Density and Viscosity Behavior of a North Sea Crude Oil, Natural Gas

Energy & Fuels 2005, 19, 1303-1313

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Density and Viscosity Behavior of a North Sea Crude Oil, Natural Gas Liquid, and Their Mixtures† Kurt A. G. Schmidt,‡ Sergio E. Quinõnes-Cisneros,§,# and Bjørn Kvamme*,‡ Department of Physics, University of Bergen, Allegate´ n 55, N-5007 Bergen, Norway, and Department of Chemical Engineering, Technical University of Denmark, Building 229, DK-2800 Lyngby, Denmark Received September 6, 2004. Revised Manuscript Received March 21, 2005

The friction theory (f-theory) for viscosity modeling, combined with a recently developed characterization procedure, which includes an accurate method to describe the fluid mass distribution, commonly used cubic equations of state, and a Peńeloux-type volume translation scheme, have been shown to accurately model the saturation pressures, densities, and viscosities of petroleum systems ranging from natural gases to heavy crude oils. The applicability of this overall modeling technique to reproduce measured bubble points, densities, and viscosities of a North Sea crude oil, a natural gas liquid, and their mixtures has been investigated. The approach has been successfully applied to the modeling of the experimental data of these fluid systems to within an acceptable accuracy.

Introduction Knowledge of the phase behavior, density, and viscosity of petroleum fluids is essential to the optimization of reservoir production systems. Often, a mixing of crude oils with lighter hydrocarbons and/or carbon dioxide (CO2) and nitrogen (N2) injection is performed, to enhance the feasibility of producing crude oil from the reservoir and/or transporting the crude oil through pipelines. In such commingling processes, the engineer must be assured that the properties of the resulting mixture are correctly predicted over the wide ranges of pressure, temperature, and blending ratios that these mixtures might experience during production and transportation. Several investigations have focused on the use of pure gases, their mixtures, and liquid diluents to reduce the viscosity of heavy oils and bitumens. However, to the authors’ knowledge, only one investigation1,2 in the open literature has focused on the properties of mixtures resulting from the blending of a real multicomponent fluid, such as a natural gas liquid (NGL), with a characterized crude oil. One desirable benefit of blending these lighter fluids with the crude oil is a reduction of the crude’s viscosity, making it more mobile. Despite the importance of the viscosity in the evaluation, simulation, and development of petroleum reservoirs, production facilities, and transport systems, the † Presented at the 5th International Conference on Petroleum Phase Behavior and Fouling. * Author to whom correspondence should be addressed. Telephone: (+47) 55 58 33 10. Fax: (+47) 55 58 94 40. E-mail address: [email protected]. ‡ University of Bergen. § Technical University of Denmark. # Present address: Institute of Physical Chemistry, University of Cologne, Luxemburger Str. 116, D-50939 Cologne, Germany. (1) Ahrabi, F.; Ashcroft, S. J.; Shearn, R. B. Chem. Eng. Res. Des. 1987, 65, 63-73. (2) Ahrabi, F.; Ashcroft, S. J.; Shearn, R. B. Chem. Eng. Res. Des. 1989, 67, 329-335.

estimation of this property has been based on models with limited applicability. These models are often unable to determine these properties for a wide range of petroleum fluids to a sufficient accuracy. Recently, an overall modeling approach, based on the friction theory3 (f-theory) one-parameter models4 for viscosity modeling and a new oil characterization method,5,6 has been developed to determine the viscosity and pressurevolume-temperature (PvT) behavior of reservoir fluids accurately.7-10 This new approach has the additional advantage of being applicable to commonly used cubic equations of state (EOSs). The f-theory has been shown to be widely versatile and accurate in the correlation and prediction of the viscosity of well-defined systems.3,4,11 In addition, when combined with the cited oil characterization procedure,5,6 the f-theory has also been shown to accurately model complex reservoir fluid systems ranging from natural gases to heavy crude oils. In this work, the applicability of this overall modeling technique to reproduce previously measured bubble points, densities, and viscosities of a North Sea crude oil, a NGL, and (3) Quinõnes-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. Fluid Phase Equilib. 2000, 169, 249-276. (4) Quinõnes-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. Fluid Phase Equilib. 2001, 178, 1-16. (5) Quinõnes-Cisneros, S. E.; Dalberg, A.; Stenby, E. H. Pet. Sci. Technol. 2004, 22, 1309-1325. (6) Quinõnes-Cisneros, S. E. Viscosity modeling and prediction of crude oils. In Proceedings of the VI Iberoamerican Conference on Phase Equilibria for Process Design (EQUIFASE 2002), Foz de Iguazu´, Brazil, October 12-16, 2002. (7) Quinõnes-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Baylaucq, A.; Boned, C. Int. J. Thermophys. 2004, 25, 1353-1366. (8) Quinõnes-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. Chem. Eng. Sci. 2001, 56, 7007-7015. (9) Quinõnes-Cisneros, S. E.; Andersen, S. I.; Creek, J. Energy Fuels 2005, 19, 1314-1318. (10) Ze´berg-Mikkelsen, C. K.; Quinõnes-Cisneros, S. E.; Stenby, E. H. Int. J. Thermophys. 2002, 23, 437-454. (11) Ze´berg-Mikkelsen, C. K.; Quinõnes-Cisneros, S. E.; Stenby, E. H. Pet. Sci. Technol. 2002, 20, 27-42.

10.1021/ef049774h CCC: $30.25 © 2005 American Chemical Society Published on Web 05/11/2005

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their mixtures1,2 will be illustrated. When possible, the results were compared to those obtained in other modeling investigations. Compositional and PvT Characterization In an oil characterization procedure, the light fraction, up to C6, is normally described in terms of 11 welldefined components: N2, CO2, H2S, CH4, C2H6, C3H8, i-C4H10, n-C4H10, i-C5H12, n-C5H12, and n-C6H12. However, the heavy (C7+) oil fraction is characterized in terms of a given number of pseudo-components for which some characteristic critical parameters are derived to correctly reproduce the phase behavior of the fluid. In following with the related characterization procedure,5 the heavy fraction of the reservoir fluid is described by a chi-squared distribution function.3-7 Note that this distribution function and the well-known exponential distribution function12 are special cases of the general gamma (Γ) distribution function,13,14 which have been used to characterize the heavy end of reservoir fluids. The use of distribution functions has also found widespread use in other continuous thermodynamics applications.15,16 The general form of the chisquared distribution function, CS(p), with p degrees of freedom, used in this approach is given by

2-p/2 -s/2 (p/2)-1 fdis ) e s Γ(p/2)

sˆ i )

0

fdis ds ) M6

MW )

Here, M6 represents the fluid total light mass fraction, up to and including the C6 fraction, and s0 is the value of s that satisfies eq 2. Accordingly, M6 may be defined as

M6 )

1

∑i xiMWi

MWT

(3)

where the summation over i covers all of the light components in the fluid sample, up to and including the C6 fraction. As such, the C7+ fraction is then characterized in m number of mass fractions, Fi, with mass fraction, fmi,

fmi )

∫ss

i

fdis ds

i-1

s‚fdis ds

(6)

MW + 1 - M6

m

fmi

∑ ˆ i)1 s

(7)

i

Tc ) -423.587 + 210.152 ln(MW)

(8)

Pc ) fc exp(9.67283 - 4.05288MW0.1)

(9)

(

ω ) exp 8.50471 -

)

15.1665 MW0.1

(10)

In these three equations, MW is expresed in units of g/mol to obtain Tc in degrees Kelvin and Pc in units of bar. The parameter fc in eq 9 represents a perturbation factor away from the value of fc ) 1, which corresponds to n-alkanes. The lumped components, up to C6, were all assigned a value of fc ) 1, and for the Fi C7+ pseudocomponents, the perturbation parameter was iteratively modified (tuned) until the saturation pressure of the fluid was matched with the EOS of interest. In this approach, only one global fc value is tuned for all the C7+ pseudo-components. The EOS parameters and molecular weights for the pure components, CH4, N2, and CO2 were taken from the DIPPR database.17 In following with the characterization technique, a Peńeloux-type volume translation parameter (c)18,19 was used to improve the fluid densities obtained from the EOS:

(4)

and molecular mass, MWi,

MWi ) MW‚sˆ i

i

i-1

In this case, MW+ represents the C7+ fraction molecular mass. After completion of the pseudoization process of the C7+ fraction into four equal mass fractions, ethane and propane and similarly n-butane, i-butane, and pentane, were lumped together. The necessary equation of state (EOS) parametersscritical temperature (Tc), critical pressure (Pc), and acentric factorsfor the lumped groups and the pseudo-components of the C7+ fraction were then estimated using eqs 8-10, which are empirical correlations that are based on the properties of normal alkanes.5

(1)

(2)

∫ss

represents the center of mass of the Fi fraction and MW is a scaling value to ensure the total mass balance is satisfied. Therefore, the average molecular weight of the fluid, in eq 5, can be determined with the combination of eq 6 and a mass balance:

where s is a molecular-weight-scaled variable that satisfies the relation

∫0s

1 fmi

v′ ) vEOS - c

(11)

∑

(12)

where

(5)

where sˆ i, which is defined as (12) Pedersen, K. S.; Fredenslund, Aa.; Thomassen, P. The Properties of Oils and Natural Gases; Contributions in Petroleum Geology & Engineering 5; Gulf Publishing Company: Houston, TX, 1989. (13) Whitson, C. H.; Brule´, M. R. Phase Behavior; SPE Monograph Series 20; Society of Petroleum Engineers: Richardson, TX, 2000. (14) Whitson C. H. SPE J. 1983, 23, 683-694. (15) Cotterman, R. L.; Bender, R.; Prausnitz, J. M. Ind. Eng. Chem. Proc. Des. Dev. 1985, 24, 194-203. (16) Cotterman, R. L.; Prausnitz, J. M. Ind. Eng. Chem. Proc. Des. Dev. 1985, 24, 434-443.

c ) Kv

i:Fi>C6

xiMWi

As can be seen, there is only one volumetric tuning (17) Rowley, R. L.; Wilding, W. V.; Oscarson, J. L.; Yang, Y.; Zundel, N. A.; Daubert, T. E.; Danner, R. P. DIPPR Data Compilation of Pure Compound Properties; Design Institute for Physical Properties; American Institute of Chemical Engineers (AIChE): New York, 2003. (18) Peńeloux, A.; Rauzy, E.; Freze, R. Fluid Phase Equilib. 1982, 8, 7-23. (19) Jhaveri, B. S.; Youngren, G. K. SPE Reservoir Eng. 1988, 3, 1033-1040.

North Sea Fluid Density and Viscosity Modeling

Energy & Fuels, Vol. 19, No. 4, 2005 1305

parameter: the constant Kv. As the summation index indicates, the shift is applied only to the heavy fluid fractions. In this approach, Kv was determined from single-phase density data, measured above the saturation curve, by a tuning process. Viscosity Characterization The friction theory (or f-theory) for viscosity modeling has been developed based on friction concepts of classical mechanics and the van der Waals theory of fluids. The theory was developed on the basis that the van der Waals repulsive and attractive pressure terms, which can be obtained from simple cubic EOSs, could be connected to the Amontons-Coulomb friction law. This connection results in a residual viscosity term that is added to a dilute gas viscosity term to estimate the viscosity of fluids over a wide range of temperature and pressure conditions. As a result, this model has been shown to be widely versatile and accurate in the correlation and prediction of the viscosity of well-defined pure, binary, and multicomponent systems.3,4,11 A detailed description of the viscosity modeling procedure for reservoir fluids using f-theory models is readily available,7,8 and only a brief summary will be given here. According to the f-theory for viscosity modeling,11 the viscosity of a fluid can be separated into a dilute gas term (η0) and a residual friction term (ηf):

η ) η0 + ηf

(13)

The model of Chung et al.20 is used to determine the dilute gas term, and the residual friction term has been shown11 to be related to the van der Waals repulsive (pr) and attractive (pa) pressure terms, via three temperature-dependent friction coefficientssκa, κr, and κrr:

ηf ) κrpr + κapa +

κrrp2r

(14)

The repulsive and attractive pressure terms can be obtained from commonly used cubic EOSs in the petroleum industry. In the case of nonpolar fluids, such as hydrocarbon mixtures and reservoir fluids, the f-theory has been developed further to a one-parameter generalized model4 (eq 15), which is dependent only on a single characteristic viscosity scaling parameter:

(

)

pr pa p2r ηf ) ηc κˆ r + κˆ a + κˆ rr 2 Pc Pc P c

(15)

Here, ηc is the characteristic viscosity scaling parameter and Pc is the critical pressure. The κˆ r, κˆ a, and κˆ rr parameters are only dependent on the reduced temperature and have been parametrized into universal constants that are related to a specific EOS.4,21 When mixtures are considered, the value of the mixture (20) Chung, T. H.; Ajlan, M.; Lee, L. L.; Starling, K. E. Ind. Eng. Chem. Res. 1988, 27, 671-679. (21) Quinõnes-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. General one-parameter friction theory viscosity model for the PatelTeja EOS. In Proceedings of the VI Iberoamerican Conference on Phase Equilibria for Process Design (EQUIFASE 2002), Foz de Iguazu´, Brazil, October 12-16, 2002.

friction coefficients is predicted using the suggested empirical mixing rules.3 For the EOS, in principle, the mixing rules that best describe the fluid-phase behavior should be used. For reservoir fluids, after the fluid has been properly characterized, the determination of the viscosity scaling parameters is similar to that for the PvT scaling parameters. That is, for well-defined light compounds (i.e., methane, ethane, etc.), reported tabulated values8 are used. For cases where some light hydrocarbon compounds (up to C6) have been lumped together, the following modified Uyehara-Watson equation5,22 (eq 16) may be used:

ηc ) 7.9483 × 10-4

xMWP2/3 c T1/6 c

(16)

where ηc is given in centipoise, Pc is given in bars, and Tc is given in degrees Kelvin. For the pseudo-components in the C7+ fraction, the characteristic viscosity scaling parameter is unknown and eq 16 can be relaxed by substituting the model constant for an adjustable common parameter Kc, which can be taken as a viscosity characterization parameter for all of the pseudo-components in the heavy fraction.

ηc ) Kc

xMWP2/3 c T1/6 c

(17)

Substitution of this expression for the C7+ pseudocomponents and eq 16 or tabulated data for well-defined components into the one-parameter f-theory model leads to the following simple expression:

η ) ηI + KcηII

(18)

where ηI and ηII are well-defined numbers.8 As can be seen, there is only one viscosity characterization parameter that is applied only to the heavy fluid fractions. In this approach, Kc was determined from single-phase viscosity data, measured above the saturation curve, by a tuning process. In this study, the original Peng-Robinson (PR) EOS,23 with the classical van der Waals mixing rules, was used in the f-theory and was used to model the PvT behavior of the reservoir fluids. Results As mentioned previously, the fluids considered for this investigation were a North Sea crude oil, a NGL, and their mixtures, with experimentally determined properties originally reported by Ahrabi et al.1,2 The molar composition of the crude oil and subsequent mixtures were reported up to C20+, and the molar composition of the NGL was reported up to C7+. The results of the fluid compositional characterization of both the crude oil and the NGL are presented in Table 1. The pseudoization process of the crude oil C7+ fraction (22) Uyehara, O. A.; Watson, K. M. Natl. Pet. News 1944, October 4th, R-714-R-722. (23) Peng, D. Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15, 59-64.

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Table 1. Main Characteristic Parameters for the Crude Oil and Natural Gas Liquid Chi-Squared Function, CS(p ) 4.0) Crude Oil N2 CO2 C1 C2-C3 C4-C5 C6 F1 F2 F3 F4

MW

mol %

MW

0.650 0.860 18.062 14.851 10.461 2.590 23.683 14.178 9.367 5.297

28.013 44.010 16.042 38.438 63.580 86.175 129.391 216.133 327.165 578.495

0.530 1.810 20.880 56.730 18.870 0.830 0.350

28.013 44.010 16.042 38.792 60.255 86.175 100.202

MWT (g/mol)a fcb a

Natural Gas Liquid, NGL

mol %

140.636 1.212

38.738 1.0

Molecular weight. b Optimal perturbation parameter.

into four equal mass fractions was performed using the optimal CS(p) function that best-described the fluid mass distribution. The p in the CS distribution function was iteratively changed until the oil mass distribution was optimally matched. An optimal value of p ) 4.0 was determined to describe the fluid mass distribution of the crude oil accurately. In the case of NGL, the C7+ fraction was considered to be a single pseudo-component with the MW of n-heptane. Also presented in Table 1 is the optimal perturbation parameter (fc) for the C7+ pseudo-components of the crude oil, which was determined by matching the available saturation pressures (at 303.2, 323.2, 353.2, and 374.8 K) of the fluid. With the optimal fc value (1.212), the overall experimental saturation pressure data could be reproduced to within an average absolute percentage deviation (AAPD) of 3.1%. (AAPD is defined as

AAPD )

100 NP

×

|(

NP ξmeasured i

∑ i)1

)|

- ζcalculated i

ξmeasured i

(19)

where ξ is the property of interest.) The obtained value of fc is reasonable with its definition of being a perturbation factor that corrects the effect that aromatic and other complex compounds, which have larger Pc values than n-alkanes, may have on the phase equilibria properties. In this study, all the crude oil hydrocarbonhydrocarbon binary interaction parameters were set to zero and the binary interaction parameters for nonhydrocarbon-hydrocarbon interactions were those suggested by Danesh,24 based on the results of Knapp and Doring25 (C7-) and those suggested by Søreide,26 which were based on the Nagy and Shirkovskiy (C7+) results.27 A temperature-dependent fc parameter improved the (24) Danesh A. PVT and Phase Behaviour of Petroleum Reservoir Fluids; Elsevier Science B. V.: Amsterdam, The Netherlands, 1998. (25) Knapp, H.; Doring, R. Vapor-Liquid Equilibria for Mixtures of Low Boiling Substances; DECHEMA Chemistry Data Series Part 1s Binary Systems; DECHEMA: Frankfurt, Germany, 1986. (26) Søreide, I. Improved Phase BehaviorsPredictions of Petroleum Reservoir Fluids from a Cubic Equation of State, Dr. Ing. Dissertation, IPT-Report 1989:4, Norwegian Institute of Technology, Department of Petroleum Technology and Applied Geophysics, Trondheim, Norway, 1989. (27) Nagy, Z.; Shirkovskiy, A. I. Presented at the 57th Annual Fall Technical Conference and Exhibition of the Society of Petroleum Engineers of AIME, New Orleans, LA, SPE Paper No. 10982, 1982.

predicted results to an AAPD value of 0.7% but was not investigated further at this point. (This temperature dependency will be pursued in future investigations.) The results are consistent with the modeling results reported by Ahrarbi.1 A perturbation parameter of fc ) 1 was used to predict the available saturation pressures of the NGL, because any changes in its magnitude had very little effect in the calculated saturation pressures. However, the introduction of methane-hydrocarbon interactions,28 in addition to those used for the crude oil, improved the predictions. As such, the saturation pressure data of the NGL could be predicted to within an AAPD of 6.7%. The saturation pressures were underpredicted with this modeling, which is consistent with the modeling results presented in the original experimental work.1 The results are presented in Figure 1 for both the crude oil and NGL. Again, the results seem to be consistent with those suggested by Ahrarbi.1 In addition to matching the saturation pressure, the Peńeloux volume translation parameter, which was described previously, was utilized to improve the fluid densities obtained from the PR-EOS. In following with the characterization technique,5 one overall volumetric tuning parameter, Kv, which was applied only to the heavy fluid fractions, was determined from the singlephase density data by a tuning process. The results of the density modeling of the crude oil and the NGL are presented in Figures 2 and 3. As can be seen in Figure 2, the calculated crude oil densities could reproduce the experimental data quite well, with the exception of the 323 K isotherm. Figure 2 also indicates that the measured densities at 323 K seem to be inconsistent with the remaining data, especially if the experimental result at 298.2 K and 166.5 bar was used to calculate the mass of the material in the cell.1 As such, the 323.2 K isotherm was not included in the optimization of the crude oil Kv. The exclusion of these data resulted in an optimal Peńeloux shift constant with a slight temperature dependence of Kv(cm3/g) ) 0.2336-0.00072 × T(K). With this form, the density data could be modeled to within an AAPD of 1.1% when all of the data were considered and to within 0.6% when the 323 K data were not included. With this volume shift parameter, the density that was used in the filling of the vessel was predicted to within an accuracy of 0.3%. The calculated densities of the crude oil obtained with this simple characterization method are better than those reported by Ahrarbi1 and by von Bergen et al.,29 which used more-complex modeling procedures. A similar inconsistency, albeit less, can also be observed with the measured density data of the NGL at 323.2 K in Figure 3. Again, the experimental result at 298.2 K and 320.6 bar was used to determine the mass of the fluid in the cell.1 This value could be determined only to within 4.3%. Figure 3 also shows that the differences in the experimental and predicted density data at 374.8 K become worse at pressures below ∼138 bar. In the case of the NGL density modeling, a volume shift was not used, because the predicted results did not significantly improve the (28) Arbabi, S.; Firoozabadi, A. SPE Adv. Technol. Ser. 1995, 3, 139145. (29) von Bergen, R.; von Bergen, Y.; Rogel, E. A. Fluid Phase Equilib. 1997, 137, 33-52.



Figure 1. Comparison of experimental (symbols) and predicted (line) saturation pressures for (]) crude oil ([) and natural gas liquid (NGL).

Figure 2. Comparison of experimental (symbols) and predicted (line) crude oil densities: ([) 298.2 K, (]) 303.2 K, (b) 323.2 K, (O) 353.2 K, and (2) 374.8 K.

predicted results. In this case, the overall high-pressure density data was predicted to within an AAPD of 4.4% and to within an AAPD of 3.0% when the low-pressure data (below 138 bar) at 374.8 K were not included. The calculated densities of the NGL presented here are slightly better than those reported by Ahrarbi.1 However, the predictions are slightly poorer than those by von Bergen et al.29 Although outside of the scope of the work here, the tuning of binary interaction parameters used to model the NGL and/or the introduction of a volume shift constant could have potentially improved the predicted results. The viscosity modeling of the crude oil is presented in Figure 4. This figure demonstrates the applicability of the f-theory to model this fluid reasonably well. However, as with the density data, one of the isotherms does not seem to be consistent with the rest of the other isotherms. When all of the data are considered, the viscosity data at 353.2 K seems to be quite high, when

compared to the modeled results. Therefore, only singlephase viscosity data from the 303.2, 323.2, and 374.8 K isotherms were used in the tuning process. The tuned value Kc of the crude oil was determined to be Kc (cP) ) {-9.266 + [0.0403 × T(K)]} × 10-4. With this form, the viscosity data could be modeled to within an AAPD of 5.1% when all of the single phase data were considered and to within 2.0% when the 353.2 K data were not included. Note that this fitting resulted in Kc values that ranged from ∼2.9 × 10-4 to 5.8 × 10-4 over the 71.6 K temperature range. These values seem to be quite low when compared to many crude oils with similar properties.5,7-9 When evaluating the Kc values at each isotherm over the full temperature range, the 353.2 K and the 374.8 K isotherms each produced Kc values of 5.8 × 10-4. These two valuessand, in fact, all of the determined Kc valuessare less than the n-alkane value of 7.9483 × 10-4 and are much less than the typical

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Figure 3. Comparison of experimental (symbols) and predicted (line) NGL densities: ([) 298.2 K, (]) 303.2 K, (b) 323.2 K and (2) 374.8 K.

Figure 4. Comparison of experimental (symbols) and predicted (line) crude oil viscosities: (]) 303.2 K, (b) 323.2 K, (O) 353.2 K, and (2) 374.8 K.

crude oil value of 8.98 × 10-4, which was determined from the empirical relation5 relating Kc and p:

Kc(cP) ) [3.1543 + 4.2045 ln(p)] × 10-4

(20)

This higher Kc value is more indicative of the crude oils studied thus far5,7-9 and puts the experimental crude oil viscosity data into question, especially at the lower temperatures. In fact,9 a low Kc value (Kc ) 4.96 × 10-4, at 360.1 K) was determined for an oil that was intentionally selected with the knowledge that there were measurement problems. Modeling results on an extensive database of reservoir fluids indicate a departure away from the n-alkanes baseline value toward moreimpeded fluid mobility (i.e., Kc > 7.9483 × 10-4) is expected. The low values obtained here suggest that the experimental viscosity is lower than expected, which may be due to problems with the experimental technique and/or insufficient calibration of the viscometer.

In terms of modeling, the reported data can be accurately reproduced by introducing a temperaturedependent form of Kc, as shown in Figures 4, 5, and 6. However, although the use of a temperature-dependent Kc term may be appropriate to improve the capabilities of the one-parameter f-theory models, the range of Kc values should increase away from the reference 7.9483 × 10-4 n-alkane value as temperature decreases. That is, it would be expected that, at low temperatures, the one-parameter f-theory model may underpredict the fluid viscosity, rather than overpredict it. However, the contrary effect is observed with the studied data. Figure 5 shows the viscosity modeling results using the temperature-dependent form of Kc and the Kc value that was fitted to the 353.2 K and the 374.8 K isotherms only. This figure illustrates the large deviations in the predictions when a value closer to the n-alkane value was used to model all the data. To further emphasize this problem, Figure 6 compares the modeling results



Figure 5. Comparison of experimental (symbols) and predicted crude oil viscosities with (s) Kc ) Kc(T) and (- - -) Kc ) 5.78 × 10-4: (]) 303.2 K, (b) 323.2 K, (O) 353.2 K, and (2) 374.8 K.

Figure 6. Comparison of experimental (symbols) and predicted crude oil viscosities with (s) Kc ) Kc(T) and (- - -) Kc ) 8.98 × 10-4: (]) 303.2 K, (b) 323.2 K, (O) 353.2 K, and (2) 374.8 K.

using the temperature-dependent correlation for Kc and a Kc value that is more representative of reservoir fluids. Yet, despite these inconsistencies, the modeling capability of the f-theory is clearly demonstrated. Furthermore, the questions raised here also open the door to the use of f-theory models as a quality assurance tool for the assessment of data consistency. Despite the low Kc values, the temperature-dependent form of Kc seems to model the viscosity data for the crude oil the best, and, as such, this form will be used in the subsequent viscosity modeling of the crude oilNGL mixtures. A temperature dependency for Kc is expected, based on the original model development, and this temperature dependency will be investigated further in subsequent studies. The results obtained with the temperature-dependent form of Kc were significantly better when compared to other modeling investigations.1,30,31

It has already been shown for light reservoir fluids, such as natural gas, that the f-theory models can accurately predict the fluid viscosity without the need of tuning any parameters,10 and, as such, the viscosity of the NGL was predicted. The viscosity modeling results of the NGL are presented in Figure 7. The experimental viscosity of the NGL seems to be in doubt, in comparison to the modeling results. This can also be seen in the figure, because the experimental 323.2 and 374.8 K isotherms cross at ∼300 bar. The other modeling investigations1,32 also had difficulties predicting the NGL viscosity to a reasonable accuracy. Thus, it should be emphasized that the predictive capabilities of the (30) Guo, X. Q.; Sun, C. Y.; Rong, S. X.; Chen, G. J.; Guo, T. M. J. Pet. Sci. Eng. 2001, 30, 15-27. (31) Xu, D.-H.; Khurana, D. Presented at the 1996 SPE Asia Pacific Oil and Gas Conference, Adelaide, Australia, SPE Paper No. 37011, 1996. (32) Elsharkawy, A. M. Pet. Sci. Technol. 2003, 21, 1759-1787.

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Figure 7. Comparison of experimental (symbols) and predicted (line) NGL viscosities: (b) 323.2 K and (2) 374.8 K.

Figure 8. Comparison of experimental (symbols) and predicted (line) mixture saturation pressures: (b) 323.2 K and (2) 374.8 K.

f-theory one-parameter models are very good when tested against the family of n-alkanes and well-defined mixtures in the modern literature. In the case of light reservoir fluids, such as natural gases, the f-theory models can deliver full viscosity prediction to practically within experimental uncertainty.10 Again, the experimental uncertainty, which has been briefly discussed above, may be an explanation for the large differences in the modeling results. As with most commingling operations in the petroleum industry, the mixed “pure” fluids have been previously individually characterized and all adjustable parameters have been tuned to achieve a satisfactory description of that fluid. Often, when the “pure” fluids (e.g., heavy oils and diluents) are mixed, no experimental data for the properties of these “mixed” fluids exist and the engineer must make important design decisions based on the calculations that ultimately involve these mixed “pure” characterized fluids. Keeping with this methodology, the saturation pressures, PvT behavior,

and the viscosities of the experimentally prepared binary mixtures2 (6.3, 11.9, 25.3, and 49.7 mass % NGL) were predicted from mixtures of the previously characterized crude oil and NGL. In the commingling process, no additional mixing rules were used in the subsequent predictions of the bubble points, densities, and viscosities of the crude oil-NGL mixtures. The previously characterized “pure” NGL and crude oil fluids were combined according to the relative amounts of NGL and crude oil, to match the specified mixture compositions (i.e., mass % NGL). The pure componentssN2, CH4, and CO2swere lumped together, because they had the same pure EOS parameters and MW; however, the other C6components and the C7+ fractions (oil and NGL) were added on a compositional basis, and no effort was made to lump these fractions into a smaller number of fractions. The predicted saturation pressures of the four mixtures, at 323.2 and 374.8 K, were quite satisfactory, with an overall AAPD of 3.9%. The predicted and experimen-



Figure 9. Comparison of experimental (symbols) and predicted (line) mixture densities at 323.2 K: ([) crude oil, (b) mixture 2, (O) mixture 3, (2) mixture 4, and (4) NGL.

Figure 10. Comparison of Experimental (symbol) and Predicted (line) Mixture Densities at 374.8 K: ([) crude oil, (]) mixture 1, (b) mixture 2, (O) mixture 3, (2) mixture 4, and (4) NGL.

tal results are compared in Figure 8. On the other hand, the prediction of the mixtures densities at 323.2 and 374.8 K was fairly reasonable, with an overall AAPD of 6.5%. The density modeling results of the mixtures at 323.2 and 374.8 K are presented in Figures 9 and 10, respectively. Predicted densities of mixture 3, especially at 374.8 K, were the worst and contributed the most to the overall AAPD. If this mixture is excluded, the data can be predicted to 4.3%. Although unexpectedly low Kc values were obtained for the crude oil, the temperature-dependent form of Kc was used in the prediction of the mixture’s viscosity. Even with these optimized values, large deviations between the modeled viscosities and the experimental viscosities resulted. As shown in Figures 11 and 12, the experimental results seem to be clearly overpredicted, although the 374.8 K results seem somewhat better. The experimental data show a more pronounced reduction in the viscosity of the crude oil, when compared with

the predicted results. Mixture single-phase viscosities could be predicted to within an AAPD of 48% and 41% when the 323.2 K data was excluded. These seemingly large deviations may again be due to experimental inconsistencies. Other blending studies with pure components or a solvent indicate a reduction of the viscosity similar to that predicted by the f-theory.33,34 When the original mole percentage blending ratios2 were considered as mass percentages (which resulted in much higher mole percentage blending ratios than reported), the predicted reduction in the viscosity of the crude oil becomes more consistent, based on our experience and those values predicted by this modeling approach. Predictions with this assumption lead to a significant decrease in the high-pressure single-phase viscosity (33) Barrufet, M. A.; Setiadarma, A. Fluid Phase Equilib. 2003, 213, 65-79. (34) Audonnet, F.; Pa´dua, A. A. H. Fluid Phase Equilib. 2004, 216, 235-244.

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Figure 11. Comparison of experimental (symbols) and predicted (line) mixture viscosities at 323.2 K: ([) crude oil, (b) mixture 2, (O) mixture 3, (2) mixture 4, and (4) NGL.

Figure 12. Comparison of experimental (symbols) and predicted (line) mixture viscosities at 374.8 K: ([) crude oil, (b) mixture 2, (O) mixture 3, (2) mixture 4, and (4) NGL.

AAPD, to 19.9%. However, the resulting predictions for the bubble points and single-phase densities increased to 11.6% and 8.0%, respectively. These increases are significantly less, when compared to the improved predictions of the viscosity. Without additional data of this type, it is difficult to determine if the exhibited rapid decrease in the viscosity of the oil is true. In addition to the modeled single-phase viscosity data, the viscosity of the liquid phases that resulted from the differential liberation experiments at the respective temperatures was predicted for the crude oil, NGL, and their mixtures. The predicted results were expected to be more similar, because the f-theory viscosity modeling approach has been shown to predict low-pressure viscosities, below the saturation pressure, and high-pressure viscosities from a Kc value tuned from viscosities of the dead oil35 quite well. The results for the singlephase region and the liquid phases below the saturation

pressure are shown in Figures 4-6, 7, and 11 and 12 for the crude oil, the NGL, and their mixtures, respectively. Predictions were also made with Kc values of 5.8 × 10-4 and 8.98 × 10-4 for the crude oil. The results were, as expected, considerably worse and, for brevity, are not presented here. Even with the unexpectedly low Kc values and the inconsistencies in the data set, the f-theory generally predicts the viscosity behavior of these mixtures to a reasonable accuracy, considering the quality of the experimental data. Curiously, the only other modeling investigation of these mixtures2 seems to be better; however, it seems that smoothed density and viscosity data were used, rather than the raw experimental data. Further work with this type of mixing process (involving characterized fluids and pure (35) Quinõnes-Cisneros, S. E.; Zeberg-Mikkelsen, C. K.; Stenby, E. H. Fluid Phase Equilib. 2003, 212, 233-243.


component solvents and other multicontact-type processes) will be pursued in other investigations. This work will certainly lead to a more thorough understanding of this data set and the capabilities of the modeling approach to model these processes. Conclusions Despite the perceived quality of the experimental data, the overall reservoir fluid PvT and viscosity modeling technique, which encompasses the friction theory (f-theory), a fluid characterization technique, and the Peng-Robinson equation of state, was used reasonably successfully to predict the measured bubble points, densities, and viscosities of a North Sea crude oil, a natural gas liquid (NGL), and their mixtures. The technique has been very successful in modeling an extensive database of reservoir fluids, ranging from natural gases to heavy oils and well-defined systems, and the problems experienced in this investigation therefore indicate some problems with this data set. Unfortunately, this data set seems to be the only publication in the open literature that involves the experimental viscosities of these types of mixtures (crude oils and natural gases with full compositional information) that can be used to evaluate the ability of this approach to model the commingling of “real” petroleum fluids. Even with the unexpectedly low Kc values and the inconsistencies in the data set, the f-theory generally predicts the viscosity of the mixtures to a practical accuracy. However, some work needs to be done, with high-quality data, to further test the ability of this modeling approach to predict the PvT and viscosity of crude oils blended with light hydrocarbon mixtures.


Nomenclature c ) Peńeloux-type volume translation parameter fc ) critical pressure perturbation factor fdis ) chi-squared distribution function fmi ) mass fraction of the C7+ fraction characterized fractions Kc ) critical viscosity constant Kv ) Peńeloux shift constant M6 ) fluid total light mass fraction MW ) molecular weight p ) degrees of freedom in the chi-squared distribution pa ) van der Waals attractive pressure term pr ) van der Waals repulsive pressure term P ) pressure Pc ) critical pressure R ) molar gas constant T ) absolute temperature Tc ) critical temperature s ) molecular-weight-scaled variable v ) molar volume x ) mole fraction z ) mass-weighted fraction Greek Letters Γ ) gamma distribution function ) mixing rule exponent η ) total viscosity η0 ) dilute gas viscosity ηc ) characteristic scaling viscosity ηf ) friction viscosity κa ) linear attractive viscous friction coefficient κr ) linear repulsive viscous friction coefficient κrr ) quadratic repulsive viscous friction coefficient ξ ) property of interest in average absolute percentage deviation (AAPD) formula ω ) Pitzer’s acentric factor EF049774H

Density and Viscosity Behavior of a North Sea Crude Oil, Natural Gas

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