340

Literature Cited

I n d . Eng. Chem. Res . 1989, 28, 340-341

Torii, K.; Iitaka, Y. The Crystal Structure of L-Isoleucine. Acta

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.

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 Caro- lina 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

Andreas K. Griewank Argonne National Laboratory, Argonne, Illinois 60439

Mobil Exploration and Producing Services, Dallas, Texas 75247

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 to 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 de- veloped 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 adjust- able 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 den- sities 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 prop-

erties are required by pipe-flow network, reservoir, and processing simulators. Fluid-phase behavior, densities, viscosities, and interfacial tensions are necessary for sim- ulation of processes which depend on the mass transfer of

* Author to whom correspondence should be addressed.

0888-5885/89/2628-0340$01.50/0 0

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 pro- cesses 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 Peng- Robinson equation, for example, predicts inaccurate liq- uid-phase densities, with errors of 10-25%, as illustrated in this repart, although a recent three-parameter modifi- cation to the Peng-Robinson equation has greatly im- proved 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 re- taining cubic equations for their fast, reliable phase equilibria predictions.

The extended corresponding states principle (ECSP) for predicting pure fluid-phase densities was introduced ori- ginally 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-

1989 American Chemical Society

Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 341

(3)

h = (Vc/V,")&T/TC, V / V c , w)(ZT,"/Zc) (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 re- duced 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 ex- tension 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 ref- erence 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 n- paraffins. However, development of new requisite refer- ence 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 ap- pear in the ECSP mixing rules can also be utilized for improving predictions, but adjustments are, in our expe- rience, 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,

f = (Tc/TT,")B(T/TC, V / V c , w) 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 res- ervoir 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 ad- justment does not improve predictions for C02 systems 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 ad- justable parameter, and we have written a computer pro- gram that automatically adjusts parameters for C7+ fractions to match experimental data.

We illustrate the accuracy of the method with five ex- amples. 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 simu- lated 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 equa- tion, reported by Ely and Hanley (1981). The corre- sponding 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.,

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 com- positional analyses and C,+ fraction characterization to find optimal critical density parameters ( p c ) for the C7+ fractions with the computer regression routine. Once op- timal parameters were determined, ECSP density pre- dictions 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 in- dividual 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 TB is the boiling point in degrees Rankine, SG60 is the specific gravity a t 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) corre- lation 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 ex- pressed as

V , = 0.036MW'.084SG60-0~580 = 0.036[MW/SG600.535] '.0a4

which differs from Hall and Yarborough's by, a t 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-com- ponent 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

Table I. Composition of the Synthetic Oil and Properties of Its Heavy Fractions

concn, concn, component mol % comDonent 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" 322 10.20 322 11.48" 322 12 .11a 322 12.11 322 11.93" 322 11.51 322 11.22 322 10.91 322 10.56 322 10.12 339 14.13 339 10.75" 339 12.37" 339 13.49" 339 13.64" 339 13.12

a Data used in fit.

12 'I 8 1

0.00 -5.3 6.9 0.00 7.0

19.87 5.6 39.73 4.4 49.69 -4.6 4.0 59.70 -4.9 4.3 69.77 5.7 74.82 6.6 79.85 -5.5 8.2 84.87 10.3 89.93 -6.1 15.1 0.00 -5.3 7.9 0.00 8.2

20.22 6.1 40.35 4.6 60.44 5.7 80.17 10.0

* P-R = Peng-Robinson EOS.

S P-R EQUATION 0 ECSP DEFAULT pc

ECSP FITTED p c

0.3 0.3 0.8 0.8 0.7 0.3 2.5 3.9 6.1 8.9

14.6 0.7 0.9 0.2 0.5 1.9 8.2

8

. .*

% ERROR 4 4 * . a vs

EXPERIMENTAL

0

1

- 8 4 1 0 20 4 0 6 0 8 0 100

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 simultane- ously. Table I1 and Figure 1 present comparisons of ex- perimental 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-

Table 111. Composition of the Light Oil and Properties of Its Heavy Fractions

comDonent concn, mol % component concn, mol %

4.22 0.25

23.48 14.49 9.38 1.01 3.96 1.49

n-C5 n-C6 heavy fr. 1 heavy fr. 2 heavy fr. 3 heavy fr. 4 heavy fr. 5 heavy fr. 6

2.46 3.35 5.74 9.90 8.10 6.20 4.47 1.50

pc, kmol/m3 heavv fr. P', MPa T c , K w MW initial fitted

1 2 3 4 5 6

DENSITY

(kglm3)

2.354 622.37 0.425 126.6 1.95 1.87 2.214 669.39 0.460 154.9 1.61 1.30 1.898 726.23 0.572 200.9 1.25 0.989 1.466 787.71 0.794 275.1 0.906 0.743 0.992 856.40 1.194 392.4 0.613 0.552 0.644 924.59 1.860 607.5 0.384 0.379

0 EXPERIMENTAL P-R EQUATION ---

,801 - ECSP

660 4

540 580L 500 3.4 5 10 15 20 25 30

PRESSURE (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 con- centrations.

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 modi- fied.

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 Peng- Robinson equation. The optimal pc parameters €or the ECSP give excellent results for density prediction with

Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989 343

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" 704.3 -10.81 0.89 10.9 19.13" 697.1 -10.90 0.95 10.8 13.96" 689.2 -11.10 1.00 10.7 11.17 693.9 -11.13 1.04 8.72 697.3 -10.17 1.95 6.98 714.1 -11.24 0.49 5.65 724.7 -11.73 -0.28

(I 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" 709.1 -10.56 -0.11 9.0 21.24" 704.2 -10.65 -0.06 8.7 18.02" 697.6 -10.69 0.14 8.7 12.39 708.2 -11.41 -0.25 8.95 721.9 -11.57 -0.50 7.12 729.7 -11.72 -0.75 5.93 731.3 -11.35 -0.45

(I Used in fit.

Table VI. Density Predictions for the Light Oil (35 mol % COZ, 377.59 K)

% error

pressure, MPa

24.17" 18.95O 14.53" 12.48 10.40 8.72 7.62

a p t 1 P , k / m 3 710.9 706.0 712.8 723.3 730.9 736.1 742.5

P-R -10.02 -11.13 -10.72 -11.38 -11.66 -11.76 -12.18

ECSP with fitted default

parameters parameters -0.53 6.4 -1.53 7.6 -0.24 7.2 -0.84 -1.11 -1.22 -1.72

" 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" 716.3 -8.55 -0.36 6.2 25.91" 712.6 -9.00 -0.27 6.1 17.79 706.9 -8.49 -1.60 14.35 738.7 -11.82 -1.81 10.52 746.0 -12.0 -1.86 7.48 750.5 -11.95 -1.80

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 ad- versely affected by increasing carbon dioxide concentra- tions, 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 COZ 0.45 heavy fr. 4 0.89 C1 61.73 heavy fr. 5 0.68 c2 12.53 heavy fr. 6 0.58 c3 6.84 heavy fr. 7 0.48 XI 1.56 heavy fr. 8 0.41 n-C, 2.15 heavy fr. 9 0.34 i-C5 0.77 heavy fr. 10 0.54 n-C, 0.67 heavy fr. 11 0.56 n-C 0.91 heavy fr. 12 0.48 heavy fr. 1 1.98 heavy fr. 13 0.33 heavy fr. 2 1.55 heavy fr. 14 0.33

heavy fr. Pc, MPa TC, K ub MW initial fitted pc, kmol/m3

1 2.935 535.33 0.343 100.20 2.44 1.69" 2 2.661 561.61 0.402 114.20 2.15 1.64" 3 2.448 585.28 0.453 128.30 1.92 1.69" 4 2.282 608.56 0.469 142.30 1.74 1.61" 5 2.138 628.33 0.507 156.30 1.59 1.59 6 2.002 646.17 0.544 170.30 1.46 1.46 I 1.883 662.56 0.577 184.40 1.34 1.34 8 1.790 677.67 0.608 198.40 1.26 1.26 9 1.717 692.11 0.648 212.40 1.18 1.18

10 1.569 711.94 0.685 233.01 1.07 1.07 11 1.412 740.50 0.765 267.38 0.928 0.876" 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

-

a 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 1.20 374 30.27 -4.4 2.50 0.50 374 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 306b 3.12" -7.7 7.90 0.10 28gb 0.10 -9.7 -2.00

" Used in fitting. Condensate phase a t surface separator con- ditions.

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 pseudocompo- nents, 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 experi- mental 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

Table X. Composition of the Volatile Oil and Properties of the Heavv Fractions

component concn, mol % component concn, mol %

CO, 0.75 n-CS 1.42 Cl 58.22 n-C6 1.86

c3 8.65 heavy fr. 2 1.24 i-C4 0.81 heavy fr. 3 1.36 n-C, 0.39 heavy fr. 4 8.14 i-C, 0.79

c2 14.36 heavy fr. 1 2.01

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

% ERROR

EXPERIMENTAL vs - 4 1

1 - 1 2 1 -16 4 4 20 25 3 0

0 5 10 15

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 com- position (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 pre- dicted 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 com- position and heavy fraction properties of a lean gas con- densate. 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

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 To and 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 corre- sponding reference fluid temperature is

To = T / f (A-2) 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.,

f i = (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)

The shape factors Oi and & are expressed as

Bi = 1 + (oi - w,)x(Ti/T,", Vj/V,") (A-5)

& = 1 + ( ~ i - wO)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,

al + bl In Ti+ + ( c , + d,/Ti+)(Vi+ - 0.5) (A-7) x(Ti/T,", Vi/V,") =

and [(Ti/T,C, V/V,") = a2(Vi' + b2) + c2(Vi' + d,) In Ti+

64-8) where

Ti+ = min (2, max (Ti/T,C, 0.5))

and Vi' = min (2, max (Vi/V,", 0.5))

The mixing rules for h and f are as follows: f CCxixjfijhij/h (A-9)

r J

h = CCxixjhij (A-10) i J

where

f i j = (fifj)1'2(1 - hij) (A-11)

h IJ = 1/8(hi1i3 + h,1/3)3(l I - Zij) (A-12)

where x i 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 sys- tems, 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

1.2 -

Table XI. Comparison of Density Predictions for the Volatile Oil

% error ECSP ECSP

Dressure, w/fitted wtdefault

0 EXPERIMENTAL ECSP -

--- PENG-ROBINSON EQUATION

temp, O F MPa parameters P-R parameters 362" 26.30 0.99 -11.1 2.43 362" 23.55 362" 20.79 297" 0.103 362 20.79 362 17.34 362 13.86 362 10.45 362 7.00 362 3.58 362 0.103 297 0.24 297 1.14

"Used in the fit.

0.75 2.20 -0.44 -13.5 1.14 -0.43 -16.0 4.12 -1.10 -0.10 0.80 0.80 -12.6 1.60 2.50 1.40 -14.0 0.40 -15.8 1.00

Table XII. Composition of the Lean Gas Condensate component composition, mol %

COP nitrogen methane ethane propane isobutane n-butane isopentane n-pentane hexanes heptanes pb

14.16 0.30

71.77 6.00 2.69 0.65 0.87 0.48 0.36 0.69 2.03

.".. t I

27.6 30 35 40 45 50

PRESSURE (MPa)

Figure 4. Comparison of compressibility factor for a lean gas con- densate 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

346 Ind. Eng. Chem. Res., Vol. 28, No. 3, 1989

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 To and p , are re- lated 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

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 To and pW By substituting the resulting values into the last two equa- tions, we then obtained another system of two nonlinear equations which can be solved with respect to To and 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 con- vergence failures which have been observed in a few cases for the successive substitution method, particularly when the mixture contains C,+ fractions.

FL['T'(>. p,,. T#r* Vir) The actual form of the inner system is

(T,c/T,)T,r - 8,(9(T,'), q(V,'))

lit\ T,,, Po, Tlr , V,') = (A-16) ( n o / P:) (z,"/z,") Vir - d,(9(T,'), q( V,'))

where F, is obtained by multiplying the reciprocal of (A-2) by T,'T,," and then subtracting 8,. H, follows analogously from (A-11,

0, and 4, are the shape factors and q ( r ) is a smoothed diferentiable approximation to the piecewise linear func- t inn tnin j2, max (7, 0.5)), namely

1 - 2 ( i - l ) T i f r < i r 2 - T i 2

The fixed value of i between 1 and 2 determines the interval (l/i,?) in which q ( r ) = 7. Outside this interval, 9 satisfies 9 ( 1 / 7 ) = 1 / 9 ( 7 ) and we have

l / q ( O ) = 2 = lim q(7 )

The choice i is not very critical. When i = 2, the function 9 is identical with the original piecewise linear function.

For optimizing the critical densities, the reduced values T,' and V,' were chosen as variables because they depend only on To, ToC, poc, ZOc, Pic, and T,C, not on the potential regression parameter p ; (critical density), as is the case for the scale factors

f, = T,'T,C/To

r- m

Due to the boundedness of Q ( T ) , eq A-16 must always have a t least one (positive) solution pair, Tt, Vi . Unfor- tunately, it has been observed that there may be several solution pairs. However, Newton's method was found to be almost always successful, and if it failed, switching to a successive substitution would usually suffice. In the very

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 To and p o are now com- puted according to the implicit function theory as the 2 X 2 matrix: a(T,', V,')/d(To, P,) =

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(To , poll (A-19)

Here the functions F = V - V and H = 3/X - V are derived from the following three expressions. First we have

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

-[a(Fi, Hi)/d(T,', v,')l-l[a(J'i, f f i) /d(To, poll (A-18)

i=l

V = (T/P)P(To,p0) / ( Topo) (A-20)

i J

where Lij = y8[(Vir/pic)1/3 + ( V { / P ~ " ) " ~ ] ~ = hij/[p<,(l - l i j ) ]

(A-22) Third, we obtain from (A-2) and (A-11) the volume mul- tiplied by T as

Q = EZ~i~jfijLij(1 - kij)(l - (A-23) 1 1

where f . . U = [TtT[T.cT,']1/2 J J = f i jTo/( l - hZj) (A-24)

So far, Newton's method has always produced a positive solution pair (To, p,) with F(T,, p,) = 0 = H(T,, p,) even though we know of no theoretical proof of its existence. The derivatives of To and p , with respect to a critical density (p i" ) can then be computed as a v o , P 0 ) / d p t = -wT,, p,)-i[m, (A-25)

The vector d[F , HI I d p i c is obtained by differentiating (A-23) with respect to pic for T t and Vir fixed and then applying the chain rule to (A-21) and (A-22).

Finally, we obtain p = 1/ V as well as its derivatives with respect to the pic from (A-20). If required, the scale factors of the mixture can be computed as

h = P o / P f = hP/Po (A-26) In order to determine the "best fit" critical density pa-

rameters for the experimental data, we use a nonlinear least-squares calculation. The predicted density and its derivatives with respect to critical density parameters are supplied to a nonlinear least-squares routine from MINPACK (Garbow, 1980). We assign a weight to each critical density to specify if and how much each should be allowed to vary.

Registry No. C1, 74-82-8; C p , 74-84-0; C B , 74-98-6; n-C4, 106- 97-8; i-C4, 75-28-5; n-C,, 109-66-0; i -C5, 78-78-4; C,, 110-54-3; Cs!

Literature Cited

142-82-5; c8, 111-65-9; coz, 124-38-9; N ~ , 7727-37-9.

Armijo, L. Minimization of Functions Having Lipschitz Continuous First Partial Derivatives. Pac. J . Math. 1966, 16, 1.

Baltatu, M. E. Prediction of Liquid Viscosity for Petroleum Frac- tions. Ind. Eng. Chem. Process Des. Deu. 1982, 21, 192-195.

Cullick, A. S.; Mathis, M. L. Densities and Viscosities of Mixtures of Carbon Dioxide and n-Decane. J . Chem. Eng. Data 1984,29, 393-396.

Ind. Eng. Chem. Res. 1989, 28, 341-355 347

Ely, J. F. Prediction of the Viscosity and Thermal Conductivity in Hydrocarbon Mixtures-Computer Program TRAPP. Presented at the 60th Annual Gas Processors Association Convention, San Antonio, TX, March 21-23, 1981.

Ely, J. F.; Hanley, H. J. M. Prediction of Transport Properties. 1. Viscosity of Fluids and Mixtures. Znd. Eng. Chem. Fundam. 1981,

Garbow, B. S., et al. MINPACK Project. Argonne National Labo- ratory, March 1980.

Goodwin, R. D. The Thermophysical Properties of Methane, from 90 to 500 K at Pressures to 700 Bar. National Bureau of Stand- ards Technical Note 653, 1974.

Hall, K. R.; Yarborough, L. New, Simple Correlation for Predicting Critical Volume. Chem. Eng. 1971, Nov 1, 76-77.

Jhaveri, B. D.; Youngren, G. K. Three-Parameter Modification of the Peng-Robinson Equation of State to Improve Volumetric Pre- dictions. SOC. Pet. Eng. Res. Eng. 1988, 3, 1033-1040.

Leach, J. W.; Chappelear, P. S.; Leland, T. W. AZChE J . 1968, 14,

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20, 323-332.

568-576.

1968, 64, 1447-1460.

Martin, J. J. Cubic Equations of State-Which?, Znd. Eng. Chem. Fundam. 1979, 18, 81-97.

Ortega, J. M.; Rheinboldt, W. C. Zterative Solution of Nonlinear Equations in Several Variables: Academic Press: New York, 1970.

Peng, D.-Y.; Robinson, D. B. A New Two-Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59-64.

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Robinson, D. B.; Peng, D.-Y. The Characterization of the Heptanes and Heavier Fractions for the GPA Peng-Robinson Programs. Gas Processors Association Research Report RR-28, March 1978.

Rowlinson, J. S.; Watson, I. D. Chem. Eng. Sci. 1969,24,1565-1573. Turek, E. A.; Metcalfe, R. S. Phase Equilibria in Carbon Dioxide-

Multicomponent Hydrocarbon Systems: Experimental Data and an Improved Prediction Technique. Presented at the 55th Annual Fall Technical Conference of the SPE, Sept 21-24, 1980; SPE 9231.

Received for review July 13, 1988 Accepted October 17, 1988

A Radially Perfused Cell for Measuring Diffusion in Compacted, Highly Sorbing Media

Frank M. Jahnket and Clayton J. Radke* Chemical Engineering Department, University of California, Berkeley, California 94720

A unique radially perfused cell designed specifically to measure the transient diffusion of species in compacted, highly adsorbing and swelling porous media is described in this paper. The perfused cell exhibits extremely high mass-transfer coefficients while resisting large swelling pressures. It also allows a simultaneous determination of the effective diffusion and sorption retardation coefficients of species from a single experiment. The concentration boundary layer resulting from the radial flow configuration is sufficiently uniform that one-dimensional, axial diffusion is studied in the sample medium. Over the range of Peclet numbers studied, from lo3 to 3 X lo5, overall Nusselt numbers range from 70 to 550, in excellent agreement with a theoretical convective dispersion prediction containing no adjustable parameters. Use of the cell is illustrated by an experiment in which cesium chloride diffuses into a moderately compacted sodium montmorillonite clay gel. We find that surface diffusion controls the migration of cesium in these clays.

1. Introduction The transient diffusion of solute species through com-

pacted, highly adsorbing porous media is of interest in such diverse areas of technology as enhanced oil recovery, ion- exchange chromatography, plant nutrition, and toxic waste disposal. We are particularly interested in the suitability of the clay mineral montmorillonite as a component of the scheme proposed to dispose of high-level nuclear wastes in underground repositories (Waste Isolation Systems Panel, 1983). Many diffusion cells have been proposed for quantifying species diffusion rates (for reviews, see Crank (1975) or Crank and Park (1968); the best allow for a simultaneous determination of the diffusion and sorption coefficients of species in the sample medium from a single experiment. None, however, have considered the unique requirements imposed by particulate and compacted, sorbing media. Specifically, a diffusion cell designed for these materials must satisfy three general but conflicting requirements.

First, the cell must be rigid enough to resist the swelling pressures the medium exerts. Common is the swelling of

* Author to whom correspondence should be addressed. +Present address: Manufacturing Research and Engineering

Organization, Research Laboratories, Eastman Kodak Company, Rochester, NY 14650.

0888- 5885 1 89 12628-0341 $0 1,501 0

materials exhibiting high species exchange capacities, such as ion-exchange resins in wetting solvents (Helfferich, 1962). Compacted clays demand particular attention to this requirement: swelling pressures of up to 10 MPa have been measured for highly compacted montmorillonite clays (Barclay and Ottewill, 1970). Next, the cell must retain the individual particles comprising the medium. Again clays prove difficult; the clay particles we study have di- ameters of but 50 nm. Finally, external mass-transfer resistances must not obfuscate a proper interpretation of the diffusion experiment. As demonstrated below, ex- periments designed to measure diffusion rates through highly sorbing media are very sensitive to the external mass-transfer resistances a diffusion cell imposes. The new radially perfused cell we describe in this paper is designed to meet these seemingly irreconcilable requirements.

The perfused cell introduces solute species a t one end of a sorbing, linear medium from a well-stirred tank of finite volume. This method has multiple advantages. The progress of the experiment is followed easily by monitoring the concentration history of the tank. If in addition the concentration profile in the medium is determined at the conclusion of an experiment, we can readily ascertain the internal consistency of an experiment by overall solute mass conservation.

Additionally, this concentration-history-with-profiling technique allows both sorption and diffusion coefficients

0 1989 American Chemical Society