SPECIAL REPORT: Study evaluates viscosity prediction of crude blends

Oct. 16, 2006
Transporting crude oil blends via pipeline requires careful estimation of their viscosities.

Transporting crude oil blends via pipeline requires careful estimation of their viscosities.

Crude transportation often requires that different crude oils be blended and transported through the same pipeline. Intentionally diluting heavy crudes by adding less viscous oil also increases the efficiency of pipelining heavy oils. Solvents injected into the reservoir for well cleaning, stimulation, fracturing and, less frequently, miscible displacement may also remain in the crude during shipment.

A mixture’s viscosity as a function of composition is extremely complex.1 Theoretical considerations have offered little help in explaining these complexities. Attempts such as McAllister’s to derive a generalized expression for viscosities of all mixtures resulted in equations with many undetermined constants.2 3 No method allows a reliable prior prediction of these constants. These methods, therefore, are purely descriptive.

The predictive methods for viscosity of liquid mixtures include semi-theoretical and empirical models. Most semitheoretical models for petroleum fractions, which have a theoretical framework but parameters determined from experimental data, are based on either the corresponding-states approach or the modified Chapman-Enskog theory. Monnery and Mehrotra have reviewed such semitheoretical models.4 5

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This article reviews the empirical viscosity models for crude oil blends, and the validity and accuracy of 14 models suited for practical engineering, using 1,577 sets of viscosity data from 22 groups of crude oil blends (Table 1).

Mathematic models

A wide variety of liquid-mixture viscosity prediction formulas use the simple-mixing-rule equation based on calculations of the weighted average of the component viscosities (Equation 1, see accompanying box). This approach is very straightforward and requires no experimental viscosity determination of crude oil blends.

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The complex nature of crude oils, however, renders such formulas ineffective and has led to numerous efforts to develop viscosity models capable of adequately predicting the viscosity of crude oil blends. Most simple viscosity-mixing formulas do not involve any viscous interaction term and can be expressed by Equation 2.

Literature suggests many forms of the viscosity function, f(µ).6-10 Choosing f(µ) = lgµ transforms Equation 2 into the Arrhenius model (Equation 3).6

Viscosity function f(µ) may also serve as the cube root equation of viscosity, as in the Kendall-Monroe model (Equation 4),7 the reciprocal viscosity6 8 (Equation 5), double-log term9 (Equation 6), and the more complex Cragoe model (Equation 7).10

The viscosity functions, however, often do not obey the linear weighting rule described in Equations 3-7, necessitating further modifications to nonlinearity. Fig. 1 illustrates this deviation for the Arrhenius model.

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Based on the Arrhenius model, Lederer developed a correlation (Equation 8) for predicting the viscosities of heavy petroleum oil with light petroleum solvent mixtures.11 For the limiting value of α = 1,the logarithm of the mixture viscosity equals the volumetric fraction of heavy oil times the logarithm of the heavy oil viscosity, plus the volumetric fraction of the solvent times the logarithm of the solvent viscosity.

These volumetric fractions are unbiased. But for a value of α lower than 1, the volumetric fraction is biased toward the solvent, and the limiting value of 0 provides a mixture viscosity identical to the solvent viscosity.

Rhames and Nelson examined this equation for mixtures with low viscosity ratios, defined as (µ12), and found this functional expression provided an excellent fit for their data.6 Shu used the same functional expression for mixtures of high-viscosity ratios typical of bitumen and solvent fractions, and correlated the parameter with characteristic properties of the individual mixture components, including the viscosity ratio and the densities of solvent and oil (Equation 9).12

Barrufet et al. used their experimental viscosity data to calibrate the Lederer model (Equation 8) and fitted the parameter α to the power-law expression shown in Equation 10.13 Predicting the viscosity of the mixture with Equations 8 and 10 requires knowing the density, molecular weight, and viscosity of each component.

Grunberg and Nissan also proposed a modification of the Arrhenius equation to predict the viscosity of liquid binary mixtures (Equation 11), but there was no method published for predicting this parameter.14 15

Irving considered this the best descriptive of a binary liquid mixture.8

A change in volume or in the force field surrounding molecules generally causes nonideal behavior. Association generally increases as dissociation reduces viscosity from the values it would normally assume.

This is not, however, always the case with liquids. The nonideality of viscosity due to mixing creates a more complicated phenomenon, and there is no general law that explains why, or the manner in which it happens, particularly for mixtures of complex liquids, such as crude oils, which are termed undefined mixtures. Al-Besharah et al. assumed that any divergence from ideality (either positive or negative) could be expressed by an excess function, leading to a modified Arrhenius equation, including an excess function (Equation 12).16

Solving a set of four simultaneous equations and using experimental mixture data provides the four parameters of Equation 13.

Since the nonideal viscosity behavior increased with an increase in the difference between the densities of component crude oils, Al-Besharah correlated G in Equation 10 with (ΔAPI)2 and proposed a new viscosity correlation (Equation 14; OGJ, Feb. 20, 1989, p. 35).

Mehrota et al. proposed and validated a viscosity-mixing rule based on the Arrhenius model for Cold Lake bitumen and its five fractions (Equation 15).17

A mixture consisting of n > 2 components allows generalized expression of Equation 11 (Equation 16).

To improve the viscosity prediction, Mehrota developed a revised form of Equations 15 and 16 and derived a new mixing formula for the blends of toluene and bitumen-bitumen fractions (Equation 17).18

The deviation between the calculated viscosities of binary crude oil blends (Equation 2) and measured viscosities reached a maximum when the components had a 1:1 ratio. Accordingly, Li proposed that Cjk in Equation 16 be determined by Equation 18.19

Calculating the viscosities of crude oil blends by Equation 16 and 18 therefore requires the viscosities of every binary component blend.

f(µ)= lgµ allows Equation 16 to be rewritten as Equation 19.

Similarly, substituting the double-log viscosity function, or Cragoe model, in Equation 16 yields the models shown in Equations 20 and 21.20 21

In addition to introducing the binary interaction parameter, Cjk, to improve the accuracy of the viscosity models, this substitution introduced a fraction modification parameter, Bjk, defined by Equation 22.19 21

Equations 23-28 list the resulting representative modified models.

The preceding review shows that only the viscosities of each component were necessary for Equations 3-7, but the viscosities of every binary component blend with equal fraction were also needed for Equations 19-21 and 23-28.

It also, however, demonstrated that the parameters either obtained from physical properties or experiments should be inputs to the other models. Equations 3-7, 19-21, and 23-28, therefore, are most practical for engineering use.

Viscosity data

This article used both literature and in-house sources to compile 1,577 sets of viscosity data from 22 groups of crude oil blends, ranging from binary to nonary. One group of blends might have contained the same components as another group, but the fractions of each component were different.

Table 1 lists the sources and the number of the components of the crude oil blends. The components included paraffin-based crude oils, asphaltic-based heavy crude oils, and light crude oils. The crude oil blends included Newtonian fluids and non-Newtonian fluids.

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Table 2 shows the viscosity data of the crude oil blends.

Model evaluation

Four indexes described the deviation of the predicted viscosity from the measured viscosity: viscosity deviation (VD, Equation 29), absolute average deviation (AAD, Equation 30), maximum deviation (MAX, Equation 31), and profile of deviation (DP).

DP consists of three groups: D1 is the proportion of viscosity deviation falling into the range 0-6%; D2 is the proportion of viscosity deviation falling into the range of 6-15%; and D3 is the proportion of viscosity deviation falling into the range 15%-∞.

The consideration that, according to the precision of the viscosity measurement (dependent on viscometer and the procedures employed), reproducibility is 6% for Newtonian crude oils and 15% for non-Newtonian crude oils provides the basis for these groupings.

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Table 3 shows both the AADs and deviation profiles of the 14 models with reference to Newtonian viscosity and non-Newtonian apparent viscosity and the overall data without distinguishing between Newtonian and non-Newtonian flow behavior.

The models provide more accurate predictions for Newtonian crude oil blends than for non-Newtonian crude oil blends. The average AAD of the 14 models for Newtonian blends is 22.0%, while the average AAD for the non-Newtonian blends is 39.6%.

The nonlinearity modification with the binary interaction parameter and the fraction modification parameter greatly improves the accuracy of viscosity prediction. For the Newtonian blends, the average AAD of the first five models without nonlinearity modification is 47.3%, but the average AAD of the 9 nonlinearly modified models (Equations 19-21 and 23-28) is only 8.0%; very good given that the measured viscosity has a 6% margin of error. Similarly, for non-Newtonian blends, the average AAD of the first five models is 74.3%, but the average AAD of the 9 nonlinearly modified models is only 20.4%.

Combining indexes

Comprehensive evaluation of these models requires combining the AAD, MAX, and DP indexes. Newtonian blends used a DP value of the deviations greater than 6%. Non-Newtonian blends took the DP value as the proportion of deviations greater than 15%. Overall data points, including both Newtonian and non-Newtonian viscosities, used two values of DP:

  • DP1 denoting the proportion of the deviations greater than 6%.
  • DP2 denoting the proportion of the deviations greater than 15%.

Calculation of the comprehensive evaluation values converted the original values of these indexes with Equation 32.

This conversion places the values of the elements of the new index matrix, Y, in a range of [0, 1].

Multiplying the index matrix Y with the matrix of weight for each of the four indexes, ωT, evaluates the models (Equation 33).

The value of the evaluation matrix, E, from larger to smaller comprehensively ranks the models.

The weight of each index influences the evaluation results. This article used two groups of weight: Evaluation 1: ωDP1 = 0.2, ωDP2 = 0.3, ωAAD = 0.4, ωMAX = 0.1; and Evaluation 2: ωDP1 = 0.3, ωDP2 = 0.2, ωAAD = 0.4, ωMAX = 0.1.

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Table 4 shows the evaluation results. The two sets of weights resulted in the same ranking of the models. Equation 21 ranks first, with Equations 20, 24, 25, 27, and 28 quite close.

Acknowledgment

Financial support from the Key Research Project (No.104118) of the Ministry of Education, the People’s Republic of China, is greatly appreciated.

References

  1. Reid, R.C., Prausnitz, J.M., and Sherwood, T.K., “The Properties of Gases and Liquids,” 3rd edition; New York: McGraw-Hill, 1977, pp. 457-463.
  2. McAllister, R.A., “The Viscosity of Liquid Mixtures,” AIChE Journal, Vol. 6 (1960), pp. 427-431.
  3. Dizechi, M., and Marshall, E., “Correlation for Viscosity Data of Liquid Mixtures,” Industrial & Engineering Chemical Research, Vol. 21 (1982), pp. 282-289.
  4. Monnery, W.D., Svrcek, W.Y., and Mehrotra, A.K. “Viscosity: a Ctitical Review of Practical Predictive and Correlative Methods,” The Canadian Journal of Chemical Engineering, Vol. 73 (1995), pp. 3-40.
  5. Mehrotra, A.K., Monnery, W.D., and Svrcek, W.Y., “A Review of Practical Calculation Methods for Viscosity of Liquid Hydrocarbons and Their Mixtures,” Fluid Phase Equilibra, 1996, pp. 344-355.
  6. Rahmes, M.H., and Nelson, W.L., “Viscosity Blending Relationships of Heavy Petroleum Oils,” Analytical Chemistry, Vol. 20 (1948), pp. 912-915.
  7. Kendall, J., and Monroe, K.P., “The Viscosity of Liquids II: The Viscosity-Composition Curve for Ideal Liquid Mixtures,” Journal of ACS, Vol. 39 (1917), pp. 1787-1806.
  8. Irving, J.B., “Viscosity of Binary Liquid Mixtures: The Effectiveness of Mixture Eq.s,” National Engineering Laboratory Report, No. 630, February 1977, East Kilbride, Glasgow, Scotland.
  9. Chen Y., and Ma, H., “The Discussion of Viscosity Blending Relationships of Heavy Oil with Light Petroleum Solvent,” Oilfield Surface Engineering, Vol. 2 (1983), No. 4, pp. 1-8.
  10. Cragoe, C.S., “Change in the Viscosity of Liquid with Temperature, Pressure, and Composition,” Proceedings, World Petroleum Congress, London, Vol. 2 (1933), pp. 529-554.
  11. Lederer, E.L., Proceedings, World Petroleum Congress, London, Vol. 2 (1933), pp. 526-528.
  12. Shu, W.R., “A Viscosity Correlation for Mixtures of Heavy Oil, Bitumen, and Petroleum Fraction,” SPEJ, June 1984, pp. 277-282.
  1. Barrufet, M.A., and 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 Equilibria, Vol. 213 (2003), pp. 65-79.
  2. Grunberg, L., and Nissan, A.H., “The Energies of Vaporization, Viscosity and Cohesion, and The Structure of Liquids,” Transactions of the Faraday Society, Vol. 45 (1949), pp. 125-137.
  3. Grunberg, L. “The Viscosity of Regular Solution-System Involving Carbon Tetrachloride, Benzene, and Cyclohexane,” Transactions of the Faraday Society, Vol. 50 (1954), pp. 1293-1303.
  4. Al-Besharah, J.M., Salman, O.A., and Akashah, S.A., “Viscosity of Crude Oil Blends,” Industrial & Engineering Chemical Research, Vol. 26 (1987), pp. 2445-2449.
  5. Mehrotra, A.K., Eastick, R.R., and Svrcek, W.Y., “Viscosity of Cold Lake Bitumen and Its fractions,” The Canadian Journal of Chemical Engineering, Vol. 67 (1989), pp. 1004-1009.
  6. Mehrotra, A.K., “Development of Mixing Rules for Predicting the Viscosity of Bitumen and Its Fractions Blended with Toluene,” The Canadian Journal of Chemical Engineering, Vol. 68 (1990), pp. 839-848.
  7. Li, C., “Rheological Behavior of Crude Oil Blends,” MS Thesis, China University of Petroleum-Beijing. 1992.
  8. Liu, T., Sun, W., Gao, Y., and Xu, C., “Study on the Ordinary Temperature Transportation process of Multi-Blended Crude,” Oil & Gas Storage and Transportation, Vol. 18 (1999), pp. 1-7.
  9. Zhang, Q., “Gel Point and Viscosity Correlations for Crude Oil Blends,” MS thesis, China University of Petroleum-Beijing, 2004.
  10. Yang, X., and Zhang, J., “Pipelining of Shangjiashi Heavy Crude Oil,” Journal of the University of Petroleum -China, Vol. 9 (1985), No. 3.
  11. “The Study of Long-Distance Pipeline Techniques for Purified Heavy Crude Oil,” Xinjiang Petroleum Administration Bureau Report, Kelamayi, China, 1990.
  12. “The Experimental Study of Pipelining Xinjiang Crude Oil at Normal Temperature,” Pipeline Research Institute Report, Langfang, China, 1991.

The authors

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Jianhua Qian is a PhD candidate at the China University of Petroleum at Beijing. He holds a BSc (1983) in petroleum storage and transportation engineering from the East China Petroleum Institute and has served Sinopec’s pipeline company as an engineer and senior engineer since 1983. Mr. Qian has been engaged in pipeline operation technologies for 23 years.

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Jinjun Zhang ([email protected]) is a professor and head of the department of petroleum storage and transportation, China University of Petroleum at Beijing. He holds the BSc (1982), MSc (1987), and PhD (1998) degrees in petroleum storage and transportation engineering from the China University of Petroleum, and has 24 years’ experience in teaching and research of crude oil theory and pipeline transportation.

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Hongying Li ([email protected]) is an associate professor of the department of petroleum storage and transportation, China University of Petroleum at Beijing. She holds a BSc (1996) and an MSc (1999) in petroleum engineering and a PhD (2002) in petroleum storage and transportation engineering from China University of Petroleum. She has 7 years’ experience in research of crude oil theory and pipeline transportation.

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Qiang Zhang is an oil and gas filtration product engineer of the department of industrial and transportation business, 3M China Ltd. Co. He holds a BSc (1996) in mechanical design and manufacture and an MSc (2004) in petroleum storage and transportation engineering from China University of Petroleum. He has 5 years’ experience in teaching and research of petroleum mechanics, Huabei Petroleum Co., CNPC, and 2 years’ experience in research and application of oil and gas filtration product at BCB Filtration Technology Ltd. Co.