Energy Fuels 2011, 25, 493–498 Published on Web 01/21/2011
: DOI:10.1021/ef101013m
Heavy Oil Viscosity Modeling with Friction Theory† Ashutosh Kumar, Amr Henni,* and Ezeddin Shirif Industrial and Petroleum Systems Engineering, University of Regina, 3737 Wascana Parkway, Regina, Saskatchewan S4S 0A2, Canada Received August 3, 2010. Revised Manuscript Received December 2, 2010
A new tuning method is proposed for the cubic equation-of-state-based friction theory viscosity model, which improves the viscosity prediction of heavy oils. With the existing tuning method, the error in viscosity prediction increases with increasing viscosity. Particularly, at pressures below the saturation pressure, the error keeps increasing as the pressure is farther away from the saturation pressure. With the proposed tuning method, whereas the single-phase error is almost the same, the two-phase error is approximately 1/3 the value of the error from the existing tuning method. Most importantly, the proposed tuning method successfully uses experimental viscosity data below the saturation pressure to reduce prediction errors.
pressure concept, which facilitated its use with all types of equations of state ranging from theoretical to highly empirical.9 The friction theory viscosity model predicted the viscosity for light oils with a good accuracy and did not require any tuning. Application of the model to lower medium and medium oils resulted in significant errors particularly in the two-phase region. However, this error was reduced by tuning the critical viscosity parameter (Kc). This one-parameter tuning method improved the viscosity prediction for medium oils. When the model with Kc tuning was applied for viscosity prediction for heavy oils, large errors were observed. The error for the two-phase viscosity was much higher compared to the single-phase region and increased with a decrease in pressure below the saturation pressure. This error was reduced by introducing another tuning parameter, i.e., the compressibility correction parameter (Kz). Thus, for heavy oils, tuning of two parameters, the critical viscosity parameter (Kc) and the compressibility correction parameter (Kz) are required. The introduction of two-parameter tuning (Kz-Kc tuning) reduced the error in heavy oil viscosity prediction to some extent but did not change the magnitude of the error in the two-phase region. It was still high and increased as the pressure became farther away from the saturation pressure. The proposed tuning method aims at reducing the overall error in viscosity prediction for heavy oils and the increase in error when the pressure is far from the saturation pressure. In the two-parameter tuning method (Kz-Kc tuning),8 a minimum of two experimental viscosity data in the one-phase region is required. The proposed tuning method requires two experimental pressure points: one above and one below the saturation pressure. This tuning method reduces the overall error to 1/3 of the initial error and contains the increase in the magnitude of the error as the pressure deceases in the two-phase region. The proposed tuning method, for the first time, successfully uses experimental viscosity data at pressures below the saturation pressure.
1. Introduction The friction theory of viscosity modeling is a semi-empirical model used to predict the viscosity. The model has been developed using Newton’s law of viscosity and classical mechanical understanding of friction forces between two surfaces under static or dynamic conditions as formulated by Coulomb.1 The model is based on 16 universal constants along with critical viscosity as an adjustable parameter. The friction theory model with cubic equation of state was applied to normal alkanes from methane to n-octadecane and also to normal alkane mixtures with a good degree of accuracy.2 The model was successfully extended to non-cubic equations of state for viscosity and density modeling of nonpolar fluids3 with an acceptable level of accuracy. Later, the model was applied to crude oil systems,3a-6 and the results showed that, with the availability of comprehensive compositional characterization, the friction theory can be used for viscosity and density modeling with a good level of accuracy. The friction theory viscosity model, as discussed above, was restricted to the use of attractive and repulsive pressure terms as defined under a van der Waals type of equation of state. Later, this restriction was removed with the use of a generalized form of attractive and repulsive pressure terms based on the internal † Presented at the 11th International Conference on Petroleum Phase Behavior and Fouling. *To whom correspondence should be addressed. E-mail: amr.
[email protected]. (1) Qui~ nones-Cisneros, S. E.; Zeberg-Mikkelsen, C. K.; Stenby, E. H. Fluid Phase Equilib. 2000, 169, 249–276. (2) Qui~ nones-Cisneros, S. E.; Zeberg-Mikkelsen, C. K.; Stenby, E. H. Fluid Phase Equilib. 2001, 178, 1–16. (3) Qui~ nones-Cisneros, S. E.; Zeberg-Mikkelsen, C. K.; Stenby, E. H. Accurate density and viscosity modeling of non-polar fluids based on the “f-theory” and a non-cubic EOS. Proceedings of the 14th Symposium on Thermophysical Properties; Boulder, CO, June 25-30, 2000. (3a) Qui~ nones-Cisneros, S. E.; Zeberg-Mikkelsen, C. K.; Stenby, E. H. Chem. Eng. Sci. 2001, 56, 7007–7015. (4) Qui~ nones-Cisneros, S. E.; Zeberg-Mikkelsen, C. K.; Stenby, E. H. Fluid Phase Equilib. 2003, 212, 233–243. (5) Qui~ nones-Cisneros, S. E.; Dalberg, A.; Stenby, E. H. Pet. Sci. Technol. 2004, 22 (9/10), 1309–1325. (6) Schmidt, K. A. G.; Qui~ nones-Cisneros, S. E.; Kvamme, B. Energy Fuels 2005, 19, 1303–1313. (7) Qui~ nones-Cisneros, S. E.; Andersen, S. I.; Creek, J. Energy Fuels 2005, 19, 1314–1318.
r 2011 American Chemical Society
(8) Qui~ nones-Cisneros, S. E.; Zeberg-Mikkelsen, C. K.; Baylaucq, A.; Boned, C. Int. J. Thermophys. 2004, 25 (5), 1353–1366. (9) Qui~ nones-Cisneros, S. E.; Deiters, U. K. J. Phys. Chem. 2006, 110, 12820–12834. (10) Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15 (1), 59–64.
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: DOI:10.1021/ef101013m
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tuning the adjustable parameter Kc. This Kc is an adjustable parameter in the modified Uyehara and Watson correlation12 for critical viscosity, given as pffiffiffiffiffiffiffiffiffiffi 2=3 MWPc ηc ¼ Kc ð10Þ Tc 1=6
2. Compositional and Pressure-Volume-Temperature (PVT) Characterization4-8 Compositional characterization involves characterizing the plus fractions using a χ2 distribution function with p degrees of freedom, S(p). This p is fitted to find the optimal mass distribution for the plus fraction. The general mathematical form of the χ2 distribution function is p p S 2- 2 e- ð 2 ÞS 2 -1 fdis ¼ ð1Þ Γ p2
where MW is the molecular weight (g/mol), Pc is the critical pressure (bar), Tc is the critical temperature (K), and ηc is the critical viscosity (mPa s). The value of Kc for a pure component is 7.9483 10-4, and for pseudo-fractions, it is tuned using experimental viscosity data. The viscosity data should be at the same temperature (Tsat) at which the saturation pressure data was measured. 3.1. Viscosity Calculation. The total viscosity at temperature Tsat is given as η = η0,mix þ ηp þ KcηII, where η0,mix is the gas limit term viscosity contribution, ηp is the friction viscosity contribution from all components in the mixture, except the pseudo-fractions, and ηII is an adjustable friction term viscosity contribution:
where s is a molecular-weight-scaled variable that satisfies the following relation: Z S0 X 1 fdis dS ¼ ð xi MWi Þ ð2Þ M6 ¼ MWT 0 M6 in eq 2 represents the mass fraction of components up to C6, and the upper limit of integration S0 is that value of S with which its definite integration in the equation is equal to M6. Similarly, if fmi is the mass fraction of the ith pseudocomponent, then Si can be found from the following equation: Z Si fdis dS ð3Þ fmi ¼ The molecular weight (MWi) of the fraction Fi is given by ð4Þ
where S~i and MW are given by the following equations: R Si Sfdis dS S~i ¼ RSiS-i 1 ð5Þ Si - 1 fdis dS ¼m MW iX fmi MW ¼ 1 - M6 i ¼ 1 S~ i
ω ¼ e
MW
Pc ¼ fc ð9:67283 - 4:05288MW0:1 Þ
ð12Þ
ð6Þ
ðMWi Ti Þ0:5 Fc, i vc, i 2=3 Ωi
ð13Þ
where MWi is the molecular weight of the ith component and T is the absolute temperature. The following empirical equation is used to estimate the reduced collision integral for a component in a mixture and can be written as follows:
ð7Þ
8:50471 - 15:1665 0:1
η ¼ ηI þ Kc ηII
η0, i ¼ 40:785
Critical parameter characterization is performed using the following correlations:7,8 Tc ¼ - 423:587 þ 210:152 lnðMWÞ
ð11Þ
Here, ηI is the sum of the gas limit term viscosity contribution, η0,mix, and the friction viscosity contribution from all components in the mixture except the pseudo-fraction, ηp. 3.2. Calculations of the Gas Limit Viscosity, η0,mix. Calculating the dilute gas limit viscosity is based on the modified Chapman-Enskog theory. The theory is applicable to the prediction of the dilute gas limit for non-polar and polar fluids. The dilute gas limit viscosity, given by Chung et al.,13 can be written for a component as follows:
Si - 1
MWi ¼ MW S~i
ηI ¼ η0, mix þ ηp
Ωi ¼
ð8Þ
1:16145 0:52487 2:16178 þ ð0:77320T Þ þ ð2:43787T Þ - 6:435 0:14874 i i e e Ti 10- 4 Ti - 0:76830 sinð18:0323Ti - 0:76830 - 7:27371Þ
ð9Þ
ð14Þ
In eqs 7-9, MW is the molecular weight of a component, Tc is the critical temperature (K), and Pc is the critical pressure (bar). In the correlation for Pc, fc is the perturbation factor, which is 1 for normal alkanes. Hence, for all fractions for C6 and lighter fractions, fc is 1, and for pseudofractions, fc is tuned to match the experimental saturation pressure to the calculated saturation pressure using the equation of state.
where Ti ¼
1:2593T Tc, i
ð15Þ
where vc,i is the critical volume of the ith component (cm3 mol-1). For nonpolar gases Fc, i ¼ 1 - 0:2756ωi
3. Viscosity Modeling3-8
ð16Þ
The dilute gas limit viscosity contribution in the total viscosity is η0,mix and is calculated using the logarithmic mixing of the dilute gas limit viscosity contribution of the
The viscosity modeling in this section is based on refs 3-8. After the mass characterization and critical property characterization were completed, the general procedure of viscosity modeling requires one experimental viscosity data for
(12) Uyehara, O. A.; Watson, K.M. A. Universal viscosity correlation. Natl. Pet. News 1944, 714–722. (13) Chung, T. H.; Ajlan, M.; Lee, L. L.; Starling, K. E. Ind. Eng. Chem. Res. 1988, 27, 671–679.
(11) Lindeloff, N.; Pedersen, K. S.; Ronningsen, H. P.; Milter, J. J. Can. Pet. Technol. 2004, 43 (9), 47–53.
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: DOI:10.1021/ef101013m
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In equations 22-27, κa,0,0, κa,1,0, κa,1,1, κa,2,0, κa,2,1, κa,2,2, κ~ca, κr,0,0, κr,1,0, κr,1,1, κr,2,0, κr,2,1, κr,2,2, κ~crr, κrr,2,1, and κ~rc are 16 universal constants, and their values are given in Table 1 of ref 3a. Pc,i and Tc,i are the critical pressure and temperature of the ith component in the mixture, respectively. Pr and Pa are the pressure repulsion and attraction terms in the Peng-Robinson cubic equation of state, respectively.10 3.4. Kc Tuning. Kc is calculated using the experimental viscosity data. If the experimental viscosity data are denoted as ηexp, then Kc is given as
Table 1. Correlations for Coefficients of Pa and Pr pure and lumped components
heavy components
ηp ¼ Ka, 1 Pa þ Kr, 1 Pr þ Krr, Ι Pr 2 ηII ¼ Ka, II Pa þ Kr, II Pr þ Krr, II Pr 2
Ka, I ¼
m X
Zi
i¼1
ηc, i Ka, i Pc, i
Ka, II ¼
n X
Zi
i¼mþ1
! MWi 0:5 Pc 2=3 Ka, i Pc, i Tc, i 1=6
Kc ¼ Kr, I ¼
m X
Zi
i¼1
Krr, I ¼
m X i¼1
Zi
ηc, i Kr, i Pc, i
Kr, II ¼
n X
Zi
i¼mþ1
ηc, i Krr, i Pc, i 2
Krr, II ¼
n X
Zi
i¼mþ1
individual components η0, mix ¼ e
ð
n P i¼1
! MWi 0:5 Pc 2=3 Kr, i Pc, i Tc, i 1=6
ð17Þ
3.3. Calculation of the Friction Term Viscosity. The friction viscosity term is calculated in two parts: the first part (ηp) for pure and lumped components, for which the critical viscosity is known, and the second part (ηII) for the heavier components, for which the critical viscosity adjustable parameter Kc is to be tuned. ηp and ηII and can be written as follows: ηp ¼ Ka, 1 Pa þ Kr, 1 Pr þ Krr, 1 Pr 2
ð18Þ
ηII ¼ Ka, II Pa þ Kr, II Pr þ Krr, II Pr 2
ð19Þ
~v ¼ v - ζ
2 ηf ¼ Kr P~r þ Ka P~a þ Krr P~r
xi ε MW i i¼1
ð21Þ
Ka, i ¼ Kca þ ΔKa, i
ð22Þ
i ¼ heavy fraction
ΔKa, i ¼ Ka, 0, 0 ðΓi - 1Þ þ ðKa, 1, 0 þ Ka, 1, 1 Ψi ÞðeðΓi - 1Þ - 1Þ þ ðKa, 2, 0 þ Ka, 2, 1 Ψi þ Ka, 2, 2 Ψi 2 Þðeð2Γi - 2Þ - 1Þ
ð23Þ
Kr, i ¼ Kcr þ ΔKr, i
ð24Þ
4. Proposed Mixing Rule14 The mixing rule used is the same as in eq 20, but in the proposed rule, the exponential factor varies with pressure. Because ε accounts for the influence that the component asymmetry may have in the mixture,5 it seems more appropriate that it varies with pressure. At lower pressure, the mole fractions of light components continue to decrease, and hence, the heavier components are dominant in defining the viscosity. Heavier components have higher asymmetry than lighter
ΔKr, i ¼ Kr, 0, 0 ðΓi - 1Þ þ ðKr, 1, 0 þ Kr, 1, 1 Ψi ÞðeðΓi - 1Þ - 1Þ þ ðKr, 2, 0 þ Kr, 2, 1 Ψi þ Kr, 2, 2 Ψi 2 Þðeð2Γi - 2Þ - 1Þ
ð25Þ
Krr, i ¼ Kcrr þ ΔKrr, i
ð26Þ
ΔKrr, i ¼ Krr, 2, 1 Ψi ðeð2Γi Þ - 1ÞðΓi - 1Þ2
ð27Þ
ð30Þ
where P~r and P~a are the repulsive and attractive pressure contributions estimated at the displaced corrected volume v~, respectively. However, for a fluid below the saturation pressure region undergoing a compositional change because of the loss of the gas content from the liquid phase, a mixing rule involving the composition of the liquid phase has been found and is given as X xi MWi 1=3 ð31Þ ζ ¼ Kz
where n X
ð29Þ
where v~ is the corrected volume, v is the volume calculated using the equation of state, and ζ is the displaced volume. After such a compressibility correction, the pressure repulsion and attraction terms are calculated at the corrected volume using the following expression:
The list of the correlation for coefficients of Pa and Pr is given in Table 1. Zi in the equations for coefficients is given as xi Zi ¼ ð20Þ MWi ε MMi
MMi ¼
ð28Þ
The above form of the viscosity model is suitable for predicting the viscosity for light and lower medium oils. The error for predicting the viscosity is low. However, in the case of medium and heavy oils, the error keeps increasing with the pressure decreasing below the saturation pressure. There is a deviation for the pressure above the saturation pressure; however, this deviation is very small compared to the deviation in the case of the pressure below the saturation pressure. Thus, viscosity modeling becomes a challenge for heavy oils with Kc tuning. Hence, some modification7,8 was performed in the model for medium and heavy oils. In addition to the adjustable parameter, Kc, one more adjustable parameter, Kz, was included in the tuning model (Kz-Kc tuning model). 3.5. Kz-Kc Tuning.7,8 The procedure involves displacing the volume calculated from the equation of state by a factor ζ in the single phase in the following way:
! MWi 0:5 Pc 2=3 Krr, i Pc, i 2 Tc, i 1=6
xi ln η0, i Þ
ηexp - ηI ηII
(14) Kumar, A. Tuning of friction theory to predict viscosity of heavy oils. M.A.Sc. Thesis, University of Regina, Regina, Saskatchewan, Canada, 2009.
In above expressions, Ψi = RTc,i/Pc,i and Γi = Tc,i/T. 495
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: DOI:10.1021/ef101013m
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Figure 1. Viscosity variation with θ at a given value of R.
Table 2. Comparison between the Two Tuning Methods for Group I Oilsa percentage of average percentage of average absolute deviation absolute deviation (% AAD) (% AAD) 8 proposed tuning Kz-Kc model
pressure above saturation pressure below saturation pressure overall
1.34 12.28b 7.50
2.13 4.04 3.18
a The total number of data for oils 2-8 is taken as 86, and the number of data below the saturation point is 47. b Calculated as 1.34 þ (the number of total data/the number of data below saturation point) (7.5 - 1.34).
components. Hence, the influence of asymmetry becomes more at pressures below the saturation pressure. At pressures above the saturation pressure, because of the presence of lighter components, the influence of asymmetry is small. With the variation kept in view of the z factor with such a symmetry influence on a particular pattern, the exponential factor (ε) is made a function of the z factor. However, there is no available correlation for this, and it is tuned for the best fit with two experimental data. The expression for ε is given as follows: R ð32Þ ε ¼ pffiffiffi ½1þð zÞ1=θ where R and θ are both dimensionless variables, tuned with two experimental viscosity data, and z is the z factor. 5. Proposed Tuning Procedure An algorithm for the proposed tuning is described in the Appendix. Let P and P be two pressures (bar) above and below the saturation pressure, respectively, and η and η be the corresponding experimental viscosity data (mPa s). The viscosity operator simply involves a set of calculations to find ηI and ηII terms as defined in eqs 12 to 28 at a given pressure and temperature. Starting with the lowest value of R (=0.9) and θ (=0.9) as recommended in the algorithm, the viscosity calculated at pressure P to match η usually starts at a lower value and increases with increasing θ at a given value of R (Figure 1), but after a certain value of θ (θmax), the rate of increase becomes very small; hence, a change in R is required. An approximate value of θmax is given by the following equation: " # MWP 2:5 θmax ¼ exp ð33Þ MWTfc
Figure 2. Comparison of viscosity prediction using Kz-Kc tuning and the new tuning for group I oils.
Similarly, with a given θmax value, the viscosity increases with increasing R, but after a certain value of R i.e. acritical, the viscosity starts decreasing and η is never achieved. If this happens, θmax can be increased by 0.5 and tuning should start with starting values of R and θ (i.e., at iteration = 0). The process is continued until the viscosity reaches η. In the process of tuning, it is important to observe that R does not reach Rcritical. However, this may happen in rare cases. 5.1. Selection of Data for Tuning. Similar to Kz-Kc tuning, for the proposed tuning method, two experimental viscosity data are required, but whereas Kz-Kc tuning requires both data at a pressure above the saturation pressure, the proposed tuning needs one data at a pressure (10-15 bar) above the saturation
where fc is the perturbation factor in eq 9, MWP is the molecular weight of the plus fraction (g/mol), and MWT is the average molecular weight (g/mol). 496
Energy Fuels 2011, 25, 493–498
: DOI:10.1021/ef101013m
Kumar et al. Table 4. R and θ Values for All Oils
Table 3. Comparison between the Two Tuning Methods for Group II Oils
pressure above saturation pressure below saturation pressure overall
percentage of average percentage of absolute deviation average absolute (% AAD) deviation (% AAD) Kz-Kc tuning proposed tuning 0.71 19.41 10.06
1.34 2.18 1.70
group I
θ
R
oil 2 oil 3 oil 4 oil 5 oil 6 oil 7 oil 8
3.149 3.400 2.876 2.234 4.349 2.254 3.341
0.970 1.532 1.766 0.900 1.325 0.900 2.220
Kc (100) group II 0.284 1.758 14.060 3.018 5.760 3.251 36.607
oil 5A oil 5B oil 5C oil 6A oil 6B oil 6C oil 7
θ
R
Kc (100)
1.071 1.995 1.024 1.057 4.034 0.979 4.970
0.900 0.900 0.900 0.900 0.901 0.900 1.351
5.845 2.656 2.267 5.220 2.349 2.147 5.975
eliminated. This is further supported by the results for group II oils and presented in Table 3. It shows that the results obtained from the proposed tuning model are consistent.
pressure and the other data at a pressure 5 to 15 bars in the twophase region.
6. Result 7. Discussion
The new tuning model was applied to oils 2-6 from ref 8 and oils 7 and 8 from ref 7. They have been referred to as oils from group I. To check the consistency of the performance of the proposed tuning method, the same tuning model was applied to oil 5 (at three temperatures of 5A, 5B, and 5C), oil 6 (at three temperatures of 6A, 6B, and 6C), and oil 7 from ref 11. They have been referred to as oils from group II. The saturation pressure data in group II oils are not available, have been derived from the respective pressure versus viscosity curve, and hence, may have some insignificant errors. Mass characterization (pseudo-fractionation) and critical parameter characterization have been performed as discussed above under Compositional and Pressure-Volume-Temperature (PVT) Characterization. However, because composition details of C7þ fractions are not available, in place of M6, M3 was used. To provide better accuracy, p in eq 1 was searched at steps of 0.1 unlike several friction-theory-based applications, where it is considered as 0.5. Also, while applying the equation of state, the number of components was kept as 12, including 4 pseudocomponents. In the case of oils in group I, the number of components was 10, including pseudo-components. Starting with the oil sample, the overall composition details, and critical parameters, liquid-phase z-factor calculation for single and two phases (differential liberation process) was performed using the Peng-Robinson equation of state10 at the desired pressure and temperature. Corresponding viscosity was predicted using the algorithm described in refs 7 and 8 with the proposed tuning method. A comparison of the results for group I oils is shown in Table 2. The graphical comparison is displayed in Figure 2. Corrected data for oils 7 and 8 were received through personal communication from the corresponding author of refs 7 and 8. The following observations can be made from the trends in Figure 2 in the case of Kz-Kc tuning: (i) The higher the viscosity, the higher the deviation. (ii) With the decrease of the pressure in the two-phase region, the deviation increases. (iii) Deviation below the saturation pressure is very high compared to that above the saturation pressure. The proposed tuning is able to attend to all of the above concerns. The results for the proposed tuning model for group II oils are listed in Table 3. Table 4 lists the R and θ values for all oils studied. From the above analysis, it is evident that the new tuning method is very effective at reducing the error not only for pressure below the saturation pressure but also when all data are taken into consideration. Figure 2 shows that, with the proposed tuning, the two major drawbacks, i.e., increase in error with a decrease in the pressure below the saturation pressure and also with the increase of viscosity, have been
The proposed tuning has been developed considering the following two important aspects. 7.1. Compressibility Correction. Unlike the Peneloux volume translation15 for density calculation, the compressibility correction to find the improved Pr and Pa terms is a further compromise with the theoretical nature of the frictiontheory-based viscosity model. The compressibility correction is similar to the Peneloux volume shift15 to calculate density. Although applying the Peneloux volume shift15 does not change the phase behavior, the pressure attraction term does not remain unchanged. The proposed tuning also retains the theoretical character of the viscosity model using the friction theory. The proposed tuning in a novel manner successfully uses the experimental viscosity data at pressures below the saturation pressure. Earlier attempts to use such data4 resulted in deviations as high as 24%. Although such large deviations may be sometimes due to inaccuracy of the two-phase experimental viscosity data, the proposed tuning method can be used to improve the overall viscosity prediction and predictions for pressure below the saturation pressure. 7.2. Mixing Rule. The mixing rule (unless it varies with pressure) does not remain effective once it is attached to a tuning. As in the case of Kz-Kc tuning, the mixing rule is static for pressure above the saturation pressure, and hence, any mixing rule is as good as eq 20. There are some changes for the pressure below the saturation pressure (as the mole fraction of individual components changes), but they are not effective enough to limit the increasing error. The proposed tuning method takes care of the above issues and makes the mixing exponential factor dynamic with pressure, which is logical, because the impact of asymmetry on viscosity changes with pressure. This resulted in the desired impact on the viscosity prediction of heavy oils. 8. Conclusion The proposed tuning has resulted in an improved viscosity prediction for heavy oils. It has been able to attend to some of the major drawbacks of the friction-theory-based viscosity model as applied to heavy oils. It is important to note that mass characterization and critical property characterization (including critical viscosity), in the case of the heavy oils, involve large uncertainties. Hence, the performance of the friction-theory-based viscosity model for heavy oils may not be the same as for mixtures with some degree of certainties for these characterization parameters. Nonetheless, tuning (15) Peneloux, A.; Rauzy, E.; Freze, R. Fluid Phase Equilib. 1982, 8, 7–23.
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: DOI:10.1021/ef101013m
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methods with the friction theory are very effective in heavy oil viscosity predictions. The friction-theory-based viscosity model, with the proposed tuning for the heavy oils, performs reasonably well.
Nomenclature fc = critical pressure perturbation factor fdis = χ2 distribution function fmi = mass fraction of the C7þ-characterized fractions Kc = critical viscosity constant M6 = fluid total light mass fraction MW = molecular weight p = degrees of freedom from eq 1 Pa = van der Waals attractive pressure term Pr = van der Waals repulsive pressure term P = pressure Pc = critical pressure R = molar gas constant T = absolute temperature Tc = critical temperature S = MW-scaled variable v = molar volume x = mole fraction Z = mass-weighted fraction from eq 20 z = z factor from eq 32
Appendix
Greek Letters R = tuning parameter ε = mixing rule exponent η = total viscosity η0 = dilute gas viscosity ηc = characteristic scaling viscosity ηf = friction viscosity κa = linear attractive viscous friction coefficient κr = linear repulsive viscous friction coefficient κrr = quadratic repulsive viscous friction coefficient θ = tuning parameter ω = Pitzer’s acentric factor Acknowledgment. The authors express their gratitude to the Petroleum Technology Research Center (PTRC) for their financial support and to the following companies: Husky Oil Operations, Ltd., BP Exploration (Alaska), Inc., Penn West Petroleum, Ltd., Total E&P Canada, Ltd., ConocoPhillips Company, Devon Energy Corporation, Canadian Natural Resources, Ltd., Nexen, Inc., Shell Canada Energy, CANMET Energy Technology Center, and Saskatchewan Energy and Resources. We also express a special thank you to Dr. S. E. Qui~ nones-Cisneros for very fruitful discussions.
Figure A1. Algorithm for the new tuning model.
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