AN AMERICAN CHEMICAL SOCIETY JOURNAL VOLUME 7, NUMBER 5
SEPTEMBER/OCTOBER 1993
0 Copyright 1993 American Chemical Society
Articles Prediction of Viscosity and Surface Tension of North Sea Petroleum Fluids by Using the Average Molecular Weight Hans Petter Ranningsen Petroleum Technical Laboratory, STATOIL a s . , Forus, N-4001 Stavanger, Norway Received December 8, 1992. Revised Manuscript Received May 11, 199P
Viscosity values of crude oils and other petroleum fluids are required in various engineering calculations, such as fluid flow in pipelines. There is thus an obvious need for correlations which can provide accurate estimates of viscosity when experimental data are lacking. Three different empirical correlations between the Newtonian viscosity of gas-free (dead or stabilized) crude oils and their molecular weights are presented in this work. The best correlation, which includes temperature as well, has been found to be valid for the great majority of North Sea crude oils and condensates in the molecular weight range from about 115 to at least 315. It is based on the Guzman-Andrade equation combined with a linear dependency of the flow activation energy and the constant on molecular weight. With the number-average molecular weight (from freezing point depression) as the only known information of the petroleum fluids, estimates of the Newtonian viscosity at temperatures from 30 to 80 O C within about 11% on average were obtained (viscosities between 0.5 and 30 mPa s). By using a viscosity measurement a t one temperature for adjustment, the average deviation is decreased to 3.4% . It is thus concluded that for most oils and for most practical purposes a quick and fairly accurate estimate of the viscosity of a stabilized oil at a range of temperatures can be obtained with a minimum amount of information about the actual fluid. It is further demonstrated that the viscosity correlation can be combined with an empirical relationship between the surface tension of crude oils and their viscosity, in order to obtain reasonable estimates of their surface tension. For a set of 10fluids of various types, the average deviation between predicted and measured surface tension at temperatures from 10 to 40 O C is 2.1 % . Since only North Sea oils are included in the data base of this work, the correlations cannot strictly be claimed to be valid for oils from other geographical areas. Introduction
The main objective of this paper is to demonstrate how the viscosity of a wide range Of petroleum fluids in the Newtonian temperature range can be correlated with a Abstract published in Aduance ACS Abstracts, September 1,1993.
minimum of analytical information, i.e., the numberaverage molecular weight of the fluid only. In connection with compositional analysis of crude oils, the molecular weight is measured routinely. The correlations are based on about 195 viscosity measurements at temperatures between 30 and 80 "C-of 35 North Sea petroleum fluids
0887-0624/93/2507-0565$04.00/00 1993 American Chemical Society
566 Energy & Fuels, Vol. 7, No. 5, 1993
with average molecular weights between 117 and 311 and viscosities between 0.5 and about 30 mPa s. In addition, surface tension measurements a t temperatures from 10 to 40 OC on 10 petroleum fluids (molecular weights between 147 and 2531, published elsewhere,1*2have been used to establish a correlation for the estimation of surface tension. The correlations presented in this work are restricted to stabilized oils at atmospheric pressure. Crude oils are highly complex mixtures of various classes of chemical compounds including saturated compounds (paraffins and naphthenes), aromatics, resins, and asphaltenes. The individual components within each class cover a very broad range of boiling points or molecular weights,from methane to high-boiling polycyclic aromatic compounds. The number-average molecular weight of a specific fluid is determined by the boiling point distribution as well as the group type distribution. All the components present somehow influence the physical properties of the fluid, such as the density, the viscosity, or the surface tension. This forms the basis for various models used to predict properties of petroleum fluids from a detailed compositional analysis combined with knowledge of the properties of individual components and a framework of constitutive equations. Typical examples are the model of Lorenz et al.3 and the corresponding state models of Ely and Hanley4 and Pedersen et a L 5 t 6 These models are founded on thermodynamic principles, are applicable to a wide range of pressures and temperatures, and generally provide quite accurate viscosity estimates for light, lowviscositymixtures but suffer from deficiencieswith regard to predicting accurately the properties of highly complex mixtures, particularlythose containing substantial amounts of high-boiling and polar material. A different approach is to develop empirical correlations based on compilations of experimental data. These correlations may not strictly have the same degree of theoretical foundation and are often restricted to a narrower range of pressure and temperature but may provide better predictions for certain mixtures. Bed7and Beggs and Robinson8have published some commonly used correlations for estimating the viscosity of gas-free oil at reservoir temperature from the API gravity of differentially stabilized oil. Beal reported an average deviation of 24.296 for 98 relatively low-gravity Californian oils while Beggs and Robinson found an average error of 114 % for 93 cases collected from the literature. Beal's correlation applied to the latter cases gave an average deviation of 378%.8 Labedie recently compared these correlations with a new one based on about 90 light Libyan crudes. His correlation, which required API gravity and reservoir temperature as input, was reported to give an average deviation for the viscosity at reservoir temperature of -2.61 96 for oils with viscosity in the range 0.66-4.79 mPa s. Bed7and Labedig also presented correlations for estimating the viscosity of (1)Pedersen, K.S.;Lund, T.; Fredenslund, Aa. J. Can.Pet. Technol. 1989,28,118-123. (2) Johansen, E. J.; Skjserve, I. M.; Lund, T.; Sjsblom, J.; Sederlund, H.; Bostrsm, G. Colloids and Surf. 1988/1989,34,353-370. (3)Lohrenz, J.; Bray, B. G.; Clark, C. R. J. Pet. Technol. 1964,Oct. 1171-1176. (4)Ely, J. F.; Hanley, J. M. Znd. Eng. Chem. Fundam. 1981,20,323332.
(5)Pedersen, K.S.;Fredenslund, Aa.; Christensen, P. L.; Thomassen, P. Chem. Eng. Sci. 1984,39,(6),1011-1116. (6) Pedersen, K.S.;Fredenslund, Aa. Chem. Eng. Sci. 1987,42,(l), IA2-18fi. - __ - - -. (7) Beal, C. Trans. AZME 1946,165,94-112. (8)Beggs, H. D.; Robinson, J. R. J.Pet. Technol. 1975,9,1140-1149. (9)Labedi, R. J. Pet. Sci. Eng. 1992,8,221-234.
Rsnningsen unstabilized oil from their dead-oil viscosity and bubble point pressure. DoolitlelO presented a simple and successful correlation between the viscosity and the average molecular weight of homologous series: = aMWb (1) where a and b are constants for a given series and MW was the molecular weight of a homologue. It will be shown below that at constant temperature, this simple relationship between the average molecular weight and the viscosity applies quite well even to highly complex mixtures such as crude oils. Doolittle and Petersonll also presented two specific correlations for pure n-alkanes in the temperature range -10-300 "C with an accuracy of about 1 % Recently, Aasen et al.12presented a somewhat more general model which included the temperature and also a composition dependency, which allowed prediction of the viscosity of mixtures of n-alkanes. Their model was described by a four-parameter equation. A standard deviation of 1.9-3.896 for predicted viscosities of n-alkane mixtures was reported. The following modified form of their equation is shown below to give reasonable estimates of the temperature variation of the viscosity. p
.
MW - 2.016 B + ( C / n 14.026 The simple two-parameter Guzman-Andrade equation: P =A&/(R~
(3) has previously been reported13to express the temperature variation of the viscosity of a series of North Sea crude oils remarkably well in the Newtonian temperature range (above about 30 "C). In eq 3, E, is viewed as the activation energy of Newtonian flow. Although the exact physical significance of the constant A is not quite clear, it is dependent on the entropy of activation of the In this work, both E, and A are shown to exhibit a regular dependency on the average molecular weight, giving rise to an empirical correlation which provides good estimates of the Newtonian viscosity of the majority of North Sea oils. A frequentlyapplied correlation for estimation of surface tension, y, is the one presented by Pelofsky:16
In y = a + b / p (4) Pedersen et al.' showed that eq 4 could be utilized to predict the surface tension of crude oil mixtures from their composition by assuming the constants a and b to be functions of aromaticity and molecular weight. In this paper, more surface tension data of North Sea oils2 are used to show that eq 4 or, even better, the following alternative empirical expression: = c + d log p (5) can be used in combination with the viscosity correlations presented, in order to obtain quite accurate estimates of y
(10)Doolittle, A. K.J. Appl. Phys. 1952,23, (4),418-426. (11)Doolittle, A. K.;Peterson, R. H. J. Am. Chem. SOC.1951,73, 2145-2151. (12)Aasen, E.;Rytter, E.; 0ye, H. A. Ind. Eng. Chem. Res. 1990,29, 1635-1640. (13)Rsnningsen, H. P.; Bjerndal, B.; Hansen, A. B.; Pedersen, W. B. Energy Fuels 1991,5, (6),895-908. (14)Bestul, A. B.;Belcher, H. V. J. Appl. Phys. 1953,24,(6),696-702. (15)Pelofsky, A. H. J. Chem. Eng. Data 1966,11,394-397.
Prediction of Viscosity and Surface Tension
oilno. O/Ca MW
Energy & Fuels, Vol. 7, No. 5,1993 567
Table I. Analytical Data of Crude Oils and Condensates Used in This Study p at 15 "C Clot* Czo+b aromaticsc asphdtenesd total waxe p (mPa e) (g/cm3) (wt %) (wt %) (wt %) (wt%) (wt%) WPTf 40°C 50°C 60°C 9.1 43.0 4.14 3.33 2.67 32.9 1.1 80.41 48.28 0.839 0.6 nm 0.54 0.52 0.46 0.0 4.98 16.3 0.746 43.00 9.7 39.0 5.29 4.05 3.25 26.7 1.4 50.70 0.855 84.44 1.36 1.08 0.95 7.6 43.0 21.4 0.7 62.55 24.04 0.808 37.0 4.12 3.37 2.74 3.8 11.5 45.48 36.6 77.81 0.849 30.5 4.68 0.6 2.2 8.17 6.16 80.80 42.4 0.884 91.20 38.5 3.59 2.92 2.31 0.2 6.7 29.8 82.54 82.54 0.842 5.46 4.11 3.38 13.8 41.0 33.8 0.1 53.83 0.844 81.88 1.25 48.0 1.79 1.52 0.2 7.5 28.85 26.7 0.819 72.88 15.6 40.0 7.88 5.94 4.73 28.7 1.2 89.87 64.70 0.861 3.35 0.4 14.8 40.0 5.54 4.26 56.15 29.0 0.856 89.12 6.03 4.44 3.61 0.7 14.5 32.0 39.4 0.857 85.43 53.40 1.33 33.0 1.81 1.52 8.5 26.3 0.0 0.816 72.16 36.09 1.79 1.52 1.29 8.8 33.0 0.4 35.44 27.0 72.3 0.817 0.93 0.84 0.74 2.7 22.0 0.0 19.03 25.8 0.788 59.42 20.0 2.25 1.89 1.59 0.1 8.4 20.0 77.42 25.80 0.817 2.06 1.72 1.54 8.5 44.0 0.6 36.77 22.0 0.805 75.34 35.5 8.31 6.18 4.68 1.5 14.1 38.6 60.57 0.875 87.79 1.34 1.89 1.62 41.0 0.4 7.0 20.3 73.38 35.68 0.806 0.90 35.0 1.25 1.02 27.8 0.0 5.7 64.59 25.11 0.803 22.0 46.60 29.30 19.30 1.5 2.7 50.1 97.39 65.00 0.933 23.0 28.90 20.20 13.80 0.6 1.8 41.3 97.80 67.50 0.907 22.0 27.60 19.00 13.30 0.7 2.2 67.10 41.3 0.908 97.40 0.55 nm 0.68 0.59 0.0 0.0 22.3 51.80 1.93 0.779 5.90 4.68 3.85 8.4 38.0 31.3 1.0 89.25 53.10 0.859 0.60 0.2 nm 0.77 0.68 0.0 3.80 21.5 0.794 60.36 nm 0.62 0.55 0.51 0.0 0.0 20.0 46.08 1.00 0.777 33.5 5.27 4.17 3.24 0.6 11.1 57.00 26.4 0.862 86.60 0.42 0.2 nm 0.54 0.46 0.0 2.67 20.1 0.750 36.50 4.25 nm 2.70 8.0 34.5 0.5 35.0 0.852 84.23 46.96 4.74 nm 2.40 7.0 38.0 35.0 1.1 0.841 80.28 45.34 2.66 33.0 4.20 3.31 0.5 5.7 39.0 0.847 81.78 47.32 1.68 1.41 1.16 35.0 0.4 4.7 22.3 70.56 29.97 0.805 32.0 1.57 1.28 1.03 0.2 6.3 18.6 69.32 29.57 0.796 6.29 39.0 11.62 8.56 18.3 37.5 4.2 59.14 0.873 88.62
at 70°C 2.26 nm 2.69 0.85 2.26 3.95 2.01 2.76 1.11 3.80 2.69 2.97 1.15 1.12 0.65 1.37 1.23 3.82 nm 1.14 13.50 10.52 10.21 nm 3.76 nm 0.45 2.67 nm 2.35 2.07 2.18 0.99 0.90 5.02
80°C 1.93 nm 2.25 0.78 2.03 3.06 1.71 2.37 0.94 3.14 2.31 2.49 1.04 0.97 nm 1.20 1.09 3.09 1.04 nm 9.88 7.89 7.49 nm 2.85 nm 0.41 2.23 nm 1.91 1.74 1.98 0.90 0.83 4.05
198 117 213 149 191 248 0 210 0 214 C 172 0 253 0 241 12 0 224 13 C 167 167 14 C 15 C 140 186 16 0 17 0 176 18 0 253 179 19 C 20 C 148 311 21 0 307 22 0 23 0 307 24 C 127 25 0 229 26 C 133 123 27 C 28 0 233 29 C 118 30 0 217 31 0 205 32 0 204 169 33 0 167 34 C 35 0 198 a Oil/condensate. b From TBP distillation. In Clo+. d Pentane insolubles. e Acetone precipitation.'S f Wax precipitation temperature (polarization microscopy).13 nm = not measured. 1
2 3 4 5 6 7 8 9 10 11
0 C 0 C 0 0
the surface tension of stabilized crude oils without knowing the detailed composition. Experimental Section Fluids Investigated. We used 35 North Sea crude oils and condensates from various fields on the Norwegian continental shelf in the viscosity part of this study. Some additional oils are included in the part on surface tension. The fluids covered a wide range of fluid compositions and physical properties including "normal" paraffinic oils, biodegraded oils, waxy oils, light oils and condensates. The common designation "oil" will mostly be used in the text. The oils were either stock tank oil or oil obtained by one-stage flash separation of separator oil to ambient conditions. In both cases they are referred to as stabilized oils. The oils were thermally preconditioned a t 80 "C in a gas-tight cell to dissolve precipitated wax.13 A summary of relevant analytical data of the oils is provided in Table I. Molecular Weight Measurement. Molecular weights were determined by freezing point depression of water saturated benzene using a petroleum cryoscope (Cryette A Model 5008, Precision Instruments). The average molecular weight MW, of a crude oil sample is given by
where Kf is the molal freezing point depression constant, which for benzene is 5.12, w, is the weight (g) of crude oil, W B is the weight of benzene, and S (= lo00 AT) and So(= 0) are the readings on the instrument for sample and water-saturated benzene, respectively. This is a number-average molecular weight as the freezing point depression depends on the number (or moles) of
solute molecules per unit weight of solvent. The instrument waa calibrated with 0.05 m solutions of n-Cl, and checked regularly with n-C9 (MW 128) and n-Cu (MW 198). Acceptable deviation is considered to be within 1 MW unit. Sample solutions are prepared with approximately the same molal concentration based on an estimated molecular weight. The repeatability of replicate measurements typically is within 0.5-1 % Interlab precision has been found to be within 1-2 % . The measured molecular weights ( m o b ) were corrected (me) for a certain content of benzene using the following equation:
.
(7) where X B is the weight fraction of benzene (typically 0.1-1.0 wt %). The reported molecular weights have not been corrected for water content in the oils (typically 0.1-0.3 wt %). Density Measurement a n d Calculation. Crude oildensities were generally measured at 15 OC using a high-precision digital frequency density meter (DMA 40 with DMA 602 M measuring cell, Anton Paar KG). The measuring principle is based on the change of the natural frequency of an oscillating hollow cylinder when filled with different fluids. The instrument was calibrated with air and air-free water. The accuracy of the measurements isk2 x 106g/cm3ifwithinO.lg/cm~ofthedensityofthestandard (water) and 5 X l(r if within 0.5 g/cm3. Densities a t temperatures other than 15 OC were calculated from measured densities at 15 "C using API standard 254O.le This table assumes a standard (16) Manual of Petroleum Measurement Standards, 1st ed.; API Standard 2540; American Petroleum Institute: Washington, DC, 1980; Vol. 111, Table 53A. (17) Knapstad, B.; Skjelsvik, P. A.; Bye, H. A. J. Chem. Eng.Data 1989,34, 37-43.
568 Energy & Fuels, Vol. 7, No. 5, 1993
h
5
u)
2
4.
u -p
Rsnningsen Table 11. Fit of Crude Oil Viscosity to Ea 1. temDerature PC) 80 70 60 50 40 a 0.2406 0.2192 0.2357 0.2472 0.2568 b 2.783 3.139 3.284 3.494 3.714 avdev (%) 10.8 10.7 11.1 12.9 14.2 std dev 0.231 0.299 0.363 0.452 0.650 (mPa s) ~
=
0.2192(!!Y;'" 100
5 , STANDARD DEVIATION 3-AVERAGEDEVIATION
o 30 m ~ sa 107%
Oils 21-23 excluded in the regression. Equation 1: p = a(MW/ 100).*
9 6 0
___________~ 150 200 250 MOLECULAR WEIGHT
Table 111. Fit of Crude Oil Viscosity to Eq 2.
300
A
B C
Figure 1. Typical relationship between viscosity and average molecular weight of a series of North Sea crude oils and condensates. The biodegraded oils 21-23 are not included. thermal expansion coefficient for oils while it is likely to depend somewhat on the composition, particularly for the volatile components. However, for some oils studied, densities at 50 "C calculated from measured densities at 15 O C have been found to agree with measured densities a t 50 "C within 0.5 % . Viscosity Measurement. Kinematic viscosity (v,centistoke cSt = 10-2 mm2/s) in the Newtonian temperature range was measured using Ubbelohde glass capillary viscometer tubes with appropriate capillary constants. Measurements were conducted at 10 "Cintervals between 30 and 80 "Cin a Lauda thermostated bath. Temperature was measured with a Fluke 2180A digital thermometer. The average was taken of three or four parallels at each temperature. Repeatability was generally within 0.51% relative. The dynamic viscosity (p, centipoise cp = mPa s) at a given temperature was calculated from the measured kinematic viscosity and the calculated density ( p , g/cm3)at that temperature (see above): /J
= UP
(8)
Surface Tension. Surface tension measurements were conducted by using an MGM Lauda ring tensiometer. For further details, see refs 1 and 2.
Results and Discussion Relationship between Crude Oil Viscosity and Molecular Weight at Constant Temperature. As mentioned above, eq 1 has been shown to be valid for homologous series such as n-alkanes at constant temperature. As crude oils are very complex mixtures containing thousands of compounds, one might anticipate such a simple relationship between viscosity and average molecular weight to fail completely for such fluids. However, eq 1was found to represent well the viscosity- molecular weight relationship at temperatures from 40 to 80 "C for the oils used in this study. A typical example (at 70 "C) is shown in Figure 1. Only some highly biodegraded, aromatic/naphthenic oils (nos. 21-23) had to be excluded. Besides, a very asphaltenic crude (15 wt ?% pentane insolubles), not included in this study, has been found to be highly underestimated by eq 1. The viscosity data of the remaining 32 oils were fitted to eq 1. The regression coefficients at each temperature are summarized in Table 11. Note that MW/100 is used instead of MW here. The average deviation between measured and calculated viscosity is 10-15%. The deviation is largest at the lowest temperature, which may be due to solid wax formation in some oils below 40 "C. Prediction of Crude Oil Viscosity in the Newtonian Temperature Range from the Molecular Weight. An (18)Rsnningsen, H. P. Energy Fuels 1992,6, 870.
0.0oO 305 4 0.8767 829.5
overall
av dev (%) 11.18 std dev (mPa s) 0.418 max dev ( % ) -33.8 40 "C avdev (%) 11.24 std dev (mPa s) 0.666 50 "C avdev(%) 10.23 std dev (mPa s) 0.441 avdev (%) 11.79 60 O C std dev (mPa 8) 0.342 70 O C avdev(%) 11.39 std dev (mPa s) 0.289 80 "C av dev (%) 11.20 std dev (mPa s) 0.276 a Oils 21-23 excluded in the regression. Equation 2: p = A[(MW - 2.016)/14.026](B+C/~.
attempt was made to include the temperature variation of crude oil viscosity by using the equation proposed by Aasen et a1.12 The parameter p in their equation was set equal to zero, giving eq 2. Again, oils 21-23 showed large deviation and were thus excluded. The constants and coefficientare summarized in Table I11together with some statistical data. The calculated and measured viscosities are compared in Figure 2. The overall average deviation, i.e., all temperatures included, is seen to be 11.2 % . There is no particular temperature dependency of the deviation. In summary, eq 2 with the coefficients in Table I11 (temperature in degrees K) could be used to estimate the Newtonian viscosity at temperatures between 40 and 80 O C . However, it did not apply to biodegraded oils, which were highly underestimated. The Arrhenius equation, eq 3, can be linearized as follows: (9)
where R is the gas constant (8.3144 J K-' mol-l). The activation energy of viscous flow is hence proportional to the slope when the logarithm of the viscosity is plotted against the reciprocal of the temperature while log A is the intercept. (10) As discussed in a previous paper,13 the onset of deviation from a straight line indicates the onset of wax precipitation and thus non-Newtonian flow behavior. Based on the viscosity data of the oils investigated (Table I), the flow activation energies and intercepts (i.e., log A ) have been calculated by linear regression. They are summarized in Table IV. Except for two of the oils, the relative standard error in the viscosity estimate is seen in most cases to be well within 10% ,on average 5.1?% . The activation energies
Energy & Fuels, Vol. 7, No. 5,1993 6669
Prediction of Viscosity and Surface Temion
9
5
Y
2 4 6 8 10 MEASURED VISCOSITY (mPe 8 )
'0
12
Table IV. Flow Activation Energies (a) and Arrhenius Constantr of Oilr Investignted* oilno. E,(kJ/mol) -1ogA RSD"(%) rp 18.4 9.1 21.8 12.6 17.2 23.0 17.6 20.6 16.0 21.1 20.4 21.9 13.6 14.1 13.9 14.9 16.1 23.8 14.4 12.6 37.4 31.0 30.8 10.1 20.7 10.6 10.2 20.6 8.9 21.1 1 8.1 ~. 17.9 14.6 14.8 24.3 ~
6.46 4.96 6.90 4.97 6.26 6.92 6.37 6.68 6.26 6.62 6.66 6.86 4.99 6.10 6.34 6.12 6.20 6.06 6.12 6.02 7.58 6.71 6.68 4.85 6.74 4.89 4.89 6.71 4.77 6.85 5.45 6.37 6.21 6.28 6.99
6.6 6.3 6.2 9.6 6.9 3.2 8.4 4.7 2.1 1.6 2.6 12.4 6.0 0.7 14.9 2.3 6.3 3.7 2.1 6.6 4.6 8.1 7.6 4.9 9.1 1.6 1.4 2.0 4.7 8.4 2.2 7.6 2.4 4.0 2.4
0.997 0.987 0.988 0.973 0.996 0.997
0.983 0.991 0.997 0.998 0.996 0.989 0.996
1.OOO 0.976 0.999
0.994 0.996 0.996
0.992 0.997 0.998 0.998 0.996 0.993 0.999
0.998 0.998 0.992 0.983
0.998 0.994 0.996 0.990
0.998
Regmaion equation: log p = a + b/Twhere a = logA and b =: EJR; T = 30-80 O C or 40-80 O C . bRelative standard error in the viecosity h a t e .
ranged from about 9 kJ/mol for the lightest condensates to about 37 kJ/mol for one of the biodegraded, highmolecular-weight oils. A correlation thus existed between the flow activation energy and the molecular weight, which is reasonable since higher energy barriers have to be overcome in order for flow to occur in the presence of large polyaromatic molecules as opposed to small paraffinic molecules. As illustrated in Figure 3,the correlation was linear to a very good approximation:
E, = -5.097 + 0.1171MW
I
125
175
225
275
315
MOLECULAR W E l M
piotve 2. Comparison of measured viecositiea and viscoeitiea calculaM using eq 2. Oils 21-23 were not included.
1 2 3 4 6 6 7 8 9 10 11 12 13 14 16 16 17 18 19 20 21 22 23 24 26 26 27 28 29 30 31 32 33 34 36
4.5
-
(11) with a correlation coefficient squared 1.2 = 0.9214, an auerage error in the E. estimate of 1.20 kJ/mol(6.29%) and a standard error of 1.76kJ/mol. The flow activation
Figure 8. Relationehip between the flow activation energy El andthemolecular-t. Aeerieaofnalkaneaared~~hdd in the plot. It can be noted that E. for light oib and ocmden" (MW < 150) ia comparable to the value for the corresponding n-alkanes,while for heavier oile, their El ia higher,probablydue to increasing content6fpolycyclicaromaticamaticswithincressiag.MW. energies of some n-alkanes are included in the same figure based on data from refs 11,12,and 17.(It should be noted that the flow activation energies of some single compounds reported in Table M of ref 13 should be multiplied by In 10,Le., 2.303. A corrected table appeared in ref 18.) It can be noted that the lightest condensateshave activation energies comparable to the n-alkanes with comparable molecular weight, while for increasing molecular weight, the crude oils have higher activation energies than the correspondingn-alkanea. This probably reflects the higher relative amount of aromatics (and naphthenes) in the heavier oils. It also means that information about the group type distribution is indirectly contained in the molecular weight. A significant correlation has in fact been found to exist between the molecular weight and the aromaticity of the Clo+ fraction. This may explain why such a close estimate of flow activation energy,ae e x p d by eq 11,can be obtainedwithout any apeciffc information about the aromaticity. There may, of course, be exceptions, as indicated by the biodegraded oil 21 which is underestimated by about 6 kJ/mol. However, it should be emphasized that nearly all kinds of oils encountered in our laboratory so far are among the 35 oils studied here. It could be mentioned that due to a close correlation between the average molecular weight and the density, a reasonably good correlation between E,and density existed as well, with an average error of about 9%. The constant term log A in eq 9 also exhibited a regular variation with molecular weight or, equivalently, with E. due to the proportionality between E . and MW (eq 11). This correlation, which is illustrated in Figure 4, could be expressed by the following linear regreasion equation: log A = -3.8587 - 0.09137E. (12) with $ = 0.9691 and a standard error in the log A estimate of 0.106. It could be noted in Figure 4 that for the lowest flow activation energies, i.e., the lowest molecular weighta, log A for the oils was comparable to the corresponding n-alkane values. For the n-alkanes, however, log A is seen to be more or less constant with increasing E .. By combining eqs 9, 11, and 12, the following nonlinear equation is obtained
+
log M = k, + kJbW k$T + (k,MW)/T (13) which gives the viscosity (in mPa 8 ) at a temperature T (in K)when the average molecular weight is known. The
Rsnningsen
570 Energy & Fuels, Vol. 7, No. 5, 1993 330
8
= I
zE 25
20
0
P 15
> n
4 =
2 -5
10 15 20 25 30 35 FLOW ACTIVATW W R G Y (kJ/mol)
40
10 5 '0
5 10 15 20 25 MEASUREDVISCOSITY (mPa 8 )
30
FigureC Relatio~be~~nthellrrbegiturcoaetantandthe Figure 6. Comparison of measured Viscosities and viacoaitiea calculated using eq 13 and correctad by means of one Viecoeity flow ndivation energy. Due to the relatisplrhip in Figure 3, the measurement (at 60 "C). Average deviation was 3.4%. conotantAiaaleotoagoodapprb.imationdefinedbytheaverage m o w weight of an oil.
T a W V. Agreement b t w e s n Munured 8pd Coloulsted
Viroorities at Various Temperatures Uring Eq 13. with and without Correctionb
no. of oils no. of data points temperature range ("0 Viscosity range (mPa 8 ) ki
ka ka kr
35 165 40-80 0.5-30 -0.07995 -0.01101 -371.8 6.215
without oonectn with correctn 11.39 3.43 av dev ( % ) std dev (mPa e) 0.498 0.200 maxdev(%) -36.2 +9.6' 40°C avdev(%) 12.3 3.8 0.772 0.284 etd dev (&a e) 11.5 2.2 50°C avdev(%) 0.571 0.159 etd dev (&a 8 ) 8OoC avdev(%) 11.6 std dev (mPa 8 ) 0.441 70 O C avdev (%) 10.5 3.1 0.326 0.190 std dev (mPa s) 10.8 5.9 80OC avdev(%) 0.289 0.205 std dev (mPa 8 ) "logp = k l + k2MW + ks/T + k&fW/T. See discuseion in text. 0 One data point (oil 21 at 40 O C ; dev 14.7%) was excluded. overall
-
CALCULATEDVS. MEASUREDVISCOSITY
n
k
gE u
25 ~-
AVERAGE DEVMTION 11 4%
/( A
20 15
p
10
43
5
3
1
STAWARD DEVIATION 0,520 mPs 6
'0
5 10 15 20 25 ~MEASUREDviscosi+i (mPa $1
30 -_
Figure 5. Comparison of measured viscosities and viscosities calculated using eq 13. Average deviation was 11.4%.
values of the constant kl and the coefficienta k z k d , obtained by fitting all the viscosity and molecular weight data in Table I to eq 13 by a nonlinear regression routine, are given in Table V. In Figure 5 the calculated (using eq 13) and measured viscosities of all 35 oils at temperatures from 40 to 80 OC are compared. It should be noted that the biodegraded
oils 21-23 are included in thb plot. With the exception of one oil (no. 21), the correlation is seen to provide quite accurate estimates in the entire viscosity range0.5-30da 8. The average deviation between measured and calcuhtad viscosities (oil 21 excluded) was 11.4%. The maximum deviation was -36%. The deviation is seen in Table V to be slightly larger at the lowest temperatures. The agreement illustrated in Figure 5, is quite remarkable, considering the comp€exityof the oils and that only the molecular weight is used to obtain the viscosity. As indicated above, a correlation might be established with the density replacing the molecular weight, due to a dependency of E, on density similar to eq 11. This was, however, found to be less accurate and is not elaborated any further here, although density is more easily obtained than molecular weight, particularly in the field. Prediction of Newtonian Viscosity by Using Eq 13 a n d One Viscosity Measurement for Correction. If one viscosity measurement, for instance at 60 O C , is used to correct the calculated viscosity, the viscosity estimates at all other temperatures can be much improved. This effectively corrects for deviation in group type distribution from an average oil. A corrected log p at temperature T is then obtained by adding to log p, calculated by eq 13, the difference between calculated and measured log p at 60 OC. Corrected calculated and measured viscosities at all temperatures are compared in Figure 6. As seen in Table V, the average deviation between measured and calculated viscosities (includingall 35 oils) was then 3.4 % , and with the exception of oil 21 at 40 O C (+14.7%), the maximum deviation was +9.6%. It can thus be concluded that very accurate estimates of the viscosity of most kinds of North Sea oilsand condensates in the temperature range 40-80 O C can be obtained if the molecular weight is known and one viscosity measurement is made (but as discussed above, quite satisfactory estimates are obtained even without a viscoeity measurement). Thie may be very useful in cases where limited amount of oil is available or when a rough estimate is required quickly. For most purpaees where oil viscosity estimatea are required, such as design of pipelines or process equipment,the calculated viscosities are believed to be accurate enough. Table VI compares measured and calculated viscosities of three oils that were not included in the set of 35 oils, one rather asphaltenic oil, one waxy oil, and one condensate. Quite satisfactory estimates in the Newtonian temperature range are obtained, particularly when the special characters of these oils are taken into account. Oils 36 and 37 would normally be considered as being among
Energy & Fuels, Vol. 7, No.5,1993 871
Prediction of Viscosity and Surface Tension
Table VI. Prediction of the Viscosity at Variour Temperature8 of Three North Sea O W Not Included in the Data Bue oil 36 oil 37 oil 38 MW 241 256 159 0.877 0.872 0.805 p at 15 "C (g/cma) 87.16 89.91 65.90 wt % ClW 13.4 24.0 5.3 total War (wt %) asphaltenes (wt % ) 5.1 0.96 0.0 ~~
viscosity (mPa s) calcd dcd calcd calcd calcd measd correctedb measd 413 correctedb measd eq13 correctadb 3.63 3.37 3.31 80 O C 3.56 3.01 0.766 0.809 0.755 70 O C 4.41 4.26 3.94 4.86 3.85 0.857 0.925 0.863 60OC 5.48 5.05 5.48 5.00 5.93 5.00 0.995 1.06 0.996 50 "C 7.28 6.60 7.15 7.83 6.60 7.27 1.12 1.24 1.16 9.71 8.76 9.60 12.8' 10.52 8.88 40 "C 1.48 1.46 1.36 30 "C 13.3' 11.9 12.8 34.2' 14.4 12.2 2.W 1.73 1.61 aO il 38 adtually was a wary condensate. b Measured viscosity at 60 "C used to correct the calculated value (see text). The oil was nonNewtmien at these temperatures (due to wax formation).
calcd eq13 3.10 3.93
STAWARD DEVIATION 0 597 mP(
s
" C ........
25 20
.........
15
' 10 c,
OO'
2
4 6 8 i o 12 14 if3 MEASURED VISCOSITY (mPa 8)
5 n i o 0 125 150 175 200 225 250 275 300 325 350
18
Figure 7. Comparison of measured viscosities at 30 O C and viscosities predicted using eq 13 and corrected by means of one viscosity measurement (at 60 "C).
the more problematic. The waxy oil (no. 37) showed pronounced non-Newtonian (shear-thinning)behavior at 30 and 40 "C. The experimental viscosities at these temperatures, which actually are the apparent viscosities a t high shear rate (500a-9, are thus seen to be higher than the calculated values, particularly at 30 OC. The Newtonian viscosities of this oil were somewhat more overestimated than normal, due to the high paraffinicity and thus higher molecular weight than an average oil with comparable boiling point distribution. As illustrated in Figure 7, in most cases good estimates of the viscosities at 30 "C are predicted by eq 13. Some data are seen to be somewhat underestimated. These are typically those which exhibit non-Newtonian behavior at this temperature. Although, strictly, the 'constants" E, and A in eq 3 are constant over limitedtemperature ranges, the correlation given by eq 13 probably can be extrapolated to also provide reasonable predictions at temperatures somewhat higher than the upper temperature included in the data base, i.e., 90-100 "C. This may be very useful since measurements at such high temperature are complicated by severe evaporation of light ends in open systems. At temperatures lower than 30 "C, the correlation is likelyto fail for many oils due tonon-Newtonianbehavior below the wax precipitation temperature. But light oils md condensates are often still Newtonain down to 10 "C and sometimes lower. The correlation should, however, be used with caution at temperatures outrside the range 40-80 "C,particularly at temperatures below 30 OC. Since only North Sea oils have been included in the data base of this work, the correlationsmay not be generally
' O O' 80 C C
' ~
MOLECULAR WEIGHT
b
4.0 3.5
3 3.0 I?
E 2.5
v
g
2.0
u)
0 1.5 0 1.0
70 *C
g
0.5 - 0 . 0
100110 120130140 150160170 180190200
MOLECULAR WEIGHT
Rgure 8. Charta based on eq 13for prediction of the viscosity of a petroleum fluid from ita average molecular weight. Figure 8b is just an expanded view of the low-viscosity part of Figure 8a.
valid for oils from other geographical areas. On the other hand, since so many kinds of oils are represented in the set, it seems likely that they would have quite general validity. Possible exceptions are very heavy oils or extremely waxy or biodegraded oils. Apart from such extreme cases, reasonable predictions are likely to be obtained in most cases. The correlation represented by eq 13 is illustrated graphically in Figure 8, parts a and b. The viscositycorrelationspresented in this paper might somehow be included as a sort of second ureferencemin addition to methane in the corresponding state models mentioned in the Introduction, in order to incorporate a better predictive capacity for stabilized and nearly stabilized oils with rather high viscosity,which are not treated satisfactorily by these models.
R~nningsen
672 Energy & Fuels, Vol. 7, No.5,1993
Tab& VII. Meamred and Cdauhted Surface Tenrion of Nine Crude Oilr (A-I) and One Condenante (J). m d surface temp calcd surface tension oil M W ( O C ) (mN/m) eq4b %dev eq6 %dev 28.7 -1.9 28.8 -1.6 29.3 A 244 10
B
210
C
202
D
244
E
254
F
226
G
253
H
241
I
196
J
147
av ( a b % )
20 30 40 10 20 30 40 20 30 40 10 20 30 40 20 40 20 30 40 20 30 40 20 30 40 20 30 40 10 20 30 40
28.2 27.7 27.0 27.2 26.3 26.9 26.3 27.4 26.1 26.2 28.9 28.5 27.8 27.0 30.0 27.7 29.9 26.8 25.2 29.0 28.2 27.7 29.0 28.3 27.7 25.9 25.3 24.9 25.4 24.3 23.7 23.3
28.3 27.7 27.0 27.3 26.5 26.6 24.5 25.9 24.8 23.7 28.7 28.3 27.7 27.0 28.6 27.5 27.5 26.8 25.9 28.6 28.1 27.5 28.2 27.6 26.9 25.3 24.2 23.0 19.3 17.8 16.2 14.7
0.3 0.1 0.1 0.4 0.8 -1.2 -3.2 -5.6 -4.8 -6.1 -0.6 -0.7 -0.2 0.1 -4.6 -0.6 -8.0 -0.1 2.7 -1.4 -0.4 -0.8 -2.8 -2.5 -3.0 -2.2 -4.3 -7.8 -23.9 -26.8 -31.6 -37.1 2.33
28.1 27.4 26.8 27.0 26.4 25.9 25.3 26.0 25.5 25.0 28.8 28.1 27.4 26.8 28.6 27.2 27.2 26.6 26.0 28.6 27.8 27.2 28.0 27.3 26.7 25.7 25.2 24.7 23.6 23.3 22.9 22.6
-0.3 -1.0 -0.8 -0.7 0.4 -0.2 0.1 -5.1 -2.4 -0.9 -0.2 -1.4 -1.3 -0.8 -4.7 -1.7 -9.0 -0.8 3.2 -1.5 -1.2 -1.9 -3.6 -3.6 -3.7 -0.7 -0.4 -0.7 -6.9 -4.2 -3.3 -3.0 2.15
0 M d data for oils A-G and J is taken from ref 1, and data for oils H-I from ref 2. b Condenaate J excluded in the regmaion.
Using the Viscosity Correlation To Obtain a n Eetimote of Crude Oil Surface Tension. For many substances, a linear relationship has been found to exist between the logarithm of the surface tension and the reciprocal of the viscosity at a given temperature, i.e., eq 4. Surface tension and viscosity measurements at temperaturea from 10 to 40 OC from refs 1 and 2 are summarized in Table VII. For ~ o m eof the oils, which beha+ dhtinctly non-Newtonian at low temperatures, only data at 30 and 40 *Cwere u d . Using these data, WB found eq 4 to be valid to a good approximation for the nine North Sea crude oils examined. The condensate, however, did not fit into the same straight line as the oils but had higher surface bnsion than predicted from ita viscosity by extrapolating the crude oil relationship. It turned out that a somewhat M e r e n t empirical relationship between aurface tension y and viscosity of the crude oils,given by eq 6, gave an equally good fit and, furthermore, included the condensate as well (except possibly the point at 10 "C). This is illustrated in Figure 9. c and din eq 6 were determined by linear regression (y in mN/m and p in mPa a): c = 37.028; d = 4.900; 1.2 = 0.9110; standard error in y estimate = 0.53 Coneidering the physical mechanisms behind surface tension, it b rather surprising that such a simple relationship to viscosity exieta. One would expect the detailed composition, particularly with regard to high-molecular-
31 ,
A-
MEASURED y (mN/m)
Figure 9. Surface temion and viscoeity data of 10 North Sea oila used to illwtrata the empiridrelationehip e x p d by eq 5. Thestraightlineiatheblinearfk Notethatthecondenaate to a good appraximation fits into the name cchd&a' n a r t h e crude oil& Combinedwiththeviecoaity correlationh 13,thie relatiomhip could be utilized to provide eatimataa of crude oil surface tension.
VISCOSITY (Pa 8 )
Figure 10. Compnrieon of meaaured surfacetenaion of the seme 10North Sea petroleum fluids aa in Figure 9 and surface tension calculated by substituting eq 13into eq 6. C = condeneabe. See Table VII. The straightline is not avian line, it mpraseatb exact correspondence between meesured and calculated aafa
weight surface-activecomponents, to be highly important. But obviously, sufficient compositional information ie hidden in the bulk viscosity, for the equation to be valid to a good approximation. Having verified empirically that the surface tension of crude oils is simplyrelated to log p and that log c is related to the molecular weight by eq 13, we are able to &date the surface tension of a variety of crude oils knowing the average molecular weight only, by eubstituting eq 13into eq 5. Since the viecosity is proportional to e raisedto the right-hand side of eq 13,the exponedial andthe logarithm in eq 5 will cancelto give an empirical,non-linsar equation for the surface h i o n of exactly the same form as that of eq 13, only with other coefficients. Measured and calculated surface teneions are compared in Figure 10. TheaveragedeviationiseeninTableWItobeonly2.16%, the maximum deviation being 9.0%. There is seen to be a slight negative bias in the calculated values. There may be various reasons for thie. First, the viscositycorrelation is a c t d y used outside its validity range at the loweat temperatures. Secondly, although them oils appeared to be Newtonian even at the lowest temperatures, there "y be small n o n - N " a n' effects causing an under&tion of the v i s d t y and hence the surfacetension. Finally, there might be a slight systematic difference between the viscosity measurements in refs 1and 2 and those umd to establishthe viscoeity-molecular weight correlation in this
Prediction of Viscosity and Surface Tension
Energy & Fuels, Vol. 7, No. 5,1993 673
provide the above equation. With no more information about the oils than their molecular weight, the average deviation between calculated and measured viscosities of 35 oils, including a wide range of fluids from light condensates to heavy biodegraded crudes, was 11.4%.By using one viscosity measurement, e.g., at 60 "C, to correct the calculated value, the predictions in the Newtonian temperature range (40-80 "C) were highly improved, reducing the average deviation to 3.4%. The equation above has been shown to provide good estimates for oils not included in the data base and also for most oils at 30 "C. At temperatures lower than 30 "C,non-Newtonian behavior (shear-thinning) caused by wax formation, will cause the correlation to fail for certain oils but may still provide reasonable estimates of the viscosityat high shear rate for moderately shear-thinning oils. It has been shown that the surface tension of a series of North Sea crude oils (and one condensate) at temperatures between 10 and 40 "C can be expressed as a linear function of the logarithm of the viscosity. Combined with the viscosity correlation above, this empirical relation provided estimates of the surface tension within about 2% on average, using the molecular weight of the oil as the only input. The use of this correlation requires the oil to be Newtonian at the actual temperature. The validity of the correlation for temperatures outaide the range 1% 40 "C has not been confirmed yet.
4
2 0 ! , ' ' ' i , 100 125 150 175 200 225 250 275 300 325 350 MOLECULAR WEIGHT ~
'
I
Figure 11. Chart based on a combination of eqs 13 and 5 for estimation of the surface tension of petroleum fluids from their averagemolecularweight. "he c u n w at 50 and 60"Careactually extrapolations outside the range of validity of the correlation.
paper. It is seen that by using eq 4 instead of eq 5, only aslighlyworse agreement was obtained for the oils (average deviation 2.3% ) while the condensate was highly underestimated (23-40 % ). The correlation established by combining eqs 13 and 5 is presented graphically in Figure 11. Thus, with a measurement of molecular weight at hand, a quick and reasonably accurate estimate of the surface tension of a crude oil can be obtained. Of come, the same possible exceptions with regard to extreme oil types apply as for the viscosity correlation. Furthermore, the correlation will not be valid for oils that behave distinctly nonNewtonian, and neither is it claimed to be valid at temperatures higher than 40 "C. Nevertheless, it can probably be extrapolated at least to 50 "C without committing a serious error. Conclusions At constant temperature in the Newtonian temperature range, Le., in general above about 30 "C,the viscosity of a wide range of North Sea petroleum fluids has been shown to exhibit a simple power-law dependency on the average molecular weight. Furthermore, the temperature dependency for a single fluid is very well represented by the Arrhenius equation, eq 3, with standard error in the viscosity estimate generally within about 8%. The flow activation energy and the Arrhenius constant have both been found to exhibit a regular, linear dependency on the average molecular weight of the oil. This gave rise to the following expression for the Newtonian viscosity of an oil (in mPa 8) at temperature T (in K) as a function of molecular weight: log p = -0.07995 - 0.01101MW - 371.8/T + (6.215MW)/T The viscosity and molecular weight data, were fitted to
a
Acknowledgment. The author would like to acknowledge the assistance by many colleagues in our laboratory over the last years in characterizingthe oils and producing data used in this paper and also Statoil for permission to publish it.
Nomenclature
E,
flow activation energy (kJ/mol) freezing point depression constant MW number-average molecular weight R gas constant (8.3144 J/K mol) S reading on petroleum cryoscope T absolute temperature (K) AT freezing point depression (K) w weight (g) x weight fraction y surface tension (mN/m) p dynamic viscosity (mPa 8) p density (g/cma) u kinematic viscosity (cSt)
Kf
