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Energy & Fuels 2005, 19, 1314-1318
Density and Viscosity Modeling and Characterization of Heavy Oils† Sergio E. Quin˜ones-Cisneros*,†,§ and Simon I. Andersen† Department of Chemical Engineering, Technical University of Denmark, Building 229, DK-2800 Kgs. Lyngby, Denmark
Jefferson Creek‡ ChevronTexaco Exploration and Production Technology Company, 2811 Hayes Road, Houston, Texas 77082-6696 Received September 8, 2004
Viscosity and density are key properties for the evaluation, simulation, and development of petroleum reservoirs. Previously, the friction theory (f-theory) was shown to be capable of delivering simple and accurate viscosity models for petroleum reservoir fluids with molecular weights up to ∼200 g/mol and viscosities up to ∼10 mPa s, under usual reservoir conditions. As a basis, the f-theory approach requires a compositional characterization procedure that is used in conjunction with a van der Waals type of equation of state (EOS). This is achieved using simple cubic EOSs, which are widely used within the oil industry. Further work also established the basis for extending the approach to heavy oils. Thus, in this work, the extended f-theory approach is further discussed with the study and modeling of a wider set of representative heavy reservoir fluids with viscosities up to thousands of mPa‚s. Essential to the presented extended approach for heavy oils is, first, achievement of accurate PvT results for the EOS-characterized fluid. In particular, it has been determined that, for accurate viscosity modeling of heavy oils, a compressibility correction in the way that the EOS is coupled to the viscosity model is required. Thus, in this work, the f-theory potential for the viscosity modeling and prediction of heavy reservoir fluids is further discussed.
Introduction In previous works, the friction theory (f-theory)1 for viscosity modeling has been applied to the accurate viscosity prediction of light reservoir fluids (natural gas)2 and the accurate modeling and prediction of denser reservoir fluids,3 with molecular weights up to ∼200 g/mol. Furthermore, the approach recently was extended to heavy oils4 with molecular weights up to >400 g/mol and reservoir conditions where the viscosity was on the order of thousands of mPa s. These applications are all based on the type of simple cubic equations of state (EOSs) that are commonly used in the oil industry. Thus, the main objective of this work is to further analyze the consistency of the heavy oil modeling results, with the intent of exploring the degree to which prediction may be possible. * Corresponding author. E-mail:
[email protected]. † Technical University of Denmark. § Present address: Institute of Physical Chemistry, University of Cologne, Luxemburger Str. 116, D-50939 Koeln, Germany. ‡ Chevron Texaco Exploration and Production Company. (1) Quin˜ones-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. Fluid Phase Equilib. 2000, 169 (2), 249-276. (2) Ze´berg-Mikkelsen, C. K.; Quin˜ones-Cisneros, S. E.; Stenby, E. H. Int. J. Thermophys. 2002, 23 (2), 437-454. (3) Quin˜ones-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. Chem. Eng. Sci. 2001, 56 (24), 7007-7015. (4) Quin˜ones-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Baylaucq, A.; Boned, C. Viscosity Modeling and Prediction of Reservoir Fluids: From Natural Gas to Heavy Oils. Int. J. Thermophys. 2004, 25 (5), 13531366.
The f-theory for viscosity modeling consists of simple models that take advantage of the repulsive and attractive pressure terms in van der Waals-type EOSs, such as the Soave-Redlich-Kwong (SRK) EOS5 or the Peng-Robinson (PR) EOS.6 Necessarily, any modeling of reservoir fluids properties that is based on this type of EOS requires a characterization procedure. In an oil characterization procedure, the light fraction, up to C6, is normally described in terms of 11 well-defined components: N2, CO2, H2S, CH4, C2H6, C3H8, i-C4, n-C4, i-C5, n-C5, and C6. However, the heavy (C7+) oil fraction is characterized in terms of a given number of pseudocomponents for which some characteristic critical parameters are derived so that the phase behavior is correctly reproduced. Consistent with previous work,4 the same characterization procedure is used here;7 this procedure is based on a generalization of previous work.8 Essentially, the method searches for a distribution function that optimally matches the laboratory-reported fluid mass distribution. Thus, for the sake of complete(5) Soave, G. S. Chem. Eng. Sci. 1972, 27, 1197-1203. (6) Peng, D.-Y.; Robinson, D. B. Ind. Eng. Chem. Fundam. 1976, 15 (1), 59-64. (7) Quin˜ones-Cisneros, S. E.; Dalberg, A.; Stenby, E. H. Pet. Sci. Technol. 2004, 22 (9&10), 1309-1325. (8) Quin˜ones-Cisneros, S. E. Viscosity Modeling and Prediction of Crude Oils. Presented at Equifase 2002, Foz de Iguazu´ (Brazil), October 12-16, 2002.
10.1021/ef0497715 CCC: $30.25 © 2005 American Chemical Society Published on Web 03/19/2005
Density and Viscosity of Heavy Oils
Energy & Fuels, Vol. 19, No. 4, 2005 1315
ness, the oil characterization procedure and the friction theory are briefly summarized in this contribution. Experimental Section Fluid Compositional and PvT Characterization. The procedure consists of characterizing the heavy oil fractions by distributing the C7+ mass fraction according to a chi-squared (χ2) distribution function with p degrees of freedom (CS(p)). The p is taken as a fitting parameter, so that an optimal mass distribution of the C7+ fraction is achieved. The determination of p is important, because it has been shown4 that a p value of ∼2 (corresponding to an exponential decay) may optimally represent a light fluid, whereas a larger value of ∼10 (a flat distribution) may be optimal for a heavy or altered fluid. The general mathematical form of the CS(p) distribution function is
2-p/2 -s/2 (p/2)-1 fdis ) e s Γ(p/2)
s0
0
fdis ds ) M6
(2)
Here, M6 represents the fluid total light mass fraction up to the C6 fraction and s0 is the value of s that satisfies eq 2. The C7+ fraction is characterized in m heavy fractions Fi with mass fraction fmi:
fmi )
∫
si f si-1 dis
v′ ) v - c
ds
(3)
The molecular weight MWi of the fraction Fi is given by
MWi ) MWsˆ i
(4)
(10)
where, in the case of reservoir fluids, c is estimated by the following mixing rule:
c ) Kv
∑x MW i
(11)
i
h.fr.
(1)
where s is a molecular-weight-scaled variable that satisfies the relation
∫
with the EOS of interest. In all of the examples presented here, ethane and propane were lumped together; similarly, n-butane, i-butane, and pentane were also lumped together. For all the remaining well-defined light compounds, tabulated values are used. In addition to matching the saturation pressure, a Pe´neloux volume translation9 is used in conjunction with this procedure.10 The volume correction delivers accurate density modeling results and consists of shifting the volume from the volume v obtained with the EOS to a translated volume v′:
In eq 11, the summation only includes the heavy fraction (h.fr.), i.e., pseudo-components corresponding to C7+, and Kv represents a characteristic volume constant for the fluid. The constant Kv is obtained by tuning against high-pressure density data measured at and above the saturation pressure Psat. All of the fluids presented here have been modeled using the PR EOS with the regular one-fluid van der Waals mixing rules (linear in b and quadratic in a) and the following binary parameters: 0.02 for N2-C1, 0.06 for N2-(C2-C3), 0.08 for N2Ci>3, 0.12 for CO2-C1, 0.15 for CO2-Ci>1, and 0 for all hydrocarbon-hydrocarbon interactions (Ci and Fi compound groups). Friction Theory. The application of the f-theory to the viscosity modeling of crude oils consists of applying f-theory models to previously characterized reservoir fluids. In the f-theory, the total viscosity (η) is separated into a dilute gas viscosity term (η0) and a residual friction term (ηf):
where sˆ i corresponds to
sˆ i )
1 fmi
∫
si
si-1
η ) η0 + ηf sfdis ds
From eq 4 and a mass balance, it can be shown that MW is given by
MW )
MW+
m
fmi
∑ sˆ 1-M 6 i)1
(12)
(5)
(6)
For the dilute gas viscosity η0, the empirical model by Chung et al.11 is recommended to be used. The residual term ηf is related to friction concepts of classical mechanics and can be approximated by
ηf ) κrpr + κapa + κrrp2r
(13)
i
where MW+ is the molar mass of the C7+ fraction. For any lumped compound groups and the m heavy fractions, the scaling parameters that are required in the EOS are estimated after empirical equations, based on the properties of normal alkanes. The empirical equations are
where pa and pr are the van der Waals attractive and repulsive pressure contributions given by a cubic EOS, such as the SRK EOS or the PR EOS, among others. In the case of reservoir fluids, the one-parameter general model12 is used. The formulation of the one-parameter general models is as follows:
Tc,i ) -423.587 + 210.152 ln(MWi)
(7)
Pc,i ) fc exp(9.67283 - 4.05288MW0.1 i )
(8)
pr pa p2r ηf ) ηc κˆ r + κˆ a + κˆ rr 2 Pc Pc P
(
ωi ) exp 8.50471 -
)
15.1665 MW0.1 i
(9)
where MW is the molecular weight (given in g/mol), Tc the critical temperature (in Kelvin), Pc the critical pressure (in bar), and ω the acentric factor for the fraction i. Here, fc represents a perturbation factor different from the fc ) 1 value that corresponds to the original fit after n-alkanes. The lumped components, up to C6, were all assigned a value of fc ) 1, and for the Fi C7+ pseudo-components, the perturbation parameter fc was tuned until the fluid saturation pressure was matched
(
c
)
(14)
where ηc is the characteristic fluid viscosity scaling parameter and Pc is the critical pressure. The κˆ r, κˆ a, and κˆ rr parameters are dependent on the reduced temperature and have been (9) Pe´neloux, A.; Rauzy, E.; Fre´ze. R. Fluid Phase Equilib. 1982, 8, 7-23. (10) Quin˜ones-Cisneros, S. E.; Dalberg, A.; Stenby, E. H. Submitted to Pet. Sci. Technol. 2003. (11) Chung, T.-H.; Ajlan, M.; Lee, L. L.; Starling, K. E. Ind. Eng. Chem. Res. 1988, 27, 671-679. (12) Quin˜ones-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. Fluid Phase Equilib. 2001, 178 (1-2), 1-16.
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Quin˜ ones-Cisneros et al.
Table 1. Main Characteristic Parameters for Two Heavy Oils (Oil 7 and Oil 8) and Two Light Oils (L Oil 1 and L Oil 2) Heavy Oils Oil 7, CS(8) component N2 CO2 C1 C2-C3 C4-C5 C6 F1 F2 F3 F4
Light Oils Oil 8, CS(6.5)
L Oil 1, CS(2.5)
L Oil 2, CS(5)
x
MW
x
MW
x
MW
x
MW
0.0007 0.0006 0.4206 0.0176 0.0075 0.0060 0.2411 0.1446 0.1001 0.0612
28.02 44.01 16.04 31.66 66.17 85.64 168.22 280.51 405.24 663.26
0.0113 0.0003 0.2786 0.0132 0.0097 0.0054 0.3029 0.1781 0.1239 0.0766
28.02 44.01 16.04 33.38 64.93 86.18 193.03 328.31 471.84 762.79
0.0002 0.0039 0.2771 0.0054 0.0105 0.0161 0.3165 0.1769 0.1205 0.0729
28.02 44.01 16.04 36.82 67.07 86.18 334.54 598.61 878.64 1451.60
0.0035 0.0004 0.2556 0.0003 0.0001 0.0002 0.3091 0.1942 0.1424 0.0941
28.01 44.01 16.04 32.07 65.74 84.00 252.56 402.02 548.12 829.31
MWT (g/mol) temp, T (K) saturation pressure, Psat (bar)
431.59 322.05 44.1
443.06 322.05 42.1
parametrized into universal constants related to a specific EOS. In the case of mixtures, the value of the mixture friction coefficients is predicted using the suggested mixing rules.12 In the case of oils, after the fluid has been properly characterized, for the well-defined light compounds (i.e., methane, ethane, etc.) reported tabulated values12 for ηc are used. If some light compounds (up to C6) have been lumped together, the following modified Uyehara-Watson equation3,13 may be used:
xMWiPc,i
2/3
ηc,i ) (7.94830 × 10-4)
1/6 Tc,i
(15)
86.57 427.6 186.2
80.9 418.15 224.3
In the case of the fluid region below the saturation pressure Psat, the composition of the liquid phase changes as it loses its gas content; therefore, a mixing rule for the foregoing corrections is required. This is an issue that is still under evaluation; however, the following mixing rule seems to provide satisfactory results:
∑x MW
ζ ) Kz
i
1/3 i
(19)
i
where the summation also applies only to the pseudocomponents of the heavy oil fraction.
Viscosity Results where the units in eq 15 are g/mol for MW, bars for Pc, and K for Tc to obtain ηc in units of mPa s. However, for the pseudocomponents corresponding to the C7+ fraction, eq 15 is relaxed by substituting the model constant for an adjustable common parameter Kc that can be taken as a viscosity characterization parameter for all of the pseudo-components in the heavy fraction, i.e.,
xMWiPc,i
2/3
ηc,i ) Kc
1/6 Tc,i
(16)
This approach has been shown to be highly accurate for the modeling and prediction of a wide range of reservoir fluids, ranging from natural gas to heavy oils.3,4,7,8,14 Compressibility Correction for Heavy Oils. Previously,4 it has been shown that, for highly dense fluids, the simple mathematical form of a van der Waals type of EOSs particularly, the repulsive termsis not adequate. However, a compressibility correction procedure that does not require a mathematical modification of the type of cubic EOSs that are used in oil applications has been devised. This procedure consists of displacing the volume by a value of ζ,
v˜ ) v - ζ
(17)
and then estimating the viscosity friction contribution, using the corrected volume. That is,
ηf ) κrp˜ r + κap˜ a + κrrp˜ 2r
(18)
where p˜ r and p˜ a are the respective repulsive and attractive pressure contributions estimated at the displaced corrected volume v˜ . (13) Uyehara, O. A.; Watson, K. M. Natl. Pet. News 1944, 36 (October 4), R-714-R-722. (14) Quin˜ones-Cisneros, S. E.; Ze´berg-Mikkelsen, C. K.; Stenby, E. H. Fluid Phase Equilib. 2003, 212 (1-2), 233-243.
As mentioned in the Introduction, this work is exploratory, in regard to the f-theory full prediction capabilities for practically the entire range of reservoir fluids that are of relevance to the oil and gas industry. For this purpose, seven different heavy oils and two light ones have been studied. The heavy oils correspond to five previously reported fluids (denoted as oils 2-6; to follow with the nomenclature in the previous work, no oil 1 is included here).4 The additional two heavy oils (oil 7 and oil 8) and two light oils (L oil 1 and L oil 2) have been characterized in a similar manner as the previously reported oils, i.e., according to the characterization procedure described here. The characterization results for these additional fluids are reported in Table 1. Also note that, in the case of oils 5 and 7, viscosity data are available for the 310.95, 322.05, and 333.15 K isotherms. For oil 8, viscosity data are also available for the 310.95 and 322.05 K isotherms. For all other fluids, the viscosity data correspond to the reservoir temperature reported in Table 1 and in the previous work for oils 2-7. Figures 1 and 2 show the viscosity modeling results obtained with the overall approach that has been recently developed4 and has been briefly described in this work. That is, by simultaneously tuning the viscosity characterization parameter Kc in eq 16 and the compressibility parameter Kz in eq 19. These modeling results go from light fluids (L oils 1 and 2) to heavy oils (oils 2-8), covering a full 4-order-of-magnitude span. Thus, these figures give a good overview of the capabilities of the method; however, some more-detailed representation is required to assess its accuracy. As previously discussed, the method is based on tuning data beyond the one-phase high-pressure region, which cor-
Density and Viscosity of Heavy Oils
Figure 1. Direct viscosity modeling results for the studied heavy oils.
Figure 2. Direct viscosity modeling results for the studied light oils.
Figure 3. Viscosity modeling results for the heavy oils onephase high-pressure region.
responds to the region where the most reliable measurements are performed, and predicting data below the saturation pressure, i.e., the area where most modeling and experimental uncertainty is observed. As such, as it is shown in Figure 3, the approach delivers highaccuracy results in the one-phase region: in Figure 3, all data and results below the saturation pressure have been omitted. As graphically demonstrated in Figure 3, the approach can follow the experimental measurements within the experimental uncertainty. In fact, for the one-phase region, the modeling results deliver an average absolute deviation (AAD) of only 1.34%, with a negligible bias of only 0.16%. In regard to the two-phase low-pressure region, the uncertainty in the viscosity measurements of heavy oils is substantial and large inconsistencies are frequently found. This is shown in
Energy & Fuels, Vol. 19, No. 4, 2005 1317
Figure 4. Illustration of viscosity modeling problems in the two-phase low-pressure region.
Figure 4, where three fluids have been selected, to show what is commonly the case. As depicted in the figure, in some cases, the viscosity predictions precisely follow the measurements (oil 3, low-temperature oil 5), whereas the measurements are underpredicted (oil 6) and overpredicted (high temperature oil 5) in other cases. However, this also reflects the intrinsic difficulties in this type of measurement, which result in substantially scattered data, even for the same sample, as it is shown in the case of oil 5. Therefore, in proposing eq 19 as a mixing rule for the compressibility correction factor, and, therefore, for the viscosity prediction of the twophase region, a compromise was sought. With the current approach, in some cases, the low-pressure viscosity measurements may be overpredicted or underpredicted, with a percentage deviation of up to two figures. However, when all the fluids discussed in this work are considered, the overall AAD is 7.5%, which is a substantial increment when only the one-phase region is considered. However, when the bias is evaluated, this turns out to be only 1.9%, which shows a rather balanced distribution of the prediction results that, for the two-phase region, this approach delivers. Next, to explore the potential of the method for prediction, the Kz parameter obtained with the direct modeling approach was analyzed in a way such that a reasonable empirical equation may be developed. For this, it was determined that the parameters that had the stronger influence were (i) the degrees of freedom (p) in the distribution function CS(p) and (ii) the molecular weight MW. In addition, some dependency on temperature was also observed and, therefore, a reduced temperature of Tr ) T/Tc was introduced, where the overall fluid critical temperature was defined as
Tc )
∑i xiTc,i
(20)
The results of this analysis are shown in Figure 5. Subsequently, using the empirical equation obtained for the compressibility parameter Kz, all of the fluids were modeled again, to obtain new values for the viscosity characterization parameter Kc. However, in this case, rather than using the overall Kc, actual ηc,i values were estimated, using eq 16, for each of the heavy fractions given by the characterization procedure (four per fluid). A residual ∆ηc,i value thenwas defined by the difference between the ηc value obtained with the
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Energy & Fuels, Vol. 19, No. 4, 2005
Figure 5. Empirical Kz correlation.
Quin˜ ones-Cisneros et al.
Figure 8. Density modeling results for the studied light oils.
proach gives quite satisfactory results for the density modeling of reservoir fluids, normally with an AAD better than 1%. No further analysis has been conducted with the density tuning parameter Kv; however, some future work in this direction will follow.
Conclusions
Figure 6. Empirical Kc correlation using ηc and eq 16.
Figure 7. Density modeling results for the studied heavy oils.
adjusted eq 16 and the ηc value that follows from the f-theory n-alkanes fit (see eq 15). Next, an empirical analysis is performed and the results are shown in Figure 6. Finally, using the empirical equations reported in Figure 2, all of the viscosity data were reproduced again. This approach resulted in an overall AAD of 21.1% with a bias of -4.4%, which is quite satisfactory for such a wide range of fluids. Density Results Some density measurements were available for the heavy oils 5-8 and the two light oils studied in this work. Figures 7 and 8 show the density modeling for all these fluids. Again, the density modeling approach is also based on tuning the one parameter in eq 11 against the one-phase high-pressure data and predicting below the saturation pressure Psat.4 Clearly, this ap-
The main result of this work indicates that the overall friction theory (f-theory) viscosity modeling approach that has been presented here, and in related references, results in characterization parameters that seem to be self-consistent and follow a pattern. Unquestionably, much remains to be studied and more heavy fluids with reliable viscosity measurements need to be analyzed. However, despite the crudeness of the empirical equations derived here, the final result is an f-theory model that can predict the viscosity of fluids that range from light ones (as the L oils 1 and 2) to all of the heavy oils studied here, within a reasonable uncertainty. The empirical equations presented in Figures 5 and 6 are derived for illustration purposes and are not, at this point, recommended. However, such empirical analysis shows a consistency of the parameters that are obtained after the directly tuning against all the fluids. Although, whenever possible, direct tuning against fluid data should be preferred for highly accurate models, it seems that the f-theory approach may be able to be further developed into a predictive tool capable of at least giving a rough estimation of the viscosity of reservoir fluids, from light to heavy ones. Therefore, this suggests that, after further analysis of a larger database, an acceptable predictive model may be feasible. In all studied cases, the density modeling results are consistently accurate. That is, above the saturation pressure Psat, the density is reproduced with an accuracy of better than 1%. Below the Psat value, the uncertainty may reach 2%-3% under dead oil conditions. At this point, the density modeling still requires the fitting of the Kv constant in eq 11. However, the development of predictive empirical equations for the Kv parameter will be the target of future work. EF0497715
