Ind. Eng. Chem. Res. 1995,34, 1873-1881
1873
Characterization of Heavy Oils. 2. Definition of a Significant Characterizing Parameter To Ensure the Reliability of Predictive Methods for PVT Calculations Jean-No451 Jaubert* and Evelyne Neau Laboratoire de Chimie Physique, Facult6 des Sciences de Luminy, 163 Avenue de Luminy, 13288 Marseille Cedex 9,France
A new and significant characterizing parameter ( q 3 0 ) which is easy to calculate is defined. This parameter can indicate in advance whether the PVT (pressure-volume-temperature) calculations performed on a crude oil using a predictive method will be in good agreement with experimental data. This parameter leads to a new classification of crude oils. When q 3 0 is small, the crude oil is considered as “classical” and all of its PVT properties can be accurately predicted. When q 3 0 is large, the crude oil is classified as “critical-like”; i.e., the prediction of some PVT properties may lead to significant deviations when compared to experimental values. In the latter case, a tuning process is proposed which in all cases improves the agreement between experimental and calculated data. Moreover, variance analysis theory is applied to estimate the experimental errors during a differential vaporization.
Introduction The design of process plants and the modeling of reservoir fluids require accurate data describing the phase behavior of oil and gas mixtures. However, in practice, PVT (pressure-volume-temperature) measurements such as density, viscosity, saturation point, shrinkage factor, swelling test, and slim tube minimum miscibility pressure are scarcely available. To estimate the properties of crude oils, it is therefore necessary to use a thermodynamic model, such as a cubic equation of state (eos), coupled with a procedure for characterizing the heavy fractions. In this respect, two estimation procedures can be described: (1)use of a predictive method such as the predictive model developed by Jaubert (1993) and Neau et al. (1993) or by Pedersen et al. (1989a,b) and (2) fitting of the heavy fraction parameters of the eos such as the critical temperature, the critical pressure, the acentric factor, or the binary interaction parameters (ku). For crude oils, the first procedure is generally preferable even if in most cases, the agreement between the measured and the calculated values is less satisfactory than using the fitting procedure. In fact, with predictive methods, reasonable accuracy is usually obtained for the saturation pressure, tank oil density, and relative volume. Tuning can be used to optimize the prediction of a given data set but may lead to less accurate results for other PVT properties not included in the tuning process. In the petroleum industry, the depletion experiments are usually performed at one temperature. As shown by Pedersen et al. (19881, tuning of the eos parameters at this temperature may lead to highly erroneous predictions at other temperatures. However, even if they are very useful for crude oils, predictive methods begin to fail for the estimation of the PVT properties of gas condensate mixtures or when the reservoir temperature is close to the critical temperature. The predictive method recently developed by Neau et al. (1993) can be used t o estimate (1)the variation of the relative volume with respect to pressure during a
* Author
to whom correspondence should be addressed.
Fax: (33)91.26.93.04. 0888-5885l95f2634-1873$09.0OfO
constant mass expansion with an average overall deviation of less than 1%, (2) the evolution of the relative volume with respect to pressure during a differential vaporization with an average overall deviation of 4.3%, and (3)the saturation pressure and the tank oil density with an average overall deviation of 2.7 and 2.5%, respectively. However, in some cases, the deviations observed on the relative volume during a differential vaporization may reach values as high as 14.7%(Neau et al., 1993, Table IV). In this paper, we focus on the limitations of the predictive methods and we define a new and significant characterizing parameter q which can indicate, before performing any experimental measurements, whether the PVT calculations performed with the predictive method of Neau et al. (1993) will be in good agreement with experimental data. Before comparing experimental and predicted values, it is necessary to estimate the errors in the experimental values using, for example, the error propagation law of Gauss. In the case where the parameter q indicates that the predictive method may lead to a significant difference between experimental and calculated values (taking into account the experimental accuracy), we have developed a method for tuning the parameters of the heavy fraction which leads to an improved correlation of the data. The main advantage of this tuning method is that the parameters obtained are not very different from the starting parameters given by the predictive method so that their use a t other temperatures should not lead to highly erroneous predictions.
Experimental Data To determine the conditions in which the predictive method leads to reliable results and to test the validity of the new characterizing parameter q , we have used the experimental data base described in detail in our previous paper (Neau et al., 1993). This data base comprises 14 crude oils from five different fields. In this paper, we have also added two new crude oils. The first one is an Indonesian crude oil, the composition of which has been published by Jaubert et al. (1995). The second one is a near-critical North Sea crude oil, the composition of which is given in Table 1. These two new oils
0 1995 American Chemical Society
1874 Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995 Table 1. Characteristics of the Near-Critical North Sea Crude Oil (Oil 16Ia reservoir oil molar weight density of the compounds (molar %) of the cuts cuts (g cm-3 HzS
NZ
coz
methane ethane propane isobutane n-butane isopentane n-pentane (36c 7
CS c9
ClO Clli
0.000 0.497 2.130 61.887 9.350 5.792 0.737 2.128 0.794 1.063 2.316 2.349 1.716 1.227 0.675 7.340
82.0 92.0 105.0 120.0 135.0 275.0
0.6795 0.7385 0.7615 0.7775 0.7830 0.8663
Depletion temperature (Tdep) = 401.05 K. Bubble point at Tdep (PSI= 374.0 bar.
Table 2. Results of Predictions Performed on Oils 15 and 16 Using the Predictive Method of Neau et al. (1993) molar ”c in methane depletion type AVrel%” AQ%~ 32.2 (oil 15) constant mass 0.52 -3.67 -0.24 differential 0.49 (-5.5 bar) vaporization 61.9 (oil 16) constant mass 0.16 -0.05 3.54 (-0.2 bar) a AVrel%is the mean absolute percent deviation of the relative volume. AP%and A@%are the percent deviations of the bubble point and standard tank oil density, respectively.
will be described as oil 15 and oil 16,respectively. The results obtained with the predictive method of Neau et al. (1993) on these two oils are summarized in Table 2. Analysis of the Accuracy of the Predictive Method of Neau et al. (1993). (a) The PVT Predictions and the Experimental Errors. As previously mentioned, the method of Neau et al. (1993) predicts the variation of the relative volume versus pressure to within an accuracy of 1%for constant mass expansions and 4.3% for differential vaporizations. This raises the question of the consistency of these deviations compared to the experimental uncertainties. The variance analysis method developed by Neau and PBneloux (1981) and applied to gas condensate mixtures by PBneloux et al. (1990)was extended to the case of differential vaporizations (Appendix). This method defines the confidence interval Z as that in which the probability of finding the “real” value of the experimental relative volume is 95%. To visualize this interval (eq A4), we have plotted the experimental values plus or minus I in Figure la,b. If the predicted values of the relative volume are in the interval Vrel & I, the deviations of the predictive method from the experimental values will be consistent with the experimental errors. If it is not the case, the predictive method is not reliable. This study was only performed on differential vaporizations since, for constant mass expansions, the predictive method calculates the relative volume to within an average deviation of less than 1%.In Table 3, we give the deviations given by the predictive method of Neau et al. (1993) on the relative volume during a differential vaporization and the values of the confidence interval 1. In the column indicates that the errors of the predictive verdict, method are lower than the experimental uncertainty, while - indicates the opposite case.
+
In half of the cases, the deviations of the predictive method are lower than the experimental uncertainties (see Figure la). When the deviations are not consistent with the experimental errors (see Figure lb), we always notice a sharp variation of the relative volume when the pressure is decreased by a few bars below the bubble point; Le., the slope of the curve is steep. For example, in the case of oil 12 (Figure lb), the relative volume varies from 4.4 to 3.5 when the pressure is decreased by only 20 bar below the bubble point. These results point out the necessity of knowing a priori whether or not a predictive method will lead to deviations larger than the experimental errors. This point will be developed in the next section. (b)The PVT Predictions and the Location of the Critical Point. It is well-known (Ahmed, 1989) that an isothermal depletion performed in the vicinity of the critical point (less than 50 K) leads to a very significant variation of the relative volume when the pressure is decreased to a few psi below the bubble point. This result is illustrated in Figure 2 where a differential vaporization has been simulated on a crude oil at a temperature 10 K lower than the critical temperature (T,)of the mixture. Under these conditions (in the vicinity of the critical point), cubic eo& produce significant errors in the prediction of liquid volume. In addition, when the deviations of the predictive method are larger than the experimental errors (see Figure lb), a sharp increase is also observed below the bubble point on the curve of relative volume versus pressure. It could therefore be expected that the depletion temperature and the critical temperature are very close. In order to check this hypothesis, we have considered all the oils whose deviations for relative volume were larger than 5 % during a differential vaporization. The critical point and the experimental slope of the relative volume curve just below the bubble point were calculated. The results obtained are given in Table 4. These results show that large deviations of the predictive method and large values for the slope of the relative volume curve are not a consequence of the vicinity of the critical point. In fact, the critical temperature is always 110 K greater than the depletion temperature. In conclusion, a depletion performed in the vicinity of the critical point will always lead to a large value for the slope of the relative volume curve below the bubble point (see Figure 2) and t o unreliable predictions of the liquid volume using a cubic eos. However, similar results may appear when the depletion temperature is far from the critical point. This effect must be explained.
Definition of a New Significant Parameter To Check the Accuracy of the Predictive Method In order to better understand why the slope of the relative volume curve (during a differential vaporization) below the bubble point remains steep (even when the critical temperature is really higher than the depletion temperature) and why the relative volume is not correctly predicted, we have assembled for each crude oil the variation of the liquid and gas volumes with respect to pressure and temperature. We have plotted inside the phase envelope the lines of constant vaporization rate u (also called “quality lines”). Each line represents the conditions of temperature and pressure for which the ratio Vga$Vtotalis constant. In this definition, V,, and Vbtd are the volume of the gas phase and the total volume, respectively. The bubble point
Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995 1875 3.0
5.0
2.0
3.0
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1.0
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vrd
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?-
__I)
0.0 300.0 600.0 0.0 300.0 600.0 Figure 1. (a) Differential vaporization at T/K= 382.05 of an oil containing 54.9 molar % of methane (oil 5;AVrel%= 1.4 and Z = 2.13%). (b) Differential vaporization at T/K= 396.15 of an oil containing 61.4 molar % of methane (oil 12;AV,1% = 3.6 and Z = 2.81%).Comparison between the predictive method of Neau et al. (-1 and the experimental data with their confidence intervals (+). Table 3. Comparison between the Errors of the fiedictive Method and the Experimental Uncertainty interval amplitude Z oil % deviationa (average value, %) verdict 1 0.3 0.95 2.01 2 1.3 3 2.2 1.89 4 5.4 2.72 2.13 5 1.4 6 1.6 2.66 1.99 7 1.6 8 5.4 2.70 14.7 3.06 9 10 no experimental data no data no data 11 2.49 9.3 2.81 12 3.6 13 5.5 2.54 no data no data 14 no experimental data 1.00 15 0.49 no data no data 16 no experimental data
12.0
=Mean absolute deviation on the relative volume during a differential vaporization using the predictive method of Neau et al.
0.0 300.0 600.0 Figure 2. Simulation of a differential vaporization at a temperature close to the critical point ofthe mixture (pluses indicate the pressures of the depletion).
+ +
+ + +
6.0
+
curve and the dew point curve correspond respectively to u = 0.0 and u = 1.0; we also note that all the quality lines coincide a t the critical point (CP). Two examples concerning oils 15 and 16 are shown in Figure 3a,b. The quality lines represent successively the vaporization rates: 0.1, 0.2, 0.3, 0.4, 0.5, 0.6,0.7, 0.8, 0.9, and 0.95. The main difference between oils 15 and 16 is the evolution of the vaporization rate with respect to pressure a t the depletion temperature. For instance, when the pressure is decreased to a value of 0.7p8,oil 16 has reached a vaporization rate equal to 0.5, whereas for oil 15, the value of u is less than 0.2. The fundamental difference between these two crude oils is therefore the swiftness of encountering the lines of equal vaporization rate when the pressure is decreased. To measure this swiftness, we define, a t a given temperature, the parameter q :
0.0
Table 4. Influence of the Vicinity to the Critical Point on the PVT Predictions oil 4 8 9 11 13
AV,l%a 5.4 5.4 14.7 9.3 5.5
experimental slopeb 12.7 24.6 25.8 27.7 9.3
Tde&
405.65 398.15 398.15 400.95 392.15
TdK 586.58 519.52 512.85 524.08 537.66
difference (Tc - Tdep)/K 180.93 121.37 114.70 123.13 145.51
AVrel%is the mean absolute percent deviation of the relative volume during a differential vaporization. "he experimental slope of the relative volume curve is calculated using the saturation pressure and the first diphasic pressure (1000[AV,~(A?'/bar)l). Depletion temperature. Calculated critical temperature.
pressure P inside the phase envelope:
lOOAv q=-z%
In this formula, Au is the variation of the vaporization rate between the bubble point (u = 0) and a diphasic
where V,, and V b d are the volume of the gas phase and the total volume at the diphasic pressure P,
1876 Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995
P’
0.7 P‘ 200.0
0.0 300.0 Tdcp 500.0 700.0 Figure 3. (a) Calculated phase envelop and quality lines (oil 15).(b) Calculated phase envelop and quality lines (oil 16).
t
300.0 pa
f P/bX
I
.... ..... . .......
T/K
4
300.0
700.0
Tdep
Figure 4. Illustration of the definition of the q parameter.
respectively. AP% is the corresponding relative variation of pressure:
AP% = 100-P - P
(3)
P
This definition is illustrated in Figure 4. We can note that at a temperature close to the critical point, the value of q may be very high. At the critical temperature itself, without decreasing the pressure (AP% 0.01, the vaporization rate u increases from u = 0 to u = 0.5 so that the value of q (eq 1)is infinite. To test the accuracy of the predictive method of Neau et al. (1993),we have systematically calculated at the depletion temperature the value of q for different values of Au (eq 1). The most significant results were obtained for a variation of the vaporization rate equal to 0.3. The corresponding value of q is described as
-
(4)
The values of q 3 0 obtained with the 16 crude oils from our data bank are reported in Table 5 . The deviations of the relative volume during a differential vaporization are also shown. This table shows that there is an obvious correlation between the value of q30 and the capability of the predictive method in estimating the relative volume during a differential vaporization. This
Table 5. Correlationbetween the Deviations of the Relative Volume during an Isothermal Depletion and the QSO Parameter dedetion t m e AVrei%a oil ann 1 0.3 constant mass 0.64 0.3 differential vaporization 2 1.o constant mass 0.95 1.3 differential vaporization 0.9 3 constant mass 0.80 2.2 differential vaporization 4 0.2 constant mass 1.36 5.4 differential vaporization 5 0.8 constant mass 1.14 1.4 differential vaporization 6 0.8 constant mass 1.70 1.6 differential vaporization 7 0.1 constant mass 1.33 1.6 differential vaporization 8 0.6 2.59 constant mass 5.4 differential vaporization 9 1.9 2.76 constant mass 14.7 differential vaporization 1.8 constant mass 10 1.06 differential vaporization no data 11 0.5 constant mass 2.59 9.3 differential vaporization 0.8 12 2.65 constant mass 3.6 differential vaporization 1.2 constant mass 13 2.03 5.5 differential vaporization 14 2.4 constant mass 3.63 no data differential vaporization 0.52 constant mass 15 0.67 0.49 differential vaporization 0.16 16 2.71 constant mass no data differential vaporization a AVr,l% is the mean absolute percent deviation of the relative volume.
feature permits the definition of a new classification for reservoir oils. (1) “Critical-like” Oils. If 4 3 0 > 1.3 at the depletion temperature, this reservoir oil can be classified as critical-like. The near-critical reservoir oils can obviously be considered as critical-like. This means that a small decrease in pressure below the bubble point leads to a large increase in the vaporization rate u as if the depletion temperature was very close to the critical temperature. In many cases, it would be quite difficult to perfectly predict the variation of the relative volume during a differential vaporization of this type of reservoir fluid. This is illustrated in Figure 5a,b. Note however that in all cases the proposed predictive method
Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995 1877
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PV*d
3.0 3.0
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Phar Phar 1.0 1.0 f 0.0 300.0 600.0 0.0 300.0 600.0 Figure 6. (a)Differential vaporization at T/K = 398.15 of an oil containing 60.2 molar % of methane (oil 8; 430 = 2.59 and AVr,l% = 5.39). (b) Differential vaporization at T/K = 392.15 of an oil containing 61.6 molar % of methane (oil 13;430 = 2.03 and AV,1% = 5.51).Comparison between experimental (+I and predicted (-1 variation of the relative volume during a differential vaporization for critical-like reservoir oils.
1
0.0 0.0 200.0 400.0 600.0 Figure 6. Constant mass expansion a t T/K = 401.05 of an oil containing 61.9 molar % of methane (oil 16).Comparison between experimental (+) and predicted (-) variation of the relative volume during a constant mass expansion for a critical-like reservoir oil.
satisfactorily predicts the bubble point, the tank oil density, and the variation of the relative volume during a constant mass expansion. This last point is illustrated in Figure 6 which refers to oil 16, the 430 value of which is one of the highest in the data bank (q30 = 2.7). (2) “Classical” Oils. If 430 1.3 a t the depletion temperature, the reservoir oil can be classified as classical. A decrease of pressure below the bubble point leads to a regular increase of the vaporization rate u . In this case, the predictive method predicts all of the PVT properties accurately as demonstrated in Figure 7a,b. (3) Discussion. In Figure 8, the variation of q 3 0 versus AT = ( T - T,) has been plotted for oil 15 and oil 16. At the depletion temperature, the first one is classical whereas the second one is critical-like. The corresponding phase envelopes are given in Figure 3. Three important points can be drawn from Figure 8: (1) When AT is close to 0, i.e. when T TS T,, the oil is critical and q30 becomes infinite. Unreliable results will be obtained with cubic equation of states for all of the oils. This is consistent with the conclusions of Ahmed (1986). (2) When AT is very large, 430 is small and all
of the oils exhibit the same behavior. The PVT properties in this case can be correctly predicted. These two first points show that the distance to the critical point and the q 3 0 parameter have the same significance when AT is close to 0 or when the critical temperature is much higher than the depletion temperature. (3) The most important difference between the two oils is observed a t moderate values of AT. For example, for AT = 150 K, 430 is equal to 1.07 for oil 15 and 1.90 for oil 16. In this case, for the same distance to the critical point, the behavior of both oils is completely different. Indeed, oil 15 is classical, whereas oil 16 is critical-like. In this case, only the 430 parameter allows us to gauge whether the predictive method will lead to good PVT predictions. In summary, it is very important to calculate this new significant parameter 430 before any other calculations or experimental determinations are conducted. If 430 < 1.3, the predictive method previously published by Neau et al. (1993) can be used to satisfactorily predict all of the PVT properties. In this case, it is not necessary to perform many experimental determinations in the laboratory. If q30 > 1.3, the reservoir oil is classified as critical-like and the predictive method may lead to significant deviations for the relative volume during a differential vaporization. It is therefore necessary to perform experimental determinations below the bubble point to analyze in more detail the swift increase of the vaporization rate with respect to pressure. These comments raise a new question, what to do when the predictive method leads to important deviations. In the case of critical-like oils, the deviations are probably due to the Peng Robinson eos and it is necessary to tune the parameters of the eos as described below.
Tuning of eos Parameters The only way to improve the agreement between the measured and the calculated results is to tune the parameters used in the eos. A totally self-acting tuning program previously developed by Gramajo (1986) and PBneloux et al. (1990) for gas condensate mixtures was extended to the case for oils. The experimental data included in the tuning process are the relative volume during a differential vaporization, the bubble point, and the tank oil density.
1878 Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995
1.0 0.0 200.0 400.0 0.0 200.0 400.0 Figure 7. (a) Differential vaporization at T/K = 380.35 of an oil containing46.4 molar % of methane (oil 2; q30 = 0.95 and AVr,l% = 1.34). (b) Differential vaporization at T/K= 360.95 of an oil containing 32.2 molar % of methane (oil 15;q30 = 0.67 and AV,1% = 0.49). Comparison between experimental (+) and predicted (-) variation of the relative volume during a differential vaporization for classical reservoir oils.
I \
3’0
I
Table 6. Results Obtained after Tuning the Parameters of the C11-Cle Pseudocomponent AVrei%=(differential oil vaporization) APO%b A@%b 1 0.42 0.19 0.00 2 0.28 0.15 0.01 0.91 3 0.42 0.00
‘,.(Oil16)
l oil 15)
’...
4 5
6 7 8 9
0.0
1
,
1 150.0
1 (Tc-T)K
___D
11 12 13 15 overall deviations
2.82 0.56 0.97 0.49 2.71 5.69 5.11 1.36 1.26 0.42 1.77
0.21 0.17 0.35 0.47 0.25 -0.08 -0.06 0.66 0.20 0.39 0.28
0.30 0.01 0.06 0.00 0.37 -1.42 -1.09 0.06 0.08 0.00 0.26
0.0 250.0 Figure 8. Correlation between q 3 0 and the distance to the critical point for a classical and for a critical-likecrude oil.
a AV-18 is the mean absolute percent deviation of the relative volume. APB%and A@%are the percent deviations of the bubble point and standard tank oil density, respectively.
The predictive method of Neau et al. (1993) describes the fluid as a mixture of nine components or pseudocomponents which are Nz, COz, C1, CZ,C3, C4, “C5-C10n, “Cll-C19”, CZO+.The best tuning results were obtained when the parameters of the Cll-C1g pseudocomponent were adjusted. We do not tune the properties of the true boiling point distillation residue CZO+since in most cases, the amount of this pseudocomponent does not exceed more than a few mole percent, whereas the amount of the Cll-C19 pseudocomponent is generally twice as much as that for CZO+.In this tuning method, the physical parameters of the pseudocomponents C5Clo and C ~ Oare + identical to those issued from the predictive method. The tuning process can be described as follows. First step: We adjust the critical temperature and the critical pressure of the Cll-Clg pseudocomponent; the acentric factor ( w ) and the Rackett compressibility factor ( Z M ) are those of the predictive method. The relative volume and the bubble point are in general very well-reproduced. Second step: We adjust the acentric factor and the ZM of the Cll-Clg pseudocomponent. The parameters obtained are in many cases very close to those calculated by the predictive method, but this procedure allows us to improve the calculation of the tank oil density. The results obtained are summarized in Table 6 and
are illustrated in Figure 9. It can be seen that the relative volumes, saturation pressures, and tank oil densities are accurately calculated. It should be pointed out that the predictive method was still used for estimating the properties of the CZO+ fraction and the parameters w and ZRAduring the first step of the tuning process. As a result, the tuned parameters were not very different from the starting values (see Figure 9a,b). Moreover, the obtained parameters were physically significant. For example, the tuned critical temperature and critical pressure of the Cll-Clg pseudocomponent were always lower and greater, respectively, than those of the CZO+ pseudocomponent. It is obvious that, in this kind of tuning process, it is absolutely not justified to tune the critical parameters or the acentric factor of the well-defined components (Nz, COz, C1, ..., CIO)since accurate measured data are available. The tuning procedure described in this paper may be used to obtain accurate results at the reservoir temperature. However, at other temperatures, the parameters of the predictive method may lead to better results. This means that the tuning process is used to improve all the correlation of the PVT properties a t the depletion temperature, but care must be taken for its use at other temperatures. The same tuning method could also be used in the case of constant mass expansions, but it is not necessary
Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995 1879 5.0
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1.o 0.0 300.0 600.0 Figure 9. (a) Differential vaporization a t T/K = 405.65 of an oil containing 51.3 molar % of methane (oil 4 9 3 0 = 1.36 and AV,1% = 2.82. Starting parameters (predictive method): TdK = 717.19, P h a r = 19.76, o = 0.5500, ZRA= 0.2529. Tuned parameters (Cll-Clg): TdK = 687.28, P h a r = 21.89, w = 0.5503, ZRA= 0.2594. (b) Differential vaporization at T/K = 392.15 of an oil containing 61.4 molar % of = 725.52, P h a r = 19.14, w = 0.5800, Z u methane (oil 12; 9 3 0 = 2.03 and AV,,1% = 1.36). Starting parameters (predictive method): = 0.2519. Tuned parameters (Cll-C19): TdK = 688.73, P h a r = 21.36, o = 0.5800, Zm = 0.2590. Comparison between experimental (+) and adjusted (-1 variation of the relative volume during a differential vaporization for different reservoir oils. 1.o
0.0
300.0
600.0
since the relative volumes are always accurately predicted (the average overall deviation on the constant mass expansions using the tuned parameter is AVr,l% = 0.3%).
Conclusion The predictive method recently developed by Neau et al. (1993)satisfactorily predicts the bubble point, the tank oil density, and the relative volume during a constant mass expansion. In more than half of the cases, the predictive method is also able to predict the variation of the relative volume versus pressure during a differential vaporization with an average error of less than 2%;in the other cases, the error is larger than 5 % and may in a few cases be as large as 15%. This paper does not improve the predictive method but instead defines a new significant characterizing parameter q 3 0 which is very easy to calculate. This parameter allows us to classify the reservoir oil under consideration as either classical or critical-like. If the crude oil is classified as classical, the predictive method will lead to a good prediction of all the experimental PVT data. If the reservoir oil is classified as criticallike, the predictive method may lead to errors larger than 5% concerning the relative volume during a differential vaporization. In this case, tuning the eos parameters of the CII-CN pseudo component is advised. This very simple approach, in all cases, leads to a very good agreement between the measured and the calculated results.
Acknowledgment The authors wish to gratefully thank Professor A. PBneloux for helpful discussions during this research and the French Petroleum Company TOTAL for financial support.
Nomenclature P = pressure P, = critical pressure Pa = bubble point of a crude oil at the depletion temperature
q 3 0 = parameter allowing to classify a crude oil as classical
or as critical-like T = temperature T, = critical temperature Tdep= depletion temperature (reservoir temperature) u = vaporization rate Vrel = relative volume during an isothermal depletion Zw = Rackett compressibility factor Greek Symbols
e = tank oil density under standard conditions (15 "C, 1 atm) w = acentric factor
Appendix I. Variance Analysis for the Estimation of the Confidence Interval I on Experimental Measurements. Experimental measurements Eo on a system include the measurement of q variables XO (for instance, the reservoir mole fraction the temperature F', and the pressure PO) and the measurement of one or several functions Q0 of these variables (like the liquid volume in the case of depletion experiments). A model O(X,A) allows for the calculation of functions when variables X and model parameters A are known. For a given set of parameters A, comparison between experimental and calculated functions is performed on the basis of the observed deviations:
e,
A@ = @' - @(P,A)
(All
This means that the estimated functions @i(XO,A) must be compared with the experimental functions @ subject t o the resulting errors a(@). In the general case, $(ai)is the diagonal term of the variance covariance matrix V,:
+
V, = VQe DyVx *Di where Vaeand VX,are the variance covariance matrices of experimental errors of measured functions and variables, respectively; DX is the matrix of partial derivatives of functions @ with respect to variables X.
1880 Ind. Eng. Chem. Res., Vol. 34, No. 5, 1995
In the case where experimental measurements Qo and
XO are independent, eq A2 leads to a diagonal matrix VQ, the elements of which are given by the error propagation law of Gauss:
For each measurement @,: the confidence interval I , at 95% in which the calculated function Qg,(XO,A) must be located is @: f Ii with
Ii= 2 d a 9
The solution of these systems at a pressure P: leads, among all the derivatives aFlaX, t o the derivative avL/ aX; the same procedure is repeated at the reference pressure P!ef, which allows us to obtain the derivatives aVredaXand therefore the derivatives OX,(eq A6). 2. Calculationsof ExperimentalErrors on Measured Relative Volumes: VR,. Due to errors on the reference volume Vref,the matrix VR, is not diagonal. The different terms are estimated as
(-44)
The previous method was applied to the estimation of the confidence intervals on the measured relative volumes: Vreli= VL/Vref= R (the relative volume is also noted R to simplifjrthe matrices notations). It was thus necessary to calculate the variance covariance matrix V R:
According to the accuracy of measurements performed in the petroleum laboratories, it was assumed that
ae(vLt) = 0.2 cm3
oe(Vref)= 0.2 cm3
(All)
3. Calculationof ExperimentalErrors on Measured Variables: VX,. No covariance was assumed with
between experimental determinations of temperature T, pressure P, and feed composition Xi,res:
where X represents the following variables: temperature T , pressure P, and reservoir compositions Xi,res. VR, and VX, are the matrices of experimental errors on the relative volumes and variables, respectively. The main difficulties encountered in these calculations are due to the fact that the use of an equation of state for flash calculations does not lead to an explicit model VL(XO,A) and also that the reservoir composition is not directly measured. The following procedure was used. 1. Calculations of Partial Derivatives: Dx. At a temperature To and pressure PO and for a given a flash calculation is perreservoir composition formed using an equation of state depending on parameters A. The calculated functions F are the liquid volume VL in the case of a single-phase system and in the case of a two-phase system, the liquid and gas volumes VL and VG, respectively, together with the liquid mole numbers n ~ If~ the . mixture contains n components, the flash calculation requires that a system n (two-phase) of p = 1 (single-phase) or p = 2 equations was solved:
However, it is necessary to estimate all the variances oe2(Xi,res) and covariances dgi,resdgj,cj,res between the mole fractions of the reservoir. For this purpose, we must recall that the reservoir feed composition is calculated from measurements performed by chromatography on the separator gas (x:,;J and on the tank oil and gas (x::z), assuming a recombination of the dif8erent fluids according to measured gas oil ratio (GORO):
+
M(XOPP)= 0
(A71
For fixed values of equation of state parameters, the derivation of equations (A71 leads to
M'AB dx)+ ATx dX = 0
(A81
(x::?)
with
and
Mai#;pk(15 "C, 1atm)
K2 = p
k liq
eair(15"C, 1atm)GORo'tank
(A151
where GORo*tank and GOR0aePare the standard gas oil ratios defined as a standard gas volume divided by a standard liquid volume
where M F and M x are the derivatives of the p equilibrium conditions (eq A7) with respect t o functions (VLor VL, VG, and n .) and variables (To,Po, respectively. It is &us possible to estimate the partial derivatives B = aF/aX of functions F with respect to variables by solving 4 systems (number of variables) of p equations (number of functions):
M'FB = -Wx
(A91
The molar weight (Mair)and the standard density of the air (@air) are 28.9784 and 1225.5614 gIm3, respectively.
Ind. Eng. Chem. Res., Vol. 34,No. 5, 1995 1881
The molar fractions xi>:, x F ~ and , x;,::~ are calculated from the measured weight fractions y: and molar weights
e:
y,pIMe
x. =‘
n
j=1
(nis the number of compounds) (A191 and co-
In order to estimate the variances Oe2(Xi,res) variances dgi,ci,resdgj,ci,resof the matrix Vx,,the variations dgi,Ci,res of the reservoir composition (eq A13) must be calculated with respect to the experimental errors 6 e ( G O R ) , de(&?) (eqs A14 and A15) but also with respect t o the errors of the weight fractions de(yi) and molar weights d e ( M i ) (eqs A17-Al9). Molar weights were assumed to be known very precisely ( d e ( M i ) = 0) for all components, except for the heavy fractions of the tank oil ( d e ( M + ) ) . The calculation of the terms of the matrix Vx, were performed assuming Oe(T) = 0.1 K, O J G O R ) = GOR x 0.005, u&) = according to the ASTM ~, norm, O J P ) = 0.1 bar, O e ( e k & = 0.0001 g ~ m - and a e ( M + ) = M + x 0.05.
Gramajo, A. Algorithmes pour le calcul des Bquilibres liquidevapeur et l’ajustement des parametres caract6risant les fractions lourdes des fluides phtroliers. Ph.D. Dissertation, The French University of Aix-Marseille 111, 1986. Jaubert, J. N. Une mBthode de caractkrisation des coupes lourdes des fluides p6troliers applicable a la prediction des propri6Gs thermodynamiques des huiles et A la rBcupBration assistbe du pBtrole. Ph.D. Dissertation. The French University of AixMarseille 111, 1993. Jaubert, J. N.; Neau, E.; PBneloux, A.; FressignB, C.; Fuchs, A. Phase Equilibrium Calculations on an Indonesian Crude Oil Using Detailed NMR Analysis or a Predictive Method To Assess the Properties of the Heavy Fractions. Znd. Eng. Chem. Res. 1996,34 (2), 640-655. Neau, E.; Peneloux, A. Estimation of model parameters. Comparison of methods based on the maximum likelihood principle. Fluid Phase Equilib. 1981,6, 1-19. Neau, E.; Jaubert, J. N.; Rogalski, M. Characterization of Heavy Oils. Znd. Eng. Chem. Res. 1993,32, 1196-1203. Pedersen, K. S.;Thomassen, P.; Fredenslund, Aa. On the dangers of “tuning“ equation of state parameters. Chem. Eng. Sci. 1988, 43,269-278. Pedersen, K.S.; Thomassen, P.; Fredenslund, Aa. Characterization of gas condensate mixtures. In C7+ Fraction Characterization; Chorn, L. G., Mansoori, G. A., Eds.; Taylor and Francis: New York, 1989a; pp 137-152. Pedersen, K. S.; Fredenslund, Aa.; Thomassen, P. Contributions in Petroleum Geology & Engineering 5. Properties of oils and natural gases; Gulf Publishing Co.: Houston, Tx, 198913; Vol. 5. PBneloux,A.; Neau, E.; Gramajo, A. Variance analysis fieri years ago and now. Fluid Phase Equilib. 1990,56, 1-16. Received for review August 17, 1994 Revised manuscript received January 12, 1995 Accepted January 30, 1995@
Literature Cited Ahmed, T. Contributions in Petroleum Geology & Engineering 7. Hydrocarbon Phase Behavior; Gulf Publishing Co.: Houston, TX, 1989; Vol. 7, pp 27-29.
IE940497J Abstract published in Advance ACS Abstracts, April 1, 1995. @
