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New Compositional Models for Calculating the Viscosity of Crude Oils Adel M. Elsharkawy,*,† Suad A. Hassan,† Yousef S. Kh. Hashim,† and Mohamed A. Fahim‡ Petroleum Engineering and Chemical Engineering Departments, Kuwait University, P.O. Box 5969, Safat 13060, Kuwait
Crude oil viscosity is an important physical property that controls and influences the flow of oil through porous media and pipelines. Hence, it is the basis of many reservoir engineering and production system calculations. In the crude oil recovery and processing, viscosity is a significant parameter because a large amount of time and money are spent in experimental measurements of viscosity. This necessitates the development of reliable viscosity models capable of predicting crude oil viscosity and reduces the expensive and cumbersome experimental measurements. Because of the complexities and varied composition of crude oils, empirical models and other predictive tools cannot replace the laboratory measurements, even if the latter is costly. This paper presents new compositional models for estimating crude oil viscosity. The new models use fluid composition, temperature, and pressure to predict the oil viscosity. The models were derived from viscosity measurements of several crudes from the Middle East, the North Sea, and others. The accuracy of the proposed models has been compared to that for several empirical correlations, corresponding state models, and equation of state based viscosity models. The comparison shows the superiority of the new models over the other methods. 1. Introduction The mechanism or theory of gas viscosity has been reasonably well-defined by the application of the kinetic theory of the gases. The theory of liquid viscosity is poorly developed because of the intermolecular forces between the molecules, which consist of short-range effects such as repulsion and hydrocarbon bonding, wide-range effects such as electrostatic effects, and longrange effects such as attractions. A further complication is the structure and degree of disorder between the molecules. Thus, there is no widely accepted simple theoretical method to calculate the viscosity of liquids.1 Petroleum crude oils and fractions are typically complex mixtures, and the physical and chemical properties vary considerably depending on the composition of the constituents. For instance, the viscosity of a crude oil or a fraction composed mostly of aromatics might have quite different temperature dependence than that composed mostly of saturates. Useful viscosity prediction methods are most conveniently based on parameters such as boiling temperatures and specific gravities that are commonly used to characterize each fraction.2 The relatively successful viscosity models available in the literature can be classified as follows: (i) empirical methods; (ii) corresponding state methods; (iii) equation of state (EOS)-based viscosity models; (iv) group contribution methods. Although numerous viscosity correlations for hydrocarbon liquids and gases are available in the literature, there are three main drawbacks in their applications:3 1. Application range and accuracy are limited. * To whom correspondence should be addressed. Tel.: +(965) 483-6059. Fax: +(965) 484-9558. E-mail: asharkawy@ kuc01.kuniv.edu.kw. † Petroleum Engineering Department. ‡ Chemical Engineering Department.
2. Because the viscosities of the liquid and gas phases are calculated by using different graphs or correlations, a smooth transition in the near-critical region cannot be achieved. 3. Density is involved in evaluating the fluid viscosity, and hence a separate density correlation is required. Recently, the efforts focused toward developing a theoretical viscosity model based on an EOS. The major advantages of such models are as follows: 1. A single model achieving a smooth transition of liquid/gas viscosity in the near-critical region can describe the viscosity of both the gas and liquid phases. 2. Both high- and low-pressure data can be correlated, and density is not involved in evaluating the fluid viscosity. 3. Using a single EOS, one can perform P-V-T, vapor-liquid equilibrium, and fluid viscosity calculations; the thermodynamic consistency in process simulation/reservoir simulation is thus improved. The state of Kuwait contains about 10% of the world proven reserve. Precise viscosity calculations are very vital to the production and transport of crude oil. Therefore, the current study is aimed at (1) investigating the accuracy of previously published methods for estimating the viscosity of Kuwaiti crude oils as well as others using the experimentally measured viscosity, (2) modifying one of the best methods available in the literature or proposing a new method for estimating the Kuwaiti crude oil viscosity, and (3) studying the effect of including the binary interaction parameter (BIP) on the EOS-based viscosity model for estimating the viscosity of Kuwaiti crudes. 2. Methods of Viscosity Estimation In general, there is no simple rule or theory for predicting the rate of change of the crude oil viscosity with temperature and pressure, or composition. The
10.1021/ie0300631 CCC: $25.00 © 2003 American Chemical Society Published on Web 07/16/2003
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only alternative is to obtain the necessary viscosity measurements and model their behavior. Usually, viscosity is measured isothermally at various pressures using the initial composition of reservoir fluids. Because the composition changes during depletion of the reservoir and obtaining representative sample during the life of the reservoir is quasi-impossible, one could imagine the following: (1) The viscosity experimental data are used to evaluate and calibrate the proposed compositional model. (2) The compositional model is used in conjunction with production data to predict the ever-changing composition. (3) The compositional model is used to update the value of the viscosity not only as a function of the changing pressure and temperature but also, most importantly, as a function of the composition. 2.1. Empirical Correlations. Previously published correlations are restricted to the prediction of the viscosity of gas-free (dead or stabilized) crude oils at atmospheric pressure; on the other hand, there are other models that cover all kinds of crude oils: dead, saturated, and undersaturated crudes. The most popular empirical models presently used are those developed by Beggs and Robinson,4 Labedi,5 Kartoatmodjo and Schmidt,6 and Elsharkawy and Alikhan.7 These correlations are a function of field measurable parameters such as temperature (T), pressure (P), separated gas gravity (γg), and tank oil gravity (API). The dead oil viscosity (µod) was expressed as a function of both oil API gravity and reservoir temperature. The saturated oil viscosity (µob) was correlated to the dead oil viscosity and solution gas-oil ratio (Rs). It is important to note that the solution gas-oil ratio is a function of the oil API gravity, gas gravity, reservoir pressure, and temperature. Thus, the accuracy of these correlations in estimating the saturated oil viscosity is largely dependent on the success of estimating the gas-oil ratio. Because no gas is being dissolved into the oil above the bubble point, pressure becomes the primary independent variable for predicting the viscosity of undersaturated oil (µo). 2.2. Corresponding State Models. The principle of corresponding states is one of the most useful methods for predicting the properties of fluids. The principle states that dimensionless properties of all fluids have the same numerical values at the same reduced conditions. This provides the most important basis for the development of correlations and estimation methods. These methods require knowledge of the properties of, at least, one reference fluid. Hanley8 developed an extended corresponding state to calculate transport properties of mixtures, which additionally requires the acentric factor and critical compressibility of the reference fluid and each component. Hanley et al.9 used the viscosity of methane as the reference fluid; the viscosity was correlated as a function of the temperature and density. For fluids that do not correspond with the reference fluid, Hanley derived a correction factor. The major difficulty in using methane as the reference fluid is that its normal freezing point is at a reduced temperature of 0.476, which is above that for many other hydrocarbons. Pedersen and Fredenslund10 used a new parameter (R) to account for the molecular size and density effects. Pedersen and Fredenslund11 proposed an improvement of the model by extending the method of Pedersen et al. to Tr < 0.4, which is below the freezing
point of methane, by modifying the equations with additional experimental viscosity data. Teja and Rice12 and Teja et al.13 proposed an alternative formulation of the corresponding state principle, based on the properties of a two-reference fluid. However, all of these methods use the acentric factor as an interpolation parameter of the reference fluid properties. Petersen et al.14 proposed a method based on the Teja-Rice method with the molecular weight as an interpolation parameter instead of the acentric factor because, for heavy petroleum fractions, the acentric factor decreases with increasing molecular weight. Because the method was developed for hydrocarbons, methane and n-decane were used as reference components. The model proposed by Petersen et al. yields accurate viscosity predictions of light petroleum fractions. However, its application to heavy fractions may give inaccurate predictions. In addition, the prediction by this model is restricted to the condition of reduced temperature Tr > 0.476. Moharam and Fahim1 used decane and eicosane as the reference fluids in predicting the viscosity of heavy crude oils and petroleum fractions of average molecular weight higher than 142. This model relaxes the limitation imposed on reduced temperature Tr > 0.476 to Tr > 0.4. 2.3. EOS-Based Viscosity Models. Two widely used procedures have been developed to calculate the viscosities of in situ reservoir gases and liquids from their composition at the desired temperature and pressure. Given is a composition expressed in methane through heptane-plus, hydrogen sulfide, nitrogen, and carbon dioxide together with the molecular weight and specific gravity of the heptane-plus fractions. The first viscosity prediction method that requires the composition of crude oil was presented by Lohrenz et al.15 This procedure based on the residual viscosity concept and the low of corresponding states. Little and Kennedy16 is the second well-known method which applies reliably to both liquids and gases, and the equation is similar in form to the van der Waals EOS. The disadvantage of the above two methods is that the predicted viscosity is very sensitive to the density, which is normally determined by a separate correlation and may be very inaccurate for high-viscosity fluids. The similarity between pressure-volume-temperature (P-V-T) and temperatureviscosity-pressure (T-µ-P) relationships was pointed out by Philips17 early in 1912. Lawal18 developed a viscosity model based on the four-parameter LawalLake-Silberberg EOS. However, it is not a predictive model because specific constants are required for each substance, and poor results were obtained when applied to the C7+ fraction containing reservoir oils. Guo et al.3 proposed two different viscosity models for pure hydrocarbons (mainly for n-alkanes). The models are developed from the three-parameter Patel-Teja (PT; eq 1) and the two-parameter Peng-Robinson (PR; eq 2) cubic EOSs. These models are applicable to both liquid and gas phases and have been extended to hydrocarbon mixtures
T)
rP a µ - b′ µ(µ + b) + c(µ - b)
(1)
T)
a rP µ - b′ µ(µ + b) + b(µ - b)
(2)
where the parameters a-c are EOS parameters. Guo reported an average absolute deviation (AAD) of 6.18%
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the previous nine gas condensates was reduced from 15% to 9.8%.
Table 1. Data Sources no. of samples
type
9 14 7 7 49
natural gas crude oil crude oil crude oil crude oil
origin
reference
Abu Dhabi North Sea Kuwait
18 18 1 10 and 11 present study
3. Current Work 3.1. Viscosity Data. A data bank of 89 fluid samples has been used in this study to check the accuracy of the published methods and develop a new compositional model. The samples have been described according to their types and sources in Table 1. Viscosity data for Kuwaiti crude oil samples are measured for each sample as a part of the differential liberation (DL) test. The DL test provides measurements of the crude oil viscosity, solution gas-oil ratio, and oil density at reservoir temperature at several pressures above and below the bubble-point pressure. The test also provides compositional analysis of the reservoir oil sample (methane through hexane, non-hydrocarbon, and heptane-plus fraction), molecular weight, and specific gravity of the heavy hydrocarbon plus fraction. The composition and viscosity measurements of the Kuwaiti crude oils at
for pure substances with the PR viscosity model, but the accuracy was reduced when both models (PR and PT) were used to estimate the viscosity of hydrocarbon mixtures. The PR viscosity model gives an AAD of 15%, whereas the PT viscosity model gives 12.6% when tested against nine gas condensates taken from Lawal.18 Also, the PT viscosity model was tested on reservoir crude oils, and an AAD of 15.07% was obtained. Because the PR method has limited accuracy, it was modified by Guo et al.19 to improve the viscosity prediction accuracy for reservoir fluids, including the highly asymmetric CO2injected enhanced oil recovery systems. The overall AAD for reservoir crude oils was 14%, whereas the AAD for
Table 2. Composition and Viscosity Measurements of Kuwaiti Crude Oils crude T no. (°F) 1 2 4 5 6 7 8 9 10 11 12 13 14 15 16 18 19 20 21 22 23 24 25 27 28 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 48 49 50 51 52 53 55
130 133 133 135 134 134 134 135 134 135 134 134 133 134 132 132 136 135 134 134 135 134 134 133 134 132 134 132 132 133 135 133 133 135 135 133 130 135 134 135 140 168 208 230 230 241 240 241 243
composition
Pb (psi)
µexp (cP)
N2
CO2
H 2S
C1
C2
C3
iC4
C4
iC5
C5
C6
C7+
γC7+
MWC7+
1365 1632 1595 1500 1615 1400 1590 1540 1399 1690 1548 1705 1653 1645 1751 1025 1640 1650 1634 1533 1548 1825 1825 1565 1580 1380 1680 1738 1920 1655 1880 1788 1805 1500 1500 1715 1670 1735 1530 1766 1730 2505 1877 2110 3730 3335 3630 3120 3180
0.510 1.120 1.250 1.350 1.240 1.950 1.630 1.160 1.600 2.810 1.150 1.300 1.500 1.010 1.610 1.280 1.000 0.660 1.250 2.300 1.740 3.520 3.390 1.560 1.910 2.010 1.900 2.020 1.770 1.860 1.000 1.540 1.520 1.740 1.910 1.050 1.550 1.480 2.100 1.990 3.010 1.300 0.790 0.490 0.250 0.440 0.430 0.450 0.460
0.0036 0.0029 0.0033 0.0035 0.0012 0.0016 0.0021 0.0011 0.0079 0.0004 0.0000 0.0045 0.0056 0.0003 0.0006 0.0000 0.0048 0.0015 0.0050 0.0068 0.0022 0.0019 0.0018 0.0044 0.0014 0.0015 0.0026 0.0020 0.0000 0.0054 0.0041 0.0020 0.0010 0.0020 0.0015 0.0049 0.0015 0.0045 0.0023 0.0019 0.0021 0.0008 0.0000 0.0000 0.0000 0.0005 0.0006 0.0004 0.0006
0.0017 0.0048 0.0022 0.0047 0.0051 0.0030 0.0015 0.0028 0.0010 0.0017 0.0017 0.0008 0.0007 0.0030 0.0081 0.0015 0.0041 0.0019 0.0087 0.0102 0.0137 0.0251 0.0246 0.0083 0.0054 0.0039 0.0126 0.0080 0.0130 0.0050 0.0065 0.0119 0.0116 0.0049 0.0048 0.0047 0.0065 0.0051 0.0106 0.0012 0.0072 0.0137 0.0112 0.0125 0.0068 0.0085 0.0094 0.0078 0.0085
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0004 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0000 0.0069 0.0021 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.2723 0.2836 0.2556 0.2652 0.2881 0.2466 0.2777 0.2753 0.2479 0.3122 0.2856 0.2935 0.2990 0.2780 0.3134 0.1950 0.3325 0.3115 0.2985 0.2549 0.2792 0.3189 0.3026 0.2775 0.2944 0.2755 0.2827 0.3142 0.3453 0.2779 0.3021 0.3226 0.3293 0.2946 0.2851 0.3155 0.3048 0.3056 0.2475 0.2862 0.3312 0.3597 0.2695 0.3335 0.4879 0.4105 0.4444 0.4091 0.4070
0.0893 0.0829 0.0687 0.0771 0.0791 0.0751 0.0768 0.0690 0.0684 0.0650 0.0863 0.0785 0.0784 0.0826 0.0742 0.0817 0.0755 0.0892 0.0733 0.0637 0.0651 0.0751 0.0730 0.0754 0.0796 0.0786 0.0600 0.0760 0.0724 0.0689 0.0689 0.0757 0.0778 0.0851 0.0817 0.0866 0.0727 0.0725 0.0678 0.0866 0.0744 0.0867 0.1076 0.0924 0.1057 0.1107 0.1076 0.1072 0.1054
0.0860 0.0738 0.0639 0.0605 0.0646 0.0692 0.0619 0.0587 0.0650 0.0475 0.0743 0.0728 0.0698 0.0673 0.0568 0.0843 0.0621 0.0738 0.0598 0.0653 0.0516 0.0659 0.0636 0.0690 0.0619 0.0630 0.0485 0.0723 0.0531 0.0634 0.0660 0.0597 0.0606 0.0663 0.0636 0.0792 0.0515 0.0728 0.0680 0.0652 0.0607 0.0594 0.0923 0.0607 0.0621 0.0707 0.0618 0.0664 0.0655
0.0168 0.0134 0.0158 0.0102 0.0091 0.0129 0.0097 0.0122 0.0150 0.0074 0.0120 0.0170 0.0198 0.0115 0.0090 0.0154 0.0107 0.0115 0.0104 0.0147 0.0086 0.0124 0.0119 0.0142 0.0107 0.0102 0.0087 0.0142 0.0087 0.0125 0.0163 0.0111 0.0108 0.0107 0.0107 0.0147 0.0069 0.0144 0.0175 0.0114 0.0094 0.0085 0.0096 0.0063 0.0073 0.0096 0.0080 0.0087 0.0087
0.0439 0.0372 0.0403 0.0301 0.0226 0.0347 0.0244 0.0320 0.0387 0.0179 0.0261 0.0433 0.0407 0.0257 0.0192 0.0446 0.0304 0.0292 0.0244 0.0368 0.0166 0.0353 0.0342 0.0319 0.0289 0.0257 0.0273 0.0346 0.0192 0.0320 0.0425 0.0274 0.0265 0.0290 0.0308 0.0401 0.0135 0.0380 0.0426 0.0364 0.0228 0.0212 0.0401 0.0177 0.0273 0.0386 0.0324 0.0357 0.0354
0.0175 0.0156 0.0167 0.0157 0.0084 0.0102 0.0071 0.0112 0.0179 0.0068 0.0096 0.0172 0.0136 0.0128 0.0065 0.0121 0.0102 0.0128 0.0089 0.0162 0.0074 0.0135 0.0140 0.0148 0.0125 0.0099 0.0153 0.0089 0.0073 0.0134 0.0177 0.0122 0.0114 0.0123 0.0143 0.0155 0.0078 0.0155 0.0160 0.0114 0.0067 0.0058 0.0090 0.0053 0.0284 0.0104 0.0091 0.0095 0.0094
0.0196 0.0186 0.0301 0.0251 0.0123 0.0163 0.0110 0.0135 0.0230 0.0103 0.0165 0.0309 0.0148 0.0280 0.0098 0.0171 0.0118 0.0220 0.0107 0.0326 0.0084 0.0216 0.0214 0.0162 0.0200 0.0159 0.0247 0.0108 0.0118 0.0195 0.0285 0.0195 0.0178 0.0214 0.0230 0.0192 0.0118 0.0221 0.0275 0.0157 0.0112 0.0092 0.0172 0.0109 0.0172 0.0185 0.0172 0.0167 0.0178
0.0255 0.0441 0.0433 0.0422 0.0325 0.0343 0.0284 0.0352 0.0367 0.0264 0.0411 0.0453 0.0266 0.0420 0.0264 0.0361 0.0288 0.0434 0.0428 0.0381 0.0259 0.0329 0.0322 0.0311 0.0384 0.0368 0.0426 0.0099 0.0348 0.0350 0.0427 0.0363 0.0370 0.0379 0.0387 0.0445 0.0276 0.0385 0.0389 0.0362 0.0305 0.0310 0.0333 0.0278 0.0280 0.0319 0.0257 0.0251 0.0281
0.4238 0.4231 0.4601 0.4657 0.4770 0.4961 0.4994 0.4890 0.4785 0.5045 0.4468 0.3962 0.4310 0.4488 0.4760 0.5122 0.4291 0.4032 0.4620 0.4603 0.5213 0.3974 0.4208 0.4572 0.4468 0.4790 0.4750 0.4491 0.4344 0.4670 0.4047 0.4216 0.4162 0.4359 0.4459 0.3751 0.4954 0.4110 0.4612 0.4478 0.4438 0.3971 0.4081 0.4329 0.2480 0.2901 0.2838 0.3134 0.3136
0.88 0.88 0.88 0.89 0.88 0.88 0.86 0.88 0.88 0.89 0.86 0.86 0.88 0.88 0.89 0.85 0.88 0.85 0.88 0.90 0.89 0.92 0.91 0.90 0.90 0.90 0.91 0.89 0.89 0.89 0.88 0.88 0.88 0.90 0.90 0.90 0.87 0.89 0.91 0.93 0.90 0.90 0.88 0.85 0.85 0.85 0.85 0.85 0.85
271 252 222 253 250 239 228 245 227 264 249 227 242 254 270 225 236 221 271 264 255 324 299 272 290 152 274 251 281 214 247 134 279 146 142 289 239 268 256 274 271 274 249 252 218 198 195 202 195
Ind. Eng. Chem. Res., Vol. 42, No. 17, 2003 4135 Table 3. Composition and Viscosity of Crude Oils from UAE crude T P no. (°F) (psi) 1 2 3 4 5 6 7
215 190 239 239 230 235 234
1261 1140 1490 1592 993 901 1190
composition
µexp (cP)
N2
CO2
H 2S
C1
C2
C3
iC4
C4
iC5
C5
C6
C7+
γC7+
MWC7+
0.897 0.668 0.673 0.579 0.750 0.813 0.675
0.0025 0.0024 0.0017 0.0032 0.0043 0.0021 0.0077
0.0219 0.0153 0.0065 0.0369 0.0347 0.0034 0.0199
0.0116 0.0060 0.0193 0.0068 0.0368 0.0000 0.0140
0.1633 0.1316 0.1259 0.2155 0.1949 0.2004 0.1738
0.0629 0.0638 0.0605 0.0860 0.0828 0.0793 0.0642
0.0748 0.0762 0.0651 0.0766 0.0685 0.0800 0.0762
0.0156 0.0169 0.0070 0.0114 0.0116 0.0193 0.0125
0.0453 0.0508 0.0356 0.0526 0.0314 0.0467 0.0437
0.0163 0.0246 0.0146 0.0219 0.0191 0.0252 0.0219
0.0273 0.0319 0.0306 0.0288 0.0227 0.0335 0.0234
0.0358 0.0637 0.0114 0.0262 0.0242 0.0508 0.0514
0.5227 0.5168 0.6218 0.4341 0.4690 0.4593 0.4913
0.8803 0.8764 0.8766 0.8687 0.8764 0.8606 0.8912
249 275 230 243 246 230 267
Table 4. Composition and Viscosity of Crude Oils from the North Sea crude T P no. (°F) (psi) 1 2 3 4 5 6 7
238 200 199 166 156 160 208
2753 3981 3926 3394 3813 2305 2952
composition
µexp (cP)
N2
CO2
H 2S
C1
C2
C3
iC4
C4
iC5
C5
C6
C7+
γC7+
MWC7+
0.380 0.404 0.320 0.425 1.120 2.100 0.299
0.0067 0.0034 0.0044 0.0090 0.0036 0.0033 0.0041
0.0211 0.0084 0.0038 0.0016 0.0106 0.0019 0.0044
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.3493 0.4923 0.4908 0.4712 0.5050 0.3542 0.4024
0.0700 0.0632 0.0760 0.0597 0.0454 0.0336 0.0769
0.0782 0.0446 0.0613 0.0462 0.0090 0.0090 0.0815
0.0122 0.0086 0.0097 0.0099 0.0055 0.0069 0.0122
0.0426 0.0218 0.0287 0.0250 0.0060 0.0026 0.0420
0.0145 0.0093 0.0102 0.0109 0.0062 0.0026 0.0142
0.0235 0.0133 0.0150 0.0146 0.0027 0.0014 0.0220
0.0304 0.0206 0.0200 0.0219 0.0160 0.0072 0.0281
0.3515 0.3145 0.2803 0.3300 0.3900 0.5773 0.3123
0.8528 0.8650 0.8564 0.8497 0.8972 0.9165 0.8448
226 230 231 217 291 255 210
Table 5. Composition and Viscosity of Other Crudesa crude T Pb no. (°F) (psi) 1 2 3 4 5 8 9 10 11 14 15 16 17 18 a
145 146 94 86 156 282 218 250 112 110 164 194 274 223
1297 1162 696 1275 1650 8590 4040 80 335 900 945 1090 2180 3665
composition
µexp (cP)
N2
CO2
H 2S
C1
C2
C3
iC4
C4
iC5
C5
C6
C7+
γC7+
MWC7+
0.660 0.550 2.620 1.620 0.660 0.110 0.530 1.490 1.130 1.530 0.750 1.370 0.440 0.450
0.0000 0.0000 0.0104 0.0147 0.0004 0.0038 0.0009 0.0010 0.0050 0.0560 0.0060 0.0060 0.0160 0.0000
0.0037 0.0013 0.0000 0.0019 0.0003 0.0164 0.0022 0.0170 0.0150 0.0040 0.0060 0.0040 0.0160 0.0010
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000
0.2289 0.2218 0.1303 0.2546 0.3018 0.7257 0.5105 0.0290 0.0690 0.1350 0.1480 0.2400 0.3510 0.5070
0.0059 0.0045 0.0834 0.0362 0.0264 0.0541 0.0345 0.0060 0.0770 0.0670 0.0610 0.0140 0.0110 0.0550
0.0141 0.0061 0.0899 0.0358 0.0123 0.0307 0.0139 0.0030 0.0820 0.0630 0.0920 0.0280 0.0080 0.0320
0.0077 0.0067 0.0118 0.0098 0.0085 0.0066 0.0053 0.0020 0.0200 0.0150 0.0190 0.0060 0.0060 0.0100
0.0071 0.0088 0.0598 0.0345 0.0064 0.0134 0.0067 0.0040 0.0630 0.0340 0.0650 0.0240 0.0140 0.0150
0.0094 0.0091 0.0188 0.0242 0.0060 0.0062 0.0041 0.0040 0.0210 0.0230 0.0320 0.0120 0.0130 0.0070
0.0012 0.0067 0.0332 0.0228 0.0046 0.0054 0.0023 0.0040 0.0260 0.0230 0.0320 0.0160 0.0170 0.0070
0.0241 0.0429 0.0539 0.0735 0.0299 0.0108 0.0218 0.0280 0.0830 0.0670 0.0650 0.0790 0.0560 0.0200
0.6979 0.6921 0.5085 0.4920 0.6034 0.1269 0.3978 0.9020 0.5410 0.5130 0.4750 0.5700 0.4920 0.3470
0.8660 0.8378 0.8628 0.8639 0.8363 0.8778 0.8692 0.9042 0.8789 0.8877 0.8670 0.8602 0.8246 0.8493
167 162 274 230 188 273 246 229 209 231 216 231 219 252
Reference 18.
bubble-point pressure are reported in Table 2. The composition and properties of other crudes used in this study are reported in Tables 3-5. 3.2. Methods for Viscosity Calculations Considered in the Study. The following methods are considered for investigating their accuracy in predicting the viscosity of crude oils: (i) empirical methods of Beggs and Robinson, Labedi, Kartoatmodjo and Schmidt, and Elsharkawy and Alikhan; (ii) corresponding state methods of Pedersen et al. and Petersen et al.; (iii) EOS-based viscosity models of Lohrenz et al., Little and Kennedy, and Guo et al. 3.3. Characterization of the C7+ Fraction. To calculate the viscosity of reservoir fluids from their composition, the critical properties of the fluids (critical pressure and temperature) and acentric factor are required. The critical properties of pure components are well documented. However, the critical properties of the heavy hydrocarbon plus fraction are usually estimated from correlations. Several correlations are available in the literature for the estimation of properties of undefined fractions. The selection of a particular correlation has a significant influence on the calculation of the crude oil viscosity because the heavy hydrocarbon fraction comprises a large portion of the composition of most crude oils. For this reason, two sets of correlations are considered in this study for characterizing the heavy hydrocarbon plus fraction of crude oils and observing the impact of the selection of Tc and Pc correlations on the viscosity prediction. The critical properties (Tc and
Pc) and acentric factor (ω) of the heptane-plus fraction were estimated in this study by the following two sets of empirical correlations. Set 1: The critical temperature (Tc) and critical pressure (Pc) of the plus fraction are estimated from the Lee-Kesler20 correlation, and the acentric factor (ω) is estimated from the Edmister21 correlation. Set 2: The critical temperature (Tc) and critical pressure (Pc) of the plus fraction are estimated from the Twu22 correlation, and the acentric factor (ω) is estimated from the Lee-Kesler correlation. 3.4. Viscosity Models. Evaluation of the various methods, which use the composition of reservoir fluids as the input parameter, considered in this study indicated that the Guo et al. method has the lowest AAD (32%) for all of the viscosity data. Therefore, many attempts have been made to improve or develop a new compositional model for predicting the viscosity of crude oils with a reasonable degree of accuracy. These attempts resulted into two models. The first is an empirical model that relates the viscosity to temperature, pressure, and composition in a simple form for all crudes considered in this study. The second is a modification of the viscosity model based on the PR EOS, which was originally proposed by Guo et al. This model is recommended for Kuwaiti crudes. 3.4.1. Empirical Model. The proposed empirical model is developed by a regression technique to fit the measured viscosity data. The database used in developing the model consists of 361 data points comprising 77
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Table 6. Summary of the Main Attempts in the Empirical Approach
a
attempt no.
Nva
function form
CORb
ARD %
AAD %
1 2 3
5 8 6
µ ) f(T,P,GL,Gm,Gh) µ ) f(T,P,GL,Gm,Gh,Gnon, γC7+,MWC7+) µ ) f(T,P,γC7+,GL,Gm,Gh)
0.751 14 0.863 282 0.859 674
8.42 4.84 5.05
31.28 24.13 25.09
Nv: no. of independent variables. b COR: coefficient of regression.
world crude oils from Kuwait, UAE, the North Sea, and others. Analysis of these data showed that the viscosity of crude oil could be expressed as a function of temperature, pressure, and several groups related to oil composition. In the proposed model, the mole fractions of the light components GL (methane and ethane) were taken as one group. The second group Gm represents the intermediate pure components (propane through hexane). The third group Gh comprises components heavier than heptane, the C7+ fraction. The fourth group Gnon consists of non-hydrocarbon components (N2, CO2, and H2S). Several attempts have been made to correlate the oil viscosity to compositional groups in addition to temperature and pressure. These attempts are summarized in Table 6. These various attempts resulted in estimation of the crude oil viscosity with an AAD that is less than that for all published methods considered in this study. The first attempt in developing a new compositional model considers the crude oil to be formed of three groups: light, medium, and heavy. This results in a model that is able to predict the crude oil viscosity with an AAD of 31.28%. Introducing the non-hydrocarbon group and the properties of the heavy hydrocarbon plus fraction, molecular weight, and specific gravity has substantially improved the accuracy of the proposed model. This results in an empirical model that is capable of predicting the crude oil viscosity with an AAD of 24%. The proposed empirical model has the following form: a7 a8 a9 6 µo ) a1Ta2Pa3GaL4 Gam5 Ganon GC7+ γC7+ MWC7+
T)
(3b)
where a1 ) 2248.089 447, a2 ) -1.278 46, a3 ) 0.117 425, a4 ) 13.197 27, a5 ) -0.324 28, a6 ) 0.066 623, and a7 ) 0.655 418. Hence, the model described by eq 3a is accounting for the presence of non-hydrocarbon components in the crude oils. However, the second form (eq 3b) is used only where the amount of non-hydrocarbon components is nil. In the above equations (3a) and (3b), the temperature is in degrees Fahrenheit, the pressure in pounds per square inch absolute, and the oil viscosity in centipoise. 3.4.2. Modified Viscosity Model Based on EOS for Kuwaiti Crudes. Guo et al. presented a viscosity model based on the PR EOS for calculating the viscosity of hydrocarbon mixtures. The model when used to
rP a µ - b′ µ(µ + b) + b(µ - b)
(4)
where P is in atmospheres, T in degrees Kelvin, and µ in micropoise.
a ) 0.45724
rc2Pc2 Tc
(5)
rcPc Tc
(6)
b ) 0.07780
r ) rcτ(Tr,Pr) rc )
µcTc PcZc
(7) (8)
where µc is the critical viscosity and is calculated by the empirical correlation proposed by Uyehara and Watson:23
µc ) 7.7Tc-1/6MW1/2Pc2/3
(9)
b′ ) bφ(Tr,Pr)
(10)
τ(Tr,Pr) ) {1 + Q1[(TrPr)0.5 - 1]}-2
(11)
(3a)
where a1 ) 13 148.237 52, a2 ) -1.290 44, a3 ) 0.112 653, a4 ) -0.339 36, a5 ) 0.058 341, a6 ) 0.018 099, a7 ) 0.638 023, a8 ) 14.011 34, and a9 ) -0.278 81. The model is improved further by reducing the number of variables by eliminating the non-hydrocarbon group and the molecular weight of the heptane-plus fraction. The databank used in this study, and crude oils in general, contains a low concentration of non-hydrocarbon components. This attempt resulted in a reduction in the number of variables in the proposed empirical model as follows: a4 a7 GaL5 Gam6 GC7+ µo ) a1Ta2Pa3γC7+
calculate the viscosity of Kuwaiti crudes resulted in an AAD of 45.7%. Therefore, the model has been modified to improve its accuracy for viscosity calculation of the Kuwaiti crudes. The model is described in the following equations:
φ(Tr,Pr) ) exp[Q2(xTr - 1)] + Q3(xPr - 1)2 (12) For hydrocarbons, carbon dioxide, and nitrogen, the three coefficients Q1, Q2, and Q3 were generalized in terms of acentric factor ω as follows, respectively:
When ω < 0.3 Q1 ) 0.829599 + 0.350857ω - 0.747680ω2
(13)
Q2 ) 1.94546 - 3.19777ω + 2.80193ω2
(14)
Q3 ) 0.299757 - 2.20855ω + 6.64959ω2
(15)
It was noticed that the functional forms of Q1, Q2, and Q3 for the C7+ fraction (i.e., ω g 0.30) play a critical role in the prediction of the hydrocarbon mixture viscosity. This is because the C7+ fraction comprises a major part of the crude oil. Hence, the following correlations were proposed in the present study for Kuwaiti crudes:
Q1 ) -0.5395ω3 + 0.9335ω2 - 0.4535ω + 18.71 (16) Q2 ) -180.71ω3 + 324.01ω2 - 175.96ω + 7.8352 (17) Q3 ) 23.96ω3 - 46.804ω2 + 28.846ω - 2.456 (18)
Ind. Eng. Chem. Res., Vol. 42, No. 17, 2003 4137
Figure 1. Error distribution by various correlations for estimating the viscosity of Kuwaiti crudes.
The EOS-based viscosity models were extended to mixtures by applying the following conventional mixing rules:
am )
∑i xiai ∑i xibi
bm ) bm b m′ )
(19) (20)
∑i ∑j xixjxbi′bj′(1 - kij)
(21)
∑i xiri
(22)
rm )
The parameters of EOS are considered to represent the attractive and repulsive forces between the molecules of different substances forming the mixture. For molecules that do not differ greatly in size or chemical structure, the binary constant kij can be set equal to zero. For binaries where both components fall into one of these categories (hydrocarbons, rare gases, permanent gases, and carbon monoxide), kij may be estimated by
kij ) 1 -
8(VciVcj)0.5 (Vci0.33 + Vcj0.33)3
(23)
4. Results and Discussion 4.1. Evaluation of the Previously Developed Models. Figure 1 shows a summary of the accuracy of four different empirical models in predicting the viscosity of Kuwaiti crudes. These models were tested using 910 experimental points from 49 Kuwaiti crudes. They are as follows: Beggs and Robinson (1975), Labedi (1992), Kartoatmodjo and Schmidt (1994), and Elsharkawy and Alikhan (1999) correlations. In these models, the composition of the crude oil is characterized by API gravity and the composition of the gas in solution is characterized by gas gravity. Figure 1 indicates that the Elsharkawy and Alikhan model has an AAD of 20.86%, which is the smallest among the other empirical models. The Beggs and Robinson model has an AAD of 26.4%, the Kartoatmodjo and Schmidt model has an
AAD of 32.11%, and the Labedi model shows the highest AAD with 33.45%. This is because Elsharkawy and Alikhan used Middle East crudes to develop their model. Those crudes have some physical and chemical similarities to the Kuwaiti crudes. Kartoatmodjo and Schmidt and Beggs and Robinson have used databank to develop their models, and Labedi used light crudes, which are quite different in physical and chemical properties from Kuwaiti crude oils. The strong weakness of these models is that paraffinic oil and aromatic oil having the same API gravity would have the same viscosity regardless of their composition. Therefore, these models cannot be used for calculations of the viscosity in compositional reservoir simulation and oils production by gas injection. Figure 2 shows a comparison between the measured and predicted viscosities by the previously mentioned models for Kuwaiti crude oil, sample no. 16. This figure also indicates that the Elsharkawy and Alikhan model closely matches the experimentally measured viscosity. None of the available models could predict the crude oil viscosity at all pressures with satisfying accuracy. Six different models, which use the composition of reservoir fluids rather than the gravity, have also been tested to predict the viscosity of 370 data points from the databank. These models are EOS-based models such as those of Lohrenz et al. (1964), Little and Kennedy (1968), and Guo et al. (1997 and 2001) and the corresponding state models by Pedersen et al. (1987) and Petersen et al. (1991). Figure 3 shows the accuracy in the estimated viscosity for 49 Kuwaiti crude oils at the bubble-point pressure using corresponding state models. Because the Pedersen model does not predict the true bubble-point pressure, it was adjusted to predict the actual bubble-point pressure. Surprisingly, the AAD by the adjusted model is higher than the original one. Figure 3 shows that the corresponding state models have relatively large AAD ranging from 39% to 51%. Table 7 shows a comparison between Guo’s work and the calculation of gas condensate and crude oil viscosities by programs made for this study. Our results agree with those reported by Guo et al. that the modified PR EOS-based viscosity model (MPR) is superior to all other methods in viscosity calculation. However, our results disagree with Guo and Lawal of viscosity calculations by Lohrenz and Little and Kennedy. Based on our
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Figure 2. Measured and predicted viscosities for Kuwaiti crude sample no. 16.
Figure 3. Error in viscosity calculations by corresponding state models for Kuwaiti crudes at bubble point. Table 7. Comparison of the Present Study to Guo’s Work for Gas Condensates and Crude Oilsa data ranges T range, °F P range, psi µexp, cP AAD %
Guo
present study
a
PR(Kij)0) MPR(Kij)0) Pedersen Lohrenz Little and Kennedy PR(Kij)0) MPR(Kij)0) MPR(Kij#0) Pedersen Lohrenz Little and Kennedy
gas condensates
crude oils
110-262 2000-5750 0.023-0.07 15.04 9.81
86-282 80-8590 0.11-2.62
18.50 16.66 16.77 15.90
16.32 33.37 167.93 47.25 58.95 17.70 17.73 27.96 26.00 29.43
Reference 18.
experience and previous studies, it is believed that our viscosity calculations by Lohrenz and Little and Kennedy are reasonable. Guo et al. and Lawal calculations of viscosity calculation using the Lohrenz and Little and Kennedy methods are questionable because they reported AADs of several hundreds. It is well-known that
the Lorenz and Little and Kennedy methods are widely used for viscosity calculations in most reservoir simulation packages. The comparison between Guo’s work and the present study indicates that the predicted results obtained by the MPR model in both studies are comparable to each other even though the characterization
Ind. Eng. Chem. Res., Vol. 42, No. 17, 2003 4139 Table 8. Summary of the Results of Estimated Oil Viscosity Using Various Modelsa data source unknownb UAE (Abu Dhabi)c North Sea crudesd Kuwaiti crudese Kuwaiti crudes at bubble point overall a
T range (°F)
P range (psi)
Np
PR(Kij)0)
MPR(Kij)0)
AAD % MPR(Kij#0)
Pedersen
Little
Lohrenz
86-282 190-239 156-238 130-243 30-243
80-8590 901-5000 115-5829 14.7-5000 1365-3730
14 53 99 146 49
58.95 67.84 62.96 70.68 73.41
17.70 24.00 16.93 45.71 35.76
17.73 24.16 17.06 45.80 36.1
27.96 36.40 36.74 42.31 38.71
26.00 35.13 174.34 103.78 30.83
29.43 72.90 67.33 51.37 43.83
361
68.06
32.20
32.34
38.87
100.13
57.03
Set 1: A Tc-Pc-ω correlation was used for the C7+ fraction. b Reference 18. c Reference 1.
Table 9. Comparison between the Proposed Model and Guo’s Worka present study (AAD %) data
Np
Guo’s model (AAD %)
empirical correlation
modified EOS-based viscosity model
Kuwait all
49 361
45.71 32.20
23.02 24.13
23.73
a
Pc, Tc, and ω for C7+ are calculated by characterization set 2.
procedures for the C7+ fraction in both studies are different. Guo’s work uses Tc and Pc correlations of Twu (1984) and the ω correlation of Kesler and Lee (1976), whereas this study uses characterization method no. 1. It is also noticed that BIP has an insignificant impact on the viscosity prediction of crude oils because the AAD is 17.7% when BIP was set to zero and the AAD is 17.73% when BIP was considered. Table 8 shows a summary of the results of viscosity calculations of 361 data points representing 77 crude oils from different parts of the world. These results indicate that the MPR EOS-based viscosity is more accurate than that of other methods. However, the model as it stands predicts the viscosity of Kuwaiti crudes with an AAD of 46%. Therefore, the model needs to be modified to predict the viscosity of Kuwaiti crude with reasonable accuracy. Results in Table 8 also indicated that the BIPs have little effect on viscosity calculations by the Guo et al. model. 4.2. Evaluation of the Proposed Models. The calculation presented in this section uses characterization method no. 2 for describing the C7+ fraction of crude
d
Reference 10. e This study.
oils to compare the performance of the proposed models presented in this study and that by Guo et al. Table 9 shows a comparison between the empirical model proposed in this study and the Guo et al. model for viscosity calculation of Kuwaiti crudes as well as others. This table indicates that the proposed empirical model has overall an AAD of 24% for all the crudes in the databank compared with 32% given by the Guo et al. viscosity model. Also, the empirical model is simple and reduces the tedious calculation involved in EOSbased viscosity correlations. On the other hand, the original MPR-viscosity model suggested by Guo et al. produces satisfactory results for gases, but produces unsatisfactory results for the 77 world crude oils (Kuwait, UAE, the North Sea, and others). The reason for good viscosity predictions by the original model is that the concentration of the heavy fraction in gases is rarely exceeding 10%. However, the hydrocarbon plus fraction forms the major portion in most crude oils (20-80%), and it is not properly defined in the model for Kuwaiti crudes. Therefore, the model was considered for modification. Calculation of the viscosity from the EOS-based viscosity model greatly depends on the critical properties of components forming the reservoir fluids. The critical properties of pure components are well documented. However, the critical properties of the heavy hydrocarbon fraction are usually estimated from correlations. The fitting parameters describing the C7+ fraction play a critical role in viscosity prediction by the EOS-based viscosity model. Therefore, these parameters have been modified to
Figure 4. Comparison between the measured and estimated viscosities for Kuwaiti crudes by the proposed empirical and Guo models.
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Figure 5. Comparison between the measured and estimated viscosities for Kuwaiti crudes by the proposed modification and original Guo models.
Figure 6. Measured and predicted viscosities for Kuwaiti crude oil sample no. 55.
properly describe the behavior of the Kuwaiti crudes. The modification has been achieved using an optimization routine. The proposed modifications resulted in a significant improvement in the viscosity prediction of Kuwaiti crudes. Table 9 also shows a comparison between the accuracy of the original EOS-based viscosity model by Guo et al. and that of the proposed modification in the model for 49 Kuwaiti crudes. The new fitting parameters have resulted in a reduction of the AAD from 46% to 23%. Figure 4 shows cross plots of measured and estimated viscosities by the empirical model presented in this study. It shows that predictions made by the proposed empirical are close to the unit slope line; however, the Guo et al. model underestimates the viscosities of Kuwaiti crudes and has a scatter below the unit slope line. Figure 5 shows another cross plot between the measured and calculated viscosities by the original Guo et al. model and the modification suggested in this study. This figure indicates that the cross plot of points
by the proposed modification is close to the 45° line and the Guo et al. model shows scatter and underestimation. Figure 6 shows a comparison between the measured and calculated viscosities as a function of pressure for one of the oil samples produced from oil fields in Kuwait. This figure depicts that the proposed empirical model and modification for Guo et al. closely match the experimentally measured viscosity. An example showing the compositional changes as a function of pressure during depletion of the oil reservoir is presented (Table 10). Figure 7 shows a comparison between the experimentally measured viscosities and the predicted ones by empirical models (eqs 3a and 3b) as well as the corresponding state model presented by Pedersen et al.11 This figure shows that both of the empirical models presented in this study successfully predicted the viscosity of the reservoir fluids as a function of the ever-changing composition at various stages of pressure depletion. Figure 7 also indicates that
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Figure 7. Example showing viscosity changes as a function of the composition during pressure depletion, sample no. 50. Table 10. Compositional Change of Crude Oil during Pressure Depletion for Sample No. 50 pressure, psi temp, °F composition, mol % CO2 C1 C2 C3 iC4 nC4 iC5 nC5 C6 C7 C8 C9 C10+ γC7+ MWC7+ MWC10+ viscosity, cP experimental empirical, eq 3a empirical, eq 3b corresponding state
3730 230
3529 230
3250 230
2750 230
2000 230
1250 230
500 230
15 230
0.668 47.894 10.376 6.096 0.717 2.68 2.788 1.688 2.749 2.455 2.208 1.985 17.697 0.84 218 259
0.668 47.894 10.376 6.096 0.717 2.68 2.788 1.688 2.749 2.455 2.208 1.985 17.697 0.84 218 259.65
0.651 45.247 10.319 6.211 0.739 2.781 2.921 1.774 2.914 2.622 2.367 2.137 19.317 0.84 218 260.19
0.616 40.081 10.175 6.433 0.784 2.983 3.191 1.948 3.25 2.959 2.688 2.442 22.452 0.84 218 260.74
0.541 31.093 9.761 6.776 0.863 3.341 3.683 2.266 3.871 3.575 3.274 2.997 27.96 0.84 218 261.09
0.425 20.278 8.821 7.022 0.951 3.758 4.298 2.666 4.673 4.375 4.035 3.715 34.982 0.84 218 261.25
0.22 7.542 6.098 6.449 0.992 4.076 5.005 3.151 5.772 5.512 5.137 4.766 45.279 0.84 218 261.36
0.003 0.063 0.165 0.408 0.113 0.583 1.265 0.925 3.112 4.386 5.301 6.173 77.503 0.84 218 264.54
0.25 0.23 0.24 0.23
0.255 0.23 0.24 0.23
0.26 0.24 0.25 0.25
0.28 0.27 0.28 0.23
0.32 0.32 0.33 0.38
0.41 0.40 0.41 0.51
0.62 0.54 0.56 0.65
1.15 1.65 1.69 1.40
the models match the experimental data better than the corresponding state model. 5. Conclusions (i) Several correlations that use oil API gravity as the input parameter and are widely used for prediction of the crude oil viscosity have been evaluated using a large databank. The results indicate that the Elsharkawy and Alikhan correlation has the best accuracy among the other correlations. However, these correlations cannot be used for compositional simulation and for oil recovery by gas injection. (ii) The current study also evaluated various EOSbased viscosity methods and corresponding state methods to calculate the viscosity of various crude oils. The results indicate that the Guo et al. model properly predicts the viscosity of gas condensates and unsatisfactorily predicts the viscosity of crude oils. (iii) The original Guo et al. model has been improved in this study by modifying its fitting parameters to
properly describe the heptane-plus fraction of the Kuwaiti crudes. The proposed modification results in a reduction of the AAD from 46% by the original model to 23% by the proposed modification. (iv) A simple compositional empirical model is also presented in this study. The new empirical model successfully predicts 361 measurements of the crude oil viscosity from a databank with an AAD of 23% for Kuwaiti crudes. The proposed model eliminates characterization of heptane plus, splitting of the heavy fraction, complex mixing rule, or inclusion of BIPs, which are needed for the EOS-based viscosity model. (v) An example is presented showing the viscosity change as a function of the ever-changing composition and pressure during the life of the oil reservoir. This example shows that the models presented in this study successfully matched the experimentally measured viscosity. (vi) Two methods for characterizing the hydrocarbon plus fraction of crude oil have been considered in this study. It was found that viscosity estimation by various
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methods is strongly dependent on the method of characterizing the heptane-plus fraction. (vii) The effect of including the BIPs in the viscosity calculation by EOS-based models has been investigated. The current work indicates that BIP has an insignificant effect on the viscosity calculation by EOS-based viscosity models. Acknowledgment The authors thank Dr. Karen Pedersen for providing the viscosity data of North Sea crudes used in this study. List of Symbols a, b, c, b′ ) equation of state parameters of eqs 1 and 2 am, bm, rm, bm′ ) mixture coefficient parameters of eqs 1 and 2 GC7+ ) mole fraction of C7+ GL ) mole fraction of light components, C1 + C2 Gm ) mole fraction of medium components C3-C6 Gnon ) mole fraction of non-hydrocarbons, N2, H2S, and CO2 kij ) binary interaction parameter MWC7+ ) molecular weight of heptane-plus fraction P ) pressure Pb ) bubble-point pressure Pc ) critical pressure Pr ) reduced temperature ) P/Pc Qi ) coefficient, i ) 1-3 r ) parameter in eqs 1 and 2 rc ) parameter in eq 8 Rs ) gas-oil ratio T ) temperature Tc ) critical temperature Tr ) reduced temperature ) T/Tc V ) molar volume Vc ) critical volume Zc ) critical compressibility Greek Symbols ω ) acentric factor τ ) functional form, eq 11 φ ) functional form, eq 12 R ) parameter for molecular size and density effect µc ) critical viscosity γC7+ ) specific gravity of C7+ µo ) viscosity of the crude oil, cP µ ) viscosity Abbreviations AAD ) average absolute deviation API ) API gravity ()141.5/γ - 131.5) ARD ) average relative error BIP ) binary interaction parameter DL ) differential liberation EOS ) equation of state MPR ) modified Peng-Robinson EOS PR ) Peng-Robinson PT ) Patel-Teja
Literature Cited (1) Moharam, H. M.; Fahim, M. A. Prediction of the viscosity of heavy petroleum fractions and crude oils using a corresponding states method. Ind. Eng. Chem. Res. 1995, 34, 4140-4144. (2) Monnery, W. D.; Svrcek, W. Y.; Mehrotra, A. K. Viscosity: A critical review of practical predictive and correlative methods. Can. J. Chem. Eng. 1995, 73, Feb, 3-40. (3) Guo, X. Q.; Wang, L. S.; Rong, S. X.; Guo, T. M. Viscosity model based on equations of state for hydrocarbon liquids and gases. Fluid Phase Equilib. 1997, 139, 405-421. (4) Beggs, H. D.; Robinson, J. R. Estimating the viscosity of crude oil system. J. Pet. Technol. 1975, 27, 1140-1149. (5) Labedi, R. Improved correlations for predicting the viscosity of light crudes. J. Pet. Sci. Eng. 1992, 8, 221-234. (6) Kartoatmodjo, T.; Schmidt, Z. Large data bank improves crude physical property correlations. Oil Gas J. 1994, 4, July, 5155. (7) Elsharkawy, A. M.; Alikhan, A. A. Models for predicting the viscosity of Middle East crude oils. Fuel 1999, 78, 891-903. (8) Hanley, H. J. M. Prediction of viscosity and thermal conductivity coefficients of mixtures. Cryogenics 1976, 16, 643-651. (9) Hanley, H. J. M.; McCarty, R. D.; Haynes, W. M. Equations for the viscosity and thermal conductivity coefficients of methane. Cryogenics 1975, July, 413-415. (10) Pedersen, K. S.; Fredenslund, A. Viscosity of crude oil. Chem. Eng. Sci. 1984, 39 (No. 6), 1011-1016. (11) Pedersen, K. S.; Fredenslund, Aa. An improved corresponding states model for the prediction of oil and gas viscosities and thermal conductivities. Chem. Eng. Sci. 1987, 42, 182-186. (12) Teja, A. S.; Rice, P. Generalized corresponding states method for the viscosities of liquid mixtures. Ind. Eng. Chem. Fundam. 1981, 20, 77-81. (13) Teja, A. S.; Thurner, P. A.; Pasumarti, B. The calculation of transport properties of mixtures for synfuels process design. Ind. Eng. Chem. Process Dev. 1985, 24, 344-394. (14) Petersen, K. A.; Knudsen, K.; Fredenslund, A. Prediction of Viscosities of Hydrocarbon Mixtures. Fluid Phase Equilib. 1991, 70, 293-308. (15) Lohrenz, J.; Bray, B. G.; Clark, C. R. Calculating viscosities of reservoir fluids from their compositions. J. Pet. Technol. 1964, 1171-1176. (16) Little, J. E.; Kennedy, H. T. A correlation of the viscosity of hydrocarbon systems with pressure, temperature and composition. Soc. Pet. Eng. J. 1968, 157-162. (17) Philips, P. Proc. R. Soc. London 1912, 87A, 48. (18) Lawal, A. S. Prediction of vapor and liquid viscosities from the Lawal-Lake-Silberberg equation of state. Presented at the 5th Symposium on Enhanced Oil Recovery, Tulsa, OK, Apr 1986; SPE/DOE Paper No. 14926. (19) Guo, X. Q.; Sun, C. Y.; Rong, S. X.; Chen, G. J.; Guo, T. M. Equation of state analogue correlations for the viscosity and thermal conductivity of hydrocarbons and reservoir fluids. J. Pet. Sci. Eng. 2001, Mar, 15-27. (20) Kesler, M. G.; Lee, B. I. Improve prediction of enthalpy of fractions. Hydrocarbon Process. 1976, 55, 153-158. (21) Edmister, W. C. Applied hydrocarbons thermodynamics Part 4. Compressibility factors and equations of state. Pet. Refin. 1958, 37, 173-179. (22) Twu, C. H. An internally consistent correlation for predicting the critical properties and molecular weights of petroleum and coal-tar liquids. Fluid Phase Equilib. 1984, 16, 137-150. (23) Uyehara, O. A.; Watson, K. M. A Universal Viscosity Correlation. Natl. Pet. News 1944, Oct, 714-723.
Received for review January 24, 2003 Revised manuscript received June 5, 2003 Accepted June 11, 2003 IE0300631
