I n d . E n g . Chem. R e s . 1989, 28, 340-341
340
Literature Cited Bunn, C. W.; Emmett, H. Crystal Growth from Solution, I. Layer Formation on Crystal Faces. Discuss. Faraday SOC.1949,5, 119. JanEiE, S.J.; Grootscholten, P. A. M. Industrial Crystallization; D. Reidel: Boston, 1984. Michaels, A. S.; Colville, A. R., Jr., The Effect of Surface Active Agents on Crystal Growth Rate and Crystal Habit. J. Phys. Chem. 1960, 64, 13. Mukerjee, P.; Mysels, K. J. Critical Micelle Concentrations of Aqueous Surfactant Systems. National Standard Reference Data System, 1971, NSRDS-NBS, p 36. Mullin, J. W. Crystallisation; Butterworth: London, 1971; p 208.
Torii, K.; Iitaka, Y. The Crystal Structure of L-Isoleucine. Acta Crystallogr. 1971, B27,2237. Zumstein, R. C. 1. Modeling, Determination, and Measurement of Growth Rate Dispersion in Crystallization. 2. The Crystallization of L-Isoleucine in Aqueous Solutions. Ph.D. Thesis, North Carolina State University, Raleigh, 1987. Zumstein, R. C.; Rousseau, R. W. Solubility of L-Isoleucine in and Recovery from Neutral and Acidic Aqueous Solutions. Ind. Eng. Chem. Res. 1989, in press.
Received for review April 25, 1988 Revised manuscript received October 12, 1988 Accepted November 2, 1988
Modified Corresponding States Method for Predicting Densities of Petroleum Reservoir Fluids Alvin S. Cullick* Mobil Research & Development Corporation, Dallas, Texas 75381
Fred N. Pebdani Mobil Exploration and Producing Services, Dallas, Texas 75247
Andreas K. Griewank Argonne National Laboratory, Argonne, Illinois 60439
An essential reservoir fluid property for reservoir engineering, fluid transport design, and reservoir simulation is the fluid density (compressibility factor). The oil density as a function of pressure is normally available from standard laboratory measurements. But in cases of enhanced oil recovery by gas injection or miscible flooding, or in multiphase transport, the dramatic changes in fluid density as a function of concentration and pressure are commonly not measured, and standard correlations are generally inaccurate. Two-parameter equations of state, e.g., Peng-Robinson, and extended corresponding states also predict densities inaccurately for complex reservoir fluids. We demonstrate computational improvements and the use of the critical volume as an adjustable parameter t o the extended corresponding states principle (ECSP) to improve predictions of petroleum reservoir fluid densities. This new method provides accurate and computationally reliable predictions for complex petroleum fluids. We illustrate with five example systems.
Summary The extended corresponding states principle (ECSP) can be used to predict fluid densities (compressibility factors). Having identified limitations with the formulation developed a t the National Bureau of Standards (Ely, 1981; Ely and Hanley, 1981) (incorporated in the TFWP program) for applying ECSP to petroleum reservoir fluids, we have modified the computational method and used an adjustable parameter to make the computations more reliable and the density predictions more accurate. Five example reservoir fluid mixtures are presented. These examples illustrate the utility of the ECSP in predicting fluid densities of petroleum reservoir fluids at reservoir or surface conditions for a lean gas condensate, a rich gas condensate, a volatile oil, and two light oils mixed with carbon dioxide. Comparisons are also made with the two-parameter Peng-Robinson equation of state. The Appendix section summarizes the theory and method. Introduction Accurate predictions of petroleum reservoir fluid properties are required by pipe-flow network, reservoir, and processing simulators. Fluid-phase behavior, densities, viscosities, and interfacial tensions are necessary for simulation of processes which depend on the mass transfer of
* Author t o whom correspondence should be addressed.
components between phases, that is, compositional effects. For example, compositional reservoir simulators and pipeline network simulators are used to simulate pressure maintenance, miscible flooding, or multiphase flow processes for which in situ fluid densities can change by an order of magnitude or more in the flow stream as a result of mass-transfer processes. Two-parameter, cubic equations of state (Martin, 19791, such as the Peng-Robinson equation (Peng and Robinson, 1976), are used throughout the petroleum industry to calculate fluid-phase properties. However, the PengRobinson equation, for example, predicts inaccurate liquid-phase densities, with errors of 10-25%, as illustrated in this repart, although a recent three-parameter modification to the Peng-Robinson equation has greatly improved its density predictions (Jhaveri and Youngren, 1988). Errors in computed formation volume factors and compressibility factors, and even viscosities and interfacial tensions that depend on densities, are directly proportional to errors in density. We require an accurate means of computing fluid densities for reservoir fluids, while retaining cubic equations for their fast, reliable phase equilibria predictions. The extended corresponding states principle (ECSP) for predicting pure fluid-phase densities was introduced originally by Leland et al. (1968) and Rowlinson and Watson (1969). Leach et al. (1968) improved the predictions for mixtures of nonpolar hydrocarbons by introducing gen-
0888-5885/89/2628-0340$01.50/0 0 1989 American Chemical Society
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 341 eralized shape factors for n-paraffins, with methane as the reference fluid. With the use of mixing rules and a BWR equation of state for the methane reference fluid, Ely (1981) and Ely and Hanley (1981) made the model available to the industry as the National Bureau of Standards' (NBS) TRAPP computer program. Although the Ely method works reasonably well for mixtures of pure fluids up to about n-pentadecane, there are four problems with the Ely formulation when applied to petroleum reservoir fluids: (1)there is no provision for C,+ oil fractions, so ECSP calculates poor densities for mixtures containing C7+ fractions; (2) the solution method often does not converge for mixtures containing C7+ fractions; (3) there is no means to utilize experimental data for improving predictions; and (4) predictions for mixtures containing high concentrations of carbon dioxide are poor. Previously, Cullick and Mathis (1984) reported on the use of mixture interaction coefficients to improve predictions for mixtures containing carbon dioxide. Interaction parameter adjustment does not improve predictions for C 0 2systems in all cases. In this paper, we report the development of a method for solving the ECSP system of simultaneous equations. This new method has eliminated convergence failures in our tests. We demonstrate that the critical molar density of C,+ fractions can be utilized as an adjustable parameter, and we have written a computer program that automatically adjusts parameters for C7+ fractions to match experimental data. We illustrate the accuracy of the method with five examples. We believe this method has value as a predictive tool as well. Two computer programs were written in this work. The first incorporates modifications to the ECSP for improved convergence. Newton's method with an Armijo (Ortega and Rheinboldt, 1970; Armijo, 1966) line search is used to compute the reference fluid corresponding state. The second program performs a parameter optimization of adjustable ECSP parameters by using a MINPACK (Garbow, 1980) routine to achieve agreement between calculated densities and laboratory measured densities. Description of the Modified ECSP ECSP is a completely predictive method for calculating fluid densities, requiring only the critical temperature, critical pressure, critical volume, and Pitzer's acentric factor for each component. As a result of our testing the NBS version of ECSP (Ely, 1981) on a number of simulated reservoir mixtures, we concluded that improvement in the method was required when a mixture contains C7+ fraction. For mixtures with C7+ fractions for which the component critical properties are sometimes ill-defined, the solution of the set of simultaneous equations often failed to converge, and ECSP often predicted much higher densities than measured when it does converge. According to the principle of corresponding states, the density, p , of a fluid can be expressed as P = P,/h
(1)
where "0" refers to the reference fluid. The reference fluid in this work is methane, as represented by a BWR equation, reported by Ely and Hanley (1981). The corresponding reference fluid temperature is
To = T / f
(2)
The parameters f and h are referred to as equivalent substance reducing ratios. These are functions of the critical properties and acentric factor of the pure fluid components, i.e.,
f = (Tc/TT,")B(T/TC, V / V c ,w) h = (Vc/V,")&T/TC,V / V c ,w)(ZT,"/Zc)
(3)
(4)
where superscript "c" refers to critical state, w is the Pitzer acentric factor, and V is the volume. The shape factors, B and 4,are continuous, piecewise linear functions of reduced volume and temperature. The shape factors were generalized for the C1-CI5 n-paraffin hydrocarbons by Leach et al. (1968). A more detailed description and extension to mixtures is in the Appendix section. As pointed out by Ely (1981) and Ely and Hanley (1981), it is not unexpected that the ECSP will be less reliable as the molecular size difference increases between the reference fluid and the mixture components. Also, Leach et al.'s (1968) generalized shape factor correlations become less applicable as the C7+ fractions deviate from the nparaffins. However, development of new requisite reference fluid equations of state and shape factor relations is an alternative, but a very tedious procedure. We solved problems of nonconvergence of ECSP by developing a mathematical technique to solve the ECSP equations more consistently (details in Appendix section). First, the piecewise shape factors were made continuous and differentiable so that a Newton procedure could be used to solve for the density rather than a successive substitution method. Second, we solve a set of two equations rather than the entire system simultaneously. We also found it necessary to improve the initial guesses for the corresponding reference fluid conditions. Ely and Hanley (1981) initiate the calculations with shape factors of unity to obtain an initial reference fluid density. For liquid phases containing C7+ fractions, that is insufficient, and it is necessary to initiate the calculations by assuming a "liquidlike" reference fluid density from which the initial reference fluid pressure is computed by the BWR equation. The liquidlike reference fluid density is obtained from an equation of state for saturated liquid densities of pure methane (Goodwin, 1974). The second problem of poor density prediction was solved by identifying and utilizing the critical volume (or its inverse, i.e., critical molar density) as an adjustable parameter to be used in fitting the ECSP to laboratory data. The adjustable parameter, which we refer to as the critical density parameter, is a good choice because the ECSP is very sensitive to the magnitude of the critical volume, and the critical volume is the least frequently measured pure component critical property. It is also the only pure component property required by ECSP but not required by many cubic equations of state, such as the Peng-Robinson equation of state (EOS). Thus, the Peng-Robinson EOS can be coupled with the ECSP in a single process simulator: adjustment of T,, P,, or w of the C7+ fractions to match K-value data will be independent of adjustment of V, (or its reciprocal p") to match density data. For mixtures, binary interaction parameters which appear in the ECSP mixing rules can also be utilized for improving predictions, but adjustments are, in our experience, not necessary for hydrocarbon binaries. Examples of Density Predictions for Reservoir Fluids Predictions of petroleum reservoir fluid densities for five mixtures are presented. The densities for each mixture range from reservoir to surface processing conditions. These fluids include a light synthetic oil with varying carbon dioxide concentrations, a light reservoir oil also with varying amounts of injected carbon dioxide, a volatile oil,
342 Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989
a rich gas condensate, and a lean gas condensate. For each example system, a set of experimental density data, typically over a range of temperatures from near ambient to reservoir, was used in conjunction with compositional analyses and C,+ fraction characterization to find optimal critical density parameters ( p c ) for the C7+ fractions with the computer regression routine. Once optimal parameters were determined, ECSP density predictions were compared with experimental density data not used in the fitting, using the ECSP. Some comparisons are for the single-phase fluid, and others are for the individual phases in two-phase systems. In some cases, those data represent fluid-phase compositions that are much different than the fluid compositions used in the fitting. Therefore, the optimal critical density parameters can be utilized for extrapolating the density predictions to fluid compositions not used in the fitting, which is essential in using this method for process design. The results from the two-parameter Peng-Robinson equation of state are presented for comparison. Initial Estimates for Properties of C7+ Fractions. The initial estimates for the critical pressure, temperature, and acentric factor of the C7+ fractions are determined by the method of Peng and Robinson (1976). For the initial critical volume estimate, two different correlations were tested in great detail during this work. The first is a correlation of average boiling point and specific gravity by Riazzi and Daubert (1980),
V , = 7.0434
X
10-7[TB2.3829][SG 60-1.683]
where TBis the boiling point in degrees Rankine, SG60 is the specific gravity at 60 OF, and V , is given in ft3/lb mol. The second is a correlation of critical volume as a function of average molecular weight and specific gravity presented by Hall and Yarborough (1971), V , = 0.025[MW/SG600.69]1.15 where MW is the component molecular weight. Both are based on the same experimental critical volumes reported in the API Research Project 44 handbook for more than 100 pure compounds, with molecular weights up to 280. The application of Riazzi and Daubert's (1980) correlation to the ECSP has been discussed by Baltatu (1982) for predicting viscosities of petroleum fluids. The average boiling point and specific gravity of each heavy fraction were supplied by Robinson and Peng's (1978) correlations. In general, the Hall and Yarborough (1971) method gave superior results for both initial estimates and final fitted values of critical density parameters for the examples studied. In principle, both methods should give similar results since Riazzi and Daubert's correlation can be expressed as
Table I. Composition of the Synthetic Oil and Properties of Its Heavy Fractions concn, concn, comDonent mol % component mol % C1 34.67 n-C, 3.06 c2 3.13 n-C, 4.95 c3 3.96 n-C, 4.97 n-C4 5.95 heavy fr. 1 (Clo) 30.21 n-C5 4.06 heavy fr. 2 (CI4) 5.04 pc, kmol/m3 heavy fr. P c . MPa T C K , wn MW initial fitted 1 (C,,,) 2.108 617.6 0.4885 142.286 1.67 1.48 2 (C14) 1.574 692.95 0.6442 198.4 1.18 1.20 ~
Acentric factor.
Table 11. Comparison of Density Predictions for Synthetic Oil % error ECSP with pressure, mol fr. default fitted temp,K MPa CO, P-Rb parameters parameters 322 15.48" 0.00 -5.3 6.9 0.3 0.00 7.0 322 10.20 0.3 322 11.48" 0.8 19.87 5.6 12.11a 322 39.73 4.4 0.8 322 12.11 0.7 49.69 -4.6 4.0 322 11.93" 59.70 -4.9 4.3 0.3 2.5 322 11.51 5.7 69.77 322 11.22 3.9 74.82 6.6 322 10.91 79.85 -5.5 8.2 6.1 322 10.56 84.87 10.3 8.9 14.6 322 10.12 89.93 -6.1 15.1 339 14.13 0.00 -5.3 7.9 0.7 339 10.75" 0.00 8.2 0.9 339 12.37" 20.22 6.1 0.2 339 13.49" 40.35 4.6 0.5 339 13.64" 60.44 5.7 1.9 339 13.12 80.17 10.0 8.2 a
Data used in fit.
'I1
= Peng-Robinson EOS.
S
P-R EQUATION 0 ECSP DEFAULT pc ECSP FITTED p c
8
12
.*
8
% ERROR
.
* . a
4 4
vs EXPERIMENTAL
0
1
V , = 0.036MW'.084SG60-0~580 = 0.036[MW/SG600.535] '.0a4 -8
which differs from Hall and Yarborough's by, at most, 5% over large ranges of molecular weights and boiling point temperatures. The differences between ECSP results seem to arise from the boiling point estimate, which is quite different for the Peng and Robinson and Riazzi and Daubert methods. Light Synthetic Oil. The composition of a 10-component synthetic oil system is tabulated in Table I with all component properties needed to reproduce the results. The densities of this system as a function of carbon dioxide concentration a t reservoir conditions were reported by Turek and Metcalfe (1980). Table I also contains a list of the properties of the two heavy components, n-decane and n-tetradecane. For the purposes of this example, the
* P-R
40
1
20
40
60
80
100
COP CONCENTRATION, MOLE PERCENT
Figure 1. Comparison of density predictions for the ECSP and Peng-Robinson equations for C02/synthetic oil mixtures a t 322 K.
decane and tetradecane were treated as C7+ fraction pseudocomponents, and their critical volumes were allowed to vary. Both critical densities were adjusted simultaneously. Table I1 and Figure 1 present comparisons of experimental and calculated densities, both before and after parameter optimization. The results of ECSP are also compared with predictions by the Peng-Robinson equation of state. Generally, for this system the Peng-Robinson EOS and unoptimized ECSP have about the same mag-
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 343 Table 111. Composition of the Light Oil and Properties of Its Heavy Fractions comDonent concn, mol % component concn, mol % 2.46 4.22 n-C5 3.35 n-C6 0.25 5.74 23.48 heavy fr. 1 heavy fr. 2 9.90 14.49 8.10 heavy fr. 3 9.38 1.01 6.20 heavy fr. 4 4.47 heavy fr. 5 3.96 1.50 heavy fr. 6 1.49 pc, kmol/m3 heavv fr. P', MPa T c ,K w MW initial fitted
Table IV. Density Predictions for the Light Oil (377.59 K ) % error ECSP with pressure, exptl p , fitted default MPa kg/m3 P-R parameters parameters 24.17" 19.13" 13.96" 11.17 8.72 6.98 5.65 (I
1 2 3 4 5 6
2.354 2.214 1.898 1.466 0.992 0.644
,801
DENSITY
(kglm3)
660
0.425 0.460 0.572 0.794 1.194 1.860
622.37 669.39 726.23 787.71 856.40 924.59
126.6 154.9 200.9 275.1 392.4 607.5
1.95 1.61 1.25 0.906 0.613 0.384
1.87 1.30 0.989 0.743 0.552 0.379
0 EXPERIMENTAL --P-R EQUATION - ECSP
704.3 697.1 689.2 693.9 697.3 714.1 724.7
-10.81 -10.90 -11.10 -11.13 -10.17 -11.24 -11.73
0.89 0.95 1.00 1.04 1.95 0.49 -0.28
10.9 10.8 10.7
Used in fit.
Table V. Density Predictions for the Light Oil (20 mol % CO,. 377.59 K ) % error ECSP with pressure, exptl p , fitted default MPa kg/m3 P-R parameters parameters 24.17" 21.24" 18.02" 12.39 8.95 7.12 5.93
4 (I
709.1 704.2 697.6 708.2 721.9 729.7 731.3
-10.56 -10.65 -10.69 -11.41 -11.57 -11.72 -11.35
-0.11 -0.06 0.14 -0.25 -0.50 -0.75 -0.45
9.0 8.7 8.7
Used in fit.
580L 540
5003 . 4 5
10
15
PRESSURE
20
25
30
(MPa)
Figure 2. Comparison of density predictions by the ECSP and Peng-Robinson equations for the light oil at 378 K.
nitude of error. After optimization, however, the ECSP gives much better results, except at the highest COz concentrations. Note that the predictions by the ECSP deteriorate as the carbon dioxide concentration increases above about 70%. At very high carbon dioxide concentrations, the critical density parameters of the heavy components have little effect on the results due to their low concentrations. We have found this problem with high COz concentrations to be common for reservoir oils. It appears either that the mixing rules (eq A-9-A-12 in the Appendix section) may need improvement for high carbon dioxide concentrations or that the critical volume used for COz should be modified. Light Oil with Varying COz Concentrations. The composition and heavy fraction properties of a light oil are tabulated in Table 111. Liquid densities were measured a t varying pressures with increasing amounts of COz to simulate changes which might occur during a COz injection process. The results of optimizing the critical density parameters ( p c ) appear in Tables IV-VI1 and in Figure 2 for the mix with no added COP The liquid-phase density decreases as the COz concentration increases up to the bubble point pressure and then increases with further pressure increase. Liquid-phase composition below the bubble point pressure was computed by using the PengRobinson equation. The optimal pc parameters €or the ECSP give excellent results for density prediction with
Table VI. Density Predictions for the Light Oil (35 mol % COZ, 377.59 K) % error ECSP with fitted default pressure, a p t 1 P, parameters parameters P-R MPa k/m3 24.17" 18.95O 14.53" 12.48 10.40 8.72 7.62
710.9 706.0 712.8 723.3 730.9 736.1 742.5
-10.02 -11.13 -10.72 -11.38 -11.66 -11.76 -12.18
-0.53 -1.53 -0.24 -0.84 -1.11 -1.22 -1.72
6.4 7.6 7.2
" Used in fit. Table VII. Density Predictions for the Light Oil (50 mol % CO,, 377.59 K ) % error ECSP with pressure, exptl p , fitted default MPa ka/m3 P-R Darameters parameters 27.39" 25.91" 17.79 14.35 10.52 7.48
716.3 712.6 706.9 738.7 746.0 750.5
-8.55 -9.00 -8.49 -11.82 -12.0 -11.95
-0.36 -0.27 -1.60 -1.81 -1.86 -1.80
6.2 6.1
Used in fit.
errors averaging less than 1% ,whereas the Peng-Robinson equation of state gives poor results, with errors averaging about 10%. ECSP predictions do not appear to be adversely affected by increasing carbon dioxide concentrations, at least up to 50 mol % COz. Only density data for the reservoir oil/C02 mixture above their bubble point pressures were used in the fitting procedure. Therefore, density predictions for the liquid phase below the bubble point pressure depend in part on the quality of the phase composition predictions, which were calculated by the
344 Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 Table VIII. Composition of the Rich Gas Condensate and ProDerties of Heavv Fractions component concn, mol % component concn, mol '7~ NZ 2.12 heavy fr. 3 1.12 0.45 heavy fr. 4 0.89 COZ 61.73 heavy fr. 5 0.68 C1 12.53 heavy fr. 6 0.58 c2 6.84 heavy fr. 7 0.48 c3 1.56 heavy fr. 8 0.41 XI 2.15 heavy fr. 9 0.34 n-C, 0.77 heavy fr. 10 0.54 i-C5 0.67 heavy fr. 11 0.56 n-C, 0.91 heavy fr. 12 0.48 n-C 1.98 heavy fr. 13 0.33 heavy fr. 1 1.55 heavy fr. 14 0.33 heavy fr. 2 pc, kmol/m3 heavy fr. P c , MPa T C K , ub MW initial fitted 1.69" 100.20 2.44 535.33 0.343 1 2.935 1.64" 561.61 0.402 114.20 2.15 2 2.661 1.69" 3 2.448 585.28 0.453 128.30 1.92 1.61" 608.56 0.469 142.30 1.74 4 2.282 1.59 628.33 0.507 156.30 1.59 5 2.138 1.46 6 2.002 646.17 0.544 170.30 1.46 1.34 I 1.883 662.56 0.577 184.40 1.34 1.26 677.67 0.608 198.40 1.26 8 1.790 1.18 9 1.717 692.11 0.648 212.40 1.18 1.07 10 1.569 711.94 0.685 233.01 1.07 740.50 0.765 267.38 0.928 0.876" 11 1.412 12 1.238 776.72 0.863 315.25 0.789 0.759" 13 1.058 817.61 1.00 377.62 0.658 0.658 14 0.825 875.39 1.25 480.27 0.512 0.512 ~~~~
~
Table X. Composition of the Volatile Oil and Properties of the Heavv Fractions component concn, mol % component concn, mol % n-CS 1.42 CO, 0.75 n-C6 1.86 Cl 58.22 c2 14.36 heavy fr. 1 2.01 c3 8.65 heavy fr. 2 1.24 i-C4 0.81 heavy fr. 3 1.36 8.14 n-C, 0.39 heavy fr. 4 i-C, 0.79 pc, kmol/m3 heavy fr. P c , MPa TC, K ua MW initial fitted 1 3.305 544.49 0.286 94.18 2.65 2.64 2 2.935 573.19 0.332 108.2 2.31 2.27 3 2.711 600.78 0.369 121.12 2.07 2.03 4 1.450 756.12 0.755 243.0 1.01 0.953 Acentric factor. P-R EQUATION 0 ECSP DEFAULT
pc
-
a
% ERROR
vs
- 4 1
EXPERIMENTAL
Used in the fitting. bAcentric factor.
Table IX. Comparison of Density Predictions for the Gas Condensate 7~error ECSP with pressure, default fitted temD. K MPa P-R parameters parameters 374 45.85' -1.7 2.94 0.91 374 43.20" -2.6 2.38 0.41 374 41.02" -3.2 1.92 0.00 374 39.16" -3.3 1.60 -0.27 374 37.61" -3.4 2.13 0.29 374 36.13 -3.9 1.87 0.07 374 32.82 -3.5 3.30 1.20 -4.4 2.50 0.50 374 30.27 374 26.82 -4.5 2.80 0.70 374 17.48 -6.5 -1.10 -2.90 374 7.48 -6.0 -5.40 -6.40 0.10 306b 3.12" -7.7 7.90 28gb 0.10 -9.7 -2.00
" Used in fitting. ditions.
Condensate phase a t surface separator con-
Peng-Robinson equation of state. Rich Gas Condensate. The compositions and heavy fraction properties of a rich gas condensate are tabulated in Table VIII. Of the 14 heavy fraction pseudocomponents, critical density parameters for 6 were used in the fitting procedures, as indicated. The dew point pressure of the fluid at 374 K is 36.13 MPa. Optimal critical density parameters appear in Table VIII, and predicted densities are compared with experimental in Table IX. Table IX shows the results when 6 of the 14 C,+ fractions were used in the fitting procedure. All 14 fractions, as well as other combinations, were tried. Using only the first four fractions, which have the largest concentrations, gave essentially the same quality of fit as using six or more fractions. The Peng-Robinson densities are listed for comparison. In this case, the Peng-Robinson equation is in error by about 3-5%, whereas the fitted
1 -12 -16
14 0
20 5
10
15
25
4
30
PRESSURE IMPa)
Figure 3. Comparison of density predictions by the ECSP and Peng-Robinson equations for the volatile oil.
ECSP predictions are accurate to within 1%at reservoir conditions. However, for the liquid condensate (oil) phase at separator conditions, the error is more than 8% for the Peng-Robinson EOS, but only 2% for the ECSP. It should be noted here that only six data points were used in the fitting, five of those having the reservoir fluid composition (above dew point pressure), whereas the data below the dew point pressure represent densities for two-phase systems. Thus, the results are in part affected by the validity of the Peng-Robinson phase composition calculations, which are very good in this case. Volatile Oil. The composition and heavy fraction properties of a blend representing a volatile reservoir oil are tabulated in Table X. The optimal critical density parameters and comparisons of experimental and predicted densities appear in Tables X and XI and Figure 3. The results of the fitting are excellent, with errors in predicted densities averaging less than 1 % , while the Peng-Robinson equation of state again gives consistently poor results with errors averaging 14%. The four fluid compositions used in the parameter fitting represent widely varying compositions because the oil's liquid-phase composition changes considerably as a function of pressure below its 28.50-MPa bubble point. Using four data points of widely varying composition provides a good test of the method. Lean Gas Condensate. Table XI1 presents the composition and heavy fraction properties of a lean gas condensate. Figure 4 compares the compressibility factors calculated by the ECSP with those measured, again dem-
Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 345 Table XI. Comparison of Density Predictions for the Volatile Oil % error ECSP ECSP Dressure, w/fitted wtdefault P-R parameters parameters MPa temp, O F 26.30 23.55 20.79 0.103 20.79 17.34 13.86 10.45 7.00 3.58 0.103 0.24 1.14
362" 362" 362" 297" 362 362 362 362 362 362 362 297 297
0.99 0.75 -0.44
-0.43 -1.10 -0.10 0.80 0.80 1.60 2.50 1.40 0.40 1.00
2.43 2.20 1.14 4.12
-11.1
-13.5 -16.0
only component critical properties and Pitzer's acentric factor, o. The model equates the compressibility factor of the mixture as well as of each individual component to that of a reference fluid at the corresponding state given by Toand po. The corresponding states are related by the substance reducing ratios f and h. The density ( p ) of a fluid, pure fluid or mixture, can be expressed as P = Po/h (A-1) where refers to the reference fluid. Also, the corresponding reference fluid temperature is To = T / f (A-2)
-12.6
The parameters f and h referred to as equivalent substance reducing ratios, for pure fluids, are functions of the critical properties (superscript "c" to equation variables) and acentric factor, i.e.,
-14.0 -15.8
fi
"Used in the fit.
= (T,"/Toc)6'i(Ti/T,", V/V,",ai)
(-4-3)
hi = (V,"/VOc)&(Ti/T,C, V/V,",wi)(Z,C/Z,") (A-4) Table XII. Composition of the Lean Gas Condensate component composition, mol %
The shape factors Oi and Bi =
14.16 0.30 71.77 6.00 2.69 0.65 0.87 0.48 0.36 0.69 2.03
COP
nitrogen methane ethane propane isobutane n-butane isopentane n-pentane hexanes heptanes pb
1+
& are expressed as
(oi - w,)x(Ti/T,", Vj/V,")
& = 1 + ( ~- iwO)l(T/T,",V/V,") (A-6) where @i = & [Z$/Z,"] as defined by Leach et al. (1968) and where 2," = P,"V,"/RT,". The ratio V,"/Z,"= RT,"/P," does not depend on p,", the eventual regression parameter, x(Ti/T,", Vi/V,") = al + bl In Ti+ ( c , d,/Ti+)(Vi+ - 0.5) (A-7)
+
+
and [(Ti/T,C,V/V,") = a2(Vi'
- --0
1.2
(A-5)
EXPERIMENTAL ECSP
+ b2) + c2(Vi' + d,)
In Ti+ 64-8)
where
Ti+= min (2, max (Ti/T,C,0.5))
PENG-ROBINSON EQUATION
and
Vi' = min (2, max (Vi/V,",0.5)) The mixing rules for h and f are as follows:
f
CCxixjfijhij/h r
J
h = CCxixjhij i
."..
t
27.6
(A-10)
J
where
I
30
(A-9)
35
40
45
50
PRESSURE (MPa)
Figure 4. Comparison of compressibility factor for a lean gas condensate fluid.
onstrating excellent agreement. In this case, there was no adjustment of parameters required.
Acknowledgment We thank the management of Mobil Oil Corporation for permission to publish this paper. This paper was first presented a t the AIChE 1988 Spring National Meeting, March 7-10, 1988, New Orleans.
Appendix: Detailed Description of the Modified Method for Densities from Extended Corresponding States The extended corresponding states model, as presented by Ely and Hanley (1981), is a completely predictive model for fluid densities and viscosities. The method requires
fij
=
(fifj)1'2(1
- hij)
h IJ = 1/8(hi1i3+ h,1/3)3(l I - Zij)
(A-11) (A-12)
where xi is the mole fraction of component i. Note also that there are two interaction coefficients, kij and lip These can be used as optimizing parameters to fit data for binary systems. Both interaction parameters are normally taken to be zero, which is valid for hydrocarbon-hydrocarbon systems. However, for carbon dioxide-hydrocarbon systems, the interaction parameters can deviate significantly from zero.5 Density Calculations. Given a mixture with molar composition ( x i ) , i = 1, ..., n, a t a fixed temperature ( T ) and pressure ( P ) ,we want to compute the resulting density ( p ) and its derivatives with respect to the mole fractions ( x i ) as well as certain critical densities (p,"). To this end, one has to determine the corresponding temperature (To) and density (p,) of a reference fluid, e.g., methane, for which the functional relationship
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Po = P(To,P,) (A-13) is explicitly known. In our program, we use Ely and Hanley's Benedict-Webb-Rubin (BWR) equation. To determine To and p,, we construct a hierarchy of smooth equations in two variables that can be solved by Newton's method. The two quantities Toand p , are related to reduced temperatures (Ti')and volumes (V:) of the fluid components i by the system of 2(n + 1)equations of the following form by
and
few remaining cases, it was assumed that Vir was at its upper bound, V; = 2, so we then solved for T; by Newton's method applied to the first equation. This procedure produced a solution, T t , V t , for all systems tested. The derivatives with respect to Toand p o are now computed according to the implicit function theory as the 2 X 2 matrix: a(T,', V,')/d(To,P,) = -[a(Fi, Hi)/d(T,', v,')l-l[a(J'i, ffi)/d(To,poll (A-18) These derivatives are then used to solve the nonlinear system (A-15) whose Jacobian with respect to To and p , according to the chain rule is given by the 2 x 2 matrix G(T,, P,) = W', W / d ( T 0 P,) , +
2 [a@', H)/d(T?, v,')l[d(Tt,v t ) / d ( T o poll , (A-19)
i=l
Instead of solving the whole system simultaneously, we prefer to use the equation pairs F, = 0 and H , = 0 to compute T,' and V; as implicit functions of Toand pW By substituting the resulting values into the last two equations, we then obtained another system of two nonlinear equations which can be solved with respect to Toand p o by Newton's method. This approach may be somewhat less efficient than solving the whole system simultaneously, but it makes it much easier to localize and remedy convergence failures which have been observed in a few cases for the successive substitution method, particularly when the mixture contains C,+ fractions. The actual form of the inner system is FL['T'(>. p,,. T#r* Vir)
(T,c/T,)T,r- 8,(9(T,'),q(V,'))
Here the functions F = V - V and H = 3/X - V are derived from the following three expressions. First we have (A-20) V = (T/P)P(To,p0) / ( Topo) which is simply the principle of corresponding states. Second, we may rewrite (A-l), using (A-lo), as P = CCXiXjhLj(1 - I,! (A-21 1 i
Lij
= y8[(Vir/pic)1/3 + ( V { / P ~ " ) =" ~hij/[p
