specific gravity The average heptanes-plus characterization factor “KC7 + ” can be estimated from the TBP distillation data by applying the following relationship: XN h K C7 + ¼

 i 1=3 Tbi =γ i mi XN m i¼1 i

i¼1

where mi is the mass of each individual cut. The Watson characterization factor “Kw” provides a qualitative measure of the paraffinicity of a crude oil. In general, Kw varies roughly from 8.5 to 13.5 as follows: n n n

For paraffinic hydrocarbon systems, Kw ranges from 12.5 to 13.5. For naphthenic hydrocarbon systems, Kw ranges from 11.0 to 12.5. For aromatic hydrocarbon systems, Kw ranges from 8.5 to 11.0.

The significance of the Watson characterization property is based on the observation that over reasonable ranges of boiling point, Kw is almost constant for C7+ distillation cuts. Whitson (1980) suggests that the Watson factor can be correlated with the molecular weight M and specific gravity γ as given by the following expression:  0:15178  M Kw  4:5579 0:84573 γ

(2.2)

Whitson pointed out that Eq. (2.2) becomes less reliable when the molecular weight “M” of the distillation cut exceeds 250. However; the limitation on the molecular weight can be removed by expressing the above relationship in the following modified form:  0:149405  M Kw  3:33475 0:935227 + 3:099934 γ

(2.2A)

82 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

If the molecular weight and specific gravity of the C7+ are available data, the Watson characterization factor is estimated from the above expression as  0:149405  M ðKw ÞC7 +  3:33475 0:935227 + 3:099934 γ C7 +

Whitson and Brule (2000) observed that a hydrocarbon mixture with multiple hydrocarbon samples, the ðKw ÞC7 + as correlated with MC7 + and γ C7 + and expressed by Eq. (2.2) often exhibits a constant value for all PVT samples and distillation cuts. The authors suggest that a plot of molecular weight versus specific gravity of the plus fractions is useful for checking the consistency of C7+ molecular weight and specific gravity measurements. Austad (1983) and Whitson and Brule (2000) illustrated this observation by plotting M versus γ for each C7+ fractions from data on two North Sea fields. Whitson and Brule’s observations are shown schematically in Figs. 2.3 and 2.4. The schematic plot of MC7 + versus γ C7 + of the volatile oil shown in Fig. 2.3 has an average ðKw ÞC7 + ¼ 11:90  0:01 for a range of molecular weights from 220 to 255. Data for the gas condensate, as shown schematically in Fig. 2.4, indicate an average ðKw ÞC7 + ¼ 11:99  0:01 for a range of molecular weights from 135 to 150. Whitson and Brule concluded

n FIGURE 2.3 Volatile oil specific gravity versus molecular weight of C7+.

Group 1. TBP 83

n FIGURE 2.4 Condensate-gas specific gravity versus molecular weight of C7+.

that the high degree of correlation for these two fields suggests accurate molecular weight measurements by the laboratory. In general, the spread in ðKw ÞC7 + values exceeds 0.01 when measurements are performed by a commercial laboratory. Whitson and Brule stated that, when the characterization factor for a field can be determined, Eq. (2.8) is useful for checking the consistency of C7+ molecular weight and specific gravity measurements. Significant deviation in ðKw ÞC7 + , such as 0.03 could indicate possible error in the measured data. Because molecular weight is more prone to error than determination of specific gravity, an anomalous ðKw ÞC7 + usually indicates an erroneous molecular weight measurement. The implication of assuming a constant Kw, eg, ðKw ÞC7 + , is that it can be used to estimate “Mi” or “γ i” of a distillation cut if they are not measured. Assuming a constant Kw for each fraction, Eq. (2.2A) is rearranged to solve for the molecular weight “Mi” and specific gravity “γ i”, to give Mi ¼ 315:5104753  106 ðKw  3:099934Þ6:6932358 γ 6:2596977 i

or γi ¼

3:624849522 Mi0:159752123 ðKw  3:099934Þ1:0692586

(2.3)

84 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

It should be noted that the molecular weight “M,” specific gravity “γ,” and the Watson characterization factor “Kw” of pseudocomponents provide the key data needed for further characterization of these fractions in terms of their critical properties and acentric factors. The following simplified two sets of correlations for estimating Tc, pc, and ω of pseudocomponents were generated by matching data from several literature published data. The first set was correlated as a function of the ratio “M/γ” while the second set was correlated as a function of “Tb/γ,” to give Set 1. Properties as a function of the ratio “M/γ”:  0:351579776   M M Tc ¼ 231:9733906  0:352767639  233:3891996 (2.3A) γ γ  0:885326626   M M  0:106467189 + 49:62573013 pc ¼ 31829 γ γ     M M ω ¼ 0:279354619 ln + 0:00141973  1:243207091 γ γ vc ¼ 0:054703719

(2.3B)

(2.3C)

 0:984070268   M M + 87:7041106  0:001007476 (2.3D) γ γ

 0:664115311   M M 379:4428973  1:40066452 Tb ¼ 33:72211  ðM=γ Þ γ γ

(2.3E)

Set 2. Properties as a function of the ratio “Tb/γ”: Tc ¼ 1:379242274

 0:967105064   Tb Tb + 0:010009512 (2.3F) + 0:012011487 γ γ

     Tb Tb  0:00867297 + 0:099138742 pc ¼ 2898:631565exp 0:002020245 γ γ (2.3G)  1:250922701   Tb Tb ω ¼ 0:000115057  0:92098982 (2.3H) + 0:00068975 γ γ    3  2 Tb vc ¼ 8749:94  106 Tγb  9:75748  106 Tγb + 0:014653 γ 595:7922076   Tb γ   0:001617123Tb  132:1481762 M ¼ 52:11038728 exp γ

4:60231684 

(2.3I)

(2.3J)

Group 1. TBP 85

If the Tb is not known, the molecular weight can be estimated from the following expression: M ¼ 0:048923 exp ð9:88378γ Þ  33:085468γ +

39:598437 γ

(2.3K)

where Tb ¼ boiling point in °R Tc ¼ critical temperature in °R vc ¼ critical volume, ft3/lbm-mol M ¼ molecular weight Notice, to express critical volume in ft3/lbm, divide vc by the molecular weight “M.” Vc ¼

vc M

The main issue that should be addressed after obtaining the distillation curve is how to break down or cut the entire boiling range into a number of cut-point ranges, which are used to define pseudocomponent. The determination of the number of cuts is essentially arbitrary; however, the number of pseudocomponents has to be sufficient to reproduce the volatility properties in fractions with wide boiling point ranges. The following provide a brief summary of industry experience and guidelines: 1. Pedersen et al. (1982) suggest selecting a small number of pseudocomponents with approximately the same weight fraction. 2. As a rule of thumb, use between 10 and 15 fractions if the TBP curve is relatively “steep” and use 5–8 fractions if the curve is relatively “flat.” 3. To better reproduce the shape of the TBP curve and to reflect the relative distribution of the light, middle, and heavy ends in pseudocomponent, Miquel et al. (1992) pointed out that the pseudocomponent breakdown can be made from either equal volumetric fractions or regular temperature intervals in the TBP curve. To reproduce the shape of the TBP curve better and reflect the relative distribution of light, middle, and heavy components in the fraction, a breakdown based on equal TBP temperature intervals is recommended. On the other hand, when equal volumetric fractions are taken, information concerning the light initial component and heavy end component could be lost. 4. Most commercial process simulators (eg, the HYSYS model) generate pseudocomponents based on boiling point ranges

86 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

representing the oil fractions. Typical boiling point ranges for pseudocomponents in a process simulator are shown below:

Boiling Point Range IBP to 800°F 800–1200°F 1200–1650°F Total no. of pseudocomponents

Suggested Number of Pseudocomponents 30 10 8 48

It should be pointed out that a complete TBP curve is rarely available for an entire range of percent distilled. In many cases, the data is obtained up to the 50% or 70% distilled point for any petroleum fractions or crude oils. This incomplete distillation curve is attributed to the fact that the petroleum fractions or crude oils contain more heavy hydrocarbons toward the end of the distillation curve; therefore it is more difficult is to achieve accurate boiling points for that fraction or crude oil. However, it is important for accurate fluid characterizations and in the process of equations of state tuning that the distillation curve should be extended to 90–95% distilled point. As conceptually shown in Fig. 2.5, Riazi and Daubert (1987) developed a

(1/b)

Temperature Tbi = To 1+

a 1 ln 1–Vi b

Tbi

To

≈50–70%

Volume distilled “Vi” n FIGURE 2.5 Mathematical representation of TBP curve.

Group 2. Stimulated Distillation by GC 87

mathematical representation of the TBP curve that can be used to complete the TBP curves up to the 95% point. The mathematical representation is given by the following equation: ( Tbi ¼ To 1 +

  a



1 ln b 1  Vi

ð1=bÞ )

where, Tbi ¼ TBP temperature and any distilled volume Vi Vi ¼ distilled volume fraction To ¼ IBP (ie, at Vi ¼ 0) a, b ¼ correlating coefficients The coefficients a and b are the two parameters that can be determined by regression or alternatively by linearizing the above equation as follows: ln

           Tbi  To 1 a 1 1 + ¼ ln ln ln To b b b 1  Vi

The above relationship indicates that a plot of ln ½Tbi  To =To  versus ln ½1=1  Vi  would produce a straight line with a slope of (1/b) and intercept of [(1/b)ln(a/b)] and used to calculate the parameters a and b. The mathematical expression can then be used to predict and complete the TBP curve up to 95% distilled volume.

GROUP 2. STIMULATED DISTILLATION BY GC Normal TBP distillation involves a long procedure and is costly. GC has been used widely as a fast and reproducible method for simulated distillation (SD) to analyze the carbon number distribution of various hydrocarbon components in the crude oil. GC is a common type of chromatography used for separating and analyzing compounds that can be vaporized without decomposition. Like all other chromatographic techniques, GC consists of a mobile and a stationary phase. The mobile phase is carrier gas comprised of an inert gas (ie, helium, argon, or nitrogen). The stationary phase consists of a packed capillary column with the packing acting as a stationary phase that is capable of absorbing and retaining hydrocarbon components for a specific time before releasing them sequentially based on their boiling points. This causes each compound to liberate at a different time, known as the retention time of the compound. GC is then similar to fractional distillation, as both processes separate the components of a mixture primarily based on boiling point.

88 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

The specific process of obtaining the SD curve from a GC experiment is based on injecting a small oil sample into a stream of carrier gas, which displaces that sample into a column packed with materials that absorbs the heavier components of the oil. The column is placed in a temperature programmable oven; as the temperature of the column is raised, the heavier components are released sequentially based on their boiling point and consequently removed by the carrier gas into a detector that senses changes in the thermal conductivity of the carrier gas. For a given program of heating, the retention time in the column is proportional to the normal boiling point of the released components. These are correlated with the retention times of a calibration curve obtained by conducting the separation under the same conditions of a known mixture of hydrocarbons, usually n-alkanes, covering the boiling range expected in the sample. The resulting calibration curve relates carbon number and boiling point to retention time and allows for the determination of boiling points for the components in the hydrocarbon sample. As shown in Fig. 2.6 and Table 2.4, all components detected by the GC between two neighboring n-alkanes are commonly grouped together and

n FIGURE 2.6 GC data calibration sample of n-alkanes.

Group 2. Stimulated Distillation by GC 89

Table 2.4 Boiling Point Data n-Alkane

Retention Time (min)

Boiling Point (°C)

C5 C6 C7 C8 C9 C10 C11 C12 C14 C15 C16 C17 C18 C20 C24 C28 C32 C36 C40

0.09764 0.11994 0.16272 0.23050 0.31882 0.41783 0.51717 0.61712 0.79714 0.87985 0.95957 1.03281 1.10226 1.23156 1.45693 1.64895 1.81618 1.96389 2.10297

21.9 54.7 84.1 111.9 136.3 159.7 181.9 201.9 220.8 239.7 256.9 273.0 288.0 301.9 376.9 416.9 451.9 481.9 508.0

reported as a pseudocomponent with a SCN equal to the higher n-alkane; eg, between n-alkanes of C8 and C9 is designated as pseudocomponent with SCN of C9. As shown in Fig. 2.7, the distillation curve resulting from the plot of GC data in terms of boiling point as a function of the retention time or SCN of n-alkane is called a SD curve and represents boiling points of compounds in a petroleum mixture at atmospheric pressure. It should be pointed out that SD curves are very close to actual boiling points shown by TBP curves. The main advantage of this SD process is that it requires smaller samples and also is less expensive than TBP distillation. The GC is used for gas and oil samples generating distillation curves as well as the composition of hydrocarbon mixtures in terms of their weight fractions “wi”. Conversion from weight fraction “wi” to mole fraction “xi” requires molecular weights “Mi” of all components and can be achieved from wi =Mi xi ¼ X ðwi =Mi Þ i

90 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Stimulated distillation curve 600

Boiling point (°C)

500 400 300 200 100 0 0

0.5

1

1.5

2

2.5

Retention time (min)

n FIGURE 2.7 Stimulated distillation curve.

Similarly, if the volume fraction vi and density ρi are available, they can be converted into mole fraction from the following relationship: vi ρ =Mi xi ¼ X i ðv ρ =Mi Þ i i i

However, it should be pointed out that a major drawback of GC analysis is that no physical fractions are collected during this chromatographic analysis test and, therefore, their molecular weights “Mi” and specific gravities are not measured. As discussed later in this chapter, to describe these fractions in terms of their critical properties and acentric factors, the molecular weight and specific gravity must be known and used as input for most of the widely used property correlations. Ahmed (2014) pointed out that without the availability of molecular weight and specific gravity of pseudocomponents, the GC stimulated distillation data could still be used to characterize these fractions. The recommended proposed methodology is based on reproducing the GC stimulated boiling point data by a corresponding set of pseudocomponents with equivalent average boiling point temperatures to that of the GC stimulated boiling point data. These pseudocomponents are identified by a set of effective carbon numbers (ECN) that can be used to estimate their physical and critical properties. This approach relies on matching the boiling point temperature “Tb” by using Microsoft Solver and regressing on the ECN as given by the following relationship: Tb ¼ 427:2959078 + 50:08577848ðECNÞ  0:88693418ðECNÞ2 + 0:00675667ðECNÞ3 

551:2778516 ðECNÞ

Group 2. Stimulated Distillation by GC 91

where Tb is expressed in °R. The physical, critical properties, and acentric factor of a pseudocomponent can then be calculated from the following set of expressions: M ¼ 131:11375 + 24:96156ðECNÞ  0:34079022ðECNÞ2 + 0:002494118ðECNÞ3 +

468:32575 ðECNÞ

γ ¼ 0:86714949 + 0:0034143408ðECNÞ  2:839627 1:1627984  105 ðECNÞ2 + 2:49433308  108 ðECNÞ3  ðECNÞ Tc ¼ 926:6022445 + 39:729363ðECNÞ  0:7224619ðECNÞ2 + 0:005519083ðECNÞ3 

1366:4317487 ðECNÞ

pc ¼ 311:236191  14:68693ðECNÞ + 0:3287671ðECNÞ2 1690:900114  0:0027346ðECNÞ3 + ðECNÞ Tb ¼ 427:295908 + 50:0857785ðECNÞ  0:8869342ðECNÞ2 + 0:00675667ðECNÞ3 

551:277852 ðECNÞ

ω ¼ 0:3142816 + 0:0780028ðECNÞ  0:00139205ðECNÞ2 + 1:02147 0:99102887  105 ðECNÞ3 + ðECNÞ vc ¼ 0:2328371 + 0:9741117ðECNÞ  0:009227ðECNÞ2 + 36:3611 0:1113515  106 ðECNÞ3 + ðECNÞ

where, ECN ¼ effective carbon number Tc ¼ critical temperature of the fraction with a carbon number “n,” °R Tb ¼ boiling point temperature of the fraction with a carbon number “n,” °R pc ¼ critical pressure of the fraction with a carbon number “n,” psia vc ¼ critical volume of the fraction with a carbon number “n,” ft3/lbm-mol M ¼ molecular weight ω ¼ acentric factor γ ¼ specific gravity It should be pointed out that the number of carbon atoms “n” that is equivalent to that of n-alkane is calculated from n ¼ INT½ðECNÞ + 1

92 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

where, ECN ¼ effective carbon number n ¼ equivalent carbon number of n-alkane INT ¼ integer As a validation of Ahmed’s approach, assume the tabulated data listed in Table 2.24 are closely approximate the TBP. The objective is to estimate the ECN for each of the listed boiling point and compare with n-alkane carbon number and to calculate the physical and critical properties of each carbon number. Step 1. Using regression (Microsoft Solver), determine the ECN from the expression Tb ¼ 427:2959078 + 50:08577848ðECNÞ  0:88693418ðECNÞ2 + 0:00675667ðECNÞ3 

551:2778516 ðECNÞ

n ¼ INT½ðECNÞ + 1

To give: n

Tb (°C)

Tb (°R)

ECN

5 6 7 8 9 10 11 12 14 15 16 17 18 20 24 28 32 36 40

21.9 54.7 84.1 111.9 136.3 159.7 181.9 201.9 220.8 239.7 256.9 273 288 310.9 376.9 416.9 451.9 481.9 508

530.82 589.86 642.78 692.82 736.74 778.86 818.82 854.82 888.84 922.86 953.82 982.8 1009.8 1051.02 1169.82 1241.82 1304.82 1358.82 1405.8

4.8 5.8 6.7 7.7 8.7 9.7 10.7 11.7 12.6 13.6 14.6 15.6 16.5 18.1 23.2 27.1 31.1 35 38.9

Group 3. The Plus Fraction Characterization Methods 93

Step 2. Calculate the physical and critical properties of each carbon number using the proposed working relationships and the values of ECN given above, to give n

M

Tc (°R)

pc (psi)

Tb (°R)

W

γ

Vc (ft3/lbm-mol)

5.000 6.000 7.000 8.000 9.000 10.000 11.000 12.000 14.000 15.000 16.000 17.000 18.000 20.000 24.000 28.000 32.000 36.000 40.000

79.151 84.980 94.678 106.586 119.791 133.750 148.123 162.688 191.849 206.271 220.515 234.547 248.342 275.171 325.661 372.107 415.049 455.213 493.416

834.591 912.423 975.996 1030.221 1077.845 1120.526 1159.332 1194.988 1258.753 1287.520 1314.526 1339.947 1363.928 1408.036 1483.330 1544.968 1596.289 1640.091 1678.898

546.140 516.176 465.156 424.744 391.568 363.599 339.539 318.518 283.332 268.402 254.890 242.602 231.383 211.673 180.771 158.115 141.146 127.974 117.046

546.140 605.461 658.000 705.768 749.899 791.089 829.797 866.342 933.821 965.074 994.834 1023.198 1050.252 1100.727 1188.915 1262.975 1325.995 1380.843 1430.278

0.240 0.271 0.309 0.350 0.393 0.436 0.479 0.522 0.604 0.643 0.681 0.718 0.753 0.820 0.938 1.038 1.122 1.194 1.257

0.651 0.693 0.724 0.747 0.766 0.782 0.796 0.807 0.826 0.835 0.842 0.849 0.855 0.866 0.885 0.900 0.912 0.922 0.931

4.434 5.306 6.162 7.002 7.826 8.633 9.424 10.200 11.704 12.433 13.147 13.846 14.530 15.855 18.338 20.610 22.685 24.577 26.298

GROUP 3. THE PLUS FRACTION CHARACTERIZATION METHODS It is well known that C7+ characterization methodology has a significant impact on EOS predictions of reservoir hydrocarbon fluid behavior. The basis for most characterization methods is the TBP data that includes mass, mole, and volume fractions of each distillation cuts with a measured molecular weight, specific gravity, and boiling point. Each distillation cut is usually treated as pseudocomponent that can be characterized by a critical pressure, critical temperature, and acentric factor. If the distillation cuts are collected within a range of neighboring normal alkanes boiling points, they are referred to as SCN groups. Correlations for estimating their critical properties are commonly used based on the molecular weight and specific gravity of the SCN group. Based on the type of existing laboratory measured data on the plus fraction, two approaches commonly are used to generate properties of the plus fractions:

94 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

n n n

Generalized correlations, PNA determination using analytical correlations, and PNA determination using graphical correlations.

These two techniques of characterizing the undefined petroleum fractions are detailed next.

Generalized Correlations To use any of the thermodynamic property prediction models, such as an equation of state, to predict the phase and volumetric behavior of complex hydrocarbon mixtures, one must be able to provide the acentric factor, ω, critical temperature, Tc, and critical pressure, pc, for both the defined and undefined (heavy) fractions in the mixture. The problem of how to adequately characterize these undefined plus fractions in terms of their critical properties and acentric factors has been long recognized in the petroleum industry. Whitson (1984) presented excellent documentation on the influence of various heptanes-plus (C7+) characterization schemes on predicting the volumetric behavior of hydrocarbon mixtures by equations of state. Numerous correlations are available for estimating the physical properties of petroleum fractions. Most of these correlations use the specific gravity, γ, and the boiling point, Tb, correlation parameters. However, coefficients of these correlations were determined from matching pure component physical and critical properties data, which might cause abnormality when applied to characterize the undefined fractions. Several of these correlations are presented next and referred to by the names of the authors, including: n n n n n n n n n n n n n n

Riazi and Daubert Cavett Kesler-Lee Winn and Sim-Daubert Watansiri-Owens-Starling Edmister Critical compressibility factor correlations Rowe Standing Willman-Teja Hall-Yarborough Magoulas-Tassios Twu Silva and Rodriguez

Group 3. The Plus Fraction Characterization Methods 95

Riazi and Daubert Generalized Correlations Riazi and Daubert (1980) developed a simple two-parameter equation for predicting the physical properties of pure compounds and undefined hydrocarbon mixtures. The proposed generalized empirical equation is based on the use of the normal boiling point and specific gravity as correlating parameters. The basic equation is θ ¼ aTbb γ c

(2.4)

where θ ¼ any physical property (Tc, pc, Vc or M) Tb ¼ normal boiling point, °R γ ¼ specific gravity M ¼ molecular weight Tc ¼ critical temperature, °R pc ¼ critical pressure, psia Vc ¼ critical volume, ft3/lbm a, b, c ¼ correlation constants are given in Table 2.5 for each property The expected average errors for estimating each property are included in the table. Note that the prediction accuracy is reasonable over the boiling point range of 100–850°F. Riazi and Daubert (1987) proposed improved correlations for predicting the physical properties of petroleum fractions by taking into consideration the following factors: accuracy, simplicity, generality, and availability of input parameters; ability to extrapolate; and finally, comparability with similar correlations developed in recent years. The authors proposed the following modification of Eq. (2.4), which maintains the simplicity of the previous correlation while significantly improving its accuracy: θ ¼ aθb1 θc2 exp ½dθ1 + eθ2 + f θ1 θ2 

Table 2.5 Correlation Constants for Eq. (2.4) Deviation (%) θ M Tc (°R) pc (psia) Vc (ft3/lb)

a

b 5

4.56730  10 24.27870 3.12281  109 7.52140  103

c

2.19620 1.0164 0.58848 0.3596 2.31250 2.3201 0.28960 0.7666

Average

Maximum

2.6 1.3 3.1 2.3

11.8 10.6 9.3 9.1

96 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

where θ ¼ any physical property a–f ¼ constants for each property Riazi and Daubert stated that θ1 and θ2 can be any two parameters capable of characterizing the molecular forces and molecular size of a compound. They identified (Tb,γ) and (M,γ) as appropriate pairs of input parameters in the equation. The authors finally proposed the following two forms of the generalized correlation. In the first form, the boiling point, Tb, and the specific gravity, γ, of the petroleum fraction are used as correlating parameters: θ ¼ aTbb γ c exp ½dTb + eγ + fTb γ 

(2.5)

The constants a–f for each property θ are given in Table 2.6. In the second form, the molecular weight, M, and specific gravity, γ, of the component are used as correlating parameters. Their mathematical expression has the following form: θ ¼ aðMÞb γ c exp ½dM + eγ + f γM

(2.6)

In developing and obtaining the coefficients a–f, as given in Table 2.7, the authors used data on the properties of 38 pure hydrocarbons in the carbon number range 1–20, including paraffins, olefins, naphthenes, and aromatics in the molecular weight range 70–300 and the boiling point range 80–650°F. Table 2.6 Correlation Constants for Eq. (2.5) θ

a

b

c

d

e

f

M Tc (°R) pc (psia) Vc (ft3/lb)

581.96000 10.6443 6.16200  106 6.23300  104

0.97476 0.81067 0.48440 0.75060

6.51274 0.53691 4.08460 1.20280

5.43076  104 5.17470  104 4.72500  103 1.46790  103

9.53384 0.54444 4.80140 0.26404

1.11056  103 3.59950  104 3.19390  103 1.09500  103

Table 2.7 Correlation Constants for Eq. (2.6) θ Tc (°R) Pc (psia) Vc (ft3/lb) Tb (°R)

a 544.40000 4.52030  104 1.20600  102 6.77857

b 0.299800 0.806300 0.203780 0.401673

c 1.05550 1.60150 1.30360 1.58262

d 4

1.34780  10 1.80780  103 2.65700  103 3.77409  103

e

f

0.616410 0.308400 0.528700 2.984036

0.00000 0.00000 2.60120  103 4.25288  103

Group 3. The Plus Fraction Characterization Methods 97

Cavett’s Correlations Cavett (1962) proposed correlations for estimating the critical pressure and temperature of hydrocarbon fractions. The correlations received wide acceptance in the petroleum industry due to their reliability in extrapolating conditions beyond those of the data used in developing the correlations. The proposed correlations were expressed analytically as functions of the normal boiling point, TbF, in °F and API gravity. Cavett proposed the following expressions for estimating the critical temperature and pressure of petroleum fractions: Tc ¼ a0 + a1 ðTbF Þ + a2 ðTbF Þ2 + a3 ðAPIÞðTbF Þ + a4 ðTbF Þ3 + a5 ðAPIÞðTbF Þ2 + a6 ðAPIÞ2 ðTbF Þ2

(2.7)

log ðpc Þ ¼ b0 + b1 ðTbF Þ + b2 ðTbF Þ2 + b3 ðAPIÞðTbF Þ + b4 ðTbF Þ3 + b5 ðAPIÞðTbF Þ2 + b6 ðAPIÞ2 ðTbF Þ + b7 ðAPIÞ2 ðTbF Þ2 (2.8)

where Tc ¼ critical temperature, °R pc ¼ critical pressure, psia TbF ¼ normal boiling point, °F API ¼ API gravity of the fraction

Note that the normal boiling point in the above relationships is expressed in °F. The coefficients of Eqs. (2.7) and (2.8) are tabulated in the table below. However, Cavett presented these correlations without reference to the type and source of data used for their development. i

ai

bi

0 1 2 3 4 5 6 7

768.0712100000 1.7133693000 0.0010834003 0.0089212579 0.3889058400  106 0.5309492000  105 0.3271160000  107

2.82904060 0.94120109  103 0.30474749  105 0.20876110  104 0.15184103  108 0.11047899  107 0.48271599  107 0.13949619  109

Kesler and Lee Correlations Kesler and Lee (1976) proposed a set of equations to estimate the critical temperature, critical pressure, acentric factor, and molecular weight of

98 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

petroleum fractions. These relationships, as documented below, use specific gravity “γ” and boiling point “Tb” as input parameters for their proposed expressions:

  0:0566 2:2898 0:11857 103 Tb  0:24244 + + γ γ γ2     3:648 0:47227 1:6977 7 2 10 1010 Tb3 + 1:4685 + T  0:42019 + + b γ γ2 γ2 (2.9)

lnðpc Þ ¼ 8:3634 

Tc ¼ 341:7 + 811:1γ + ½0:4244 + 0:1174γ Tb +

½0:4669  3:26238γ 105 (2.10) Tb

 M ¼ 12272:6 + 9486:4γ + ½4:6523  3:3287γ Tb + 1  0:77084γ  0:02058γ 2      720:79 107

181:98 1012 + 1  0:80882γ  0:02226γ 2 1:8828  1:3437  Tb Tb Tb Tb3 (2.11)

The molecular weight equation was developed by regression analysis to match laboratory data with molecular weights ranging from 60 to 650. In addition, Kesler and Lee developed two expressions for estimating the acentric factor that use the Watson characterization factor and the reduced boiling point temperature as correlating parameters. The two correlating parameters are defined by the following two parameters: n n

the Watson characterization factor “Kw”; ie, Kw ¼ (Tb)1/3/γ the reduced boiling point “θ” as defined by θ ¼ Tb/Tc

where the boiling point Tb and critical temperature Tc are expressed in °R. Kessler and Lee proposed the following two expressions for calculating the acentric factor that are based on the value of the reduced boiling point: For θ > 0:8 : ω ¼ 7:904 + 0:1352Kw  0:007456ðKw Þ2 + 8:359θ +  ln For θ < 0:8 : ω ¼

where

1:408  0:01063Kw θ

(2.12)

pc 6:09648 + 1:28862 ln ½θ  0:169347θ6  5:92714 + θ 14:7 15:6875 15:2518   13:4721 ln ½θ + 0:43577θ6 θ (2.13)

pc ¼ critical pressure, psia Tc ¼ critical temperature, °R Tb ¼ boiling point, °R

Group 3. The Plus Fraction Characterization Methods 99

ω ¼ acentric factor M ¼ molecular weight γ ¼ specific gravity Kesler and Lee stated that Eqs. (2.9) and (2.10) give values for pc and Tc that are nearly identical with those from the API data book up to a boiling point of 1200°F.

Winn and Sim-Daubert Correlations Winn (1957) developed convenient nomographs to estimate various physical properties including molecular weight and the pseudocritical pressure for petroleum fractions. The input data in both nomographs are boiling point (or Kw) and the specific gravity (or API gravity). Sim and Daubert (1980) concluded that the Winn’s monograph is the most accurate approach for characterizing petroleum fractions. Sim and Daubert developed analytical relationships that closely matched the monograph graphical data. The authors used specific gravity and boiling point as the correlating parameters for calculating the critical pressure, critical temperature, and molecular weight. Sim and Daubert relationships as given below can be used to estimate the pseudocritical properties of the undefined petroleum fraction: pc ¼ 3:48242  109 Tb2:3177 γ 2:4853

 Tc ¼ exp 3:9934718T 0:08615 γ 0:04614 b

(2.15)

M ¼ 1:4350476  105 Tb2:3776 γ 0:9371

(2.16)

(2.14)

where pc ¼ critical pressure, psia Tc ¼ critical temperature, °R Tb ¼ boiling point, °R

Watansiri-Owens-Starling Correlations Watansiri et al. (1985) developed a set of correlations to estimate the critical properties and acentric factor of coal compounds and other undefined hydrocarbon components and their derivatives. The proposed correlations express the critical and physical properties of the undefined fraction as a function of the fraction normal boiling point, specific gravity, and molecular weight. These relationships have the following forms: lnðTc Þ ¼ 0:0650504  0:0005217Tb + 0:03095 ln ½M + 1:11067 lnðTb Þ h i + M 0:078154γ 1=2  0:061061γ 1=3  0:016943γ (2.17)

100 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

lnðvc Þ ¼ 76:313887  129:8038γ + 63:1750γ 2  13:175γ 3 + 1:10108 ln ½M + 42:1958 ln ½γ  (2.18)    0:8   Tc M Tb  0:08843889  8:712 lnðpc Þ ¼ 6:6418853 + 0:01617283 Vc M Tc (2.19) 8 9 > 5:12316667  104 Tb + 0:281826667ðTb =MÞ + 382:904=M > > > > > > > > > 2 5 4 > > > + 0:074691  10 ðTb =γ Þ  0:12027778  10 ðTb ÞðMÞ > > > > > > > > > 2 4 > > > > < + 0:001261ðγ ÞðMÞ + 0:1265  10 ðMÞ =5T  b ω¼ 1=3 ð T Þ > > 9M b 2 > > > > + 0:2016  104 ðγ ÞðMÞ  66:29959 > > > > M > > > > > > > > 2=3 > > > > T > > b > > : 0:00255452 2 ; γ (2.20)

where ω ¼ acentric factor pc ¼ critical pressure, psia Tc ¼ critical temperature, °R Tb ¼ normal boiling point, °R vc ¼ critical volume, ft3/lb-mol The proposed correlations produce an average absolute relative deviation of 1.2% for Tc, 3.8% for vc, 5.2% for pc, and 11.8% for ω.

Edmister’s Correlations Edmister (1958) proposed a correlation for estimating the acentric factor, ω, of pure fluids and petroleum fractions. The equation, widely used in the petroleum industry, requires boiling point, critical temperature, and critical pressure. The proposed expression is given by the following relationship: ω¼

3½ log ðpc =14:70Þ 1 7½ðTc =Tb  1Þ

where ω ¼ acentric factor pc ¼ critical pressure, psia Tc ¼ critical temperature, °R Tb ¼ normal boiling point, °R

(2.21)

Group 3. The Plus Fraction Characterization Methods 101

If the acentric factor is available from another correlation, the Edmister equation can be rearranged to solve for any of the three other properties (providing the other two are known).

Critical Compressibility Factor Correlations The critical compressibility factor is defined as the component compressibility factor calculated at its critical point by using the component critical properties; ie, Tc, pc, and Vc. This property can be conveniently computed by the real gas equation of state at the critical point, or Zc ¼

p c vc RTc

(2.22)

where R ¼ universal gas constant, 10.73 psia ft3/lb-mol, °R vc ¼ critical volume, ft3/lb-mol If the critical volume, Vc, is given in ft3/lbm, Eq. (2.22) is written as Zc ¼

pc Vc M RTc

where M ¼ molecule weight Vc ¼ critical volume, ft3/lb The accuracy of Eq. (2.22) depends on the accuracy of the values of pc, Tc, and vc. The following table presents a summary of the critical compressibility estimation methods that have been published over the years: Method Haugen Reid, Prausnitz, and Sherwood Salerno et al. Nath

Year

Zc

Equation No.

1959 1977

Zc ¼ 1=ð1:28ω + 3:41Þ Zc ¼ 0:291  0:080ω

(2.23) (2.24)

1985 1985

Zc ¼ 0:291  0:080ω  0:016ω2 Zc ¼ 0:2918  0:0928ω

(2.25) (2.26)

Rowe’s Characterization Method Rowe (1978) proposed a set of correlations for estimating the normal boiling point, the critical temperature, and the critical pressure of the heptanes-plus fraction C7+. The prediction of the C7+ properties is based on the assumption that the plus fraction behaves as a normal paraffin hydrocarbon fraction that has a number of carbon atoms “n” with identical critical properties to that of

102 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

the plus fraction. Rowe proposed the following set of formulas for characterizing the C7+ fraction in terms of the critical temperature, critical pressure, and boiling point temperature that are equivalent to properties of a normal paraffin hydrocarbon fraction with a number of carbon atoms “n” that can be calculated from the molecular weight of the C7+ fraction by the following relationship: n¼

MC7 +  2:0 14

ðTc ÞC7 + ¼ 1:8½961  10a 

(2.27) (2.28)

The coefficient “a” as defined by a ¼ 2:95597  0:090597n2=3

The author correlated the critical pressure and the boiling point temperature of the C7+ fraction in terms of the critical temperature and number of carbon atoms by the following expressions: ðpc ÞC7 + ¼

10ð4:89165 + Y Þ ðTc ÞC7 +

(2.29)

with the parameter “Y” as given by Y ¼ 0:0137726826 n + 0:6801481651 ðTb ÞC7 + ¼ 0:0004347ðTc Þ2C7 + + 265

(2.30)

where ðTc ÞC7 + ¼ critical temperature of C7+, °R ðTb ÞC7 + ¼ boiling point temperature of C7+, °R ðpc ÞC7 + ¼ critical pressure of the C7+ in psia MC7 + ¼ molecular weight of the heptanes-plus fraction Soreide (1989) proposed an alternative methodology of calculating the boiling point that was generated from the analysis of 843 TBP fractions taken from 68 reservoir C7+ samples. Soreide proposed the boiling point temperature correlation is expressed as a function of molecular weight and specific gravity of the plus fraction: Tb ¼ 1928:3 

 

 1:695  105 γ 3:266 exp 4:922  103 M  4:7685γ + 3:462  103 Mγ M0:03522

where Tb is expressed in °R.

Group 3. The Plus Fraction Characterization Methods 103

Standing’s Correlations Matthews et al. (1942) presented graphical correlations for determining the critical temperature and pressure of the heptanes-plus fraction. Standing (1977) expressed these graphical correlations more conveniently in mathematical forms as follows: h i h i ðTc ÞC7 + ¼ 608 + 364log ðMÞC7 +  71:2 + 2450log ðMÞC7 +  3800 log ðγ ÞC7 + (2.31) h i ðpc ÞC7 + ¼ 1188  431log ðMÞC7 +  61:1 n h ih io + 2319  852 log ðMÞC7 +  53:7 ðγ ÞC7 +  0:8

(2.32)

where ðMÞC7 + and ðγ ÞC7 + are the molecular weight and specific gravity of the C7+.

Willman-Teja Correlations Willman and Teja (1987) proposed correlations for determining the critical pressure and critical temperature of the n-alkane homologous series. The authors used the normal boiling point and the number of carbon atoms of the n-alkane as a correlating parameter. The authors used a nonlinear regression model to generate a set of relationships that best match the critical properties data reported by Bergman et al. (1977) and Whitson (1980). Willman and Teja introduced the ECN parameter “n” into their proposed correlations and suggested that this correlating parameter can be calculated by matching the boiling point of the undefined petroleum fraction. Ahmed (2014) suggested that the ECN can be best estimated by regression (eg, Microsoft Solver) to match the boiling point temperature “Tb” as given by following expression:   Tb ¼ 434:38878 + 50:125279n  0:9097293n2 + 7:0280657 103 n3  601:85651=n

where Tb is expressed in °R. The critical temperature and pressure can then be calculated from the following expressions: h i Tc ¼ Tb 1 + ð1:25127 + 0:137252nÞ0:884540633 Pc ¼

339:0416805 + 1184:157759n ½0:873159 + 0:54285n1:9265669

where n ¼ effective carbon number Tb ¼ average boiling point of the undefined fraction, °R

(2.33) (2.34)

104 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Tc ¼ critical temperature of the undefined fraction, °R Pc ¼ critical pressure of the undefined fraction, psia

Hall and Yarborough Correlations Hall and Yarborough (1971) proposed a simple expression to determine the critical volume of a fraction from its molecular weight and specific gravity: vc ¼

0:025M1:15 γ 0:7935

(2.35)

where vc is the critical volume as expressed in ft3/lb-mol. Note that, to express the critical volume in ft3/lb, the relationship is given by vc ¼ MVc

where M ¼ molecular weight Vc ¼ critical volume in ft3/lb vc ¼ critical volume in ft3/lb-mol The critical volume also can be calculated by applying the real gas equation of state at the critical point of the component as pV ¼ Z

m M

RT

Applying real gas equation at the critical point, gives Vc ¼

Zc RTc pc M

Magoulas and Tassios Correlations Magoulas and Tassios (1990) correlated the critical temperature, critical pressure, and acentric factor with the specific gravity, γ, and molecular weight, M, of the fraction as expressed by the following relationships: Tc ¼ 1247:4 + 0:792M + 1971γ  lnð pc Þ ¼ 0:01901  0:0048442M + 0:13239γ +

27000 707:4 + M γ

227 1:1663  + 1:2702 lnðMÞ M γ

ω ¼ 0:64235 + 0:00014667M + 0:021876γ 

where Tc ¼ critical temperature, °R pc ¼ critical pressure, psia

4:559 + 0:21699 ln ðMÞ M

Group 3. The Plus Fraction Characterization Methods 105

Twu’s Correlations Twu (1984) developed a suite of critical properties, based on perturbationexpansion theory with normal paraffins as the reference states, which can be used for determining the critical and physical properties of undefined hydrocarbon fractions, such as C7+. The methodology is based on selecting (finding) a normal paraffin fraction with a boiling temperature TbP identical to that of the hydrocarbon-plus fraction “TbC + ,” such as C7+. The methodology requires the availability of the boiling point temperature of the plus fraction “TbC + ,” molecular weight of the plus fraction “MC + ,” as well as its specific gravity “γ C + .” If the boiling point temperature is not available, it can be estimated from the correlation proposed by Soreide (1989):  a

 TbC + ¼ a1 + a2 ðMC + Þa3 γ C + 4 exp a5 MC + + a6 γ C + + a7 MC + γ C +

where a1 ¼ 1928.3 a2 ¼ 1.695(105) a3 ¼ 0.03522 a4 ¼ 3.266 a5 ¼ 4.922(103) a6 ¼ 4.7685 a7 ¼ 3.462(103) The Twu approach is performed by applying the following two steps: Step 1. Calculate the properties of the normal paraffins, ie, TcP, pcP, γ P, and vcP, from the following set of working expressions: n Critical temperature of normal paraffins, TcP, in °R: " TcP ¼ TbC + A1 + A2 TbC +

2 + A3 TbC +

3 + A4 TbC +

+

A5 ðA6 TbC + Þ13

where A1 ¼ 0.533272 A2 ¼ 0.191017(103) A3 ¼ 0.779681(107) A4 ¼ 0.284376(1010) A5 ¼ 0.959468(102) A6 ¼ 0.01 n

Critical pressure of normal paraffins, pcP, in psia:

 2 4 2 pcP ¼ A1 + A2 α0:5 i + A3 αi + A4 αi + A5 αi

#1

106 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

With the parameter αi as defined by the expression αi ¼ 1 

TbC + TcP

where A1 ¼ 3.83354 A2 ¼ 1.19629 A3 ¼ 34.8888 A4 ¼ 36.1952 A5 ¼ 104.193 n

Specific gravity of normal paraffins, γ P: γ P ¼ A1 + A2 αi + A3 α3i + A4 α12 i

where A1 ¼ 0.843593 A2 ¼ 0.128624 A3 ¼ 3.36159 A4 ¼ 13749.5 n

Critical volume of normal paraffins, vcP, in ft3/lbm-mol:

8 vcP ¼ 1 + A1 + A2 αi + A3 α3i + A4 α14 i

where A1 ¼ 0.419869 A2 ¼ 0.505839 A3 ¼ 1.56436 A4 ¼ 9481.7 Step 2. Calculate the properties of the plus fraction by applying the following relationships: n Critical temperature of the plus fraction, TC + , in °R:  TC + ¼ TcP

1 + 2fT 1  2fT

2

With the function “fT” as given by 

   fT ¼ exp 5 γ p  γ C +  1

"

# !

   A1 A3  exp 5 γ p  γ C +  1 + A2 + 0:5 0:5 TbC TbC + +

where A1 ¼ 0.362456 A2 ¼ 0.0398285 A3 ¼ 0.948125

Group 3. The Plus Fraction Characterization Methods 107

n

Critical volume of the plus fraction, vC + , in ft3/lbm-mol:  vC + ¼ vcP

1 + 2fv 1  2fv

2

with ! " # h  i o A h  i o A3 n 1 2 2 2 2 exp 4 γ p  γ C +  1 fv ¼ exp 4 γ p  γ C +  1 + A2 + 0:5 0:5 TbC TbC + + n

where A1 ¼ 0.466590 A2 ¼ 0.182421 A3 ¼ 3.01721 n

Critical pressure of the plus fraction, pC + , in psia:  pC + ¼ pcP

TC + TcP



vcP vC +



1 + 2fp 1  2fp

2

with n

 i o fp ¼ exp 0:5 γ p  γ C +  1 n

h

h  i o exp 0:5 γ p  γ C +  1

! A2 A1 + 0:5 + A3 TbC + TbC + !!

! A5 + A4 + 0:5 + A6 TbC + TbC +

where A1 ¼ 2.53262 A2 ¼ 46.19553 A3 ¼ 0.00127885 A4 ¼ 11.4277 A5 ¼ 252.14 A6 ¼ 0.00230535

Silva and Rodriguez Correlations Silva and Rodriguez (1992) suggested the use of the following two expressions to estimate the boiling point temperature and specific gravity from the molecular weight:    M Tb ¼ 460 + 447:08723 ln 64:2576

Use the preceding calculated value of Tb to calculate the specific gravity of the fraction from the following expression: γ ¼ 0:132467 ln ðTb  460Þ + 0:0116483

where the boiling point temperature Tb is expressed in °R.

108 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Sancet Correlations Sancet (2007) presented a set expressions to estimate the critical properties and boiling point from the molecular weight “M”; these correlations have the following forms: pc ¼ 653 exp ð0:007427MÞ + 82:82 Tc ¼ ½383:5 ln ðM  4:075Þ  778:5 h i Tb ¼ 0:001241ðTc Þ1:869 + 194

where the boiling point temperature Tb and Tc are expressed in °R.

Comparison of Molecular Weight Methods Correlations Mudgal (2014) compare several methodologies for predicting the molecular weight from the boiling point temperature data. The author estimated the molecular weight by using Lee-Kesler, Riazi-Daubert, Winn nomographs, Goossens, and Twu methods. Mudgal concluded that Goossens and Twu methods yielded appropriate estimations of the molecular weight. In addition, Ahmed (2014) applied the concept of the ECN and equivalent carbon number “n” to match Mudgal’s boiling point temperature “Tb”; as given previously in the chapter by Tb ¼ 427:2959078 + 50:08577848ðECNÞ  0:88693418ðECNÞ2 + 0:00675667ðECNÞ3 

551:2778516 ðECNÞ

with ‘n’ as given by n ¼ INT½ðECNÞ + 1

The equivalent carbon number “n” is used to estimate the molecular weight from M ¼ 131:11375 + 24:96156n  0:34079022n2 + 0:002494118n3 + ð468:32575=nÞ

The above expression shows a good agreement with Twu’s methodology; the tabulated comparison is shown in Table 2.8 and presented in a graphical form in Fig. 2.8.

Group 3. The Plus Fraction Characterization Methods 109

Table 2.8 Comparison Molecular Weight Correlations LeeKesler

RiazDaubert

Winn

Twu

Goossens

Ahmed

Fractions

Tb (°K)

Tb (°R)

M

M

M

M

M

n

M

1 2 3 4 5 6 7 8 9 10 Residue

363.15 403.15 453.15 473.15 498.15 548.15 583.15 593.15 613.15 644.35 764.83

653.67 725.67 815.67 851.67 896.67 986.67 1049.67 1067.67 1103.67 1159.83 1376.694

87.78 106.15 131.79 143.32 155.18 190.82 219.15 225.15 241.19 268.56 434.2

92.68 117.51 149.16 160.25 178.52 217.03 248.41 256.75 274.79 305.92 450.25

91.02 112.55 139.77 148.29 163.66 194.13 219.9 225.48 239.71 265.62 414.18

94.3 115.35 144.77 155.62 173.02 209.98 240.97 249.21 267.45 300.34 474.49

92 112 143.5 156 175 225.5 242 250 251 270 345

7 8.4 10.6 11.6 12.9 15.7 18 18.7 20.2 22.7 36.5

94.67761 111.7485 142.3398 156.8496 175.8379 216.263 248.3421 257.8502 277.7957 309.7129 460.0781

Molecular weight vs. boiling point Lee-Kesler

Riaz-Daubert

Winn Nomogram

Twu

Goossens

Ahmed

Silva-Rodriguez

1300

1400

600

500

400

M 300

200

100

0 600

700

800

900

n FIGURE 2.8 Comparison of molecular weight estimation methods.

1000 1100 Tb (°R)

1200

1500

110 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

EXAMPLE 2.1 Estimate the critical properties and the acentric factor of the heptanes-plus fraction, that is, C7+, with a measured molecular weight of 150 and specific gravity of 0.78 by using: n n

the Riazi-Daubert equation ((2.6)) and Edmister’s equation for calculating the acentric factor Eqs. (2.3A)–(2.3E)

Solution Applying the Riazi-Daubert equation:

n

θ ¼ aðMÞb γ c exp ½dM + eγ + f γM

 Tc ¼ 544:2ð150Þ0:2998 ð0:78Þ1:0555 exp 1:3478  104 ð150Þ  0:61641ð0:78Þ + 0 ¼ 1139:4°R pc ¼ 4:5203  104 ð150Þ0:8063 ð0:78Þ1:6015

 exp 1:8078  103 ð150Þ  0:3084ð0:78Þ + 0 ¼ 320:3 psia

Vc ¼ 1:206  102 ð150Þ0:20378 ð0:78Þ1:3036 exp 2:657  103 ð150Þ + 0:5287ð0:78Þ2:6012  103 ð150Þð0:78Þ ¼ 0:06035 ft3 =lb

Tb ¼ 6:77857ð150Þ0:401673 ð0:78Þ1:58262 exp 3:77409  103 ð150Þ + 2:984036ð0:78Þ  4:25288  103 ð150Þð0:78Þ ¼ 825:26°R

Use Edmister’s equation (Eq. (2.21)) to estimate the acentric factor: ω¼

3½ log ðpc =14:70Þ 1 7½Tc =Tb  1

ω¼

3½ log ð320:3=14:7Þ  1 ¼ 0:5067 7½1139:4=825:26  1

Applying Eqs. (2.3A)–(2.3E)  0:351579776   M M  233:3891996 ¼ 1170°R Tc ¼ 231:9733906  0:352767639 γ γ n

 0:885326626   M M pc ¼ 31829  0:106467189 + 49:62573013 ¼ 331:6 psi γ γ     M M + 0:00141973  1:243207091 ¼ 0:499 ω ¼ 0:279354619 ln γ γ    0:984070268 M vc ¼ 0:054703719 Mγ + 87:7041  106 γ 0:001007476 ¼ 9:69 ft3 =lbm  mol ¼ 0:0646 ft3 =lbm

Group 3. The Plus Fraction Characterization Methods 111

 0:664115311   M M 379:4428973 Tb ¼ 33:72211  1:40066452  ¼ 837°R γ γ ðM=γ Þ n

Silva and Rodriguez Correlations

Boiling point temperature and specific gravity of the cut are not available:    M ¼ 840°R Tb ¼ 460 + 447:08723 ln 64:2576 γ ¼ 0:132467 ln ðTb  460Þ + 0:0116483 ¼ 0:7982 The following table summarizes the results for this example. Method

Tc (°R)

pc (psia)

Vc (ft3/lb)

Tb (°R)

ω

Riazi-Daubert Silva and Rodriguez Sancet Eqs. (2.3A)–(2.3E)

1139 – 1133 1170

320 – 297 331

0.0604 – – 0.0646

825 840 828 837

0.507 – – 0.499

EXAMPLE 2.2 Estimate the critical properties, molecular weight, and acentric factor of a petroleum fraction with a boiling point of 198°F (658°R) and specific gravity of 0.7365 by using the following methods: (a) (b) (c) (d) (e) (f) (g) (h)

Riazi-Daubert (Eq. (2.5)) Riazi-Daubert (Eq. (2.5)) Cavett Kesler-Lee Winn and Sim-Daubert Watansiri-Owens-Starling Willman and Teja Eqs. (2.3F)–(2.3J)

Solution (a) Using Riazi-Daubert (Eq. (2.4)) θ ¼ aTbb γ c M ¼ 4:5673  105 ð658Þ2:1962 ð0:7365Þ1:0164 ¼ 96:4 Tc ¼ 24:2787ð658Þ0:58848 ð0:7365Þ0:3596 ¼ 990:67°R pc ¼ 3:12281  109 ð658Þ2:3125 ð0:7365Þ2:3201 ¼ 466:9 psia Vc ¼ 7:5214  103 ð658Þ0:2896 ð0:7365Þ0:7666 ¼ 0:06227 ft3 =lb

112 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Solve for Zc by applying Eq. (2.22): Zc ¼

pc Vc M ð466:9Þð0:06227Þð96:4Þ ¼ ¼ 0:26365 RTc ð10:73Þð990:67Þ

Solve for ω by applying Eq. (2.21): ω¼

3½ log ðpc =14:70Þ 1 7½Tc =Tb  1

ω¼

3½ log ð466:9=14:7Þ  1 ¼ 0:2731 7½ð990:67=658Þ  1

(b) Riazi-Daubert (Eq. (2.5)) θ ¼ aTbb γ c exp ½dTb + eγ + fTb γ  Applying the above expression and using the appropriate constants yields:

  M ¼ 581:96ð658Þ0:97476 0:73656:51274 exp 5:4307 104 ð658Þ   + 9:53384ð0:7365Þ + 1:11056 103 ð658Þð0:7365Þ M ¼ 96:911

  Tc ¼ 10:6443ð658Þ0:810676 0:73650:53691 exp 5:1747 104 ð658Þ   0:54444ð0:7365Þ + 3:5995 104 ð658Þð0:7365Þ Tc ¼ 985:7°R Similarly, pc ¼ 465:83 psia Vc ¼ 0:06257 ft3 =lb The acentric factor and critical compressibility factor can be obtained by applying Eqs. (2.21) and (2.22), respectively. ω¼

3½ log ðpc =14:70Þ 3½ log ð465:83=14:70Þ 1¼  1 ¼ 0:2877 7½Tc =Tb  1 7½ð986:7=658Þ  1 Zc ¼

pc Vc M ð465:83Þð0:06257Þð96:911Þ ¼ ¼ 0:2668 RTc 10:73ð986:7Þ

(c) Cavett’s Correlations: Step 1. Solve for Tc using Eq. (2.7) with the coefficients i

ai

bi

0 1 2 3 4 5 6 7

768.0712100000 1.7133693000 0.0010834003 0.0089212579 0.3889058400  106 0.5309492000  105 0.3271160000  107

2.82904060 0.94120109  103 0.30474749  105 0.20876110  104 0.15184103  108 0.11047899  107 0.48271599  107 0.13949619  109

Group 3. The Plus Fraction Characterization Methods 113

Tc ¼ a0 + a1 ðTb Þ + a2 ðTb Þ2 + a3 ðAPIÞðTb Þ + a4 ðTb Þ3 + a5 ðAPIÞðTb Þ2 + a6 ðEÞ2 ðTb Þ2

to give Tc ¼ 978.1°R Step 2. Calculate pc with Eq. (2.8): log ðpc Þ ¼ b0 + b1 ðTb Þ + b2 ðTb Þ2 + b3 ðAPIÞðTb Þ + b4 ðTb Þ3 + b5 ðAPIÞðTb Þ2 + b6 ðAPIÞ2 ðTb Þ + b7 ðAPIÞ2 ðTb Þ2 to give pc ¼ 466.1 psia. Step 3. Solve for the acentric factor by applying the Edmister correlation (Eq. (2.21)): ω¼

3½ log ð466:1=14:7Þ  1 ¼ 0:3147 7½ð980=658Þ  1

Step 4. Compute the critical compressibility by using Eq. (2.25): Zc ¼ 0:291  0:080ω  0:016ω2 Zc ¼ 0:291  ð0:08Þð0:3147Þ  0:016ð0:3147Þ2 ¼ 0:2642 Step 5. Estimate vc from Eq. (2.22): vc ¼

Zc RTc ð0:2642Þð10:731Þð980Þ ¼ ¼ 5:9495 ft3 =lb  mol pc 466:1

Assume a molecular weight of 96; estimate the critical volume: Vc ¼

5:9495 ¼ 0:06197 ft3 =lb 96

(d) Kesler and Lee Correlations: Step 1. Calculate pc from Eq. (2.9):

 lnðpc Þ ¼ 8:3634  0:0566=γ  0:24244 + 2:2898=γ + 0:11857=γ 2 103 Tb

 7 2 2 + 1:4685 + 3:648=γ + 0:47227=γ 10 Tb

  0:42019 + 1:6977=γ 2 1010 Tb3 to give pc ¼ 470 psia. Step 2. Solve for Tc by using Eq. (2.10); that is, Tc ¼ 341:7 + 811:1γ + ½0:4244 + 0:1174γ Tb +

½0:4669  3:26238γ 105 Tb

to give Tc ¼ 980°R. Step 3. Calculate the molecular weight, M, by using Eq. (2.11):

 M ¼ 12,272:6 + 9,486:4γ + ½4:6523  3:3287γ Tb + 1  0:77084γ  0:02058γ 2      720:79 107

181:98 1012 1:3437  + 1  0:80882γ  0:02226γ 2 1:8828  Tb Tb Tb Tb3 to give M ¼ 98.7

114 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Step 4. Compute the Watson characterization factor K and the parameter θ: ð658Þ1=3 ¼ 11:8 0:7365 658 θ¼ ¼ 0:671 980 K¼

Step 5. Solve for acentric factor by applying Eq. (2.13):

ω¼

 ln

h p i 6:09648 c  5:92714 + + 1:28862 ln ½θ  0:169347θ6 14:7 θ ¼ 0:306 15:6875 15:2518   13:4721 ln ½θ + 0:43577θ6 θ

Step 6. Estimate for the critical gas compressibility, Zc, by using Eq. (2.26): Zc ¼ 0:2918  0:0928ω Zc ¼ 0:2918  ð0:0928Þð0:306Þ ¼ 0:2634 Step 7. Solve for Vc by applying Eq. (2.22): Vc ¼

Zc RTc ð0:2634Þð10:73Þð980Þ ¼ ¼ 0:0597 ft3 =lb pc M ð470Þð98:7Þ

(e) Winn-Sim-Daubert Correlations: Step 1. Estimate pc from Eq. (2.14):   γ 2:4853 pc ¼ 3:48242  109 2:3177 Tb pc ¼ 478:6 psia Step 2. Solve for Tc by applying Eq. (2.15):

 Tc ¼ exp 3:9934718Tb0:08615 γ 0:04614 Tc ¼ 979:2°R Step 3. Calculate M from Eq. (2.16):   Tb2:3776 M ¼ 1:4350476  105 0:9371 γ M ¼ 95:93 Step 4. Solve for the acentric factor from Eq. (2.21): ω¼

3½ log ðpc =14:70Þ 1 7½ðTc =Tb  1Þ ω ¼ 0.3280

Group 3. The Plus Fraction Characterization Methods 115

Solve for Zc by applying Eq. (2.24): Zc ¼ 0:291  0:080ω Zc ¼ 0:291  ð0:08Þð0:3280Þ ¼ 0:2648 Step 5. Calculate the critical volume Vc from Eq. (2.22): Vc ¼

Vc ¼

Zc RTc pc M

ð0:2648Þð10:731Þð979:2Þ ¼ 0:06059 ft3 =lb ð478:6Þð95:93Þ

(f) Watansiri-Owens-Starling Correlations: Step 1. Because Eqs. (2.17) through (2.19) require the molecular weight, assume M ¼ 96. Step 2. Calculate Tc from Eq. (2.17): lnðTc Þ ¼ 0:0650504  0:00052217Tb + 0:03095 ln ½M + 1:11067 lnðTb Þ h i + M 0:078154γ 1=2  0:061061γ 1=3  0:016943γ Tc ¼ 980:0°R Step 3. Determine the critical volume from Eq. (2.18) to give lnðVc Þ ¼ 76:313887  129:8038γ + 63:1750γ 2  13:175γ 3 + 1:10108 ln ½M + 42:1958 ln ½γ Vc ¼ 0:06548 ft3 =lb Step 4. Solve for the critical pressure of the fraction by applying Eq. (2.19) to produce  0:8     Tc M Tb  0:08843889  8:712 lnðpc Þ ¼ 6:6418853 + 0:01617283 Vc M Tc pc ¼ 426:5 psia Step 5. Calculate the acentric factor from Eq. (2.20) 8 9 5:12316667  104 Tb + 0:281826667ðTb =MÞ + 382:904=M > > > > > > > > 2 > > 5 4 > > ð T =γ Þ  0:12027778  10 ð T Þ ð M Þ + 0:074691  10 > > b b > > > > > > 2 4 > > > > ð Þ ð Þ ð Þ M + 0:001261 γ M + 0:1265  10 < =5T  b ω¼ 1=3 ðTb Þ > > 9M 2 4 > > > > > + 0:2016  10 ðγ ÞðMÞ  66:29959 M > > > > > > > > > > > 2=3 > > T > > b > > : 0:00255452 2 ; γ to give ω ¼ 0.2222

116 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Step 6. Compute the critical compressibility factor by applying Eq. (2.24): Zc ¼ 0:2918  0:0928ω ¼ 0:2918  0:0928ð0:2222Þ ¼ 0:27112 (g) Willman and Teja Correlations: Step 1. Using Microsoft Solver, determine the ECN “n” by matching the given boiling point “Tb”   Tb ¼ 434:38878 + 50:125279n  0:9097293n2 + 7:0280657 103 n3  601:85651=n   658 ¼ 434:38878 + 50:125279n  0:9097293n2 + 7:0280657 103 n3  601:85651=n To give n ¼ 7 Step 2. Calculate Tc from Eq. (2.33): h i Tc ¼ Tb 1 + ð1:25127 + 0:137252nÞ0:884540633 h i Tc ¼ Tb 1 + ð1:25127 + 0:137252ð7ÞÞ0:884540633 ¼ 983:7°R Step 3. Calculate pc from Eq. (2.34): Pc ¼ Pc ¼

339:0416805 + 1184:157759n ½0:873159 + 0:54285n1:9265669 339:0416805 + 1184:157759ð7Þ ½0:873159 + 0:54285ð7Þ1:9265669

¼ 441:8 psia

(h) Eqs. (2.3F) through (2.3J):  0:967105064   Tb Tb + 0:010009512 ¼ 996°R Tc ¼ 1:379242274 + 0:012011487 γ γ   Tb pc ¼ 2898:631565 exp ½0:002020245 ðTb =γ Þ  0:00867297 + 0:099138742 γ ¼ 469 psi  1:250922701   Tb Tb ω ¼ 0:000115057  0:92098982 ¼ 0:26 + 0:00068975 γ γ vc ¼ 8749:94  106

 3 Tb γ

 9:75748  106

 2 Tb γ

  Tb + 0:014653 γ

595:7922076   ¼ 6:273 ft3 =lbm  mol Tb γ   0:001617123Tb M ¼ 52:11038728exp  132:1481762 ¼ 89 γ 4:60231684 

Group 3. The Plus Fraction Characterization Methods 117

The following table summarizes the results for this example. Method

Tc (°R) pc (psia) Vc (ft3/lb) M

ω

Zc

Riazi-Daubert no. 1 Riazi-Daubert no. 2 Cavett Kesler-Lee Winn Watansiri Willman-Teja Eqs. (2.3F)–(2.3J)

990.67 986.70 978.10 980.00 979.20 980.00 983.7 996

0.2731 0.2877 0.3147 0.3060 0.3280 0.2222 – 0.26

0.26365 0.26680 0.26420 0.26340 0.26480 0.27112 – –

466.90 465.83 466.10 469.00 478.60 426.50 441.8 469

0.06227 0.06257 0.06197 0.05970 0.06059 0.06548 – 0.0689

96.400 96.911 – 98.700 95.930 – – 89

EXAMPLE 2.3 If the molecular weight and specific gravity of the heptanes-plus fraction are 216 and 0.8605, respectively, calculate the critical temperature and pressure by using (a) Rowe’s correlations. (b) Standing’s correlations. (c) Magoulas-Tassios’s correlations. (d) Eqs. (2.3A) and (2.3B). Solution (a) Rowe’s correlations Step 1. Calculate the number of carbon atoms of C7+ from Eq. (2.28) to give n¼

MC7 +  2:0 216  2:0 ¼ ¼ 15:29 14 14

Step 2. Calculate the coefficient a: a ¼ 2:95597  0:090597ðnÞ2=3 a ¼ 2:95597  0:090597ð15:29Þ2=3 ¼ 2:39786 Step 3. Solve for the critical temperature from Eq. (2.27) to yield ðTc ÞC7 + ¼ 1:8½961  10a 

 ðTc ÞC7 + ¼ 1:8 961  102:39786 ¼ 1279:8°R Step 4. Calculate the coefficient Y: Y ¼ 0:0137726826n + 0:6801481651 Y ¼ 0:0137726826ð2:39786Þ + 0:6801481651 ¼ 0:647123

118 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Step 5. Solve for the critical pressure from Eq. (2.29) to give ðpc ÞC7 + ¼

10ð4:89165 + Y Þ 10ð4:89165 + 0:647123Þ ¼ ¼ 270 psi ðTc ÞC7 + 1279:8

(b) Standing’s correlations Step 1. Solve for the critical temperature by using Eq. (2.31) to give h i h i ðTc ÞC7 + ¼ 608 + 364 log ðMÞC7 +  71:2 + 2450log ðMÞC7 +  3800 log ðγ ÞC7 + ðTc ÞC7 + ¼ 1269:3°R Step 2. Calculate the critical pressure from Eq. (2.32) to yield h i h h i ðpc ÞC7 + ¼ 1188  431 log ðMÞC7 +  61:1 + 2319  852 log ðMÞC7 +  53:7 h i ðγ ÞC7 +  0:8 ðpc ÞC7 + ¼ 270 psia (c) Magoulas-Tassios correlations Tc ¼ 1247:4 + 0:792M + 1971γ 

27,000 707:4 + M γ

27, 000 707:4 + ¼ 1317°R 216 0:8605 227 1:1663  + 1:2702 lnðMÞ lnðpc Þ ¼ 0:01901  0:0048442M + 0:13239γ + M γ

Tc ¼ 1247:4 + 0:792ð216Þ + 1971ð0:8605Þ 

lnðpc Þ ¼ 0:01901  0:0048442ð216Þ + 0:13239ð0:8605Þ 227 1:1663  + 1:2702 ln ð216Þ ¼ 5:6098 216 0:8605 pc ¼ exp ð5:6098Þ ¼ 273 psi +

(d) Eqs. (2.3A) and (2.3B)  0:351579776   M M Tc ¼ 231:9733906  0:352767639  233:3891996 ¼ 1294°R γ γ

pc ¼ 31829

 0:885326626   M M  0:106467189 + 49:62573013 ¼ 262 psi γ γ

Summary of results is given below: Method

Tc (°R)

pc (psia)

Rowe Standing Magoulas-Tassios Eqs. (2.3A) and (2.3B)

1279 1269 1317 1294

270 270 273 262

Group 3. The Plus Fraction Characterization Methods 119

EXAMPLE 2.4 Calculate the critical properties and the acentric factor of C7+ with a measured molecular weight of 198.71 and specific gravity of 0.8527. Employ the following methods: (a) Rowe’s correlations. (b) Standing’s correlations. (c) Riazi-Daubert’s correlations. (d) Magoulas-Tassios’s correlations. (e) Eqs. (2.3A) through (2.3E). Solution (a) Rowe’s correlations Step 1. Calculate the number of carbon atoms, n, and the coefficient a of the fraction to give n¼

M  2 198:71  2 ¼ ¼ 14:0507 14 14

a ¼ 2:95597  0:090597n2=3 a ¼ 2:95597  0:090597ð14:0507Þ2=3 ¼ 2:42844 Step 2. Determine Tc from Eq. (2.28) to give ðTc ÞC7 + ¼ 1:8½961  10a 

 ðTc ÞC7 + ¼ 1:8 961  102:42844 ¼ 1247°R Step 3. Calculate the coefficient Y: Y ¼ 0:0137726826n + 0:6801481651 Y ¼ 0:0137726826ð2:42844Þ + 0:6801481651 ¼ 0:6467 Step 4. Compute pc from Eq. (2.29) to yield ðpc ÞC7 + ¼

10ð4:89165 + 0:6467Þ ¼ 277 psi 1247

Step 5. Determine Tb by applying Eq. (2.30) to give ðTb ÞC7 + ¼ 0:0004347ðTc Þ2C7 + + 265 ¼ 0:0004347ð1247Þ2 + 265 ¼ 941°R Step 6. Solve for the acentric factor by applying Eq. (2.21) to give ω¼

3½ log ðpc =14:70Þ 1 7½ðTc =Tb  1Þ

ω ¼ 0:6123

120 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

(b) Standing’s correlations Step 1. Solve for the critical temperature of C7+ by using Eq. (2.31) to give h i h i ðTc ÞC7 + ¼ 608 + 364 log ðMÞC7 +  71:2 + 2450log ðMÞC7 +  3800 log ðγ ÞC7 + ðTc ÞC7 + ¼ 1247:73°R Step 2. Calculate the critical pressure from Eq. (2.32) to give h i h h i ðpc ÞC7 + ¼ 1188  431 log ðMÞC7 +  61:1 + 2319  852 log ðMÞC7 +  53:7 h i ðγ ÞC7 +  0:8 ðpc ÞC7 + ¼ 291:41 psia (c) Riazi-Daubert correlations θ ¼ aðMÞb γ c exp ½dM + eγ + f γM Tc ¼ 544:2ð198:71Þ0:2998 ð0:8577Þ1:0555

 exp 1:3478  104 ð198:71Þ  0:61641ð0:8577Þ ¼ 1294°R pc ¼ 4:5203  104 ð198:71Þ0:8061 ð0:8577Þ1:6015

 exp 1:8078  103 ð198:71Þ  0:3084ð0:8577Þ ¼ 264 psi Determine Tb by applying Eq. (2.6) to give Tb ¼ 958.5°R. Solve for the acentric factor from Eq. (2.21) to give ω¼

3½ log ðpc =14:70Þ 1 7½ðTc =Tb  1Þ

ω ¼ 0:5346 (d) Magoulas-Tassios correlations Tc ¼ 1247:4 + 0:792M + 1971γ 

27,000 707:4 + M γ

Tc ¼ 1247:4 + 0:792ð198:71Þ + 1971ð0:8527Þ  lnðpc Þ ¼ 0:01901  0:0048442M + 0:13239γ +

27, 000 707:4 + ¼ 1284°R 198:71 0:8527

227 1:1663  + 1:2702 lnðMÞ M γ

lnðpc Þ ¼ 0:01901  0:0048442ð198:71Þ + 0:13239ð0:8527Þ + 

1:1663 + 1:2702 ln ð198:71Þ ¼ 5:6656 0:8527 pc ¼ exp ð5:6656Þ ¼ 289 psi

227 198:71

Group 3. The Plus Fraction Characterization Methods 121

4:559 + 0:21699 ln ðMÞ M 4:559 ω ¼ 0:64235 + 0:0014667ð198:71Þ + 0:021876ð0:8527Þ 198:71 + 0:21699 ln ð198:71Þ ¼ 0:531

ω ¼ 0:64235 + 0:0014667M + 0:21876γ

(e) Eqs. (2.3A) through (2.3E)  0:351579776   M M Tc ¼ 231:9733906  0:352767639  233:3891996 ¼ 1259°R γ γ  0:885326626   M M + 49:62573013 ¼ 280 psi  0:106467189 γ γ     M M ω ¼ 0:279354619 ln + 0:00141973  1:243207091 ¼ 0:61 γ γ

pc ¼ 31829

 0:984070268   M M + 87:7041  106 vc ¼ 0:054703719  0:001007476 γ γ ¼ 11:717 ft3 =lbm-mol ¼ 0:0589 ft3 =lbm  0:664115311   M M 379:4428973  Tb ¼ 33:72211  1:40066452 ¼ 931°R γ γ ðM=γ Þ

The following table summarizes the results for this example. Method

Tc (°R)

pc (psia)

Tb (°R)

ω

Rowe’s Standing’s Riazi-Daubert Magoulas-Tassios Eqs. (2.3A)–(2.3E)

1247 1247 1294 1284 1259

277 291 264 289 280

941 – 950 – 931

0.612 – 0.534 0.531 0.61

Recommended Plus Fraction Characterizations As discussed in Chapter 5, numerous forms of cubic equations of state that are widely used in the petroleum industry (eg, Peng-Robinson EOS) require the critical properties and acentric factor for each of the defined and undefined petroleum fractions in the hydrocarbon system. The characterizations of the undefined petroleum fractions rely on applying generalized correlations with their parameters, which were developed by regression to match critical and physical properties of pure components. Selected generalized correlations, often with significantly diverging results among themselves, are used in equations of state applications that

122 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

commonly produce erroneous EOS results. As a result, equations of state are not predictive and require adjusting or “tuning” the critical properties of the undefined petroleum fractions by matching the available laboratory PVT data. Recognizing the fact that tuning EOS requires the adjustment of the critical properties of undefined hydrocarbon groups, the need is to select a generalized property correlation (or correlations) that is capable of producing a logical and consistent properties trend of the hydrocarbon groups (eg, Tc and ω are increasing and pc is decreasing with increasing molecular weight of the hydrocarbon group). The Riazi-Daubert critical properties and Kesler-Lee acentric factor generalized correlations provide the suggested required properties trend and, therefore, are recommended for use in EOS applications. As of the determination of the exact composition and critical characteristics of the heavy-end plus fraction (eg, C7+) is a nearly impossible task, a useful type of compositional analysis is to determine the relative amounts of paraffins “P,” naphthenes “N,” and aromatics “A” for pseudofractions distilled by the TBP analysis. Detailed PNA analysis could provide accurate estimation of the physical and critical properties of these petroleum fractions. The crude oil assay testing could include such measurements; however, analysis could be costly and time consuming. This author agrees with Whitson and Brule (2000) that PNA determination has a limited usefulness for improving equations of state fluid characterizations. However, detail of the PNA determination methodologies is summarized below.

PNA Determination The vast number of hydrocarbon compounds making up naturally occurring crude oil has been grouped chemically into several series of compounds. Each series consists of those compounds similar in their molecular makeup and characteristics. Within a given series, the compounds range from extremely light, or chemically simple, to heavy, or chemically complex. In general, it is assumed that the heavy (undefined) hydrocarbon fractions are composed of three hydrocarbon groups: n n n

paraffins (P) naphthenes (N) aromatics (A)

The PNA content of the plus fraction (the undefined hydrocarbon fraction) can be estimated experimentally from distillation or a chromatographic analysis. Both types of analysis provide information valuable for use in characterizing the plus fractions. Bergman et al. (1977) outlined the chromatographic analysis procedure by which distillation cuts are characterized by the density

Group 3. The Plus Fraction Characterization Methods 123

and molecular weight as well as by weight average boiling point (WABP). It should be pointed out that when a single boiling point is given for a plus fraction, it is given as its volume average boiling point (VABP). Generally, five methods are used to define the normal boiling point for the plus fraction (eg, C7+) these are 1. VABP — This is defined mathematically by the following expression: X Vi Tbi

VABP ¼

(2.36)

i

where Tbi ¼ boiling point of the distillation cut i, °R Vi ¼ volume fraction of the distillation cut i 2. WABP — This property is defined by the following expression: WABP ¼

X wi Tbi

(2.37)

i

where wi ¼ weight fraction of the distillation cut i 3. Molar average boiling point (MABP) — The MABP is given by the following relationship: MABP ¼

X xi Tbi

(2.38)

i

where xi ¼ mole fraction of the distillation cut i 4. Cubic average boiling point (CABP) — The relationship is defined as " #3 X 1=3 CABP ¼ xi Tbi

(2.39)

i

5. Mean average boiling point (MeABP) — This is defined by MeABP ¼

MABP + CABP 2

(2.40)

The MeABP (Tb) is recommended as the most representative boiling point of the plus fraction because it allows better reproduction of the properties of the entire mixture. As indicated by Edmister and Lee (1984), the above five expressions for calculating normal boiling points result in values that do not differ significantly from one another for narrow boiling petroleum fractions. The three parameters that are commonly employed to estimate the PNA content of the heavy hydrocarbon fraction include:

124 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

n n n

molecular weight specific gravity VABP or WABP

Hopke and Lin (1974), Erbar (1977), Bergman et al. (1977), and Robinson and Peng (1978) used the above parameters and PNA concept to characterize the undefined hydrocarbon fractions in terms of their critical properties and acentric factors. Two of the most widely used PNA based characterization methods are n n

Peng-Robinson Method Bergman’s Method

Details of these two methods are summarized below

Peng-Robinson’s Method Robinson and Peng (1978) proposed a detailed procedure for characterizing heavy hydrocarbon fractions. The procedure is summarized in the following steps. Step 1. Calculate the PNA content (XP, XN, XA) of the undefined fraction by solving the following three rigorously defined equations: X

Xi ¼ 1

(2.41)

½Mi Tbi Xi  ¼ ðMÞðWABPÞ

(2.42)

i¼P, N, A

X i¼P, N, A

X

½Mi Xi  ¼ M

(2.43)

i¼P, N, A

where XP ¼ mole fraction of the paraffinic group in the undefined fraction XN ¼ mole fraction of the naphthenic group in the undefined fraction XA ¼ mole fraction of the aromatic group in the undefined fraction WABP ¼ weight average boiling point of the undefined fraction, °R M ¼ molecular weight of the undefined fraction (Mi) ¼ average molecular weight of each cut (ie, PNA) (Tb)i ¼ boiling point of each cut, °R Eqs. (2.41) through (2.43) can be written in a matrix form as follows: 2

1

1

1

3 2

XP

3

2

1

3

7 7 6 7 6 6 6 ½MTb P ½MTb N ½MTb A 7 6 XN 7 ¼ 6 MðWABPÞ 7 5 5 4 5 4 4 ½M P ½M N ½M A M XA

(2.44)

Group 3. The Plus Fraction Characterization Methods 125

Robinson and Peng pointed out that it is possible to obtain negative values for the PNA contents. To prevent these negative values, the authors imposed the following constraints: 0  XP  0:90 XN  0:00 XA  0:00

Solving Eq. (2.44) for the PNA content requires the WABP and molecular weight of the cut of the undefined hydrocarbon fraction. Robinson and Peng proposed the following set of working expressions to estimate the boiling point temperature (Tb)P, (Tb)N, and (Tb)A and the molecular waiting (M)P, (M)N, and (M)A: Paraffinic group :

ln ðTb ÞP ¼ ln ð1:8Þ +

6 h i X ai ðn  6Þi1

(2.45)

i¼1

Naphthenic group :

ln ðTb ÞN ¼ ln ð1:8Þ +

6 h i X ai ðn  6Þi1

(2.46)

i¼1

Aromatic group :

ln ðTb ÞA ¼ ln ð1:8Þ +

6 h i X ai ðn  6Þi1

(2.47)

i¼1

Paraffinic group : ðMÞP ¼ 14:026n + 2:016

(2.48)

Naphthenic group : ðMÞN ¼ 14:026n  14:026

(2.49)

Aromatic group : ðMÞA ¼ 14:026n  20:074

(2.50)

where n ¼ number of carbon atoms in the undefined hydrocarbon fraction, eg, 7, 8, …, etc. ai ¼ coefficients of the equations, which are given below

Coefficient

Paraffin, P

Napthene, N

Aromatic, A

a1 a2 a3 a4 a5 a6

5.83451830 5.85793320 5.86717600 0.84909035  101 0.79805995  101 0.80436947  101 0.52635428  102 0.43098101  102 0.47136506  102 0.21252908  103 0.14783123  103 0.18233365  103 0.44933363  105 0.27095216  105 0.38327239  105 0.37285365  107 0.19907794  107 0.32550576  107

126 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Step 2. Having obtained the PNA content of the undefined hydrocarbon fraction, as outlined in step 1, calculate the critical pressure of the fraction by applying the following expression: pc ¼ XP ðpc ÞP + XN ðpc ÞN + XA ðpc ÞA

(2.51)

where pc ¼ critical pressure of the heavy hydrocarbon fraction, psia. The critical pressure for each cut of the heavy fraction is calculated according to the following equations: Paraffinic group : ðpc ÞP ¼ Naphthenic group : ðpc ÞN ¼

206:126096n + 29:67136 ð0:227n + 0:340Þ2

206:126096n  206:126096

Aromatic group : ðpc ÞA ¼

ð0:227n  0:137Þ2 206:126096n  295:007504 ð0:227n  0:325Þ2

(2.52)

(2.53)

(2.54)

Step 3. Calculate the acentric factor of each cut of the undefined fraction by using the following expressions: Paraffinic group : ðωÞP ¼ 0:432n + 0:0457

(2.55)

Naphthenic group : ðωÞN ¼ 0:0432n  0:0880

(2.56)

Aromatic group : ðωÞA ¼ 0:0445n  0:0995

(2.57)

Step 4. Calculate the critical temperature of the fraction under consideration by using the following relationship: Tc ¼ XP ðTc ÞP + XN ðTc ÞN + XA ðTc ÞA

(2.58)

where Tc ¼ critical temperature of the fraction, °R. The critical temperatures of the various cuts of the undefined fractions are calculated from the following expressions: (

)

 3 log ðPc ÞP  3:501952

 Paraffinic group : ðTc ÞP ¼ S 1 + ð T b ÞP 7 1 + ðωÞP ( Naphthenic group : ðTc ÞN ¼ S1 ( Aromatic group : ðTc ÞA ¼ S1

(2.59)

)

 3 log ðPc ÞN  3:501952

 1+ ðTb ÞN (2.60) 7 1 + ðωÞP

)

 3 log ðPc ÞA  3:501952

 1+ ðTb ÞA 7 1 + ðωÞA

(2.61)

Group 3. The Plus Fraction Characterization Methods 127

where the correction factors S and S1 are defined by the following expressions: S ¼ 0:99670400 + 0:00043155n S1 ¼ 0:99627245 + 0:00043155n

Step 5. Calculate the acentric factor of the heavy hydrocarbon fraction (residue) by using the Edmister’s correlation (Eq. (2.21)) to give ω¼

3½ log ðPc =14:7Þ 1 7½ðTc =Tb Þ  1

(2.62)

where ω ¼ acentric factor of the residue heavy fraction Pc ¼ critical pressure of the heavy fraction, psia Tc ¼ critical temperature of the heavy fraction, °R Tb ¼ average weight boiling point, °R EXAMPLE 2.5 Using the Peng and Robinson approach, calculate the critical pressure, critical temperature, and acentric factor of an undefined hydrocarbon fraction that is characterized by the laboratory measurements: n n n

molecular weight ¼ 94 specific gravity ¼ 0.726133 WABP ¼ 195°F, ie, 655°R

Solution Step 1. Calculate the boiling point of each cut by applying Eqs. (2.45) through (2.47) to give ( ) 6 h i X i1 ¼ 666:58°R ðTb ÞP ¼ exp ln ð1:8Þ + ai ðn  6Þ ) 6 h i X i1 ¼ 630°R ln ð1:8Þ + ai ðn  6Þ

(

) 6 h i X ln ð1:8Þ + ai ðn  6Þi1 ¼ 635:85°R

ðTb ÞN ¼ exp ðTb ÞA ¼ exp

i¼1

(

i¼1

i¼1

Step 2. Compute the molecular weight of various cuts by using Eqs. (2.48) through (2.50) to give Paraffinic group : ðMÞP ¼ 14:026n + 2:016 ðMÞP ¼ 14:026  7 + 2:016 ¼ 100:198 Naphthenic group : ðMÞN ¼ 14:026n  14:026 ðMÞN ¼ 14:0129  7  14:026 ¼ 84:156

128 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Aromatic group : ðMÞA ¼ 14:026n  20:074 ðMÞA ¼ 14:026  7  20:074 ¼ 78:180 Step 3. Solve Eq. (2.44) for XP, XN, and XA: 3 2 3 2 1 1 1 XP 7 6 7 6 6 ½MTb P ½MTb N ½MTb A 7 6 XN 7 ¼ 5 4 5 4 ½M P ½M N ½M A XA 2

1

1

1

3 2

2

3

1

7 6 6 MðWABPÞ 7 5 4 M

XP

3

2

1

3

7 6 7 7 6 6 6 66689:78 53018:28 49710:753 7 6 XN 7 ¼ 6 61570 7 5 4 5 5 4 4 199:198 84:156 78:180 94 XA to give XP ¼ 0:6313 XN ¼ 0:3262 XA ¼ 0:04250 Step 4. Calculate the critical pressure of each cut in the undefined fraction by applying Eqs. (2.52) and (2.54): ðpc ÞP ¼ ðpc ÞN ¼ ðpc ÞA ¼

206:126096  7 + 29:67136 ð0:227  7 + 0:340Þ2

¼ 395:70 psia

206:126096  7  206:126096 ð0:227  7  0:137Þ2 206:126096  7  295:007504 ð0:227  7  0:325Þ2

¼ 586:61 ¼ 718:46

Step 5. Calculate the critical pressure of the heavy fraction from Eq. (2.51) to give pc ¼ XP ðpc ÞP + XN ðpc ÞN + XA ðpc ÞA pc ¼ 0:6313ð395:70Þ + 0:3262ð586:61Þ + 0:0425ð718:46Þ ¼ 471 psia Step 6. Compute the acentric factor for each cut in the fraction by using Eqs. (2.55) through (2.57) to yield ðωÞP ¼ 0:432  7 + 0:0457 ¼ 0:3481 ðωÞN ¼ 0:0432  7  0:0880 ¼ 0:2144 ðωÞA ¼ 0:0445  7  0:0995 ¼ 0:2120

Group 3. The Plus Fraction Characterization Methods 129

Step 7. Solve for (Tc)P, (Tc)N, and (Tc)A by using Eqs. (2.59) through (2.61) to give S ¼ 0:99670400 + 0:00043155  7 ¼ 0:99972 S1 ¼ 0:99627245 + 0:00043155  7 ¼ 0:99929  3 log ½395:7  3:501952 ðTc ÞP ¼ 0:99972 1 + 666:58 ¼ 969:4°R 7½1 + 0:3481  3 log ½586:61  3:501952 ðTc ÞN ¼ 0:99929 1 + 630 ¼ 974:3°R 7½1 + 0:2144  3 log ½718:46  3:501952 ðTc ÞA ¼ 0:99929 1 + 635:85 ¼ 1014:9°R 7½1 + 0:212 Step 8. Solve for (Tc) of the undefined fraction from Eq. (2.58): Tc ¼ XP ðTc ÞP + XN ðTc ÞN + XA ðTc ÞA ¼ 964:1°R Step 9. Calculate the acentric factor from Eq. (2.62) to give ω¼

3½ log ð471:7=14:7Þ  1 ¼ 0:3680 7½ð964:1=655Þ  1

Bergman’s Method Bergman et al. (1977) proposed a detailed procedure for characterizing the undefined hydrocarbon fractions based on calculating the PNA content of the fraction under consideration. The proposed procedure originated from analyzing extensive experimental data on lean gases and condensate systems. In developing the correlation, the authors assumed that the paraffinic, naphthenic, and aromatic groups have the same boiling point. The computational procedure is summarized in the following steps. Step 1. Estimate the weight fraction of the aromatic content in the undefined fraction by applying the following expression: wA ¼ 8:47  0:7Kw

(2.63)

where wA ¼ weight fraction of aromatics Kw ¼ Watson characterization factor, defined mathematically by the following expression Kw ¼ ðTb Þ1=3 =γ

γ ¼ specific gravity of the undefined fraction Tb ¼ WABP, °R

(2.64)

130 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Bergman et al. imposed the following constraint on the aromatic content: 0:03  wA  0:35

Step 2. Bergman proposed the following set of expressions for estimating specific gravities of the three groups, ie, γ P, γ N, and γ A γ P ¼ 0:583286 + 0:00069481ðTb  460Þ      0:7572818 106 ðTb  460Þ2 + 0:3207736 109 ðTb  460Þ3

(2.65)

γ N ¼ 0:694208 + 0:0004909267ðTb  460Þ      0:659746 106 ðTb  460Þ2 + 0:330966 109 ðTb  460Þ3

(2.66)

  γ A ¼ 0:916103  0:000250418ðTb  460Þ + 0:357967 106 ðTb  460Þ2  9   0:166318 10 ðTb  460Þ3 ð2:67Þ

Bergman suggested a minimum paraffin content of 0.20 was set by Bergman et al. To ensure that this minimum value is met, the estimated aromatic content that results in negative values of wP is increased in increments of 0.03 up to a maximum of 15 times until the paraffin content exceeds 0.20. They pointed out that this procedure gives reasonable results for fractions up to C15. Step 3. With the estimate of the aromatic content as outlined in step 1 and specific gravity of the three PNA groups as calculated in step 2, determine the weight fractions of the paraffinic and naphthenic cuts by solving the following system of linear equations simultaneously: wP + wN ¼ 1  wA

(2.68)

wP wN 1 wA + ¼  γP γN γ γA

(2.69)

where wP ¼ weight fraction of the paraffin cut wN ¼ weight fraction of the naphthene cut γ ¼ specific gravity of the undefined fraction γ P, γ N, γ A ¼ specific gravity of the three groups at the WABP of the undefined fraction Combining Eq. (2.68) with (2.69) and solving for weight fraction of the aromatic group gives   γ P γ N wA γ P γ N   ð1  wA Þγ P γ γA wP γN  γP wN ¼ 1  ðwA + wP Þ

Group 3. The Plus Fraction Characterization Methods 131

Step 4. Calculate the critical temperature, the critical pressure, and acentric factor of each cut from the following expressions. For paraffins, ðTc ÞP ¼ 735:23 + 1:2061ðTb  460Þ  0:00032984ðTb  460Þ2

(2.70)

ðpc ÞP ¼ 573:011  1:13707ðTb  460Þ + 0:00131625ðTb  460Þ2    0:85103 106 ðTb  460Þ3

(2.71)

  ðωÞP ¼ 0:14 + 0:0009ðTb  460Þ + 0:233 106 ðTb  460Þ2

(2.72)

For naphthenes, ðTc ÞN ¼ 616:8906 + 2:6077ðTb  460Þ    0:003801ðTb  460Þ2 + 0:2544 105 ðTb  460Þ3

(2.73)

  ðpc ÞN ¼ 726:414  1:3275ðTb  460Þ + 0:9846 103 ðTb  460Þ2    0:45169 106 ðTb  460Þ3

(2.74)

ðωÞN ¼ ðωÞP  0:075

(2.75)

Bergman et al. assigned the following special values of the acentric factor to the C8, C9, and C10 naphthenes: C8 ðωÞN ¼ 0:26 C9 ðωÞN ¼ 0:27 C10 ðωÞN ¼ 0:35

For the aromatics, ðTc ÞA ¼ 749:535 + 1:7017ðTb  460Þ    0:0015843ðTb  460Þ2 + 0:82358 106 ðTb  460Þ3

(2.76)

ðpc ÞA ¼ 1184:514  3:44681ðTb  460Þ + 0:0045312ðTb  460Þ2    0:23416 105 ðTb  460Þ3

(2.77)

ðωÞA ¼ ðωÞP  0:1

(2.78)

Step 5. Calculate the critical pressure, the critical temperature, and acentric factor of the undefined fraction from the following relationships: pc ¼ wP ðpc ÞP + wN ðpc ÞN + wA ðpc ÞA

(2.79)

Tc ¼ wP ðTc ÞP + wN ðTc ÞN + wA ðTc ÞA

(2.80)

ω ¼ wP ðωÞP + wN ðωÞN + wA ðωÞA

(2.81)

Whitson (1984) suggested that the Peng-Robinson and Bergman PNA methods are not recommended for characterizing reservoir fluids containing fractions heavier than C20.

132 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

EXAMPLE 2.6 Using Bergman’s approach, calculate the critical pressure, critical temperature, and acentric factor of an undefined hydrocarbon fraction that is characterized by the laboratory measurements: n n n

molecular weight ¼ 94 specific gravity ¼ 0.726133 WABP ¼ 195°F, ie, 655°R

Solution Step 1. Calculate Watson characterization factor: Kw ¼

ðTb Þ1=3 ð655Þ1=3 ¼ ¼ 11:96 γ 0:7261

Step 2. Estimate the weight fraction of the aromatic content from wA ¼

0:847 0:847 ¼ ¼ 0:070819 Kw 11:96

Step 3. Estimate specific gravities of the three groups, ie, γ P, γ N, and γ A, by applying Eqs. (2.65)–(2.67):   γ P ¼ 0:583286 + 0:00069481ð195Þ  0:7572818 106 ð195Þ2   + 0:3207736 109 ð195Þ3 ¼ 0:692422   γ N ¼ 0:694208 + 0:0004909267ð195Þ  0:659746 106 ð195Þ2   + 0:330966 109 ð195Þ3 ¼ 0:767306   γ A ¼ 0:916103  0:000250418ð195Þ + 0:357967 106 ð195Þ2   0:166318 109 ð195Þ3 ¼ 0:879652 Step 4. Estimate the weight fraction of the paraffinic and naphthenic contents from   γ P γ N wA γ P γ N  ð1  wA Þγ P  γ γA wP ¼ 0:607933 γN  γP wN ¼ 1  ðwA + wP Þ ¼ 1  ð0:079819 + 0:607933Þ ¼ 0:321248 Step 5. Calculate the critical temperature, the critical pressure, and acentric factor of each cut from the following expressions: For paraffins, ðTc ÞP ¼ 735:23 + 1:2061ð195Þ  0:00032984ð195Þ2 ¼ 957:9°R   ðpc ÞP ¼ 573:011  1:13707ð159Þ + 0:00131625ð195Þ2  0:85103 106 ð195Þ3 ¼ 395 psi   ðωÞP ¼ 0:14 + 0:0009ð195Þ + 0:233 106 ð195Þ2 ¼ 0:3244

Group 3. The Plus Fraction Characterization Methods 133

For naphthenes,

  ðTc ÞN ¼ 616:8906 + 2:6077ð195Þ  0:003801ð195Þ2 + 0:2544 105 ð195Þ3 ¼ 999:8°R     ðpc ÞN ¼ 726:414  1:3275ð195Þ + 0:9846 103 ð195Þ2  0:45169 106 ð195Þ3 ¼ 501:6 psi ðωÞN ¼ ðωÞP  0:075 ¼ 0:2494 For the aromatics,

  ðTc ÞA ¼ 749:535 + 1:7017ð195Þ  0:0015843ð195Þ2 + 0:82358 106 ð195Þ3 ¼ 1027°R   ðpc ÞA ¼ 1184:514  3:44681ð195Þ + 0:0045312ð195Þ2  0:23416 105 ð195Þ3 ¼ 667 psi ðωÞA ¼ ðωÞP  0:1 ¼ 0:2244 Step 6. Calculate the critical pressure, the critical temperature, and acentric factor of the undefined fraction from the following relationships: pc ¼ wP ðpc ÞP + wN ðpc ÞN + wA ðpc ÞA ¼ 449 psi Tc ¼ wP ðTc ÞP + wN ðTc ÞN + wA ðTc ÞA ¼ 976°R ω ¼ wP ðωÞP + wN ðωÞN + wA ðωÞA ¼ 0:2932

Graphical Correlations Several mathematical correlations for determining the physical and critical properties of petroleum fractions have been presented. These correlations are readily adapted to computer applications. However, it is important to present the properties in graphical forms for a better understanding of the behavior and interrelationships of the properties.

Boiling Points Numerous graphical correlations have been proposed over the years for determining the physical and critical properties of petroleum fractions. Most of these correlations use the normal boiling point as one of the correlation parameters. As stated previously, five methods are used to define the normal boiling point:

134 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

1. 2. 3. 4. 5.

VABP WABP MABP CABP MeABP.

Figure 2.91 schematically illustrates the conversions between the VABP and the other four averaging types of boiling point temperatures. The following steps summarize the procedure of using Fig. 2.9 in determining the desired average boiling point temperature. Step 1. On the basis of ASTM D-86 distillation data, calculate the volumetric average boiling point from the following expressions: VABP ¼ ðt10 + t30 + t50 + t70 + t90 Þ=5

(2.82)

where t is the temperature in °F and the subscripts 10, 30, 50, 70, and 90 refer to the volume percent recovered during the distillation. Step 2. Calculate the 10–90% “slope” of the ASTM distillation curve from the following expression: Slope ¼ ðt90  t10 Þ=80

(2.83)

Enter the value of the slope in the graph and travel vertically to the appropriate set for the type of boiling point desired.

Correction to be added to VABP to obtain other boiling points

Slope = 2

1

3

2

t90 – t10 80 4 5

6

7

8

9

0 -2 -4 -6 -8

n FIGURE 2.9 Correction to volumetric average boiling points.

1

The original conversion chart can be found in the API Technical Data Book.

Group 3. The Plus Fraction Characterization Methods 135

Step 3. Read from the ordinate a correction factor for the VABP and apply the relationship: Desired boiling point ¼ VABP + correction factor

(2.84)

The use of the graph can best be illustrated by the following examples.

EXAMPLE 2.7 The ASTM distillation data for a 55° API gravity petroleum fraction is given below. Calculate WAPB, MABP, CABP, and MeABP. Cut

Distillation % Over

Temperature (°F)

1 2 3 4 5 6 7 8 9 10 Residue

IBP 10 20 30 40 50 60 70 80 90 EP

159 178 193 209 227 253 282 318 364 410 475

IBP, initial boiling point; EP, end point.

Solution Step 1. Calculate VABP from Eq. (2.82): VABP ¼ ð178 + 209 + 253 + 318 + 410Þ=5 ¼ 273°F Step 2. Calculate the distillation curve slope from Eq. (2.83): Slope ¼ ð410  178Þ=80 ¼ 2:9 Step 3. Enter the slope value of 2.9 in Fig. 2.9 and move down to the appropriate set of boiling point curves. Read the corresponding correction factors from the ordinate to give Correction factors for WABP ¼ 6°F Correction factors for CABP ¼ 7°F Correction factors for MeABP ¼ 18°F Correction factors for MABP ¼ 33°F

136 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Step 4. Calculate the desired boiling point by applying Eq. (2.84): WABP ¼ 273 + 6 ¼ 279°F CABP ¼ 273  7 ¼ 266°F MeABP ¼ 273  18 ¼ 255°F MABP ¼ 273  33 ¼ 240°F

Molecular Weight Fig. 2.10 shows a convenient graphical correlation for determining the molecular weight of petroleum fractions from their MeABP and API gravities. The following example illustrates the practical application of the graphical method.

n FIGURE 2.10 Relationship between molecular weight, API gravity, and MeABPs. (Courtesy of the Gas Processors Suppliers Association. Published in the GPSA Engineering

Data Book, tenth edition, 1987.)

Group 3. The Plus Fraction Characterization Methods 137

EXAMPLE 2.8 Calculate the molecular weight of the petroleum fraction with an API gravity and MeABP as given in Example 2.7, ie, API ¼ 55° MeABP ¼ 255°F Solution Enter these values in Fig. 2.10 to give MW ¼ 118.

Critical Temperature The critical temperature of a petroleum fraction can be determined by using the graphical correlation shown in Fig. 2.11. The required correlation parameters are the API gravity and the MABP of the undefined fraction. EXAMPLE 2.9 Calculate the critical temperature of the petroleum fraction with physical properties as given in Example 2.7. Solution From Example 2.7, API ¼ 55° MABP ¼ 240°F Enter the above values in Fig. 2.6 to give Tc ¼ 600°F.

Critical Pressure Fig. 2.12 is a graphical correlation of the critical pressure of the undefined petroleum fractions as a function of the MeABP and the API gravity. The following example shows the practical use of the graphical correlation. EXAMPLE 2.10 Calculate the critical pressure of the petroleum fraction from Example 2.7. Solution From Example 2.7, API ¼ 55° MeABP ¼ 255°F Determine the critical pressure of the fraction from Fig. 2.12, to give pc ¼ 428 psia.

138 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

n FIGURE 2.11 Critical temperature as a function of API gravity and boiling points. (Courtesy of the Gas Processors Suppliers Association. Published in the GPSA Engineering

Data Book, tenth edition, 1987.)

Group 3. The Plus Fraction Characterization Methods 139

n FIGURE 2.12 Relationship between critical pressure, API gravity, and MeABPs. (Courtesy of the Gas Processors Suppliers Association. Published in the GPSA Engineering Data Book, tenth edition, 1987.)

140 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

SPLITTING AND LUMPING SCHEMES All forms of equations of state and their modifications require the critical properties and acentric factor of each component in the hydrocarbon mixture. For pure components, these required properties are well defined; however, most hydrocarbon fluids contain hundreds of different components that are difficult to identify and characterize by laboratory separation techniques. These large number of components heavier than C6 are traditionally lumped together and categorized as the “plus fraction”; eg, heptanes-plus “C7+” or undecanes-plus “C11+.” A significant portion of naturally occurring hydrocarbon fluids contain these hydrocarbon-plus fractions, which could create major problems when applying equations of state to model the volumetric behavior and predict the thermodynamic properties of hydrocarbon fluids. These problems result from the difficulty of properly characterizing the plus fractions (heavy ends) in terms of their critical properties and acentric factors. As presented earlier in this chapter, many correlations were developed to generate the critical properties and acentric factor for the plus fraction based on the molecular weight and specific gravity. However, routine laboratory experiments on the plus fraction (eg, C7+) provide description of the plus fraction in terms of its molecular weight and specific gravity, noting that the measured molecular weight of the plus fraction can have an error of as much as 20%. In the absence of detailed analytical TBP distillation or chromatographic analysis data for the plus fraction in a hydrocarbon mixture, erroneous predictions and conclusions can result if the plus fraction is categorized as one component and directly used in EOS phase equilibria calculations. As an example, errors could occur when calculating the saturation pressure for a rich gas condensate sample. The EOS calculation will, at times, predict a bubble point pressure instead of a dew point pressure. These problems associated with treating the plus fraction as a single component can be substantially reduced if the plus fraction is split into a manageable number of pseudofractions. Splitting refers to the process of breaking down the plus fraction into a certain number; an optimum number of SCN fractions. Generally, with a sufficiently large number of pseudocomponents used in characterizing the heavy fraction of a hydrocarbon mixture, a satisfactory prediction of the PVT behavior by the equation of state can be obtained. However, in compositional models, the cost and computing time can increase significantly with the increased number of components in the system. Therefore, strict limitations are set on the maximum number of components that can be used in compositional models and the original components have to be lumped into a smaller number of pseudocomponents for equation-of-state calculations.

Splitting and Lumping Schemes 141

The characterization of the fraction (eg, C7+) generally consists of the following three steps: (a) Splitting the plus fraction into pseudocomponents (eg, C7 to C45+) (b) Lumping the generated pseudocomponents into an optimum number of SCN fractions (c) Characterizing the lumped fraction in terms of their critical properties and acentric factors The problem, then, is how to adequately split a C7+ fraction into a number of pseudocomponents characterized by mole fraction, molecular weight, and specific gravity. These characterization properties, when properly combined, should match the measured plus fraction properties; that is, (M)7+ and (γ)7+.

Splitting Schemes Splitting schemes refer to the procedures of dividing the heptanes-plus fractions into hydrocarbon groups with a SCN (C7, C8, C9, etc.), described by the same physical properties used for pure components. It should be pointed out that plotting the compositional analysis of any naturally occurring hydrocarbon system will exhibit a discrete compositional distribution for all components lighter than heptanes “C7” and show continuous compositional distribution for all components heavier than hexane fraction, as shown in Fig. 2.13. Mole % 50.000

Discrete distribution

Continuous distribution

5.000

0.500 C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 n FIGURE 2.13 Discrete and continuous compositional distribution.

142 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Based on the continuous compositional distribution observation, several authors have proposed different schemes for extending the molar distribution behavior of C7+ in terms of mole fraction as a function of molecular weight or number of carbon atoms. In general, the proposed schemes are based on the observation that lighter systems, such as condensates, usually exhibit exponential molar distribution, while heavier systems often show left-skewed distributions. This behavior is shown schematically in Fig. 2.14. Three important requirements should be satisfied when applying any of the proposed splitting models: 1. The sum of the mole fractions of the individual pseudocomponents is equal to the mole fraction of C7+. 2. The sum of the products of the mole fraction and the molecular weight of the individual pseudocomponents is equal to the product of the mole fraction and molecular weight of C7+. 3. The sum of the product of the mole fraction and molecular weight divided by the specific gravity of each individual component is equal to that of C7+. These requirements can be expressed mathematically by the following relationship: n+ X z n ¼ z7 +

(2.85)

n¼7

Compositional distribution

6 5

Mole %

4 3 Norm Left-ske w al an d he ed distrib avy c u rude tion oil sy stem Cond Expo s n ensa te an ential dis d ligh tribu tion t hyd roca rbon syste ms

2 1

0

7

8

9

10

11

12 13 14 15 16 Number of carbon atoms

n FIGURE 2.14 The exponential and left-skewed distribution functions.

17

18

19

20

21

Splitting and Lumping Schemes 143

n+ X ½zn Mn  ¼ z7 + M7 +

(2.86)

n¼7 n+ X zn M n n¼7

γn

¼

z7 + M7 + γ7 +

(2.87)

Eqs. (2.86) and (2.87) can be solved for the molecular weight and specific gravity of the last fraction after splitting, as

Mn + ¼

γn + ¼

z7 + M7 + 

Xðn + Þ1 n¼7

½ zn M n 

zn +

zn + M n + z7 + M7 + Xðn + Þ1 zn Mn  n¼7 γ7 + γn

where z7+ ¼ mole fraction of C7+ n ¼ number of carbon atoms n+ ¼ last hydrocarbon group in the C7+ with n carbon atoms, such as 20 + zn ¼ mole fraction of pseudocomponent with n carbon atoms M7+, γ 7+ ¼ measured molecular weight and specific gravity of C7+ Mn, γ n ¼ molecular weight and specific gravity of the pseudocomponent with n carbon atoms Several splitting schemes have been proposed. These schemes, as discussed next, are used to predict the compositional distribution of the heavy plus fraction.

Modified Katz’s Method Katz (1983) presented an easy-to-use graphical correlation for breaking down into pseudocomponents the C7+ fraction present in condensate systems. The method was originated by plotting the extended compositional analysis of only six condensate systems as a function of the number of carbon atoms on a semilog scale and drawing a curve that best fit the data. The limited amount of data used by Katz were not sufficient to develop a generalized and relatively accurate expression that can be used to extend the compositional analysis of the Heptanes-plus fraction from its mole fraction; ie, z7+. To improve Katz graphical correlation, more experimental

144 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

compositional analysis was added to the Katz data to develop the following expression: zn ¼ 1:269831zC7 + exp ð0:26721nÞ + 0:0060884zC7 + + 10:4275  106 (2.88)

where z7+ ¼ mole fraction of C7+ in the condensate system n ¼ number of carbon atoms of the pseudocomponent zn ¼ mole fraction of the pseudocomponent with the number of carbon atoms of n Eq. (2.88) is applied repeatedly until Eq. (2.85) is satisfied. The molecular weight and specific gravity of the last pseudocomponent can be calculated from Eqs. (2.86) and (2.87), respectively. The computational procedure of using Eq. (2.88) is best explained through the following example.

EXAMPLE 2.11 A naturally occurring condensate-gas system has the following composition: Component

zi

C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+

0.7393 0.11938 0.04618 0.0124 0.01544 0.00759 0.00703 0.00996 0.00896 0.00953 0.00593 0.00405 0.00268 0.00189 0.00168 0.00131 0.00107 0.00082 0.00073 0.00063 0.00055 0.00289

Splitting and Lumping Schemes 145

Properties of C7+ and C20+ in terms of their molecular weights and specific gravities are given below: Group

Mole %

Weight %

Total fluid C7+ C10+

100.000 4.271 1.830

100.000 22.690 13.346

MW

SG

27.74 147.39 202.31

0.412 0.789 0.832

Using the modified Katz splitting scheme as applied by Eq. (2.88), extend the compositional distribution of C7+ to C20+ and calculate M, γ, Tb, pc, Tc, and ω of C20+. Compare with laboratory extend compositional analysis Solution Applying Eq. (2.88) with z7+ ¼ 0.04271 gives the following results. zn ¼ 1:269831zC7 + exp ð0:26721nÞ + 0:0060884zC7 + + 10:4275  106 zn ¼1:269831ð0:04271Þexp ð0:26721nÞ+0:0060884ð0:04271Þ+10:4275106

Component

zi

Eq. (2.88)

C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+

0.00896 0.00953 0.00593 0.00405 0.00268 0.00189 0.00168 0.00131 0.00107 0.00082 0.00073 0.00063 0.00055 0.00289

0.008229 0.006478 0.005104 0.004026 0.00318 0.002516 0.001995 0.001587 0.001267 0.001017 0.000822 0.000669 0.00055 0.00467a

a

This value was obtained by applying Eq. (2.85); ie,

0:04271 

X19

z n¼7 n

¼ 0:00467

The procedure for characterizing of C20+ is summarized in the following steps: Step 1. Calculate the molecular weight and specific gravity of C20+ by solving Eqs. (2.86) and (2.87) for these properties.

146 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

n

z006E

Mn

znMn

γn

znM/γ n

7 8 9 10 11 12 13 14 15 16 17 18 19 20+ P

0.00862 0.00667 0.00517 0.00402 0.00314 0.00247 0.00195 0.00156 0.00126 0.00102 0.00085 0.00071 0.00061 0.00467

96 107 121 134 147 161 175 190 206 222 237 251 263 –

0.827940631 0.713207474 0.625078138 0.538412919 0.461470288 0.397115715 0.341529967 0.295904773 0.258658497 0.227464594 0.200922779 0.17881299 0.160107188 – 5.226625952

0.727 0.749 0.768 0.782 0.793 0.804 0.815 0.826 0.836 0.843 0.851 0.856 0.861 –

1.138845434 0.952212916 0.813903826 0.688507569 0.581929746 0.493925019 0.419055174 0.358238224 0.309400117 0.269827514 0.236101973 0.208893679 0.185954922 – 6.656796111

Solving for the molecular weight of the C20+, gives X19 z7 + M7 +  ½Z M  n¼7 n n M20 + ¼ Z20 + M20 + ¼

ð0:04271Þð147:39Þ  5:22662 ¼ 228:67 0:00467

With a specific gravity of γ 20 + ¼

γ 20 + ¼

z20 + M20 + z7 + M7 + X19 zn Mn  n¼7 γ γ7 + n ð0:00467Þð228:67Þ ¼ 0:808 ½0:04271  147:39=0:789  6:656796

Step 2. Calculate the boiling points, critical pressure, and critical temperature of C20+, using the Riazi-Daubert correlation (Eq. (2.6)), to give θ ¼ aðMÞb γ c exp ½dMeγ + f γM Tc ¼ 544:4ð228:67Þ0:2998 ð0:808Þ1:0555

 exp 1:3478  104 ð228:67Þ  0:6164ð0:808Þ ¼ 1305°R pc ¼ 4:5203  104 ð228:67Þ0:8063 ð0:808Þ1:6015

 exp 1:8078  103 ð228:67Þ  0:3084ð0:808Þ ¼ 207 psi

Splitting and Lumping Schemes 147

Tb ¼ 6:77857ð228:67Þ0:401673 ð0:808Þ1:58262 exp

3:7709  103 ð228:67Þ  2:984036ð0:808Þ 4:25288  103 ð228:67Þð0:808Þ ¼ 1014°R Step 3. Calculate the acentric factor of C20+ by applying the Edmister correlation, to give solve for the acentric factor from Eq. (2.21) to give ω¼

ω¼

3½ log ðpc =14:70Þ 1 7½ðTc =Tb  1Þ

3 log ð207=14:7Þ  1 ¼ 0:712 7 ð1305=1014Þ  1

To ensure continuity and consistency in characterizing the heavy end (eg, C20+) in terms of critical properties and acentric factor, the reader should consider applying Eqs. (2.3A) through (2.3E), ie, M 228:67 ¼ ¼ 283 γ 0:808 Tc ¼ 231:9733906 ¼ 1355°R

 0:351579776   M M  0:352767639  233:3891996 γ γ

 0:885326626   M M  0:106467189 + 49:62573013 ¼ 234:3 psi pc ¼ 31829 γ γ

ω ¼ 0:279354619 ln

    M M + 0:00141973  1:243207091 ¼ 0:7357 γ γ

Lohrenz’s Method Lohrenz et al. (1964) proposed that the heptanes-plus fraction could be divided into pseudocomponents with carbon number ranges from 7 to 40. They mathematically stated that the mole fraction zn is related to its number of carbon atoms n and the mole fraction of the Hexane fraction z6 by the expression zn ¼ z6 eAðn6Þ

2

+ Bðn6Þ

(2.89)

148 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

where z6 ¼ mole fraction of the Hexane component n ¼ number of carbon atoms of the pseudocomponent zn ¼ mole fraction of the pseudocomponent with number of carbon atoms of n A and B ¼ correlating parameters Eq. (2.89) assumes that individual C7+ components are distributed from the hexane mole fraction “C6” and tail off to an extremely small quantity of heavy hydrocarbons. Lohrenz’s expression can be linearized to determine the coefficient A and B; however, the method requires a partial extended analysis of the hydrocarbon system (eg, C7 to C11+) to be able to determine the values of the coefficients A and B. In addition, the validity and applicability of the equation could not be verified when testing to extend the molar distribution on several condensate samples.

Pedersen’s Method Pedersen et al. (1982) proposed that for naturally occurring hydrocarbon mixtures, an exponential relationship exists between the mole fraction of a component and the corresponding carbon number. They expressed this relationship mathematically in the following exponential form: zn ¼ eðAn + BÞ

(2.90)

where A and B ¼ constants n ¼ number of carbon atoms, eg, 7, 8, 9, …, etc. For condensates and volatile oils, the authors suggested that A and B can be determined by a least squares fit to the molar distribution of the lighter fractions; that is, the methodology requires partial extended compositional analysis of the plus fraction. Pedersen’s expression can also be expressed in the following linear form to determine the coefficients of the equation, as lnðzn Þ ¼ An + B

Eq. (2.90) can then be used to calculate the molar content of each of the heavier fractions by extrapolation. The classical constraints as given by Eqs. (2.85) through (2.87) also are imposed.

Splitting and Lumping Schemes 149

EXAMPLE 2.12 Rework Example 2.11 by using the Pedersen splitting correlation. Assume that the partial molar distribution of C7+ is only available from C7 through C10. Extend the analysis to C20+ and compare with the experimental data. Component

zi

C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+

0.7393 0.11938 0.04618 0.0124 0.01544 0.00759 0.00703 0.00996 0.00896 0.00953 0.00593 0.00405 0.00268 0.00189 0.00168 0.00131 0.00107 0.00082 0.00073 0.00063 0.00055 0.00289

Group

Mole %

Weight %

Total fluid C7+ C10+

100.000 4.271 1.830

100.000 22.690 13.346

MW

SG

27.74 147.39 202.31

0.412 0.789 0.832

Solution Step 1. Calculate the coefficients A and B graphically by plotting ln(zn) versus “n” using the compositional analysis from C7 to C10 as shown in Fig. 2.15 to give the slope “A” and intercept “B” as A ¼ 0:252737 B ¼ 2:98744 Step 2. Predict the mole fraction of C10 through C20 by applying Eq. (2.90), as shown below. zn ¼ eðAn + BÞ

150 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

– 4.00000 – 4.20000 – 4.40000 y = −0.252737x − 2.987440

– 4.60000

Ln(zn)

– 4.80000 – 5.00000 – 5.20000 – 5.40000 – 5.60000 – 5.80000 – 6.00000

6

7

8 9 10 Number of carbon atoms “n”

11

12

n FIGURE 2.15 Calculating the coefficients A and B for Example 2.12.

Component

zi

Eq. (2.90)

C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+

0.00896 0.00953 0.00593 0.00405 0.00268 0.00189 0.00168 0.00131 0.00107 0.00082 0.00073 0.00063 0.00055 0.00289

0.00896 0.00953 0.00593 0.00405 0.00268 0.00246 0.00191 0.00149 0.00116 0.00090 0.00070 0.00054 0.00042 0.00199

Ahmed’s Method Ahmed et al. (1985) devised a simplified method for splitting the C7+ fraction into pseudocomponents. The method originated from studying the molar behavior of 34 condensate and crude oil systems through detailed laboratory compositional analysis of the heavy fractions. The only required

Splitting and Lumping Schemes 151

data for the proposed method are the molecular weight and the total mole fraction of the heptanes-plus fraction. The splitting scheme is based on calculating the mole fraction, zn, at a progressively higher number of carbon atoms. The extraction process continues until the sum of the mole fraction of the pseudocomponents equals the total mole fraction of the heptanes-plus (z7+). z n ¼ zn +

  Mðn + 1Þ +  Mn + Mð n + 1 Þ +  M n

(2.91)

where zn ¼ mole fraction of the pseudocomponent with a number of carbon atoms of n Mn ¼ molecular weight of the hydrocarbon group with n carbon atoms Mn+ ¼ molecular weight of the n+ fraction as calculated by the following expression: Mðn + 1Þ + ¼ M7 + + Sðn  6Þ

(2.92)

where n is the number of carbon atoms and S is the coefficient of Eq. (2.92) with the values given below.

No. of Carbon Atoms

Condensate Systems

Crude Oil Systems

n8 n>8

15.5 17.0

16.5 20.1

The stepwise calculation sequences of the proposed correlation are summarized in the following steps. Step 1. According to the type of hydrocarbon system under investigation (condensate or crude oil), select the appropriate values for the coefficients. Step 2. Knowing the molecular weight of C7+ fraction (M7+), calculate the molecular weight of the octanes-plus fraction (M8+) by applying Eq. (2.92). Step 3. Calculate the mole fraction of the heptane fraction (z7) by using Eq. (2.91). Step 4. Repeat steps 2 and 3 for each component in the system (C8, C9, etc.) until the sum of the calculated mole fractions is equal to the mole fraction of C7+ of the system. The splitting scheme is best explained through the following example.

152 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

EXAMPLE 2.13 Rework Example 2.12 using Ahmed’s splitting method. Component

zi

C1 C2 C3 i-C4 n-C4 i-C5 n-C5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+

0.7393 0.11938 0.04618 0.0124 0.01544 0.00759 0.00703 0.00996 0.00896 0.00953 0.00593 0.00405 0.00268 0.00189 0.00168 0.00131 0.00107 0.00082 0.00073 0.00063 0.00055 0.00289

Group

Mole %

Weight %

MW

SG

Total fluid C7+ C10+

100.000 4.271 1.830

100.000 22.690 13.346

27.74 147.39 202.31

0.412 0.789 0.832

Solution Step 1. Extract the mole fraction of z7; ie, n ¼ 7   Mðn + 1Þ +  Mn + zn ¼ zn + Mðn + 1Þ +  Mn   M8 +  M7 + z7 ¼ z 7 + M8 +  M7 n

Calculate the molecular weight of C8+ by applying Eq. (2.92). n¼7 Mðn + 1Þ + ¼ M7 + + Sðn  6Þ M8 + ¼ 147:39 + 15:5ð7  6Þ ¼ 162:89

Splitting and Lumping Schemes 153

n

Solve for the mole fraction of heptane (z7) by applying Eq. (2.91).   Mðn + 1Þ +  Mn + z n ¼ zn + Mð n + 1 Þ +  M n z 7 ¼ z7 +

    M8 +  M7 + 162:89  147:39 ¼ 0:04271 ¼ 0:0099 M8 +  M7 162:89  96

Step 2. Extract the mole fraction of z8; ie, n ¼ 8 z n ¼ zn +

  Mðn + 1Þ +  Mn + Mð n + 1 Þ +  M n 

z8 ¼ z8 +

n

M9 +  M8 + M9 +  M8



Calculate the molecular weight of C9+ from Eq. (2.92). n¼8 Mðn + 1Þ + ¼ M7 + + Sðn  6Þ M9 + ¼ 147:39 + 15:5ð8  6Þ ¼ 178:39

n

Determine the mole fraction of C8+ z8 + ¼ z7 +  z7 ¼ 0:04271  0:0099 ¼ 0:0328

n

Determine the mole fraction of C8 from Eq. (2.91). z8 ¼ z8 + ½ðM9 +  M8 + Þ=ðM9 +  M8 Þ z8 ¼ ð0:0328Þ½ð178:39  162:89Þ=ð178:39  107Þ ¼ 0:0074

Step 3. Extract the mole fraction of z9; ie, n ¼ 9 z n ¼ zn +

  Mðn + 1Þ +  Mn + Mð n + 1 Þ +  M n

  M10 +  M9 + z9 ¼ z9 + M10 +  M9 n

Calculate the molecular weight of C9+ from Eq. (2.92). n¼9 Mðn + 1Þ + ¼ M7 + + Sðn  6Þ M10 + ¼ 147:39 + 17ð9  6Þ ¼ 198:39

n

Determine the mole fraction of z9+ z9 + ¼ z7 +  z7  z8 ¼ 0:04271  0:0099  0074 ¼ 0:0254

154 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

n

Determine the mole fraction of C9 from Eq. (2.91).   Mðn + 1Þ +  Mn + zn ¼ zn + Mðn + 1Þ +  Mn     M10 +  M9 + 198:29  178:39 ¼ 0:0254 ¼ 0:0065 z9 ¼ z9 + M10 +  M9 198:29  121

Step 4. Repeat the extracting method outlined in the preceding steps, to give the following results. N

Mn+

Mn

zn

7 8 9 10 11 12 13 14 15 16 17 18 19 20+

147.39 162.89 177.39 198.39 215.39 232.39 249.39 266.39 283.39 300.39 317.39 334.39 351.39 –

96 107 121 134 147 161 175 190 206 222 237 251 263 –

0.00990 0.00740 0.00650 0.00395 0.00298 0.00230 0.00180 0.00143 0.00116 0.00094 0.00076 0.00061 0.00048 0.00250

Step 5. The boiling point, critical properties, and the acentric factor of C20+ are determined using the appropriate methods, to give N

zn

Mn

znMn

γn

znM/γ n

7 8 9 10 11 12 13 14 15 16 17 18 19 20+ P

0.00990 0.00740 0.00650 0.00395 0.00298 0.00230 0.00180 0.00143 0.00116 0.00094 0.00076 0.00061 0.00048 0.00250

96 107 121 134 147 161 175 190 206 222 237 251 263 –

0.9504 0.7918 0.7865 0.529266249 0.437822572 0.371019007 0.315026462 0.272443701 0.239056697 0.209010771 0.179601656 0.152316618 0.126282707 – 5.360546442

0.727 0.749 0.768 0.782 0.793 0.804 0.815 0.826 0.836 0.843 0.851 0.856 0.861 –

1.307290234 1.057142857 1.024088542 0.67681106 0.55210917 0.461466427 0.386535537 0.32983499 0.285952987 0.247936858 0.211047775 0.177939974 0.146669811 – 6.864826221

Splitting and Lumping Schemes 155

Solving for the molecular weight of the C20+, gives z7 + M7 + 

M20 + ¼

X19 n¼7

½Zn Mn 

Z20 +

M20 + ¼

ð0:04271Þð147:39Þ  5:36054 ¼ 374 0:00250

With a specific gravity of γ 20 + ¼

γ 20 + ¼

z20 + M20 + z7 + M7 + X19 zn Mn  n¼7 γ γ7 + n ð0:0025Þð374Þ ¼ 0:839 ½0:04271x147:39=0:789  6:864826

Apply Eqs. (2.3A) through (2.3E), which gives M 228:67 ¼ ¼ 283 γ 0:808 Tc ¼ 231:9733906

 0:351579776   M M  0:352767639 233:3891996 ¼1355°R γ γ

 0:885326626   M M pc ¼ 31829  0:106467189 + 49:62573013 ¼ 234 psi γ γ

ω ¼ 0:279354619 ln

    M M + 0:00141973  1:243207091 ¼ 0:736 γ γ

Whitson’s Method The most widely used distribution function is the three-parameter gamma probability function as proposed by Whitson (1983). Unlike all previous splitting methods, the gamma function has the flexibility to describe a wider range of distribution by adjusting its variance, which is left as an adjustable parameter. Whitson expressed the function in the following form:

pðMÞ ¼

ðM  ηÞα1 exp f½M  η=βg β α Γ ð ηÞ

(2.93)

156 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

where Γ ¼ gamma function η ¼ the minimum molecular weight included in the distribution; eg, MC7 α and β ¼ adjustable parameters that describe the shape of the compositional distribution M ¼ molecular weight The key parameter defines the shape of the distribution “α” with its value usually ranging from 0.5 to 2.5. Fig. 2.16 illustrates Whitson’s model for several values of the parameter α: n n

n n

For α ¼ 1, the distribution is exponential For α < 1, the distribution model gives accelerated exponential distribution For α > 1, the distribution model gives left-skewed distributions In the application of the gamma distribution to heavy oils, bitumen, and petroleum residues, it is indicated that the upper limit for α is 25–30, which statistically approaches a log-normal distribution

Whitson indicates that the parameter η can be physically interpreted as the minimum molecular weight found in the Cn+ fraction. Whitson recommended η ¼ 92 as a good approximation if the C7+ is the plus fraction, for other plus fractions (ie, Cn+) the following approximation is used: η  14n  6

(2.94)

and β¼

a =1

a = 10

MCn +  η α

(2.95)

a = 15 a = 25

zi

Molecular weight n FIGURE 2.16 Gamma distributions for C7+.

Splitting and Lumping Schemes 157

It should be pointed out that when the parameter α ¼ 1, the compositional distribution is exponential and probability function becomes

pðMÞ ¼

exp f½M  η=βg β

Rearranging, gives   exp ðη=βÞ pðMÞ ¼ exp ðM=βÞ β

(2.96)

with β ¼ MC n +  η

The cumulative frequency of occurrence “fi” for a SCN group, such as the C7 group, C8 group, C9 group, …, etc. having molecular weight boundaries between Mi+1 and Mi is given by fn ¼

ð Mn

PðMÞdM ¼ PðMn Þ  PðMn1 Þ

Mn1

Integrating the above expression, gives

fn ¼  exp

      η Mn Mn1  exp exp β β β

(2.97)

The mole fraction zn for each SCN group is given by multiplying the cumulative frequency of occurrence fn of the SCN times the mole fraction of the plus fraction, that is, zn ¼ zC7 + fn

Whitson and Brule indicated that, when characterizing multiple samples simultaneously, the values of Mn, η, and β must be the same for all samples. Individual sample values of MC7 + and α, however, can be different. The result of this characterization is one set of molecular weights for the C7+ fractions, while each sample has different mole fractions zn (so that their average molecular weights MC7 + are honored).

158 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

EXAMPLE 2.14 Rework Example 2.13 using Whitson’s exponential splitting method Component

zi

C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+

0.01 0.009 0.0095 0.0059 0.0041 0.0027 0.0019 0.0017 0.0013 0.0011 0.0008 0.0007 0.0006 0.0006 0.0029

Solution Step 1. Calculate η using η  86 Step 2. Calculate β using β ¼ MCn +  η β ¼ 147:39  86 ¼ 61:39 Step 3. Calculate the mole fraction zn for each SCN group:       η Mn Mn1  exp exp fn ¼  exp β β β       86 Mn Mn1  exp fn ¼  exp exp 61:39 61:39 61:39 zn ¼ zC7 + fn        86 Mn Mn1  exp zn ¼ zC7 +  exp exp 61:39 61:39 61:39  zn ¼ ð0:17335Þ

 exp

   Mn Mn1  exp 61:39 61:39

Splitting and Lumping Schemes 159

Step 4. Calculate the mole fraction for each SCN group with values as tabulated below: N

Mn

zn

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 +

84 96 107 121 134 147 161 175 190 206 222 237 251 263 379

0.0078344 0.0059532 0.0061861 0.0046089 0.0037294 0.0032244 0.0025669 0.0021723 0.0018007 0.0013876 0.0010103 0.0007443 0.0005159 0.0009755

Group

Mole %

Weight %

MW

SG

C7+ C10+

4.271 1.83

22.69 13.346

147.39 202.31

0.789 0.832

Ahmed Modified Method The gas rate from unconventional reservoirs typically exhibits a left-skewed distribution; similar to what continues hydrocarbon compositional distribution. Ahmed (2014) adopted the methodology of describing flow-rate methodology to extend compositional analysis of the plus fraction. The methodology, as illustrated by the following steps, requires partial extended analysis of the plus fraction: normalize the Step 1. Given the partial extended compositional analysis “zn,”X composition by calculating and tabulating the ratio zn = zn for each component starting from the C7. Data below illustrates step 1:

n

zn 7 8 9

0.0583 0.0552 0.0374

X

zn

0.0583 0.1135 0.1509

zn =

X zn

1 0.486344 0.247846

160 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

10 11 12 13 14 15 16

0.0338 0.0257 0.0202 0.0202 0.0165 0.0148 0.0116

0.1847 0.2104 0.2306 0.2508 0.2673 0.2821 0.2937

0.182999 0.122148 0.087598 0.080542 0.061728 0.052464 0.039496

X Step 2. Plot the ratio zn = zn versus number of carbon atoms as defined by “n  6” on a log-log scale, as shown in Fig. 2.17. Step 3. Express the best fit of the data by a straight-line fit as described by the power function form: z Xn ¼ aðn  6Þm zn

(2.98)

And determine the parameters “m” and “a.” The slope “m” must be treated as positive in all calculations. Step 4. Locate the component with the highest mole fraction “zN” in the observed extended molar analysis and “N” designated as its number of carbon atoms. For example, if the observed extended analysis shows that C8 has the highest mole fraction, then set N ¼ 8.

10

1

a = 1.18799 and m = 1.43978

zn / Σzn

y = 1.18799x-1.43978

0.1

0.01 1

10 n–6

n FIGURE 2.17 The power function representation.

100

Splitting and Lumping Schemes 161

Step 5. Calculate zn as a function of “n” from the following expressions:

 zn ¼ zN aðn  6Þm eX ðN  6Þ

(2.99)

with X¼

 a h i ðn  6Þ1m  ðN  6Þ1m 1m

(2.100)

Step 6. Adjust the coefficients “a and m” to match the observed zn; use Microsoft Solver to optimize the two coefficients. Step 7. Using the regressed values of “a and m” from step 5, extend the analysis by assuming different values of n in the above equation and calculate the corresponding mole fraction of the component. EXAMPLE 2.15 Rework Example 2.14 using the modified Ahmed splitting method. Use only the data from C7 to C11. Component

zi

C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+

0.009 0.0095 0.0059 0.0041 0.0027 0.0019 0.0017 0.0013 0.0011 0.0008 0.0007 0.0006 0.0006 0.0029

Solution X Step 1. Normalize compositional analysis “zn” by calculating the ratio zn = zn for each component starting from C7 to C11. n

zn

X zn

zn =

n26

7 8 9 10 11

0.00896 0.00953 0.00593 0.00405 0.00268

0.00896 0.01849 0.02442 0.02847 0.03115

1 0.515414 0.242834 0.142255 0.086035

1 2 3 4 5

X zn

162 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

10

y = 1.17556x-1.52272 a = 1.1755 and m = 1.52272

zn / Σzn

1

0.1

0.01 1

10 n–6

n FIGURE 2.18 Dimensionless ratio vs. number of carbon atoms.

X Step 2. Plot zn = zn versus number of carbon atoms “n  6” on a log-log scale, as shown in Fig. 2.18 and fit the data with power function to determine the parameters “a” and “m,” to give a ¼ 1.175558 and m ¼ 1.52272 (notice the coefficient “m” must be positive). Step 3. Identify the component C8 as the fraction with the highest mole fraction in the partial analysis and set: zN ¼ 0:00953 and N ¼ 8 Step 4. Apply Eqs. (2.99) and (2.11) and compare with laboratory data:  a h i ðn  6Þ1m  ðN  6Þ1m X¼ 1m   i 1:175558 h ðn  6Þ11:52272  ð8  6Þ11:52272 ¼ 1  1:52272



 zn ¼ zN aðn  6Þm eX ðN  6Þ ¼ ð0:00953Þð1:175558Þðn  6Þm eX ð8  6Þ

N

Lab. zn

a 5 1.175558 m 5 1.52272 X

zn

7 8 9 10 11

0.00896 0.00953 0.00593 0.00405 0.00268

0.68354 0.00000 0.29897 0.47579 0.59575

0.01131 0.00780 0.00567 0.00437 0.00351

Splitting and Lumping Schemes 163

Step 5. Regress on the coefficients a and m to match composition from C7 through C11 to give the following optimized values: a ¼ 1:988198 and m ¼ 1:992053

N

Lab. zn

a 5 1.988198 m 5 1.992053 X

zn

7 8 9 10 11

0.00896 0.00953 0.00593 0.00405 0.00268

0.99653 0.00000 0.33370 0.50102 0.60161

0.01399 0.00953 0.00593 0.00395 0.00280

Step 6. Extend the molar distribution to C12 and apply the equation through C19.

n

Lab. zn

7 8 9 10 11 12 13 14 15 16 17 18 19 20 +

0.00896 0.00953 0.00593 0.00405 0.00268 0.00189 0.00168 0.00131 0.00107 0.00082 0.00073 0.00063 0.00055 0.00289

a 5 1.9881981 m 5 1.9920531 X

zn

0.66945 0.71789 0.75431 0.78271 0.80547 0.82412 0.83969 0.85289 -

0.00896 0.00953 0.00593 0.00405 0.00268 0.00208 0.00161 0.00128 0.00104 0.00086 0.00073 0.00062 0.00054 0.00281

Results of the methodology are shown graphically in Fig. 2.19. Ahmed (2014) also suggested the use of the logistic growth function (LGF), as given below, for extending the analysis of the C7+: zn ¼

zC7 + m a nm1 ½a + nm 2

Where the coefficients “a and m” are correlating, that must be adjusted to match the observed partial extended analysis. The value of the coefficient “m” will impact the shape of the compositional distribution, that is, exponential, left-skewed, …, etc.

164 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

0.016 0.014 0.012 0.01 zi 0.008

0.006 0.004 0.002 0

6

8

10

12

14

16

18

20

n

n FIGURE 2.19 Example 2.16 using LGF.

Reworking Example 2.15 using LGF and matching the partial extended analysis by adjusting the coefficients “a and m”, to give a ¼ 0.2292 and m ¼ 1.9144: zn ¼

zC7 + m a nm1 ½a + nm 2

¼

ð0:04272Þð1:9144Þð0:2292Þn1:91441 ½0:2292 + n1:9144 2

The extended composition analysis is shown in Fig. 2.20. 0.01200

0.01000

0.00800

zi

0.00600

0.00400

0.00200

0.00000

7

8

9

10

11

12

13 n

n FIGURE 2.20 Mole fraction vs. number of carbon atoms.

14

15

16

17

18

19

Splitting and Lumping Schemes 165

Lumping Schemes Equations-of-state calculations frequently are burdened by the large number of components necessary to describe the hydrocarbon mixture for accurate phase-behavior modeling. Often, the problem is either lumping together the many experimentally determined fractions or modeling the hydrocarbon system when the only experimental data available for the C7+ fraction are the molecular weight and specific gravity. The term lumping or pseudoization then denotes the reduction in the number of components used in equations-of-state calculations for reservoir fluids. This reduction is accomplished by employing the concept of the pseudocomponent. The pseudocomponent denotes a group of pure components lumped together and represented by a single component with a SCN. Essentially, two main problems are associated with “regrouping” the original components into a smaller number without losing the predicting power of the equation of state: 1. How to select the groups of pure components to be represented by one pseudocomponent each. 2. What mixing rules should be used for determining the physical properties (eg, pc, Tc, M, γ, and ω) for the new lumped pseudocomponents. Several unique published techniques can be used to address these lumping problems, notably the methods proposed by n n n n n n n

Lee et al. (1979) Whitson (1980) Mehra et al. (1980) Montel and Gouel (1984) Schlijper (1984) Behrens and Sandler (1986) Gonzalez et al. (1986)

Several of these techniques are presented in the following discussion.

Whitson’s Lumping Scheme Whitson (1980) proposed a regrouping scheme whereby the compositional distribution of the C7+ fraction is reduced to only a few multiple carbon number (MCN) groups. Whitson suggested that the number of MCN groups necessary to describe the plus fraction is given by the following empirical rule: Ng ¼ Int½1 + 3:3 log ðN  nÞ

(2.101)

166 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

where Ng ¼ number of MCN groups Int ¼ integer N ¼ number of carbon atoms of the last component in the hydrocarbon system n ¼ number of carbon atoms of the first component in the plus fraction; that is, n ¼ 7 for C7+ The integer function requires that the real expression evaluated inside the brackets be rounded to the nearest integer. The molecular weights separating each MCN group are calculated from the following expression:   MN + I=Ng MI ¼ MC7 MC7

where

(2.102)

(M)N+ ¼ molecular weight of the last reported component in the extended analysis of the hydrocarbon system MC7 ¼ molecular weight of C7 I ¼ 1, 2, …, Ng For example, components with molecular weight of hydrocarbon groups MI1 to MI falling within the boundaries of these molecular weight values are included in the Ith MCN group. Example 2.16 illustrates the use of Eqs. (2.101) and (2.102). EXAMPLE 2.16 Given the following compositional analysis of the C7+ fraction in a condensate system, determine the appropriate number of pseudocomponents forming in the C7+: Component

zi

C7 C8 C9 C10 C11 C12 C13 C14 C15 C16+ M16+

0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671 259

Splitting and Lumping Schemes 167

Solution Step 1. Determine the molecular weight of each component in the system. Component

Zi

Mi

C7 C8 C9 C10 C11 C12 C13 C14 C15 C16+

0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

96 107 121 134 147 161 175 190 206 259

Step 2. Calculate the number of pseudocomponents from Eq. (2.102). Ng ¼ Int½1 + 3:3 log ð16  7Þ Ng ¼ Int½4:15 Ng ¼ 4 Step 3. Determine the molecular weights separating the hydrocarbon groups by applying Eq. (2.109).

I=4 MI ¼ 96 259 96 MI ¼ 96½2:698I=4 where the values for each pseudocomponent, when MI ¼ 96[2.698]I/4, are Pseudocomponent Pseudocomponent Pseudocomponent Pseudocomponent

1 ¼ 123 2 ¼ 158 3 ¼ 202 4 ¼ 259

These are defined as Pseudocomponent 1: The first pseudocomponent includes all components with molecular weight in the range of 96–123. This group then includes C7, C8, and C9. Pseudocomponent 2: The second pseudocomponent contains all components with a molecular weight higher than 123 to a molecular weight of 158. This group includes C10 and C11. Pseudocomponent 3: The third pseudocomponent includes components with a molecular weight higher than 158 to a molecular weight of 202. Therefore, this group includes C12, C13, and C14. Pseudocomponent 4: This pseudocomponent includes all the remaining components, that is, C15 and C16+.

168 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

These are summarized below. Group I

Component

zi

zI

1

C7 C8 C9 C10 C11 C12 C13 C14 C15 C16+

0.00347 0.00268 0.00207 0.001596 0.00123 0.00095 0.00073 0.000566 0.000437 0.001671

0.00822

2

3 4

0.002826 0.002246

0.002108

It is convenient at this stage to present the mixing rules that can be employed to characterize the pseudocomponent in terms of its pseudophysical and pseudocritical properties. Because the properties of the individual components can be mixed in numerous ways, each giving different properties for the pseudocomponents, the choice of a correct mixing rule is as important as the lumping scheme. Some of these mixing rules are given next.

Hong’s Mixing Rules Hong (1982) concluded that the weight fraction average, wi, is the best mixing parameter in characterizing the C7+ fractions by the following mixing rules. Defining the normalized weight fraction of a component, i, within the set of the lumped fraction, that is, I 2 L, as zi M i w∗i ¼ XL ZM i2L i i

The proposed mixing rules are

XL

w i pci XL Pseudocritical temperature TcL ¼ w∗ T i2L i ci XL Pseudocritical volume VcL ¼ w∗ v i2L i ci XL Pseudoacentric factor ωL ¼ w∗ ω i2L i i XL Pseudomolecular weight ML ¼ v∗ M i2L i i XL XL   Binary interaction coefficient kkL ¼ 1  w∗ w∗ 1  kij i2L j2L i j Pseudocritical pressure PcL ¼

i2L

Splitting and Lumping Schemes 169

where wi* ¼ normalized weight fraction of component i in the lumped set kkL ¼ binary interaction coefficient between the kth component and the lumped fraction The subscript L in this relationship denotes the lumped fraction.

Lee’s Mixing Rules Lee et al. (1979), in their proposed regrouping model, employed Kay’s mixing rules as the characterizing approach for determining the properties of the lumped fractions. Defining the normalized mole fraction of a component, i, within the set of the lumped fraction, that is, I 2 L, as zi z∗i ¼ XL

z i2L i

The following rules are proposed: XL

z∗ M i2L i i

(2.103)

XL

 z∗ M =γ i2L i i i

(2.104)

XL

 z∗ M V =ML i2L i i ci

(2.105)

pcL ¼

XL

 z∗ p i2L i ci

(2.106)

TcL ¼

XL

 z T i2L i ci

(2.107)

ωL ¼

XL

 z∗ ω i2L i i

(2.108)

ML ¼ γ L ¼ ML = VcL ¼

EXAMPLE 2.17 Using Lee’s mixing rules, determine the physical and critical properties of the four pseudocomponents in Example 2.16. Solution Step 1. Assign the appropriate physical and critical properties to each component, as shown below. Group Component zi 1

C7 C8 C9

0.00347 0.00268 0.00207

zI 0.00822

Mi

γi a

Vci a

pci a

Tci a

ωi a

96 0.272 0.06289 453 985 0.280a 107 0.749 0.06264 419 1036 0.312 121 0.768 0.06258 383 1058 0.348

170 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

2

3 4

C10 C11 C12 C13 C14 C15 C16+

0.001596 0.002826 134 0.782 0.06273 351 1128 0.385 0.00123 147 0.793 0.06291 325 1166 0.419 0.00095 0.002246 161 0.804 0.06306 302 1203 0.454 0.00073 175 0.815 0.06311 286 1236 0.484 0.000566 190 0.826 0.06316 270 1270 0.516 0.000437 0.002108 206 0.826 0.06325 255 1304 0.550 0.001671 259 0.908 0.0638b 215b 1467 0.68b

a

From Table 2.2. From Table 2.2.

b

Step 2. Calculate the physical and critical properties of each group by applying Eqs. (2.103) through (2.108) to give the results below. Group

zI

ML

γL

VcL

pcL

TcL

ωL

1 2 3 4

0.00822 0.002826 0.002246 0.002108

105.9 139.7 172.9 248

0.746 0.787 0.814 0.892

0.0627 0.0628 0.0631 0.0637

424 339.7 288 223.3

1020 1144.5 1230.6 1433

0.3076 0.4000 0.4794 0.6531

Behrens and Sandler’s Lumping Scheme Behrens and Sandler (1986) used the semicontinuous thermodynamic distribution theory to model the C7+ fraction for equation-of-state calculations. The authors suggested that the heptanes-plus fraction can be fully described with as few as two pseudocomponents. A semicontinuous fluid mixture is defined as one in which the mole fractions of some components, such as C1 through C6, have discrete values, while the concentrations of others, the unidentifiable components such as C7+, are described as a continuous distribution function, F(I). This continuous distribution function F(I) describes the heavy fractions according to the index I, chosen to be a property of individual components, such as the carbon number, boiling point, or molecular weight. For a hydrocarbon system with k discrete components, the following relationship applies: C6 X zi + z7 + ¼ 1:0 i¼1

The mole fraction of C7+ in this equation is replaced with the selected distribution function, to give ðB C6 X zi + FðIÞdI ¼ 1:0 i¼1

A

(2.109)

Splitting and Lumping Schemes 171

where A ¼ lower limit of integration (beginning of the continuous distribution, eg, C7) B ¼ upper limit of integration (upper cutoff of the continuous distribution, eg, C45) This molar distribution behavior is shown schematically in Fig. 2.21. The figure shows a semilog plot of the composition zi versus the carbon number n of the individual components in a hydrocarbon system. The parameter A can be determined from the plot or defaulted to C7; that is, A ¼ 7. The value of the second parameter, B, ranges from 50 to infinity; that is, 50  B  1. However, Behrens and Sandler pointed out that the exact choice of the cutoff is not critical. Selecting the index, I, of the distribution function F(I) to be the carbon number, n, Behrens and Sandler proposed the following exponential form of F(I): FðnÞ ¼ DðnÞeαn dn

(2.110)

with AnB

in which the parameter α is given by the following function f(α): 1 ½A  BeBα f ðαÞ ¼  cn + A  Aα ¼0 α e  eBα

(2.111)

Compositional distribution Discrete distribution

Continuous distribution

Mole fraction

Beginning of the continuous distribution

Upper cutoff of the continuous distribution

Norm Left-skew al an e d he d distrib avy c u rude tion oil sy stem

s

Expo Light nential d istrib hydr ocar bon ution syste ms A

7

Number of carbon atoms

n FIGURE 2.21 Schematic illustration of the semicontinuous distribution model.

B 45

172 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

where cn is the average carbon number as defined by the relationship cn ¼

MC7 + + 4 14

(2.112)

Eq. (2.111) can be solved for α iteratively by using the method of successive substitutions or the Newton-Raphson method, with an initial value of α as α ¼ ½1= cn   A

Substituting Eq. (2.112) into Eq. (2.111) yields ðB C6 X zi + DðnÞeαn dn ¼ 1:0 A

i

or z7 + ¼

ðB

DðnÞeαn dn

A

By a transformation of variables and changing the range of integration from A and B to 0 and c, the equation becomes z7 + ¼

ðc

DðrÞer dr

(2.113)

0

where c ¼ ðB  AÞα r ¼ dummy variable of integration

(2.114)

The authors applied the “Gaussian quadrature numerical integration method” with a two-point integration to evaluate Eq. (2.113), resulting in z7 + ¼

2 X Dðri Þwi ¼ Dðr1 Þw1 + Dðr2 Þw2

(2.115)

i¼1

where ri ¼ roots for quadrature of integrals after variable transformation and wi ¼ weighting factor of Gaussian quadrature at point i. The values of r1, r2, w1, and w2 are given in Table 2.9. The computational sequences of the proposed method are summarized in the following steps: Step 1. Find the endpoints A and B of the distribution. As the endpoints are assumed to start and end at the midpoint between the two carbon numbers, the effective endpoints become A ¼ ðstarting carbon numberÞ  0:5

(2.116)

B ¼ ðending carbon numberÞ + 0:5

(2.117)

Splitting and Lumping Schemes 173

Table 2.9 Behrens and Sandler Roots and Weights for Two-Point Integration c

r1

r2

w1

w2

c

r1

r2

w1

w2

0.30 0.40 0.50 0.60 0.70 0.80 0.90 1.00 1.10 1.20 1.30 1.40 1.50 1.60 1.70 1.80 1.90 2.00 2.10 2.20 2.30 2.40 2.50 2.60 2.70 2.80 2.90 3.00 3.10 3.20 3.30 3.40 3.50 3.60 3.70 3.80 3.90 4.00 4.10 4.20 4.30

0.0615 0.0795 0.0977 0.1155 0.1326 0.1492 0.1652 0.1808 0.1959 0.2104 0.2245 0.2381 0.2512 0.2639 0.2763 0.2881 0.2996 0.3107 0.3215 0.3318 0.3418 0.3515 0.3608 0.3699 0.3786 0.3870 0.3951 0.4029 0.4104 0.4177 0.4247 0.4315 0.4380 0.4443 0.4504 0.4562 0.4618 0.4672 0.4724 0.4775 0.4823

0.2347 0.3101 0.3857 0.4607 0.5347 0.6082 0.6807 0.7524 0.8233 0.8933 0.9625 1.0307 1.0980 1.1644 1.2299 1.2944 1.3579 1.4204 1.4819 1.5424 1.6018 1.6602 1.7175 1.7738 1.8289 1.8830 1.9360 1.9878 2.0386 2.0882 2.1367 2.1840 2.2303 2.2754 2.3193 2.3621 2.4038 2.4444 2.4838 2.5221 2.5593

0.5324 0.5353 0.5431 0.5518 0.5601 0.5685 0.5767 0.5849 0.5932 0.6011 0.6091 0.6169 0.6245 0.6321 0.6395 0.6468 0.6539 0.6610 0.6678 0.6745 0.6810 0.6874 0.6937 0.6997 0.7056 0.7114 0.7170 0.7224 0.7277 0.7328 0.7378 0.7426 0.7472 0.7517 0.7561 0.7603 0.7644 0.7683 0.7721 0.7757 0.7792

0.4676 0.4647 0.4569 0.4482 0.4399 0.4315 0.4233 0.4151 0.4068 0.3989 0.3909 0.3831 0.3755 0.3679 0.3605 0.3532 0.3461 0.3390 0.3322 0.3255 0.3190 0.3126 0.3063 0.3003 0.2944 0.2886 0.2830 0.2776 0.2723 0.2672 0.2622 0.2574 0.2528 0.2483 0.2439 0.2397 0.2356 0.2317 0.2279 0.2243 0.2208

4.40 4.50 4.60 4.70 4.80 4.90 5.00 5.10 5.20 5.30 5.40 5.50 5.60 5.70 5.80 5.90 6.00 6.20 6.40 6.60 6.80 7.00 7.20 7.40 7.70 8.10 8.50 9.00 10.00 11.00 12.00 14.00 16.00 18.00 20.00 25.00 30.00 40.00 60.00 100.00 1

0.4869 0.4914 0.4957 0.4998 0.5038 0.5076 0.5112 0.5148 0.5181 0.5214 0.5245 0.5274 0.5303 0.5330 0.5356 0.5381 0.5405 0.5450 0.5491 0.5528 0.5562 0.5593 0.5621 0.5646 0.5680 0.5717 0.5748 0.5777 0.5816 0.5836 0.5847 0.5856 0.5857 0.5858 0.5858 0.5858 0.5858 0.5858 0.5858 0.5858 0.5858

2.5954 2.6304 2.6643 2.6971 2.7289 2.7596 2.7893 2.8179 2.8456 2.8722 2.8979 2.9226 2.9464 2.9693 2.9913 3.0124 3.0327 3.0707 3.1056 3.1375 3.1686 3.1930 3.2170 3.2388 3.2674 3.2992 3.3247 3.3494 3.3811 3.3978 3.4063 3.4125 3.4139 3.4141 3.4142 3.4142 3.4142 3.4142 3.4142 3.4142 3.4142

0.7826 0.7858 0.7890 0.7920 0.7949 0.7977 0.8003 0.8029 0.8054 0.8077 0.8100 0.8121 0.8142 0.8162 0.8181 0.8199 0.8216 0.8248 0.8278 0.8305 0.8329 0.8351 0.8371 0.8389 0.8413 0.8439 0.8460 0.8480 0.8507 0.8521 0.8529 0.8534 0.8535 0.8536 0.8536 0.8536 0.8536 0.8536 0.8536 0.8536 0.8536

0.2174 0.2142 0.2110 0.2080 0.2051 0.2023 0.1997 0.1971 0.1946 0.1923 0.1900 0.1879 0.1858 0.1838 0.1819 0.1801 0.1784 0.1754 0.1722 0.1695 0.1671 0.1649 0.1629 0.1611 0.1587 0.1561 0.1540 0.1520 0.1493 0.1479 0.1471 0.1466 0.1465 0.1464 0.1464 0.1464 0.1464 0.1464 0.1464 0.1464 0.1464

174 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Step 2. Calculate the value of the parameter α by solving Eq. (2.111) iteratively. Step 3. Determine the upper limit of integration c by applying Eq. (2.114). Step 4. Find the integration points r1 and r2 and the weighting factors w1 and w2 from Table 2.9. Step 5. Find the pseudocomponent carbon numbers, ni, and mole fractions, zi, from the following expressions: For the first pseudocomponent, r1 +A α z1 ¼ w1 z7 +

n1 ¼

(2.118)

For the second pseudocomponent, r2 +A α z2 ¼ w2 z7 +

n2 ¼

(2.119)

Step 6. Assign the physical and critical properties of the two pseudocomponents from Table 2.2. EXAMPLE 2.18 A heptanes-plus fraction in a crude oil system has a mole fraction of 0.4608 with a molecular weight of 226. Using the Behrens and Sandler lumping scheme, characterize the C7+ by two pseudocomponents and calculate their mole fractions. Solution Step 1. Assuming the starting and ending carbon numbers to be C7 and C50, calculate A and B from Eqs. (2.116) and (2.117): A ¼ ðstarting carbon numberÞ  0:5 A ¼ 7  0:5 ¼ 6:5 B ¼ ðending carbon numberÞ + 0:5 B ¼ 50 + 0:5 ¼ 50:5 Step 2. Calculate cn from Eq. (2.112): MC7 + + 4 14 226 + 4 cn ¼ ¼ 16:43 14

cn ¼

Step 3. Solve Eq. (2.111) iteratively for α, to give 1 ½A  BeBα ¼0  cn + A  Aα e  eBa α 1 ½6:5  50:5e50:5α ¼0  16:43 + 6:5  6:5α e  e50:5α α

Splitting and Lumping Schemes 175

Solving this expression iteratively for α gives α ¼ 0.0938967. Step 4. Calculate the range of integration c from Eq. (2.114): c ¼ ðB  AÞα c ¼ ð50:5  6:5Þ0:0938967 ¼ 4:13 Step 5. Find integration points ri and weights wi from Table 2.9: r1 ¼ 0:4741 r2 ¼ 2:4965 w1 ¼ 0:7733 w2 ¼ 0:2267 Step 6. Find the pseudocomponent carbon numbers ni and mole fractions zi by applying Eqs. (2.118) and (2.119). For the first pseudocomponent, r1 n1 ¼ + A α 0:4741 n1 ¼ + 6:5 ¼ 11:55 0:0938967 z1 ¼ w1 z7 + z1 ¼ ð0:7733Þð0:4608Þ ¼ 0:3563 For the second pseudocomponent, r2 n2 ¼ + A α 2:4965 n2 ¼ + 6:5 ¼ 33:08 0:0938967 z2 ¼ w2 z7 + z2 ¼ ð0:2267Þð0:4608Þ ¼ 0:1045 The C7+ fraction is represented then by the two pseudocomponents below. Pseudocomponent

Carbon Number

Mole Fraction

1 2

C11.55 C33.08

0.3563 0.1045

Step 7. Assign the physical properties of the two pseudocomponents according to their number of carbon atoms using the Katz and Firoozabadi generalized physical properties as given in Table 2.2 or by calculations from Eq. (2.6). The assigned physical properties for the two fractions are shown below. Pseudocomponent n 1 2

Tb (°R) γ

11.55 848 33.08 1341

M

Tc (°R) pc (psia) ω

0.799 154 1185 0.915 426 1629

314 134

0.437 0.921

176 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Lee’s Lumping Scheme Lee et al. (1979) devised a simple procedure for regrouping the oil fractions into pseudocomponents. Lee and coworkers employed the physical reasoning that crude oil fractions having relatively close physicochemical properties (such as molecular weight and specific gravity) can be accurately represented by a single fraction. Having observed that the closeness of these properties is reflected by the slopes of curves when the properties are plotted against the weight-averaged boiling point of each fraction, Lee et al. used the weighted sum of the slopes of these curves as a criterion for lumping the crude oil fractions. The authors proposed the following computational steps: Step 1. Plot the available physical and chemical properties of each original fraction versus its weight-averaged boiling point. Step 2. Calculate numerically the slope mij for each fraction at each WABP, where mij ¼ slope of the property curve versus boiling point i ¼ l, …, nf j ¼ l, …, np nf ¼ number of original oil fractions np ¼ number of available physicochemical properties  ij as defined: Step 3. Compute the normalized absolute slope m  ij ¼ m

mij max i¼1, …, nf mij

(2.120)

 i for each fraction as follows: Step 4. Compute the weighted sum of slopes M Xnp i ¼ M

j¼1

np

 ij m

(2.121)

 i represents the averaged change of physicochemical where M properties of the crude oil fractions along the boiling point axis.  i for each fraction, group those Step 5. Judging the numerical values of M  i values. fractions that have similar M Step 6. Using the mixing rules given by Eqs. (2.103) through (2.108), calculate the physical properties of pseudocomponents.

EXAMPLE 2.19 The data in Table 2.10, as given by Hariu and Sage (1969), represent the average boiling point molecular weight, specific gravity, and molar distribution of 15 pseudocomponents in a crude oil system. Lump the given data into an optimum number of pseudocomponents using the Lee method.

Splitting and Lumping Schemes 177

Table 2.10 Characteristics of Pseudocomponents in Example 2.18 Pseudocomponent

Tb (°R)

M

γ

zi

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

600 654 698 732 770 808 851 895 938 983 1052 1154 1257 1382 1540

95 101 108 116 126 139 154 173 191 215 248 322 415 540 700

0.680 0.710 0.732 0.750 0.767 0.781 0.793 0.800 0.826 0.836 0.850 0.883 0.910 0.940 0.975

0.0681 0.0686 0.0662 0.0631 0.0743 0.0686 0.0628 0.0564 0.0528 0.0474 0.0836 0.0669 0.0535 0.0425 0.0340

Solution Step 1. Plot the molecular weights and specific gravities versus the boiling points and calculate the slope mij for each pseudocomponent as in the following table. Pseudocomponent

Tb

mi1 5 @M/@Tb

mi2 5 @γ/@Tb

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

600 654 698 732 770 808 851 895 938 983 1052 1154 1257 1382 1540

0.1111 0.1327 0.1923 0.2500 0.3026 0.3457 0.3908 0.4253 0.4773 0.5000 0.6257 0.8146 0.9561 1.0071 1.0127

0.00056 0.00053 0.00051 0.00049 0.00041 0.00032 0.00022 0.00038 0.00041 0.00021 0.00027 0.00029 0.00025 0.00023 0.00022

178 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

i Table 2.11 Absolute Slope mij and Weighted Slopes M Pseudocomponent

 i1 ¼ mij =1:0127 m

 i2 ¼ mij =0:00056 m

 i ¼ ðm  i1 + m  i2 Þ=2 M

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

0.1097 0.1310 0.1899 0.2469 0.2988 0.3414 0.3859 0.4200 0.4713 0.4937 0.6179 0.8044 0.9441 0.9945 1.0000

1.0000 0.9464 0.9107 0.8750 0.7321 0.5714 0.3929 0.6786 0.7321 0.3750 0.4821 0.5179 0.4464 0.4107 0.3929

0.55485 0.53870 0.55030 0.56100 0.51550 0.45640 0.38940 0.54930 0.60170 0.43440 0.55000 0.66120 0.69530 0.70260 0.69650

Step 2. Calculate the normalized absolute slope mij and the weighted sum of  i i using Eqs. (2.121) and (2.122), respectively. This is shown in slopes M Table 2.11. Note that the maximum value of mi1 is 1.0127 and for mi2 is 0.00056.  i , the pseudocomponents can be lumped into Examining the values of M three groups: Group 1: Combine fractions 1–5 with a total mole fraction of 0.3403. Group 2: Combine fractions 6–10 with a total mole fraction of 0.2880. Group 3: Combine fractions 11–15 with a total mole fraction of 0.2805. Step 3. Calculate the physical properties of each group. This can be achieved by computing M and γ of each group by applying Eqs. (2.103) and (2.104), respectively, followed by employing the Riazi-Daubert correlation (Eq. (2.6)) to characterize each group. Results of the calculations are shown below. Group I

Mole Fraction (zi)

M

γ

Tb

Tc

pc

Vc

1 2 3

0.3403 0.2880 0.2805

109.4 171.0 396.5

0.7299 0.8073 0.9115

694 891 1383

1019 1224 1656

404 287 137

0.0637 0.0634 0.0656

Splitting and Lumping Schemes 179

The physical properties M, γ, and Tb reflect the chemical makeup of petroleum fractions. Some of the methods used to characterize the pseudocomponents in terms of their boiling point and specific gravity assume that a particular factor, called the characterization factor, Cf, is constant for all the fractions that constitute the C7+. Soreide (1989) developed an accurate correlation for determining the specific gravity based on the analysis of 843 TBP fractions from 68 reservoir C7+ samples and introduced the characterization factor, Cf, into the correlation, as given by the following relationship: γ i ¼ 0:2855 + Cf ðMi  66Þ0:13

(2.122)

Characterization factor Cf is adjusted to satisfy the following relationship:   zC M C γ C7 + exp ¼ C 7 + 7 + (2.123) N+ X zi Mi i¼C7

γi

  where γ C7 + exp is the measured specific gravity of the C7+. Combining Eq. (2.122) with Eq. (2.123) gives   zC7 + MC7 + # γ C7 + exp ¼ C " N+ X zi Mi 0:2855 + Cf ðMi  66Þ0:13

i¼C7

Rearranging, f ðCf Þ ¼

CN + X i¼C7

"

#

zi Mi

0:2855 + Cf ðMi  66Þ0:13

zC MC   7+  7+ ¼ 0 γ C7 + exp

(2.124)

The optimum value of the characterization factor can be determined iteratively by solving this expression for Cf. The Newton-Raphson method is an efficient numerical technique that can be used conveniently to solve the preceding nonlinear equation by employing the relationship   f Cnf n+1 n   Cf ¼ Cf  @f Cnf =@Cnf with

2   CN + X6 @f Cnf ¼ 4 @Cnf i¼C7

3 ðMi  66Þ0:13 0:2855 + Cnf ðMi  66Þ

0:13

7 2 5

The method is based on assuming a starting value Cnf and calculating an improved value Cnf + 1 that can be used in the second iteration. The iteration process continues until the difference between the two, Cnf + 1 and Cnf , is small, say, 106. The calculated value of Cf then can be used in Eq. (2.122) to determine specific gravity of any pseudocomponent γ i. Soreide also developed the following boiling point temperature, Tb, correlation, which is a function of the molecular weight and specific gravity of the fraction

180 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Tb ¼ 1928:3 

  1:695 105 Aγ 3:266 M0:03522

with A ¼ exp ½0:003462Mγ  0:004922M  4:7685γ  where Tb ¼ boiling point temperature, °R M ¼ molecular weight

EXAMPLE 2.20 The molar distribution of a heptanes-plus,  as given by Whitson and Brule, is listed below, where MC7 + ¼ 143 and γ C7 + exp ¼ 0:795. Calculate the specific gravity of the five pseudocomponents. C7+ Fraction (i)

Mole Fraction (zi)

Molecular Weight (Mi)

1 2 3 4 5

2.4228 2.8921 1.2852 0.2367 0.0132

95.55 135.84 296.65 319.83 500.00

Solution Step 1. Solve Eq. (2.124) for the characterization factor, Cf, by trial and error or Newton-Raphson method, to give Cf ¼ 0:26927 Step 2. Using Eq. (2.122), calculate the specific gravity of the five, shown in the following table, where γ i ¼ 0:2855 + Cf ðMi  66Þ0:13 γ i ¼ 0:2855 + 0:26927ðMi  66Þ0:13 C7+ Fraction (i)

Mole Fraction (zi)

Molecular Weight (Mi)

Specific gravity (γ i 5 0.2855 + 0.26927(Mi 2 66)0.13)

1 2 3 4 5

2.4228 2.8921 1.2852 0.2367 0.0132

95.55 135.84 296.65 319.83 500.00

0.7407 0.7879 0.8358 0.8796 0.9226

Splitting and Lumping Schemes 181

Characterization of Multiple Hydrocarbon Samples When characterizing multiple samples simultaneously for use in the equation of state applications, the following procedure is recommended: 1. Identify and select the main fluid sample among multiple samples from the same reservoir. In general, the sample that dominates the simulation process or has extensive laboratory data is chosen as the main sample. 2. Split the plus fractions of the main sample into several components using Whitson’s method and determine the parameters MN, η, and β*, as outlined by Eqs. (2.93) and (2.94). 3. Calculate the characterization factor, Cf, by solving Eq. (2.124) and calculate specific gravities of the C7+ fractions by using Eq. (2.122). 4. Lump the produced splitting fractions of the main sample into a number of pseudocomponents characterized by the physical properties pc, Tc, M, γ, and ω. 5. Assign the gamma function parameters (MN, η, and β) of the main sample to all the remaining samples; in addition, the characterization factor, Cf, must be the same for all mixtures. However, each sample can have a different MC7 + and α. 6. Using the same gamma function parameters as the main sample, split each of the remaining samples into a number of components. 7. For each of the remaining samples, group or lump the fractions into pseudocomponents with the condition that these pseudocomponents are lumped in a way to give similar or the same molecular weights to those of the lumped fractions of the main sample. 8. The results of this characterization procedure are n n

One set of molecular weights for the C7+ fractions Each pseudocomponent has the same properties (ie, M, γ, Tb, pc, Tc, and ω) as those of the main sample but with a different mole fraction

It should be pointed out that, when characterizing multiple samples, the outlier samples must be identified and treated separately. Assuming that these multiple samples are obtained from different depths and each is described by laboratory PVT measurements, the outlier samples can be identified by making the following plots: n n n n

Saturation pressure versus depth C1 mole % versus depth C7+ mole % versus depth Molecular weight of C7+ versus depth

Data deviating significantly from the general trend, as shown in Figs. 2.22– 2.25, should not be used in the multiple samples characterization procedure. Takahashi et al. (2002) point out that a single EOS model never

182 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

Saturation pressure

X X True vertical depth

X X

X

X X Outlier (removed) data

n FIGURE 2.22 Measured saturation pressure versus depth.

Mole % of methane

X

True vertical depth

X

X

X X

X

X Outlier (removed) data

n FIGURE 2.23 Measured methane content versus depth.

Splitting and Lumping Schemes 183

Molecular weight of heptanes-plus fraction

True vertical depth

X

X

X

X Outlier (removed) data

n FIGURE 2.24 Measured C7+ molecular weight versus depth.

Mole % of heptanes-plus fraction

X

True vertical depth

X X

X

X

n FIGURE 2.25 Measured mole % of C7+ versus depth.

X Outlier (removed) data

184 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

predicts such outlier behavior using a common characterization. Also note that systematic deviation from a general trend may indicate a separate fluid system, requiring a separate equation-of-state model, while randomlike deviations from clear trends often are due to experimental data error or inconsistencies in reported compositional data as compared with the actual samples used in laboratory tests.

PROBLEMS 1. A heptanes-plus fraction with a molecular weight of 198 and a specific gravity of 0.8135 presents a naturally occurring condensate system. The reported mole fraction of the C7+ is 0.1145. Predict the molar distribution of the plus fraction using (a) Katz’s correlation (b) Ahmed’s correlation Characterize the last fraction in the predicted extended analysis in terms of its physical and critical properties. 2. A naturally occurring crude oil system has a heptanes-plus fraction with the following properties: M7+ ¼ 213.0000 γ 7+ ¼ 0.8405 x7+ ¼ 0.3497 Extend the molar distribution of the plus fraction and determine the critical properties and acentric factor of the last component. 3. A crude oil system has the following composition: s

xi

C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 C11 C12 C13+

0.3100 0.1042 0.1187 0.0732 0.0441 0.0255 0.0571 0.0472 0.0246 0.0233 0.0212 0.0169 0.1340

The molecular weight and specific gravity of C13+ are 325 and 0.842, respectively. Calculate the appropriate number of pseudocomponents necessary to adequately represent these components using

Problems 185

4.

5. 6.

7.

8.

9.

(a) Whitson’s lumping method (b) Behrens-Sandler’s method Characterize the resulting pseudocomponents in terms of their critical properties. If a petroleum fraction has a measured molecular weight of 190 and a specific gravity of 0.8762, characterize this fraction by calculating the boiling point, critical temperature, critical pressure, and critical volume of the fraction. Use Riazi-Daubert’s correlation. Calculate the acentric factor and critical compressibility factor of the component in problem 4. A petroleum fraction has the following physical properties: API ¼ 50° Tb ¼ 400°F M ¼ 165 γ ¼ 0.79 Calculate pc, Tc, Vc, ω, and Zc using the following correlations: (a) Cavett (b) Kesler-Lee (c) Winn-Sim-Daubert (d) Watansiri-Owens-Starling An undefined petroleum fraction with 10 carbon atoms has a measured average boiling point of 791°R and a molecular weight of 134. If the specific gravity of the fraction is 0.78, determine the critical pressure, critical temperature, and acentric factor of the fraction using (a) Robinson-Peng’s PNA method (b) Bergman PNA’s method (c) Riazi-Daubert’s method (d) Cavett’s correlation (e) Kesler-Lee’s correlation (f) Willman-Teja’s correlation A heptanes-plus fraction is characterized by a molecular weight of 200 and specific gravity of 0.810. Calculate pc, Tc, Tb, and acentric factor of the plus fraction using (a) Riazi-Daubert’s method (b) Rowe’s correlation (c) Standing’s correlation Using the data given in problem 8 and the boiling point as calculated by Riazi-Daubert’s correlation, determine the critical properties and acentric factor employing (a) Cavett’s correlation (b) Kesler-Lee’s correlation Compare the results with those obtained in problem 8.

186 CHAPTER 2 Characterizing Hydrocarbon-Plus Fractions

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