SPE-16941-PA

A Systematic and Consistent Approach To Determine Binary Interaction Coefficients for the Peng-Robinson Equation of State Carsten SlotPetersen, * SPE, Dansk Olie- & Gasproduktion A/S

Summary. A new technique to determine binary interaction coefficients for a cubic equation of state (EOS) is introduced. The tech-nique is physically consistent and easy to apply when an EOS is tuned to laboratory-measured PVT data. The method's simplicity provides a rational way of evaluating the sensitivity of binary interaction coefficients to fluid composition and temperature. Laboratory-measured expansion data are demonstrated to be crucial for proper determination of binary interaction coefficients for near-critical fluids.

The ability of the technique has been demonstrated by matching laboratory-measured constant-composition-expansion curves for six fluid compositions, including black-oil, volatile-oil, rich-gas-condensate, and lean-gas-condensate systems. In addition, the near-critical fluid behavior has been evaluated by matching constant-composition-expansion data for a fluid at four temperatures in the near-critical range, two on either side of the critical temperature. The literature reports that the Peng-Robinson EOS (PR-EOS) cannot predict the retrograde phase behavior of the rich-gas-condensate fluids applied in this study. This paper shows that this is because insufficient and/or improper binary interaction coefficients were used in the fluid characterization.

The method enables tuning of near-critical fluid behavior in addition to normal bubblepoint and dewpoint system behavior. Use of the proposed procedure to assign binary interaction coefficients may allow matching of renascent condensation for a near-critical bubble-point system.

Introduction Use of a cubic EOS to predict phase behavior became standard after the invention of the digital computer. The EOS provides continuity and smoothness of PVT -dependent fluid properties. The first cubic EOS, expressing the continuity from gaseous to liquid states, was derived by van der Waals in 1873.' van der Waals' work formed the foundation for similar EOS's such as the Redlich-Kwong,2 the Soave-Redlich-Kwong,3 and the PR-EOS.4

None of the EOS's accurately predict the phase behavior of a complex hydrocarbon mixture at reservoir conditions unless fine tuning of the EOS parameters has taken place. 5,6 During the tuning process, the critical properties (Pc and Tc) of the high-boiling-point fractions are normally adjusted within reasonable ranges to obtain a good density match. In addition, binary interaction coefficients may be assigned to reflect the attractive forces between light and heavy molecules. Many correlations estimate the critical properties of the high-boiling-point fractions in a hydrocarbon mixture. Whitson6 studied the effect of using various critical property corre-lations in EOS predictions.

A universally adequate method of determining binary interaction coefficients has yet to be proposed. Katz and Firoozabadi 7 as-signed interaction coefficients between methane and heavier hydrocarbon components based on minimizing the error in the cal-culated bubblepoint pressure of the binary mixture of methane with the heavier component in question. In this fashion, a linear rela-tionship was found between the methane binary interaction coeffi-cients and the density of the heavier components at standard conditions. As this study shows, however, the binary interaction coefficients at reservoir pressures are strongly dependent on fluid composition. Additionally, it seems that temperature has some effect as well. Hence, binary interaction coefficients found on the basis of binary mixtures alone are not adequate when dealing with complex multicomponent mixtures at elevated temperatures and pressures.

The latest innovation in EOS tuning is that of nonlinear regression, which minimizes the error between EOS-calculated and laboratory-observed data. 5 This procedure, however, often does not preserve the physical significance of the regression parameters. The final calculated numerical values of the regression parameters, even though they generally give a reasonable match to laboratory data, appear to be independent of molecular weight. In addition, the com-puting time required for nonlinear regression may be significant.

Tuning an EOS for a near-critical fluid has generally been con-sidered difficult or even, in some cases, impossible. Firoozabadi

'Now at Dansk Operat~rselskab liS Copyright 1989 Society of Petroleum Engineers

488

et al. 8 tuned the PR-EOS to match exactly the dewpoint for Kilgren's9 rich-gas-condensate laboratory data. In spite of this, the PR-EOS predicted a bubblepoint system when the fluid was flashed. Whitson6 reported a similar phenomenon when he used both the PR-EOS and Yarborough's version of the modified Zudkevich-Joffe-Redlich-Kwong lO EOS on the rich-gas-condensate data reported by Jacoby. 11 In this case, the laboratory data indicated dewpoint systems, whereas the "tuned" PR-EOS predicted bubblepoint systems. This study shows that the apparent inability of the PR-EOS to predict the retrograde behavior of the Jacoby gas condensate properJy6 is caused by the use of insufficient and/or improper binary interaction coefficients. It is believed that the similar phe-nomenon observed by Firoozabadi et al. 8 can be explained in the same way.

The purposes of this paper are (1) to propose a simple, systematic, and physically consistent way to determine binary interaction coeffi-cients for the PR-EOS; (2) to demonstrate that the PR-EOS is capable of predicting the retrograde behavior of near-critical gas-condensate fluids; and (3) to examine the sensitivity of the binary interaction coefficients to mixture composition and/or temperature.

Jacoby et al." presented data on recombined samples of sepa-rator oil and gas from a retrograde gas-condensate reservoir. The recombination was performed for GOR's between 2,000 and 25,000 scf/STB [360 and 4460 std m3/stock-tank m3] and represented eight related synthetic reservoir fluid mixtures (Fluids S-1 through S-8). Constant-composition-expansion data were available for Fluids S-l through S-6; hence, only these six fluids were examined in this study. Compositional data were available up to a normal boiling point of 475F [246C] corresponding to a C 13 + fraction. In ad-dition, the liquid density at standard conditions, 14.7 psia and 60F [10 1 kPa and 15.5 C], and the molecular weight of each boiling-point fraction were given. Table 1 contains the available data for the six fluids evaluated and adjusted acentric factors, pseudocritical pressures, and pseudocritical temperatures for the high-boiling-point fractions F6 through F 13+'

Method Description The PR-EOS has the following form 4:

P=Pr+Pa, ....................................... (1)

where the repulsion pressure term is given by

Pr=RT/(V-b) .................................... (2)

SPE Reservoir Engineering, November 1989

TABLE 1-COMPOSITIONS OF SYNTHETIC FLUID MIXTURES STUDIED (Jacoby et a/. 11 )

Fluids

S-l S-2 S-3 S-4 S-5 S-6 Liquid Density at (mole (mole (mole (mole (mole (mole 60F and 14.7 psia Molecular Weight Pc Tc

fraction) fraction) fraction) fraction) fraction) fraction) (g/cm 3) (Ibm/Ibm mOl) (psia) ~ w --N2 0.0128 0.0153 0.0158 0,0166 0.0177 0.0191 C , 0.5636 0.6526 0.6715 0.6995 0.7401 0.7906 CO 2 0.0139 0.0156 0.0159 0.0165 0.0172 0.0182 C 2 0.0709 0.0701 0.0699 0.0697 0.0693 0.0689 C 3 0.0509 0.0427 0.0410 0.0384 0.0347 0.0301 iC 4 0.0164 0.0123 0.0114 0.0101 0.0082 0.0058 nC 4 0.0376 0.0285 0.0266 0.0238 0.0197 0.0145 iC s 0.0208 0.0149 0.0137 0.0118 0.0092 0.0058 nC s 0.0249 0.0177 0.0162 0.0139 0.0107 0.0066 150F NBP 0.0365 0.0255 0.0232 0.0197 0.0147 0.0085 0.6793 89 437.49 934.85 0.3180 200F NBP 0.0282 0.0195 0.0177 0.0150 0.0110 0.0061 0.7175 98 433.93 983.62 0.3460 250F NBP 0.0174 0.0121 0.0109 0.0092 0.0068 0.0037 0.7396 110 409.60 1,023.71 0.3453 300F NBP 0,0120 0.0083 0.0075 0.0063 0.0046 0.0025 0.7535 121 386.93 1,053.70 0.3851 350F NBP 0.0201 0.0139 0.0125 0,0106 0.0077 0.0042 0.7621 131 367.61' 1,077.32 0.3974 400F NBP 0.0201 0.0138 0.0125 0,0105 0.0077 0.0042 0.7742 144 346.31 1,107.90 0.4111 450F NBP 0.0323 0.0223 0.0202 0.0170 0.0124 0.0067 0.7852 165 315.69 1,148.27 0.4259 475F NBP 0.0216 0.0149 0.0135 0.0114 0.0083 0.0045 0.8133 216 264.51 1,239.04 0.5126 --- --- --- --- --- ---

Total 1.0000 1.0000 1.0000 1.0000 1.0000

P sat at 163F 3,194 3,809 3,941 4,183 4,520 Stock-tank

gravity, API 63.3 n/a 63.0 n/a 62.5 Nominal separator

GOR, scf/STB 2,000 3,500 4,000 5,000 7,500

NBP = normal bOiling point.

and the attraction pressure term is given by

Pa = -al[V(V+b)+b(V-b)] . ........................ (3)

The parameter bin Eq. 2 is related to the molecular volume. The parameter a in Eq. 3 is related to the energy of intermolecular in-teraction.

Binary interaction coefficients are used to model the inter-molecular interaction through empirical adjustment of the a factor in the attraction pressure term. Peng and Robinson4 found that the interaction coefficients were dependent on the difference in molecular size of the components in a binary system. The method proposed in this paper is to assign hydrocarbon/hydrocarbon binary interaction coefficients based on the following hypotheses.

1. The interaction between two hydrocarbon components in-creases as the relative difference between their molecular weights increases-i.e. ,

kij

k(1+1)-F13+ Ri l ki-F13+

Ri ~ 1.

Fig. 1-Blnary interaction coefficient relationship.

5000

Lab PR-EOS Fluid Binary function Ratio I data prediction '1 .,

'3 I

I 5-1 k ij = l!Il (Mjl 9+dl O. O. o.

1:,. 5-' k ij ... l!I2 (Mj l 5+d 2 .. .. .. 5-3 k ij = l!I3 (Mj) 5+d3 .. . . 0 5-' k ij ~ l!I4 (Mj) 4+d4

. , ., o . 5-5 k ij - l!Is (Mj) 3+dS O. O. o. I 5-' k ij = l!I6 (Mj) 1+d6 O. O. o.

00 10 20 30 40 50 60 70 80 90 100 VOLUME PERCENT LIQUID

Fig. 2-Constant-composltlon expansion of Synthetic Fluid Mixtures S-1 through S-6 at 163F.

number' of components in a systematic and consistent way while varying only a few parameters, such as (1) the value of the power n, (2) the number of nonzero binary interaction coefficients, and (3) the numerical value of the ratios Rcl , RC2 ....

The Appendix gives a practical example of how to generate the binary interaction coefficients for Fluid S-3.

Discussion of Results Fig. 2 shows the PR-EOS constant-composition-expansion predic-

490

tions for Synthetic Fluid Mixtures Sol through S-6. The binary in-teraction coefficients have been assigned by use of the binary functions and the Ri values between methane, ethane, propane, and butane binaries as indicated in Fig. 2. It is evident that the binary interaction coefficients are strongly dependent on the fluid compo-sition. In this respect, it is worthwhile to note the following.

1. For a black-oil system (Fluid S-l), the most essential binary interaction coefficients are between methane and the highest-boiling-point fractions. All other binaries may be set to zero.


5000

4000t===~===-,,=-:;_..ill._

3000

p (pSIAl

2000

1000 Lab PR-EOS Temp. Binary f unct ion Rat 10 data prediction of R, R, RJ

tJ. 96.5 k .. ,., ml(Mj) J+d1 1. 1. 'J * 131.8 k ij .. m2 (Mj) 3+d2 1. 1. 1. 163.5 k .. "'" In3 (Mj) 5+d ) .8 .8 .8 'J 207. J Kij ,., m4 (Mj) 4+d4 .7 .7 .7

10 20 30 40 50 60 70 80 90 100 VOLUME PERCENT L1aUID

Fig. 3-Constant-composltlon expansion of Synthetic Fluid Mixture S-3 at several temper-atures.

TABLE 2-BINARY INTERACTION COEFFICIENTS USED TO MATCH LABORATORY DATA FOR FLUID S-3 AT 163F

Component;

Component i --"L ~ CO, ~ ~ ~ ~ ~ N, 0.0000 0.1000 0.0120 0.1000 0.1000 0.0000 0.0000 0.1000 C, 0.0000 0.1000 0.0000 0.0000 0.0000 0.0000 0.0000 CO, 0.0000 0.1000 0.1000 0.0000 0.0000 0.1000 C, 0.0000 0.0000 0.0000 0.0000 0.0000 CJ 0.0000 0.0000 0.0000 0.0000 iC4 0.0000 0.0000 0.0000 nC. 0.0000 0.0000 iCs 0.0000 nC, F, F, F. F. FlO F" F12 F'3+

2. For near-critical fluids (Fluids S-2 through S-4), it is crucial to include binary interaction coefficients between more of the light components and all of the intermediate and heavy fractions.

3. For lean-gas condensates (Fluids S-5 and S-6), only methane binaries are of importance.

Fig. 3 illustrates that the binary interaction coefficients to some extent are temperature dependent. A proper match of the laboratory data in this near-critical temperature range requires assignment of nonzero binary interaction coefficients between the light compo-nents (methane, ethane, propane, isobutane, and normal butane) and the intermediate and heavier fractions. The temperature de-pendence, however, does not seem to be quite as systematic as the compositional dependence.

The binary interaction coefficients that give the optimum match for the rich-gas condensate (Fluid S-3) at the near-critical temper-ature, 163F [73C], are listed in Table 2. Figs. 4 through 6 show the effect (on the rich-gas-conqensate match) of varying either the


~ ~ ~ ~ ~ ~ ~ ~ ~ 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000 0.0000 0.0027 0.0044 0.0079 0.0127 0.0190 0.0304 0.0601 0.2310 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000 0.1000 0.0000 0.0022 0.0035 0.0063 0.0102 0.0152 0.0243 0.0481 0.1848 0.0000 0.0017 0.0028 0.0050 0.0081 0.0121 0.0194 0.0384 0.1478 0.0000 0.0012 0.0021 0.0039 0.0064 0.0096 0.0154 0.0306 0.1183 0.0000 0.0012 0.0021 0.0039 0.0064 0.0096 0.0154 0.0306 0.1183 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

0.0000 0.0000 0.0000

power n, the number of binary interaction coefficients with a nu-merical value of zero, or the ratios Rcl , Rc2 , and Rc3 . For two sets of binary interaction coefficients (Curve 3 on Fig. 5 and Curve 1 on Fig. 6), the PR-EOS predicts renascent condensation. ("At a constant temperature below the critical ... the quantity of the liquid phase decreases, then increases in an abnormal manner, and there-after decreases normally." 13) Note that in this case all physical conditions (e.g., temperature) and parameters (e.g., fluid compo-sition and component critical properties) are kept constant. The only variable is the relationship between the binary interaction coefficients-i.e., the binary function. This means that the EOS-predicted phase behavior of a fluid can be altered significantly simply by varying the relationship between the binary interaction coeffi-cients.

Whitson6 included phase-behavior predictions for the Jacoby Synthetic Fluid Mixtures S-1 through S-7. Even though numerous attempts were made to match the experimental results, Whitson pre-

491

492

5000

Fig. 4-Effect of the numerical value of n on the PR-EOS flash prediction.

5000

4oo0t-________________ ~~ ____ ~

3000

p (PSIA)

2000

1000

00 10

Lab data:

PR-E08 prediction

j 20 30

40

(Fluid 8-3 at 1630 F)

Binary function

k ij = ml(Mj)5+d1 k ij m2(Mj)5+d 2 k ij m3(Mj)5+d 3 k ij m4 (Mj) 5+d4

50 60 70

VOLUME PERCENT LIQUID

Ratio Rl R2 R3 O. O. o.

.8 O. o.

.8 .8 o.

.8 .8 .8

80 90 100

Fig. 5-Effect of nonzero binary Interaction coefficients on the PR-EOS flash prediction.


5000

~t-----------~=:~-=~~~--------------------------~

3000

p

(PSIA)

2000

Lab data: (Fluid 5-3 at 163F) PR-E05 Binary functiori Ratio

1000 prediction Rl R2 R3

I k ij = ml(Mj)5+d1 .6 .6 .6 k ij = m2(Mj)5+d2 .7 .7 .7 k ij = m3(Mj)5+d 3 .8 .8 .8 k ij = m4(Mj)5+d4 .9 .9 .9

10 20 30 40 50 60 70 80 90 100 VOLUME PERCENT LIQUID

Fig. 6-Effect of the numerical value of Rc, , Rco, and RCa on the PR-EOS flash prediction.

dicted a critical fluid composition (GOR) of 5,000 to 7,500 scf/STB [890 to 1340 std m3/stock-tank m3] and critical temperature of 260F [127C], which are considerably different from those measured experimentally (critical GOR between 3,500 and 4,000 scf/STB [630 and 720 std m3 /stock-tank m3] and critical temper-ature between 132 and 163F [56 and 73C)). In his study, Whitson apparently applied only one nonzero binary interaction coefficient (using a value for that between methane and the C6+ fraction of 0.0536).

It is evident from Figs. 4 through 6 that binary interaction coeffi-cients for the rich-gas condensate, Fluid S-3, at 163F [73C] can be determined properly only if laboratory-measured expansion data are available. Figs. 4 through 6 also illustrate that the saturation pressure for a fluid may be matched exactly, independent of the binary function used. Therefore, tuning the EOS to match measured properties at the saturation pressure is not sufficient (at least in the case of near-critical fluids).

Conclusions 1. The binary interaction coefficients required in the PR-EOS

for an acceptable match to laboratory-measured data consistently are strongly sensitive to mixture composition.

2. The interaction coefficients also appear to be influenced by temperature. The temperature dependence, however, seems to be less pronounced and more random than the compositional de-pendence.

3. Use of improper and/or insufficient binary interaction coeffi-cients in an EOS may result in the prediction of a bubblepoint system even though laboratory measurements show a rich-gas-condensate system.

4. Laboratory-measured expansion data (Le., constant-composition expansion, differential liberation, or constant-volume depletion) are crucial to determine the proper binary interaction coefficients for near-critical fluid conditions. Without expansion data, there is no control on the EOS-predicted phase behavior during pressure depletion.


5. Tuning an EOS to match only the saturation pressure does not give an unambiguous determination of the binary interaction coeffi-cients. A large number of combinations of binary interaction coeffi-cients can be found to give a match of the saturation pressure.

6. A power function relating the binary interaction coefficients to the molecular weight of the components was found to be an ex-cellent and systematic way to determine binary interaction coeffi-cients for the PR-EOS.

Nomenclature a = EOS parameter, psia(ft3/lbm mol)2

[kPa(m3/kmol)2] b = EOS parameter, ft3/lbm mol [m3/kmol]

di = intercept constant in power function for Component i, dimensionless

F6 = heavy fraction (corresponding to C6) F7 = heavy fraction (corresponding to C7) mi = slope constant in power function for Component i,

(Ibm mol/lbm)n [(kmol/kg)n] kij = binary interaction coefficient between Components

i and j, dimensionless MFm+ = molecular weight of the heaviest fraction Fm+,

Ibm/Ibm mol [kg/kmol] Mi = molecular weight of Component i, Ibm/Ibm mol

[kg/kmol] Mj = molecular weight of Component j, Ibmllbm mol

[kg/kmol] n = exponent in power function, dimensionless P = pressure, psia [kPa]

Pa = attraction pressure, psia [kPa] Pc = critical (fi"essure, psia [kPa] Pr = repulsion pressure, psia [kPa] R = universal gas constant, 10.73 (psia-ft3)/(lbm mol-

OR) [8.3145 (kPam 3)/(kmolK)]

493

RCI = ratio between kCZFm+ and kCIFm +' dimensionless Ri = ratio between k(i+ I)Fm+ and kiFm+' dimensionless T = absolute temperature, oR [K]

Tc = critical temperature, oR [K] V = molar volume, ft 3 /lbm mol [m 3 /kmol]

VL = liquid volume, fraction w = Pitzer's acentric factor

Acknowledgments I thank Dansk Olie- & Gasproduktion A/S for their financial support, Scientific Software Intercomp for providing their PVT program, and the Petroleum Engineering Dept. at the Colorado School of Mines for the use of computing resources.

References I. van der Waals, J.D.: "On the Continuity of the Liquid and Gaseous

State," PhD dissertation, Sigthoff, Leiden, Holland (1873). 2. Redlich, O. and Kwong, J.N.S.: "On the Thennodynamics of Solutions.

V. An Equation of State. Fugacities of Gaseous Solutions," Chem. Rev. (1949) 44, 233-44.

3. Soave, G.: "Equilibrium Constants From a Modified Redlich-Kwong Equation of State," Chem. Eng. Sci. (1972) 27, No.6, 1197-1203.

4. Peng, D. and Robinson, D.B.: "A New Two-Constant Equation of State," Ind. & Eng. Chem. Fundamentals (1976) 15, No.1, 59-64.

5. Coats, K.H. and Smart, G.T.: "Application ofa Regression-Based EOS PVT Program to Laboratory Data," SPERE (May 1986) 277-99; Trans., AIME,281.

6. Whitson, C.H.: "Effect of C 7+ Properties on Equation-of-State Pre-dictions," SPEl (Dec. 1984) 685-96.

7. Katz, D.L. and Firoozabadi, A.: "Predicting Phase Behavior of Con-densate/Crude-Oil Systems Using Methane Interaction Coefficients," JPT (Nov. 1978) 1649-55; Trans., AIME, 265.

8. Firoozabadi, A., Hekim, Y., and Katz, D.L.: "Reservoir Depletion Calculations for Gas Condensates Using Extended Analyses in the Peng-Robinson Equation of State," Cdn. J. Chem. Eng. (Oct. 1978) 56, 610-15.

9. Kilgren, K.H.: "Phase Behavior of a High-Pressure Condensate Reservoir Fluid," JPT(Aug. 1966) 1001-05; Trans., AIME, 237.

10. Yarborough, L.: "Application of a Generalized Equation of State to Petroleum Reservoir Fluids," Proc. , ACS Symposium on Equation of State in Engineering and Research, K.C. Chao and R.L. Robinson Jr. (eds.), American Chemical Soc., Washington, DC (1979) 385-435, Ser. 182.

II. Jacoby, R.H., Koeller, R.C., and Berry, V.J.: "Effect of Composition and Temperature on Phase Behavior and Depletion Perfonnance of Rich Gas-Condensate Systems," Trans., AIME (1959) 216, 406-11.

12. Slot-Petersen, C.: "A Comparison of the Recoveries Obtainable by Depletion, Gas Recycling and Waterflooding of a Retrograde Gas Con-densate Reservoir," MS thesis, Colorado School of Mines, Golden (1986).

13. Katz, D.L-etal.: "Handbook of Natural Gas Engineering," McGraw-Hill Book Co. Inc., New York City (1959).

Appendix-Determination of Binary Interaction Coefficients for Fluid 5-3 at 163F [73C] Binary interaction coefficients for Fluid S-3 at 163 OF [73C] were determined according to the 10 steps outlined in the text. Physical parameters are given for all heavy fraction components in Table 1.

Step 1. Numerical power value: n=5. Step 2. Nonzero hydrocarbon binary interaction coefficients:

kij*O, where i=C 1, Cz, C3, iC4 , nC4 andj=F6' F7 .F13+. Default PVT program values chosen for N z and CO2 binary in-teraction coefficients (see Table 2).

Step 3. Ratios Ri : RCI =0.8, Rcz =0.8, and RC3 =0.8. Step 4. Estimate methanel F 13 + fraction binary interaction

coefficient: kCI - FJ3 + =0.2310. Hence,

kcz -FJ3+ = (0.8)(0.2310) =0.1848,

kC3 -FI3+ =(0.8)(0.1848)=0.1478,

k iC4 -FJ3+ =knC4'-FJ3+ =kC4 -FI3+'

and kC4 -Fl3+ =(0.8)(0.1478) =0.1183. Step 5. Calculate mi and d i :

mCI =0.2310/[(216)5 -(16.043)5]

=4.9130xlO- 13 (Ibm molllbm)5

[4.913xlO- 13 (kmollkg)5],

494

TABLE A-1-COMPARISON OF LABORATORY AND SIMULATED VOLUMETRIC BEHAVIOR

P Laboratory PR-EOS Deviation' (psia) VL __ V_L_ (%) 3,940 0.0 0.0 0.0 3,900 0.3828 0.3242 15.3 3,860 0.4037 0.3772 6.6 3,814 0.4146 0.4041 2.5 3,774 0.4214 0.4168 1.1 3,684 0.4201 0.4296 -2.3 3,590 0.4171 0.4319 -3.5 3,507 0.4157 0.4292 -3.2 3,410 0.4095 0.4228 -3.2 3,233 0.3936 0.4056 -3.0 3,026 0.3728 0.3805 -2.1 2,629 0.3263 0.3256 0.2 2,235 0.2748 0.2674 2.7 1,820 0.2159 0.2059 4.6 1,416 0.1636 0.1484 9.3

918 0.0976 0.0841 13.8

[(VL. lab - VlPR.EOS)lVllabI100.

dCI = -(16.043)5 x4.9130x 10-13 = -5.2212 X 10-7 ,

mC2 =0.1848/[(216)5 -(30.070)5]

=3.9306x 10- 13 (Ibm molllbm)5

[3.93 x 10 -13 (kmol/kg)5],

dC2 = -(30.070)5 x3.9306x 10-13 = -9.6632 X 10-6 ,

mC3 =0.1478/[(216)5 -(44.097)5]

=3.1446xlO- 13 (Ibm molllbm)5

[3.145 x 10 -13 (kmol/kg)5],

dC3 = -(44.097)5 x3.1446x 10 -13 = -5.2433 x 10 -5,

mC4 =0.1183/[(216)5 -(58.124)5]

=2.5196 x 10 -13 (Ibm molllbm)5

[2.52 x 10 -3 (kmollkg)5],

and dC4 = -(58.124)5 x2.5196 x 10 -13 = -1.6715 X 10-4 .

Step 6. Calculate remaining binary interaction coefficients:

kCI -FlO =4.9130 x 10 -13(131)5 -5.2212 X 10-7 =0.0190,

kC2 -FlO =3.9306x 10- 13 (131)5 -9.6632 x 10-6 =0.0152,

kC3 -FlO =3.1446 X 10- 13 (131)5 -5.2433 X 10-5 =0.0121,

kiC4-FIO =2.5196x 10 -13(131)5 -1.6715 x 10 -4 =0.0096,

and knC4-FIO =2.5196x 10 -13(131)5 -1.6715 x 10 -4 =0.0096.

The remaining binary interaction coefficients are calculated in a similar manner. The symmetric matrix is given in Table 2.

Step 7. Predict the saturation pressure with the PR-EOS:

PR-EOS: Pdew =3,940 psia [27.166 MPa]

Laboratory: Pdew=3,941 psia [27.173 MPa].

Step 8. Compare calculated and predicted saturation pressure: excellent match. Continue.

Step 9. Simulate constant-composition expansion (see Table A-I). Step 10. Compare PR-EOS expansion simulation and the

laboratory-measured data: reasonable agreement. The binary in-teraction coefficient matrix has been determined properly.

51 Metric Conversion Factors OF (OF-32)/1.8 C psi x 6.894 757 E+OO kPa

SPERE

Original SPE manuscript received lorreview Sept. 27. 1987. Paper accepted lor publication April 12. 1989. Revised manuscript received Feb. 22. 1989. Paper (SPE 16941) lirst presented at the 1987 SPE Annual Technical Conlerence and Exhibition held in Dallas. Sept. 27-30.


Discussion of A Systematic and Consistent Approach To Determine Binary Interaction Coefficients for the Peng-Robinson Equation of State Ingolf S.relde, SPE, and Curtis H. Whitson, SPE, Norwegian Inst. of Technology

In "A Systematic and Consistent Approach To Detennine Binary Interaction Coefficients for the Peng-Robinson Equation of State" (Nov. 1989 SPERE, Pages 488-94), Slot-Petersen proposes a tech-nique for determining binary interaction coefficients (BIC's) for the Peng-Robinson equation of state (PR-EOS).1 He applies this technique to represent phase and volumetric behavior of the Jaco-by synthetic mixtures2 denoted Fluids S-l through S-6. The match to the measured data for these mixtures obtained by Slot-Petersen is quite good, as shown in his Figs. 2 and 3. The fluid characteri-zation included a single set of C6+ pseudocomponent properties 'Now at Norsk Hydro AlS.

used for all mixtures, but different sets of BIC's for each specific mixture. For Fluid S-3, different sets of BIC's were determined at each specific temperature. From these results, he concludes that the composition dependence on BIC's is more important than the temperature dependence.

The Jacoby synthetic mixtures were recombined in various ra-tios of companion separator oil and gas samples produced from a gas-condensate reservoir. The result is a family of seven related mixtures composed of the same components. Therefore, the seven mixtures should be characterized with a universal set of C6+ pseu-docomponent properties and BIe's.

TABLE D-l-Cs+ COMPOSITION OF THE JACOBY SYNTHETIC MIXTURES

Cs+ Composition (mol%) Pseudocomponent 8-1 8-2 8-3 8-4 8-5 8-6

Cas " 3.65100 2.55312 2.32232 1.97252 1.47181 0.84761 C 7 + (1) 2.22647 1.54438 1.50906 1.18379 0.86007 0.45150 C 7 + (2) 6.51009 4.49074 3.99399 3.42740 2.51872 1.38996 C 7 + (3) 4.92918 3.39510 3.00203 2.58750 1.89959 1.04507 C 7 + (4) 1.39537 0.96192 0.89698 0.73322 0.53305 0.28167 C 7 + (5) 0.11919 0.08244 0.08694 0.06295 0.04483 0.02174 Total 18.8313 13.0277 11.8113 9.9674 7.3281 4.0375

'Cas is composed of 50 mol% hexane, 25 mol% 2methylpentane, and 25 mol,o 3-methylpentane.

TABLE D-2-PR-EOS C 7 + CHARACTERIZATION FOR THE JACOBY SYNTHETIC MIXTURES

M Tc Pc Tb VT* BIC"" Pseudocomponent (Ibm/mol) ~ (psia) _w __ "1_ ~ s=c/b C 1 through C 7 +

C 7 + (1) 95.4 938.6 497.7 0.2537 0.7135 617.1 - 0.1789 0;026653 C 7 +(2) 119.1 1,006.1 433.1 0.34020.7424 685.0 -0.1781 0.033949 C 7+(3) 164.0 1,125.0 348.2 0.46020.7863 801.1 -0.1314 0.046276 C 7 + (4) 235.7 1,272.7 273.1 0.6187 0.8393 949.9 - 0.081 0 0.060644 C 7+(5) 350.0 1,438.6 187.3 1.44030.9010 1,190.8 0.0660 0.081902

'c=EOS-calculated molar volume minus the correct molar volume and b=repulsive EOS constant. "C 1 through Cas =0.02192.

TABLE D-3-NONHYDROCARBON/HYDROCARBON BIC'S AND VT PARAMETERS FOR THE JACOBY

SYNTHETIC MIXTURES

TABLE D-4-COMPARISON OF MEASURED AND CALCULATED SATURATION PRESSURES FOR THE

JACOBY SYNTHETIC MIXTURES

260

Component

N2 CO 2 C 1 C2 C3 iC 4 nC 4 iC 5 nC 5 Cas C7 +

~ 0.0 0.0 0.025 0.010 0.090 0.095 0.100 0.110 0.125 0.125 0.125

BIC

CO 2 0.0 0.0 0.105 0.130 0.125 0.115 0.115 0.115 0.111 0.115 0.115

VT" s=c/b

-0.1927 -0.0817 -0.1595 -0.1134 -0.0863 -0.0844 -0.0675 -0.0608 -0.0390 -0.0026

'Pure-component VT parameters are obtained from a method proposed by S(I)reide. 3

"See Table 0-2 for C7+ VT parameters.

Fluid

8-1 8-2 8-3 8-3 8-3 8-3 8-4 8-5 8-6

Temperature (oF)

163.5 164.0 96.5

131.8 163.5 207.3 163.5 163.5 163.5

8aturation Pressure, P. (psia)

Experimental

3,194 3,809 3,750 3,840 3,941 4,000 4,183 4,520 4,964

Calculated

3,213.7 3,811.3 3,745.3 3,896.5 3,973.5 3,990.0 4,142.7 4,530.5 4,965.5

Deviation (%)

0.61 0.06

-0.12 1.47 0.82

-0.24 -0.96

0.23 0.03

SPE Reservoir Engineering, May 1990

4000

---. 0 'iii 3000 a. '-'

Q) '-:::l

~ 2000 Q) '-

Cl. PR EOS S?stem 1 -1 0 2 S-2

1000 0 3 S-3

Author's Reply to Discussion of A Systematic and Consistent Approach To Determine Binary Interaction Coefficients for the Peng-Robinson Equation of State Carsten Slot Petersen, SPE, Dansk Operat~rselskab liS

S~reide and Whitson contend that my characterization of the Jaco-by synthetic fluid mixtures 1 is inconsistent. They also state that S~reide2 obtained a consistent Peng-Robinson equation of state (PR-EOS)3 characterization for the six mixtures.

Consistency is defmed as keeping to the same principles. 4 In my paper, binary interaction coefficients (BIC's) were determined on the basis of specific hypotheses about intermolecular interaction. The sets ofBIe's given in my paper are all compatible and in agree-ment with these governing principles; hence, the procedure is ful-ly consistent.

The Jacoby synthetic mixtures are recombined from companion separator oil and gas samples. S~reide and Whitson argue that characterization of all the fluid mixtures should therefore use a universal set of heavy-fraction pseudocomponent properties and BIC's to have a consistent fluid characterization.

I agree with S~reide and Whitson that a universal set of heavy-fraction critical properties should be used under these circumstances. This was actually done (see Table 1 of my paper). In fact, the pos-sibility of using a universal critical property characterization for all mixture compositions was one of the main reasons the Jacoby mixtures were used. They form a representative fluid system for evaluating possible composition and/or temperature effects of the BIC's. Another main reason for using the Jacoby synthetic fluid mixtures was that the reported laboratory data included a full heavy-end fractional distillation analysis. A perturbation technique minimizing the necessary adjustment of the critical properties cal-culated from correlations for the high-boiling-point fractions was used to fine tune the EOS to predict confidently the high-boiling-point laboratory-measured specific gravities. 5

BIC's are used in cubic EOS's to model intermolecular interac-tion. Fig. R-I is a representation of the relative molecular distri-bution in three Jacoby synthetic fluid systems, Fluids S-1 (black oil), S-3 (rich gas condensate), and S-6 (lean gas condensate). In these systems, all individual-component physical properties are the same. The only difference is the overall composition-Le., the rela-tive distribution between light, intermediate, and heavy components.

0.3

Black 011 (S-1)

.

Lean Gas Condensate (S-6)

Critical Fluid (S-3)

Light Molecule

Intermediate Molecule

Heavy Molecule

Fig. R-1-Relatlve molecular distribution.

Because of interference from the surrounding molecules, the in-termolecular interaction between specific heavy and light molecules (as modeled by a BIC) is not necessarily the same in these three systems. Therefore, from a physical point of view, BIC's are not dictated to be independent of fluid composition.

FILid Mixture %C,

0 5-1 56.36 c 5-2 & 5-3 65.26 - 67.15 .4 S-4 69.95

5-5 74.01 A 5-6 79.06 Sereide & Whitson (table 0-21

__________________ -e

0.0 +-~;:a::::::;=::::::::~---___r----..,_---_,----. 80 100 150 200 250 300 350

Molecular Weight UbI mol)

Fig. R-2-Methane BIC's vs. molecular weight and composition at 163F.

262 SPE Reservoir Engineering, May 1990

0.3 Temperature of

207.3 183.5 .. 131.8

C a 98.5 "ll =i 0 0 0.2 !5 .., () .. S .5

~ ill .. 0.1 c: .. li ~

0.0 +-.a:::;!i==~---"-------'--------r-------r------', 80 100 150 200 250 300 350

Molecular Weight Ub/moU

Fig. R-3-Methane BIC's vs. molecular weight and temperature for Fluid S-3.

TABLE R-1-COMPARISON OF MEASURED AND CALCULATED SATURATION PRESSURES

Fluid

S-1 S-2 S-3 S-3 S-3 S-3 S-4 S-5 S-6

Temperature (oF)

163.5 163.5 96.5

131.8 163.5 207.3 163.5 163.5 163.5

Experimental!

3,194 3,809 3,735 3,847 3,941 4,000 4,183 4,520 4,964

Predicted bubblepoint (laboratory data indicate dewpoint).

Fig. R-2 shows the different BIC relationships between methane and the high-boiling-point fractions I used to match the constant-composition expansion (CCE) of the six mixtures (Fluids S-l through S-6) at 163F [72.8C]. The compositional dependence seems obvious as discussed in my paper. Note that the methane BIC's for the near-critical Fluids S-2 (volatile oil) and S-3 (rich gas condensate) are identical. The only difference between the BIC's for these two fluid mixtures is that the ethane, propane, and butane BIC's used to characterize Fluid S-2 are reduced by some 25% com-pared with those for Fluid S-3 (see Fig. 2 of my paper). Similarly, Fig. R-3 shows the methane BIC relationships used to match the CCE of Fluid S-3 at different temperatures. In this case, the BlC's seem more closely related because they fall within a fairly narrow range and follow a similar trend. Again, slight differences exist in the ethane, propane, and butane binaries (see Fig. 3 of my paper).

S~reide and Whitson show that S~reide2 obtained a PR-EOS characterization of the six mixtures with a universal set of heavy-fraction pseudocomponent properties and a single set of BIC's. For comparison, these BIC's are shown in Fig. R-2. The CCE match-es obtained by S~reide and Whitson are shown in their Figs. D-l and D-2. Table R-l compares my calculated saturation pressures with those obtained by S~reide.

My characterization of the Jacoby synthetic mixtures differs from S~reide's in several ways. First, when characterizing the high-boiling-point fractions (corresponding to C6 through C 13 +), I paid due attention to the measured fractional distillation analysis and had confidence in the given compositional split between the distillation


Saturation Pressure (psia)

Slot-Petersen 7

3,196 3,808 3,732 3,850 3,940 4,003 4,179 4,517 4,964

Calculated

SI/lI'eide 2

3,325 3,399 3,884 4,039 4,122' 4,150' 4,294 4,462 4,381

Sq,reide 2 Regressed

3,214 3,811 3,745 3,897 3,974 3,990 4,143 4,531 4,966 .

cuts and their respective measured molecular weights and specific gravities. Hence, I did not perform any artificial lumping and/or splitting of the eight heavy fractions. Second, I used a perturbation technique to minimize the required adjustment of the critical prop-erties for the PR-EOS to predict liquid densities at standard condi-tions properly before evaluating the influence of the BIC's on predicted phase and volumetric behavior. Third, my fluid charac-terization was performed manually.

S~reide2 characterized the fluids by (1) using a PR-EOS with volume translation, (2) lumping and splitting the eight C6+ laboratory-measured high-boiling-point fractions into six heavy pseudocomponents by use of a statistical molar distribution func-tion, and (3) using nonlinear regression.

The lumping and splitting technique applied by S~reide results in a molar distribution of the heavy ends different from that ob-tained from measured fractional distillation analysis (see Table 1 and Table D-l). S~reide first lumped the seven distillation cuts with a normal boiling point greater than 200F [93.3 0c] into an overall C7 + fraction. Next, a probabilistic gamma distribution function was used to model the molar distribution of the lumped C7+ frac-tion as a continuous function of molecular weight. The gamma dis-tribution parameters IX (characterizing the form of the distribution function) and 11 (minimum molecular weight of any component with-in the lumped C7+ fraction) were determined as 1.518 and 90, re-spectively. Finally, the continuous gamma distribution function was discretized into five pseudocomponents (see Table D-2) by a gen-eralized discretization method that used Gaussian quadrature.

263

Methane high-boiling-point BIC's were determined from a modi-fied Chueh-Prausnitz correlation.

After S~ide had gone through all these steps, the EOS predicted a too-high mixture critical temperature, as indicated by bubblepoint predictions instead of dewpoints for Fluid S-3 at 163.5 and 207.3 of [73.1 and 97.4C] (see Table R-l). S~reide then used nonlinear regression in an attempt to force fit to measured data the EOS-predicted saturation pressures and CCE liquid volume fractions for all six fluid mixtures simultaneously. Regression parameters con-sisted of heavy-pseudocomponent critical and boiling-point prop-erties and methane BIC's. In addition, S~reide manually adjusted the gamma distribution parameter ex for Fluids S-3, S-5, and S-6 (by three different factors-i.e., 0.95, 1.01, and 1.05) to "match experimental phase and volumetric data ... 2 I believe that by han-dling the molar distribution in this way, S~reide introduces incon-sistency in his fluid characterization.

S~reide and Whitson state that it is impossible to apply my fluid characterization in a compositional reservoir simulator to model a reservoir with a compositional gradient. It is true that simulation of reservoirs exhibiting a compositional gradient may be impracti-cal with the currently available compositional reservoir simulator models and composition-dependent BIC's. This deficiency might be overcome if BIC relationships could be related to composition through the mole fraction of a controlling component-e.g., methane concentration. Nevertheless, I disagree with S~eide and Whitson's conclusion that BIC's must be independent of fluid composition be-cause of the limitations of currently available computer models.

I obtained extraordinarily good matches by my characterization technique and documented in my paper that the apparent inability of the PR-EOS to predict the retrograde behavior of the Jacoby fluid mixtures properly reported by Whitson6 could be a result of inap-propriate BIC's in his characterization at that time.

264

S~reide and Whitson's present characterization, even after non-linear regression, still has problems with the CCE liquid dropout for Fluid S-3 at different temperatures, and also for Fluids S-5 and 8-6 at 163.soF.

S~reide and Whitson used compositionally independent BIC's to obtain a fair match after some mathematical manipulation, includ-ing nonlinear regression and alteration and extrapolation of the molar distribution of the high-boiling-point cuts. I obtained an excellent match through a relatively simple manual tuning process with com-positionally dependent BIC's and without altering composition, spe-cific gravities, and molecular weights found from the true-boiling-point distillation.

Reference. 1. Jacoby, R.H., Koeller, R.C., and Berry, V.J.: "Effect of Composition

and Temperature on Phase Behavior and Depletion Performance of Rich Gas-Condensate Systems," Trans., AIME (1959) 216, 406-11.

2. S~reide, I.: "Improved Phase Behavior Predictions of Petroleum Reser-voir Fluids From a Cubic Equation of State," PhD dissertation, Nor-wegian Inst. of Technology, Trondheim (1989).

3. Peng, D. and Robinson, D.B.: "A New Two-Constant Equation of State," Ind. & Eng. Chern. Fund. (1976) IS, No. I, 59-64.

4. Hornby, A.S.: "Oxford Advanced Leamer's Dictionary of Current En-glish," Oxford U. Press, New York City (1974).

5. Slot-Petersen, C.: "A Comparison of the Recoveries Obtainable by Deple-tion, Gas Recycling and Waterflooding of a Retrograde Gas Conden-sate Reservoir," MS thesis, Colorado School of Mines, Golden (1986).

6. Whitson, C.H.: "Effect of C7+ Properties on Equation-of-State Pre-dictions," SPE/ (Dec. 1984) 685-96.

7. Slot-Petersen, C.: "A Systematic and Consistent Approach To Deter-mine Binary Interaction Coefficients for the Peng-Robinson Equation of State," SPERE (Nov. 1989) 488-94.

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