Energy & Fuels 2000, 14, 43-51
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Effect of Precipitated Wax on ViscositysA Model for Predicting Non-Newtonian Viscosity of Crude Oils Karen S. Pedersen*,† and Hans P. Rønningsen‡ Calsep A/S, Gl. Lundtoftevej 7, DK-2800 Lyngby, Denmark, and STATOIL, N-4035 Stavanger, Norway Received June 10, 1999. Revised Manuscript Received October 14, 1999
Viscosities of 18 North Sea oils (API gravity 23.8 to 47.6) have been measured at temperatures between 40 and 0 °C and shear rates ranging from 30 to 500 s-1. Precipitated wax has a pronounced effect on the viscosity and rheological behavior of these oils. At low temperatures where wax precipitation is most extensive, the oils typically behave like pseudoplastic or viscoplastic fluids. A shear-rate-dependent viscosity model is presented. It is based on a correspondence between viscosity and volume fraction of precipitated wax and further uses the Casson rheological fluid model. It contains a Newtonian and two shear-rate-dependent terms. The Newtonian term is similar to the type of viscosity models used for oil/water emulsions. The model correlates 713 measured viscosity data points with an average absolute deviation of 48%. The model has been tested on three oils not included in the data basis. The non-Newtonian viscosities of these oils (176 data points) were predicted with an average absolute deviation of 47%.
Introduction At sufficiently high temperatures, typically above 40 °C, most crude oils, despite their enormous compositional complexity, behave like simple Newtonian liquids with a unique viscosity at any given temperature. The Newtonian viscosity can be predicted quite accurately using, for example, corresponding states models1,2,3 or correlations in measurable physical properties such as density.4,5 The temperature dependence is normally well described by an Arrhenius type of equation including a flow activation energy term,6 meaning that there is linear relationship between the logarithm of the viscosity and the inverse temperature. Transport of waxy oil and gas condensate mixtures in sub-sea pipelines often gives rise to wax precipitation. The precipitation commenses when the temperature reaches the wax appearance temperature (WAT). Some wax will deposit at the inner side of the wall and some will precipitate in the bulk oil phase as solid particles. The wax particles in the bulk will increase the apparent viscosity of the oil and give rise to an increased pressure drop in the pipeline. This is important in the design of new oil field developments. When present in sufficiently high concentrations, the wax particles will gradually * Corresponding author. † Calsep A/S. ‡ STATOIL. (1) Ely, J. F.; Hanley, J. M. Ind. Eng. Chem. Fundam. 1981, 20, 323-332. (2) Pedersen, K. S.; Fredenslund, Aa.; Christensen, P. L.; Thomassen, P., Chem. Eng. Sci. 1984, 39, 1011-1016. (3) Pedersen, K. S.; Fredenslund, Aa. Chem. Eng. Sci. 1987, 42, 182-186. (4) Beal, C. Trans. AIME 1946, 165, 94-112. (5) Beggs, H. D.; Robinson, J. R. J. Petr. Technol. 1975, 9, 11401149. (6) Rønningsen, H. P. Energy Fuels 1993, 7, 565-573.
change the flow properties of the oil/wax suspension from Newtonian to non-Newtonian behavior. This transition typically occurs about 10-15 °C below the WAT, but the temperature may vary depending on the waxiness of the oil. The transition seems to correspond with a solid wax fraction of 1-2 wt %. Ultimately, when cooled further toward the pour point, the oil turns into a gelled, solidlike state exhibiting highly non-Newtonian behavior. This typically occurs when the weight percent of solid wax reaches about 4-5%.7 No satisfactory model or correlation exists for predicting the shear-ratedependent viscosity of crude oils in the non-Newtonian temperature range. That is the background for the present study with the objective of developing a first version of such a correlation model. Rheological Models For a Newtonian fluid the shear stress (τxy) is directly proportional to the shear rate, or velocity gradient (dvx/ dy)
dvx dy
τxy ) η
(1)
The viscosity is defined as the ratio of shear stress to shear rate. As may be seen from eq 1, the viscosity of a Newtonian fluid is independent of shear rate. Figure 1 illustrates Newtonian and three kinds of non-Newtonian flow behavior. Several rheological models or rheological equations of state have been proposed in order to describe the nonlinear flow curves of non-Newtonian fluids. Two of the flow curves illustrated in Figure 1, (7) Rønningsen, H. P.; Bjørndal, B.; Hansen, A. B.; Pedersen, W. B. Energy Fuels 1991, 5, 895-908.
10.1021/ef9901185 CCC: $19.00 © 2000 American Chemical Society Published on Web 12/11/1999
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higher than about 200 s-1. It has also been shown that waxy crude oils, cooled to a gelled, solidlike state under static conditions, e.g., during a pipeline shut-down, behave very much like a time-dependent Bingham plastic fluid during a restart situation where a constant shear stress (i.e., pumping pressure) is imposed on the fluid.10 The breakdown of the gelled structure is caused primarily by a breakdown of the Bingham yield stress. Recently, this rheological model has been utilized in a numerical pipeline start-up model for waxy crudes.11 Like the Herschel-Bulkley model, another common rheological fluid model, the Casson model,12 takes into account both the nonlinearity of the flow curve and the existence of a yield stress, often observed with waxy oils. According to Barry,9 at lower shear rates (below 200 s-1) a non-Newtonian oil with suspended wax can be mathematically represented by the Casson equation
x
xτxy ) a + b Figure 1. Newtonian and Non-Newtonian flow behaviors.
the pseudoplastic and dilatant fluids, both exhibit a nonlinear relationship between shear rate and shear stress. The viscosity of a pseudoplastic fluid decreases with increasing shear rate, while the opposite is the case for a dilatant fluid. A pseudoplastic fluid without a yield stress is most often described by a simple two-parameter power function, i.e., the well-known power-law model. If a yield stress exists, it may just be added to the power term, giving the so-called Herschel-Bulkley model8
( )
τxy ) τ0 + K
dvx dy
n
(2)
where n is a shear rate index and τ0 the yield stress. The non-Newtonian fluids with a yield stress are collectively called viscoplastic materials and include the Bingham plastic and Casson type of fluids described below. Waxy crude oils very often exhibit viscoplastic behavior at low temperatures. A Bingham plastic fluid is similar to a Newtonian fluid in the sense that there is a linear relationship between shear stress and shear rate. However, a Bingham plastic fluid differs by requiring a finite shear stress (pressure) to initiate any flow at all. This threshold shear stress is called the yield stress of the fluid. The behavior of a Bingham plastic fluid can be mathematically expressed as
dvx τxy ) τ0 + ηP dy
(3)
where τ0 is the Bingham yield stress. The constant ηP is referred to as the plastic viscosity. Like the pseudoplastic fluids, the Bingham fluid exhibits decreasing viscosity with increasing shear rate (shear thinning). Barry9 has found that an oil with suspended wax behaves like a Bingham plastic fluid at shear rates (8) Herschel, H.; Bulkley, R. Proc. Am. Soc. Test. Mater. 1926, 26 (II), 621-633. (9) Barry, E. G. J. Inst. Pet. 1971, 57, 74-85.
dvx dy
(4)
where a and b are constants. The Casson model has also been found to represent the equilibrium flow properties of waxy oils very well at higher shear rates (up to 700 s-1).10 Using the definition of the viscosity, the Casson equation may be rewritten as
η)A+
B
x
dvx dy
+
C dvx dy
(5)
where A, B, and C are other constants. Samples and Experimental Methods A total of 18 different North Sea crude oils have been used in this study. The oils are mainly of a rather waxy nature, but nevertheless cover quite a broad spectrum of oil types with API gravities from 23.8° to 47.6°. Table 1 gives an overview of some important analytical data of the oils included in the data basis for the viscosity model. Table 2 gives corresponding data for three other oils not included in the data basis, but used for validation of the model. The oil samples were in general obtained by a single-stage flash to atmospheric conditions from preheated pressurized separator oil bottles. After the flash the samples were transferred to a 1 L gas-tight stainless steel cell and subjected to a standardized thermal pretreatment (beneficiation) consisting of heating to 80 °C for at least 2 h in order to dissolve all wax and erase the memory of previous wax formation, followed by cooling at a rate of 9 °C/h (0.11 °C/min) to 40 or 35 °C. This condition was the starting point of all the viscosity measurements with the rotational viscometer. Kinematic viscosities were measured at 10 °C intervals in the Newtonian temperature range (from 80 to 40 °C) using Ubbelohde glass capillary tubes with appropriate capillary constants. An average of three parallels was taken at each temperature. Repeatability was generally within 1% relative. Dynamic viscosity was calculated from the kinematic viscosity by multiplying with the density at each temperature. Thermally beneficiated oil was transferred with a 10 mL pipet to a Haake RV12 concentric cylinder rotational viscometer equipped with double cap measuring geometry (NV1, radii (10) Rønningsen, H. P. J. Pet. Sci. Eng. 1992, 7, 177-213. (11) Chang, C.; Ngyuen, D.; Rønningsen, H. P. Isothermal start-up of pipeline transporting waxy crude oil. J. Non-Newtonian Fluid Mech. submitted. (12) Casson, N. Br. Soc. Rhel. September, 1957.
Predicting Non-Newtonian Viscosity of Crude Oils
Energy & Fuels, Vol. 14, No. 1, 2000 45
Table 1. Analytical Data of the Oils Included in the Data Basis oil no. 1
2
3
4
5
6
7
8
density (15 °C, 866 856 858 860 891 911 841 851 kg/m3) °API 31.9 33.8 33.4 33.0 27.3 23.8 36.8 34.8 no. av molecular 263 241 253 228 256 292 201 210 weight, g/mol C10+ total, wt % 90.15 89.12 87.64 87.50 86.75 91.89 77.54 82.77 C10+ aromatics, 28.8 29.0 28.4 31.2 52.9 47.1 32.7 28.3 wt % (of C10+) total wax, wt % 15.6 14.8 24.1 10.4 17.4 18.1 10.3 12.6 purified wax, wt % 10.4 11.3 17.7 7.8 4.9 11.4 6.0 10.4 C5-asphaltenes, 1.2 0.4 1.0 0.19 4.9 3.2 1.1 0.59 wt % wax appearance 40 40 47 38 33 40 40 39 temp, °C pour point, °C as received 26 25 32 10 -4 24 2 12 minimum 16 13 26 8 -30 -2 -20 -50 kinematic viscosity, c St 80 °C 3.84 2.86 3.74 2.68 7.90 8.15 2.50 2.52 50 °C 7.07 5.12 7.39 4.64 17.1 18.4 4.80 4.30
9
10
11
12
13
14
15
811
907
879
857
841
878
790
43.0 162
24.5 307
29.5 239
33.6 224
36.8 201
29.7 258
47.6 145
70.96 25.8 6.1 5.8 0.18
97.8 41.3 1.8 1.2 0.58
85.21 51.3
85.43 39.4
78.77 32.0
91.27 36.9
62.61 17.6
13.0 4.0 2.1
14.5 8.0 0.73
11.4 7.1 1.5
13.2 6.3 1.5
4.3 4.2 0.26
38
23
34
32
35
37
39
-14 -16
-42 -
-4 -34
-8 -8
6 6
10 -30
-20 -30
1.26 1.75
9.12 23.0
4.27 7.81
3.07 5.37
2.43 3.87
3.87 7.65
0.96 1.24
Table 2. Analytical Data of Test Oils not Included in the Data Basis oil no. kg/m3)
density (15 °C, °API no. av molecular weight, g/mol C10+ total, wt % C10+ aromatics, wt % (of C10+) total wax, wt % purified wax, wt % C5-asphaltenes, wt % wax appearance temp, °C pour point, °C as received minimum kinematic viscosity, cSt 80 °C 50 °C
16
17
18
798 45.8 154 66.38 22.8 4.7 4.3 0.0 40
847 35.6 226 86.68 26.9 21.1 15.8 0.76 50
849 35.2 215 82.92 35.9 8.9 4.9 1.21 38
-22 -22 1.01 1.41
18 12 2.71 4.88
10 10 2.64 4.22
ratio 1.02) and M150 or M500 torque head. Apparent viscosity was measured in temperature scans from 40 or 35 °C to 1 °C at different constant shear rates typically in the range from 30 to 500 s-1. Cooling rate was 12.5 °C/h (0.21 °C/min). The repeatability was somewhat dependent on temperature but on the order of (10% relative, somewhat higher for the most waxy oils. Figures 2 and 3 show typical viscosity plots for two of the oils. Wax appearance temperatures (WAT) have been measured for each of the oils. The WATs ranged from 23 to 50 °C. For Oils 1 and 2 experimental data further exist for the amount of precipitated wax as a function of temperature as determined by pulsed NMR. The data reproduced from Pedersen et al.13 are given in Table 3. The pulsed NMR method is based on the fact that protons in a dissolved (liquid) state and a solid state have highly different relaxation times after being excited with a pulse of radio frequency radiation. By analyzing the relaxation pattern, the ratio of solid to liquid phase can be deduced. It is routinely applied in the food industry for measuring solid fat content.14 After careful calibration it was found to be a good method also for measuring solid wax in a broad range of North Sea crude oils. The data presented by Pedersen et al.13 have been used by others, e.g., Pedersen15 and Lira-Galeana et al.16 in modeling wax formation. (13) Pedersen, W. B.; Hansen, A. B.; Larsen E.; Nielsen, A. B.; Rønningsen, H. P. Energy Fuels 1991, 5, 908-913. (14) IUPAC method 2.323; Pure Appl. Chem. 1982, 54, 2766-2774. (15) Pedersen, K. S. SPE Production Facilities February 1995, 4649. (16) Lira-Galeana, C.; Firoozabadi, A.; Prausnitz, J. M. AIChE J. 1996, 42, 239.
Figure 2. Measured viscosities for Oil 1. With reference to Tables 1 and 2, the methods used are described by Rønningsen et al.7 Briefly, density was measured with a frequence density meter, molecular weight by freezing point depression (of benzene), C10+ weight percent by gas chromatography, C10+ aromatics by preparative liquid chromatography (silica), total wax content by acetone precipitation (at about -25 °C), purified wax by hexane filtration of total wax through a silica column, asphaltenes by pentane precipitation (40× excess) at room temperature, wax appearance temperature by crossed polar microscopy, and pour point by a modified ASTM D-87 including thermal beneficiation (minimum pour point).
Viscosity Model Figures 4 and 5 show measured viscosities of Oils 1 and 2 at four different shear rates plotted against the fraction of solid wax. They clearly illustrate the strong dependence of the viscosity on the amount of precipitated wax. Considering an estimated (10% uncertainty on the solid wax fraction,13 an exponential relationship between viscosity and amount of solid wax seems to be a reasonable approximation. This observation is utilized in the viscosity model presented below.
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Figure 5. Measured viscosity of Oil 2 as a function of measured solid wax weight percent.
Figure 3. Measured viscosities for Oil 4. Table 3. Measured Solid Wax Fraction (wt %) of Oils 1 and 2 as a Function of Temperature13 temp (°C)
oil 1
oil 2
temp (°C)
oil 1
oil 2
45 40 35 30 25 20 15
0.3 0.5 0.4 1.1 2.1 4.4 5.9
0.2 0.4 0.1 0.5 0.9 2.3 3.2
10 5 0 -5 -10 -15 -20
8.2 8.6 9.8 10.0 10.4 9.9 11.8
4.9 4.9 5.5 5.7 5.8 5.8 6.9
Figure 4. Measured viscosity of Oil 1 as a function of measured solid wax weight percent.
The relation given in eq 5 is used as the basis for a shear-rate-dependent viscosity model. According to this expression the viscosity becomes to a good approximation equal to the constant A for high shear rates. This constant gives the Newtonian contribution to the viscosity and is modeled using an analogy from water/oil emulsions. According to the model of Richardson17 the (17) Richardson, E. G. Kolloid-Z. 1933, 65 (1), 32-37.
viscosity of an oil/water emulsion can be represented by a simple exponential expression of the type
η ) ηc exp(DΦ)
(6)
where D is a constant, ηc the viscosity of the continuous phase, and Φ the volume fraction of the dispersed phase. In the present work, by considering the viscosity versus solid wax relationship discussed above, the volume fraction of solid (precipitated) wax, Φwax, replaces Φ and the viscosity of the liquid phase, ηliq, with no wax suspended, replaces ηc. The parameters B and C are assumed to be functions of Φwax. As may be seen from Figures 2 and 3 the viscosity for high wax contents (low temperatures) decreases rapidly with increasing shear rate. For lower wax contents (higher temperatures) the decrease is less pronounced. It was found that this behavior could be modeled by making the parameter C proportional to ηliqΦ4wax. For higher shear rates the third term in eq 5 will gradually cancel out and the shear rate dependence will arise mainly from the second term. At these shear rates the viscosity is seen to be less influenced by the amount of solid wax and it was found appropriate to make the parameter B proportional to ηliqΦwax, giving the following overall semiempirical equation:
[
η ) ηliq exp(DΦwax) +
EΦwax
x
dvx dy
+
]
FΦ4wax dvx dy
(7)
The liquid viscosity was calculated using the corresponding states model of Pedersen et al.2,3 tuned to the measured dynamic viscosity data above the WATs. For Oils 1 and 2 the measured amounts of precipitated wax were used to determine the solid wax volume fraction. For the remaining oils the volume fraction of precipitated wax was modeled using the wax model of Røn-
Predicting Non-Newtonian Viscosity of Crude Oils
ningsen et al.18 The volume fraction of wax is calculated from the weight fraction and the simulated composition of the precipitated wax. It is assumed that the density of solid wax is 10% higher than the density of the same composition in liquid form.19 The compositions of the oils are given to either C10 or to C20. The plus fraction is extrapolated to maximum C80 by assuming a linear relationship between the logarithm of the mole fraction and the carbon number.20 For six of the oils the experimental and simulated wax appearance temperatures deviated by more than 10 °C. For these oils the assumed enthalpies of fusion of the wax-forming components were adjusted to match the measured wax appearance temperature. This is the standard tuning procedure in the wax model. On the basis of experience with similar oils where the wax fraction was measured using pulsed NMR,13 it is expected that the calculated wax fraction with this adjustment is accurate to within (50% at a given temperature. By a parameter fit to the viscosity data for Oils 1-15 the constants in eq 7 were determined to be (viscosities in mPa s and shear rates in s-1)
D ) 37.82 E ) 83.96 F ) 8.559 × 106 In summary, the present model for calculation of shear-rate-dependent viscosities of oils below their WAT, is based on a combination of four elements: (1) Measurement or calculation of the amount of solid wax present at any temperature. A thermodynamic wax model is used. (2) Calculation of the viscosity of the Newtonian wax-free base liquid (in which wax particles are suspended). A corresponding states viscosity model is tuned to measured data above WAT. (3) An analogy with emulsion viscosity describing the effect of the suspended wax on the viscosity of the base liquid. An exponential dependence on wax volume fraction is assumed. (4) A nonlinear rheological fluid model representing the shear rate dependence of the viscosity, i.e., the non-Newtonian behavior. The Casson rheological equation is used. Basically, to calculate the viscosity of a waxy oil at a given temperature and shear rate with the present correlation model, the only input required is a standard composition to C10+ (or even C7+). An extended composition to higher carbon number, e.g., a TBPdistillation to C20+, may improve the fluid characterization and thus the viscosity and wax precipitation calculations. In addition, an experimental WAT can be used to tune and thus improve the wax calculations. Of course, if available, experimental wax precipitation data (amount vs temperature) can replace the values obtained by a thermodynamic wax model. Finally, experimental viscosities in the Newtonian temperature range (above WAT) may be used to tune the base liquid (18) Rønningsen, H. P.; Sømme, B. F.; Pedersen, K. S. An Improved Thermodynamic Model for Wax Precipitation: Experimental Foundation and Application. Paper presented at 8th International Conference on Multiphase ’97, Cannes, France, 18-20 June, 1997. (19) Templin, P. R. Ind. Eng. Chem. 1956, 48, 154-161. (20) Pedersen, K. S.; Blilie, A. L.; Meisingset, K. K. IEC Res. 1992, 31, 1378-1384.
Energy & Fuels, Vol. 14, No. 1, 2000 47
Figure 6. Measured Newtonian viscosities of Oil 1 (solid squares) fitted to an exponential Arrhenius equation and extrapolated to lower temperatures. Table 4. Molar Composition of Oil 1 component
mol %
molecular weight
liquid density g/cm3
C3 iC4 nC4 iC5 nC5 C6 C7 C8 C9 C10 C11 C12 C13 C14 C15 C16 C17 C18 C19 C20+
0.010 0.040 0.331 0.601 1.212 3.466 8.484 11.139 8.504 7.212 5.930 5.009 5.139 4.167 4.057 3.236 3.306 2.524 2.685 22.769
92.5 105.0 119.0 134.0 148.0 161.0 175.0 189.0 203.0 216.0 233.0 248.0 260.0 466.0
0.735 0.763 0.788 0.790 0.793 0.806 0.821 0.833 0.838 0.844 0.839 0.842 0.852 0.928
Table 5. Molar Composition of Oil 4 component
mol %
molecular weight
liquid density g/cm3
C1 C2 C3 iC4 nC4 iC5 nC5 C6 C7 C8 C9 C10+
0.281 0.448 1.171 0.503 1.198 0.809 0.933 1.989 5.981 9.859 7.160 69.668
89.0 100.7 115.5 282.1
0.758 0.789 0.808 0.875
viscosity estimates. As a simple alternative to the corresponding states viscosity model for estimating the base liquid viscosity, an extrapolation of Newtonian viscosities above WAT, fitted to an Arrhenius equation, could be applied as a reasonable approximation. Figure 6 shows an example with Oil 1. Results and Discussion Tables 4 and 5 show two examples of compositions of the oils studied. Viscosity data for the same two oils are shown in Tables 6 and 7. To give an overview of the diversity in viscosities presented by the different oils, Table 8 shows viscosity data for all 18 oils at a shear rate of 100 s-1. The viscosity plots in Figures 2 and 3 clearly illustrate the highly shear-thinning nature of these oils at low temperatures. Slight shear thinning
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Table 6. Viscosity Data for Oil 1 temp (°C)
dynamic viscosity (mPa s)
temp (°C)
dynamic viscosity (mPa s)
80 70 60
3.14 3.80 4.73
50 40
5.94 7.88
viscosity (mPa s) at shear rate (s-1) 30 70 100 300
temp (°C)
20
40 30 20 15 10 5
14.5 36.0 540 1180 2150 3670
13.9 33.7 385 870 1600 2800
14.0 31.0 250 510 860 1460
14.0 27.6 200 390 650 1070
500
13.8 23.5 110 195 310 495
13.5 21.9 105 180 235 380
Table 7. Viscosity Data for Oil 4 temp (°C)
dynamic viscosity (mPa s)
temp (°C)
dynamic viscosity (mPa s)
80 70 60
2.33 2.60 3.11
50 40
3.88 4.94
temp (°C)
viscosity (mPa s) at shear rate (s-1) 30 100 300 500
34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2
6.44 13.0 20.3 27.5 31.6 36.0 43.0 53.9 73.2 104 152 212 283 369 470 575 725
6.2 9.2 11.7 14.1 16.4 19.4 23.9 32.4 48.2 70.5 100 134 172 210 267 326 395
5.9 6.8 8.0 9.4 10.9 13.4 19.2 29.5 42.7 57.7 75.0 89.8 105 126 150 177 200
6.7 7.9 9.0 10.9 12.9 16.1 22.8 33.0 43.3 55.3 69.9 78.6 89.0 104 114 128 148
is evident even at 30 °C, which is about 15 °C above the minimum pour point. The apparent viscosity is seen to approach a constant value with increasing shear rate. Figure 7 shows the corresponding plot of shear stress versus shear rate for one of the oils, illustrating the gradually increasing nonlinearity of the flow curve as the temperature decreases and more wax precipitates. The plot suggests that the waxy oils behave like pseudoplastic fluids with a yield point (i.e., Casson or Herschel-Bulkley) at low temperatures, as Bingham plastic fluids at higher temperatures and as Newtonian fluids when approaching the WAT. The pronounced increase in viscosity with decreasing temperature seen for most of the oils is closely related to the amount of wax precipitated. This was seen in Figures 4 and 5, and is clearly also illustrated in Figure 8. It shows the calculated viscosity of Oil 4 at a shear rate of 500 s-1 in comparison with the volume percent of precipitated wax calculated by the thermodynamic wax model. At temperatures above the WAT the viscosity only increases slowly with decreasing temperature. Below the WAT there is a marked increase in viscosity, which for this high shear rate is an almost linear function of the volume percent of wax in accordance with the second term on the right-hand side of eq 7. These observations seem to justify an approach based on a coupling between a thermodynamic wax model and a viscosity model.
The non-Newtonian viscosity data for Oils 1-15 comprise a total of 713 data points, which were correlated, with an absolute average deviation of 48% using eq 7. As typical examples, Figures 9 and 10 compare experimental and calculated viscosities of Oils 1 and 4 at some selected temperatures. The model is seen to provide reasonable estimates of the viscosities and also a reasonably correct representation of the shear rate dependence. The viscosities measured for Oil 1 are very high, and for the lowest temperature the deviations between the measured and the calculated viscosities exceed 100%. In general the deviations are lower than that and the results obtained for Oil 4 are more typical. For practical purposes, viscosity estimates of this accuracy may very often be satisfactory, especially in an early design phase, and definitely better than pure extrapolation from the Newtonian temperature range without taking into account the effect of precipitated wax. This may highly underestimate the low-temperature viscosity of waxy oils. Viscoplastic equations other than the Casson equation, e.g., the Bingham or Herschel-Bulkley models, or even a simple power-law representation without a yield point, probably could have been chosen as a basis without significant loss of accuracy in the outcome of the overall model. Oils 16 to 18 were not included in the data basis. The non-Newtonian viscosities of these three oils comprised 176 data points, which were predicted with an average absolute deviation of 47%. This is comparable to the oils contained in the data basis. As an example, Figure 11 shows measured and calculated viscosities for Oil 17 as a function of temperature for shear rates of 30 and 500 s-1. It should be emphasized that the viscosity as well as the pour point of a non-Newtonian waxy crude oil is not only a function of temperature and shear rate, but also highly dependent on the thermal history (i.e., pretreatment temperature, cooling rate, temperature cycling, etc.) and shear history of the oil.7,21 One example of this phenomenon was shown in Figure 2 of Rønningsen et al.,7 where the viscosity at a given shear rate (100 s-1) and temperature was twice as high for a nonbeneficiated sample (i.e., a sample not preheated to 80 °C) as for a beneficiated one. Moreover, reducing the cooling rate from 12.5 to 2 °C/h was found to increase the viscosity considerably below 10 °C. Another example of the effect of thermal history is shown in Table 1 of the present paper, where a large difference in minimum and maximum (“as received”) pour point is noticed for some of the oils. Since the pour point also is a good indicator of the onset of severe nonNewtonian behavior, this alone suggests that the thermal history is important when trying to model the nonNewtonian viscosity. However, so far we are not able in any reasonable way to predict the effect of thermal history quantitatively. Therefore, our approach has been to use viscosity data that are the most well-defined, and in our opinion also the most relevant in most cases, namely the viscosities referring to a continuous cooling from well above the wax appearance temperature without any temperature cycling (i.e., cooling-reheating-cooling). This is believed to be the case in most situations with pipeline transfer of crude oil, although (21) Wardaugh, L. T.; Boger, D. V. AIChE J 1991, 37 (6), 871-885.
Predicting Non-Newtonian Viscosity of Crude Oils
Energy & Fuels, Vol. 14, No. 1, 2000 49
Table 8. Viscosity Data (mPas) for All Eighteen Oils at a Shear Rate of 100 s-1 temp (°C) 40 34 32 30 28 26 24 22 20 18 16 15 14 12 10 8 6 5 4 2 1 0
1
2
3
4
5
6
7
20.0 32.1 56.5 94.4 146 197 260 340
6.2 9.2 11.7 14.1 16.4 19.4 23.9 32.4 48.2 70.5
22.8 24.9 27.5 30.3 33.3 37.1 38.8 43.0 48.9 59.9
41.2 45.9 54.9 65.8 82.4 104 140 187 263 353
8.86 7.15 8.47 9.88 11.1 12.1 13.4 16.3 20.8 25.1
oil no. 9 10
11
6.8 7.52 8.19 9.57 11.6 13.0 15.3 17.5
3.24 3.54 3.75 3.80 4.47 5.11 5.29 5.61
21.1
5.58
8
12
13
14
15
16
17
18
16.4 18.1 20.9 24.6 26.6 31.0 33.9 40.4 45.9
11.0 13.2 14.6 16.9 19.7 22.9 24.7
10.6 10.2 11.0 11.4 14.9 17.3 18.0 23.0 27.0 33.4
11.5 12.9 13.2 14.2 14.4 18.3 19.6 23.9 27.7 33.4
2.14 2.23 2.39 2.55 3.11 3.55 3.62 3.70
7.54 8.39 8.74 9.41 10.9 13.5 17.6 22.9 32.2 45.7
1.41 1.47 1.54 1.60 1.71 1.77 1.87 1.98
8.18 10.4 12.6 16.0 21.0 28.9 49.3
53.4 65.7 80.8 98.7 121
27.7 30.0 33.6 37.2 42.6
41.2 52.5 61.9 73.3 87.0
41.6 49.3 63.5 78.9 95.8
2.09 3.98 71.2 2.24 4.04 110 2.37 5.00 170 2.53 250 2.77 340
150 184
49.0 106 53.4 130 59.9 161
14.0 27.6
10.1
200
36.0
390
95.0
650
195
1070
325 500
70.8 96.1 122
448 595 852 1050 1300
100 134 172 210 267
69.8 473 83.9 618 103 782 129 967 165 1183
1430 1770 1850
326 395
217 285
30.4 38.0 47.8 57.8 71.6
167 241
1426 1708
88.7 115
360
Figure 7. Shear stress versus shear rate for Oil 1.
there may be exceptions where for instance a cold stream is reheated before entering a pipeline. Hence, all viscosities used in this work are based on oil samples preheated to 80 °C in a gastight cell prior to measurement and cooled in a controlled way. This, of course, is automatically reflected in the viscosity predictions obtained by the presented model, due to the parameter fitting. Since transport of unprocessed, live wellstream fluids at low subsea temperatures is becoming increasingly common, it would be highly beneficial to have a nonNewtonian viscosity model for live oils. The wax model applied in this study has been verified against wax appearance temperatures of a number of live oil systems,18 and is believed to provide reasonable estimates of solid wax content in live oils. The present viscosity model, however, is based solely on viscosity data of stabilized (gas-free) oils. Since the apparent viscosity as predicted by eq 7 is a function of the viscosity of the continuous liquid phase with no compositional terms,
120 148 179
3.07 3.54
429 520
50.7 63.9 78.6 93.1 108 130.9 157 176
618
Figure 8. Calculated viscosity and vol % wax for Oil 4 as a function of temperature. The viscosity is for a shear rate of 500 s-1. The vol % wax was calculated using the thermodynamic wax model.17
the basic features of the model should in fact be valid also when a certain amount of gas is dissolved in the oil. Oil 4 originates from a reservoir fluid with a saturation pressure of about 23 MPa. Flash of this reservoir fluid to 40 °C and four different pressures (15, 10, 5, and 0.1 MPa) was simulated. Calculated WATs at the different pressure stages were 32, 33, 35, and 38 °C, respectively. Calculations of wax content and viscosity as a function of temperature were made for the liquid phase from each flash. The viscosity simulations are presented in Figure 12 for temperatures ranging from 5 to 40 °C at a shear rate of 100 s-1. Viscosity simulations for the 5 MPa liquid phase are shown in Figure 13 at four different shear rates and temperatures from 5 to 40 °C. The simulation results clearly illustrate the considerable reduction of viscosity with increasing amount of dissolved gas, and the non-Newtonian behavior of the live oil below about 30 °C. Both effects are
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Energy & Fuels, Vol. 14, No. 1, 2000
Figure 9. Measured and calculated viscosities for Oil 1 at 5 and 15 °C as a function of shear rate.
Figure 10. Measured and calculated viscosities for Oil 4 at 5, 10, and 20 °C as a function of shear rate.
qualitatively reasonable, although it is difficult to give a qualified estimate of the uncertainty. Since the correlation has been developed on the basis of data for stable oils, a somewhat larger uncertainty must be expected when the model is used for live oils. Viscosity data for live oils at low temperatures are needed to verify the model for live conditions and, if needed, to revise the present model parameters in order to improve live oil predictions. Taking into account the broad variety of oil types included in the data basis, the model presented is believed to have quite general application, although validation against oils from areas other than North Sea has not been conducted in this study. Conclusion Solid wax particles in an oil phase has a great impact on the apparent viscosity of the oil. For high contents
Pedersen and Rønningsen
Figure 11. Measured and calculated viscosities of Oil 17 for shear rates of 30 and 500 s-1. Oil 17 was not included in the data basis used to develop the model.
Figure 12. Calculated viscosities for live oils originating from the same reservoir fluid as Oil 4. The viscosities are for a shear rate of 100 s-1, and the saturation pressures (Psat) refer to a temperature of 40 °C. No experimental data exist for these conditions.
of solid wax the fluid behavior is non-Newtonian and can be classified as either pseudoplastic or Bingham plastic. The Casson rheological model is a reasonable compromise taking into account the nonlinear flow behavior and the existence of a yield point. A database consisting of a total of 889 viscosity data points has been presented for a broad set of 18 different gas-free North Sea oils. The viscosities cover about 3 orders of magnitude, ranging from 3 to 4100 mPa s. A model correlating the experimental data is presented. Minimum input is a standard fluid composition. Extended composition and experimental data for tuning may improve the predictions. The model is based on a combination of four elements: (1) Calculation of the amount of solid wax present at any temperature using
Predicting Non-Newtonian Viscosity of Crude Oils
Energy & Fuels, Vol. 14, No. 1, 2000 51
dependence of the viscosity, i.e., the non-Newtonian behavior. The model correlates 713 measured viscosity data points with an average absolute deviation of 48%. It has been tested on three oils, which were not included in the data basis. The non-Newtonian viscosities of these oils (176 data points) were predicted with an average absolute deviation of 47%. Notation A a B b C K dvx/dy n WAT
Figure 13. Calculated viscosities at four different shear rates for live oil originating from the same reservoir fluid as Oil 4. The live oil has a saturation pressure of 5 MPa at 40 °C. The calculated WAT of this oil was 35 °C.
a thermodynamic wax model; (2) Calculation of the viscosity of the Newtonian wax-free base liquid (in which wax particles are suspended) using a corresponding states viscosity model; (3) An analogy with emulsion viscosity describing the effect of suspended wax on the viscosity of base liquid, and (4) A nonlinear Casson type rheological fluid model representing the shear rate
constant constant constant constant constant constant in Herschel-Bulkley model ( eq 2) shear rate shear rate index wax appearance temperature
Greek Letters η apparent viscosity τ0 yield stress Φ volume fraction τxy shear stress Sub and P liq wax
Superscripts plastic liquid wax
Supporting Information Available: Eighteen tables each of compositional data and viscosity data. This material is available free of charge via the Internet at http://pubs.acs.org. EF9901185
