Chemical Geology 275 (2010) 148–160
Contents lists available at ScienceDirect
Chemical Geology
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / c h e m g e o
A model for single-phase PVTx properties of CO2–CH4–C2H6–N2–H2O–NaCl fluid
mixtures from 273 to 1273 K and from 1 to 5000 bar
Shide Mao a,b, Zhenhao Duan c,⁎, Jiawen Hu d, Dehui Zhang a,b
a
State Key Laboratory of Geological Processes and Mineral Resources, China University of Geosciences Beijing, 100083, China
School of Earth Sciences and Resources, China University of Geosciences, Beijing 100083, China
Key Laboratory of the Earth's Deep Interior, Institute of Geology and Geophysics, Chinese Academy of Sciences, Beijing 100029, China
d
College of Resources, Shijiazhuang University of Economics, Shijiazhuang 050031, China
b
c
a r t i c l e
i n f o
Article history:
Received 18 August 2009
Received in revised form 6 May 2010
Accepted 6 May 2010
Editor: D.B. Dingwell
Keywords:
CO2–CH4–C2H6–N2–H2O–NaCl
Fluid mixture
Volume
Equation of state
PVTx
a b s t r a c t
A thermodynamic model explicit in Helmholtz free energy is constructed to calculate the single-phase PVTx
properties of the CO2–CH4–C2H6–N2–H2O fluid mixtures. Parameters of the binary CO2–H2O, CH4–H2O, C2H6–
H2O and N2–H2O mixtures are regressed from assessed experimental data. On the basis of the binary mixture
parameters, the model can be used to predict the single-phase volumes of the CO2–CH4–C2H6–N2–H2O–NaCl
fluid mixtures with a simple approach. Comparison with a large number of experimental data shows that the
model can reproduce the single-phase PVTx properties of the CO2–CH4–C2H6–N2–H2O–NaCl fluid mixtures
from 273 to 1273 K and from 1 to 5000 bar, with or close to experimental accuracy. Online calculations can
be made on the website: www.geochem-model.org/.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
CO2–CH4–C2H6–N2–H2O–NaCl mixtures are typical geological
fluids in the Earth's crust. Thermodynamic properties of the mixtures,
especially vapor–liquid phase equilibria (Duan et al., 1995; Duan and
Sun, 2003; Mao et al., 2005; Duan and Mao, 2006; Mao and Duan,
2006) and pressure–volume–temperature–composition (PVTx) properties (Bakker, 1999; Duan et al., 1996, 2003; Mao and Duan, 2008),
are fundamental in the quantitative interpretation of boiling,
immiscibility, gas solubility, and fluid migration, and also in the
studies of fluid inclusions (Diamond, 2001; Dubessy et al., 1999, 2001;
Guillaume et al., 2003; Vasyukova and Fonarev, 2006). Due to highly
non-ideal mixing properties of the gas–H2O–NaCl fluid mixtures, it is
difficult to predict both the phase equilibria and volumetric properties
simultaneously with a single equation of state within experimental
uncertainty over a large temperature–pressure–composition region,
such as 273–1273 K, 1–5000 bar and 0–1 xNaCl (mole fraction of NaCl).
However, it is possible to use an equation of state to calculate the
volume of the gas–H2O–NaCl mixtures, and use another model to
predict the vapor–liquid phase equilibria. The main interest of the
⁎ Corresponding author.
E-mail address: [email protected] (Z. Duan).
0009-2541/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.chemgeo.2010.05.004
paper is to predict the molar volume of the CO2–CH4–C2H6–N2–H2O–
NaCl fluid mixtures, from which homogenization volume (or density)
and isochores can be obtained in the studies of the fluid inclusions.
Previously, Lemmon and Jacobsen (1999) developed a generalized
equation of state explicit in Helmholtz free energy to predict the
thermodynamic properties of mixtures containing CH4, C2H6, C3H8, nC4H10, i-C4H10, C2H4, N2, Ar, O2 and CO2 within the estimated accuracy
of experimental data. In that model, equations of state of pure fluid are
from those NIST recommends. However, H2O, as one of important
natural fluids, is not included in that model. Several years later, Paulus
and Penoncello (2006) extended the generalized model (Lemmon and
Jacobsen, 1999) to calculate the single-phase thermodynamic properties of CO2–H2O mixture. However, this model has systematic
volumetric deviations at low temperatures by comparison to experimental volumetric data, as discussed below. Recently, Driesner (2007)
presented a volumetric correlation for NaCl–H2O fluid covering a large
temperature–pressure–composition region of 273–1273 K, 0–
5000 bar and 0–1 xNaCl (mole fraction of NaCl). However, a volumetric
model of the multi-component gas–water–salt fluid mixtures like the
CO2–CH4–C2H6–N2–H2O–NaCl from 273 to 1273 K and pressures up to
5000 bar is still lacking.
In this study, the generalized equation of state (Lemmon and
Jacobsen, 1999) is extended to calculate the volume of the CO2–CH4–
C2H6–N2–H2O mixtures, and new parameters for the binary CO2–H2O,
CH4–H2O, C2H6–H2O and N2–H2O mixtures are evaluated. Combining
S. Mao et al. / Chemical Geology 275 (2010) 148–160
149
Table 1
Experimental measurements for the volumetric properties of the binary CO2–H2O, CH4–H2O, C2H6–H2O and N2–H2O fluid mixtures.
CO2–H2O mixture
References
T (K)
P (bar)
xCO2
Nd
Franck and Tödheide (1959)
Greenwood (1969)
Gehrig (1980)
Shmulovich et al. (1980)
Zawisza and Malesińska (1981)
Zakirov (1984)
Wormald et al. (1986)
Patel et al. (1987)
Patel and Eubank (1988)
Nighswander et al. (1989)
Crovetto et al. (1991)
Sterner and Bodnar (1991)
Crovetto and Wood (1992)
King et al. (1992)
Ohsumi et al. (1992)
Brodholt and Wood (1993)
Fenghour et al. (1996a)
Hnedkovsky et al. (1996)
Frost and Wood (1997)
Teng et al. (1997)
Seitz and Blencoe (1999)
Song et al. (2003a,b)
Hebach et al. (2004)
Li et al. (2004)
Duan and Zhang (2006)
673.15–1023.15
723.15–1073.15
673.15–773.15
673.15–773.15
373.15–473.15
573.15–673.15
473.2–773.2
323.15–498.15
323.15–498.15
352.85–471.25
651.1–725.51
673.15–973.15
622.75–642.7
288.15–298.15
276.15
662–1580
415.36–699.3
298.15–705.38
1473.15–1673.15
278–293
673.15
273.25–284.15
283.8–333.19
332.15
673.15–1273.15
300–2000
50–500
100–600
1000–4500
2.08–33.04
50–1800
10–120
1–100
0.8553–102.37
20.4–102.1
279.8–380.5
2000–6000
196.4–281.3
60.8–243.2
347.545
2250–38,590
58.84–345.77
10–350
9500–19,400
64.4–294.9
99.4–999.3
7–125
10.8–306.6
4.5–286.2
10,000
0.2–0.8
0–1
0.1–0.9
0–0.6232
0.0653–0.879
0.2–0.805
0.5
0.5–0.98
0.5–0.98
0.0022–0.0166
0.0013–0.01387
0.1234–0.8736
0.048–0.08745
0.02445–0.0307
0.001769–0.006164
0.25–0.75
0.2087–0.9388
0.002785–0.003322
0.108–0.787
0.025–0.0349
0.1–0.9
0.006195–0.02327
Saturated values
0–0.02428
0.25–0.75
303
869
198
46
142
159
115
423
297
33
94
92
77
27
5
38a
162
32
19
24
95
109
203
36
33a
References
T (K)
P (bar)
xCH4
Nd
Welsch (1973)
Christotoforakos (1985)
Sretenskaya et al. (1986)
Joffrion and Eubank (1988, 1989)
Shmonov et al. (1993)
Abdulagatov et al. (1993a,b)
Fenghour et al. (1996b)
Hnedkovsky et al. (1996)
Zhang (1997)
Zhang et al. (2007)
648.15–698.15
653.15–673.15
648.15–723.15
398.15–498.15
653–723
523.15–653.15
429.9–698.66
298.15–705.45
667.15–873.15
673–2573
400–2500
100–1000
250–2000
0.72–113.27
100–2000
22.4–632.4
74.91–303.82
280–350
1000–3010
500–100,000
0–1
0.4–0.9
0.1134–0.7477
0.5–0.9
0–1
0–1
0.324–0.924
0.0020–0.0025
0.051–0.166
0.2–0.8
Graph
Graph
189
169
126
162
87
31
42
990a
CH4–H2O mixture
C2H6–H2O mixture
References
T (K)
P (bar)
xC2H6
Nd
Lancaster and Wormald (1987)
473.2–773.2
10–120
0.5
115
References
T (K)
P (bar)
xN2
Nd
Japas and Franck (1985)
Abdulagatov et al. (1993a)
Fenghour et al. (1993)
480–673
523.15–663.15
428.74–697.89
155–2775
20.8–692.8
82.08–309.19
0.1–0.866
0.1173–0.9415
0.3593–0.9501
111
55
101
N2–H2O mixture
Note: T, P, and x refer to temperature, pressure and mole fraction, respectively, so is the same in Tables 5–6; Nd: Number of measurements.
a
Data are from molecular simulation.
the volumetric models of the NaCl–H2O (Driesner, 2007) and CO2–
CH4–C2H6–N2–H2O fluid mixtures with a simple method, the model
can be used to predict the single-phase molar volume of the CO2–CH4–
Table 2
Coefficients and exponents of mixture Eq. (4).
k
Nk
dk
tk
1
2
3
4
5
6
7
8
9
10
−2.45476271425D-2
−2.41206117483D-1
−5.13801950309D-3
−2.39824834123D-2
2.59772344008D-1
−1.72014123104D-1
4.29490028551D-2
−2.02108593862D-4
−3.82984234857D-3
2.69923313540D-6
1
1
1
2
3
4
5
6
6
8
2
4
−2
1
4
4
4
0
4
−2
C2H6–N2–H2O–NaCl fluid mixtures up to 1273 K and 5000 bar. The
framework of the article is as follows. First, the volumetric model in
Helmholtz free energy for the CO2–CH4–C2H6–N2–H2O fluid mixtures
is given in Section 2. Then in Section 3, a predictive volumetric model
for the CO2–CH4–C2H6–N2–H2O–NaCl mixtures is proposed, and its
accuracy is demonstrated by comparison with experimental data.
Table 3
Critical parameters of pure fluids.
i
Tci (K)
ρci (mol dm−3)
CO2
CH4
C2H6
N2
H2O
304.1282
190.564
305.322
126.192
647.096
10.624978698
10.139342719
6.87085454
11.1839
17.87371609
150
S. Mao et al. / Chemical Geology 275 (2010) 148–160
2. Volumetric model for the CO2–CH4–C2H6–N2–H2O fluid mixtures
Table 4
Parameters of the mixture Eqs. (6)–(8).
Binary
mixture
Fij
ζij
ςij (K)
(dm3 mol−1)
CO2–H2O
1.196
0.0108
C2H6–H2O
0.452
0.0108
N2–H2O
0.630
0.0060
CH4–H2O
1.309
0.0150
CH4–C2H6
1.0
0.0
CH4–CO2
0.808546 0.0
0.634182 0.00381045
CH4–N2
C2H6–N2
1.021463 0.00964861
C2H6–CO2 −0.154127 0.00951999
N2–CO2
2.780647 0.00659978
−223.33
−1715.00
−239.39
−265.76
0.0
−37.271180
−17.818676
−17.864694
−63.629672
−31.149300
βij
References
1.0
1.0
1.0
1.0
1.0
1.0
1.0
1.233477
1.0
1.0
This study
Lemmon
and Jacobsen
(1999)
Note: Subscript i refers to the first component and subscript j refers to the second
component, respectively.
2.1. Review of experimental volumetric data
On the experimental volumetric data of CO2–CH4–C2H6–N2 fluid
mixtures, detailed review can be seen from the study of Kunz et al.
(2007). Here we focus on the experimental volumetric data of the binary
aqueous systems: CO2–H2O, CH4–H2O, C2H6–H2O and N2–H2O (Table 1).
2.1.1. CO2–H2O system
From Table 1, it can be seen that there are many experimental
volumetric measurements on the CO2–H2O fluid mixtures. Over 3500
data points have been reported for this system, covering a large T–P–
xCO2range from 273 to 1673 K, 1 to 19,400 bar and 0 to 1xCO2. Most data
are consistent with each other. Experimental precision is about 0.1–
Table 5
Calculated volumetric deviations from experimental data.
CO2–H2O mixture
References
T (K)
P (bar)
xCO2
Nd
AAD (%)
MAD (%)
Greenwood (1969)
Gehrig (1980)
Shmulovich et al. (1980)
Zawisza and Malesińska (1981)
Zakirov (1984)
Wormald et al. (1986)
Patel et al. (1987)
Patel and Eubank (1988)
Nighswander et al. (1989)
Crovetto et al. (1991)
Sterner and Bodnar (1991)
King et al. (1992)
Ohsumi et al. (1992)
Fenghour et al. (1996a)
Hnedkovsky et al. (1996)
Frost and Wood (1997)
Teng et al. (1997)
Seitz and Blencoe (1999)
Song et al. (2003a,b)
Hebach et al. (2004)
Li et al. (2004)
Duan and Zhang (2006)
723.15–1073.15
673.15–773.15
673.15–773.15
373.15–473.15
573.15–673.15
473.2–773.2
323.15–498.15
323.15–498.15
352.85–471.25
651.1–725.51
673.15–973.15
288.15–298.15
276.15
415.36–699.3
298.15–705.38
1473.15–1673.15
278–293
673.15
273.25–284.15
283.8–333.19
332.15
673.15–1273.15
50–500
100–600
1000–4500
2.08–33.04
50–1800
10–120
1–100
0.8553–102.37
20.4–102.1
279.8–380.5
2000–6000
60.8–243.2
347.545
58.84–345.77
10–350
9500–19,400
64.4–294.9
99.4–999.3
7–125
10.8–306.6
4.5–286.2
10,000
0–1
0.1–0.9
0–0.6232
0.0653–0.879
0.2–0.805
0.5
0.5–0.98
0.5–0.98
0.0022–0.0166
0.0013–0.01387
0.1234–0.8736
0.02445–0.0307
0.001769–0.006164
0.2087–0.9388
0.002785–0.003322
0.128–0.787
0.025–0.0349
0.1–0.9
0.006195–0.02327
0.0032–0.03156a
0–0.02428
0.25–0.75
869
198
46
142
159
115
423
297
33
94
92
27
5
162
32
17
24
95
109
203
36
33
0.58
1.94
1.30
1.15
2.21
0.78
0.06
0.05
2.11
0.42
1.33
0.10
0.03
0.58
0.14
1.11
0.64
0.60
0.05
0.07
0.08
1.28
3.41
19.14
8.63
10.86
6.68
2.31
0.95
0.46
3.04
1.44
5.58
0.26
0.05
5.07
0.52
2.18
1.06
1.94
0.13
0.33
0.35
3.14
CH4–H2O mixture
References
T (K)
P (bar)
xCH4
Nd
AAD (%)
MAD (%)
Sretenskaya et al. (1986)
Joffrion and Eubank (1988, 1989)
Shmonov et al. (1993)
Abdulagatov et al. (1993a,b)
Fenghour et al. (1996b)
Zhang (1997)
Hnedkovsky et al. (1996)
Zhang et al. (2007)
648.15–723.15
398.15–498.15
653–723
523.15–653.15
429.9–698.66
667.15–873.15
298.15–705.45
673–1273
250–2000
0.72–113.27
100–2000
22.4–632.4
74.91–303.82
1000–3010
280–350
500–10,000
0.1134–0.7477
0.5–0.9
0–1
0–1
0.324–0.924
0.051–0.166
0.0020–0.0025
0.2–0.8
189
169
122
156
87
39
31
201
1.60
0.14
0.98
1.81
0.41
1.63
0.16
2.76
10.58
0.62
3.79
6.93
0.84
4.86
0.58
7.48
MAD (%)
C2H6–H2O mixture
References
T (K)
P (bar)
xC2H6
Nd
AAD (%)
Lancaster and Wormald (1987)
473.2–773.2
10–120
0.5
115
0.80
References
T (K)
P (bar)
xN2
Nd
AAD (%)
Japas and Franck (1985)
Abdulagatov et al. (1993a)
Fenghour et al. (1993)
480–673
523.15–663.15
428.74–697.89
155–2775
20.8–692.8
82.08–309.19
0.1–0.866
0.1173–0.9415
0.3593–0.9501
111
55
101
1.37
1.61
0.58
5.70
N2–H2O mixture
MAD (%)
7.06
7.03
4.58
AAD: Average absolute deviations calculated from this model; MAD: Maximum absolute deviations calculated from this model; Nd: Number of data points.
a
Solubility of CO2 is calculated from the model of Duan and Sun (2003).
Fig. 1. Comparisons with experimental data of liquid CO2–H2O mixture at low temperatures: Left side volume is calculated from the parameters of this study and right side volume is
calculated from the parameters of Paulus and Penoncello (2006). Vcal refers to calculated molar volume from this model, and Vexp refers to experimental molar volume. Vcal and Vexp
in Figs. 2–8 and 12 denote the same meaning.
S. Mao et al. / Chemical Geology 275 (2010) 148–160
151
152
S. Mao et al. / Chemical Geology 275 (2010) 148–160
Fig. 2. Volumetric deviations of this model from experimental data for CO2–H2O fluid
mixtures at high temperatures and pressures.
0.5% below 647 K, above which experimental uncertainties increase to
1%. Hu et al. (2007) reviewed in detail the volumetric data of the
binary system up to 647 K. It should be pointed out that the volumetric data of Nighswander et al. (1989) and Teng et al. (1997) below
647 K have large uncertainties. Above 647 K, data of Franck and
Tödheide (1959) are inconsistent with others' experimental measurements at the same T–P–xCO2 range, with about average deviation of 4%
from others. The data of Brodholt and Wood (1993) and Duan and
Zhang (2006) are from molecular-dynamics simulation and cover a
very large T–P–xCO2 space. All volumetric data listed in Table 1 but
those (Franck and Tödheide, 1959; Nighswander et al., 1989; Brodholt
and Wood, 1993; Frost and Wood, 1997; Teng et al., 1997; Duan and
Zhang, 2006) are used in the parameterization. The data of Duan and
Zhang (2006) and Frost and Wood (1997) at high temperatures and
pressures are not used in the parameterization but as a test of the
extrapolation ability of the volumetric model.
2.1.2. CH4–H2O system
Experimental volumetric data sets for the CH4–H2O fluid mixture are
also listed in Table 1. The data of Welsch (1973) and Christotoforakos
(1985) are in form of graphs and are not included in the parameterization. The data (Joffrion and Eubank, 1988, 1989; Fenghour et al.,
1996b; Hnedkovsky et al., 1996) are of high accuracy. The data of Zhang
et al. (2007) are from molecular simulation and cover a large T–P–xCH4
region. The PVTx data near critical region are difficult to measure and
contain large uncertainties. The data of Shmonov et al. (1993) at
1400 bar have a very large deviation. We compared the data to equation
of state of pure CH4 (Setzmann and Wagner, 1991), and found that the
deviation is over 8% at 1400 bar. Hence the data at 1400 bar are
corrected according to the equation of state of pure CH4 (Setzmann and
Wagner, 1991). The experimental volumetric data (Sretenskaya et al.,
1986; Joffrion and Eubank, 1988, 1989; Abdulagatov et al., 1993a,b;
Shmonov et al., 1993; Fenghour et al., 1996b; Hnedkovsky et al., 1996;
Zhang, 1997) are used in the parameterization. The data of Zhang et al.
(2007) are not used in the parameterization but as a test of extrapolation
ability of the volumetric model.
2.1.3. C2H6–H2O system
Only one experimental volumetric data set (Lancaster and
Wormald, 1987) is found for the C2H6–H2O mixtures up to 773.2 K
and 120 bar. Therefore, these data points (115) are used in the
parameterization of the model. Apparently, future experimental
measurements are needed for this binary system.
Fig. 3. Volumetric deviations of this model from experimental data for CH4–H2O mixtures.
2.1.4. N2–H2O system
Experimental volumetric data of the N2–H2O system (Japas and
Franck, 1985; Abdulagatov et al., 1993a; Fenghour et al., 1993) are not as
extensive as those of the CO2–H2O and CH4–H2O fluid mixtures. These
experimental data cover a T–P–xN2 range of 429–698 K, 21–2775 bar and
0.1–0.95, and are consistent with each other, so they are all used in the
parameterization.
2.1.5. Ternary systems
Two experimental data sets of CH4–C2H6–CO2 fluid mixtures (Hou
et al., 1996; McElroy et al., 2001) and one experimental data set of
CH4–CO2–N2 mixtures (Seitz and Blencoe, 1996) have been found in
literature. These data will be used as a test of the model predictability
in Section 2.3. For CO2–CH4–H2O mixture and other multi-component
systems, no PVTx data are found and experimental measurements are
needed for the test of the model.
2.2. Volumetric model in Helmholtz free energy
The volumetric model of the CO2–CH4–C2H6–N2–H2O fluid mixtures is in terms of dimensionless Helmholtz free energy α, defined as
α=
A
RT
ð1Þ
where A is molar Helmholtz free energy, R is molar gas constant
whose value is 8.314472 J mol−1 K−1, and T denotes temperature in K.
So are the same in the following equations.
S. Mao et al. / Chemical Geology 275 (2010) 148–160
τ=
153
Tc
T
ð5Þ
where ρ is the density of the mixtures, and ρc and Tc are defined as
"
n
ρc = ∑
i=1
n
n−1
n
xi
+ ∑ ∑ xi xj ζij
ρci
i=1 j=i + 1
n−1
Tc = ∑ xi Tci + ∑
i=1
n
∑
i=1 j=i + 1
β
#−1
xi ij xj ςij
ð6Þ
ð7Þ
where ρci and Tci are the critical density and critical temperature of the
component i, respectively, xj denotes mole fraction of the component
Fig. 4. Volumetric deviations of CH4–H2O mixtures in high temperature–pressure region.
The dimensionless Helmholtz free energy α of the mixture is
represented by
id
E
α = αm + α
ð2Þ
where αid
m is the dimensionless Helmholtz free energy of an ideal
mixture and αE is the excess dimensionless Helmholtz free energy. αid
m
comes directly from the fundamental equations of pure fluids and can
be written as
n
r
0
αid
m = αm ðδ; τ; xÞ + ∑ xi αi ðδ; τÞ
i=1
h
i
n
n
0
r
= ∑ xi αi ðδ; τÞ + lnðxi Þ + ∑ xi αi ðδ; τÞ
i=1
ð3Þ
i=1
where α0m is an ideal-gas part of dimensionless Helmholtz free energy
of the mixture, α0i is the ideal-gas part of dimensionless Helmholtz
free energy of component i, αri is a residual part of dimensionless
Helmholtz free energy of component i, xi is mole fraction of the
component i. id in superscript denotes ideal mixing. 0 and r in the
superscripts denote the ideal-gas part and the residual part of
dimensionless Helmholtz free energy, respectively. i and m in
subscripts denote the component and mixtures, respectively. δ and
τ are reduced parameters, which are defined by
δ=
ρ
ρc
ð4Þ
Fig. 5. Volumetric deviations of this model from experimental data for C2H6–H2O fluid
mixtures.
Fig. 6. Volumetric deviations of this model from experimental data for N2–H2O fluid mixtures.
154
S. Mao et al. / Chemical Geology 275 (2010) 148–160
Values of the binary parameters (ζij, ςij, βij and Fij) in the above
equations for the CO2–CH4–C2H6–N2 mixtures are from the model
(Lemmon and Jacobsen, 1999), and the parameter values of the binary
CO2–H2O, CH4–H2O, C2H6–H2O and N2–H2O mixtures are determined
by a regression to experimental volumetric data analyzed above. In
this article, equations of state for pure CO2, CH4, C2H6, N2 and H2O
fluids are from references (Setzmann and Wagner, 1991; Span and
Wagner, 1996; Span et al., 2000; Wagner and Pruß, 2002; Bucker and
Wagner, 2006). These equations of state are all explicit in dimensionless Helmholtz energy and are considered the best equations of
these pure fluids. Critical parameters of the pure CO2, CH4, C2H6, N2
and H2O fluids are listed in Table 3.
2.3. Parameterization and comparisons
As mentioned above, the values of ζij, ςij, βij and Fij for the CO2–H2O,
CH4–H2O, C2H6–H2O and N2–H2O mixtures are determined by a nonlinear regression to experimental volumetric data. Regressed parameters of the CO2–H2O, CH4–H2O, C2H6–H2O and N2–H2O mixtures
and those of the CO2–CH4–C2H6–N2 mixtures are listed in Table 4. The
molar volume of the CO2–CH4–C2H6–N2–H2O mixtures can be
calculated from Eq. (9) with the Newton iterative method. If the
CO2–CH4–C2H6–N2–H2O fluid mixtures are in vapor or supercritical
state, we can set the initial value of density for that of ideal gas. If the
CO2–CH4–C2H6–N2–H2O fluid mixtures are in liquid state, saturated
liquid density value of the pure component with the highest critical
temperature can be set as the initial density value.
r ∂α
P = ρRT 1 + δ
∂δ τ
ð9Þ
n
where P is pressure, and αr = ∑ xiαri (δ,τ) + αE(δ,τ,x).
i=1
Fig. 7. Volumetric deviations from experimental data for CH4–C2H6–CO2 fluid mixtures.
j, and ζij, ςij, and βij are mixture-dependent binary parameters of the
components of i and j.αE of equation of (2) is given by
E
n−1
α = ∑
n
∑
i=1 j=i + 1
10
d
t
xi xj Fij ∑ Nk δ k τ k
k=1
ð8Þ
where values of parameters Nk, dk and tk are independent of the fluid
and are from the general model (Lemmon and Jacobsen, 1999)
(Table 2), Fij is a mixture-dependent binary parameter of the components of i and j.
Fig. 8. Volumetric deviations from experimental data for CH4–CO2–N2 fluid mixtures.
With the above parameters, the molar volumes of the CO2–H2O,
CH4–H2O, C2H6–H2O and N2–H2O mixtures can be calculated. Table 5
shows the average and maximum absolute deviations of the model
from each data set. Fig. 1 shows the comparisons between experimental results and model predictions for the CO2–H2O system at
low temperatures. It can be seen in Fig. 1 that the molar volumes
calculated from the parameters of the model (Paulus and Penoncello,
2006) have systematic negative deviations. In order to test the model
predictive ability, we compared the model to the data (Frost and
Wood, 1997; Duan and Zhang, 2006) up to 1673 K and 19,400 bar
(Fig. 2), which shows average absolute deviation of about 2%,
indicating the good extrapolation ability of the model. Figs. 3–4
show the volumetric deviations from experimental data for the CH4–
H2O mixture. Figs. 5–6 show the volumetric deviations from experimental data for the C2H6–H2O and N2–H2O mixtures, respectively. For
the CH4–H2O and N2–H2O mixture, some deviations from experimental data near the critical regions are very large, where experimental
measurements are difficult to do and volumetric data in themselves
contain large uncertainties. So the model prediction is also poor in
these regions. In addition, experimental volumetric data of the CH4–
C2H6–CO2 (Hou et al., 1996; McElroy et al., 2001) (Fig. 7) and CH4–
CO2–N2 (Seitz and Blencoe, 1996) (Fig. 8) mixtures are also compared,
which show good agreement. As seen from these figures, the molar
volume of the CO2–CH4–C2H6–N2–H2O mixtures up to 1273 K and
10,000 bar can be reproduced by this generalized model within or
near experimental uncertainties.
In the study of fluid inclusions, isochores are important for
obtaining the temperature and pressure of fluids trapped in minerals,
which can be calculated from above model. Figs. 9–11 show the
calculated isochores of the CO2–H2O, CH4–H2O and CH4–CO2 systems,
respectively. From Figs. 9–10, it can be seen that the isochores of the
CO2–H2O and CH4–H2O mixtures at low temperatures are a bit curved
and almost linear in other regions. Experimental isochores of the CO2–
S. Mao et al. / Chemical Geology 275 (2010) 148–160
H2O mixture (Sterner and Bodnar, 1991) are compared in Fig. 9a,b. It
can be seen that the experimental isochores, whose valid temperature–pressure–composition range is 673–973 K, 2000–6000 bar, and
0–1 mole fraction of CO2 are in good agreement with our calculated
results. The iso-Th lines, which can be approximated as isochores,
reported by Lin and Bodnar (2010) for the CH4–H2O mixture are also
plotted for comparison in Fig. 10a,b. The experimental equation of isoTh lines is valid from the bubble point to 773 K, 3000 bar and
compositions ≤4 mol% CH4. It can be seen from Fig. 10a, b that the
calculated isochores from this study are quite close to the iso-Th lines.
From Fig. 11, it can be seen that the isochores of the CH4–CO2 mixture
from 273 to 1273 K are curved similar to those of pure CO2 fluid.
3. Volumetric model for the single-phase
CO2–CH4–C2H6–N2–H2O–NaCl fluid mixtures
H ONaCl
where V is molar volume of the gas–H2O–NaCl fluid mixtures, VH2ONaCl
is the molar volume of the binary H2O–NaCl solution calculated
from the model (Driesner, 2007), xH2ONaCl denotes the mole fraction
of the H2O–NaCl components, xgas denotes the mole fraction of gas
(pure CO2, CH4, C2H6, N2 or their mixtures) in gas–H2O–NaCl fluids,
2 ONaCl
and VH
refers to apparent molar volume of gas in the H2O–NaCl
gas, ϕ
fluids.
Usually, gas (CO2, CH4, C2H6, N2) solubility in water is not big. If
NaCl is added, the gas solubility will decrease rapidly with increasing
salinity. Therefore, the possibilities of gas–Na+ and gas–Cl− pairs
appearing in fluids are very small. In this case, the effect of NaCl on
2 ONaCl
2 ONaCl
VH
can be neglected. That is, VH
can be approximated as the
gas, ϕ
gas, ϕ
2O
apparent molar volume of the gas in water (VH
gas, ϕ). Eq. (10) is then
changed for
H O
CO2–H2O–NaCl and CH4–H2O–NaCl are the most frequently
encountered gas–water–salt natural systems. However, Up to now,
no a volumetric model of the gas–H2O–NaCl mixtures is valid for the
T–P–xNaCl region of 273–1273 K, 1–5000 bar and 0–1 xNaCl. When the
gas–H2O–NaCl mixtures are in vapor state, NaCl content is very small.
Hence, the fluids can be approximated as salt-free mixtures, and the
molar volume can be calculated from above equations. When the gas–
H2O–NaCl mixtures are in liquid or in supercritical state, the molar
volume is obtained from the following equation:
2
V = xH2 ONaCl VH2 ONaCl + xgas Vgas;ϕ
155
ð10Þ
2
V = xH2 ONaCl VH2 ONaCl + xgas Vgas;ϕ
ð11Þ
2O
where VH
gas, ϕ is calculated from the above salt-free PVTx model. For the
CO2–H2O–NaCl mixture where xCO2b0.05, the approximation proves
to be very good below 647 K (Duan et al., 2008). In order to test the
validity of Eq. (11), the experimental single-phase volume data of the
CO2–H2O–NaCl mixture (Gehrig et al., 1986; Nighswander et al., 1989;
Johnson, 1992; Schmidt et al., 1995; Teng and Yamasaki, 1998; Li et al.,
2004; Song et al., 2005) are compared with the model. Table 6 lists the
average and maximum absolute deviations and Fig. 12 shows the
volumetric deviations of Eq. (11) from experimental data of the CO2–
H2O–NaCl mixtures. It can be seen from Table 6 and Fig. 12 that the
agreement is excellent. Fig. 13 shows that the experimental single-
Fig. 9. Isochores of CO2–H2O system: Isochores are calculated from this model; Bubble point curve and dew point curve above 523 K are from the model of Mao et al., 2009, and those
below 523 K are from the model of Duan and Sun, 2003; ● is from experimental isochores (Sterner and Bodnar, 1991); unit of Vm is cm3 mol−1.
156
S. Mao et al. / Chemical Geology 275 (2010) 148–160
Fig. 10. Isochores of the CH4–H2O system: Isochores are calculated from this model; Bubble point curve and dew point curve above 523 K are from the model of Mao et al., 2009, and
those below 523 K are from the model of Duan and Mao, 2006; ■ is from experimental iso-Th lines (Lin and Bodnar, 2010); unit of Vm is cm3 mol−1.
phase volume data of the CH4–H2O–NaCl fluid mixture (Krader and
Franck, 1987) at 800 K are also in good agreement with the model.
However, when compared to the saturated density of the CO2–H2O–
NaCl (Gehrig et al., 1986), CH4–H2O–NaCl (Krader and Franck, 1987)
and C2H6–H2O–NaCl (Michelberger and Franck, 1990) mixtures, large
deviations are found. For example, the average absolute deviation of
the saturated experimental density data (total 138 data points) is
3.85% for the CH4–H2O–NaCl system (Krader and Franck, 1987).
Therefore, Eq. (11) can only be used to predict the single-phase
volume of the gas–H2O–NaCl fluid mixtures. Fig. 14 shows the
isochores and the lines of equal homogenization temperature at
constant compositions (Schmidt and Bodnar, 2000) for the CO2–H2O–
NaCl fluid mixtures. Apparently, the results from Eq. (11) deviate
significantly from those of Schmidt and Bodnar (2000) in some
regions. Therefore, the isochores calculated from Eq. (11) may be only
used as an approximation in these regions, but the trend of the
isochores is correct. Because no experimental volumetric data of other
single-phase gas–H2O–NaCl mixtures have been found, future
experimental work for those mixtures is needed to further validate
Eq. (11).
Limitation of the model: The above discussion suggests that
although the model extrapolates well, we should pay attention to the
applicable range of the model. For the CO2–CH4–C2H6–N2–H2O
mixtures, it can be safely used to calculate the single-phase molar
volumes in the temperature–pressure region that equation of state of
pure fluids cover, beyond which it can be used up to 1273 K and
10,000 bar with slightly lower accuracy. For the CO2–CH4–C2H6–N2–
H2O–NaCl mixtures, it can only be used to calculate the single-phase
PVTx properties of gas–H2O–NaCl mixtures from 273 to 1273 K and
pressures to 5000 bar.
4. Conclusions
A generalized thermodynamic mixture model based on Helmholtz
free energy proposed by Lemmon and Jacobsen (1999) is extended to
calculate the molar volume of the CO2–CH4–C2H6–N2–H2O fluid
mixtures. With a simple approximate method, the model can be
used to predict the single-phase volumes or densities of the CO2–CH4–
C2H6–N2–H2O–NaCl fluid mixtures. It shows that the model can
reproduce the volume of the CO2–CH4–C2H6–N2–H2O–NaCl fluid
mixtures from 273 to 1273 K and from 1 to 5000 bar with or close
to experimental accuracy. The model for the CO2–CH4–C2H6–N2–H2O–
NaCl fluid mixtures established here is very useful for calculating the
isochores of the corresponding gas–water–salt fluid inclusions.
Acknowledgements
We thank the anonymous reviewers for the constructive suggestions. This work is supported by Zhenhao Duan's “Key Project Funds”
(#90914010) awarded by the National Natural Science Foundation of
China, “Major Development Funds” (#:kzcx2-yw-124) by the Chinese
Academy of Sciences, the Open Foundation of the State Key Laboratory
of Oil and Gas Reservoir Geology and Exploitation (PLC200701) and
the Natural Science Foundation of Hebei Province (D2008000535).
S. Mao et al. / Chemical Geology 275 (2010) 148–160
Fig. 11. Isochores of the CH4–CO2 system: Unit of Vm is cm3 mol−1.
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