a improving predictions of equation of state by modifying its parameters for super critical...

EUlIILlmA ELSEVIER Fluid Phase Equilibria 112 (1995) 45-61

Improving predictions of equation of state by modifying its parameters for super critical components of hydrocarbon reservoir

fluids

A. Danesh *, D.-H. Xu, D.H. Tehrani, A.C. Todd

Department of Petroleum Engineering, Heriot-Watt University, Riccarton, Edinburgh EH I 4 4AS, UK

Received 4 November 1994; accepted 28 April 1995

Abstract

The parameters of equations of state of van der Waals type have been generally correlated by matching the properties of pure substances, and extended to mixtures by employing mixing rules commonly with binary interaction parameters. It is proposed to use equilibrium data on binary systems to determine the attraction term in equations of state (EOS) for super critical components instead of data of pure substances. The proposed method was applied to the Peng-Robinson EOS, as an example, resulting in a significant improvement in predicting the phase behaviour of different types of fluids. As no binary interaction parameters are required in this method the computational requirement for flash calculations is drastically reduced for fluids with many components. The proposed method is particularly advantageous for predicting fluid phase equilibria in compositional reservoir simulators, where the reservoir fluid can be described with any desirable number of components without any significant increase in computational time.

Keywonts: Theory; Equation of state; Cubic; Mixing rules; Vapour-liquid equilibria; Super critical; Hydrocarbons

1. Introduction

Cubic equations of state (EOS) are commonly used in the petroleum industry, particularly in compositional reservoir simulators, to predict the phase behaviour and volumetric properties of hydrocarbon reservoir fluids. These equations are based on the classical van der Waals equation, but with more complex parameters to account for the attraction and repulsion forces between the molecules. The parameters have been generally determined and correlated by matching measured and predicted properties of pure substances.

The success of these semi-empirical EOS in predicting fluid properties depends to a large extent on the type of data used to correlate their parameters. Almost all the equations which are used to predict

* Corresponding author.

0378-3812/95/$09.50 1995 Elsevier Science B.V. All rights reserved SSD10378-3812(95)02782-3

46 A. Danesh et al. / Fluid Phase Equilibria 112 (1995) 45-61

the phase behaviour of hydrocarbon mixtures, have employed vapour pressure data of pure compounds for this purpose. For example, whereas saturation pressures of pure substances at the reduced temperature of 0.7 were used to develop the parameters of Soave-Redlich-Kwong EOS (Soave, 1972), vapour pressure data from the normal boiling point to the critical point were used by Peng and Robinson (1976). The experimental equilibrium data used in the development of these equations have, therefore, been limited to conditions below the critical point of pure compounds. Only few investigators (e.g. Kubic, 1982) have considered using data in supercritical regions.

EOS are applied to mixtures by introducing mixing rules to relate their parameters to properties and concentrations of their constituents. The performance of EOS for mixtures is improved by introducing binary interaction parameters in the mixing rules, expressing the interaction between pairs of non-similar molecules. As the interaction parameters are determined by matching the predicted values with experimental data, they should be considered primarily as fitting parameters for a particular EOS than rigorous physical parameters. Hence, the interaction parameters for a pair of substances developed for various EOS are generally different even with the same mixing rules.

In this work, we propose a different approach to determine the parameters of semi-empirical EOS for phase behaviour studies of multicomponent systems, particularly for petroleum reservoir fluids. The crux of the proposed method is to employ phase behaviour data of binary systems, instead of those of pure substances, to determine the parameters of EOS for super critical components. The advantages of the method are as follows: 1. Light components of reservoir fluids, particularly methane which constitutes a large fraction of

reservoir fluids, are generally at temperatures well above their critical points, where no vapour pressure data exist to be used for developing EOS parameters. The use of binary data extends the temperature range of relevant data, and it is only logical to expect more reliability when employing any correlation within its correlated domain.

2. As binary data are used in correlating the parameters of EOS, the interaction between pairs of non-similar molecules and/or the deficiencies of EOS for binary systems, are taken into account to some extent. The need for the use of binary interaction parameters is, therefore, reduced. This would allow for a significant simplification in phase behaviour calculations resulting in a major reduction in the computational requirement, particularly for mixtures described by a large number of components. The reduction of time spent in flash calculations is highly desirable in compositional reservoir simulators, where many hundred thousand flashes may be performed in a study. The proposed approach was applied in this work to the Peng-Robinson EOS, as the most widely

used equation in the industry. A large data bank of binary data was employed to revise the correlation expressing the temperature effect on the attraction term for super critical hydrocarbon components. The equation with the revised parameter has been used to predict the phase behaviour and volumetric properties of a wide range of fluids. The reliability of the proposed method and the saving in computational time for phase behaviour studies of reservoir fluids are demonstrated in this work.

2. Parameters of equation of states

Almost all van der Waals type EOS which have received significant attention for industrial applications can be expressed by the following general form:

RT a P= - - (1)

v -b v2+uv+w

A. Danesh et al. / Fluid Phase Equilibria 112 (199.5) 45-61 47

where P, u, T and R are the pressure, molar volume, temperature and universal gas constant respectively; and a and b represent the molecular attraction and repulsion terms in EOS. In 2-parameter forms of the equation, u and w are related to b, whereas in a 3-parameter form u and w are related to b and a third parameter e or some properties such as the acentric factor.

The parameters of an EOS can be presented by the following general equations:

R2Tc 2 = (2)

Pc R<

b = < (3)

RL c = n c - (4) <

In the original van der Waals equation and some earlier two-parameter modifications, such as the Redlich-Kwong EOS (1948), the values of g2 a and ~'~b are constants and can be simply determined by invoking the requirement of zero values of the first and second derivatives of pressure relative to volume at the critical point. These parameters are, however, mostly component-dependent and may vary with the temperature in more successful modifications. Correlations have, therefore, been developed to determine these parameters by matching predicted values for pure compounds to measured data.

The equations used in the modelling of petroleum reservoir fluids have been generally developed by equating the calculated fugacities of saturated liquid and vapour phases. Hence, only data below the critical conditions of pure compounds have been used in their development, whereas light compounds such as methane in reservoir fluids are almost always at super critical conditions. The majority of equations assume that the developed correlations below the critical temperature are also applicable at conditions above the critical points of the mixture constituents. Zudkevitch and Joffe (1970), however, assumed that the two parameters of the Redlich-Kwong equation remain constant above the critical point as the equation explicitly includes the square root of temperature in the attraction term. It is, therefore, logical to question the validity of these assumptions and to search for an alternative approach which would include data at more relevant conditions in developing the correlations.

To fix ideas we select the Peng-Robinson equation (PR EOS), as the most popular equation, for our discussion and application of the proposed method. Nevertheless the concept and the method are applicable to other semi-empirical EOS.

The PR EOS is obtained by making u = 2b, and w = -b 2 in Eq. (1), RT a

,P = - - b2 (5) u - b u 2 + 2by -

and the parameters are given by,

RaTc 2 el = 0 .45724c~ (6)

Pc RL

b = 0 .07780- - (7) Pc


where a expresses the temperature dependency of the attraction term. Peng and Robinson (1976) used vapour-pressure data from the normal boiling point to the critical point to determine o~ as,

2 (8)

where T~ is the reduced temperature, and m is a constant characteristic of each substance. The value of m was correlated with the acentric factor w, and later modified to improve predictions for heavier components (Robinson and Peng, 1978),

m = 0.3796 + 1.485w - 0.1644w 2 + 0.01667w 3 (9)

3. Binary interaction parameter

The application of EOS has been extended to multicomponent systems by defining mixing rules to relate the mixture parameters to those of its constituents. Although different mixing rules have been proposed, the classical van der Waals mixing rules, also called the random mixing rules, are the accepted forms for reservoir hydrocarbon mixtures:

N N 0.5 a= Y'~ yiYj(aiaj) (1 -k i j ) (10)

j= l i= l

N

b= ~., Yibi (11) j= l

where Yi and yj are the mole fractions of the component i and j, and kij is the binary interaction parameter (BIP) expressing the interaction between pairs of non-similar molecules. As the interaction parameter is determined by matching the predicted values for binary mixtures with experimental data, it could be considered as a fitting parameter and not strictly a rigorous physical parameter. Generalised correlations as well as tables of calculated/estimated values have been reported for different equations. In this study we use the parameters determined by Knapp and Doting (1982), and Robinson and Peng (1978) for PR EOS. No attempt was made to adjust the BIP in this work.

A comparative study of ten EOS (Danesh et al., 1991) concluded that the Patel and Teja equation as modified by Valderrama (Valderrama, 1990) without any BIP was more successful in modelling phase behaviour of reservoir hydrocarbon fluids than others with pertinent BIP. The success of the above equation with no BIP strengthens the view that these parameters, at least for hydrocarbon mixtures, mostly cover the deficiencies of EOS than accounting for the interaction between molecules of different compounds. Furthermore, the above study also showed that the performance of the relatively simple equation of state of Redlich and Kwong (1948) surpassed those of more advanced equations when its parameters were determined by matching saturation data of pure compounds, as suggested by Zudkevitch and Joffe (1970). An improvement to an EOS, such as employment of more

A. Danesh et al. / Fluid Phase Equilibria 112 (1995) 45-61

Table 1 Compositions and properties of synthetic multi-component mixtures

4t4

Fluid Mixture 1 Mixture 2 Mixture 3 source Danesh et al. (1991) this work Danesh et al. (1991) component mol.(%) mol.(%) tool.(%)

C l 46.80 82.05 C 2 8.77 C3 7.44 8.95 nC 4 4.01 nC 5 2.56 5.00 nC 6 1.77 Met cycl pent 2.25 Cyc hex 2.20 nC7 0.46 Met cycl hex 2.36 Toluene 0.72 nC 8 1.02 o-Xylene 1.79 nC,~ 1.66 nC 1~ 2.73 1.99 nC it 2.37 nC ,2 2.04 nC l ~ 1.77 nC 1,1 1.53 nC 15 1.34 nCt6 1.15 2.01 nC ,7 0.99 nC ,~ 0.87 nC,~ 0.75 nC 2o 0.65 Temperature C 100.0 80.0 Sat. Press (MPa) 20.28 31.98 Sat. Den (g cm -3) 0.541 0.326

69.82 13.09 ll.10 5.99

reliable values for its parameters, should therefore reduce the need to use BIP for hydrocarbon fluids which do not contain complex compounds.

As an example, the values o f /2a and Ob in PR EOS were determined by matching the properties of pure saturated compounds at the prevailing temperature, similarly to the method of Zudkevitch- Joffe, instead of the generalised expressions given by Eqs. (6) and (7). The equation was then used to predict the dew point of a gas condensate identified by Mixture 3 in Table 1. The predicted dew points by the above method without any BIP as well as those by the original PR EOS with and without BIP are compared with experimental data in Fig. 1. Note that the predictions by the Peng-Robinson EOS with revised values of a and b (Rev PR), but without any BIP, are very close to those by PR EOS with binary interaction parameters, PR(kij), whilst PR without any BIP is inferior to others. (The predictions identified by mPR in Fig. 1 will be discussed later). This example is not given to suggest that the predicted values of PR EOS as revised above and without BIP are always to


35-

32-

=~ 30-

~= 28- r~

25-

20

o Exp.

. . . . . . . . . . PR(kij) - - ' mPR

. . . . . . . Rev PR . . . . . . . PR

10 I I I 60 80 100

Temperature (C)

Fig. I. Dew point pressure of Mixture 3.

120

be similar to those of the original PR EOS with BIP, but merely to highlight the significance of the method used to determine EOS parameters and also the contribution of BIP as a fitting parameter.

4. Modif icat ion

The above example demonstrated the value and significance of the data used to determine the parameters of EOS. The limitation of correlating the parameters for super critical components of a mixture using experimental data can be eliminated by employing binary data at prevailing temperatures. It is acknowledged that the parameters so obtained for a substance include various contributions of the second component, and hence may be less reliable for that substance as a pure fluid. For practical purposes, this is not really a limitation as although cubic EOS are developed and evaluated for pure substances, they are hardly ever used to predict the behaviour and properties of pure systems.

The above approach is similar to the use of Henry's constant instead of pure compound's fugacity for predicting gas solubility in liquids. The fugacity of the component in the mixture can be simply evaluated as it is proportional to its concentration according to Henry's law, whereas it does not provide reliable information for the compound at the pure condition. Indeed, the aforementioned limitation has been well accepted as the price for simplicity and reliability achieved at working conditions. A similar argument is presented in support of using binary data to determine the parameters of EOS. In addition to expected improvement in reliability due to using more relevant data, the need for BIP should be eliminated resulting in a major reduction of computational time as will be shown in the next section.

Over 5,000 vapour-liquid equilibrium experimental data of binary systems containing a super critical component with hydrocarbons ranging from C 1 to nC12 (Knapp and Doting, 1982) were used in this study. The bubble point pressure of the liquid phase and the composition of the equilibrated vapour phase as predicted by PR EOS were matched to experimental data by adjusting the parameter a of the super critical component. The optimum value of a for super critical hydrocarbon components was found to be reasonably expressed by Eq. (8) replacing m with m',

m' = 1.21m (12)

A. Danesh et al. / Fluid Phase Equilibria 112 (1995) 45-61

Table 2 Compositions and properties of tested real reservoir fluids

51

Fluid source Fluid l Fluid 2 Fluid 3 component McCain (1990) Pedersen et al. (1989) Whitson and Torp (1983)

Mol.% MW SG Mol.% MW SG Mol.% MW SG

N 2 0.16 CO 2 0.91 C I 36.47 C, 9.67 C ~ 6.95 i-C 4 1.44 nC~ 3.93 i-C 5 1.44 nC 5 1.41 C~, 4.33 C 7 33.29 Cs

CIo Ctl

C ~2 C t3 C14 Cj5

(~ 17 Ci8 C19 C 20 C21 C22 Temperature (C) 104.44 Sat press (MPa) 18.17 Sat den (g cm -3) 0.656

0.31 2.37

52.00 73.19 3.81 7.80 2.37 3.55 0.76 0.71 0.96 1.45 0.69 0.64 0.51 0.68

85 0.666 2.06 1.09 85 0.666 218 0.852 2.63 99. 0.749 8.21 184 0.816

2.34 110. 0.758 2.35 121. 0.779

29.52 221. 0.852

93.4 137.8 46.64

MW, molecular weight; SG, specific gravity at 15.56 C. The last component of each fluid is the plus fraction.

More complex expressions could possibly be derived, but the above simple modification was found to be adequate.

5. Results

Methane and ethane are almost always at super critical conditions in reservoir fluids. The concentration of these components, particularly methane, cover a wide range depending on the type of the fluid. In order to evaluate the reliability of the proposed method, different fluids covering black oil to lean gas condensate samples were studied. The compositions and properties of the tested fluids are given in Tables 1 and 2.


100

80'

g 60'

~ 40'

20"

Michelsen

.......... Conventional

..,,..."''' o/"

o,o' oOO'/

000000 ."

o.."

7 `

. . . . . . . . . . . . . . . . '

5 10 15

Number of Component

i 20 25

Fig. 2. CPU time of calculating multiple forward contact of gas Mixture 4 with oil Mixture 1 at 100.0 C and 20.79 MPa.

5.1. Computational requirement

The reduction of computational time of flash calculations is an essential element in compositional reservoir simulation. This is commonly achieved by grouping fluid components (Danesh et al., 1992) and hence describing the fluid by a few pseudo components as the computational time decreases sharply by reducing the number of components which leads to fewer equations to be solved. The main draw back of the method is the lack of sufficient compositional information on produced reservoir fluids required in the design and operation of fluid processes at the surface.

As Michelsen (1986) has shown the number of equations to be solved in flash calculations can be reduced, to three, without additional complexity, regardless of the number of components when no BIP is used in the mixing rules. Since our proposed approach to determine the parameters of EOS eliminates the need for binary interaction parameter, it can significantly save the computational requirement of compositional reservoir simulation along with the Michelsen method (see Appendix).

Fig. 2 shows the variation of the computer CPU time vs. the number of components for predicting the equilibrium condition of a black oil (Mixture 1) when contacted with a rich gas (Mixture 3) at 100.0 C and 20.79 MPa in 4 consecutive stages simulating the injected gas advancement in a reservoir. The number of components describing the 25 component mixture was reduced by the grouping method proposed by Danesh et al. (1992). The figure demonstrates that the CPU time required by the conventional method is about two orders of magnitude higher than the Michelsen's method with no BIP when the mixture is described by 25 components Note that the computational time decreases sharply for the conventional method when the number of components describing the fluid decreases, whereas the reduction for the Michelsen's method is insignificant. The lack of BIP also reduces the computational time greatly at near critical point where many iterations are generally required. The above calculations were performed using a Newton type method to solve the equations. Similar observations were also made using a successive substitution numerical method to solve the equations.

A. Danesh et al. / Fluid Phase Equilibria 112 (1995) 45-61 53

e~

16"

14" . , ' / ~i eeo o / ' f/e/" q~ , 12 .,

/

I0 ., /"

8 S / o Vapour ,"4~///t L iquid q

"a~'" . . . . . PR(k i j= .0515) 4" /

/ PR O'

2 . .

0 , i t J

0 20 40 60 8o Mole Fraction of N2,%

Fig. 3. Phase envelope of N2-C2 at -78.89 C.

100

5.2. Reliability

The performance of PR EOS as modified in this work, without any BIP, was compared with that of the original form for a number of fluids. Examples of the results are presented in this report. The volume translation concept, proposed originally by Peneloux et al. (1982), and adopted by Jhaveri and Youngren (1988) for PR EOS was implemented in both methods to improve the predicted volumes.

Binary systems The proposed method was evaluated for a few binary mixtures whose data were not used to modify

EOS. The modification had clearly improved the prediction of PR EOS in general. Fig. 3 shows the phase envelope of the C2/N 2 system at -79.89 C (Stryjek et al., 1974) indicating that the modification is also applicable to mixtures containing nitrogen. As both PR EOS with BIP and mPR EOS without BIP have used binary data to develop their parameters, kij in PR, and m' in mPR, they are bound to generally give more reliable results than PR EOS without BIP.

Synthetic mixtures The advantage of evaluating phase behaviour models using experimental data on synthetic

multi-component mixtures, with similar component distributions as those of real reservoir fluids, is that the composition and properties of their components are more reliably known than real reservoir fluids.

Fig. 4 is the ternary diagram of the C1/nC4/nC10 mixture at 71.1 C and 20.68 MPa reported by Reamer et al. (1949). The modified method predicted the phase envelope as reliably as the original PR EOS with BIP and more reliably than the EOS without BIP.

The predicted dew-point pressures of the five-component synthetic gas condensate system (Mixture 3) by mPR are also shown in Fig. 1. It is quite evident that the modification significantly improved the prediction of dew point pressures compared with those by PR EOS with and without BIP.

Fig. 5 shows predicted saturation pressures in a swelling test where a rich gas (Mixture 3) was incrementally added to a synthetic black oil (Mixture 1) at 100.0 C. Both PR EOS with BIP and mPR


1.0" CI 1.0

0.6

0.4

0.2 / . . . . . . . . . PR(k i j )

/ ' . .... ' , ,

0.0 0.2 0.4 0.6 0.8 1.0 nC 10 nC4

Fig. 4. Ternary diagram of C 1-nC4-nC 10 at 71.1 C and 20.68 MPa.

predicted nearly the same saturation pressures for the system and much closer to the measured value than PR EOS without BIP. Methane was injected into a near critical volatile oil at 100.0 C to simulate gas injection into a reservoir by multiple contact tests (Danesh et al., 1991). The average absolute deviations of predicted saturation pressures of the above test by mPR, PR with BIP and without BIP were respectively 3%, 5%, and 11%, with deviations equal to 2%, 3% and 2% in saturation densities.

Reservoirfluids In this study the critical properties of pseudo-components of real reservoir fluids were estimated by

using the Twu (1984) correlation The acentric factors were calculated by the Lee and Kesler (1975) correlation. For the literature data where the fluid heavy end was reported only as a C 7 + fraction, the gamma distribution function was used to describe the heavy end, as proposed by Whitson (1983), and

28'

26' g,

- 24 '

~ 22"

"~ 20"

18

O ....~, ...... O

, - " . . . . . PR(k i j )

- - " PR

|6 . B . i - i . i - i

0.00 0.25 0.50 0.75 1.00 1.25 1.50

Injected CraaOil Mole Ratio

Fig. 5. Bubble point pressure of oil Mixture I and added gas Mixture 4 in swelling test at I00.0 C.


200'

o Exp.

E. ~ mPR / - / /

"~ 150' p~, / o - - " PR

1~ 100'

2 4 6 8 10 12 14 16 18 20

Pressure, MPa

Fig. 6. Solution gas/oi l ratio of black oil (Fluid 1) in differential liberation test at 104.44 C.

split it into a number of pseudo-components, up to C20. Due to lack of proper binary interaction parameters for pseudo-components, the modified equation (mPR) was compared only with PR EOS without BIP for real reservoir fluids. The compositions and properties of reservoir fluids tested in this study are given in Table 2.

Fig. 6 shows the measured and predicted solution gas/oil ratio of a black oil (Fluid 1 ) determined by the differential liberation test as reported by McCain (1990). The measured bubble point pressure at 104.4 C was 18.17 MPa, compared with 15.65 MPa ( - 13.9%) predicted by the original PR EOS and 17.24 MPa (-5.1%) predicted by mPR. The measured (Pedersen et al., 1989) and predicted liquid mole fractions of a more volatile oil (Fluid 2) during a constant composition expansion test at 93.4 C are given in Fig. 7. All the reported results for oil systems show that the modification improved the prediction of PR EOS.

Fluid 3 is a North Sea rich gas-condensate reported by Whitson and Torp (1983). The measured and predicted results of a constant volume depletion test at 137.8 C are shown in Fig. 8. The original

1.0'

0.9 '

0 0.8

'~ 0,7

0.6

0.5

0.4

o Exp. / /

mPR / / / / /~

- - " PR j / /

/ J o

/4 o / /

j i i i

0 5 10 15 20 25 Pressure, MPa

Fig. 7. Liquid fraction of volatile oil (Fluid 2) in constant composition expansion test at 93.4 C.

56 A. Danesh et al. // Fluid Phase Equilibria 112 (1995) 45-61

lOO

80- 8

$

~ 40-

~ 2o I

0 50

PR

"""%' Dew Point

i i i ! -

10 20 30 40

Pressure ,MPa

Fig. 8. Cumulative gas production of gas condensate (Fluid 3) in constant volume depletion test at 137.8 C.

PR EOS significantly under predicted the dew-point pressure whereas the mPR prediction matched the measured value of 46.7 MPa. The modification also improved predictions of the gas compressibi l- ity factor, cumulative gas production and produced gas composition.

Table 3 Comparison of predicted and experimental dew point pressures for Starling's gas condensates

Fluid Experiment 1 Dew point pressure (MPa)

Original This work SBWR

Fluid 4 at 101.11 C 26.17 21.00 24.58 25.61 Fluid 4 at 121.11 C 24.24 22.10 23.88 23.55 Fluid 5 at 93.89 C 26.49 22.59 24.38 > 35.27

Table 4 Comparison of predicted and experimental liquid/gas ratios for Starling's gas condensates

Pressure (MPa) liquid/gas yields (m3/std m 3 X 106)

Experimental Original This work SBWR

Fluid 4 at 101.11 C

Fluid 4 at 121.11 C

Fluid 5 at 93.89 C

13.89 880.33 1213.99 1164.21 10.45 877.52 1068.62 1038.55 7.00 822.96 903.88 889.00 3.55 648.58 688.49 684.21

13.89 672.20 894.51 874.11 8.72 711.02 763.03 750.35

20.10 51.19 33.98 48.84 17.34 69.75 62.99 71.44 13.89 87.75 87.01 90.61 10.45 95.63 99.64 100.82 7.00 95.06 102.13 102.36 3.55 84.94 92.20 92.25

1056.96 990.02 872.46 673.33 830.83 755.46 63.56 77.63 90.56 98.44 97.31 83.81


The compositional analysis of two gas condensate fluids (Fluids 4 and 5) as reported by Starling (1966) are given in Table 2. As no information on the specific gravity and molecular weight of the single carbon groups were provided, the physical properties of relevant normal paraffin hydrocarbons were used in this study. This also allowed a comparison with the Starling model--the modified Benedict-Webb-Rubin (SBWR) EOS with 11 parameters. The predicted and measured dew point pressures and liquid yields for the two fluids at various temperatures are given in Tables 3 and 4 respectively. The average absolute deviations between experimental and predicted dew-point pressures by mPR, PR, and SBWR EOS are respectively 5.17%, 14.42% and 12.69% with 12.77%, 17.12% and 11.03% deviations in predicted liquid yields. It should be noted that Starling used measured equilibrium ratios of the these fluids to determine parameters of SBWR EOS prior to making the above predictions. The equilibrium ratios at different pressures predicted by mPR also agreed more closely with the experimental data than those by the original PR EOS. The above results clearly indicate the improvement in PR EOS due to the modification. Using the generalised single carbon properties (Katz and Firoozabadi, 1978) instead of normal alkane values for the above fluids resulted to the same conclusion.

6. Conclusions

The proposed method which uses equilibrium data on binary systems to determine the attraction term in EOS for super critical components can improve the prediction of the phase behaviour of fluids containing these components. As binary data are used to develop the parameters of EOS, the equation can be used without any binary interaction parameters for simple fluid mixtures such as hydrocarbon reservoir fluids. This would lead to a simple set of equations for flash calculations where only three equations need to be solved simultaneously regardless of the number of components of the mixture. The proposed method is particularly advantageous for predicting fluid phase equilibria in compositional reservoir simulators, where many hundred thousands of equilibrium flashes may be performed in simulation of a large reservoir. As it uses data in supercritical regions to determine the parameters of EOS, the method may well also lead to more reliable results in supercritical extractions.

The proposed method was applied to the Peng-Robinson EOS as an example. A simple modification of the correlation relating the attraction term to the temperature was found to be adequate to describe the behaviour of super critical components of binary systems. The modified equation was evaluated for different types of fluids and was found to be superior to the original EOS without binary interaction parameters, and comparable, and in most cases more reliable, than the equation with binary interaction parameters. A reduction in computational time by more than one order of magnitude was achieved for mixtures described by many components using the simplified flash algorithm developed by Michelsen without binary interaction parameters as suggested in the proposed modification. The method is applicable to any semi-empirical cubic equation of state.

7. List of symbols

a, b, c EOS parameters K equilibrium ratio


kij ! m,m

P R T u

V x

Y Z

binary interaction parameter between components i and j component dependence factor of the temperature coefficient in EOS pressure gas constant temperature molar volume vapour phase molar fraction mole fractions in liquid phase mole fractions mole fractions in feed

Greek letters ce temperature dependent in EOS o9 acentric factor g2 parameters of EOS ~b fugacity coefficient

Subscripts C critical F feed i,j component i,j L liquid phase r reduced property V vapour phase

Acknowledgements

The authors wish to thank K. Bell, K. Malcolm and A. Reid for performing the experiments. This study was part of a research project sponsored by the UK Department of Trade and Industry, Bow Valley Petroleum, British Gas Exploration and Prod. Ltd., BP Exploration Operating Company Ltd., Chevron Petroleum (UK) Ltd., Conoco (UK) Ltd., Elf (UK) Ltd., Elf Enterprise, Marathon Interna- tional (GB) Ltd., Mobil (North Sea) Ltd., Phillips Petroleum Co. (UK) Ltd., Texaco Britain Ltd., and Total Oil Marine Plc which is gratefully acknowledged.

Appendix A

Michelsen's Flash calculation method

Let one mole of the feed be flashed at pressure P and temperature T into V moles of vapour and (1 - V) moles of liquid. The material balance for the component i in the mixture is

z~=Vyi+(1-V)x i i=1 ,2 ..... N (A1)


where z~, Yi and x i are mole fractions of the component i in the feed, vapour and liquid phases respectively, and

Y'~ (y , -x~) = 0 (A2)

The equilibrium condition is given by

K~=- -= i=1,2 .... N (A3)

where K is the equilibrium ratio and ~b is the fugacity coefficient determined by the use of EOS. The above 2N + 1 equations can be reduced to three equations as follows, when all binary

interaction parameters are equal to zero. Eq. (10) is converted to

a= Y', Y'~ y iY j (a ia j ) -- Y ' ,Y ia{ /2 =(a ' ) 2 (A4) i= l j= l i= l

, i/2 and ~N , where a i = a i i= 1 Y ia i Substituting the above in Eq. (5), the fugacity coefficient can be expressed as

In cbi = qo + q]a'i = q2bi (A5)

where q0, q~, q2 depend on a', b, T and P only;

qo=- In -~(v -b

a

q' = v~(bRT)

_

q2 = -~ RT 1

O f

+ Gql

If Eq. (AI) is multiplied by a' i, then

E = VE y,.'i + (1 - V) E x,a',

or

a'F= Va v + (l -- V)a L (A6) and similarly

b F = Vb v + (1 - V)b e (A7)

Hence, for a set of a' v and b v, the parameters for the liquid phase can be determined from

a' F - Va v b~- Vb v ~l~ = 1 - - V 'bE = 1 -- V (A8)


Substituting the K value in Eq. (A1), we obtain:

Zi Zi g i

x i= 1 4- V(K i - 1) 'Y i= 1 -t- V(K i - 1) (A9)

The computational procedure, as suggested by Michelsen, which involves only the three indepen- dent variables a v, b v and V is as follows: 1. At the given T and P, evaluate the a' i and b i for each component and calculate a' F and b F.

Estimate the values of a v, b v and V. 2. Calculate aL and b L from Eq. (A8) and evaluate the fi L and fi v and the K values from Eq. (A5)

and (A3). Calculate x i and Yi from Eq. (A9). 3. Evaluate check functions

el = E ( Y i - x i )

e2 = E ( yia'i - - a'v )

e3= ~,(yibi-bv)

If the above values are all less than a given tolerance, the phase distribution at equilibrium has been determined. If not, perform an iterative correction of a'v, b v and V, then return to step (2).

References

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Jhaveri, B.S. and Youngren, G.K., 1988. Three-parameter modification of the Peng-Robinson equation of state to improve volumetric predictions, SPE Res. Eng. (Aug.): 1033-1040.

Katz, D.L. and Firoozabadi, A., 1978. Predicting phase behaviour of condensate/crude oil systems using methane interaction coefficients, Trans., AIME, 265: 1649-55.

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