viscosity bergman sutton spe 110194 pa p

December 2009 SPE Reservoir Evaluation & Engineering 815

A Consistent and Accurate Dead-Oil-Viscosity Method

David F. Bergman, BP America, and Robert P. Sutton, Marathon Oil Company

Copyright © 2009 Society of Petroleum Engineers

This paper (SPE 110194) was accepted for presentation at the SPE Annual Technical Conference and Exhibition, Anaheim, California, USA, 11–14 November 2007, and revised for publication. Original manuscript received for review 2 August 2007. Revised manuscript received for review 31 December 2008. Paper peer approved 14 February 2009.

SummaryThe calculation of pressure drop resulting from the flow of oil through porous media or pipes requires the evaluation of viscosity. This is the single most important transport property necessary to calculate pressure drop accurately. The basis for oil-viscosity cal-culations using a traditional black-oil approach is the determination of dead- or gas-free-oil viscosity.

A total of 23 dead-oil-viscosity calculation methods have been identified from the literature and evaluated in this paper. A large database consisting of data from conventional pressure/volume/temperature (PVT) reports, crude-oil assays, and the literature was compiled from more than 3,000 samples from around the world. The number of actual viscosity measurements exceeded 9,800. An evaluation of the correlations yielded unacceptable results largely because of the failure of the methods to properly account for the physics of the problem. In general, this results from the methods’ failure to properly account for the change in viscosity with temperature and to address the chemical nature of the oil. A significant improvement in results can be realized through the use of the Watson characterization factor in addition to oil API grav-ity and temperature in the correlation of viscosity. This work has identified the character of the crude to have a significant effect on oil viscosity, especially for oils with gravities less than 25°API.

Methods have been proposed in the literature that use the Watson characterization factor; however, these have been largely ignored in the upstream oil industry. Therefore, a new method has been devel-oped that shows significant improvement over existing methods. At reservoir conditions, a 2- to 13-fold reduction in average absolute error was noted when compared with the error observed from traditional methods. At surface process conditions, this improve-ment ranged 3- to 60-fold. In addition, an updated correlation for Watson characterization has been developed. The ASTM density correction for varying temperature has been examined. Revised coefficients were developed that enhance the method’s accuracy for both oils and pure components and provide a suitable means to convert kinematic viscosity to absolute viscosity.

IntroductionNumerous relationships have been proposed over the years that relate dead-oil viscosity and temperature to commonly measured oil properties. Table 11 summarizes many of these relationships and supplements the information presented by Lake (Lake 2006). A review of these methods shows that the correlations can be placed into one of the four functional categories.

Table 1 shows Category 1 is the most popular equation choice for correlating viscosity, and traditionally this has been the approach used in the upstream sector of the oil and gas industry. It is also the least accurate approach for calculating viscosity and the change in viscosity with temperature.

A further note on Table 1—some methods substitute specific gravity for API gravity. These properties are directly related from the following well-known equation:

��

=+

141 5

131 5

.

. API

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

Additionally, other methods require the boiling point, to provide an additional characterization parameter for the oil. This quantity can be derived from the definition of the Watson characterization factor (Watson and Nelson 1933; Watson et al. 1935) as follows:

T Kb w= ( )�3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

In this paper, the Watson characterization factor (or Watson K fac-tor) is used to further describe the character of the oil.

Methods using the functional form offered by Category 4 require knowledge of density at temperature to convert from kine-matic viscosity to absolute viscosity.

� � �= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)

Therefore, an examination of auxiliary relationships between density and temperature is appropriate and is presented in a later section of this paper. Furthermore, the basic relationship of dead-oil viscosity and temperature is investigated to provide a sound physical basis to explore the accuracy of existing methods and to provide a foundation for the development of a more robust viscosity-calculation method.

DatabaseIn order to ascertain the accuracy of the published dead-oil-viscos-ity methods and the supporting auxiliary relationships, a database was developed. Data from conventional oil PVT reports, from crude-oil assays, and pure-component data were included in the database, which encompassed the following ranges of properties shown in Table 2 (Amin and Beg 1994; Amin and Maddox 1980; API 42 1966; BHP; BP; Chevron; Exxon; Petronas; Petrosky 1990; GeoMark 2006; Rønningsen 1995; Rossini et al. 1953; Santos; Statoil; SPR; Total; Watkins 1979).

A total of 9,837 viscosity measurements from 3,047 fluid samples are present in the database, representing samples from many of the major producing basins around the world. For many of the fluid samples, viscosity was measured at several tempera-tures, which helps to validate the measurement and ensure that this behavior is correlated properly.

Auxiliary RelationshipsThe development of a relationship to calculate dead-oil viscos-ity requires methods to reliably determine fluid properties, such as density, that change with temperature. Correlations that relate kinematic viscosity to temperature must include the evaluation of density at temperature to convert results to absolute viscosity. The Watson K factor is defined from the average boiling-point tempera-ture (BPT) and specific gravity. BPT is rarely reported for crude oils and, therefore, must be derived from other more commonly measured liquid properties. Furthermore, the basic relationship of viscosity and temperature must be explored to provide a realistic metric to evaluate both data and correlations.

Viscosity/Temperature RelationshipsDead-oil viscosity and temperature have a relationship in which viscosity decreases with increasing temperature for any given hydrocarbon liquid. Using the proper technique, a linear relation-ship can be established. The relationship is valid over a wide range of temperatures from the wax-appearance temperature (WAT) to the BPT of the sample. At temperatures below the WAT, the viscosity/

816 December 2009 SPE Reservoir Evaluation & Engineering

temperature slope trend increases as wax crystals appear in the liquid. This phonomenon does not occur at the WAT but usually 20–30°F below the WAT (as defined by rigorous methods such as cross-polar microscopy) when sufficient wax crystals are formed that perturb the normal viscosity/temperature relationship. At tem-peratures between the WAT and BPT, deviation from the straight line is an indication of inconsistencies in the measurement.

Over the years, several techniques have been proposed within the industry for linearizing viscosity with temperature. ASTM (Wright 1969) provides such a relationship between kinematic viscosity and temperature. The resulting equation is a modification of an earlier work dating to 1921 that related kinematic viscosity and temperature by using a double-log-form equation.

ln ln lnZ A B T( )⎡⎣ ⎤⎦ = − ( )abs , . . . . . . . . . . . . . . . . . . . . . . . . . (4)

where (Manning 1974)

Z = + + − − −( )� � �0 7 1 47 1 84 0 51 2. exp . . . . . . . . . . . . . . . . . (5)

The exponential term in Eq. 5 is significant only for kinematic viscosity less than 2 cSt. For larger viscosities, the exponential term goes to 0 and is, therefore, insignificant to this method. The ASTM method is used routinely throughout the oil industry to relate changes in kinematic viscosity with temperature.

Andrade (1930) proposed a method to relate dead-oil viscosity to absolute temperature:

ln ln�od AB

T( ) = ( )+

abs

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (6)

which is also sometimes expressed as

�od

BTAe= ( )abs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (7)

The method reportedly results in a linear relationship above the freezing-point temperatures to reduced temperatures of approxi-mately 0.7 (Reid et al. 1977).

Bergman (Whitson and Brulé 2000) demonstrated that dead-oil absolute viscosity could be expressed in a linear form using the fol-lowing relationship. The Bergman relationship was developed from an examination of pure-component behavior with temperature.

ln ln ln�od A B T+( )⎡⎣ ⎤⎦ = + +( )1 310 . . . . . . . . . . . . . . . . . . . (8)

The use of the ASTM and Bergman’s method is discussed further in Table 12.

A review of the database developed for this project was con-ducted for samples with viscosity measurements at three or more temperatures. A total of 1,301 samples were identified containing 6,614 measurements. As a supplement to Bergman’s method, the coefficients (310 and 1.0) in Eq. 8 were allowed to vary in a non-linear regression to determine an optimized function that reduced equation error. Results (labeled Bergman and Sutton) showed a slight variation (302.7 and 0.974) from Bergman’s original values, confirming the robustness of the method. The overall results are summarized in Table 3.

Despite its acceptance by the industry, the Andrade method performed poorly. The ASTM and updated Bergman and Sutton relationships performed comparably but only slightly better than the original Bergman method. Further correlation work with these three methods resulted in essentially the same statistical results. From a practical standpoint, the ASTM method using kinematic viscosity requires a density at each temperature to convert to absolute viscos-ity. At extreme temperature conditions, the adjustment of density for temperature effects can be a potential source of error. Therefore, the original Bergman method, which uses absolute viscosity directly, is recommended on the basis of its accuracy and simplicity.

Fig. 1 shows the linear relationship of viscosity with tempera-ture. Pure-component data from the n-paraffin, aromatic, cyclohex-ane, olefin, and naphthalene hydrocarbon families are depicted in the plot. In addition, actual crude-oil data are included to illustrate the behavior over a wide range of hydrocarbon liquids. In general, the data form a linear relationship with well-behaved slopes. This presentation is useful in the determination of consistent viscosity measurements.

Density/Temperature RelationshipsDead-oil-viscosity correlations are developed to determine kine-matic or absolute viscosity. Kinematic viscosity is related to absolute viscosity through the following relationship:

��

�= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (9)

The use of Eq. 9 requires the knowledge of density at the tem-perature of interest. Since dead-oil-viscosity correlations require

TABLE 1—SUMMARY OF DEAD-OIL-VISCOSITY CORRELATING METHODS

Category Equation Type Method

1 µod = f ( API,T ) Beal; Beggs and Robinson (1975); Glasø (1980); Labedi; Egbogah and Ng; Kaye (1985); Al-Khafaji et al. (1987);

Petrosky (1990); Kartoatmodjo and Schmidt (1994); De Ghetto (1994); Bennison (1998); Elsharkawy (1999); Bergman; Dindoruk and Christman (2001); Hossain et al.

(2005); and Naseri et al. (2005) 2 µod= f ( µref , Tb , T ) Orbey and Sandler 3 µod = f ( API , Kw , T ) Standing 4 v = f ( API , Tb , T ) Twu, Fitzgerald

TABLE 2—SUMMARY OF OIL PROPERTIES IN DATABASE

Property Minimum Maximum

Oil gravity (°API) 0.45 135.9 Watson characterization factor 10.8 14.25 Density (at temperature, g/cm3) 0.389 1.061 Dead oil viscosity (cp) 0.0596 1.357 × 1012 Temperature (°F) –40 500

TABLE 3—ACCURACY OF METHODS FOR VISCOSITY/TEMPERATURE EXTRAPOLATION METHODS

Method

% Absolute

Error (AE)

Standard Deviation

(SD)

% Average Absolute

Error (AAE) SD

ASTM –0.01 1.13 0.77 0.83 Andrade 0.19 6.10 3.85 4.74 Bergman –0.01 1.31 0.93 0.93 Bergman

and Sutton 0.00 1.20 0.84 0.85


oil API gravity as a correlating parameter, the density at 60°F is known.

��

o600 999012

141 5

131 5=

+.

.

. API

, . . . . . . . . . . . . . . . . . . . . . . . (10)

where the value 0.999012 g/cm3 is the density of water at 60°F, and the remainder of the equation is the standard relationship between specific gravity and API gravity.

ASTM (Petroleum Measurement Tables 1980) has developed a procedure for adjusting oil density to different temperature condi-tions. The thermal-expansion coefficient with a base temperature of 60°F is calculated from

��

�60

0 1

260

60

=+( )K K o

o

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11a)

The coefficients K0 and K1 are determined experimentally for the liquid of interest. Table 4 shows the values for generalized crude oils through lubricating oils provided by the ASTM method.

The density is calculated from

� ��

o o

T T

Te= − +( )⎡⎣ ⎤⎦

60

60 601 0 8� �., . . . . . . . . . . . . . . . . . . . . . . . (11b)

where

�T T= −60 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11c)

ASTM publishes tables of volume correction factors (VCFs) for various types of fluids and ranges of temperatures. The VCF is determined as follows

VCF = = − +( )⎡⎣ ⎤⎦1 60 601 0 8

Be

o

T T� �� .. . . . . . . . . . . . . . . . . . . . . (11d)

Therefore, the resulting density as affected by a temperature change can be determined from the density at 60°F and the VCF:

� �o oT=

60VCF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

Gomez (1992) published a method to determine VCF accounting for both liquid density and Watson K factor. The resulting equa-tion is

Y a a K a Kwa

oa T

wa

oa T= + + ( )0 1 1 8 4 1 8

22 3 5 6� �abs abs

. .

++ ( ) +a K a Kwa

oa T

wa

oa T

7 1 8

3

10 1 88 9 11 12� �abs abs

. .(( )+ ( )

4

13 1 8

514 15a Kw

aoa T� abs

.

, . . . . . . . . . . . (13a)

Viscosity of Pure Hydrocarbons by Family

0.1 cp

0.3 cp

0.5 cp

1 cp

3 cp

10 cp

100 cp

1000 cp10,000 cp

0 °F 50 °F 100 °F 150 °F 200 °F 250 °F 300 °F 400 °F–3

–2

–1

0

1

2

3

5.6 5.8 6 6.2 6.4 6.6

Ln (T + 310)

Ln L

n (V

isco

sity

,cp

+ 1)

n-ParaffinsAromaticsCyclohexanesNaphthalenesOlefinsCrude Oils

InconsistentMeasurement

Wax

Fig. 1—Relationship between viscosity and temperature for pure components in comparison to crude oil.

TABLE 4—COEFFICENTS FOR THERMAL-EXPANSION-COEFFICIENT EQUATION

Category K0 K1

Generalized crude oils 3.410957 x 10–4 0.0 Gasoline and naphthenes 1.924571 x 10–4 2.438 x 10–4 Jet fuels and kerosene 3.303010 x 10–4 0.0 Diesels, heating oils, and fuel oils 1.038720 x 10–4 2.701 x 10–4 Lubricating oils 1.440427 x 10–4 1.896 x 10–4 Generalized crude oils (BS update) 2.5042 x 10–4 8.302 x 10–5 Pure component (BS) 3.4175 x 10–4 –4.542 x 10–5


X a a K a K

a K a

wa

oa

wa

oa

wa

oa

= + +

+ +16 17 18

19

2 3 5 6

8 9

� �

� 220 2111 12 14 15K a Kw

aoa

wa

oa� �+

, . . . . . . . . . . . . . (13b)

VCF = Y

X, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13c)

where

a0 = 0.3569622 a11 = −7.061760 a1 = −0.1399101 a12 = −8.500160 a2 = −1.765440 a13 = −9.508922 × 10−6

a3 = −2.125040 a14 = −8.827200 a4 = 0.03861655 a15 = −10.62520 a5 = −3.530880 a16 = 0.3566053 a6 = −4.250080 a17 = −40.35242 a7 = −5.385089 × 10−3 a18 = 3,215.505 a8 = −5.296320 a19 = −129,456.5 a9 = −6.375120 a20 = 2,517,668 a10 = 3.627542 × 10−4 a21 = −19,053,400.

Eq. 13 is unnecessarily complex but offers a comparison to the ASTM procedure and adds Watson K factor as a correlating parameter.

The VCF methods offered by ASTM and Gomez were tested further against crude-oil data (1,516 measurements) derived from differential-liberation studies and data published by ASME (224). A total of 1,740 measurements were evaluated, with the results summarized below and in Fig. 2. Despite the addition of another correlating parameter, the results show the ASTM method to be more accurate than the Gomez method. It is interesting to note

that the ASTM options for kerosenes and lubricating oils yielded more accurate results than the general crude-oil option for this data set. To ensure that the errors were minimized for the viscos-ity calculations in this paper, the coefficients were updated. The accuracy of the updated ASTM method is depicted graphically in Fig. 3. These results show that in general, errors should be on the order of ±1.0% over the temperature range of interest. Errors do increase with temperature above the base temperature of 60°F. For most cases below 300°F, measured data yielding errors greater than 1% should be considered suspect (Table 5).

Pure-component data (2,121 measurements) reported by API was also evaluated (Table 6). The accuracy of the Gomez method improved slightly for the pure components, while the accuracy of the ASTM method declined. However, the equation coefficients offered by the ASTM method are calibrated for crude oils and refinery products. A nonlinear regression was used to develop coef-ficients suitable for the pure-component data. Results are shown in Fig. 4. As with the crude-oil data, the accuracy of the ASTM method is generally within ±1.0% of the temperature range of interest, with errors increasing at extreme temperatures. Therefore, it is desirable to limit the required temperature range to minimize the introduction of error. As a matter of practicality, dead-oil-vis-cosity measurements are considered to have an accuracy of 5–15% (Mehrotra et al. 1996; Twu 1985), so the procedure for determining VCF is more than adequate.

Crude-Oil CharacterizationWatson characterization factors are useful because they remain reasonably constant for chemically similar hydrocarbons. The Watson characterization factor provides a means of determining the paraffinicity or character of a crude oil or hydrocarbon component. A characterization factor of 12.5 or greater indicates a hydrocarbon compound predominately paraffinic in nature. Lower values of this

Volume Correction Factor Error(ASTM D 1250 General Crudes)

–5

–4

–3

–2

–1

0

1

2

3

4

5

0 100 200 300 400 500

Temperature, °F

VCF

Erro

r, %

Fig. 2—Accuracy of ASTM VCF calculation for crude oil using general crude values.

Volume Correction Factor Error(Bergman and Sutton)

–5

–4

–3

–2

–1

0

1

2

3

4

5

0 100 200 300 400 500

Temperature, °F

VCF

Erro

r, %

Fig. 3—Accuracy of ASTM VCF calculation for crude oil using updated coefficients.

TABLE 5—VCF METHOD ACCURACY (CRUDE OILS)

Method % AE Standard Deviation % AAE

Standard Deviation

ASTM—General crude oil –0.24 0.60 0.36 0.53 ASTM—Gasolines and naphthenes –1.01 1.07 1.08 1.00 ASTM—Kerosenes 0.11 0.56 0.31 0.48 ASTM—Diesels –0.19 0.58 0.34 0.51 ASTM—Lube oils 0.17 0.55 0.32 0.48 ASTM—API weighted combination –0.21 0.59 0.35 0.52 Gomez 0.39 0.66 0.52 0.55 BS update to ASTM –0.01 0.54 0.28 0.46


factor indicate hydrocarbons with higher amounts of naphthenic or aromatic components. Highly aromatic hydrocarbons exhibit values of 10.0 or less. For crude oils, the following ranges were observed by Nelson (1958) (see Table 7).

As defined, the Watson characterization factor is a function of BPT and specific gravity:

KT

wb

o

=1 3/

�. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

Using equations developed by Riazi-Daubert (1980) relating molecular weight to BPT and specific gravity, Whitson (1983) re-expressed Eq. 14 as a function of molecular weight and specific gravity:

K Mw o o= −4 5579 0 15178 0 84573. . .� . . . . . . . . . . . . . . . . . . . . . . . . (15)

The equation used by Whitson was developed from data taken from pure-component hydrocarbons from C5 to C20 and has a limited range of applicability to compounds with molecular weights from 70 to 300.

Riazi (2005) later updated the equation relating BPT to molecu-lar weight and specific gravity, increasing the upper limit to molecular weights ranging from 300 to 700; however, it was stated that this method also could be used for molecular weights as low as 70. This equation, modified to calculate Watson K factor, is

K aa M a

a MMw

o o

o ooa

oa=

++

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦1

2 3

4

5 6exp�

�� ⎥⎥

a

oa

7

8� , . . . . . . . . . . . (16)

where

a1 = 16.80642 a5 = 0.5369 a2 = 1.6514 × 10−4 a6 = −0.7276 a3 = 1.4103 a7 = 0.3333 a4 = −7.5152 × 10−4 a8 = −1.0.

Some of the oil and fraction data from the crude-oil assays reported specific gravity, molecular weight, and the Watson characterization factor. A total of 561 data points were identified, as shown in Fig. 5. The data range is shown in Table 8.

For comparison, the pure-component data are also presented on the plot. It should be noted that the pure-component data cover a wider range of Watson-characterization-factor values than the oil

TABLE 6—VCF METHOD ACCURACY (PURE COMPONENTS)

Method % AE Standard Deviation % AAE

Standard Deviation

ASTM—General crude oil –0.63 0.86 0.75 0.76 ASTM—Gasolines and naphthenes –1.42 1.57 1.51 1.49 ASTM—Kerosenes –0.44 0.71 0.59 0.59 ASTM—Diesels –0.20 0.62 0.47 0.45 ASTM—Lubricating oils 0.21 0.73 0.49 0.58 ASTM—API weighted combination –0.84 1.35 0.97 1.26 Gomez 0.03 0.67 0.48 0.47 BS fit to pure component data 0.00 0.59 0.39 0.43

Volume Correction Factor Error (Bergman and Sutton—Pure Component)

–5

–4

–3

–2

–1

0

1

2

3

4

5

–100 0 100 200 300 400 500 600

Temperature, °F

VCF

Erro

r, %

Fig. 4—Accuracy of ASTM VCF calculation for pure compo-nents using Bergman and Sutton method coefficients.

TABLE 7—RANGE OF CRUDE-OIL CHARACTERIZATION FACTORS

Crude-Oil Base Watson K Factor

Paraffinic 12.2–12.9 Intermediate 11.5–12.2 Naphthenic 10.5–11.5

Watson Characterization Factor(Pure Components, Oils, and Fractions)

9.0

9.5

10.0

10.5

11.0

11.5

12.0

12.5

13.0

13.5

14.0

0 20 40 60 80 100

API Gravity

Wat

son

K F

acto

r

Oils and FractionsAPI Project 44

Fig. 5—Reported Watson characterization factors for pure com-ponents, crude oils, and petroleum fractions.

TABLE 8—RANGE OF OIL PROPERTIES USED TO TEST WATSON K FACTOR RELATIONSHIPS

Property Minimum Maximum

API gravity 0.8 82.7 Specific gravity 0.661 1.070 Molecular weight 92 1320 Watson K factor 10.16 12.94


and fraction data, with the latter typically constrained to 10.8 < Kw < 13.5. This range agrees with the ranges for crude oils reported by Nelson (1958). Figs. 6 and 7 show the accuracy of the method proposed by Whitson. As expected, the accuracy degrades at molecular weights greater than 300, with values of Watson char-acterization factor overpredicted. The updated equation from Riazi (Figs. 8 and 9) shows increased accuracy up to molecular weights of approximately 700.

It will be demonstrated later that the Watson characterization factor can be an important parameter in the correlation of oil vis-cosity. Therefore, Eq. 16 was updated using nonlinear regression techniques to minimize the error in predicted Watson K factor resulting in the following equation:

K bb M b

b MMw

o o

o oob

ob=

++

⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦1

2 3

4

5 6exp�

�� ⎥⎥

b

ob

7

8� , . . . . . . . . . . . . (17)

where

b1 = 2012.84 b5 = 0.589485 b2 = −1.8519 × 10−3 b6 = 3.36211 b3 = −3.70833 b7 = 0.3333 b4 = 1.31441 × 10−3 b8 = −1.0.

The accuracy of Eq. 17 is shown in Figs. 10 and 11. It should be noted that Eqs. 15 and 16 were developed from lower-molecu-lar-weight, pure-component data. Eq. 17 was developed using higher-molecular-weight oil and fraction data and is accurate over a wide range of properties, as evidenced in Figs. 10 and 11. However, Eq. 17 is not suitable for lighter pure components with API gravity greater than 60, a specific gravity less than 0.74, and a molecular weight less than 150. Table 9 summarizes the results of the three methods.

For the purposes of this paper, the Watson K factor can be estimated from a TBP analysis using techniques discussed by UOP (UOP Method 375-88 1986). Alternatively, it can be calculated

Whitson Method

–10

–5

0

5

10

15

0 20 40 60 80 100

API Gravity

% E

rror

Fig. 6—Error in Whitson method for Watson characterization factor.

Whitson Method

–10

–5

0

5

10

15

0 200 400 600 800 1000 1200 1400

Mole Weight

% E

rror

Fig. 7—Error in Whitson method for Watson characterization factor.

Riazi Method

–10

–5

0

5

10

15

0 20 40 60 80 100

API Gravity

% E

rror

Fig. 8—Error in Riazi method for Watson characterization factor.

Riazi Method

–10

–5

0

5

10

15

0 200 400 600 800 1000 1200 1400

Mole Weight

% E

rror

Fig. 9—Error in Riazi method for Watson characterization factor.

Bergman and Sutton Method

–10

–5

0

5

10

15

0 20 40 60 80 100

API Gravity

% E

rror

Fig. 10—Error in Bergman and Sutton method for Watson characterization factor.


from the molecular weight and specific gravity using Eq. 17. These values can be obtained for the oil itself or derived from the plus fraction values reported in PVT reports. Fig. 12 shows the values derived for the entire database used in this study. This plot should provide guidance for typical values of Watson K factor with oil API gravity.

Dead-Oil-Viscosity CorrelationsA review of viscosity measurements shows common evaluation temperatures of 100 and 210°F. The measured data used in the Viscosity/Temperature Relationships section of this paper were used to determine the viscosity at a constant temperature of 100°F. The results are shown as a function of API gravity and Watson characterization factor in Fig. 13. The authors note that the lines in this plot are not meant to be correlations but are placed simply as a reference to aid in visualizing the trends. The plot clearly shows the effect of the Watson characterization factor on viscosity. As the characterization factor increases (i.e., the oil becomes more paraf-finic, containing long-chain paraffin molecules), viscosity increases for a given oil gravity. Furthermore, as the API gravity decreases, the effect of the characterization factor becomes more important. Fig. 14

illustrates the change in viscosity for a hypothetical set of oils with a base Watson K of 11.5. The change in viscosity with Watson K factor is greater as API gravity decreases. It is also noted that the change is less significant at higher temperatures.

Correlations proposed by Twu (1985), Fitzgerald (1994), Orbey and Sandler (1993), and Standing (1981) evaluate viscosity using the Watson K factor (or derived BPT) as a correlating parameter, while other methods ignore this parameter. Fig. 15 illustrates this result. The Beggs and Robinson (1975) method is plotted to illustrate the problem of not accounting for the effect of crude-oil character on viscosity. By eliminating this key property, one cannot hope to correlate viscosity accurately.

The Orbey-Sandler and Fitzgerald methods display anomalous behavior at low gravity and high values of Watson K factor. Refer-ring to Fig. 15, this behavior could have an impact on the method’s accuracy as the anomaly occurs within a range of expected proper-ties. Standing’s method tends to overpredict viscosity at the higher

Bergman and Sutton Method

–10

–5

0

5

10

15

0 200 400 600 800 1000 1200 1400

Mole Weight

% E

rror

Fig. 11—Error in Bergman and Sutton method for Watson char-acterization factor.

Fig. 12—Relationship between the Watson characterization factor and oil API gravity.

Trends in Viscosity with Watson K at 100°F

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

10.5 11 11.5 12 12.5 13 13.5Watson K Factor

Visc

osity

, cp

15 API20 API30 API40 API50 API

Fig. 13—Relationship between API gravity, Watson K factor, and viscosity.

Change in Viscosity with Watson Kat 100°F

1

10

100

1000

11.4 11.5 11.6 11.7 11.8 11.9 12 12.1Watson K Factor

% C

hang

e in

Vis

cosi

ty

15 API

15 API and 210°F

20 API

30 API

40 API

50 API

50 API and 210°F

1

Fig. 14—Change in viscosity with Watson characterization factor for constant API gravity.

TABLE 9—ACCUARACY OF WATSON K FACTOR CORRELATIONS

Method % AE SD % AAE SD

Whitson 1.27 2.20 1.52 2.04 Riazi –0.11 1.15 0.70 0.92 Bergman and Sutton –0.15 0.85 0.50 0.71


values of Watson K while the Twu method is more reasonably behaved.

On the basis of observations of the performance of the Twu method, it was decided to use it as a basis for an updated dead-oil-viscosity correlation. Various forms from Twu were tested against the database using a nonlinear regression technique to minimize the error in calculated oil viscosity. The final resulting equations are

T T

T

Tco

b

b

=

+ ×

+ ×

−

−

0 533272 1 91017 10

7 79681 10

4

8

. .

. bb

b

b

T

T

2

11 3

27 13

2 84376 10

9 59468 10

− ×

+ ×

⎛

⎝

⎜⎜⎜ −

−

.

.⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

−1

, . . . . . . . . . . . . (18a)

� = −1 TT

b

co , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18b)

ln . . .

.

� �

�

2 0 152995 2 40219 9 59688

3 45656 2

−( ) = −

+ −1143 632 4. �, . . . . . . . . (18c)

ln . . ln

. ln

�

�

1 0 701254 1 38359 2

0 103604 2

( ) = + ( )+ (

v

))⎡⎣ ⎤⎦2

, . . . . . . . . . . . . . . . . (18d)

� �

� �

oo = −

− −

0 843593 0 128624

3 36159 13749 53 12

. .

. ., . . . . . . . . . . . . . . . . . . . . (18e)

�� o o oo= − , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18f)

x Tb= −2 68316 62 0863 0 5. . . , . . . . . . . . . . . . . . . . . . . . . . (18g)

f x To o b22 0 547 6033= −� �� . . , . . . . . . . . . . . . . . . . . . . . (18h)

ln.

ln.

� �210

232 442

2

232 442+⎛⎝⎜

⎞⎠⎟

=+

⎛⎝⎜

⎞⎠⎟T Tb b

11 2

1 22

2

2+−

⎛

⎝⎜⎞

⎠⎟f

f, . . . . . . (18i)

f x To o b12 0 50 980633 47 6033= −. . .� �� , . . . . . . . . . . . . (18j)

ln.

ln.

� �100

232 442

1

232 442+⎛⎝⎜

⎞⎠⎟

=+

⎛⎝⎜

⎞⎠⎟T Tb b

11 2

1 21

1

2+−

⎛

⎝⎜⎞

⎠⎟f

f, . . . . . . (18k)

� �o o VCF100 60

0 999012 100= . , . . . . . . . . . . . . . . . . . . . . . . . . (18l)

� � �od o100 100100= , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18m)

� �o o210 600 999012 210= . VCF , . . . . . . . . . . . . . . . . . . . . . . . (18n)

� � �od o210 210210= . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (18o)

As a point of further discussion, Eqs. 18a through k follow the method proposed by Twu. Eqs. 18c, 18d, and 18g through k have been modified from the original Twu equations using a nonlinear regression routine designed to minimize the error in the calculated viscosity. The 9,837 data points included in the regression came from measurements from oils, petroleum fractions identified in crude assays, and pure-component data from API 42 (1966) and API 44. Because the goal of this work is the accurate simulation of crude-oil viscosity, the pure-component data were limited to data with Watson characterization factors of 10.8 to 13.0—a range con-sistent for crude oils. Furthermore the data included API-gravity

Trends in Viscosity with Watson K at 100°F

1E-01

1E+00

1E+01

1E+02

1E+03

1E+04

1E+05

1E+06

10.5 11 11.5 12 12.5 13 13.5Watson K Factor

Visc

osity

, cp

15 API20 API30 API40 API50 APIBeggs and RobinsonTwuOrbey and SandlerFitzgeraldStanding

Fig. 15—Characteristics of correlations for modeling viscosity behavior, 20°API oil, and variable Watson K factor.

Trends in Viscosity with Watson K at 100°F(Bergman and Sutton)

1.0E-01

1.0E+00

1.0E+01

1.0E+02

1.0E+03

1.0E+04

1.0E+05

1.0E+06

10.5 11 11.5 12 12.5 13 13.5Watson K Factor

Visc

osity

, cp

15 API20 API30 API40 API50 API

Fig. 16—Bergman and Sutton method for modeling viscosity behavior, variable Watson K factor, and 15–50�API.

TABLE 10—CORRELATIONS THAT PROPERLY MODEL VISCOSITY CHANGE WITH TEMPERATURE

Method 20°API 30°API 40°API

Twu X X X Orbey and Sandler X X X Fitzgerald X X X Bennison (1998) X Bergman X X X Dindoruk and Christman (2001)

X

Hossain X Bergman and Sutton X X X


WAT or boiling-point limit is beyond the scope of this paper. The data included in the regression included measured data over the temperature range −40 to 500°F, which should easily cover the range of expected temperature conditions.

The resulting correlation was tested to ensure it met physical-behavior criteria from real fluids. Fig. 16 shows the behavior of viscosity with the Watson characterization factor. Over the range of available data, the method is well behaved. Turning to the change in viscosity with temperature, Figs. 17 and 18 compare the behavior


0.1 cp

0.3 cp

0.5 cp

1 cp

3 cp

10 cp

100 cp

1000 cp10,000 cp

400 °F0 °F 50 °F 100 °F 150 °F 200 °F 250 °F 300 °F–3

–2

–1

0

1

2

3

5.6 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Ln (T + 310)

Ln L

n (V

isco

sity

,cp

+ 1)

n-ParaffinsAromaticsCyclohexanesNaphthalenesOlefinsBeal 1 Beal 2Beggs and Robinson Glaso Labedi-Libya Labedi-Nigeria/AngolaNg and Egbogah Kaye Al-Khafaji Petrosky Kartoatmodjo and SchmidtDe Ghetto De Ghetto-Agip Bennison Elsharkawy BergmanDindoruk and Christman Hossain Naseri

Fig. 17—Accuracy of Category 1 correlations to model viscosity change with temperature for 30°API oil.


0.1 cp

0.3 cp

0.5 cp

1 cp

3 cp

10 cp

100 cp

1000 cp10,000 cp

400 °F0 °F 50 °F 100 °F 150 °F 200 °F 250 °F 300 °F–3

–2

–1

0

1

2

3

5.6 5.7 5.8 5.9 6 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Ln (T + 310)

Ln L

n (V

isco

sity

,cp

+ 1)

n-ParaffinsAromaticsCyclohexanesNaphthalenesOlefinsTwu Orbey and Sandler Fitzgerald StandingBergman and Sutton

Fig. 18—Accuracy of Category 2–4 correlations to model viscosity change with temperature for 30°API 11.5 Kw oil.

ranges from 0.45 to 135.9 to maintain the integrity of the correla-tion over the target area of interest (5 to 80°API). The method determines oil viscosity at temperatures of 100 and 210°F, which are standard temperatures historically used in viscosity correlations and product specifications. The viscosity at the temperature of interest is then determined using the linear relationship determined from the Bergman method described in Table 12. The practical limits of this technique approximately constrain temperatures to the WAT and boiling point of the oil. The determination of the


TAB

LE 1

1a—

SUM

MA

RY O

F PU

BLI

SHED

DEA

D-O

IL-V

ISC

OSI

TY M

ETH

OD

S

Aut

hor

Cor

rela

tion

Orig

in

No.

of

Dat

a P

oint

s µ o

d R

ange

(c

p)

T R

ange

(°

F)

AP

I Ran

ge

AE

(%

) S

D

(%)

AA

E

(%)

Bea

l 1 (1

946)

(

Bea

l 197

0)

(S

tand

ing

1981

)

US

75

3 0.

865–

1,55

0 98

–250

10

.1–5

2.5

24.2

na

na

Bea

l 2 (1

946)

(

Bea

l 197

0)

(P

ipef

low

198

4)

w

here

US

75

3 0.

865–

1,55

0 98

–250

10

.1–5

2.5

24.2

na

na

C

1 = 0

.105

4399

× 1

02 C

2 = –

0.44

5214

2 C

3 = 0

.664

7024

× 1

0–2

C4 =

–0.

3359

724

× 10

–4

C5 =

–0.

4705

169

× 10

–1

C6 =

0.8

3155

74 ×

10–4

C

7 = –

0.78

9049

9 ×

10–7

C

8 = 0

.135

1673

× 1

0–2

C9 =

–0.

1144

95 ×

10–4

C

10 =

–0.

9550

553

× 10

–6

Beg

gs

and

Rob

inso

n (1

975)

na

460

na

70–2

95

16.0

–58.

0 0.

64

13.5

3 na

Gla

sø (1

980)

Nor

th S

ea

29

0.61

6–39

.10

50–3

00

20.1

–48.

1 na

na

na

Labe

di

(19

82 a

nd 1

992)

Liby

a 91

0.

66–4

.79

100–

306

32.2

–48.

0 2.

61

23.0

6 na

Labe

di (1

982)

Nig

eria

and

A

ngol

a 29

0.

72–2

1.15

10

4–22

1 25

.5–4

5.5

5.87

33

.03

na

Egb

ogah

a

nd N

g (1

983)

(E

gbog

ah

and

Ng

1990

)

na

39

4 na

59

–176

5.

0–58

.0

5.13

55

.51

na


TAB

LE 1

1b—

SUM

MA

RY O

F PU

BLI

SHED

DEA

D-O

IL-V

ISC

OSI

TY M

ETH

OD

S (C

ON

TIN

UED

)

Aut

hor

Cor

rela

tion

Orig

in

No.

of D

ata

Poi

nts

µ od R

ange

(c

p)

T R

ange

(°

F)

AP

I Ran

ge

AE

(%)

SD

(%

) A

AE

(%

)

Twu

(198

5)*

Pur

e C

ompo

nent

s (C

6–C

44)

Pet

role

um

Frac

tions

563

0.25

–290

cS

t at

100

°F

0.

33–1

,750

cS

t at

210

°F

100

and

210

–4.0

–93.

1 na

na

7.

85

Kay

e (1

986)

(K

aye

1985

)

Offs

hore

C

alifo

rnia

na

na

14

3–28

2 6.

6–41

.1

na

na

na

Al-K

hafa

ji (1

987)

na

350

na

60–3

00

15.0

–51.

0 2.

4 4.

8 3.

2

Pet

rosk

y (1

990)

(P

etro

sky

1990

; P

etro

sky

and

Fars

had

1995

)

Gul

f of

Mex

ico

118

0.72

5–10

.25

114–

288

25.4

–46.

1 3.

48

16.4

12

.38

Kar

toat

mod

jo a

nd

Sch

mid

t (19

91)

(Kar

toat

mod

jo a

nd

Sch

mid

t 199

4)

In

done

sia,

Nor

thA

mer

ica,

Mid

dle

Eas

t, an

d La

tinA

mer

ica

661

0.50

6–68

2.0

80–3

20

14.4

–59.

0 13

.16

na

39.6

1

* S

ee A

STM

met

hod

in T

able

12

to fi

nd

and

µod

at t

empe

ratu

re.

ν

Δ

ΔΔ

ΔΔ


TAB

LE 1

1c—

SUM

MA

RY O

F PU

BLI

SHED

DEA

D-O

IL-V

ISC

OSI

TY M

ETH

OD

S (C

ON

TIN

UED

)

Aut

hor

Cor

rela

tion

Orig

in

No.

of

Dat

a P

oint

s µ o

d R

ange

(c

p)

T R

ange

(°

F)

AP

I R

ange

A

E (%

) S

D (%

) A

AE

(%

)

Orb

ey

and

San

dler

(1

993)

Pur

e co

mpo

nent

da

ta n

C1–

nC20

olef

ins

and

cycl

ic

hydr

ocar

bons

fro

m A

PI p

roje

ct

44

Sel

ecte

d sa

mpl

es

from

AP

I pro

ject

42

na

0.2–

7,83

0 –3

10–6

44

6.3–

147.

6 na

na

6.

4

De

Ghe

tto (1

994)

Med

iterr

anea

n B

asin

, Afri

ca,

Per

sian

Gul

f, an

d N

orth

Sea

195

0.46

–1,

386.

9 81

–342

6.

0–56

.8

°AP

I<10

17

.4

10<°

AP

I<22

.3

37.8

22

.3<°

AP

I<31

.1

35.1

°A

PI>

31.1

21

.6

A

gip

30.7

°AP

I<10

8.

9 10

<°A

PI<

22.3

21

.9

22.3

<°A

PI<

31.1

22

.8

°AP

I>31

.1

15.6

Agi

p 20

.0

na


TAB

LE 1

1d—

SUM

MA

RY O

F PU

BLI

SHED

DEA

D-O

IL-V

ISC

OSI

TY M

ETH

OD

S (C

ON

TIN

UED

)

Aut

hor

Cor

rela

tion

Orig

in

No.

of D

ata

Poi

nts

µ od R

ange

(c

p)

T R

ange

(°

F)

AP

I R

ange

A

E

(%)

SD

(%

) A

AE

(%

)

Fitz

gera

ld (1

997)

* (A

PI T

DB

199

7 an

d Fi

tzge

rald

19

94)

Sau

di A

rabi

a, Ir

an,

Iraq,

Kuw

ait,

Liby

a,

Nor

th S

ea, U

S,

Indo

nesi

a, S

ovie

t U

nion

, Rom

ania

, an

d S

outh

Am

eric

a

7,26

7 0.

3–30

,000

cS

t –3

0–50

0 –2

–71.

5 3.

77

na

14.0

8

Ben

niso

n (1

998)

N

orth

Sea

16

6.

4–8,

396

39–3

00

11.1

–19.

7 na

na

16

.0

Els

hark

awy

(199

9)

Mid

dle

Eas

t 25

4 0.

6–33

.7

100–

300

19.9

–48

2.5

25.8

19

.3

Ber

gman

(20

00)

(Whi

tson

and

Bru

lé

2000

)

Wor

ldw

ide

454

0.5–

500,

000

40–4

00

12–6

0 –4

.65

38.5

2 27

.23

Sta

ndin

g (2

000)

(W

hits

on a

nd B

rulé

20

00)

na

na

na

na

na

na

na

na

* S

ee A

STM

met

hod

in T

able

12

to fi

nd ν

and

µ od

at t

empe

ratu

re.


TAB

LE 1

1e—

SUM

MA

RY O

F PU

BLI

SHED

DEA

D-O

IL-V

ISC

OSI

TY M

ETH

OD

S (C

ON

TIN

UED

)

Aut

hor

Cor

rela

tion

Orig

in

No.

of D

ata

Poi

nts

µ od R

ange

(c

p)

T R

ange

(°

F)

AP

I R

ange

A

E

(%)

SD

(%

) A

AE

(%

)

Din

doru

k an

d C

hris

tman

(200

1)

Gul

f of M

exic

o 95

0.

896–

62.6

3 12

1–27

6 17

.4–4

0.0

2.86

16

.74

12.6

2

Hos

sain

(200

5)

na

18

4 12

–451

32

–215

7.

1–22

.3

na

na

27.4

Nas

eri (

2005

)

Ira

n 25

0 0.

75–5

4 10

5–29

5 17

–44

6.61

na

7.

77 of pure components and crude oil with published correlations

assuming an oil gravity of 30°API and a Watson K factor of 11.5. It was established in Fig. 1 that viscosity should plot linearly and roughly parallel to the pure-component data. Methods that provide physically realistic results will honor these criteria. Noticeable anomalies are detected in the behavior of many of the methods. These anomalies are typically an abnormal change in the slope of the line, primarily at extreme temperatures (below 100°F or above 200°F). Some methods will even predict a decrease in viscosity with decreasing temperature, which is physically impossible.

As a further test, calculations were performed using all of the methods. A test matrix including oil API gravities of 20, 30, and 40 over a temperature range of 35–350°F was established. Using measured pure-component data as a standard, the methods were checked for linear behavior and response similar to the pure-com-ponent data. Table 10 indicates the methods that honored these criteria. Many methods though were found to provide inconsistent results. Methods not listed in Table 10 did not meet the criteria for the conditions tested.

The statistical accuracy of the correlations is summarized in Tables 13 through 16 and Figs. 19 through 42. It is important to note both the accuracy and consistency of the methods, which can be seen in the average-absolute-error and standard-deviation columns in the Tables 13 through 16 and can also be visualized in the plots of Figs. 19 through 42. Because the range of data is rather large, histograms were constructed to examine correlation accuracy over selected ranges of temperature, API gravity, and Watson K factor. These results are depicted in Figs. 43 through 49. Fig. 44 examines correlation accuracy over selected temperature ranges. Measured data were available to a minimum value of −40°F. Several correlations are not designed to evaluate viscosity at a temperature less than 0°F. This range was included only to illustrate the consistency and range of the new method. A more conventional (35–100°F) temperature range was included for a comparison of all of the methods, in Fig. 44.

Dead-oil-viscosity correlations are used to model oils at sur-face transportation/process temperatures, intermediate temperatures between the wellhead and reservoir, and at the reservoir temperature. Temperature conditions representative of each stage of the produc-tion process were selected to investigate correlation performance further. Table 14 provides a statistical summary for the methods over the temperature range 35–100°F. Fig. 45 graphically depicts this summary while providing insight over ranges of oil API grav-ity. Table 15 and Fig. 46 summarize correlation performance over the temperature range 100–200°F, while Table 16 and Fig. 47 cover the range 200–300°F. Fig. 48 provides a statistical summary for temperature conditions greater than 300°F. A detailed examination of these charts and tables reveals inconsistencies in many of the vis-cosity methods because of their failure to properly model viscosity changes with temperature over the range of oil gravity. The proposed method offers increased accuracy and consistency over the range of temperature and API gravity.

Conclusions1. A large database of crude-oil, petroleum-fraction, and pure-

component properties was created with the purpose of evaluating existing dead-oil-viscosity correlations and developing a new consistent and accurate method.

2. The existing dead-oil-viscosity correlations were categorized into four groups. Most of the methods fall into Category 1, which uses oil gravity and temperature to estimate absolute viscosity. This is the traditional approach and is also the least accurate approach. Improvements are seen in the Category 2 and 3 approaches; however, the Category 2 method lacks flexibility, while Category 3 is physically inconsistent at lower tempera-tures. Category 4 methods add an additional parameter, which further characterizes the oil and offers increased accuracy in the calculated viscosity.

3. The determination of the Watson characterization factor was evaluated using standard industry-accepted techniques. These were found to be in error when used to characterize the heavier


TABLE 13—STATISTICAL ACCURACY OF VISCOSITY METHODS *

Method # Pts % AE Standard Deviation % AAE

Standard Deviation

> 10% Error Count

Beal 1 8,950 18.6 102.8 59.6 85.8 7,684 Beal 2 9,024 34.5 120.9 68.1 105.7 7,862 Beggs and Robinson 9,024 222.9 2,693.0 248.7 2,690.7 8,203 Glasø 9,024 13.2 144.8 52.2 135.7 7,647 Labedi-Libya 9,024 87.5 523.4 121.6 516.5 8,087 Labedi-Nigeria/Angola 9,024 47.4 153.0 85.8 135.3 7,890 Egbogah and Ng 9,024 29.1 95.5 61.8 78.4 7,893 Twu 9,024 –9.6 28.7 20.4 22.4 5,228 Kaye 9,024 32.3 216.0 72.6 205.9 7,758 Al-Khafaji 8,973 17.4 342.4 66.5 336.3 7,883 Petrosky 9,024 19.4 141.5 57.1 130.9 7,830 Kartoatmodjo and Schmidt 9,024 18.8 245.7 60.3 238.9 7,600 Orbey and Sandler 9,024 –32.9 30.7 34.6 28.9 6,793 De Ghetto 9,024 26.2 93.7 60.0 76.6 7,852 De Ghetto-Agip 9,024 15.7 85.1 55.2 66.6 7,886 Fitzgerald 9,024 –9.1 25.9 19.4 19.5 4,939 Bennison 9,024 1.8 502.7 125.6 486.8 8,668 Elsharkawy 9,024 49.9 129.4 73.7 117.4 8,015 Bergman 9,024 33.8 130.3 61.6 119.7 7,730 Standing 9,024 174.3 1.5 E+04 202.4 1.5 E+04 5,839 Dindoruk and Christman 9,024 –1.7 85.3 45.3 72.3 7,792 Hossain 9,024 –50.4 105.2 81.8 83.1 8,681 Naseri 9,024 –15.2 91.8 55.7 74.6 8,153 Bergman and Sutton 9,024 –5.1 21.4 16.6 14.5 4,992

* API gravity range: 5–80; temperature range: 35–500°F.

TABLE 12—SUMMARY OF METHODS RELATING VISCOSITY AND TEMPERATURE AND PROCEDURE TO CALCULATE VISCOSITY AT ANY TEMPERATURE

Method Calculation Procedure

ASTM The ASTM method is defined

the slope, B, is determined from known viscosity at two temperatures, 100 and 210°F

and the viscosity at any temperature, T, can then be determined

convert kinematic viscosity to absolute viscosity

Note: for clarification

Bergman Bergman’s method is defined

the slope, B, is determined from known viscosity at two temperatures, 100 and 210°F

and the viscosity at any temperature, T, can then be determined


TABLE 14—STATISTICAL ACCURACY OF VISCOSITY METHODS*


Standard Deviation

> 10% Error Count

Beal 1 1,440 63.1 158.2 91.2 143.8 1,258 Beal 2 1,442 68.0 201.4 90.5 192.3 1,251 Beggs and Robinson 1,442 1,150.0 6,660.0 1,150.0 6,660.0 1,413 Glasø 1,442 19.1 130.5 61.2 116.8 1,242 Labedi-Libya 1,442 –17.4 82.9 52.1 66.8 1,285 Labedi-Nigeria/Angola 1,442 184.1 287.5 209.2 269.7 1,346 Egbogah and Ng 1,442 12.3 120.4 58.4 106.0 1,247 Twu 1,442 –11.7 40.3 25.8 33.2 975 Kaye 1,442 39.2 390.0 91.8 381.0 1,262 Al-Khafaji 1,433 55.2 144.6 84.3 129.8 1,256 Petrosky 1,442 9.6 101.5 60.5 82.0 1,293 Kartoatmodjo and Schmidt 1,442 43.2 152.8 76.4 139.3 1,268 Orbey and Sandler 1,442 –38.8 33.2 40.9 30.6 1,174 De Ghetto 1,442 32.1 150.4 74.4 134.5 1,272 De Ghetto-Agip 1,442 –2.2 102.2 54.4 86.5 1,282 Fitzgerald 1,442 –15.5 25.7 22.4 19.9 924 Bennison 1,442 –80.0 130.7 97.7 118.1 1,425 Elsharkawy 1,442 72.5 233.2 95.5 224.8 1,286 Bergman 1,442 52.8 235.6 80.9 227.4 1,229 Standing 1,442 11.5 545.0 70.8 540.5 1,109 Dindoruk and Christman 1,442 2.8 117.4 55.7 103.3 1,276 Hossain 1,442 –76.5 149.6 98.5 136.1 1,428 Naseri 1,442 36.1 150.6 77.6 134.0 1,279 Bergman and Sutton 1,442 –9.3 21.9 18.1 15.3 847




Standard Deviation

> 10% Error Count

Beal 1 4,442 30.8 97.8 61.6 82.0 3,761 Beal 2 4,462 45.5 107.4 70.0 93.3 3,880 Beggs and Robinson 4,462 73.9 126.8 96.9 110.2 4,060 Glasø 4,462 7.5 123.8 49.8 113.6 3,762 Labedi-Libya 4,462 23.4 129.6 60.9 116.8 3,867 Labedi-Nigeria/Angola 4,462 42.0 101.5 77.4 77.9 3,962 Egbogah and Ng 4,462 14.2 77.5 52.0 59.1 3,829 Twu 4,462 –9.4 27.3 20.6 20.3 2,654 Kaye 4,462 13.1 154.9 58.7 144.0 3,735 Al-Khafaji 4,433 38.8 468.0 69.7 464.4 3,833 Petrosky 4,462 6.3 99.6 49.3 86.7 3,792 Kartoatmodjo and Schmidt 4,462 10.2 189.3 52.8 182.0 3,766 Orbey and Sandler 4,462 –34.5 30.3 35.9 28.6 3,423 De Ghetto 4,462 14.1 73.7 52.9 53.2 3,848 De Ghetto-Agip 4,462 1.9 69.0 47.1 50.4 3,827 Fitzgerald 4,462 –9.5 25.9 19.8 19.2 2,503 Bennison 4,462 –69.7 65.0 82.3 48.1 4,341 Elsharkawy 4,462 36.3 95.4 64.1 79.4 3,903 Bergman 4,462 37.0 112.8 63.3 100.4 3,819 Standing 4,462 338.1 2.1 E+04 367.1 2.1 E+04 2,959 Dindoruk and Christman 4,462 –4.6 76.9 45.5 62.1 3,853 Hossain 4,462 –68.5 77.2 84.9 58.7 4,376 Naseri 4,462 –20.3 60.0 47.6 41.7 3,964 Bergman and Sutton 4,462 –5.0 22.3 17.6 14.6 2,657





Standard Deviation

> 10% Error Count

Beal 1 2,404 –13.6 53.7 40.5 37.9 2,044 Beal 2 2,434 15.7 73.7 54.6 51.9 2,103 Beggs and Robinson 2,434 10.0 67.0 50.1 45.6 2,100 Glasø 2,434 16.0 188.6 50.7 182.4 2,037 Labedi-Libya 2,434 161.8 348.2 174.6 342.0 2,296 Labedi-Nigeria/Angola 2,434 –5.4 56.1 41.6 37.9 2,063 Egbogah and Ng 2,434 51.0 95.8 73.3 80.1 2,224 Twu 2,434 –8.9 24.0 18.1 18.1 1,318 Kaye 2,434 49.8 158.0 77.1 146.6 2,121 Al-Khafaji 2,421 –21.2 125.4 49.9 117.0 2,121 Petrosky 2,434 32.8 176.2 59.0 169.2 2,105 Kartoatmodjo and Schmidt 2,434 14.1 347.9 56.9 343.5 2,000 Orbey and Sandler 2,434 –29.5 29.2 31.2 27.4 1,778 De Ghetto 2,434 34.3 82.7 60.5 65.9 2,109 De Ghetto-Agip 2,434 37.0 86.9 63.5 69.8 2,175 Fitzgerald 2,434 –5.5 26.1 18.2 19.6 1,239 Bennison 2,434 –6.4 86.4 63.5 58.8 2,218 Elsharkawy 2,434 54.4 98.2 75.6 83.1 2,236 Bergman 2,434 22.3 73.8 52.1 56.9 2,072 Standing 2,434 15.9 102.8 28.3 100.1 1,462 Dindoruk and Christman 2,434 –1.0 85.7 42.2 74.6 2,075 Hossain 2,434 –30.3 76.4 64.4 51.2 2,260 Naseri 2,434 –33.2 64.7 52.9 49.9 2,273 Bergman and Sutton 2,434 –3.6 20.2 15.2 13.8 1,253


Fig. 19—Accuracy of Beal 1 method. Fig. 20—Accuracy of Beal 2 method.

crude oils with API gravities less than 20 –30°API. A new method was developed that offers accurate results to 10°API.

4. Methods to depict linear trends in viscosity with temperature were evaluated. The Andrade method, which has been widely used in the industry, was found to be inaccurate. The accuracy of the ASTM and Bergman methods was found to be comparable. The Bergman method is recommended over the ASTM method because it is easier to apply.

5. Methods to correct density for temperature changes were evaluated. A modified ASTM approach was found to be suitably accurate for both crude oils and pure components.

6. A chart illustrating the effect of the Watson characterization factor on oil viscosity was prepared to emphasize the impor-tance of proper oil characterization to the accurate correlation of viscosity.

7. Existing correlations were compared for the consistency of the calculated viscosity with changing temperature using measured data trends as a metric. Several of the correlations failed this consistency test.

8. A new method was developed from data collected from the data-base. The new method (referred to as the Bergman and Sutton method) provides for accurate and consistent results over a wide


Fig. 21—Accuracy of Beggs and Robinson method. Fig. 22—Accuracy of Glasø method.

Fig. 23—Accuracy of Twu method. Fig. 24—Accuracy of Labedi (Libya) method.

Fig. 25—Accuracy of Labedi (Nigeria/Angola) method. Fig. 26—Accuracy of Egbogah and Ng method.


Fig. 27—Accuracy of Kaye method. Fig. 28—Accuracy of Al-Khafaji method.

Fig. 29—Accuracy of Petrosky method. Fig. 30—Accuracy of Kartoatmodjo and Schmidt method.

Fig. 31—Accuracy of Orbey and Sandler method. Fig. 32—Accuracy of De Ghetto method.


Fig. 33—Accuracy of De Ghetto-Agip method.

Fig. 35—Accuracy of Bennison method.

Fig. 38—Accuracy of Standing method.Fig. 37—Accuracy of Bergman method.

Fig. 36—Accuracy of Elsharkawy method.

Fig. 34—Accuracy of Fitzgerald method.


Fig. 39—Accuracy of Dindoruk and Christman method.

Fig. 41—Accuracy of Naseri method. Fig. 42—Accuracy of Bergman and Sutton method.

Fig. 40—Accuracy of Hossain method.

Effect of Oil API GravityAll Temperatures

0

10

20

30

40

50

60

70

80

90

100

Beal 1

Beal 2

Beggs

and R

obins

onGlas

ø

Labe

di-Lib

ya

Labed

i-Nigeri

a/Ang

ola

Egoba

h and

Ng

TwuKay

e

Al-Kha

faji

Petros

ky

Kartoa

tmod

jo an

d Sch

midt

Obey and S

andler

De Ghe

tto

De Ghe

tto-A

gip

Fitzge

rald

Bennis

on

Elshark

awy

Bergman

Standin

g

Dindoru

k and

Chri

stman

Hossa

inNas

eri

Bergman

and S

utton

Ave

rage

Abs

olut

e Er

ror,

%

5<API<80API<1010<API<2020<API<3030<API<4040<API<50API>50

Fig. 43—Summary of dead-oil-viscosity methods by API gravity for temperatures ranging 35–500°F.


Effect of Temperature(All API)

0

10

20

30

40

50

60

70

80

90

100

Ave

rage

Abs

olut

e Er

ror,

%

35-500°F-40-100°F 35-100°F100-200°F200-300°F>300°F

Beal 1

Beal 2

Beggs

and R

obins

onGlas

ø

Labe

di-Lib

ya

Labed

i-Nigeri

a/Ang

ola

Egoba

h and

Ng

TwuKay

e

Al-Kha

faji

Petros

ky

Kartoa

tmod

jo an

d Sch

midt

Obey and S

andler

De Ghe

tto

De Ghe

tto-A

gip

Fitzge

rald

Bennis

on

Elshark

awy

Bergman

Standin

g

Dindoru

k and

Chri

stman

Hossa

inNas

eri

Bergman

and S

utton

Fig. 44—Summary of dead-oil-viscosity methods by temperature for all ranges of API gravity.

Effect of Oil API GravityTemperature 35-100°F

0

10

20

30

40

50

60

70

80

90

100

Ave

rage

Abs

olut

e Er

ror,

%

5<API<80API<2020<API<3030<API<4040<API<50API>50

Beal 1

Beal 2

Beggs

and R

obins

onGlas

ø

Labe

di-Lib

ya

Labed

i-Nigeri

a/Ang

ola

Egoba

h and

Ng

TwuKay

e

Al-Kha

faji

Petros

ky

Kartoa

tmod

jo an

d Sch

midt

Obey and S

andler

De Ghe

tto

De Ghe

tto-A

gip

Fitzge

rald

Bennis

on

Elshark

awy

Bergman

Standin

g

Dindoru

k and

Chri

stman

Hossa

inNas

eri

Bergman

and S

utton

Fig. 45—Summary of dead-oil-viscosity methods by API gravity for temperatures ranging 35 –100°F.



0

10

20

30

40

50

60

70

80

90

100

Ave

rage

Abs

olut

e Er

ror,

%


Beal 1

Beal 2

Beggs

and R

obins

onGlas

ø

Labe

di-Lib

ya

Labed

i-Nigeri

a/Ang

ola

Egoba

h and

Ng

TwuKay

e

Al-Kha

faji

Petros

ky

Kartoa

tmod

jo an

d Sch

midt

Obey and S

andler

De Ghe

tto

De Ghe

tto-A

gip

Fitzge

rald

Bennis

on

Elshark

awy

Bergman

Standin

g

Dindoru

k and

Chri

stman

Hossa

inNas

eri

Bergman

and S

utton



0

10

20

30

40

50

60

70

80

90

100

Ave

rage

Abs

olut

e Er

ror,

%


Beal 1

Beal 2

Beggs

and R

obins

onGlas

ø

Labe

di-Lib

ya

Labed

i-Nigeri

a/Ang

ola

Egoba

h and

Ng

TwuKay

e

Al-Kha

faji

Petros

ky

Kartoa

tmod

jo an

d Sch

midt

Obey and S

andler

De Ghe

tto

De Ghe

tto-A

gip

Fitzge

rald

Bennis

on

Elshark

awy

Bergman

Standin

g

Dindoru

k and

Chri

stman

Hossa

inNas

eri

Bergman

and S

utton



Effect of Oil API GravityTemperature >300°F

0

10

20

30

40

50

60

70

80

90

100A

vera

ge A

bsol

ute

Erro

r, %


Beal 1

Beal 2

Beggs

and R

obins

onGlas

ø

Labe

di-Lib

ya

Labed

i-Nigeri

a/Ang

ola

Egoba

h and

Ng

TwuKay

e

Al-Kha

faji

Petros

ky

Kartoa

tmod

jo an

d Sch

midt

Obey and S

andler

De Ghe

tto

De Ghe

tto-A

gip

Fitzge

rald

Bennis

on

Elshark

awy

Bergman

Standin

g

Dindoru

k and

Chri

stman

Hossa

inNas

eri

Bergman

and S

utton

Fig. 48—Summary of dead-oil-viscosity methods by API gravity for temperatures ranging >300°F.

Effect of Watson K Factor(35 < T <500°F)

0

10

20

30

40

50

60

70

80

90

100

Ave

rage

Abs

olut

e Er

ror,

%

All KwKw<11.511.5<Kw<1212<Kw<12.5Kw>12.5

Beal 1

Beal 2

Beggs

and R

obins

onGlas

ø

Labe

di-Lib

ya

Labed

i-Nigeri

a/Ang

ola

Egoba

h and

Ng

TwuKay

e

Al-Kha

faji

Petros

ky

Kartoa

tmod

jo an

d Sch

midt

Obey and S

andler

De Ghe

tto

De Ghe

tto-A

gip

Fitzge

rald

Bennis

on

Elshark

awy

Bergman

Standin

g

Dindoru

k and

Chri

stman

Hossa

inNas

eri

Bergman

and S

utton

Fig. 49—Summary of dead-oil-viscosity methods by Watson K factor for temperatures ranging 35–500°F.


range of conditions. All of the viscosity methods were tested against the database, and the results were reported. These results can aid in the selection of suitable methods for engineering cal-culations that require viscosity over the wide range of conditions encountered in production and processing applications.

Nomenclature Bo = oil formation volume factor, bbl/STB Kw = Watson characterization factor Rs = solution-gas/oil ratio, scf/STB T = temperature, °F Tb = average BPT, °R Tc = critical temperature, °R Tabs = temperature, °R � = reduced BPT �60 = coeffi cient of thermal expansion at a base tempera-

ture of 60°F � = kinematic viscosity, cSt x, f1, f2,�1,�2 = correlating parameters � = absolute viscosity, cp �od = dead-oil viscosity, cp � = density, g/cm3

�o = oil density, g/cm3

�o60 = oil density at 60°F, g/cm3

�oT = oil density at temperature T, g/cm3

�API = oil API gravity �o = oil specifi c gravity

Subscripts 100 = property at 100°F 210 = property at 210°F

Superscripts ° = n-alkanes property

AcknowledgmentsThe authors would like to thank the management of Marathon Oil Company and BP America for permission to publish this paper. Finally, the primary author would like to thank his wife, Nancy. Without her patience and understanding, this would have never been written.

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Statistical QuantitiesAE = average error, %

AE calc meas

meas

=−

=∑100

1N

X X

Xi i

ii

N

AAE = average absolute error, %

AAE calc meas

meas

=−

=∑100

1N

X X

Xi i

ii

N

S = standard deviation

SX X

N

ii

N

=−( )

−=∑ 2

1

1

X = generic dependent variableN = number of observations

SI Metric Conversion Factors °API 141.5/(131.5+°API) = g/cm3

bbl × 1.589 873 E – 01 = m3

cp × 1.0* E – 03 = Pa�s ft3 × 2.831 685 E – 02 = m3

°F (°F – 32)/1.8 = °C °F (°F + 459.67)/1.8 = K psi × 6.894 757 E + 00 = kPa

*Conversion factor is exact.

Rob Sutton is a senior technical consultant for Marathon Oil Company in Houston where he works in the reservoir perfor-mance group under the upstream technology organization. He joined Marathon in 1978 after earning his BS degree in petro-leum engineering from Marietta College. Sutton also holds an MS degree in petroleum engineering from the U. of Louisiana at Lafayette. Dave Bergman joined Amoco as a research engi-neer in 1976 after earning his PhD degree in chemical engi-neering from the U. of Michigan. He joined BP at the merger in 1999 until he retired in 2008. During that time, Bergman was active in laboratory analyses and procedures and equation of state modeling of reservoir fluids. In his retirement he is still a consultant for fluid properties and other PVT areas.

2010 SPE Reservoir Evaluation & Engineering 1

Discussion of A Consistent and Accurate Dead-Oil-Viscosity Method

Faruk Civan, SPE, University of Oklahoma

SPE 110194-DS.

SummaryThe Vogel-Tammann-Fulcher equation (VTF), which allows for determination of the high-temperature limit of viscosity, critical-limit temperature, and activation energy is proven to be simpler, and more accurate and advantageous than Bergman’s equation for correlation of temperature dependence of oil viscosity.

Introduction Bergman and Sutton (2009) demonstrated by correlating numerous experimental viscosity data of various petroleum fluids that the viscosity equation of Bergman given below performs better than all other equations investigated in their article:

ln ln ln� +( )⎡⎣ ⎤⎦ = + +( )1 A B T To . . . . . . . . . . . . . . . . . . . . . . . (1)

We can write Eq. 1 as either of the following two forms:

� = − + + +( )⎡⎣ ⎤⎦{ }1 exp exp lnA B T To , . . . . . . . . . . . . . . . . . (2)

� = − + ( ) +( )⎡⎣

⎤⎦ =1 2 71828exp , . ...e T T e

A

o

B, . . . . . . . . . . . . (3)

where � represents temperature-dependent viscosity (cp), T is temperature (°F), To = 310, and A and B are empirical fitting constants. Further, for best data regression, Bergman and Sutton (2009) adjusted the constants of this equation from 1 and 310 to 0.974 and 302.7. However, they did not provide any comparison with the Vogel-Tammann-Fulcher equation (VTF) (Vogel 1921; Tammann and Hesse 1926; Fulcher 1925). Given the values of parameters A and B, calculation of viscosity should be made using Eq. 2, not Eq. 3.

Note that the units used in the above equation by Bergman and Sutton (2009) are inconsistent. All equations discussed here should be expressed in consistent units, as indicated in the Nomenclature. However, we retain the units considered by Bergman and Sutton (2009) in the following discussion for direct comparison of our results with theirs.

As explained here, Bergman’s equation (Eq. 1) performs well in correlation of temperature dependence of viscosity because it is a good approximation to the VTF equation in the range of the correlated viscosity data. However, it is mathematically complex and, therefore, requires tedious numerical processing of experi-mental data. In contrast, the mathematically simple VTF equation is practical, and it can produce results with comparable or better accuracy.

FormulationThe VTF equation has been proved to perform very well for many systems. It is an asymptotic exponential function expressed by (Civan 2006, 2007, 2008a, 2008b):

ln�

�c c

C

T T

⎛⎝⎜

⎞⎠⎟

=−

, . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (4)

where � represents temperature-dependent viscosity (Pa.s); �c is a pre-exponential coefficient (Pa.s); T and Tc are the actual and the critical-limit absolute temperatures (K) respectively; and C ≡ E/R,

where E is the activation energy (J/kmol) and R is the universal gas constant [J/(kmol·K)]. The best-estimate values of the three parameters of Eq. 4, �c, C, and Tc, can be determined uniquely using the least-squares method developed by Monkos (2003). An Excel™ spreadsheet prepared for this purpose is provided else-where by Civan (2008a).

It is not surprising that Bergman’s equation (Eq. 1) performs well because it can be manipulated as in the following to show that it is an approximation to the VTF equation. We carry out the follow-ing approximate mathematical analysis only for the case of � < 1. This condition can be satisfied by scaling the numerical values of viscosity in a suitable manner. After all, the order of magnitude of numerical values of viscosity depends on its unit such as centi-poise, poise, or Pa·s. Hence, comparison of its numerical value to 1.0 is meaningless because it depends on its unit. This is a problem of practical importance associated with Bergman’s equation. The VTF equation does not have such a problem. Obviously, for high numerical values, we can apply as an approximation � + 1 ≅ �.

Consider the following relationships given by Potter and Gold-berg (1987):

exp!

...,x xx

x( ) = + + + < ∞12

2

, . . . . . . . . . . . . . . . . . . . . . . . (5)

1

11 12

−= + + + <

xx x x..., , . . . . . . . . . . . . . . . . . . . . . . . . . (6)

ln( ) ...,12 3

1 12 3

+ = − + − + − ≤ <x xx x

x . . . . . . . . . . . . . . . . . (7)

We neglect the second- and higher-order terms in the following approximate analysis.

Applying Eq. 5, Eq. 3 can be approximated as

� ≅ ( ) +( ) = ( ) +⎛⎝⎜

⎞⎠⎟

e T T e TT

TA

o

B A

oB

o

B

1 . . . . . . . . . . . . . . . . . . (8)

Taking a logarithm of Eq. 8 and applying Eqs. 6 and 7 yields the following for |T/To| < 1:

ln ln ln ln� ≅ + ( ) + +⎛⎝⎜

⎞⎠⎟

≅ + ( ) − −A B T BT

TA B T B

T

Too

o1oo

o

o

A B T BTT

A

⎛⎝⎜

⎞⎠⎟

≅ + ( ) − − ++

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

≅ln 11

1++ ( ) + + −

+B T B

BT

T Too

o

ln

. . . . . . . . . . . . . . . . . . . . . . . . (9)

Thus, Eq. 9 can be cast in the form of a VTF-type equation as

ln ln ,

ln ln ,

� �

�

≅ ++

= + ( ) + = −

oo

o o o

C

T T

A B T B C BT

. . . . . . . . . . . . . . . . . . . . (10)

We see by comparing Eqs. 4 and 10 that �c = �o and Tc = −To. Hence, this exercise proves that Bergman’s equation can be approximated to match the form of the VTF equation.

2 2010 SPE Reservoir Evaluation & Engineering

does not give the best fit of data. The best-estimate values of the parameters of the VTF equation and, thus, the accurate correlation of temperature dependence of n-paraffin and crude-oil viscosity can be obtained (as shown in Figs. 3 and 4, respectively) using the spreadsheet program prepared by Civan (2008a) based on the Monkos (2003) least-squares regression method. The best-estimate parameter values for n-paraffin were determined with a coefficient of regression of R2 = 0.9993 (very close to 1.0) as �c, pre-exponential coefficient of viscosity, 0.0064 cp; Tc, critical-limit temperature of viscosity, −632.47 °F; and C, activation energy of viscosity/univer-sal gas constant, 2720.49°R. The best-estimate parameter values for crude oil were determined with a coefficient of regression of R2 = 0.9998 (very close to 1.0) as �c, pre-exponential coefficient of viscosity, 0.0211 cp; Tc, critical-limit temperature of viscosity, −129.16°F; and C, activation energy of viscosity/universal gas con-stant, 2492.86°R. This exercise demonstrates that the VTF equation can represent the viscosity data very accurately when the equation’s parameters are determined by the method of Monkos (2003).

Note that the correlation of various experimental data by Berg-man and Sutton (2009) using Eq. 1 indicates that B<0 (See Fig. 1). Thus, the limits of Bergman’s equation can be derived from Eq. 3, as indicated by Eqs. 11 and 12, as

� = → ∞0,T , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (11)

� → ∞ → −,T To. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (12)

Application, Comparison, and DiscussionWe do not have the actual numerical values of the viscosity data used by Bergman and Sutton (2009), but we can generate them from their figures by digitizing. Obviously, the values extracted this way involve uncertain amounts of digitizing errors in addition to the possible experimental errors associated with the data. How-ever, this is sufficient for our purpose of proving that Bergman’s equation is an approximation to the VTF equation in the range of data.

For example, Figs. 1 and 2 show the straight-line plots of the typical data extracted from Bergman and Sutton (2009) accord-ing to Bergman’s equation and the VTF equation, respectively. Both equations describe the data accurately, with coefficients of regression very close to 1.0. This exercise demonstrates that both Bergman’s equation and the VTF equation yield comparable-qual-ity correlations in the range of data used here. However, Bergman’s equation involves a double logarithm in the ordinate ln[ln(�+1)] and a single logarithm in the abscissa ln(T+To) in order to accom-plish a straight-line plotting of data. It is, therefore, more-compli-cated and -tedious than the VTF equation, for which straight-line plotting of data can be accomplished on a semilog coordinate

system by plotting ln� vs. 1

T To+.

The correlation presented by the VTF equation in Fig. 2 used the value of To = 310 according to Bergman’s equation. This value

n-Paraffiny = –1.8479x+9.6534

R² = 0.9992

Crude Oily = –2.7791x+18.667

R² = 0.9997

–2

–1

0

1

2

3

5.8 5.9 6 6.1 6.2 6.3

ln[ln

(µµ+1

)], µ

cp

ln(T+310), T °F

n-Paraffin

Crude Oil

Fig. 1—Correlation of temperature dependence of viscosity according to Bergman’s equation.

n-Paraffiny=861.18x−3.4646

R²=0.9956

Crude Oily=8018.7x−12.485

R²=0.9986

–2

0

2

4

6

8

10

0.0018 0.0022 0.0026 0.003

lnµµ ,

µ c

p

1/(T+310), T °F

n-ParaffinCrude Oil

Fig. 2—Correlation of temperature dependence of viscosity according to the VTF equation.

y=2716.6x−5.0529R²=0.9993

–2.0

–1.8

–1.6

–1.4

–1.2

–1.0

–0.8

0.0011 0.0013 0.0015 0.0017

ln (µµ

, cp)

1/(T–Tc), 1/°F

n-Paraffin

Fig. 3—Correlation of temperature dependence of n-paraffin viscosity according to the VTF equation by Monkos (2003) least-squares regression.

y=2492x−3.8549R²=0.9998

5.0

5.5

6.0

6.5

7.0

7.5

8.0

8.5

9.0

0.0035 0.0040 0.0045 0.0050 0.0055

ln (µµ

, cp)

Crude Oil

1/(T–Tc), 1/°F

Fig. 4—Correlation of temperature dependence of crude-oil vis-cosity according to the VTF equation by Monkos (2003) least-squares regression.


Recall that, for best data regression, Bergman and Sutton (2009) adjusted the constants of this equation from 1 and 310 to 0.974 and 302.7. If the adjusted values are considered, then the lower limit of viscosity will be −0.974+1.0 = 0.026.

On the other hand, noting that B<0 (see Fig. 1) and, thus, C>0 (see Figs. 2 through 4), the limits of the VTF equation can be derived from Eq. 10, as indicated by Eqs. 13 and 14, as

� �= → ∞o T, , . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (13)

� → ∞ → −,T To. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (14)

Therefore, the lower limit of the VTF equation �o is determined depending on the fluid type as illustrated above for n-paraffin and crude oil, respectively. Extensive examples are provided elsewhere by Civan (2006, 2007, 2008a), who determined the best-estimate values of the parameters To, �o, and C for the oils as a function of gravity (°API) or for the brines as a function of the dissolved salt type and concentration.

ConclusionIt is concluded that the VTF equation is simpler (Eq. 4), and more accurate and advantageous than Bergman’s equation (Eq. 1). The VTF equation should be tested further using the actual numerical values of all the data used by Bergman and Sutton (2009) to avoid the digitizing errors. However, the best-estimate values of To, �o, and C should be determined according to Monkos (2003) using the spreadsheet given by Civan (2008a).

Nomenclature A, B = empirical constants, dimensionless C = derived parameter equal to the E/R ratio, K E = activation energy, J/kmol R = universal gas constant, J/(kmol·K), or coeffi cient of

regression, dimensionless

T = actual absolute temperature, K Tc, To = critical-limit absolute temperature, K x = a general variable � = temperature-dependent viscosity, Pa.s �c, �o = pre-exponential coeffi cient, Pa.s

ReferencesBergman, D.F. and Sutton, R.P. 2009. A Consistent and Accurate Dead-Oil-

Viscosity Method. SPE Res Eval & Eng 12 (6): 815–840. SPE-110194-PA. doi: 10.2118/110194-PA.

Civan, F. 2006. Viscosity-temperature correlation for crude oils using an Arrhenius-type asymptotic exponential function. Petroleum Science and Technology 24 (6): 699–706. doi: 10.1081/LFT-200041178.

Civan, F. 2007. Brine Viscosity Correlation with Temperature Using the Vogel-Tammann-Fulcher Equation. SPE Drill & Compl 22 (4): 341–355. SPE-108463-PA. doi: 10.2118/108463-PA.

Civan, F. 2008a. Predicting Brine Viscosity With Temperature and Concentration Using the Vogel-Tammann-Fulcher (VTF) Equation (Addendum to SPE 108463). Paper SPE 118750 available from SPE, Richardson, Texas, USA.

Civan, F. 2008b. Use Exponential Functions to Correlate Temperature Dependence. Chemical Engineering Progress 104 (7): 46–52.

Fulcher, G.S. 1925. Analysis of Recent Data of the Viscosity of Glasses. J. of the American Ceramic Society 8 (6): 339–355. doi: 10.1111/j.1151-2916.1925.tb16731.x.

Monkos, K. 2003. A Method of Calculations of the Parameters in the Vogel-Tammann-Fulcher’s Equation: An Application to the Porcine Serum Albu-min Aqueous Solutions. Current Topics in Biophysics 27 (1–2): 17–21.

Potter, M.C. and Goldberg, J. 1987. Mathematical Methods, second edition. Englewood Cliffs, New Jersey, USA: Prentice Hall.

Tammann, G. and Hesse, W. 1926. Die Abhängigkeit der Viscosität von der Temperatur bie unterkühlten Flüssigkeiten. Zeitschrift für anor-ganische und allgemeine Chemie 156 (1): 245–257. doi: 10.1002/zaac.19261560121.

Vogel, H. 1921. Das Temperature-abhängigketsgesetz der Viskosität von Flüssigkeiten. Phys. Zeit. 22: 645–646.


Reply to Discussion of A Consistent and Accurate Dead-Oil-Viscosity Method

David F. Bergman, Consultant and Robert P. Sutton, Marathon Oil Company

SPE 110194-RP.

Faruk Civan’s discussion of our 2009 (Bergman and Sutton 2009) paper focuses on a small but important part of viscosity modeling, namely the accurate simulation of the temperature effect on oil vis-cosity. As discussed in our paper, many correlations fail to model the change in dead-oil viscosity with temperature adequately. We demonstrated the accuracy of industry accepted methods that establish linear relationships of dead-oil viscosity with tempera-ture. The methods evaluated came from ASTM (Wright 1969), Andrade (1930) and Bergman (Whitson and Brulé 2000). A data set of 6,614 viscosity measurements from 1,301 samples was assembled to evaluate and test method accuracy. We did not include an evaluation of Vogel’s method in our paper.

DiscussionAs reported by Poling et al. (2000), Andrade’s equation from 1930 provides a linear relationship of the natural logarithm of viscosity with reciprocal absolute temperature, as follows:

ln.

� = ++

AB

T 459 67. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)

Vogel’s equation appeared earlier in the literature in 1921 and was largely unnoticed (Scherer 1992). Later publications by Tammann and Hesse (1926) and Fulcher (1925) brought focus to this equa-tion form, which is currently known as the VTF method in honor of its contributors. Essentially, the equation is of the same form as Andrade’s equation with the exception of a variable temperature offset instead of the constant used by Andrade to obtain absolute temperature:

ln� = ++

AB

T C. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (2)

The form of Eq. 2 may also be recognized as the Antoine equa-tion, which has gained use in the correlation of vapor pressure. Poling et al. (2000) state that the Antoine equation is usable over a limited pressure range and should never be extrapolated beyond data limits used to develop the equation coefficients (A, B, and C). Examples will be shown to illustrate that these concepts also apply to the VTF equation for the correlation of the viscosity/temperature relationship.

Civan claims the units used in Bergman’s equation (Eq. 3) are inconsistent, which is false. Further claims are made that the Bergman and VTF equation forms are approximately the same. Bergman, along with 19 of the 23 literature dead-oil-viscosity methods identified in Table 11 from our paper, used viscosity and temperature units of cp and °F, respectively. Bergman modified a form of the MacCoull-Walther equation long used by ASTM to develop linear kinematic viscosity/temperature relationships (Wright 1969). Table 1 compares the development (Stachowiak and Batchelor 2005) of both the Bergman and VTF equation forms. These results show that the two equation forms are different.

Bergman empirically developed the equation by examining relationships that linearize absolute viscosity/temperature data while using common oilfield units, resulting in the following equation:

ln ln ln� +( )⎡⎣ ⎤⎦ = + +( )1 310A B T . . . . . . . . . . . . . . . . . . . . . . (3)

The original equation used by ASTM added 0.8 to the kinematic-viscosity term, which limited the equation’s use to certain higher-viscosity lubricating oils. In the equation’s current form, ASTM varies the constant from 0.7 to 0.863, depending on kinematic viscosity. Bergman’s equation uses a value of 1, effectively making the equation suitable for all ranges of dead-oil absolute viscosity. The relationship was originally developed primarily from pure-component viscosity and temperature data, as indicated on the Bergman plot in Fig. 1. The relationship linearizes data from not just one sample but uses the aggregate data set, and further work by Bergman and Sutton confirmed this relationship using a more diverse sample data set. Several uses of this relationship and the Bergman plot have subsequently been identified, which include

• Pure-component paraffin, aromatic, cyclohexane, naphthalene, and olefin liquid hydrocarbons exhibit viscosity/temperature behav-ior trends that are approximately linear and parallel in nature.

• Express dead-oil viscosity as a linear function of temperature.• Provide a basis for determining the consistency of measured

crude-oil viscosity and temperature data.• Identify non-Newtonian liquid viscosity behavior.In his discussion, Civan prepared plots from two data sets that

were digitized from the Bergman plot. This data included measure-ments from n-hexane and a 16.9°API crude oil. Civan’s Fig. 1 plots the data using Bergman’s coordinates and illustrates the linear and approximate parallel nature of these significantly different liquids. Civan’s Fig. 2 recreates the plot using VTF equation coordinates and arbitrarily uses a temperature offset of 310. The resulting relationships are linear over the limited temperature interval but deviate from linear as the temperature range is expanded. Fur-thermore, the slopes of the lines differ by an order of magnitude, which complicates their use in verifying measurement consistency. To further test the applicability of the VTF equation, the data set used previously to validate Bergman’s relationship was used to find an average temperature offset (C) for the VTF equation. This was determined to be 165.51. The resulting equation accuracy is compared to our prior work in Table 2.

Optimizing the temperature offset results in improved method accuracy over Andrade’s method, but the accuracy lags results from ASTM, Bergman, and Bergman and Sutton. Fig. 2 was developed using the VTF form equation and a constant temperature offset of 165.51. Note that the temperature relationships are not linear over a wide temperature range and the significant change in slope for different samples hinders checks for data quality and consistency. This illustrates the problem of using an average temperature offset with a diverse set of samples.

Optimal usage of the VTF equation is achieved by specifically fitting the equation to individual fluids, as indicated by Civan in the development of his Figs. 3 and 4. The temperature offsets used in these figures are approximately 634 (Civan Fig. 3) and 129 (Civan Fig. 4), which results in widely varying ranges for each plot’s x-axis. Combining the results on a single plot provides results that plot near opposite ends of the x-axis, which creates difficulty in the comparison and validation of samples. As a further test of using the VTF equation to aid in sample validation, the example from the Bergman plot labeled “wax” was investigated. This sample had viscosity measured at temperatures above and below the wax appearance temperature. The change in viscosity character (i.e., slope change) is clearly apparent in the Bergman plot because of a non-Newtonian behavior resulting from the presence of wax crystals in the oil at lower temperatures. Fig. 3 shows this effect is effectively muted by the tuned VTF equation.


TABLE 1—DEVELOPMENT OF BERGMAN FORM AND VTF FORM EQUATIONS

MacCoull-Walther Form Equation Vogel (VTF) Form Equation

Bergman modified the MacCoull-Walther equation as follows

Assuming d=2.718 281 828 and taking the natural logarithm on each side of the equation

.

.

The natural logarithm is taken again to develop a linear equation

.

Using pure component hydrocarbon viscosity/temperature data from several families of compounds routinely found in crude oil, Bergman empirically determined a=1 and e=310, which results in the final equation

.

Taking the natural logarithm on each side of the equation gives the following result

,

which simplifies to the following linear equation

For the temperature offset c, Andrade used c=459.67 as his method preceded the more general VTF method. For mineral oils over a wide range of temperature, Cameron (1945) determined c=139. For crude oils, Civan (2006) recommended c=0. Bergman and Sutton determined a value of c=165.51 from their database.

µ

µ

µ

µ

µ

µ

µ

µ( )

[

[ ]

[ ] ( )

]


0.1 cp

0.3 cp

0.5 cp

1 cp

3 cp

10 cp

100 cp

1000 cp10,000 cp

0°F 50°F 100°F 150°F 200°F 250°F 300°F 400°F–3

–2

–1

0

1

2

3

5.6 5.8 6 6.2 6.4 6.6

Ln (T, °F + 310)

Ln L

n (V

isco

sity

,cp

+ 1)

n-ParaffinsAromaticsCyclohexanesNaphthalenesOlefinsCrude Oils


Wax8.6 API

16.9 API

16.1 API39.9 API

25.9 API

32.4 API

35.0 API

Linear Trend

Fig. 1—Bergman plot with crude-oil examples.

TABLE 2—ACCURACY OF METHODS FOR VISCOSITY/TEMPERATURE EXTRAPOLATION

Method %

Average Error Standard Deviation

% Average Absolute Error

Standard Deviation

ASTM 0.01 1.13 0.77 0.83 Andrade (Eq. 1) 0.19 6.10 3.85 4.74 Bergman (Eq. 3) 0.01 1.31 0.93 0.93 Bergman and Sutton 0.00 1.20 0.84 0.85 VTF (Eq. 2, C=165.51) 0.07 3.88 2.59 2.89


The primary use in our paper for Bergman’s Eq. 3 is to deter-mine viscosity at different temperatures. The Bergman and Sutton method determines dead-oil viscosity at 100 and 210°F and then, using Eq. 3, evaluates viscosity at a specific temperature of interest. Since the VTF equation is a three-parameter equation, it cannot be used in this scenario because it cannot be defined properly as a fully tuned equation from only two data points. Therefore, further discussion of the VTF method is only for academic interest.

Viscosity data for pure n-decane was derived at a pressure of 100 psia using methods available from NIST (Lemmon et al. 2010). This pressure was selected to increase the upper temperature limit to a comparable value investigated by Bergman and Sut-ton. The resulting data are plotted in Fig. 4, in which a smooth

Bergman Plot Using VTF Formulation

–4

–2

0

2

4

6

8

10

12

0.001 0.002 0.003 0.004 0.005 0.006 0.007

1/(T, °F + 165.51)

Ln (V

isco

sity

, cp)

n-Paraffins

Aromatics

Cyclohexanes

Naphthalenes

Olefins

Crude Oils

16.1 API

8.6 API

16.9 API

25.9 API

35.0 API32.4 API

39.9 API


Wax

Fig. 2—Check of VTF method for linear viscosity/temperature behavior.

Wax Example

y=433.76x – 0.9791R2=0.9971

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014

1/(T, °F + C)

Ln(V

isco

sity

, cp)

Wax appears as measurement variation

Fig. 3—Failure of VTF method to detect nonlinear behavior resulting from wax crystals.

relationship is established over the temperature range −20 to 500 °F. This large temperature variation is required because viscosity models may be used in cold environments (deep water or arctic) or in hot environments (deep reservoir or thermal). Furthermore, this approximates the temperature range of −100 to 700°F reported by Wright (1969) for the MacCoull-Walther equation. Results dis-playing method error with temperature show the Bergman method to be more accurate overall when compared with the VTF method (Fig. 5). The average temperature offset of 165.51 determined from our larger data set was used in the development of Fig. 5. The use of a generalized temperature offset adversely affects the accuracy of the VTF method; however, improved accuracy can be obtained when the equation is tuned using linear regression, as described


by Civan (2008), where all three coefficients are optimized. The significantly improved results from the tuned VTF equation are compared with Bergman’s equation in Fig. 6.

For data encompassing large temperature intervals, the tuned VTF method displays a characteristic trend to underpredict viscosity at low temperatures and overpredict viscosity at high temperatures. For application with the Bergman and Sutton method, viscosity is calculated at two temperatures, as previously discussed. Results for Bergman’s equation were determined using only data at these two temperatures. In order to test the tuned VTF equation for extrapo-lation accuracy, a third intermediate point was added so that the equation’s three coefficients could be determined. These results are provided in Fig. 7. The error resulting from Bergman’s equation is only slightly larger than the error determined from fitting the entire data set. The VTF results show error similar to that in the evaluation in Fig. 6 at low temperature. For the intermediate tem-perature range used to fit the equation, the results are comparable to the results from Bergman’s equation. At higher temperatures, the

error becomes excessive. This test confirms the observation from Poling et al. (2000) that data fitted with an Antoine style equation are modeled accurately over a limited (temperature) interval, and the tuned equation should not be extrapolated.

Civan recommends the use of correlations that were developed to use oil API gravity to determine the constants A and B used in the VTF equation. In that work, Civan (2006) recommended a constant temperature offset (C = −460) where the equation was defined with temperature units of degrees Rankine. The problems previously discussed with an average temperature offset went unnoticed because of a limited temperature range of the viscosity data and only four measurements available for each sample. Fur-thermore, the method is limited to oils with API gravity ranging from 20 to 50. The slope term, B, is correlated with a quadratic equation, which can be problematic if used outside the specified range. As a side note with reference to Fig. 4, the values of the VTF constants, A, B, and C vary with the portion of the viscosity/temperature relationship fitted with the VTF equation. Therefore,

n-Decane Viscosity at 100 psia

0.0

0.5

1.0

1.5

2.0

2.5

3.0

–100 0 100 200 300 400 500 600

Temperature, °F

Visc

osity

, cp

VTF CoeffsA=–3.49548B=1,370.5C=333.73

VTF CoeffsA=–5.24354B=3,582.2C=632.85

VTF CoeffsA=10.771B=40,738C=–3,641.6

VTF Coefficients for all dataA=–5.0834, B=3,091.8, C=546.36

Fig. 4—n-decane viscosity and VTF equation coefficients.

Comparison of Viscosity Method Error(Decane)

–20

–10

0

10

20

30

40

–100 0 100 200 300 400 500 600

Temperature, °F

% E

rror

BergmanVTF

Fig. 5—Comparison of method errors where VTF method uses a constant temperature offset (165.51).


general correlations using only a parameter such as API gravity are insufficient to define these parameters adequately.

Civan’s viscosity equation does not include a term (such as the Watson K factor) to address the chemical nature of the crude oil. As demonstrated by Bergman and Sutton (2009), neglecting this important parameter often results in inconsistencies and excessive model error. Results from Civan’s viscosity equation are added to the Bergman plot displayed in Fig. 8. In general, the results do not follow normal viscosity trends with temperature. This inconsistency was traced to the data used to develop the model that is displayed on the Bergman plot in Fig. 9. Civan identified an inconsistency with the data from the 22°API oil sample; however, a review of Fig. 9 shows that all of the data for samples lighter than 30°API are inconsistent and do not follow established viscosity trends with temperature. This example illustrates the power of the Berg-man plot to aid in the determination of data quality. A subset (oils

with API gravities consistent with Civan’s data) of our database of 9,837 viscosity measurements from 3,047 samples was used to investigate the accuracy of Civan’s viscosity model. The statisti-cal results are compared with the Bergman and Sutton method and are presented in Table 3. The inclusion of oils with �API < 20 or �API >50 in this analysis results in significantly larger error for Civan’s method, while the accuracy of the Bergman and Sutton method is essentially the same as the results given in Table 3 (see Table 13 in the original paper). On the basis of these results, we do not concur with Civan’s recommendation to use his method.

In conclusion, the Bergman equation has been shown to be superior to the VTF equation for general broad application and recommendations made in our original paper stand. Civan goes to great length in his discussion to equate the VTF equation with Bergman’s equation; however, as can be seen from our discussion, it is not equivalent; it cannot be applied generally and is actually


–10

–5

0

5

10

15

20

25

–100 0 100 200 300 400 500 600

Temperature, °F

% E

rror

BergmanVTF

Fig. 6—Comparison of method errors with equations fit over entire data range.


–10

–5

0

5

10

15

20

25

–100 0 100 200 300 400 500 600

Temperature, °F

% E

rror

BergmanVTF

Fig. 7—Comparison of method errors with equations fit to data at 100 and 210°F.


Bergman Plot with Civan Viscosity Method

10,000 cp1000 cp

100 cp

10 cp

3 cp

1 cp

0.5 cp

0.3 cp

0.1 cp

400°F300°F250°F200°F150°F100°F50°F0°F–3

–2

–1

0

1

2

3

5.6 5.8 6 6.2 6.4 6.6

Ln (T, °F + 310)

Ln L

n (V

isco

sity

,cp

+ 1)

n-ParaffinsAromaticsCyclohexanesNaphthalenesOlefinsCivan

20 API

40 API

30 API

50 API

Fig. 8—Behavior of Civan (2006) viscosity equation.

Bergman Plot with Viscosity Data from Civan (2006)

1000 cp

100 cp

10 cp

3 cp

1 cp

0.5 cp

0.3 cp

300°F250°F200°F150°F100°F50°F–2

–1.5

–1

–0.5

0

0.5

1

1.5

2

5.8 5.9 6 6.1 6.2 6.3 6.4 6.5

Ln (T, °F + 310)

Ln L

n (V

isco

sity

,cp

+ 1)

n-ParaffinsAromaticsCyclohexanesNaphthalenesOlefinsHashim & Hassaballah Data

20 API

50 API

30 API

40 API

Oil data ranges 20-50 API with 2 API increments

Fig. 9—Consistency of data used in the development of Civan’s viscosity equation.

TABLE 3—ACCURACY OF VISCOSITY METHODS FOR 20 < API < 50 AND 38 < T < 500°F

Method # Pts %

Average Error Standard Deviation

% AverageAbsolute Error

Standard Deviation

Civan 7,054 332 3,922 353 3,920 Bergman and Sutton 7,054 5.4 21.5 17.0 14.3


less accurate than Bergman’s equation. Civan states that the VTF equation is simpler and easier to use than Bergman’s equation. While the VTF equation correlates viscosity using a single loga-rithm rather than the double logarithm used in Bergman’s equation, the VTF equation also requires a linear-regression routine to tune the equation’s constants. As shown by the example for a waxy-crude sample, this can lead to erroneous conclusions. Bergman’s equation does not carry the complexity of using a regression rou-tine to optimize the equation. Civan explores the use of constants (0.907 and 302.7) derived by Bergman and Sutton; however, these were developed only to demonstrate the soundness of Bergman’s original equation and were not recommended for use by Bergman and Sutton. We recommend the use of the original Bergman equa-tion on the basis of its simplicity and demonstrated accuracy.

Nomenclature a, b, c = empirical constants a’, b’, c’ = empirical constants A, B, C = empirical constants T = temperature, °F Tabs = temperature, °R � = absolute viscosity, cp � = kinematic viscosity, cSt

ReferencesAndrade, E.N. da C. 1930. The Viscosity of Liquids. Nature 125: 309–310.

doi: 10.1038/125309b0.

Bergman, D.F. and Sutton, R.P. 2009. A Consistent and Accurate Dead-Oil-Viscosity Method. SPE Res Eval & Eng 12 (6): 815–840. SPE-110194-PA. doi: 10.2118/110194-PA.

Cameron, A. 1945. The Determination of the Pressure-Viscosity Coefficient and Molecular Weight of Lubrication Oils by Means of the Temperature-Viscosity Equations of Vogel and Eyring. J. Inst. Pet. 31: 401–414.

Civan, F. 2006. Viscosity-temperature correlation for crude oils using an Arrhenius-type asymptotic exponential function. Petroleum Science and Technology 24 (6): 699–706. doi: 10.1081/LFT-200041178.

Civan, F. 2008. Predicting Brine Viscosity With Temperature and Concentration Using the Vogel-Tammann-Fulcher (VTF) Equation (Addendum to SPE 108463). Paper SPE 118750 available from SPE, Richardson, Texas.

Lemmon, E.W., McLinden, M.O., and Friend, D.G. 2008. Thermophysical Properties of Fluid Systems. In the NIST Chemistry WebBook, ed. P.J. Lin-strom and W.G. Mallard. National Institute of Standards and Technology (NIST), http://webbook.nist.gov/chemistry/ (retrieved March 12, 2010).

Poling, B.E., Prausnitz, J.M., and O’Connell, J.P. 2000. The Properties of Gases and Liquids, fifth edition (hardback), 7.4 and 9.57–9.58. New York: McGraw-Hill Professional.

Scherer, G.W. 1992. Editorial Comments on a Paper by Gordon S. Fulcher. J. of the American Ceramic Society 75 (5): 1060–1062. doi: 10.1111/j.1151-2916.1992.tb05537.x.

Stachowiak, G.W. and Batchelor, A.W. 2005. Engineering Tribology, third edition, Sec. 2, 14–15. Burlington, Massachusetts: Elsevier.

Whitson, C. and Brulé, M. 2000. Phase Behavior, Chap. 3. Monograph Series, SPE, Richardson, Texas, USA 20.

Wright, W.A. 1969. An Improved Viscosity-Temperature Chart for Hydro-carbons. J. of Materials (JMLSA) 4 (1): 19–27.

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