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New models for calculating the viscosity of mixed oil
Fuel 95 (2012) 431–437
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Fuel journal homepage: www.elsevier.com/locate/fuel
New models for calculating the viscosity of mixed oil Yuanping Gao a, Kewen Li b,c,⇑ a
Energy and Resources Dept., College of Engineering, Peking University, Beijing 100871, China School of Energy Resources, China University of Geosciences, Beijing 100083, China c Energy Resources Dept., Stanford University, CA 94305, USA b
a r t i c l e
i n f o
Article history: Received 1 August 2011 Received in revised form 19 December 2011 Accepted 20 December 2011 Available online 30 December 2011 Keywords: Viscosity mixing rules Modified models Rheological properties Experimental viscosity data
a b s t r a c t It is necessary to reduce the viscosity of heavy crude oil in many cases such as transportation and production of crude oil. One of the methods to reduce the viscosity is the addition of lighter oils. There have been many models to calculate the viscosity of the mixed oil. Unfortunately, there is no universal model for the computation. In this paper, five frequently-used models were chosen and evaluated. Totally 20 mixed oil samples were prepared with different ratios of light to crude oil from different oil wells but the same oil field. The viscosities of the mixtures under the same shear rates of 10 s1 were measured using a rotation viscometer at the temperatures ranging from 30 °C to 120 °C. After comparing all of the experimental data with the corresponding model values, the best one of the five models for this oil field was determined. Using the experimental data, one model with a better accuracy than the existing models was developed to calculate the viscosity of mixed oils. Another model was derived to predict the viscosity of mixed oils at different temperatures and different values of mixing ratio of light to heavy oil. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Viscosity is an important physical parameter of crude oil, and it is closely related with the whole processes of production and transportation, especially for heavy crude oil, reducing the viscosity of heavy oil is an effective way to increase production and facilitate transportation. Addition of light oil and heating of heavy oil are traditional methods used to reduce oil viscosity in petroleum and other industries. In some heavy oil reservoirs, crude oil can flow in the formation but may not flow through the wellbore because the viscosity becomes very high due to the low temperature around the wellbore as the crude oil flows up to the surface. Injecting light oil is an important recovery method which is being utilized in petroleum industry [1]. In this case, calculating the viscosity of the mixed oil accurately turns to be indispensable to decide how much light oil should be added. In order to determine the mixture ratio, it is needed to calculate the viscosity of the mixed oil accurately. Currently, about a dozen empirical formulas, semi-empirical formulas and calculating charts of evaluating the mixture viscosity have been reported [2–4]. Essentially, most of these formulas are derived from the regression analysis of experimental data instead of the physical mechanisms [5–9]. Arrhenius [10] proposed a viscosity computation rule for mixed liquids based on ideal solution. The Arrhenius model was applica⇑ Corresponding author at: School of Energy Resources, China University of Geosciences, Beijing 100083, China. E-mail address: [email protected] (K. Li). 0016-2361/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.fuel.2011.12.043
ble to calculate the viscosity of pure hydrocarbon mixtures but was not suitable for mixed crude oils. However the Arrhenius model has a great impact on later studies. Many subsequent models were based on it, for example, Lederer [11] added the correction factors on the volume fraction of light oil and heavy oil. Bingham [12] proposed a new model to calculate the viscosity of mixed liquids, which was also based on ideal solution. A few years later, Kendall and Monroe [13] proposed a model for estimating the viscosity of hydrocarbon mixtures. Walther [14] derived a mixing rule using the form of double logarithmic. In the case where the viscosity ratio of heavy to light oil is greater than 100, Cragoe [15] reported another model. These models were useful within a specific range. For example, the several component viscosity models can calculate the viscosity of mixed components accurately in some cases. However, mixed crude oil is a complex mixture with many components, which often made these models inapplicable. In recent years, many researchers investigated the mixing rules. Dolmatov et al. [16] reported that the Walther model fit their experimental data best. Barrufet and Setiadarma [17] evaluated several mixing rules of heavy oil using their experimental data at many solvent proportions and for temperatures from ambient to 177 °C. They concluded that the most promising mixing rule follow a methodology proposed originally by Lederer [11]. Mago et al. [18] demonstrated the importance of mixing rules when these models were used in performing reliable modeling studies of fluid flow in the reservoir. They summarized that the Arrhenius model and Kendall–Monroe model could not provide
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Nomenclature a aA aB b bA bB C n R2 T TA TB
the constant of ASTM model the heavy oil’s coefficient fitted by ASTM model the light oil’s coefficient fitted by ASTM model the constant of ASTM model the heavy oil’s coefficient fitted by ASTM model the light oil’s coefficient fitted by ASTM model the constant of Walther model the index of Kendall–Monroe model which is generally equal to 3 the goodness of fitting the Fahrenheit temperature the temperature of heavy oil the temperature of light oil
enough flexibility to describe the viscosity of extra heavy oil with temperature and provided a more flexible model. Centeno et al. [19] examined seventeen mixing rules reported in the literature used for predicting kinematic viscosity of petroleum and its fractions for accuracy by comparing the estimated values with the experimental viscosities of four crude oils. They concluded that no rule was capable of estimating viscosity for all the crude oils and pointed out that predicting viscosity is still a challenging task. Although there is no universal model that can calculate the viscosity of mixed oil accurately, it is usually necessary to find the most suitable model in the specific application for an oil field. In this study, 20 mixed oil samples were prepared with different ratios of light to crude oil from different oil wells but the same oil field. The viscosities of the mixtures were measured at the same shear rate but different temperatures. Five models were examined using the experimental viscosity data to find the most suitable viscosity model for this oil field. Finally, one modified model was proposed to calculate the viscosity of mixed oils with a better accuracy than the existing models. Another model was developed to predict the viscosity of mixed oils at different temperatures and different values of mixing ratio of light to heavy oil.
2. Theoretical background The five frequently-used models (the Arrhenius model, the Walther model, the Kendall–Monroe model, the Bingham model, and the Cragoe model) have different characteristics. Arrhenius model is appropriate for high viscosity ratio; it can calculate the viscosity of mixture satisfactorily when very heavy oil (or bitumen) is mixed with light oil. Walther model is not suitable for mixed crude oil which showing the behavior of non-Newtonian fluid. Kendall–Monroe model is recommended by American Petroleum Institute to calculate the viscosity of mixture of pure hydrocarbon at low pressure, providing that each component with similar molecular weight and similar character [20]. So the scope of Kendall–Monroe model’s application is rather limited. Bingham model is proposed based on ideal solution, and it has large error for calculating the viscosity of mixed oil [21]. The expressions of these five models are as follows: Arrhenius model [10]:
ln l ¼ V A ln lA þ V B ln lB
ð1Þ
Walther model [14]:
ln lnðl þ CÞ ¼ V A ln lnðlA þ CÞ þ V B ln lnðlB þ CÞ
ð2Þ
XA XB Xi VA VB
the the the the the the the the the the the the
a l lA lB
lic lie l e
corrected volume fraction of heavy oil corrected volume fraction of light oil relative error volume fraction of heavy oil volume fraction of light oil correction coefficient viscosity of mixed oil viscosity of heavy oil viscosity of light oil viscosity calculated by model viscosity of experiment average value of viscosities of experiment
Kendall–Monroe model [13]: 1=n
l
1=n ¼ V A l1=n A þ V B lB
ð3Þ
Bingham model [12]: 1 l1 ¼ V A l1 A þ V B lB
ð4Þ
Cragoe model [15]:
1= lnð2000lÞ ¼ V A = lnð2000lA Þ þ V B = lnð2000lB Þ
ð5Þ
where VA is the volume fraction of heavy oil; VB is the volume fraction of light oil; l is the viscosity of mixed oil; lA is the viscosity of heavy oil; lB is the viscosity of light oil; C is a constant which is set to zero in this study; and n is an index which is equal to 3 in many cases. 3. Experiments In this study, the oil samples included light oil, heavy oil samples A–E. These six oil samples were from the same oil field but different oil wells. Oil samples A–E were mixed with light oil using different proportions (the volume ratio of heavy oil to light oil was 4:1, 2:1, 1:1 and 1:2) respectively. Totally there were 20 mixed oil samples. The value of each sample’s viscosity was very different. The wide range of viscosity data may make the selected model more general. Anton Paar rotation viscometer was used to measure the viscosity of oil samples at the same temperature but different shear rates, or at the same shear rate but different temperatures. Most heavy crude oils were non-Newtonian fluid, and their viscosities will change with the shear rate. So, it is necessary to measure the viscosity of all oil samples at a fixed shear rate to make the experimental data comparable. For light oil, oil samples A–E, we measured the relationship between the viscosity and the shear rate at the temperatures of 30 °C, 70 °C and 120 °C respectively. The values of shear rate in the formation and wellbore were calculated according to the oil production data. The results showed that the shear rate ranged from 1 to 150 s1, and most of the shear rates were in the range from 1 to 10 s1. Even so, the viscosity of all the oil samples was measured at the shear rates ranging from 1 to 150 s1. The time interval of the two experimental points was 5 s. 4. Results 4.1. Rheological properties of oil samples Fig. 1 shows the rheological properties for the six oil samples at different temperatures. Generally speaking, the heavier oil samples (such as oil samples B–E) have shown the clear characteristics of non-Newtonian fluid. The non-Newtonian behavior follows the
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shear-thinning rule, which means the viscosity decreases as the shear rate increases. However, the lighter oil samples (such as light oil and oil sample A) basically show the characteristics of Newtonian fluid, and their viscosities do not change significantly, especially when the shear rate is greater than 10 s1. At low temperatures, the heavier oil samples (such as oil samples B–E) do not approach a fixed value of viscosity as the shear rate increases. But at the temperature of 120 °C, they basically do. One can see from Fig. 1 that the rheological data are fluctuant for shear rates less than 10 s1, even when the oil sample’s viscosity is small. The reason might be because the time interval of two experimental points was only 5 s and the oil samples were unstable, especially at high temperatures. Note that the values of shear rate in the formation and wellbore calculated according to the oil production data were mostly in the
Viscosity, mPa⋅s
105 Light oil Sample A Sample B Sample C Sample D Sample E
104
103
102 101 100
range from 1 to 10 s1. Considering this and the rheological data shown in Fig. 1 (viscosity data were stabilized at shear rates greater than 10 s1), the shear rate was set at 10 s1 for the following experiments. 4.2. The relationship between viscosity and temperature Walther [14] proposed a viscosity–temperature formula, which was recommended as the standard of the relationship between viscosity and temperature by ASTM [22]. It is named as ASTM model here and is expressed as follows:
log logðl þ 0:7Þ ¼ a þ b log T
where T is Fahrenheit temperature; a and b are constants. There have been other forms of viscosity–temperature models [23,24], but ASTM model was the only one used in this study. The viscosity of all the oil samples was measured at different temperatures and the experimental data are shown in Fig. 2. Fig. 2a plots the effect of temperature on viscosity and Fig. 2b shows the results of ASTM model fitting. Table 1 shows the goodness of fitting of the six oil samples fitted by ASTM model and the values of a and b. Overall, the viscosity will decrease as the temperature increases; all the relationships between viscosity and temperature are basically consistent with ASTM model, and the fitting results of the lighter oil samples (such as light oil and oil sample A) are better. Meanwhile, the experimental results of mixed oils are similar to those of crude oils. 4.3. Comparison of the five frequently-used models
101
102
103
Shear Rate, s-1
In order to describe the error of the five models conveniently, the mean absolute error, the sum of squared residuals (SSRs) and goodness of fitting are defined as follows: 105
Light oil Sample A Sample B Sample C Sample D Sample E
103
102
101
Light oil Sample A Sample B Sample C Sample D Sample E
104
Viscosity, mPa⋅s
Viscosity, mPa⋅s
104
100 100
ð6Þ
103 102 101
101
102
100 20
103
40
60
100
80
140
120
Temperature, °C
Shear Rate, s-1
(a) viscosity-temperaturecurve 0.8 Light oil Sample A Sample B Sample C Sample D Sample E
60 40 20
Light oil Sample A Sample B Sample C Sample D Sample E Light oil (linear) Sample A (linear) Sample B (linear) Sample C (linear) Sample D (linear) Sample E (linear)
0.6
log log (μ+0.7)
Viscosity, mPa⋅s
80
0.4 0.2 0
0 100
101
102
Shear Rate,
103
s-1
-0.2 1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
log T
(b) ASTM model fitting Fig. 1. Rheological curves of the six oil samples: (a) 30 °C, (b) 70 °C, and (c) 120 °C.
Fig. 2. The viscosity–temperature characteristics of the six oil samples.
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Table 1 ASTM model fitting results of the six oil samples. Oil sample
Light oil
Sample A
Sample B
Sample C
Sample D
Sample E
Values of a Values of b Goodness of fitting R2
1.8790 0.8405 0.9904
1.7740 0.7856 0.9930
2.3190 0.8993 0.9920
1.6759 0.6298 0.9772
2.7873 1.0731 0.9780
2.5582 0.9886 0.9870
SSR ¼
k X i¼1
lic lie
105
k 1X jX i j k i¼1
ð7Þ
2
ð8Þ i 2
Pk li le R2 ¼ 1 Pki¼1 c i 2 i¼1 le le
ð9Þ
2
where Xi is the relative error; R is the goodness of fitting; l is the e viscosity calculated by model; lie is the viscosity of experiment; l is the average value of viscosities of experiment. In order to avoid the larger viscosity data having too strong impact on the value of SSR calculated by Eq. (8) and balance the weight of each viscosity data relatively, the logarithm values of lic and lie are used in Eq. (8). Fig. 3 shows the viscosities of different mixing ratios of light to heavy oil for oil sample B at a temperature of 100 °C and the results calculated using the above five different models. One can see that the viscosity of mixed oil decrease as the volume fraction of light oil increases, as expected. Relatively speaking, the Arrhenius model and the Cragoe model are more accurate than the other models while the viscosity data computed with the Cragoe model are less than the true experimental values and those computed with the Arrhenius model are greater than the true experimental values. Totally there were 200 viscosity values of the 20 mixed oil samples at 10 different temperatures. After comparing these experimental values with the values calculated using the five models, the best model can be determined. Fig. 4 shows the comparison between the Arrhenius model and the experimental data. One can see from this figure that the Arrhenius model data are consistent with the experimental data. The value of the goodness of fitting, R2, is equal to 0.9876. When the viscosities of mixed oil are less than about 1000 mPa s, the results of the Arrhenius model fitting is better, and the calculated values are just slightly greater than the experimental values. When the viscosities are greater than 1000 mPa s, the calculated values are mostly smaller.
Model Viscosity, mPa⋅s
Mean absolute error ¼
i c
Arrhenius model Equivalent line
103 102 101 100 100
101
102
103
104
105
Experimental data, mPa⋅s Fig. 4. The comparison between the Arrhenius model and experimental data (R2 = 0.9876).
Fig. 5 demonstrates the comparison between the Walther model and the experimental data. It is similar to the Arrhenius model that the model fitting is better when the viscosities of mixed oil are less than about 1000 mPa s. But the Walther model data are mostly smaller than the experimental values. The greater the viscosities of mixed oil are the larger the errors are. The value of the goodness of fitting, R2, is equal to 0.9679 for the Walther model. The comparison between the values calculated by Kendall– Monroe model and the experimental data is shown in Fig. 6. The viscosity calculated by the Kendall–Monroe model is the largest among the five models at the same conditions. As it can be seen from Fig. 6, the calculated values are basically greater than the experimental viscosity data. Even when the viscosities of mixed oil are small, the errors are big. The value of the goodness of fitting, R2, is equal to 0.8922. So it is clear that the Kendall–Monroe model is not suitable to calculate the viscosity of mixed oil from this oil field. Fig. 7 plots the comparison between the Bingham model and the experimental data. Opposite to the Kendall–Monroe model, the viscosity data calculated by the Bingham model are the 105
45 40
Experimental data Arrhenius model Walther model Kendall-Monroe model Bingham model Cragoe model
35 30 25
Model Viscosity, mPa⋅s
Viscosity, mPa⋅s
104
20 15
104
Walther model Equivalent line
103 102 101
10 5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Volume fraction of light oil, fraction Fig. 3. The viscosities of different mixing ratios of light to heavy oil (oil sample B, 100 °C).
100 100
101
102
103
104
105
Experimental data, mPa⋅s Fig. 5. The comparison between the Walther model and experimental data (R2 = 0.9679).
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105 Kendall-Monroe model Equivalent line
104
Model Viscosity, mPa⋅s
Model Viscosity, mPa⋅s
105
103 102 101 100 100
101
102
103
104
104 103 102 101 100 100
105
Cragoe model Equivalent line
101
Experimental data, mPa⋅s
102
103
104
105
Experimental data, mPa⋅s
Fig. 6. The comparison between the Kendall–Monroe model and experimental data (R2 = 0.8922).
Fig. 8. The comparison between the Cragoe model and experimental data (R2 = 0.9761).
where
Model Viscosity, mPa⋅s
105
XA ¼ Bingham model Equivalent line
104
aV A aV A þ V B
XB ¼ 1 XA
ð11Þ ð12Þ
minimum among the five models at the same conditions. Almost all of the Bingham model data are smaller than the experimental values. The value of the goodness of fitting is very small and equal to 0.5642. It is also clear that the Bingham model is not suitable to calculate the viscosity of mixed oil from this oil field. The comparison between the Cragoe model and experimental data is shown in Fig. 8. The calculated values of the Cragoe model are slightly smaller than the experimental data when the viscosities of the mixed oil are greater than about 1000 mPa s. The value of the goodness of fitting is equal to 0.9761. Table 2 shows the error analysis of the five frequently-used models. One can see that the Arrhenius model has the smallest mean absolute error and SSR and the biggest goodness of fitting R2 among the five existing models.
In Eq. (11) the correction coefficient, a, can be determined by minimizing the sum of the square of the absolute error for each oil sample at each temperature. For our experimental data, the values of a of modified model I for different oil samples at different temperatures are shown in Table 3. The comparison between the experimental and the theoretical data calculated using the modified model I is shown in Fig. 9. The error of the modified model is significantly smaller than the five existing frequently-used models (also see Table 2). The results show that the mean absolute error of the modified model I is 6.19%, and the goodness of fitting R2 is 0.9950. Although the accuracy of the modified model I is better than the five existing models, it is based on a large amount of regression of experimental data to obtain the value of a. It may be difficult to use in some practical cases where experimental data are not available. Therefore, if accuracy is not required extremely high, the value of a could be set at 1 for the sake of simplicity. In this case, the comparison between the experimental and the theoretical data calculated using the modified model I with a = 1 is shown in Fig. 10. One can see that the simplified model could still work satisfactorily with a value of the goodness of fitting of 0.9867. The experimental data reported by Meng [21] were used to further test the modified model I. The comparison between the experimental and the theoretical data calculated using the modified model I (with regression) and the Arrhenius model are shown in Fig. 11, and the error analysis are shown in Table 2. Meng [21] demonstrated that the Arrhenius model is the best for her experimental data. Fig. 11 and Table 2 show that the modified model I is better than the Arrhenius model.
4.4. Modified model I
4.5. Modified model II
Fig. 3 as well as Figs. 4 and 8 show that the calculated values of the Arrhenius model are greater and the Cragoe model are smaller than the experimental data. This observation inspired us to build a new model by combining the two models with a correction factor added. The modified model is proposed as follows:
It is known that the viscosity is a function of temperature. One question arises: how to calculate the viscosity of the mixed oil at different temperatures when the temperature and the volume fractions of light oil and heavy oil are avalaible? Another model was proposed in this study to answer this question. The previous experimental results show that the relationships between viscosity and temperature for all of the oil samples are closely consistent with ASTM model. The Arrhenius model is the best one among the five existing models for the experimental data from this oil field. Considering the above analysis, one can obtain from Eq. (6):
103 102 101 100 100
101
102
103
104
105
Experimental data, mPa⋅s Fig. 7. The comparison between the Bingham model and experimental data (R2 = 0.5642).
ln l ¼
1 ðX A ln lA þ X B ln lB Þ 2 1 lnð2000lA Þ lnð2000lB Þ þ lnð2000Þ 2 X A lnð2000lA Þ þ X B lnð2000lB Þ
ð10Þ
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Table 2 Error analysis of models of calculating viscosity of mixed oil. Our experimental data
Arrhenius model Walther model Kendall–Monroe model Bingham model Cragoe model Modified model I Modified model I (a = 1) Modified model II
Experimental data by Meng [21]
Mean absolute error (%)
SSR
Goodness of fitting R
13.26 18.01 57.72 38.24 14.37 6.19 11.27 19.13
1.19 3.06 10.28 41.56 2.28 0.48 1.27 2.15
0.9876 0.9679 0.8922 0.5642 0.9761 0.9950 0.9867 0.9775
Table 3 Values of a of modified model I.
30 40 50 60 70 80 90 100 110 120
SSR
Goodness of fitting R2
1.899 – – – – 0.112 – –
0.5819 – – – – 0.9754 – –
103
Values of a Sample A
Sample B
Sample C
Sample D
Sample E
3.30 3.90 4.22 5.83 5.21 2.98 1.38 0.74 0.30 0.16
0.98 0.97 0.96 0.95 0.96 0.98 1.01 0.99 0.91 0.86
0.50 0.51 0.52 0.52 0.56 0.64 0.67 0.83 1.03 1.40
1.23 1.13 1.07 1.04 1.04 1.05 1.08 1.09 1.02 0.94
1.17 1.12 1.10 1.07 1.06 1.07 1.08 1.09 1.08 1.05
Model Viscosity, mPa⋅s
Temperature (°C)
2
Arrhenius model Modified model I Equivalent line
102
101
100 100
101
102
103
Experimental data, mPa⋅s Fig. 11. The comparison between the experimental and theoretical data calculated using modified model I (R2 = 0.9754) and the Arrhenius model (R2 = 0.5819) (experimental data from Meng [21]).
104
Modified model I Equivalent line
105 3
10
102 1
10
0
10 100
101
102
103
104
105
Experimental data, mPa⋅s Fig. 9. The comparison between the modified model I and experimental data (R2 = 0.9950).
Model Viscosity, mPa⋅s
Model Viscosity, mPa⋅s
105
Modified model II Equivalent line
104 103 102 101 100 100
101
102
103
104
105
Experimental data, mPa⋅s Fig. 12. The comparison between the modified model II and experimental data (R2 = 0.9775).
105
Model Viscosity, mPa⋅s
ðaA þbA log T A Þ
10
4
10
3
lA ¼ 1010
Modified model I (α=1) Equivalent line
10ðaB þbB log T B Þ
lB ¼ 10
ð13Þ
0:7
ð14Þ
where aA and bA are the heavy oil’s coefficients fitted by ASTM model; TA is the temperature of heavy oil; aB and bB are the light oil’s coefficients fitted by ASTM model; TB is the temperature of light oil. The values of aA, bA, aB and bB can be obtained from Table 1. Substitute Eqs. (13) and (14) into Eq. (1), the modified model II is obtained:
102 101 100 100
0:7
h
ðaA þbA log T A Þ
l ¼ exp V A lnð1010 101
102
103
104
105
Experimental data, mPa⋅s Fig. 10. The comparison between the modified model I (a = 1) and experimental data (R2 = 0.9867).
0:7Þ þ V B lnð1010
ðaB þbB log T B Þ
0:7Þ
i
ð15Þ From Eq. (15) we can calculate the viscosities of mixtures which blended heavy oil with light oil at any temperatures and volume fractions, which is convenient and helpful in many cases.
Y. Gao, K. Li / Fuel 95 (2012) 431–437
The comparison between the values calculated by the modified model II and the experimental data are shown in Fig. 12. The model data are very close to the experimental data. The mean absolute error is 19.13% and the goodness of fitting R2 is 0.9775. 5. Conclusions The following conclusions may be drawn according to the present study: 1. The viscosity of the lighter oil sample approaches a fixed value as the shear rate increases, which shows the characteristics of the Newtonian fluid. For the heavier oil samples, the viscosity decreases as shear rate increases, which follows the shearthinning rule of non-Newtonian fluid. 2. The viscosities of all crude oils and mixed oils decrease as the temperature increases, and the relationships between viscosity and temperature are basically in line with ASTM model. 3. The Arrhenius model is the best one among the five existing models for the oil samples studied. 4. One modified model (the modified model I) was proposed and has a better accuracy than the Arrhenius model for the oil samples studied. 5. Another model (the modified model II) was derived in order to calculate the viscosity of the mixed oils at any temperatures and volume fractions of light and heavy oils.
Acknowledgements This research was conducted with partial financial support from the National Natural Science Foundation of China under Grant 50974005, the contributions of which are gratefully acknowledged. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.fuel.2011.12.043. References [1] Yu Y, Li K. Method for calculating temperature profile in heavy oil wells with injection of light oil diluent. J Pet Sci Technol; 2012. accepted for publication.
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[2] Beggs HD, Robinson JF. Estimating the viscosity of crude oil systems. J Pet Technol 1975;27:1140–1. [3] Miadonye A, Singh B, Puttagunta VR. One-parameter correlation in the estimation of crude oil viscosity. SPE 26206; 1992. [4] Orbey H, Sandler SI. The prediction of the viscosity of liquid hydrocarbons and their mixtures as a function of temperature and pressure. Can J Chem Eng 1993;71(3):437–46. [5] Shu WR, Mobil R, Crop D. A viscosity correlation for mixtures of heavy oil, bitumen, and petroleum fractions. SPE 11,280-PA, SPE J 1984;24(3): 277–82. [6] Quan ZY. The problem of calculating the viscosity of diluted heavy oil. Storage Transport Oil Gas 1985;4(5):8–12 [in Chinese]. [7] Liu TY, Xu C. The study and improvement of methods of calculating the viscosity of diluted heavy oil. Storage Transport Oil Gas 1992;11(2):56–60 [in Chinese]. [8] Li CW. The rheological properties of mixed crude oils and their compatibility laws [dissertation]. China University of Petroleum Beijing; 1996 [in Chinese]. [9] Anil KM, Wayne DM, William YS. A review of practical calculation methods for the viscosity of liquid hydrocarbons and their mixtures. Fluid Phase Equilib 1996;117:344–55. [10] Arrhenius SA. Uber die Dissociation der in Wasser gelosten Stoffe. Z Phys Chem 1887;1:631–48. [11] Lederer EL. Viscosity of mixtures with and without diluents. Proc World Pet Cong London 1933;2:526–8. [12] Bingham EC. The viscosity of binary mixtures. J Phys Chem 1914;18:157–65. [13] Kendall J, Monroe K. The viscosity of liquids II. The viscosity-composition curve for ideal liquid mixtures. J Am Chem 1917;9:1787–802. [14] Walther C. The evaluation of viscosity data. Erdol Teer 1931;7:382. [15] Cragoe CS. Changes in the viscosity of liquids with temperature, pressure and composition. Proc World Pet Cong London 1933;2:529–41. [16] Dolmatov LV, Kutukov EG, Kutukov IE. Adequacy of mathematical models for calculating the viscosity of mixed liquid petroleum products. Chem Technol Fuels Oils 2001;37:201–5. [17] Barrufet MA, Setiadarma A. Reliable heavy oil–solvent viscosity mixing rules for viscosities up to 450 K, oil–solvent viscosity ratios up to 4 105, and any solvent proportion. Fluid Phase Equilib 2003;213:65–79. [18] Mago AL, Barrufet MA, Nogueira MC. Assessing the impact of oil viscosity mixing rules incyclic steam stimulation of extra-heavy oils. In: SPE 95,643, paper presented at SPE annual technical conference and exhibition, Dallas, Texas, 9–12 October; 2005. [19] Centeno G, Sánchez-Reyna G, Ancheyta J, Munoz JAD, Cardona N. Testing various mixing rules for calculation of viscosity of petroleum blends. Fuel 2011. doi:10.1016/j.fuel.2011.02.028. [20] Degiorgis GL, Maturano S, Garay M, Galliano GR, Fornes A. Oil mixture viscosity behavior: use in pipeline design. In: SPE 69,420, paper presented at SPE latin american and caribbean petroleum engineering conference. Buenos Aires, Argentina, 25–28 March; 2001. [21] Meng QP. An available viscosity calculation model for mixed crude oil. Oil Gas Storage Transport 2007;26(10):22–24,34 [in Chinese]. [22] ASTM Designation: Part 23. Annual Book of ASTM Standards. The American Society for Testing and Materials, Philadelphia, PA; 1981. [23] Guzman JD. Relation between fluidity and heat fusion. Anales Soc Espan Fia Y Quim 1913;11:353. [24] Egbogah EO, Jack NT. An improved temperature–viscosity correlation for crude oil systems. J Pet Sci Eng 1990;4(3):197–200.
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Fuel journal homepage: www.elsevier.com/locate/fuel
New models for calculating the viscosity of mixed oil Yuanping Gao a, Kewen Li b,c,⇑ a
Energy and Resources Dept., College of Engineering, Peking University, Beijing 100871, China School of Energy Resources, China University of Geosciences, Beijing 100083, China c Energy Resources Dept., Stanford University, CA 94305, USA b
a r t i c l e
i n f o
Article history: Received 1 August 2011 Received in revised form 19 December 2011 Accepted 20 December 2011 Available online 30 December 2011 Keywords: Viscosity mixing rules Modified models Rheological properties Experimental viscosity data
a b s t r a c t It is necessary to reduce the viscosity of heavy crude oil in many cases such as transportation and production of crude oil. One of the methods to reduce the viscosity is the addition of lighter oils. There have been many models to calculate the viscosity of the mixed oil. Unfortunately, there is no universal model for the computation. In this paper, five frequently-used models were chosen and evaluated. Totally 20 mixed oil samples were prepared with different ratios of light to crude oil from different oil wells but the same oil field. The viscosities of the mixtures under the same shear rates of 10 s1 were measured using a rotation viscometer at the temperatures ranging from 30 °C to 120 °C. After comparing all of the experimental data with the corresponding model values, the best one of the five models for this oil field was determined. Using the experimental data, one model with a better accuracy than the existing models was developed to calculate the viscosity of mixed oils. Another model was derived to predict the viscosity of mixed oils at different temperatures and different values of mixing ratio of light to heavy oil. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Viscosity is an important physical parameter of crude oil, and it is closely related with the whole processes of production and transportation, especially for heavy crude oil, reducing the viscosity of heavy oil is an effective way to increase production and facilitate transportation. Addition of light oil and heating of heavy oil are traditional methods used to reduce oil viscosity in petroleum and other industries. In some heavy oil reservoirs, crude oil can flow in the formation but may not flow through the wellbore because the viscosity becomes very high due to the low temperature around the wellbore as the crude oil flows up to the surface. Injecting light oil is an important recovery method which is being utilized in petroleum industry [1]. In this case, calculating the viscosity of the mixed oil accurately turns to be indispensable to decide how much light oil should be added. In order to determine the mixture ratio, it is needed to calculate the viscosity of the mixed oil accurately. Currently, about a dozen empirical formulas, semi-empirical formulas and calculating charts of evaluating the mixture viscosity have been reported [2–4]. Essentially, most of these formulas are derived from the regression analysis of experimental data instead of the physical mechanisms [5–9]. Arrhenius [10] proposed a viscosity computation rule for mixed liquids based on ideal solution. The Arrhenius model was applica⇑ Corresponding author at: School of Energy Resources, China University of Geosciences, Beijing 100083, China. E-mail address: [email protected] (K. Li). 0016-2361/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.fuel.2011.12.043
ble to calculate the viscosity of pure hydrocarbon mixtures but was not suitable for mixed crude oils. However the Arrhenius model has a great impact on later studies. Many subsequent models were based on it, for example, Lederer [11] added the correction factors on the volume fraction of light oil and heavy oil. Bingham [12] proposed a new model to calculate the viscosity of mixed liquids, which was also based on ideal solution. A few years later, Kendall and Monroe [13] proposed a model for estimating the viscosity of hydrocarbon mixtures. Walther [14] derived a mixing rule using the form of double logarithmic. In the case where the viscosity ratio of heavy to light oil is greater than 100, Cragoe [15] reported another model. These models were useful within a specific range. For example, the several component viscosity models can calculate the viscosity of mixed components accurately in some cases. However, mixed crude oil is a complex mixture with many components, which often made these models inapplicable. In recent years, many researchers investigated the mixing rules. Dolmatov et al. [16] reported that the Walther model fit their experimental data best. Barrufet and Setiadarma [17] evaluated several mixing rules of heavy oil using their experimental data at many solvent proportions and for temperatures from ambient to 177 °C. They concluded that the most promising mixing rule follow a methodology proposed originally by Lederer [11]. Mago et al. [18] demonstrated the importance of mixing rules when these models were used in performing reliable modeling studies of fluid flow in the reservoir. They summarized that the Arrhenius model and Kendall–Monroe model could not provide
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Nomenclature a aA aB b bA bB C n R2 T TA TB
the constant of ASTM model the heavy oil’s coefficient fitted by ASTM model the light oil’s coefficient fitted by ASTM model the constant of ASTM model the heavy oil’s coefficient fitted by ASTM model the light oil’s coefficient fitted by ASTM model the constant of Walther model the index of Kendall–Monroe model which is generally equal to 3 the goodness of fitting the Fahrenheit temperature the temperature of heavy oil the temperature of light oil
enough flexibility to describe the viscosity of extra heavy oil with temperature and provided a more flexible model. Centeno et al. [19] examined seventeen mixing rules reported in the literature used for predicting kinematic viscosity of petroleum and its fractions for accuracy by comparing the estimated values with the experimental viscosities of four crude oils. They concluded that no rule was capable of estimating viscosity for all the crude oils and pointed out that predicting viscosity is still a challenging task. Although there is no universal model that can calculate the viscosity of mixed oil accurately, it is usually necessary to find the most suitable model in the specific application for an oil field. In this study, 20 mixed oil samples were prepared with different ratios of light to crude oil from different oil wells but the same oil field. The viscosities of the mixtures were measured at the same shear rate but different temperatures. Five models were examined using the experimental viscosity data to find the most suitable viscosity model for this oil field. Finally, one modified model was proposed to calculate the viscosity of mixed oils with a better accuracy than the existing models. Another model was developed to predict the viscosity of mixed oils at different temperatures and different values of mixing ratio of light to heavy oil.
2. Theoretical background The five frequently-used models (the Arrhenius model, the Walther model, the Kendall–Monroe model, the Bingham model, and the Cragoe model) have different characteristics. Arrhenius model is appropriate for high viscosity ratio; it can calculate the viscosity of mixture satisfactorily when very heavy oil (or bitumen) is mixed with light oil. Walther model is not suitable for mixed crude oil which showing the behavior of non-Newtonian fluid. Kendall–Monroe model is recommended by American Petroleum Institute to calculate the viscosity of mixture of pure hydrocarbon at low pressure, providing that each component with similar molecular weight and similar character [20]. So the scope of Kendall–Monroe model’s application is rather limited. Bingham model is proposed based on ideal solution, and it has large error for calculating the viscosity of mixed oil [21]. The expressions of these five models are as follows: Arrhenius model [10]:
ln l ¼ V A ln lA þ V B ln lB
ð1Þ
Walther model [14]:
ln lnðl þ CÞ ¼ V A ln lnðlA þ CÞ þ V B ln lnðlB þ CÞ
ð2Þ
XA XB Xi VA VB
the the the the the the the the the the the the
a l lA lB
lic lie l e
corrected volume fraction of heavy oil corrected volume fraction of light oil relative error volume fraction of heavy oil volume fraction of light oil correction coefficient viscosity of mixed oil viscosity of heavy oil viscosity of light oil viscosity calculated by model viscosity of experiment average value of viscosities of experiment
Kendall–Monroe model [13]: 1=n
l
1=n ¼ V A l1=n A þ V B lB
ð3Þ
Bingham model [12]: 1 l1 ¼ V A l1 A þ V B lB
ð4Þ
Cragoe model [15]:
1= lnð2000lÞ ¼ V A = lnð2000lA Þ þ V B = lnð2000lB Þ
ð5Þ
where VA is the volume fraction of heavy oil; VB is the volume fraction of light oil; l is the viscosity of mixed oil; lA is the viscosity of heavy oil; lB is the viscosity of light oil; C is a constant which is set to zero in this study; and n is an index which is equal to 3 in many cases. 3. Experiments In this study, the oil samples included light oil, heavy oil samples A–E. These six oil samples were from the same oil field but different oil wells. Oil samples A–E were mixed with light oil using different proportions (the volume ratio of heavy oil to light oil was 4:1, 2:1, 1:1 and 1:2) respectively. Totally there were 20 mixed oil samples. The value of each sample’s viscosity was very different. The wide range of viscosity data may make the selected model more general. Anton Paar rotation viscometer was used to measure the viscosity of oil samples at the same temperature but different shear rates, or at the same shear rate but different temperatures. Most heavy crude oils were non-Newtonian fluid, and their viscosities will change with the shear rate. So, it is necessary to measure the viscosity of all oil samples at a fixed shear rate to make the experimental data comparable. For light oil, oil samples A–E, we measured the relationship between the viscosity and the shear rate at the temperatures of 30 °C, 70 °C and 120 °C respectively. The values of shear rate in the formation and wellbore were calculated according to the oil production data. The results showed that the shear rate ranged from 1 to 150 s1, and most of the shear rates were in the range from 1 to 10 s1. Even so, the viscosity of all the oil samples was measured at the shear rates ranging from 1 to 150 s1. The time interval of the two experimental points was 5 s. 4. Results 4.1. Rheological properties of oil samples Fig. 1 shows the rheological properties for the six oil samples at different temperatures. Generally speaking, the heavier oil samples (such as oil samples B–E) have shown the clear characteristics of non-Newtonian fluid. The non-Newtonian behavior follows the
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shear-thinning rule, which means the viscosity decreases as the shear rate increases. However, the lighter oil samples (such as light oil and oil sample A) basically show the characteristics of Newtonian fluid, and their viscosities do not change significantly, especially when the shear rate is greater than 10 s1. At low temperatures, the heavier oil samples (such as oil samples B–E) do not approach a fixed value of viscosity as the shear rate increases. But at the temperature of 120 °C, they basically do. One can see from Fig. 1 that the rheological data are fluctuant for shear rates less than 10 s1, even when the oil sample’s viscosity is small. The reason might be because the time interval of two experimental points was only 5 s and the oil samples were unstable, especially at high temperatures. Note that the values of shear rate in the formation and wellbore calculated according to the oil production data were mostly in the
Viscosity, mPa⋅s
105 Light oil Sample A Sample B Sample C Sample D Sample E
104
103
102 101 100
range from 1 to 10 s1. Considering this and the rheological data shown in Fig. 1 (viscosity data were stabilized at shear rates greater than 10 s1), the shear rate was set at 10 s1 for the following experiments. 4.2. The relationship between viscosity and temperature Walther [14] proposed a viscosity–temperature formula, which was recommended as the standard of the relationship between viscosity and temperature by ASTM [22]. It is named as ASTM model here and is expressed as follows:
log logðl þ 0:7Þ ¼ a þ b log T
where T is Fahrenheit temperature; a and b are constants. There have been other forms of viscosity–temperature models [23,24], but ASTM model was the only one used in this study. The viscosity of all the oil samples was measured at different temperatures and the experimental data are shown in Fig. 2. Fig. 2a plots the effect of temperature on viscosity and Fig. 2b shows the results of ASTM model fitting. Table 1 shows the goodness of fitting of the six oil samples fitted by ASTM model and the values of a and b. Overall, the viscosity will decrease as the temperature increases; all the relationships between viscosity and temperature are basically consistent with ASTM model, and the fitting results of the lighter oil samples (such as light oil and oil sample A) are better. Meanwhile, the experimental results of mixed oils are similar to those of crude oils. 4.3. Comparison of the five frequently-used models
101
102
103
Shear Rate, s-1
In order to describe the error of the five models conveniently, the mean absolute error, the sum of squared residuals (SSRs) and goodness of fitting are defined as follows: 105
Light oil Sample A Sample B Sample C Sample D Sample E
103
102
101
Light oil Sample A Sample B Sample C Sample D Sample E
104
Viscosity, mPa⋅s
Viscosity, mPa⋅s
104
100 100
ð6Þ
103 102 101
101
102
100 20
103
40
60
100
80
140
120
Temperature, °C
Shear Rate, s-1
(a) viscosity-temperaturecurve 0.8 Light oil Sample A Sample B Sample C Sample D Sample E
60 40 20
Light oil Sample A Sample B Sample C Sample D Sample E Light oil (linear) Sample A (linear) Sample B (linear) Sample C (linear) Sample D (linear) Sample E (linear)
0.6
log log (μ+0.7)
Viscosity, mPa⋅s
80
0.4 0.2 0
0 100
101
102
Shear Rate,
103
s-1
-0.2 1.9
2
2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
log T
(b) ASTM model fitting Fig. 1. Rheological curves of the six oil samples: (a) 30 °C, (b) 70 °C, and (c) 120 °C.
Fig. 2. The viscosity–temperature characteristics of the six oil samples.
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Table 1 ASTM model fitting results of the six oil samples. Oil sample
Light oil
Sample A
Sample B
Sample C
Sample D
Sample E
Values of a Values of b Goodness of fitting R2
1.8790 0.8405 0.9904
1.7740 0.7856 0.9930
2.3190 0.8993 0.9920
1.6759 0.6298 0.9772
2.7873 1.0731 0.9780
2.5582 0.9886 0.9870
SSR ¼
k X i¼1
lic lie
105
k 1X jX i j k i¼1
ð7Þ
2
ð8Þ i 2
Pk li le R2 ¼ 1 Pki¼1 c i 2 i¼1 le le
ð9Þ
2
where Xi is the relative error; R is the goodness of fitting; l is the e viscosity calculated by model; lie is the viscosity of experiment; l is the average value of viscosities of experiment. In order to avoid the larger viscosity data having too strong impact on the value of SSR calculated by Eq. (8) and balance the weight of each viscosity data relatively, the logarithm values of lic and lie are used in Eq. (8). Fig. 3 shows the viscosities of different mixing ratios of light to heavy oil for oil sample B at a temperature of 100 °C and the results calculated using the above five different models. One can see that the viscosity of mixed oil decrease as the volume fraction of light oil increases, as expected. Relatively speaking, the Arrhenius model and the Cragoe model are more accurate than the other models while the viscosity data computed with the Cragoe model are less than the true experimental values and those computed with the Arrhenius model are greater than the true experimental values. Totally there were 200 viscosity values of the 20 mixed oil samples at 10 different temperatures. After comparing these experimental values with the values calculated using the five models, the best model can be determined. Fig. 4 shows the comparison between the Arrhenius model and the experimental data. One can see from this figure that the Arrhenius model data are consistent with the experimental data. The value of the goodness of fitting, R2, is equal to 0.9876. When the viscosities of mixed oil are less than about 1000 mPa s, the results of the Arrhenius model fitting is better, and the calculated values are just slightly greater than the experimental values. When the viscosities are greater than 1000 mPa s, the calculated values are mostly smaller.
Model Viscosity, mPa⋅s
Mean absolute error ¼
i c
Arrhenius model Equivalent line
103 102 101 100 100
101
102
103
104
105
Experimental data, mPa⋅s Fig. 4. The comparison between the Arrhenius model and experimental data (R2 = 0.9876).
Fig. 5 demonstrates the comparison between the Walther model and the experimental data. It is similar to the Arrhenius model that the model fitting is better when the viscosities of mixed oil are less than about 1000 mPa s. But the Walther model data are mostly smaller than the experimental values. The greater the viscosities of mixed oil are the larger the errors are. The value of the goodness of fitting, R2, is equal to 0.9679 for the Walther model. The comparison between the values calculated by Kendall– Monroe model and the experimental data is shown in Fig. 6. The viscosity calculated by the Kendall–Monroe model is the largest among the five models at the same conditions. As it can be seen from Fig. 6, the calculated values are basically greater than the experimental viscosity data. Even when the viscosities of mixed oil are small, the errors are big. The value of the goodness of fitting, R2, is equal to 0.8922. So it is clear that the Kendall–Monroe model is not suitable to calculate the viscosity of mixed oil from this oil field. Fig. 7 plots the comparison between the Bingham model and the experimental data. Opposite to the Kendall–Monroe model, the viscosity data calculated by the Bingham model are the 105
45 40
Experimental data Arrhenius model Walther model Kendall-Monroe model Bingham model Cragoe model
35 30 25
Model Viscosity, mPa⋅s
Viscosity, mPa⋅s
104
20 15
104
Walther model Equivalent line
103 102 101
10 5
0.2
0.3
0.4
0.5
0.6
0.7
0.8
Volume fraction of light oil, fraction Fig. 3. The viscosities of different mixing ratios of light to heavy oil (oil sample B, 100 °C).
100 100
101
102
103
104
105
Experimental data, mPa⋅s Fig. 5. The comparison between the Walther model and experimental data (R2 = 0.9679).
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Y. Gao, K. Li / Fuel 95 (2012) 431–437
105 Kendall-Monroe model Equivalent line
104
Model Viscosity, mPa⋅s
Model Viscosity, mPa⋅s
105
103 102 101 100 100
101
102
103
104
104 103 102 101 100 100
105
Cragoe model Equivalent line
101
Experimental data, mPa⋅s
102
103
104
105
Experimental data, mPa⋅s
Fig. 6. The comparison between the Kendall–Monroe model and experimental data (R2 = 0.8922).
Fig. 8. The comparison between the Cragoe model and experimental data (R2 = 0.9761).
where
Model Viscosity, mPa⋅s
105
XA ¼ Bingham model Equivalent line
104
aV A aV A þ V B
XB ¼ 1 XA
ð11Þ ð12Þ
minimum among the five models at the same conditions. Almost all of the Bingham model data are smaller than the experimental values. The value of the goodness of fitting is very small and equal to 0.5642. It is also clear that the Bingham model is not suitable to calculate the viscosity of mixed oil from this oil field. The comparison between the Cragoe model and experimental data is shown in Fig. 8. The calculated values of the Cragoe model are slightly smaller than the experimental data when the viscosities of the mixed oil are greater than about 1000 mPa s. The value of the goodness of fitting is equal to 0.9761. Table 2 shows the error analysis of the five frequently-used models. One can see that the Arrhenius model has the smallest mean absolute error and SSR and the biggest goodness of fitting R2 among the five existing models.
In Eq. (11) the correction coefficient, a, can be determined by minimizing the sum of the square of the absolute error for each oil sample at each temperature. For our experimental data, the values of a of modified model I for different oil samples at different temperatures are shown in Table 3. The comparison between the experimental and the theoretical data calculated using the modified model I is shown in Fig. 9. The error of the modified model is significantly smaller than the five existing frequently-used models (also see Table 2). The results show that the mean absolute error of the modified model I is 6.19%, and the goodness of fitting R2 is 0.9950. Although the accuracy of the modified model I is better than the five existing models, it is based on a large amount of regression of experimental data to obtain the value of a. It may be difficult to use in some practical cases where experimental data are not available. Therefore, if accuracy is not required extremely high, the value of a could be set at 1 for the sake of simplicity. In this case, the comparison between the experimental and the theoretical data calculated using the modified model I with a = 1 is shown in Fig. 10. One can see that the simplified model could still work satisfactorily with a value of the goodness of fitting of 0.9867. The experimental data reported by Meng [21] were used to further test the modified model I. The comparison between the experimental and the theoretical data calculated using the modified model I (with regression) and the Arrhenius model are shown in Fig. 11, and the error analysis are shown in Table 2. Meng [21] demonstrated that the Arrhenius model is the best for her experimental data. Fig. 11 and Table 2 show that the modified model I is better than the Arrhenius model.
4.4. Modified model I
4.5. Modified model II
Fig. 3 as well as Figs. 4 and 8 show that the calculated values of the Arrhenius model are greater and the Cragoe model are smaller than the experimental data. This observation inspired us to build a new model by combining the two models with a correction factor added. The modified model is proposed as follows:
It is known that the viscosity is a function of temperature. One question arises: how to calculate the viscosity of the mixed oil at different temperatures when the temperature and the volume fractions of light oil and heavy oil are avalaible? Another model was proposed in this study to answer this question. The previous experimental results show that the relationships between viscosity and temperature for all of the oil samples are closely consistent with ASTM model. The Arrhenius model is the best one among the five existing models for the experimental data from this oil field. Considering the above analysis, one can obtain from Eq. (6):
103 102 101 100 100
101
102
103
104
105
Experimental data, mPa⋅s Fig. 7. The comparison between the Bingham model and experimental data (R2 = 0.5642).
ln l ¼
1 ðX A ln lA þ X B ln lB Þ 2 1 lnð2000lA Þ lnð2000lB Þ þ lnð2000Þ 2 X A lnð2000lA Þ þ X B lnð2000lB Þ
ð10Þ
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Table 2 Error analysis of models of calculating viscosity of mixed oil. Our experimental data
Arrhenius model Walther model Kendall–Monroe model Bingham model Cragoe model Modified model I Modified model I (a = 1) Modified model II
Experimental data by Meng [21]
Mean absolute error (%)
SSR
Goodness of fitting R
13.26 18.01 57.72 38.24 14.37 6.19 11.27 19.13
1.19 3.06 10.28 41.56 2.28 0.48 1.27 2.15
0.9876 0.9679 0.8922 0.5642 0.9761 0.9950 0.9867 0.9775
Table 3 Values of a of modified model I.
30 40 50 60 70 80 90 100 110 120
SSR
Goodness of fitting R2
1.899 – – – – 0.112 – –
0.5819 – – – – 0.9754 – –
103
Values of a Sample A
Sample B
Sample C
Sample D
Sample E
3.30 3.90 4.22 5.83 5.21 2.98 1.38 0.74 0.30 0.16
0.98 0.97 0.96 0.95 0.96 0.98 1.01 0.99 0.91 0.86
0.50 0.51 0.52 0.52 0.56 0.64 0.67 0.83 1.03 1.40
1.23 1.13 1.07 1.04 1.04 1.05 1.08 1.09 1.02 0.94
1.17 1.12 1.10 1.07 1.06 1.07 1.08 1.09 1.08 1.05
Model Viscosity, mPa⋅s
Temperature (°C)
2
Arrhenius model Modified model I Equivalent line
102
101
100 100
101
102
103
Experimental data, mPa⋅s Fig. 11. The comparison between the experimental and theoretical data calculated using modified model I (R2 = 0.9754) and the Arrhenius model (R2 = 0.5819) (experimental data from Meng [21]).
104
Modified model I Equivalent line
105 3
10
102 1
10
0
10 100
101
102
103
104
105
Experimental data, mPa⋅s Fig. 9. The comparison between the modified model I and experimental data (R2 = 0.9950).
Model Viscosity, mPa⋅s
Model Viscosity, mPa⋅s
105
Modified model II Equivalent line
104 103 102 101 100 100
101
102
103
104
105
Experimental data, mPa⋅s Fig. 12. The comparison between the modified model II and experimental data (R2 = 0.9775).
105
Model Viscosity, mPa⋅s
ðaA þbA log T A Þ
10
4
10
3
lA ¼ 1010
Modified model I (α=1) Equivalent line
10ðaB þbB log T B Þ
lB ¼ 10
ð13Þ
0:7
ð14Þ
where aA and bA are the heavy oil’s coefficients fitted by ASTM model; TA is the temperature of heavy oil; aB and bB are the light oil’s coefficients fitted by ASTM model; TB is the temperature of light oil. The values of aA, bA, aB and bB can be obtained from Table 1. Substitute Eqs. (13) and (14) into Eq. (1), the modified model II is obtained:
102 101 100 100
0:7
h
ðaA þbA log T A Þ
l ¼ exp V A lnð1010 101
102
103
104
105
Experimental data, mPa⋅s Fig. 10. The comparison between the modified model I (a = 1) and experimental data (R2 = 0.9867).
0:7Þ þ V B lnð1010
ðaB þbB log T B Þ
0:7Þ
i
ð15Þ From Eq. (15) we can calculate the viscosities of mixtures which blended heavy oil with light oil at any temperatures and volume fractions, which is convenient and helpful in many cases.
Y. Gao, K. Li / Fuel 95 (2012) 431–437
The comparison between the values calculated by the modified model II and the experimental data are shown in Fig. 12. The model data are very close to the experimental data. The mean absolute error is 19.13% and the goodness of fitting R2 is 0.9775. 5. Conclusions The following conclusions may be drawn according to the present study: 1. The viscosity of the lighter oil sample approaches a fixed value as the shear rate increases, which shows the characteristics of the Newtonian fluid. For the heavier oil samples, the viscosity decreases as shear rate increases, which follows the shearthinning rule of non-Newtonian fluid. 2. The viscosities of all crude oils and mixed oils decrease as the temperature increases, and the relationships between viscosity and temperature are basically in line with ASTM model. 3. The Arrhenius model is the best one among the five existing models for the oil samples studied. 4. One modified model (the modified model I) was proposed and has a better accuracy than the Arrhenius model for the oil samples studied. 5. Another model (the modified model II) was derived in order to calculate the viscosity of the mixed oils at any temperatures and volume fractions of light and heavy oils.
Acknowledgements This research was conducted with partial financial support from the National Natural Science Foundation of China under Grant 50974005, the contributions of which are gratefully acknowledged. Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at doi:10.1016/j.fuel.2011.12.043. References [1] Yu Y, Li K. Method for calculating temperature profile in heavy oil wells with injection of light oil diluent. J Pet Sci Technol; 2012. accepted for publication.
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