Kelvin equation for a non-ideal multicomponent mixture

HUIDPHiIS[ EQUIUHIA ELSEVIER

Fluid Phase Equilibria 134 (1997) 87-101

Kelvin equation for a non-ideal multicomponent mixture A l e x a n d e r A. Shapiro *, Erling H. Stenby Engineering Research Center IVC-SEP, Department of Chemical Engineering, Technical Universi O' of Denmark, Building 229, DK 2800 Lyngby, Denmark Received 26 March 1996; accepted 22 January 1997

Abstract

The Kelvin equation is generalized by application to a case of a multicomponent non-ideal mixture. Such a generalization is necessary to describe the two-phase equilibrium in a capillary medium with respect to both normal and retrograde condensation. The equation obtained is applied to the equilibrium state of a hydrocarbon mixture in a gas-condensate reservoir. © 1997 Elsevier Science B.V. Keywords: Kelvin equation; Capillary; Porous medium; Hydrocarbons; Vapor-liquid equilibria; Inteffacial tension

1. I n t r o d u c t i o n

Since the time of its discovery Ill, the Kelvin equation has become one of the most useful tools for the study of vapor-liquid equilibria in capillary media [2]. It has been extensively used for testing different kinds of porous medium, and for the evaluation of their internal surface and of the size distributions of macro- and mesopores. However, the application of the Kelvin equation to the modelling of phase equilibria is highly conjectural for a majority of the industrial and natural processes that deal with porous media. The reason is that the derivation of this equation is based on the following three assumptions: 1. the fluid is single component in nature; 2. the vapor phase is ideal; 3. the liquid phase is incompressible. These assumptions are violated at high pressures when the vapor phase becomes non-ideal and in the cases where the multicomponent nature of the mixture cannot be neglected.

*

Corresponding author.

0378-3812/97/$17.00 © 1997 Elsevier Science B.V. All rights reserved. PII S0378-3812(97)00045-9

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A.A. Shapiro, E.H. Stenby / Fluid Phase Equilibria 134 (1997) 87 101

The extreme case of 'non-Kelvin behavior' is the behavior of hydrocarbon mixtures in o i l - g a s condensate reservoirs. Although the Kelvin equation is applied to tests of the porous media of the reservoirs [3,4], it cannot be used for the modelling of equilibria in such reservoirs--not only because of the high pressure and multicomponent composition of the mixture, but also because this mixture exhibits retrograde behavior when the liquid phase precipitates as the pressure decreases. Such a behavior cannot, in principle, be described in terms of a single-component model. Therefore, several semi-analytical and numerical methods were developed to solve the system of equations for chemical potentials and to find the saturation of the dispersed condensate [5-9]. However, the results show high instability, as a result of the singularity of the governing system to the capillary pressure as a small parameter [6]. In this paper, we try to overcome the above-mentioned shortcomings of the Kelvin equation and to derive its analog for a non-ideal multicomponent mixture. Unlike the original equation, the equation obtained is only approximate and valid in some neighborhood of the dew point. However, the capillary forces may quite often lead to liquid precipitation only when the vapor is close to its dew point. Additionally, the equation obtained is in good agreement with physical intuition and does not lose meaning even when the distance to the dew point increases. The main distinction between the generalized Kelvin equation and the classical equation is the presence of a new thermodynamic parameter called 'the mixed volume'. This parameter arises from the multicomponent nature of the mixture and cannot be reduced by any assumption about mixture ideality. The relation of the mixed volume to the liquid molar volume is different in the regions of normal and retrograde condensation, which makes it possible to apply the generalized Kelvin equation for both regions. It is proven that, in both cases, capillary condensation is possible only for the wetting liquid phase.

2. Kelvin equation for a single-component non-ideal fluid Let us consider a vapor-liquid equilibrium in an isothermal system, with no tensio-active species in the mixture. If the system is subject to the action of capillary forces, for example, in a porous medium, then equilibrium between the gas and liquid phases is possible for vapor pressures Pv that are different from the dew point pressure Pj (Pj is defined for a given vapor composition without any curvature of the interface). The general Kelvin equation must establish a relation between the relative pressure X = Pv/Pd and the capillary pressure Pc. Then, on the basis of the Laplace equation, the curvatures of menisci that separate the phases may be found, and the distribution of the phases in the medium may be determined by geometrical considerations. In this section, we discuss the form of the single-component Kelvin equation that is appropriate for modelling high-pressure capillary condensation. Separate consideration of a single-component case is necessary to distinguish between the modifications that should be introduced into the Kelvin equation as a result of non-ideality and as a result of the multicomponent nature of the mixture. Let us follow the standard procedure of derivation of the Kelvin equation [10], omitting some assumptions which are usually made when this procedure is followed. We consider a non-ideal single-component gas in a capillary medium at a pressure Pv. Under the action of capillary forces, the liquid phase is formed in thin capillaries of the medium. The pressure in this liquid phase is denoted

A.A. Shapiro, E.H. Stenby / Fluid Phase Equilibria 134 (1997)87-101

89

by Pi, so that the difference Pc = P~ - Pv is the capillary pressure. The condition of equilibrium for the two phases is ~v(P~) = ~ , ( P , ) where /x v and /it are the chemical potentials of the vapor and liquid at a fixed temperature. At the dew point pressure Pd, corresponding to the phase transition, the chemical potentials of both phases become equal. We have /zv ( P d ) = / z , ( P d ) The difference of the last two equations is # v ( P v ) - #, (Pa) = / z , ( P , ) - / z , ( P o )

(l)

Each chemical potential /~i may be represented as

fV~(P)dP where Vi is the corresponding molar volume (i = 1, g). Assuming that the liquid is incompressible and that its molar volume V1 is invariable, we transform Eq. 1 to obtain

fe'v~( P) dP = V,( P, - Pd) ej

or, using the definition of the capillary pressure, we can write

fP'V~( P ) d P = V,Pc + VI(P v - - Pd)

(2)

Pd

Further transformation of the left-hand side of Eq. (2) demands knowledge of an equation of state. Note that any such equation may be written in the form

PV,,

-RT

(3)

Z

where z is the vapor compressibility factor. At constant temperature, z = z(P). For the sake of generality, further transformation will be performed on the basis of the general equation of state Eq.

(3). Let us change from the vapor pressure P~ to the relative pressure X = PJPd, and from z(Pv) to

Z(X) = z(Pv)/Z(Po). Eq. (2) is transformed to

f ~ Z( x')

x:

V,Pc dx'-

V,

Vv(Pa)po + V~(pa-~----~(X-1)

(4)

In the standard derivation of the Kelvin equation, it is assumed that the last term of Eq. (4) may be neglected, which is validated by the fact that, at normal pressure, the inequality Vv >> V~ holds. However, this inequality is not valid at high pressures, when the vapor density is comparable with the liquid density, and the last term of Eq. (4) cannot be omitted.

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A.A. Shapiro, E.H. Stenby / Fluid Phase Equilibria 134 (1997) 87-101

To simplify the left-hand side of Eq. (4), we assume that the values of Z ( X ) may be substituted by Zav as

d x ' -~ Za,,( x ) I n X

(5)

where Zav(X) is some characteristic average ratio of vapor compressibilities for the pressures between Pd and Pv. Such a substitution is validated as follows. If Pv is close to Po, then the ratio of compressibilities may be neglected: Z~v ~ 1. Note that the situation X -~ 1 is characteristic for the liquid precipitation in macroporous rocks of petroleum reservoirs (or, in their laboratory samples). Additionally, let us note that Eq. (5) becomes exact if we set

z v(x) = -j[z(

X') dln X' In X

(6)

Thus, Z~v(X) is a logarithmic mean of the value Z( X)- From Eq. (6), it may be derived that, as X ~ 1, the deviation of Zav from unity is the second-order correction to the whole term Zav(X) In XFor example, in a neighborhood of the dew point, it is reasonable to approximate the value Z ( X ) by a linear dependence, i.e. Z ( X ) = 1 + a ( X - 1). Eq. (6) then gives Z~v(X) = 1 - c~ +

o~(x-- 1) o~ = I ( X In X 2- 1 )

(7)

Consider also the case of X << 1. Because the logarithm tends to infinity in this limit, the major contribution to the integral

f z(x') dln x' belongs to the values Z ( X ' ) with X ' = X- Thus, it is reasonable to substitute Z ( X ) = z(Pv)/z(Pd) instead of Z~v(X). At the limit of low pressures, we also have z(P,,) -~ 1 and Z ( X ) -~ 1/z(Pa). Note that, in both limits X -- 1 and X << 1, it is possible to express Z ( X ) through the values which depend only on P~ and Pj. In the general case, Eq. (6) in principle allows us to evaluate Z~ by a known equation of state. After the substitution of Eq. (5) into Eq. (4), we obtain the modified Kelvin equation in the form Za~ In X =

v,P

v,

+ - - ( X Vv(Pd)P,t V~(Pd)

1)

or

Pc = I'd

[Vv(ed)Z.vlnX-X+ 1] V,

(8)

There are two distinctions between this equation and the standard Kelvin equation. One distinction is the presence of the multiplier Zav, as a result of the vapor compressibility. This distinction is not sufficient, because, as was shown above, the value of Z~v may be set to unity. The other distinction is

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91

the term X - 1, which cannot be omitted at high pressures when the molar volume of vapor is comparable with that of liquid. An important feature of Eq. (8) is that it contains only known functions of X (especially, if Zav is neglected) and characteristics of vapor and liquid at the dew point. Thus, Eq. (8) is as convenient for calculations as the original Kelvin equation. It should be emphasized that Eq. (8) becomes incorrect in the critical region. First, in this region, the liquid cannot be considered as being incompressible (this shortcoming can be corrected by introducing the liquid compressibility factor). Secondly, in this region, Vv approaches Vl, and the right-hand side of Eq. (8) becomes of the second order by ( X - 1), which makes all the approximations questionable. This is considered in more detail in the next section, for the case of the multicomponent mixture. Assuming that the pores are cylindrical and that the capillary pressure is expressed through the surface tension coefficient cr and the wetting angle 0 by the Laplace equation 2 cr cos 0 Pc -

(9) rK

we obtain the value of the Kelvin radius r K as 2 o- cos 0 r K = Po{[Vv(Pd)/VI]Zav In X - X+ 1} For the pores of more complex shapes, the value of r K expresses inverse curvature of the surface that separates the liquid and vapor phases.

3. Kelvin equation for a multicomponent mixture As was mentioned in the Introduction, the Kelvin formula for a single-component fluid cannot, in principle, be used for the estimation of the capillary condensation of a multicomponent mixture when it exhibits retrograde behavior. To take into account this phenomenon, it is necessary to modify the Kelvin equation so that it can be used for a multicomponent mixture. In this section, such an equation will be derived. Unlike the previous section, where the whole pressure range was considered, we discuss here the case when the pressure Pv is close to the dew point pressure Pd" We consider a vapor phase with the composition x v = ( x .I. . . . . Xvn - l ) where n is the number of components. Without capillary forces, the precipitation of the liquid phase takes place at the dew point pressure Pd' The composition of this liquid phase, i.e. Xld = (X~d. . . . . X~,d- l), and the value of Pd depend on the vapor composition x v through the equalities of the chemical potentials in both phases. Assume that, under the action of capillary forces, the liquid condensate is formed in thin capillaries. The pressure in this capillary condensate is P1 and its composition is x 1 = ( x I . . . . . x/'-l). The chemical potentials of components in both phases obey the conditions of equilibrium similar to Eq. (1). We have

t z i ( p v , X v ) _ /Zv(P i d , x v ) = / z i ( P , , x , ) - / z { ( P d . X,d. ), . i = . 1,.

n

(10)

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92

Transforming to integrals, we find that

P',zit p, fPi ~ p i ( p ' x v ) d e :

f

Pd

Vl ~"

x, O#i(P,, x ) Xld)de

-~- fx

i~

OX

dx

(11)

Here, the values of V~i and 1/1" (i = 1. . . . . n) are partial volumes of the components in the vapor and the liquid phases. By O#i/Ox, we denote the n-vector that consists of the components Otzl/OxJ ( j = 1. . . . . n - 1). The designation Otzl/Ox d x stands for the scalar product of the corresponding vectors. The integration

is carried out along the straight line that connects the points x~j and x~. Let us multiply each of n Eqs. (11) by Xld,i including n-I n Xld = 1 -

i

Y'~ Xld

i=1

and add up all these equations. As a result, we obtain v

Pd

VvI(P, x v ) d P =

PJ

P, Xld)dP +

i

~ i=

X,a

Olzi( Pl, x ) dx Ox

(12)

Here, V~ is the molar volume of the liquid phase of the composition Xld. We have

v,(P, Xld)=

' i( P, x d) xldvl i=1

In the left-hand side of Eq. (12), we introduced the designation

Vv,(P, Xv) = ~ x,i~vvi(p, xv)

(13)

i=1 i The value of Vvl depends only on P and x v, because, as was mentioned above, the values of xtd are functions of x v. We shall call it the 'mixed volume'. Previously, a similar characteristic of the vapor-liquid equilibrium was used in the multicomponent Clausius-Clapeyron equation [11,12]. Note that the value of VI(P, Xld) is also dependent on x v, just as Vvl. We assumed that the pressure Pv is close to the dew point pressure Pd" According to this assumption, we will omit the terms of the second order to the difference Pv - Pd and the terms of the second order to x I -Xld. We will show that the last integral in Eq. (12) may be neglected in that neighborhood of the dew point. This integral represents the contribution of the compositional shift to the Kelvin equation. The fact that it may be neglected means that, close to a dew point, the compositional shift makes an insignificant contribution to the value of the capillary pressure and, therefore, to the value of saturation of the condensate. However, it should be stressed that the compositional shift itself may be of the first order and may not be neglected when the composition of the capillary condensate is evaluated.

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93

Let us split the integral

x,

I = fx

a

i(P,, x )

i Xld

Id i = 1

dx

OX

(14)

into the two addenda as

fxX' ~ ,°

i

,

0#l(P , . x )

dx+

i=,

fxll i= ~l xi O~'Li(OxP''dx x)

The last term disappears in view of the Gibbs-Duhem equation, i.e. at constant pressure, we have

~

x i d/~i = 0

i=1

Let us note now that the term

S;; I,d

~-

i= 1

"'I

dx 0X

i _ xl), and the integration is is of the second order to x~ -X~d, because it includes the difference (x~d from xl to x~d. In the chosen approximation, this term may be neglected. Thus, we have proven that the whole integral in Eq. (14) may be omitted in Eq. (12). This equation is reduced to

fei'Vv,( P, Xv)dP= fpP'Vl( d P, X,d)dP

(15)

Assume now that, at a fixed composition, the liquid is pressure incompressible: the value of V~ is independent of P. As other assumptions of the present derivation, this one might be too strong for the liquid in the near-critical region. Retaining the liquid compressibility would lead to the factor of the type Zav for the liquid phase. Because the liquid phase is usually less compressible than the gas phase, this factor appears to be less significant than the corresponding factor for the gas phase. At a distance from the critical region, both factors disappear as X tends to unity. In the assumption above, the right-hand side of Eq. (15) is simplified similarly to Eq. (2):

f"'v,( P, Xv)dP

= V,P~ + V,( Pv - Pd)

(16)

Pd

To find the integral in the left-hand side of Eq. (16). we must assume some form of the vapor compressibility. We take the form to be

Vv,(n.x.) Za.(x.) = - (17) Vv,(t'd, x.) X where X = P/Pd, as usual, and Zav(X v) is the ratio of average compressibility of the value Vv~ for the pressures between Pd and Pv to the compressibility at the pressure Pd. The assumption in Eq. (17) is less validated than the similar assumption for the single-component fluid, because the value of Vv~

94

A.A. Shapiro, E.H. Stenbv / l-Tuid Phase Equilibria 134 (1997)87 I01

has no such clear physical meaning as for the vapor volume Vv. However, assuming that the partial volumes V,,* (or, at least, the partial volumes of the most valuable components) obey the equations similar to Eq. (3), we can still validate Eq. (17). To do so, reasoning similar to that of the previous section can be used, invoking the fact that Vvj is the linear combination of Vvi with coefficients independent of the pressure. Additionally, because we consider the case of Pv ~ Pd, we may assume that Zav ~ 1. After substitution of Eq. (17) into Eq. (16) and transformations such as Eqs. (2)-(4), we obtain the generalization of the Kelvin equation in the form

&

Vv,(&, Xv)

&

v,(xv)

Z ,v(Xv)ln x - x + I

(18)

This equation allows us to find the value of the capillary pressure and, therefore, the Kelvin radius from the value of the relative pressure and the composition of the vapor phase. Again, the capillary pressure is expressed in terms of the relative pressure X and the characteristics at the dew point, within the term Zav(Xv), which may be omitted close to the dew point. The main distinction between the Kelvin equation presented as Eq. (18) for the multicomponent mixture and its form in Eq. (8) for a single-component gas phase is in the presence of the mixed volume Vv~ instead of the vapor volume Vv. This difference arises solely because of the multicomponent nature of the mixture. It cannot be reduced by any assumption about ideality of the mixture (ideal compressibility or ideality of mixing). Thus, the capillary condensation from a 'sufficiently' multicomponent mixture cannot be treated on the basis of a single-component Kelvin equation. Note that we cannot neglect the second term on the right-hand side of Eq. (18), and that the reason is more important than that in the single-component case. For a multicomponent mixture, the value of V,,j differs from the vapor volume V, and may be close to the liquid volume VI. Moreover, in a hydrocarbon mixture, partial volumes of some heavy components may become negative. The fractions of such components in the liquid phase, i.e. Xld, i are usually much higher than those in the vapor ' Hence, the volume phase, i.e. x v. gvl = E x , aiV , , i

may be much less compared with the vapor volume

Vv = Ex(, V,' In the next section, we show that, for retrograde condensation, the inequality Vvj < VI holds. Expansion of the right-hand side of Eq. (18) up to the second-order terms by X - 1 gives (19) This expansion shows that Eq. (18) becomes invalid in the neighborhood of the critical point. In this neighborhood, the volumes V,q and V~ become equal, the term with ( X - 1) 1 is eliminated and the value of PJPa turns out to be of the order ( X - 1) 2, meaning that the integral in Eq. (14) cannot be omitted in Eq. (12).

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95

4. Conditions of capillary condensation In this section, we will prove that, in the neighborhood of a dew point, capillary condensation is possible only for the wetting phase. This statement is in agreement with physical intuition, because the wetting liquid is 'attracted' from the vapor phase by the walls of the capillaries. However, to the best of our knowledge, it has not been proven thermodynamically. We present such a proof based on Eq. (18), which is not restrictive, because this equation becomes exact as X--+ 1. As a basic case, we consider retrograde condensation of a wetting fluid. Normal condensation can be considered in a similar way. Because the value of P,. is negative for the wetting liquid, to make retrograde capillary condensation possible, the right-hand side of Eq. (18) must be negative for X > 1. Note that Z~v ~ 1 as X --+ 1. Also, the inequality In X < ( X - 1) holds at X > 1. Thus, the right-hand side of Eq. (18) is non-positive if Vvl(Pd, I v ) ~ Vl(Xv)

(20)

The condition in Eq. (20) makes retrograde capillary condensation possible for a wetting liquid. The inverse inequality would be sufficient for retrograde capillary condensation of a non-wetting liquid. Let us show that, for retrograde condensation, Eq. (20) is always valid. Probably the shortest way to prove it is to use the Gibbs tangent plane condition for stability [13] (it can also be proven on the basis of the analysis of the multicomponent Clausius-Clapeyron equation). Let us consider the vapor phase of the composition x v at the pressure P >__Pd" Such a phase must be stable and obey the Gibbs condition for stability. For any composition x, we have H

E x'[

x)-

Iv)] >__0

i-I

where /xi denotes 'universal' expressions for the chemical potentials coinciding with the /Xiv and #i for the vapor and liquid phases respectively. Consider the function f(P)=

~X(d[ /zi(P,Xld)-,i,(P, i

Xv)]

1

According to the Gibbs tangent plane condition, f ( P ) >_ 0 when P >_ Pd" However, at P = Pd, the equality f ( P ) = 0 holds, as a result of the equalities of the chemical potentials in both phases. Thus, we have df(P) dP

P=

>0 eL,

At the same time, at P = Pd, the derivative d J ~ d P is equal to n

df(P) dP

P=

= Pd

,

[

E XId[

i O~l(e,

0

Xld )

OP

i /-£v( P , X v )

OP

i= 1

=

P= Pd

XldW] i=1

(ed , X I d )

--

XldVv ( e d , i

1

X v ) = V I -- Vvl

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96

Comparison of the last two equations proves the inequality of Eq. (20). For normal condensation, the condition in Eq. (20) must be changed to the opposite condition, i.e. gvl(Pd, .'~v) ~ gl(xv)

(21)

This inequality can be proven similarly to Eq. (20). Its application to the modified Kelvin equation (Eq. (18)) in a similar manner shows that capillary condensation of the non-wetting liquid is again impossible. Note that the conditions in Eqs. (20) and (21) have been proven in a strictly thermodynamic way; therefore, they are more general than just conditions of the capillary condensation. For example, they can be used for checking whether a point on the phase diagram corresponds to the retrograde or to the normal condensation.

5. Sample calculations All the thermodynamic calculations were performed using the program LNGFLASH developed in the Engineering Research Center IVC-SEP. The program is based on the cubic (Peng-Robinson) equation of state and its results of calculations are in good agreement with experimental data for hydrocarbon mixtures [14]. Calculations of the surface tensions were carried out using the modified scaling law, with the assistance of the program IFT written by A. Dandekar. It is shown by Danesh et al. [15] that the modified scaling law is in a good agreement with experimental data on the surface tension of hydrocarbon mixtures.

5.1. Binary mixture Let us analyze the behavior of the binary mixture methane-n-butane in a capillary medium. The phase diagram for this mixture at a temperature of 300 K is shown in Fig. 1 (see also Table 1). The critical point corresponds to a pressure of 137 bar (13.7 MPa) and to the molar fraction of methane 0.77. If the molar fraction of methane exceeds this value, then the mixture shows retrograde behavior. The partial volume of butane Vv2 is negative in the region of retrograde condensation. The dependence of the liquid volume and of the mixed volume Vv~ on the pressure is shown in Fig. 2. This dependence allows us to define precisely the dew point pressure at which the retrograde P, bar 140 120

.......

1oo

V

80 P

6O 40 0.2

,.i.,,i.,,i.,.i..,i,..i.,.i... 0.4 0.6 0.8

XI 1.0

Fig. 1. Phase diagram for the system C 1- nC4.

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97

Table 1 Composition of the vapor phase and of the equilibrium liquid phase for the gas-condensate mixture Component

Gas molar fraction

Liquid molar fraction

C1 C2

0.7606 0.0708

0.2560 0.0614

C3 nC4

0.0272 0.0148

0.0469 0.0504

nC5 C6

0.0102 0.0040

0.0669 0.0494

C7

0.0176

0.4063

N2 CO 2 H2S

0.0081 0.0546 0.0321

0.0014 0.0271 0.0342

condensation is changed to the normal condensation, i.e. P,r = 68 bar (6.8 MPa) (point A in the phase diagram). Note that this value is difficult to define directly from the phase diagram, because the corresponding branch of the dew point pressure curve is sharply inclined to the xl-axis. As predicted in the previous section, Vv~ > VI in the region of normal condensation, while, in the region of retrograde condensation, the value of Vv~ is smaller than I/1, reaching its minimum at P = 100 bar. At the same time, the value of Vv always exceeds both Vvl and Vl: for example, Vv = 290 cm 3 mol-1 at 70 bar and 185 cm 3 mol-1 at 100 bar. The difference between all three volumes disappears as the mixture approaches the critical point. Fig. 3 shows the dependence of the capillary pressure on the relative pressure X, with the value of Zav equal to unity. The capillary pressure P i - P v is negative, because the liquid is wetting. The non-monotonic dependence of the capillary pressure on the dew point pressure Pd is explained by the fact that the dependence VvI/V ~ is also non-monotonic (see Fig. 2). The lowest capillary pressures correspond to the neighborhood of the critical point. The dependence of the Kelvin radius for the porous medium calculated by Eq. (9) is represented in Fig. 4. Unlike the capillary pressure, the Kelvin radius depends monotonically on the dew point pressure, because it is mostly affected by the values of the surface tension o-. These values exhibit considerable variation, being equal to 1.579 mN m ~ for Pd = 80 bar, 0.734 mN m-~ for Pd = 100 bar, and 0.206 mN m 1 for Pj = 120 bar. The variation of the ratio V v l / V 1 is not very marked. For a pressure of Pd = 80 bar and X < 1.1, the Kelvin radius is of the order of the macropore size ( 1 0 - 6 - 1 0 -8 m). This size range is comparable with the sizes of pores which determine flows in

60

V

'

cm3/mol

i

140 ~ V f . . i . . . . . . . . . V v i . . . . . ........................ ~........................

40~ . . . . . . 40 60

J,,, 80

i ...... 100 120

P, bar 140

Fig. 2. Molar volumes for the system C 1 - n C 4 .

A.A. Shapiro, E.H. S t e n b y / Fhdd Phase Equilibria 134 ( 1 9 9 7 ) 8 7 - 1 0 1

98

P~, bar .....................i...................i...................

-5 -10

"........... i..................

-15

.........

-20 -25

3C

Pg ~Pd

10

11

12

13

14

15

Fig. 3. Dependence of the capillary pressure o11 the relative pressure for the mixture C I - n C 4 . The dew point pressures are (1) 80 bar, (2) 100 bar and (3) 120 bar.

Iog~o rK, log~o m -6 _7 !

1

2

3

-8 -9

-~0 -]]

.............. i ~ 7 7 ~ = . . . . . . . . . . . . . . . . . . . . . . !.................................

-12 1.0

,,

i,,, 1.1

i,,,, 12

~............. ~

m

rgre

. . . . . . 13 14

1

Fig. 4. Dependence of the logarithm of the Kelvin radius on the relative pressure for the mixture C I - n C 4 . The dew point pressures for the mixture are (1) 80 bar, (2) 100 bar and (3) 120 bar.

P~, bar

-20

:

.,o ............ -60

: ............................................

....................................

] O0

Fig. 5. D e p e n d e n c e

105

1.10

i ~

.........

i 115

Pg/P~ 1 20

of the capillary pressure on the relative pressure lor the m u R i c o m p o n e n t

mixture.

l o g 1 0 rr~, log~o m -6

-7

-8

-9

!

i 1.00

1.05

i 1.10

1.15

,

Pg/Pd 120

Fig. 6. Dependence of the logarithm of the Kelvin radius on the relative pressure for the multicomponent mixture.

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low-permeable hydrocarbon reservoirs. For higher relative pressures values, the Kelvin radius becomes comparable with the sizes of meso- and micropores. Because the mechanism of adsorption in micropores differs from capillary condensation, the value of the Kelvin radius cannot be used in this region. For pressures of P j - - 1 0 0 and 120 bar, the Kelvin radius is comparable with the sizes of macropores only in the neighborhood of the phase diagram, X = 1. It then moves into the meso- and micropore region. We calculated the capillary pressures and the Kelvin radii for relative pressures up to X = 1.5. For such high relative pressures, the assumptions made for the derivation of the modified Kelvin equation are no longer valid. Nevertheless, the equation exhibits physically correct behavior, even at such high values of X. 5.2. Multicomponent mixture Sample calculations of the Kelvin radius were performed for a multicomponent gas-condensate reservoir mixture the composition of which is shown in Table 1. The dew point pressure of this mixture is Pa = 780 bar and the reservoir temperature is T = 350 K. The liquid and the vapor densities are 476 and 70 kg m-3 respectively. The liquid and the mixed molar volumes are V1= 132 cm 3 mol-1 and Vv~ = 88 cm 3 mol-~. The vapor molar volume Vv = 325 cm 3 tool-L greatly exceeds the value of V~. The major difference between V,,I and Vv is explained by the fact that the partial molar volumes of heavy hydrocarbons are negative for the mixture considered. The surface tension, calculated using the modified scaling law, is cr = 2.740 mN m -~. The curves of the capillary pressure and of the Kelvin radius vs. relative pressure are shown in Figs. 5 and 6. The shape of these curves is similar to that of the binary mixture represented by curve 2 in Figs. 3 and 4. The reason is similar ratios of Vvl/V 1 and o-/Po for these two mixtures. The Kelvin radius varies in the macropore scale (10-6-10 -s m), because the value of X does not exceed 1.025. The radius then moves into the mesopore scale and stays there until approximately X = 1.1. After this point, the Kelvin radius moves to the region of micropores and below it, where it is physically meaningless. Thus, capillary condensation is possible and sufficient only in the neighborhood of the dew point.

6. Concluding remarks The modified Kelvin equation can be used for modelling the behavior of capillary condensation in different areas of chemical and reservoir engineering. It makes it possible to avoid cumbersome flash calculations related to the evaluation of the capillary equilibrium. In combination with the percolation theory methods, the equation proposed allows us to estimate the quantity of the dispersed liquid phase on the basis of a pore size distribution.

Acknowledgements The authors would like to acknowledge Prof. Michael L. Michelsen for his notes about the multicomponent Clausius-Clapeyron equation, and Dr. Klaus Potsch for the idea behind Eq. (7). Dr.

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Kim Knudsen and Dr. Abhijit Dandekar are kindly acknowledged for useful discussions and for supplying software for calculating phase equilibria and surface tension coefficients. The authors thank the Danish Technical Research Council for financial support.

Appendix A. List of symbols n P R r T V x Z z

number of components pressure gas constant radius temperature molar volume molar fraction compressibility ratio compressibility

Greek letters /x X o0

chemical potential relative pressure surface tension wetting angle

Superscripts i, j

component identities

Subscripts av c d K 1 v

average capillary dew point Kelvin liquid vapor

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A.I. Brusilovsky, SPE Paper 20180, 1990. S.-T. Lee, SPE Paper 19650, 1989. R.J. Wheaton, SPE Reservoir Eng. May (1991) 239-244. L.I. Kheifets and A.V. Neimark, Multiphase Processes in Porous Media, Khimia, Moscow, 1982 (in Russian). R. Haase, Termodynamik der Mischphasen, Springer, Berlin, 1956. W. Malesifiski, Azeotropy and Other Theoretical Problems of Vapour-Liquid Equilibrium, Wiley, New York, 1965. M.I. Michelsen, Fluid Phase Equilibria 9 (1982) 1-19. K. Knudsen, Phase equilibria and transport of multiphase systems, PhD Thesis, Technical University of Denmark, 1992. [15] A.S. Danesh, A.Y. Dandekar, A.C. Todd and R. Sakar, SPE Paper 22710, 1991.