Comparative Study of Eight Equations

of State for Predicting Hydrocarbon

Volumetric Phase Behavior
Tarek H. Ahmed, SPE, Montana C. of Mineral Science & Technology

Summary. The objective of this study is to present a review of eight equations of state (EOS's) and compare their ability to
predict the volumetric and phase equilibria of gas-condensate systems. Included in the study are the Peng-Robinson (PR), the
Soave-Redlich-Kwong (SRK), the Schmidt-Wenzel (SW), the Usdin-McAuliffe (UM), the Heyen, the Kubic, the Adachi-Lu (AL),
and the Patel-Teja (PT) EOS's. The SW equation exhibits a superior predictive capability for volumetric properties of cOl1densate
systems. The PR equation is found to represent the phase equilibrium behavior of condensate systems accurately. In terms of
compressibility factors, the SW and PT equations give better predictions than other equations.

Introduction
Since the proposal of the van der Waals equation, many EOS's have and
been proposed for the representation of fluid volumetric, ther-
modynamics, and phase equilibrium behavior. These equations, b=nbRTc/pc' ..................................... (6)
many of them a modification of the van der Waals EOS, range in
complexity from simple expressions containing two. or three con- where na and nb are the SRK dimensionless pure-component pa-
stants to complicated forms containing more than 30. Although the rameters. For the unmodified SRK equation, na =0.42747 and
complexity of any EOS presents no computational problem, most nb =0.08664 are independent of temperature, pressure, composi-
apthors prefer to retain the simplicity found in the van der Waals tion, or particular component.
cubic equation while improving its accuracy through modifications. Eq. I can be expressed in terms of the compressibility factor as
Semiempirical EOS's set forth in recent years have retained the
van der Waals thermal repulsive term. Differences exist in the ex- Z3 -Z2 +(A-B-B2)Z-AB=0, ...................... (7)
pression of the attractive pressure term. The most significant mile-
stone in the development of cubic EOS's is the publication by where Z is the compressibility facto~ and A and B are defined by
Soave I of a modification in the evaluation of the parameter a in
the attractive term of the Redlich-Kwong equation. A =aap/(RT)2 ............................ .' ....... (8)
This suggestion by Soave has prompted an enormous increase
in activity on the part of scientists and engineers who are interested and
in the use of EOS's to calculate fluid properties.
The objective of this study is to present a review of developments B=bp/(RT) . ...................................... (9)
in cubic EOS's and to compare their predictive capabilities for volu-
metric and vapor/liquid equilibrium (VLE) predictions. In the VLE calculations for mixtures, Soave suggested the fol-
lowing mixing rules:
Review of Recent Developments in Cubic EOS's
SRK EOS and Modifications. The SRK I EOS has the form
(aa)m = 2: 2:ziz/aiapia)O.5(1-Kij) ............... (10)
p=RT/(V -b) -aa/[V(V +b)], ........................ (I) i j

where the dimensionless factor a is a function of temperature: and

a=[1+m(I-Tr 05)]2 . ............................. (2)

b m = 2:z ib i , ..................................... (II)
Soave correlated the slope, m, against acentric factor, w, by the
generalized relationship
where Zi is the mole fraction of the phase, gas or liquid, and the
m =0.480 + 1.574w -0.176w 2 . . ...................... (3) values of Kij are binary interaction coefficients that are usually set
equal to zero for regular solutions.
Graboski 2 used a regression program to re-evaluate the term m The fugacity coefficient, cPi. of a component in a mixture is
on the basis of a detailed set of hydrocarbon vapor pressure data given by
with the purpose of improving pure-component vapor pressure pre-
dictions by the SRK EOS. They proposed the following expres-
sion for m:
-(A/B)(2t/J/t/J-b/bm)ln(1 +B/Z), .............. (12)
m=0.48508+ 1.55171w-0.15613w 2 . . ................ (4)
where
For any pure component, the constants a and b are found from
the critical properties:
t/J i = 2:z/aiapiaj)O.5(I-Kij) ...................... (13)
a=naR2Tc2/pc ................................... (5) j
Copyright t 988 Society of Petroleum Engineers

SPE Reservoir Engineering, february 1988 337

and Peng and Robinson adopted Soave's approach for calculating a
as given by Eq. 2. They also used", as the correlating parameter
for the slope, m, as given by
1/;= ~ ~z;z/a;ap;a)0.5(1- Kij)' .................. (14)
; j m=0.37464+ 1.54226",-0.26992",2.

Graboski 2 suggests that no interaction parameters are required for This expression was later expanded by the investigators to give the
hydrocarbon systems. With nonhydrocarbons present, however, relationship
binary interaction can greatly improve the equilibria predictions.
Vidal 3 and Elliot and Daubert 4 presented a theoretic background m=O.3796+ 1.485",-0.1644",2 +0.01667",3.
to the meaning of the interaction coefficients and the quadratic mix-
ing rules of the Soave equation. Introducing the z factor into Eq. 15 yields
Correlations of optimal Kij proposed by Elliot and Daubert are
for N2 systems, Kij =0.107089+2.9776kij; for CO 2 systems, Kij
=0.08058-0.77215kij -1.8407(kij)2; and for methane systems
with compounds of 10 carbons or more, Kij =0. 17985+2.6958kij The fugacity coefficient of Component j in a mixture is calculated
+ 10. 853(kij )2, where from

kij =[ -(E;-Ej)2]/(2E;E) In <P; =b;(Z-I)/b m -In(Z-B)-A/(2.82843B)(21/;;l1/;-b;lb m )

and xln[(Z+ 2.414B)/(Z-0.414B)]. . ................ (17)

The mixture parameters b m , A, B, 1/;;, and 1/; in Eqs. 16 and 17

were defined previously.
Accuracy of the SRK EOS and its modifications is restricted, how- Stryjek and Vera 9 ,10 observed large deviations between ex-
ever, to the estimation of vapor pressure and VLE calculations. perimental and calculated vapor pressures as a function of reduced
The volumetric estimates obtained through this two-parameter EOS temperature given by the PR equation. They further stated that the
are not as accu rate. observed deviations are large at all temperatures for compounds
Peneloux et ai. 5 developed a method of improving the volumet- with large acentric factors and that the errors increase rapidly at
ric predictions by introducing a volume-correction parameter, c;, low reduced temperatures for all compounds. Stryjek and Vera pro-
into the SRK EOS. The third parameter does not change the VLE posed that a major improvement can be obtained with the following
conditions determined by the unmodified SRK equation, but modi- expression for a:
fies the liquid and gas volumes by effecting the following transla-
tion along the volume axis:

with

kO=0.378893 + 1.4897153",-0.17131848",2 +0.0196554",3

where V is the molal volume of the hydrocarbon phase (gas or liq- and kl being an adjustable parameter characteristic of each pure
uid) as calculated by Eq. 1. The authors proposed six different compound.
schemes for calculating c;. For petroleum fluids and heavy Ihaveri and Youngren II found in their application of the PR
hydrocarbons, Peneloux et at. suggested that the best correlating equation to reservoir fluids that the error in the prediction of the
parameter for c; is the Rackett 6 compressibility factor ZRA: gas-phase compressibility factors ranged from 3 to 5 % and the error
in the liquid density predictions from 6 to 12 %. Following the proce-
dure proposed by Peneloux et at. ,5 Ihaveri and Youngrcn intro-
duced the volume-correction parameter, c;, to the PR EOS.
Gibbons and Laughton 7 introduced two adjustable substance- Because the proposed third parameter has the same units as the scc-
dependent parameters, X and Y, for the expression a of the SRK ond paramcter of the unmodified PR equation, b i , they defined a
equation: dimensionless shift parameter, s;, as S i = c;l b i' They then presented
values of the shift paramcters for the well-defined lighter hydrocar-
a= I +X(Tr -l)+ Y(Tr 0.5 -1). bons. They also outlined the characterization procedure of the shift
. parameter for thc heptanc-plus fraction. Ihaveri and Youngren con-
Values of X and Yare chosen by minimizing the root-mean-square cluded that a significant improvement in the volumctric predictions
error in the calculated vapor pressures for the whole rangc of thc by thc modified PR equation can be obtained.
liquid/vapor-pressure curve. However, the authors were unable to
generalize a correlation for X and Y. UM EOS. Usdin and McAuliffe 12 introduced a new coefficient,
d, to replace b in the second term of the SRK equation:

PR EOS and Modifications. Peng and Robinson 8 proposed a p =RT/(V -b) -aa/[V(V+d)] . ....................... (18)
modified Redlich-Kwong EOS capable of predicting the liquid den-
sity as well as vapor pressure to improve VLE predictions further. They contended that the two terms of the SRK equation are inter-
Thcir equation is given by connected by cocfficient b, forcing the conclusion that all materials
possess a value of critical compressibility of 0.333. This conclu-
p=RT/(V-b)-aa/[V(V+b)+b(V-b)], ............... (15) sion cffccts a substantial depression in the values of liquid density.
They stated that severing the tie created by the shaping of the coeffi-
with cient band rcplacing it with d would produce accurate liquid den-
sities.
a=0.457235(RTY/pc Defining

and

b = O. 077796RTc/Pc'

338 SPE Reservoir Engineering, February 1988

and Stein observed that to define a given system it suffices to make
each 11a and 11b a variable function rather tljan the Redlich-Kwong
constants, and to introduce another variable Kij for multicompo-
nent systems. These parameters were evaluated first for pure com-
Usdin and McAuliffe proved the coefficients 11 a , 11 b , and 11d can ponents and then for binary mixtures. The coefficient of ~hese
be related to the critical compressibility parameter, Z~, by the fol- parameters was generalized in terms of lITr and w by a multiple
lowing expressions: regression technique. This resulted in a best fit to the expressions

11j +(6Z~-1)113+[12(Z1)2 -3Z1]11" + (8Z1-3)(Z1)2 =0, 11 b =0.08664+[(lOO-15l9w)1O- 50ITr -l)], ......... (20)

11a(lITr) -1.38 = 11b [A I +A 2(IITr -1)+A 3(IITr -1)2

and +A 3(lITr -l)3 +A 5(IITr -1)w+A 6 (IITr -l)w 2

+A7(lITr-I)3 w 3], ................... (21)

They presented a graphic illustration and tabulated values of Z~ for and

several components. They also adopted Soave's formulation of Ci
(Eq. 2) with

m = 0 .48049 + 4. 516wZ[ where Vm is the molar volume. Stein also presented the values of
the constants A 1 through A7 in a tabular form. In terms of com-
+ [0.67713(w -0.35) -0.02](Tr -0.7), for Tr 5, 0.7 pressibility factor and fugacity coefficient, Eq. 19 corresponds to
Eqs. 7 and 8, respectively, with if;i and if; defined as
and

m=0.48049+4.516wZ~ if;i= L,z/aiaj)O.5(1-Kij)

j
+ [37. 7846w(Z~)3 +0. 78662](Tr -0.7)2, for 0.75, Tr 5, 1.0.
and
Eq. 18 can be rearranged into compressibility factor form to give

Z3 +(D-B-l)Z2 +(A-D-BD)Z-AB=O. if; = L, L,Ziz/a;aj)0.50- Kij)'

I J
The mixture parameters (A, B, and D) are defined by
Stein suggested that Eqs. 20 and 21 can be used to calculate 11a
and 11b for heavy hydrocarbons having w > 0.487 but warned that
this should be done with caution beyond w=0.74.

Heyen EOS. To reproduce experimental Zc' Heyen 14 suggested

a third parameter, c, to be included in the attractive term. He pro-
posed the following form of three-con$tant EOS:

p=RTI(V-b)-al(V2 +(b+c)V-bc), ................ (22)

and
where

D= L,11diZiPr/Tri,
i

where the reduced temperature Tri and reduced pressure Pri are
defined by the expressions Tri=T1Tci and Pri=plPci'
In their verification of the proposed equation and in comparison and
with the SRK EPS, Usdin and McAuliffe suggested that a better
pure-component density prediction could be obtained with accura-
cy similar to that of the SRK equation for VLE computations.
Heyen identified the parameters k, n, m, and 0 for any compo-
Stein-Redlich-Kwong EOS. Stein 13 stated that the deficiency in nent by forcing the EOS to match experimental vapor pressure and
the Redlich-Kwong two-constant equation is that the number ofvari- saturated liquid volume. This led to the following relationships for
abIes in Eq. 19 is less than the number of degrees of freedom; hence these parameters, with w as the correlating factor:
the calculation system is underdefined. The Redlich-Kwong equation
is defined by k=0.49164+0.43882w-0.0882Iw 2 ,

p=RTI(V-b)-al[V(V+b)To5], .................... (19) n= 1.637 + 1.387w,

where 0=7.2562+ 14.153w+ l.33137w 2 ,

and

and m=0.23333 -0.06737w+0.4911w 2 ,

with

where 11a =0.42747 and 11b =0.08664.

SPE Reservoir Engineering, February 1988 339
and In these equations,

where nb is the smallest positive real root of the equation and where

a O= -0. 1514Tr +0.7895 +0.3314ITr +0.029ITr2 +0.OOI5ITr 7

Introducing the compressibility factor, Z, to Eq. 22 results in and

a 1= -0.237Tr -0.7846ITr + l.OO261T,2 +0.0l9/T'7;

Z3 +(C-l)Z2 +(A-2BC-B-C-B2)Z+(BC+B2C-AB)=0.
b =(0.082 -0 .0713w')RTelp e;
................................... (23)
. and
For mixtures, Heyen suggested the classic quadratic mixing rules,

where
am = I; I;ZiZj(aia)o.S(l-Ki), ..................... (24)
I J
')'0 =4.275051- 8.8788891Tr + 8.5089321Tr 2

b m = I;Zibi, ..................................... (25) - 3.4814081Tr 3 +0.576312ITr4,

')' 1 = 12.856404- 34.7441251Tr + 37.433095IT,2

and
-18.05942I1Tr 3 +3.51405ITr 4 ,
Cm = I;ZiCi, ..................................... (26) and

with w' =0.000756+0.90984w+0.16226w 2 +0.14549w 3 .

A=ampl(Rn, .................................. (27a) For mixtures, the classic mixing rules, Eqs. 24 through 24, are
suggested. The fugacity coefficient is calculated from
B=bmPI(RT), .................................. (27b)
In </>j= -In{Z-B+[pl(RT)](b/Z-B)} - [pl(RT)]2[2>/;/(Z+C)]
and
+ [pl(RT)]Ac/(Z+ C)2.
C=cmpl(RT). . ................................. (27c) Kubic pointed out that the ratio blVe varies as a function of the
acentric factor in the modified Martin equation, which improves
The expression for the fugacity coefficient is given by the accuracy of the equation over a wider range of fluids and im-
proves its ability to handle asymmetric mixtures. The investigator
In cPi= -In(Z-B)+ {bi/[(Z-B)->/;/RTd]ln(Q+d)/Q-d} concluded that the predictions given by this modification are as good
as or better than those given by the PR EOS for VLE.
+0.05A[(b j +cj)/(Q" -d 2 )] +0.125A[cj(3B+C)

+h j(3C+B)]{ln[(Q+d)/(Q-d)]-2Qdl(Q2 -d 2 )} AL EOS. Adachi and Lu 17 suggested that identical results in VLE

calculations can be obtained from various cubic EOS's, including
x [p/(RT)], the simplest cubic EOS, the van der Waals equation. They indicat-
ed that the parameter values capable of predicting vapor pressures
where of pure substances would generally be suitable for VLE calcula-
tions by treating na [which is related to the parameter a by:
a=n a (RTc )2IPc] as temperature-dependent with all other param-
>/;i = I;z/aiaj)o.S(l- Kij)'
j
eters kept constant at their critical point. The authors developed
the following expression for the parameter a in the van der Waals
Q=Z+0.5(B+C),
. EOS:
and p=RTI(V-b)-a/V 2 , ............................. (28)
d=lBC+0.25(B+C)2]OS. where a=n,,(RTe2)lpc with na=(27/64)lOm(I~Tr) and b=
0.125RTclpc' with
Heyen concluded that the proposed equation produces a satisfactory
agreement between the experimental VLE data and predictions. m=0.228165+0.791981w-0.648552w 2 +0.654505w 3 .

Watson et al. 18 re-evaluated the AL m correlation and proposed

Kubic EOS. Kubic IS presented a modification of Martin's 16 EOS
the expression
for calculating VLE. The modified equation predicts a variable crit-
ical compressibility factor for accurate description of real fluids. m=0.23489+0.67662w-0.13832w 2 . ................ (29)
Kubic proposed the following three-parameter cubic equation:
Watson et al. suggcsted that bctter vapor pressure predictions could
p=RTI(V-b)-al(V+c)2, be obtained with Eq. 29.
In terms of compressibility factor, Eq. 28 can be rewritten as
or in terms of Z,
Z3 +( -B-1)Z2 +AZ-AB=O,
Z3 +(2C-B-l)Z2 +(C2 -2CB-2C+A)Z
with the fugacity coefficient as
_(BC2 +C2 +AB)=O. In cPj=-ln(Z-B)+b;l(Z-B)+2AIZ.

340 SPE Reservoir Engineering, February 1988

 Adachi and Lu concluded that the modified van der Waals equa- For the PT EOS, the fugacity coefficients are given by
tion is capable of achieving the same satisfactory results for VLE
calculations as those obtained from the currently popular cubic EOS. RT In(cf>;) = -RTln(Z-B)+RTbJ(V-b)

PT EOS. Patel and Teja]9 proposed the following form of EOS: -(1PJd)ln[(Q+d)/(Q-d)] +am(b i +ci)/[2(Q2 -d 2 )]

p=RTI(V-b)-aal[V(V+b)+cev-b)], ............... (30) + [a ml(8d 3 )][Ci(3b m+cm)

where a, b, and c, are given by +b i(3c m+bm)]{ln[(Q+d)/(Q-d)] -2Qdl(Q2 -d 2)},

a=fla(RTc)2Ipc> ................................. (31) where

b=flbRTclpc, .................................... (32) V=ZRTlp,

and

and
where flc=I-3rc and fla=3r~+3(1-2rJflb+flb+(1-3U,
with rc=pcVcl(RTe)' and where fib is the smallest positive root
of the cubic
Patel and Teja concluded that the proposed EOS was capable of
yielding accurate VLE predictions while reproducing with sufficient
accuracy liquid- and vapor-phase densities.
An approximation value of fib is given by
SW EOS. Schmidt and Wenzel 20 proposed the following form of
fib =O.32429re -0.022005. EOS:

Finally, a is obtained from p=RTlev-b)-aa/[V2 +(\ +3w)bV-3wb 2 ], .......... (33)

where parameters a and b are defined by Eqs. 31 and 32 with

The PT equation therefore requires a knowledge offour charac-

r
terization constants Te , PC' e , and F for any fluid in interest. The
authors pointed out that rc is not equal to the experimental critical and
compressibility factor, but is obtained from one or more liquid den-
sity data points and F is obtained from the vapor pressure of the
pure fluids. They proposed the following generalized expressions
for the parameters: where (3(' is given by the smallest root of the equation

F=OA52413 + 1.30982w -0.295937w 2 (6w+ 1)(3? + 3(3~ + 3(3" -1 =0

and and

re =0.329032 -0.076799w+0.0211947w 2 .

Eq. 30 can be used for the calculation of mixture properties if An approximation value of (3 c is given by
the constants a, b, and c are replaced by the mixture constants am'
b m , and c m as follows: (3e =0.25989-0.0217w+O.00375w 2 .

The dimensionless factor a is a function of temperature and is given

am = L; L;Ziz/aiapia)0.5 (1- Kij), by Eq. 2. Schmidt and Wenzel propose the following expressions
I ] for the slope, m: for w:::;OA, m=m], and for w2:0.55, m=m2'
and for intermediate range (004 < w < 0.55),

m=[(w-OA)/0.15)m2 + [(0.55 -w)/0.15jm],

and
with

m] =mo+(1170)(5Tr - 3mo _1)2,

If we define m2 =mo+0.71(Tr-0.779)2,

mo=OA65+1.347w-0.528w 2 , for w:::;0.3671;

and
and mo=0.5361 +0.9593w, for w>0.3671.
C=cmPI(RT),
For supercritical compounds, the temperature dependence of the
Eq. 30 can be rearranged in terms of Z to produce parameter a is also dependent on w:

Z3 +(C-l)Z2 +(A - 2BC - B- C - B2)Z+(BC + B2 C - AB) =0. a=I-(OA774+1.328w)ln T" for Tr>l.

SPE Reservoir Engineering, February 1988 341

 TABLE 1-MOLAR COMPOSITIONS OF TABLE 2-H'tDROCARBON ANALYSIS OF
AVE CONDENSATE SYSTEMS LEAN-GAS SAMPLE

Temperature (OF) COmponent mol%

200 280 232 CO 2 o
N2 o
Component Cases 1 and 2 Case 3 Case 4 Cl 94.68
CO 2 0.0121 0.02371 0.0864 C2 5.27
N2 0.0194 . 0.0031 0.0071 C3 0.05
Cj 0.6599 0.7319 0]086 C4+ o
C2 0.0869 0.0780 0.0853
C3 0.0591 0.0355 .0.0495
i-C 4 0.0239 0.0071 0.0075 They expressed the parameters U and W in terms of w:
n-C 4 0.0278 0.0145 0.0125
i-C s 0.0157 0:0064 0.0041 U=1+3w
n-C s 0.0112 0.0068 0.0040
C6 0.018.1 0.0109 0.0046 and
C7 0.0144 0.0821 0.0061
Cs 0.0150 0.0071 W=-3w.
Cg 0.0105 0.0039
C1Q 0.0073 0.0028 In terms of Z. Eq. 33 can be written in the form:
C ll 0.0049 0.0020
C 12 0.0138 0.0015 Z3 +(UB-B-I)Z2 +(WBL UB2 -UB+A)Z
C 13 0.0011
C 14 0.0010
_(WB3 + WB2 +AB)=O.
C 15 0.0007
C 16 0.0005 where A and B aTe defined by Eqs. 8 and 9.
C 17 + 0.0037 Finally, they developed the following expression for the equilib-
rium ratios (K factors):

For mixtures, Sthmidt and Wenzel used the classic mixing rules
(Eqs. 10 and 11) for evaluating am and b m . The authors adopted where
the following mixing rules for the acentric factor: F(Zi) =RT In[RT(Z- B)lp] - [pbJ(Z- B)]

+ (2-.f;JD-am{3JD2)ln(GIH)
+ [(Ei +{3i)/G-(Ei -(3;)IH][a m /(2D)],

•
D EXP
o P-R
+ P-T
)C HEYEN
¢ KUBIC
V s-w

.. • SRI<.

... 1...
l~--------~----------'----------T----------~---------'
• ,...
aCPIBBL OJt DBW .oINT GAS
'I'" " ..
Fig. 1-Swollen volumes vs. cumulative gas injected during sw.elling with lean gas at 20QoF.

342 SPE Reservoir Engineering, February 1988

where
....
....
(3j =sb j + [9R/wj -w)(1 +w)bm]/(zjS), .H.

S =(iJ2 +4iJ-4)o.s, ~

f ....
G=ZRTlp+(iJ+s)b m I2, JIi
.: n ..
;:
f'"
D=Sb m •
H"
and
'"
Z
f
.... a EXP
o
; '"'
P-l
•t SIU<
ELF
Ie P-T
Schmidt and Wenzel concluded that the proposed EOS can rep- o HEYEN
• CHEVRON
lH.
resent the VLE of mixtures with accuracy similar to that of other • s-w
* PETEX
established EOS's and with superior accuracy for volumetric pre-
dictions.
...
Equations Studied and Data Sources •• HI lH' ,... UN ...
SCP/IIBL 0., DlIW PODfT GA.
The eight equations evaluated in this study are the PR, SRK, PT, .
SW, Heyen, Kubic, AL, and the UM EOS's. The selection ofthese Fig. 2-Dewpoint pressure vs. cumulative gas injected during
equations was influenced by my familiarity with certain equations swelling with lean gas at 200°F.
and interest in evaluating their predictive capability in comparison
with the established EOS's.
Ten condensate-hydrocarbon systems with experimental PVT 4. The predictive capabilities of selected EOS's were tested
measurements were used as a data base. The following procedure against measured PVT data for each condensate system. Predic-
was adopted throughout the study. tion accuracy for PVT properties was calculated for average absolute
1. The Riazi and Daubert 21 correlation was used to calculate deviations (AAD). Graphic presentation of results has been em-
physical properties of the plus-fraction. phasized throughout the study, and tabulations are given where ap-
2. Katz-and-Firoozabadi 22 -tabulated physical properties were propriate .
. assigned for all components between n-C s and the plus-fraction.
3. The interaction coefficient between methane and the heaviest Results and Discussion
component was then adjusted until a match of the measured dew- The following discussion is by no means a comprehensive study
point pressure was obtained. of the broad field of EOS' s as applied to complex multicomponent

• D EXP
0 P-R
A. P-T
+ s-w
X A-L

-.. ,
~
•
~•
" t

• , ... ,... I...

.+-------,--------r-------r------~------~------~
nBSSUItB. "IG
.... .... " ..
Fig. 3-Pressure/volume relations of reservoir fluid at 200°F, constant-companion expansion.

SPE Reservoir Engineering, February 1988 343

 . ••
.. o
am
p-it . a
o
EXP
P-it

..
A P-T 6 P-T
+)( + s-w
.
HEYEN
s-w ~ P-R and A-K
P-T and A-I
S-W and A-K

M .. to

I·
M

i
~ .
= II

= II

11
11

.,+---------,---------,---------~--~--~
• t ...
" .
.....SlID •••••
. .. ...
..
• 1111 ,...
.+---------,---------,---------.---~--~
.... ..
....SSlJlt••••••
Fig. 4-Graphic representation of constant-volume depletion
at 200°F. Fig. 5-Saturation vs. pressure, constant-volume depletion.

TABLE 3-EXPERIMENTAL AND PREDICTED K VALUES AT 700 psig AND 200°F, CASE 1

Component Experimental PR SRK PT Heyen Kubic SW AL

14.5
----
7.42
--
10.08
----
8.02 11.30
---
8.2
----
6.46
--
10.63
N2
C1 5.58 4.02 5.15 4.35 5.79 4.3 6.33 5.50
CO 2 3.0 2.06 3.3 2.1 2.49 2.2 2.8 2.5
C2 1.83 1.54 1.81 1.61 1.78 1.65 2.1 1.88
C3 0.88 0.80 0.86 0.81 0.87 0.85 0.99 0.90
i-C 4 0.54 0.48 0.51 0.48 0.52 0.51 0.55 0.51
n-C 4 0.44 0.40 0.41 0.40 0.44 0.43 0.45 0.42
i-C s 0.26 0.24 0.24 0.23 0.26 0.25 0.25 0.24
n-C s 0.22 0.21 0.20 0.20 0.23 0.22 0.21 0.20
Ca 0.12 0.12 0.11 0.11 0.13 0.13 0.11 0.11
C7 0.07 0.07 0.06 0.06 0.075 0.07 0.06 0.06
Ca 0.04 0.04 0.03 0.04 0.04 0.04 0.03 0.03
Cg 0.02 0.02 0.02 0.02 0.02 0.02 0.02 0.02
C1Q 0.01 0.01 0.01 0.01 0.01 0.01 0.01 0.01
Cn 0.002 0.002 0.001 0.003 0.004 0.004 0.001 0.001

hydrocarbon systems. Rather, the predictive capabilities of selected

1M equations have been tested and documented when used to predict
volumetric equilibrium and VLE of condensate systems.
Discussion of the predicted PVT properties of four condensate!
~2 hydrocarbon systems are presented in the following sections. Origi-
10
•o nal compositions of the fivc condensate systems are given in Table 1.

Case I-Rich-Gas Condensate: Third SPE Comparative Solution

Project. The first casc presented is a rich-gas condensate whose
measured and simulated phase behaviors are fully documented 23
in figures and tables by nine companies. The reported PVT data
c, include hydrocarbon samplc analyses, constant-expansion data,
'.1 C
o
EXP
constant-volume depletion data, and swelling data of four mixtures
f-R
I
••" c,
.to SRI<
+ P-T of reservoir gas with lean gas. The hydrocarbon analysis of the lean
'1- HEYEN
¢ KlfBIC
• gas is shown in Table 2.
.... V
•
s-w
A-L Figs. I and 2 show the swelling of reservoir gas with increasing
volumes of lean gas and the dewpoint pressure of these different
blends as predicted by various equations.
D
Fig. 2 also includes the graphic presentation of the swelling test
....II+-----,----"'T'----,---__,r_--"'T'----~--__,r_... as predicted by three different companies. Significant discrepancies
are seen in the predicted results, especially at higher lean-gas volume
Fig. 6-K values at 200°F and 714.7 psia. 22 injected. All equations tend to underestimate the dewpoint pressure
and overestimate the swollen volume. The SW EOS appears to give
the best prediction.
Fig. 3 shows pressure!volume data in constant-composition ex-
pansion of the reservoir gas at 200°F [93°C) as predicted by the

344 SPE Reservoir Engineering, February 1988

 TABLE 4-COMPARISON OF EXPERIMENTAL AND PREDICTED VAPOR MOLAL
COMPOSITIONS AND COMPRESSIBILITY FACTORS, CASE 1
Table Values Represent Percent AAD* in Compositions and Z

Z
3.2 1.1 4.5 1.44 1.1 7.7 2.1

Component PR PT Heyen Kubic SW AL SRK

CO 2 1.3 1.3 1.6 2.1 1.3 1.4 1.0
N2 0.8 0.9 1.4 1.5 1.1 1.5 1.2
C1 0.3 0.3 0.9 1.1 0.4 1.1 0.9
C2 0.52 0.7 0.6 0.5 0.54 0.5 0.5
C3 0.95 1.0 2.3 1.3 1.5 1.6 1.B
i-C 4 1.76 1.B 2.5 2.2 2.9 2.6 2.4
n-C 4 1.0 1.0 3.6 2.2 1.5 3.B 4.9
i-C 5 2.31 1.B 5.B 3.3 0.7 7.5 6.9
n-C 5 1.6 1.1 5.1 3.7 1.5 7.3 6.9
Cs 4.0 3.7 6.7 9.3 4.3 7.2 6.2
C7 4.7 4.5 6.1 13.2 4.8 12.6 10.3
Cs 5.5 4.8 10.0 14.1 5.5 14.3 11.6
Cg
C 10
C 11

'Number of data pOints =6.

"AAO stated only if experimental mol% > 0.5.

eight equations. All equations performed equally well for the pres- is summarized in Table 4. Overall, the PR, PT, and SW equations
sure/volume test and predicted the measured data accurately. give the best predicted VLE. PT and SW equations show excellent
Fig. 4 compares experimental and predicted liquid dropout by compressibility factor prediction with an average deviation of 1.1 %.
various EOS's. With the exception of the SW equation, all equa-
tions showed a large increase in predicted liquid volumes with pres-
sure just below the satUration point. For this spccific case, the SW Case 2-Rich-Gas Condensate: Moses-Wilson Hydrocarbon Sys-
equation exhibited an excellent agreement with the experimental tem. The hydrocarbon system reported in Case 1 was used by Moses
data. An attempt was made to recalculate the liquid-phase volumes and Wilson 25 to study experimentally and to evaluate the effec-
with the Alani-Kennedy24 density equation; the results are shown tiveness of N2 in maintaining reservoir pressure and displacing
in Fig. 5. The Alani-Kennedy equation is seen to yield better pre- condensate from the retrograde gas-condensate reservoir. A swelling
diction results when applied in conjunction with the EOS. test with N2 was conducted on a gas-condensate sample. The dew-
Experimental and predicted equilibrium ratios at 200°F and 714.7 point pressure was observed and the initial portion of the retrograde
psia [93°C and 4.93 MPa] are compared in Table 3 and presented curve was measured for each addition of N 2' The predicted and
graphically in Fig. 6. The K values predicted by various equations measured dewpoint pressures are shown in Fig. 7. The PT and SW
are all quite close to each other and to the experimental data. A EOS's give excellent predictions in comparison with experimental
moderate discrepancy can be observed between the various pre- data.
dicted K values of N2 and the plus fraction. The SW equation was selected to reproduce the measured retro-
For the hydrocarbon system considered in this study and for a . grade curves. As shown in Fig. 8 in this study and Fig. 3 in the
pressure up to 3,500 psia [24 MPa], the comparison in terms of Moses-Wilson 25 report, an excellent agreement with the ex-
the absolute average deviations in VLE and compressibility factors perimental retrograde curves was obtained.

.... ••
.. c
o
£xP. tOO
MIX #1 151 SCF /B8L

.... .,
~
+j/..
MIX 112
MIX #3
Mix 114
395 SCF/BBL
638 SCF /BBL
940 SCF /BBL

e .
"Ii
I 5.
0

I I . .' e
o
EXP
P-R
..:
E
",

§ n .. A
+
x.
P-T
KUBIC
s-w
II

¢ A-L
V SRI(
tI

II. .

, m _ _ _ rn _ _ _ _ ~
,+---,---,----r---r---r--~--_.--_.--_.r__,

SCJI/BBL U DBW PODIT G....

Fig. 7-Dewpoint elevation with N 2 • • 1. . . 'u•• ,... ....

.+-----~----~----~~~~----T4~--~----,~

PIl••SU••• ,.1,
" .. •••• . ..
Fig. 8-Retrograde liquid condensation for mixtures of con-
densate reservoir fluid and N 2' constant-volume depletion.

SPE Reservoir Engineering, February 1988 345

 III

.
.... +

. 11

n.'
II
!
M
tr.'
!
5Ie If

::> o EXP
~ 1t.' o P-R o EXP
A WHITSON o P-R
'" II
+
'1-
¢
P-T
KUBIC
s-w
a
+
P-T
KUBIC C7+
¥
... VSRl(
<J
s-w
SRI<
D
+

'.11 •
...
• +-----r----,----~----_r----,_~--~--~
• ".. '1M' nit
,... ..nl
. . .1111".••••••
_HI .... ....I+-------,...----.....----___---r-----.-------.
Fig. 9-Liquid dropout for North Sea gas condensate, Fig. 10-K values of the North Sea fluid at 280°F and 714.7
constant-volume depletion. psia.

Case 3-North Sea Condensate System: Whitson's Hydrocarbon Case 4-Pedersen-Thomassen-Fredenslund Condensate System.
Condensate Gas. This North Sea gas-condensate system was chosen Pedersen et al. 27 presented methods of characterizing heavy frac-
by Whitson 26 to illustrate the application of the two-parameter PR tions in hydrocarbon systems. They also reported the results of pres-
EOS to the hydrocarbon-condensate system. Whitson attempted sure/temperature flash calculations and three PVT experiments
more than 20 different adjustments of the C7+ characterization commonly performed on oil and gas mixtures. The experimental
procedure to improve liquid volume predictions. The author stated results are simulated by the authors with the SRK EOS. Because
that none of these adjustments were helpful in improving liquid the SRK equation produces severe errors for the liquid-phase
dropout predictions. volumes, they used various correlations for the liquid-phase molar
Fig. 9 shows liqUid dropout curves as predicted by the various volumes. Their correlations include: Peneloux et al. .5 Lee and
equations. Whitson's predictions are included for the sake of com- Kesler,28 Standing and Katz,29 and the Alani-Kennedy24 equation.
parison. In Fig. 9, the SW EOS shows a good volumetric match Results of Pedersen et al. 's simulation model for the constant-
with the experimental retrograde curve. The graphic presentation composition expansion test and predictions of selected equations
of the predicted and measured Kvalues is presented in Fig. 10 and (PR, PT, and SW) are presented in Fig. 12. The SW equation shows
tabulated in Table 5. For this hydrocarbon system, the SW equa- an excellent predictive capability for this system.
tion gives consistently higher predicted equilibrium ratios than other Table 7 summarizes the overall AAD in predicted vapor compo-
equations. Table 6 gives a summary of the average etrors in the sitions and compressibility factors for 10 condensate systems with
predicted vapor compositions and compressibility factors. In terms a total of 62 data points for vapor compositions and 69 data points
of AAD, the PT and SW equations give accurate predictions for for compressibility factors. A complete description of the program
the compressibility factors over the range of pressure considered and results of the study can be obtained from the author.
in the study (up to 7,000 psi r48 MPa]). All equations predicted
vapor composition equally well except for the C 7 + component.
Considerable deviation for the predicted molal fraction of C7+
TABLE 6-COMPARISON OF EXPERIMENTAL AND
could be attributed to its incorrect characterization in terms of critical
PREDICTED VAPOR MOLAL COMPOSITIONS AND
properties and acentric factor. COMPRESSIBILITY FACTORS FOR THE
Fig. 11 compares experimental and predicted North Sea gas NORTH SEA FLUID, CASE 3
FVF's as a function of pressure. The graphic presentation shows Table Values Represent Percent AAD*
an excellent match with experimental data for all equations. in Compositions and Z

Z
4.4 2.0 1.9 2.3 3.1
TABLE 5-EXPERIMENTAL AND PREDICTED K VALUES
AT 700 psig AND 280°F, CASE 3 Component PR SW PT SRK Kubic
--
Component Experimental PR PT Kubic SW SRK CO 2 0.50 0.5 0.40 1.2 1.2
-- -- -- -- -- N2 2.0 1.2 1.5 2.7 1.7
CO 2 4.86 4.53 4.75 3.38 5.03 5.74 C1 0.9 0.8 0.7 1.0 1.4
N2 19.41 18.23 19.97 12.82 23.35 21.57 C2 1.3 1.6 1.20 1.2 0.4
C1 7.27 9.12 10.18 6.50 11.54 9.08 C3 1.8 2.7 1.9 1.3 1.7
C2 2.70 3.32 3.61 2.56 4.03 3.30 i-C 4 2.3 2.6 2.5 2 2.7
C3 1.35 1.75 1.84 1.38 2.06 1.66 n-C 4 1.7 3.1 2.0 1.2 3.4
i-C 4 0.85 1.06 1.09 0.84 1.24 1.06 i-C s 2.1 4.0 2.4 2.1 4.9
n-C 4 0.74 0.89 0.90 0.72 1.0 0.84 n-C s 2.30 2.8 2.5 2.7 5.7
i-C s 0.45 0.52 0.52 0.44 0.58 0.52 Cs 3.9 3.9 3.9 5.5 8.1
n-C s 0.41 0.46 0.46 0.39 0.50 0.45 38.3 37.8 32.2 40.7 30.6
C7 +
Cs 0.34 0.25 0.25 0.23 0.27 0.24
C7+ 0.029 0.01 0.01 0.02 0.01 0.01 * Number of data points = 6.

346 SPE Reservoir Engineering, February 1988

 11
1.111'

1.12f1 C EXP
o ALL EQCATIO:.lS

1.1:11

I.ml

t.1111

t.""
I .•tst

o"EX>
I.n"

A
'-1.
sn
1.'l1t + r-t
II 'III'ZLOIJI.
<) su. aDd. L-1.
I.Utl ... s-w
• su. a" A-l
:*SUaIl4S-K
1.'Mt

I.'HI

'-H,. t,--,'"',''"=",--.,T"",,:----:,":':..':'",--.:':'..'"=",-----c.. T""

..- -• ..,.
. .- ,- - , ' ".'.,
·+.------IIT""H-----.-.~
••-----u.~'-----.H~.-----"T""••--~-.~•••
~SIlV.1I •••1. ...••uo.••••
Fig. 11-Gas FVF vs. pressure for the North Sea gas at 280°F. Fig. 12-Constant-composition expansion.

Conclusions ml,m2,{Je = constants for each component in the SW

The following conclusions and recommendations are drawn from EOS
studying the results of applying various EOS's to predict the volu- M, W,w = mixture parameters in the SW EOS
metric and phase behavior of 10 gas-condensate systems. P = pressure, psi [kPa]
1. The SW EOS exhibits a superior predictive capability for volu- Pc = critical pressure, psi [kPaj
metric properties of gas-condensate systems. Pr = reduced pressure, dimensionless
2. The PT and SW equations are found to give reliable gas com- R = universal gas constant
pressibility factor predictions. Sj = shift parameter
3. In terms ofVLE predictions, the PR, PT, and SW equations T = temperature, oR [Kj
all perform equally well.
Te = critical temperature, oR [K]
Tr = reduced temperature. dimensionless
Nomenclature V = molal volume
a,b = EOS constants Vcorr = corrected molal volume, ft 3 l1bm-mol
A,B,C.D = EOS constants [m 3 /kg-mol]
AI through A7 = constants in the Stein-Redlich-Kwong Vm = molar volume
EOS V sat = saturated volume, ft 3 lm 3 ]
ci = volume correction factor for Component i Vsat,orig = original saturated volume, ft3 lm 3 ]
d, Q = terms in the PT and the Heyen EOS' s xi = mole fraction of component in the liquid
Ei,G,H,s,{Ji = parameters in the SW EOS phase
F = variable in the PT EOS Yi = mole fraction of component in the gas
k,n,fJ = parameters in the Heyen EOS phase
kij = Daubert correlating parameter Zi = mole fraction of Component i in the gas
ko,kl ,X,Y = adjustable parameters or liquid phase
Kij = binary interaction coefficient Z = gas compressibility factor
m = characteristic constant Zc = critical compressibility factor

TABLE 7-0VERALL PERCENT AAD IN VAPOR COMPOSITIONS

AND COMPRESSIBILITY FACTORS

Z
2.4 1.5 1.6 2.3 2.4 4.8 10.7 5.3

Component PR SW PT SRK Kubic Heyen AL UM

-- - - -- -- -- --
CO 2 1.5 1.5 1.5 3.1 1.6 1.9 3.7 3.2
N2 2.6 1.8 2.3 4.6 1.6 4.9 2.4 3.8
C1 1.2 0.9 1.1 1.9 1.2 2.6 1.2 2.1
C2 1.10 1.5 1.2 0.9 0.9 1.5 1.7 1.9
C3 1.7 2.1 1.9 2.2 1.5 2.1 2.2 2.4
i-C 4 2.4 3.2 2.4 3.2 2.5 2.9 3.8 3.5
n-C 4 2.3 2.5 2.4 4.6 2.8 2.9 4.9 2.7
i-C s 3.9 2.8 3.9 7.3 4.1 4.3 8.6 5.3
n-C s 3.7 3.8 3.8 7.3 4.3 4.3 8.5 6.4
Cs 5.8 6.2 5.8 7.8 7.7 8.1 8.75 9.6
C7+ 7.2 7.4 7.1 8.9 9.2 9.7 10.3 10.7

SPE Reservoir Engineering, February 1988 347

 t = critical compressibility parameters
Z~ , c 14. Heyen, G.: "A Cubic Equation of State With Extended Range of Ap-
ZRA = Rackett compressibility factor plication," Proc., Second World Congress Chern. Eng., Montreal (Oct.
Ci = correction factor for EOS constant a
4-9, 1983).
15. Kubic, W.L. Jr.: "A Modification of the Martin Equation of State For
CiO,Ci I ,'Y0,'Y I ,w' = parameters in the Kubic EOS Calculating Vapor-Liquid Equilibria," Fluid Phase Equilibria (1982)
Ei = correlating parameter 9,79-97.
<Pi = fugacity coefficient of Component i 16. Martin, J.J.: "Cubic Equation of State-Which?," Ind. Eng. Chern.
1/; = mixture parameter as defined by Eq. 14 Fund. (1979) 18, 81-98.
17. Adachi, Y. and Lu, B.: "Simplest Equation of State for Vapor-Liquid
1/;i = parameter in the fugacity formula and Equilibrium Calculation: A Modification of the van der Waals Equa-
defined by Eq. 13 tion," AIChE J. (Nov. 1984) 30, No.6, 991-93.
na,nb'{lc,{ld = EOS constants 18. Watson, P. et al.: "Prediction of Vapor Pressures and Saturated Molar
w = acentric factor Volumes With a Simple Cubic Equation of State: Part II: The VAN
DER WAALS-711 EOS," Fluid Phase Equilibria Inti. J. (June 10,
1986) 27, 35-52.
Subscripts
19. Patel, N. and Teja, A.: "A New Equation of State For Fluids and Fluid
c = critical Mixtures," Chern. Eng. Sci. (1982) 37, No.3, 463-73.
i = component identifier 20. Schmidt, G. and Wenzel, H.: "A Modified van der Waals Type Equa-
j = component identifier tion of State," Chern. Eng. Sci. (1980) 135, 1503-12.
m = mixture 2!. Riazi, M.R. and Daubert, T.E.: "Prediction ofthe Composition ofPe-
troleum Fractions," Ind. Eng. Chern. Process Des. Dev. (1980) 19,
References 289-94.
22. Katz, D.L. and Firoozabadi, A.: "Predicting Phase Behavior of Con-
I. Soave, G.: "Equilibrium Constants from a Modified Redlich-Kwong densate/Crude-Oil Systems Using Methane Interaction Coefficients,"
Equation of State," Chern. Eng. Sci. (1972) 27, 1197-1203. JPT (Nov. 1978) 1649-55; Trans., ArME, 228.
2. Graboski, M. S.: •• A Modified Soave Equation of State for Phase Equi- 23. Kenyon, D.E. and Behie, A.: "Third SPE Comparative Solution Project:
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443. 24. Alani, G.H. and Kennedy, H.T.: "Volume of Liquid Hydrocarbons
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Eng. Sci. (1978) 33, 787. 288-95.
4. Elliot, J. and Daubert, T.: "Revised Procedure for Phase Equilibrium 25. Moses, P. and Wilson, K.: "Phase Equilibrium Considerations in Using
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OF (OF-32)/1.8
tion of State with New Mixing Rules for Strongly Nonideal Mixtures,"
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the Peng-Robinson Equation of State to Improve Volumetric Predic-
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12. Usdin, E. and McAuliffe, J.: "A One Parameter Family of Equations
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librium Calculations," Ind. Eng. Chern. Process Des. Dev. (1982) 21, ed at the 1986 SPE Annual Technical Conference and Exhibition held in New Orleans,
564-69. Oct. 5-6.

348 SPE Reservoir Engineering, February 1988