Ind. Eng. Chem. Process Des. Dev. 1985, 2 4 , 948-954
948
Literature Cited Bromely, L. A. A I C M J. 1973, 19, 313. Chen. C. C.: Britt. H. I.: Boston. J. F.: Evans, L. B. AIChE J. 1982, 28. 588. Gamble, D. ‘S.can. J.. Chem. ‘1967; 45, 2805. a u q u e , w. F.; h n u n g , E. w.; Kunzler, J, E,;Rubin, T, R, J , A ~ them, , Soc. 1980, 82, 62. Gordon, J. E. “The Organic Chemlstry of Electrolytes”; Wiley-Interscience: New York, 1975; Chapter 1. Hanson, C.; Ismail, H. A. M. J. Appi. Chem. Biotechnol. 1975, 25, 319. Kirk, R, E,;mmr,D. F, ~n “ ~ ~ ~ ~of ~Chemical l o ~Technow-, e d ~ 2nd ed.; Wlley: New York, 1969; Val. 19, p 441. Librovich, N. 8.; Zarakhani. N. G.; Vlnnik, M. I.Russ. J. Phys. Chem. 1971, 45, 1257. Libr, M, “Reaction ~chenisms in Sulfwic Acid and Saong Acid Solutions”; Academlc Press: New York, 1971; p 19. Moeller, T. “Inorganic Chemistry”; Wlley: New York, 1952: p 532. Munro, D. C. Appl. Specwosc. 1968, 22, 199. PRzer, K. S.;Roy, R. N.; Sylvester, L. F. J. Am. Chem. SOC. 1977, 99,
Robinson, D. R.; Jencks, W. P. J. A m . Chem. SOC. 1965, 8 7 , 2462. Rudakov, E. S.;Lutsyk, A. I . Russ. J. Phys. Chem. 1979, 53, 731. Setschenows J. z . .‘ Stoechiom. Verwandfschaftsl 1889* 4 , .1-
III.
Wagman, D. D.; Evans, W. H.; Parker, V. 8.: Schumm, R. H.; Halow, I.; Bailey, S. M.; Churney, K. L.; Nutail, R . J. J. Phys. Chem. Ref. Data 1982, 1 1 , 1. Young, T. F.; Maranviile, L. F.; Smith, H. M. I n “The Structure of Electrolytic Solutions”; Hamer, W. J., Ed.; Wiley: New York, 1959; Chapter 4. Zarakhani, N. G.;Vinnik, M. I.RUSS. J . PhYS. Chem. 1963, 37,260. Zemaitis, J. F. I n “Thermodynamics of Aqueous Solutions with Industrial Applications”, ACS Symposium Series No. 133; Newman, S., Ed.; American Chemical Soclety: Washington, D.C.. 1980; Chapter 10. Zemaitis, J. F. Chem. Solve., Inc., Morristown, NJ, personal communication, 1982.
Received for review March 7 , 1983 Accepted October 25, 1984
AQ3n
Ridd&’J. A.; Bunger, W. 6. “Organic Solvents, Physical Properties and ~ e t h o d sof Purification”, 3rd ed.;Wiley-Interscience: New york. 1970; p
78.
Presented at the AIChE Annual Meeting, LOSAngeles, CA, Nov 1982. Sessions on Computers in Process Design and Analysis.
Thermodynamics of Petroleum Mixtures Containing Heavy Hydrocarbons. 3. Efficient Flash Calculation Procedures Using the SRK Equation of State Karen Schou Pedersen,t S Per Thomassen,§ and Aage Fredenslund Instftuttet for Kemiteknik, Danmarks Teknlske H ojskole, DK-2800 Lyngby, Denmark, and STA TOIL, Den Norske Stats Oljeselskap a s , N-400 7 Stavanger, Norway
Two methods for drastically reducing oil/gas flash calculation computing times using the SRK equation of state are described. (1)The characterization procedure given in parts 1 and 2 of this series has been extended with a procedure for grouping hydrocarbon fractions. The predictions of the phase behavior using a total of only six hydrocarbon fractions (C1-C30+)are as accurate a s when 40 hydrocarbon fractions are used. (2) The abovementioned characterization procedure uses binary interaction coefficients (kt values) equal to zero for all hydrocarbon-hydrocarbon interactions. In naturally occurring oil and gas systems, the contents of non-hydrocarbons (mainly N, and COP)are often below 10 mol % . For such mixtures it is found that using kt = 0 for all interactions (also with the non-hydrocarbons) has virtually no effect on the calculated results. Explicit use of the assumption of zero ki/ values leads to substantial savings in the flash calculation computer time.
Introduction The first two articles of this series (Pedersen et al., 1984a,b) presented a characterization method for heavy hydrocarbons that, when applied with the SRK equation of state, permits accurate predictions of the phase behavior of the phase equilibria in gas condensates and heavy oils. About 40 hydrocarbon fractions (pseudocomponents)were used to represent a petroleum mixture. However, in many practical applications, such as oil reservoir computer simulation studies and two-phase flow calculations, it is not possible to include such a large number of components in the calculations. In this work, methods for reducing the number of pseudocomponents are investigated. Not only computer storage facilities but also computing time may be limiting factors. Michelsen and Heidemann (1981) have shown that the computing time needed to calculate a critical point by using a cubic, two-constant Instituttet for Kemiteknik. *Present address: Calsep Aps, Lyngby Hovedgade 29, DK-2800 Lyngby, Denmark. 8 STATOIL.
0196-4305/85/1124-0948$01.50/0
equation of state can be considerably reduced if all the binary interaction coefficients ( k i jvalues) are equated to zero. Similar computational advantages exist in two-phase flash calculations (Michelsen, 1983). The consequences of using kij values equal to zero are examined with respect to computing time and accuracy for oil/gas systems.
Data In order to develop the procedure for reducing the number of pseudocomponents and to study the effect of that and the effect of equating the kij values to zero, it is necessary to have available reliable analytical, volumetric, and phase equilibrium data for many different gas condensates and heavy oils. Such data and procedures for obtaining the data are given by Pedersen et al. (1984a,b). Two additional analytical techniques have recently been introduced. By use of vacuum distillation at a pressure of 130 N/m2, the end point of the TBP distillation has been extended approximately from CZ0to Cs0. Enough carbon number fraction is collected to permit measurements of specific gravity and molecular weight. In addition, the PNA analysis has been improved. A liquid-liquid chromatographic method which on a preparative scale 0 1985 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985 949
splits each carbon number fraction into a paraffinic (P) naphthenic (N) fraction and an aromatic (A) fraction is used. Refractive indices are used to determine the relative amounts of P and N. (For further details see Pedersen et al., 1984a.) Three analytical data sets on North Sea reservoir fluids are listed in Table VII. Sample A is a typical gas condensate for which TBP data are available up to C30+. Sample B is a heavy oil and sample C a heavy gas condensate. The Cg-Cm fractions of sample C have been split in a P N and an A fraction. The data given in Table VI1 may be used to characterize the reservoir fluids as recommended by Pedersen et al. (1984a). Recommended Characterization Procedure for t h e SRK Equation of State The components of a reservoir fluid may be classified as follows: defined components for which there exists a quantitative GC analysis and for which T,, P,, and w are known; TBP fractions covering the true boiling point fractions between C7 and ca. Cs0, each containing many different components. Careful analysis has shown that for the North Sea oils, the best results were obtained by using the Cavett (1964) relations for T, and P, T , = 768.071 1.7134T50- (0.10834 X 10-2)TW2+ (0.3889 x 10-6)T503- (0.89213 X 10-2)T50API (0.53095 x 10*)T5,2API (0.32712 x 10-7)T502AP12 (1)
+
+
+
+
+
log P, = 2.829 + (0.9412 X 10-3)T5~ (0.30475X 10-5)T502+ (0.15141 X 10-8)T503- (0.20876 X 104)T5,API + (0.11048 x 10-7)T502API+ (0.1395 X 10-9)T502AP12 - (0.4827 X 10-7)T50AP12(2) and the Lee-Kesler relations (Kesler and Lee, 1976) for 0:
w =
[In P B-~5.92714 + 6.09648/T~,+ 1.28862 h T B , - 0.169347T~,e]/[15.251815.6875/T~,- 13.4721 In T B r -k 0.43577T~,6](3)
(for TB, < 0.8) w = -7.904 0.1352K - 0.007465@ + 8.359T~,+ (1.408 - 0.001063K)TBr (4)
+
(for TBr > 0.8). In eq 1and 2, API = 141.5/SG - 131.5, where SG is the 60 OF/60 O F specific gravity. The critical temperature, T,, and the midvolume boiling point of the fraction, T50, are given in OF and P, is psia. In eq 3 and 4, TBr is the reduced boiling point, TB/T,, and P B , is the reduced pressure, PB/Pc,where P B is the pressure a t which the boiling point, TB, has been measured. K is the Watson characterization factor that equals TB1/3/sG,where TB is in O R and SG is as above. The results are improved if the PNA distribution of each boiling point fraction is taken into consideration as suggested by Erbar (1977). T B P Residue. In the case of gas condensate, the molar composition of the TBP residue is estimated by assuming a logarithmic relationship between the molar content, z N , of a fraction and the corresponding carbon number, CN, for C N L 7. C N = A1 + B1 In Z N (5) A , and B1are constants determined by a least-squares fit to the experimental TBP data for the C,-C, fractions. Cm is the heaviest carbon number fraction considered, and molar fractions below 5 X lo4 are neglected. If the measured T B P residue is larger than the one calculated
Table I. Input to Flash Program for Sample A (Gas Condensate) components mol % wt % T,,K P,,atm N Z
co2 C1 c2-c3 c4-c6
c&10
cll-c15 c16-c48
0.64 9.16 68.80 13.54 3.94 2.30 0.94 0.68
0.63 4.19 38.83 16.85 9.20 8.57 5.13 6.60
126.2 304.2 190.6 335.7 450.7 565.0 653.3 766.2
33.5 72.8 45.4 45.2 34.3 31.6 25.1 14.8
w 0.040 0.225 0.008 0.123 0.225 0.408 0.656 1.085
Table 11. Input to Flash Program for Sample B (Heavy Oil) comDonents mol % wt % T,.,K P,.atm w N2
COZ
C1 c2-c3 c4-c6
CTc15 cl6-c26
C27--C76
0.34 0.84 49.13 10.76 7.34 19.69 7.80 4.10
0.10 0.41 8.69 4.26 5.69 29.87 23.56 26.31
126.2 304.2 190.6 338.2 464.6 620.3 759.4 939.0
33.5 72.8 45.4 45.0 33.2 25.3 14.9 9.8
0.040 0.225 0.008 0.126 0.242 0.628 1.039 1.488
from the obtained A, and B1values, A , and B1are changed accordingly as explained in part 1 of this series. For the heavy oil systems, a different characterization procedure is used. The TBP distillation curve is extrapolated to 100 wt % distilled off by fitting a fifth degree polynomial to the experimental TBP data plus the following two artificially generated data points: - 10-0.2127T 1.103SG-0.6495 (6)
50wt% 101.083T 0.7097~~0.6717 TlciJwt%5Gwt%
T80wt%
(7)
Tsowt%, Tam%,and TlmW% are the temperatures in K at which 50, 80, and 100 wt % have been distilled off. SG is the specific gravity of the TBP sample that roughly corresponds to the C7+fraction of the total mixture. The specific gravities of the subfractions of the TBP residue are obtained by assuming a logarithmic dependence of SG against carbon number. For the boiling points, the values of Katz and Firoozabadi (1978) are used up to C45. For gas condensates, the relation TB (K) = 97.58MW0.3323SG0.0460g
(8)
is used for the fractions heavier than C45. For the heavy oils, the weight-based TBP curve is related to carbon number fractions assuming 6K boiling point intervals per carbon number for the C45+fractions. The characterization procedure described above yields equation of state input parameters for typically 80 components or pseudocomponents. Obviously, for extensive process calculations it is necessary to drastically reduce this number. Reduction of t h e Number of Components A procedure has been developed whereby all the hydrocarbon segments of the C7+ fractions are given approximately equal importance. This is accomplished via a grouping of carbon number fractions on weight basis. Typical groupings are shown in Tables I and 11, from which it may be seen that Nz, COz, and CH4 (C,) are given separate status, ethane and propane are grouped together (C2-C3), and so are c4’s, c i s , and c6’s. The c7+ fraction is divided into three (or more) groups, which on a weight basis are of approximately equal size. The group critical properties are computed as weightmean values:
950
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
T,i = Pci =
x;MWjT,j/&jMW;
J
J
Cx;MWjPC;/CxjMW; J
J
wi
(9)
= Cx;MWjwj/Cx;MWj I
J
(11)
Index i is a group of carbon number fractions containing, e.g., C1.(methane)only, or Cz and CB,or C4,C5,and C6,etc. Index J runs from the lowest to the highest carbon number within a group i. For the C,+ carbon number fractions, Pcj, Tcj,and wj in eq 9-11 are computed by using eq 1-4. Tables I and I1 give examples of input parameters to SRK-flash calculations using six hydrocarbon fractions corresponding to, respectively, sample A and B of Table VII. Such inputs plus overall compositions, temperatures, and pressures are all that is required for the flash calculations.
Flash Calculations Using the SRK Equation of State with Interaction Coefficients Equal to Zero The Soave-Redlich-Kwong equation of state may be written as follows (Soave, 1972): RT a p=--u - b U(U + b) For mixtures, the constants a and b, are given by the mixing rules a = CCYgjaij I
(13)
J
b = CYibi i
Indices i and j indicate summation over all components and/or groups of carbon number fractions. Usually, aij for i # j is given by aij = (aiiajj)li2(1- kij)
(15)
where the binary interaction coefficients kij are small, typically zero for hydrocarbon-hydrocarbon interactions and in the range 0-0.15 for others. The pure-component parameters aii and bi are given by aii = (0.42748R2Tci2/Pci)X [l + mi(l - (T/Tci)'/2)]2 (16) bi
= 0.07780RTci/Pci
(17)
In eq 16, mi is given in terms of the acentric factor, wi. mi = 0.480 + 1.574wi - 0.176wi2
(18)
If the characterization procedure described in the previous two sections is used in connection with the SRK equation, all hydrocarbon-hydrocarbon kij values (eq 15) are equal to zero. If in addition ki; values for the nonhydrocarbon-hydrocarbon and non-hydrocarbon-nonhydrocarbon interactions may be assumed equal to zero. Michelsens's (1983) simplified flash calculation procedure described briefly below may be applied. We shall later see that k , = 0 for all i - j interactions is a good approximation for reservoir fluids even if they contain up to 10 mol % N2 plus COz. When k , = 0, for all i and j , the mixing rule of eq 13 may be written (19)
a = a2
with (y
= Cyi(yi, 1
(y. I
=
11
(20)
Table 111. Deviations between Measured and Calculated Saturation Points for 15 North Sea Reservoir Fluids" 3CH 6CH 10CH 20CH 40CH fractns fractns fractns fractns fractns AAD 18.1 5.6 5.0 5.2 5.0 BIAS -0.1 -3.2 -2.2 -2.5 -2.3
" CH:
hydrocarbon; AAD: average absolute percent deviation;
BIAS: percent bias. Table IV. Relative Computer Time for Flash Calculations on Petroleum Mixtures" 3CH 6CH 10CH 20CH 40CH program fractns fractns fractns fractns fractns 0.46 1.00 0.16 0.25 kij Z 0 0.08 0.06 0.08 0.09 0.15 kij = 0 0.05 "CH: hydrocarbon; kij # 0: flash program with some of the binary interaction coefficients different from zero; ki, = 0: flash program with all binary interaction coefficients equal to zero.
With all kij = 0 the fugacity coefficient for component i is found to be given by In
+i =
"( RT b
- 1)- In (&(u
-
b)) -
Equation 21 is valid for both vapor and liquid phases. For a vapor phase, vapor-phase properties uv, bv, and av are used, and similarly uL, bL, and aLare used for the liquid phase. The equilibrium ratio, Ki = yi/xi, is given by
Ki = +?/c$?
(22)
From eq 21 it may be seen that Ki = Ki(T,P,aV,aL,bV,bL) (all i ) (23) The volume, u , does not appear in eq 23 as u is given implicitly by a , b, T , and P (see eq 12). As may be readily shown (Michelsen 1983), a and b for the feed (F), vapor (V), and liquid (L) are interrelated. ffF = P f f V + (1- 6 ) a L (24) b~ = 6bv + (1- 6)bL (25) P is the fraction of moles in the feed that appears in the vapor phase. a F and bF are known. Thus, in a flash calculation where T and P are specified, Ki is a function of only three variables:
Ki = Ki(aV,bv,P)
(all i) (26) This means that, irrespective of the number of components i, T, P, flash calculation procedures involving only three variables can now be formulated. For multicomponent mixtures such as reservoir fluids, this observation may be used to improve drastically the efficiency of flash algorithms. One such algorithm, i.e., the one used in this work, is described in detail by Michelsen (1983). Results We examine separately the effect of reducing the number of components and of equating all kij values to zero. Reduction of the Number of Components. Dew and bubble point calculations have been carried out for 15 reservoir fluids (the samples for which molar compositions are given in the supplements to parts 1and 2 of this series and samples A-C of this work). The samples cover petroleum mixtures ranging from light gas condensates to heavy oils. Table I11 lists the deviations between experimental and calculated saturation points using 3,6, 10, 20,
Ind. Eng. Chem. Process Des. Dev.,Vol. 24, No. 4, 1985 951
-.-
Experiment 8 CH-fract.(kij#O) 8 CH-frlct. (kij - 0 )
:>
-
P
9
100
200
300
400 Pressure ( a t m )
Figure 1. Constant volume depletion results for sample A at T = 121 OC. Vapor and liquid densities have been determined by the method of PBneloux et al. (1982).
100
200
300
400
500
Pressure (atm)
Figure 2. Constant mass expansion results for sample C at T = 129 "C. The SRK equation is used for the vapor densities and the Alani-Kennedy equation (1960) for the liquid densities.
and 40 groups of carbon number fractions. The relative flash calculation computer time is shown in Table IV. The computer time (cpu s) needed to perform one flash calculation for a mixture of eight components is given in Table VI1 for five different computers. Tables V and VI show measured and calculated phase compositions of a gas condensate (Table VII, sample A) and a heavy oil (Table VII, sample B) using 6 and 40 groups of hydrocarbon fractions. Figures 1-4 present results of simulations of three PVT experiments described in part 2 of these series using 6 and 40 hydrocarbon fractions. The pseudocomponents defined for the total initial mixture are maintained for both the gas and liquid phases for any further process calculations on each separate phase. Figure 1 shows measured and predicted liquid dropout curves of a constant volume depletion process on sample A. In this case the differences between the results for 6 and 40 CH fractions are so small that only one curve is shown. Figure 2 shows results for a constant mass expansion process on sample C. Figures 3 and 4 present
952
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
Table VII. Analytical Data for Samdes A. B. and C" sp grav comDonents
mol %
(15 " C i 1 5 "C)
0.64 9.16 68.80 8.43 5.11 0.81 1.45 0.52 0.53 0.63 0.83 0.95 0.52 0.26 0.20 0.17 0.16 0.15 0.11 0.086 0.078 0.068 0.050 0.046 0.035 0.025 0.034 0.023 0.017 0.018 0.014 0.012 0.013 0.047
0.741 0.780 0.807 0.819 0.810 0.828 0.849 0.857 0.868 0.872 0.859 0.854 0.866 0.873 0.876 0.876 0.875 0.877 0.876 0.878 0.882 0.886 0.889 0.908
P
MW Sample A (Gas Condensate)*
96 107 121 134 147 161 175 190 206 222 237 251 263 339 G o +
PNA distribution, mol % N
A
0.50 0.45 0.48 0.47 0.56 0.55 0.54 0.49 0.52 0.55 0.57 0.70 Cia+
0.42 0.38 0.27 0.30 0.27 0.24 0.22 0.27 0.20 0.19 0.20 0.11
0.08 0.17 0.25 0.23 0.17 0.21 0.24 0.24 0.28 0.26 0.23 0.19
0.56 0.58 0.58 0.57 0.62 0.59 0.57 0.57 0.53 0.53 0.56 0.58 0.57 0.56
0.39 0.33 0.30 0.33 0.31 0.33 0.32 0.31 0.32 0.31 0.30 0.28 0.28 0.20
0.05 0.09 0.12 0.10 0.07 0.08 0.11 0.12 0.15 0.16 0.14 0.14 0.15 0.24
Sample B (Heavy Oil)*
c 19
c20+
0.34 0.84 49.23 6.32 4.46 0.86 2.18 0.93 1.33 2.06 3.33 4.06 2.76 1.33 1.79 1.70 1.81 1.46 1.49 1.08 1.13 0.99 0.88 7.64
0.7395 0.7518 0.7756 0.7930 0.7902 0.8060 0.8203 0.8311 0.8446 0.8515 0.8542 0.8561 0.8663 0.9350
99 106 120 139 146 160 174 194 205 218 234 248 265 465
sp . grav _ (15 "C/15 "C) components N2 CO2 C1 c2
c3 i-C, n-C, i-C5
n-C5 c6
mol % 0.12 2.49 76.43 7.46 3.12 0.59 1.21 0.50 0.59 0.79
total fractn
P+N A MW Sample C (Heavy Gas Condensate)
PNA distribution mol 70 P N A
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
953
Table VI1 (Continued) sp grav (15 OC/15 "C) components C,
mol %
total fractn
0.95 1.08 0.78 0.592 0.467 0.345 0.375 0.304 0.237 0.208 0.220 0.169 0.140 0.833
0.726 0.747 0.769 0.781 0.778 0.785 0.802 0.815 0.817 0.824 0.825 0.831 0.841 0.873
P+N
0.738 0.756 0.765 0.777 0.785 0.792 0.877 0.805 0.807 0.812 0.820 0.827
PNA distribution mol % N A
A
MW
P
0.862 0.874 0.883 0.898 0.953 0.934 0.987 0.951 0.949 0.968 0.975 0.962
95 106 116 133 152 164 179 193 209 218 239 250 264 377
0.564 0.113 0.483 0.530 0.681 0.757 0.709 0.635 0.729 0.624 0.668 0.675 0.652 0.519
0.361 0.611 0.311 0.275 0.193 0.123 0.183 0.209 0.168 0.232 0.185 0.192 0.190 0.320
0.076 0.277 0.206 0.195 0.126 0.120 0.108 0.156 0.103 0.144 0.147 0.133 0.158 0.161
"P, paraffinic compounds; N, naphthenic compounds; A, aromatic compounds. bThe PNA analyses of samples A and B are only based on measurements of refractive index. Table VIII. T B P Data for S a m d e B" temp, cum wt temp, temp, cum wt cum wt "C % distilled "C % distilled "C % distilled 100 127 152 176 197
u
; 5 1.4
. >
El
->5 1.3
-
-
..........
0
1.1
l'*
I
, 300
200
100
I
0 Pressure (atm)
Figure 3. Differential liberation results for sample B at T = 93.3". The SRK equation is used for the vapor densities and the AlaniKennedy equation (1960) for the liquid densities.
-.0
.........
Exparimen1 40 CH-traction1 6 CH-lractioni
I
4 00
300
218 237 255 272
28.43 32.48 36.13 40.06
288 304 318 332
43.10 46.52 49.67 52.67
"specific gravity of TBP sample = 0.8499 (15 "C/15 "C).
Experiment 4 0 CH-traction. 6 CH-fractions
1 1 400
10.54 14.54 19.17 21.55 24.92
I
I
I
200
100
0 Pressure ( a t m )
Figure 4. Differential liberation results for sample B at T = 93.3 "C. The oil density has been determined by using the Alani-Kennedy equation (1960).
relative and absolute liquid volumes of sample B during a differential liberation process. Calculation with All Binary Interaction Coefficients Equal to Zero. As already indicated, we normally use kij values equal to zero for interactions between hydrocarbons and nonzero values for interactions between
non-hydrocarbons and between hydrocarbons and nonhydrocarbons. The influence on the predicted saturation points of setting all kij values equal to zero has been checked for mixtures containing COz and N2 If COP+ N2 constitutes less than 1.5 mol % of the total mixture, the calculated saturation points deviate less than 170from those calculated with kij # 0. For sample A, a gas condensate that contains about 10 mol % non-hydrocarbons, the effect of kij = 0 on the calculated dew points is 4.0%. Relative computer times for flash calculations with the kij = 0 version are given in Table IV. Calculated phase compositions of samples A and B a t separator conditions using the kj.= 0 version are shown in Tables V and VI. The simdations of PVT experiments mentioned in the previous section have been repeated with the kij = 0 version. The liquid dropout curve obtained for sample A by using six hydrocarbon fractions is shown in Figure 1. The results of the constant mass expansion calculation on sample C and those of the differential liberation calculation on sample B are not shown because they are almost indistinguishable from the results with k, # 0.
Discussion It is evident from Table IV that much computer time may be saved in oil/gas flash calculations if the kij values (see eq 15) are equated to zero for all i-j interactions. The calculations performed on gas condensates and heavy oils show that the kij = 0 approximation has little influence on the results for North Sea reservoir fluids, which seldom contain more than 10 mol % CO, + N,. If the content of COz and N, is less than 1.5 mol % , the influences on the predicted phase behavior and saturation points are negligible; only the equilibrium ratio for CO,, KCo2,is affected (see Table VI). If the content of COz + Nz approaches 10 mol 70,the ki, = 0 and k , # 0 versions of the programs give saturation pressures that differ by 4%. This is the case for sample A of Table VII. Only minor differences are in this case observed in the two-phase region, except for the calculated CO, concentrations.
954
Ind. Eng. Chem. Process Des. Dev., Vol. 24, No. 4, 1985
Table IX. Computer Time (cpu 8 ) Needed for One One-Phase and One Two-Phase Flash Calculation on Different ComputersD(Eight Component Mixture) program one-phase two-phase comwter version region region 0.021 0.029 IBM 3081 k,j f 0 k, =0 0.012 0.010 Burrough B-7800 k,, # 0 0.080 0.125 k,, = 0 0.030 0.060 VAX ll/750 k,, f 0 0.326 0.203 0.279 UNIVAC ll00/62 k, # 0 0.374 0.530 0.340 HP9020 kLJ # 0.150 k, = 0 0.250 The two-phase calculation corresponds to the data and results of Table V.
The above results indicate that one may take advantage of the k,, = 0 approximation in many applications where flash calculations are needed for petroleum mixtures. This conclusion does not apply to situations where mixtures with appreciable contents of non-hydrocarbons are present, e.g., COz injection studies. Also, reducing the number of components (or pseudocomponents) greatly reduces the computer time needed for flash calculations (see Table IV). The results obtained show that a total of only six groups of carbon number fractions gives saturation points and PVT properties that are sufficiently accurate for many practical purposes. As shown in Tables V and VI, the only discrepancy is the content of heavy components in the vapor phase. The liquid dropout curves predicted for samples A and C deviate somewhat from the measured ones as seen in Figures 1 and 2. Very accurate predictions are obtained for the dew point of sample A, but there are some problems with the liquid volumes. As the liquid phase only constitutes a few volume percent of the total volume, our results are satisfactory for most practical purposes. Sample C is a heavy gas condensate for which we calculate the following critical properties: T,= 43.8 "C; P, = 416 atm. The reservoir temperature is 129 "C, which means that the mixture a t the dew point in Figure 2 is nearly critical. The near-critical condition is also reflected in the calculated phase compositions, which slightly below the dew point were nearly identical. Problems with predictions of densities are to be expected in the critical region, and we attribute most of the discrepancies shown in Figure 2 hereto. It is worth noting that no computational problems were encountered in these flash calculations.
Conclusions A simplified SRK calculation procedure (Michelsen, 1983) enables considerable savings in computer time for mixtures for which all k,, values can be set to zero. The computer time is only a weak function of the number of components, and the storage requirements are drastically reduced. Irrespective of the number of components, only three variables need to be stored between the iterations. For mixtures of hydrocarbons, COz,and N2 in which the COz + N2 content does not exceed 10 mol % , all k , values can be set to zero without significant influence on the predicted phase behavior. In addition, a new procedure has been developed that allows reservoir fluids to be represented by a total of six hydrocarbon pseudocomponents only. It has been shown
that six pseudocomponents are sufficient to represent densities and saturation points as well as what was previously reported in parts 1 and 2 of this series using 40 hydrocarbon fractions. Acknowledgment We thank Arne Hole of Statoil and M. L. Michelsen of Instituttet for Kemiteknik for their continued interest in this work. Nomenclature a SRK parameter defined in eq 1 2 parameter defined in eq 5 A1 API API specific gravity (=141.5/SG-131.5) b SRK parameter defined in eq 12 parameter defined in eq 5 R, carbon number CN binary interaction coefficient k, K Watson characterization factor (TB'/3/sG) equilibrium factor, component i K, m SRK parameter defined in eq 18 MW molecular weight P pressure R gas constant SG specific gravity T temperature normal boiling point TB midvolume boiling point T50 midweight boiling point T50 wt% liquid-phase mole fraction of component i XI vapor-phase mole fraction of component i Y1 feed mole fraction of component i 21 Greek Symbols CY parameter defined in eq 20 vapor mole fraction P W acentric factor fugacity coefficient of component i 41 Subscripts Br reduced boiling point critical property C F feed component or carbon number group indices i, j liquid L V vapor Registry No. C02, 124-38-9; N2, 7127-31-9.
Literature Cited Alani, H. G.; Kennedy, H. T. Pet. Trans., Am. Inst. Min. Eng. 1960, 219, 200. Cavett, R. H. "Physical Data for Distillation Calculations - Vapor-Liquid Equiiibrla", presented at the 27th Midyear Meeting, API Division of Refining, San Francisco, May 15, 1964. Erbar, J. H. Gas Processors Association, Tulsa, OK, 1977, Research Report 13. Katz, D. L.; Firoozabadi, A. Pet. Techno/. 1978, 2 0 , 1649. Kesler, M. G.; Lee, B. I. Hydrocarbon Process. 1978, 5 5 , 153. Michelsen, M. L.; Heidemann, R. A. AIChEJ. 1981, 2 7 , 521. Michelsen, M. L. "Simplified Flash Calculations for Cubic Equations of State", SEP 8313; Instituttet for Kemiteknik: Lyngby. Denmark, 1983. Pedersen, K. S.; Thomassen, P.; Fredenslund, Aa. a. Ind. Eng. Chem. Process D e s . Dev. 1984, 2 3 , 163-170. b. Ind. Eng. Chem. Process Des. Dev., 1984, 23, 566-573. PEneloux, A,; FrBze. P. Nuid Phase Equilib. 1982, 8 . 7. Soave, G. Chem. Eng. Sci. 1972, 2 7 , 1197.
Received for reuiew October 3, 1983 Revised manuscript received October 2, 1984 Accepted October 23, 1984
