Chemical Geology 347 (2013) 43–58
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Chemical Geology
journal homepage: www.elsevier.com/locate/chemgeo
An improved model for calculating CO2 solubility in aqueous NaCl
solutions and the application to CO2–H2O–NaCl fluid inclusions
Shide Mao ⁎, Dehui Zhang, Yongquan Li, Ningqiang Liu
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
a r t i c l e
i n f o
Article history:
Received 6 July 2012
Received in revised form 17 March 2013
Accepted 21 March 2013
Available online 29 March 2013
Editor: D.B. Dingwell
Keywords:
Equation of state
Solubility
CO2–H2O–NaCl
Fluid inclusion
Isochore
Homogenization pressure
a b s t r a c t
To determine compositions, homogenization pressures and isopleths of CO2–H2O–NaCl fluid inclusions, an improved activity–fugacity model is developed to calculate CO2 solubility in aqueous NaCl solutions. The model
can predict the CO2 solubility in aqueous NaCl solutions from 273.15 K to 723.15 K, from 1 bar to 1500 bar
and from 0 to 4.5 mol kg−1 of NaCl, within or close to experimental uncertainties. Compared to a large number
of reliable experimental solubility data available, the average absolute deviation is 4.62% for the CO2 solubility in
aqueous NaCl solutions. In the near-critical region, the calculated CO2 solubility deviations increase to over 10%.
The CO2 solubility model, together with the updated volumetric model of CO2–H2O–NaCl fluid mixtures, is
applied to calculate the CO2 contents, homogenization pressures, molar volumes and volume fractions of the
CO2–H2O–NaCl fluid inclusions by an iterative method based on the assumption that the volume of an inclusion
keeps constant during heating and cooling. Calculation program code of the CO2 solubility in aqueous NaCl solutions can be obtained from Chemical Geology or the correspondence author ([email protected]).
© 2013 Elsevier B.V. All rights reserved.
1. Introduction
Fluid inclusions approximated by the system CO2–H2O–NaCl are
common in many different geologic environments, e.g., hydrothermal
ore deposits (Roedder, 1984; Bodnar, 1995; Roedder and Bodnar, 1997;
Yoo et al., 2011) and metamorphic rocks (Crawford, 1981; Touret,
1981, 2001; Cuney et al., 2007). In the studies of CO2–H2O–NaCl fluid inclusions, isochores of CO2–H2O–NaCl inclusions can be calculated from
PVTx models (Brown and Lamb, 1989; Bakker, 1999; Mao et al., 2010),
but the inclusion compositions and homogenization pressures must be
known before constructing isochores. This information is often obtained
by combining experimental microthermometric and Raman analysis
with thermodynamic phase-equilibrium models (Diamond, 2003). For
three-phase (liquid CO2 + vapor CO2 + liquid H2O) CO2–NaCl–H2O
inclusions at room temperatures, their salinities (NaCl contents) can be
satisfactorily calculated from the equation of Chen (1972) or Darling
(1991). For two-phase (liquid or vapor CO2 + liquid H2O) CO2–NaCl–
H2O inclusions at room temperatures, their salinities can be obtained
from the equations of Diamond (1992) or from combining Raman
analysis (Fall et al., 2011; Wang et al., 2011) with thermodynamic
models (Bakker et al., 1996; Bakker, 1997, 2003; Duan and Sun, 2006).
However, how to determine the CO2 contents and homogenization
⁎ Corresponding author at: State Key Laboratory of Geological Processes and Mineral
Resources, China University of Geosciences, Beijing, 100083, China.
E-mail address: [email protected] (S. Mao).
0009-2541/$ – see front matter © 2013 Elsevier B.V. All rights reserved.
http://dx.doi.org/10.1016/j.chemgeo.2013.03.010
pressures of CO2–H2O–NaCl inclusions is still a challenge. Several
workers have shown recently that it is possible to determine the CO2
(or CH4) content of volatile-bearing fluid inclusion using a combination
of microthermometry, Raman analysis, and equations of state to uniquely define the fluid inclusion composition (Azbej et al., 2007; Becker et al.,
2010; Fall et al., 2011; Mao et al., 2013). To obtain the CO2 contents and
homogenization pressures of CO2–H2O–NaCl inclusions, it requires an
accurate CO2 solubility model covering a large temperature, pressure,
and ionic strength (T − P − mNaCl) range. Up to now, accurate models
are still lacking for calculating CO2 solubility in aqueous NaCl solutions
over a wide T − P − mNaCl space, and those available are inconvenient
to use for the inclusion researchers.
During the past century, numerous thermodynamic models have been
published to calculate CO2 solubility in aqueous NaCl solutions (Barton
and Chou, 1993a, 1993b; Duan and Sun, 2003; Akinfiev and Diamond,
2010), which will not be listed here in detail. In more recent years, several
models have been reported to calculate CO2 solubility in aqueous NaCl solutions (Sorensen et al., 2002; Duan and Sun, 2003; Dubessy et al., 2005; Ji
et al., 2005; Portier and Rochelle, 2005; Spycher and Pruess, 2005, 2010;
Duan et al., 2006; Marin and Patroescu, 2006; Bahadori et al., 2009;
Akinfiev and Diamond, 2010; Darwish and Hilal, 2010). Table 1 lists the
valid T − P − mNaCl range of each model. Sorensen et al. (2002) modeled
CO2 solubility in pure water (348–623 K and 16–1400 atm) and aqueous
NaCl solutions (298–523 K, 1–1382 atm and 0.46–5.70 mol kg−1). However, the average deviation of the calculated CO2 solubility from experimental measurements is up to 37.0% for the CO2–H2O system and up to
44
S. Mao et al. / Chemical Geology 347 (2013) 43–58
Table 1
Thermodynamic models for calculating CO2 solubility in aqueous NaCl solutions since
2000.
References
T (K)
P (bar)
mNaCl(mol kg−1)
Sorensen et al. (2002)
Duan and Sun (2003, 2006)
Dubessy et al. (2005)
Ji et al. (2005)
Portier and Rochelle (2005)
Spycher and Pruess (2005, 2010)
Marin and Patroescu (2006)
Bahadori et al. (2009)
Akinfiev and Diamond (2010)
Darwish and Hilal (2010)
298–523
273–533
273–543
298–373
273–573
285–573
273–433
300–400
251–373
300–500
1–1382
1–2000
1–300
1–200
1–300
1–600
1–90
50–700
1–1000
50–2000
0.46–5.70
0–4.3
0–6.28
0–6
0–3
0–6
0–6
0–4
0–6
1–4
20.3% for the CO2–H2O–NaCl system. Duan and Sun (2003) and Duan
et al. (2006) developed a model to predict CO2 solubility in aqueous
NaCl solutions (273–533 K, 0–2000 bar and 0–4.3 mol kg−1 NaCl),
which has been widely used in CO2 sequestration. This model has limited
applicability to studies of CO2–H2O–NaCl inclusions because many such
inclusions homogenize above 533 K (260 °C). Dubessy et al. (2005)
presented an unsymmetric thermodynamic model for the liquid–vapor
equilibria of CO2–H2O–NaCl system up to 543 K and high ionic strength.
However, the equation is only valid to 300 bar owing to the use of a
cubic equation of state for the vapor phase. Ji et al. (2005) used SAFT1RPM approximations to calculate the phase equilibrium and density of
CO2–H2O–NaCl system. The valid P-T conditions of the model are relevant
to geological CO2 sequestration environments. Spycher and Pruess (2005,
2010) developed a phase-partitioning model for CO2-brine mixtures at elevated temperatures and pressures. The model is valid for geological CO2
sequestration but is only valid for homogenization pressures ≤600 bar.
Portier and Rochelle (2005) presented a thermodynamic model to calculate CO2 solubility in aqueous NaCl solutions with a valid T − P − mNaCl
range of 273–573 K, 1–300 bar and 0–3 mol kg−1. This model is also
valid for CO2 sequestration environments, similar to other models listed
above. Marin and Patroescu (2006) presented a semi-empirical model
to calculate CO2 solubility in aqueous NaCl solutions up to 6 molality of
NaCl, but the calculated solubility deviations are around 10% up to
433 K and 90 bar. Bahadori et al. (2009) established a polynomial equation to predict CO2 solubility in aqueous NaCl solutions, covering a valid
T − P − mNaCl range of 300–400 K, 50–700 bar and 0–4 mol kg−1.
Owing to the valid PTx range, this model has only limited applicability
to fluid inclusion studies. Akinfiev and Diamond (2010) developed a
semi-empirical model to calculate thermodynamic properties of aqueous
liquid in the ternary CO2–H2O–NaCl system. However, the highest valid
temperature of this model is 373 K, which is applicable to CO2 sequestration environments, but of only limited use for interpreting fluid inclusion
measurements. Darwish and Hilal (2010) developed a simple model
to predict CO2 solubility in the H2O–NaCl system that is valid from
300–500 K, 50–2000 bar and 1–4 mol kg−1. Because the highest applicable temperature of the model is 500 K, the model has only limited
applicability to fluid inclusions, and it is more appropriate for the CO2
sequestration environments. The models described above can be used
to predict properties of CO2–H2O–NaCl fluids, but most of them have
only limited applicability to fluid inclusions owing to the relatively
low temperatures and/or pressures to which they are valid. Many
CO2–H2O–NaCl fluid inclusions homogenize at temperatures in excess
of 300 °C (573.15 K), and none of the listed models are valid above
this temperature.
To overcome the deficiencies of the previous models in applications
to fluid inclusions, we present an improved model to calculate CO2 solubility in pure water and aqueous NaCl solutions (273.15–723.15 K,
1–1500 bar and 0–4.5 molality of NaCl) by improving the theoretical
approach and using updated experimental data. The framework of the
model is as follows: First, the experimental solubility data are briefly
reviewed. Then a thermodynamic model for CO2 solubility in aqueous
NaCl solutions is presented. Finally, the applications of the model to
CO2–H2O–NaCl fluid inclusions are discussed in detail.
2. CO2 solubility in aqueous NaCl solutions
2.1. Review of the CO2 solubility data
The CO2 solubility in pure water and aqueous NaCl solutions has been
measured over a wide T − P − mNaCl range (Tables 2 and 3). The solubility measurements of CO2 in water are extensive. Since 1900, over 80
data sets have been reported for the CO2–H2O system (Table 2). Carroll
et al. (1991) surveyed the solubility data of CO2 in water at low pressures
(b10 bar) before 1991. Diamond and Akinfiev (2003) provided detailed
summary of solubility data of CO2 in water from −1.5 to 100 °C and from
0.1 to 100 MPa. Duan and Sun (2003) and Duan et al. (2006) also evaluated the solubility data of CO2 in pure water and aqueous NaCl solutions.
We find that most of the data sets are consistent with each other. A
few data points of Drummond (1981) measured during the pressure
Table 2
CO2 solubility measurements in water since 1900.
References
T (K)
P (bar)
Nd
Hantzsch and Vagt (1901)
Findlay and Creighton (1910)
Findlay and Shen (1912)
Sander (1911–1912)
Findlay and Williams (1913)
Findlay and Howell (1915)
Kunerth (1922)
Buch (1928)
Morgan and Pyne (1930)
Morgan and Maass (1931)
Kobe and Williams (1935)
Kritschewsky et al. (1935)
Shedlovsky and MacInnes (1935)
Orcutt and Seevers (1937)
Zel'venskii (1937)
Curry and Hazelton (1938)
Wiebe and Gaddy (1939)
Van Slyke (1939)
Wiebe and Gaddy (1940)
Markham and Kobe (1941)
Wiebe and Gaddy (1941)
Harned and Davis (1943)
Koch et al. (1949)
Morrison and Billett (1952)
Bartholome and Friz (1956)
Dodds et al. (1956)
Ellis (1959)
Malinin (1959)
Bartels and Wrbitzky (1960)
Novák et al. (1961)
Austin et al. (1963)
Ellis and Golding (1963)
Tödheide and Franck (1963)
Takenouchi and Kennedy, (1964)
Yeh and Peterson (1964)
Takenouchi and Kennedy (1965)
de Khanikof and Louguinine (1967)
Vilcu and Gainar (1967)
Matous et al. (1969)
Power and Stegall (1970)
Stewart and Munjal (1970)
Barton and Hsu (1971)
Li and Tsui (1971)
Murray and Riley (1971)
Weiss (1974)
Zawisza and Malesińska (1981)
Cramer (1982)
Shagiakhmetov and Tarzimanov (1982)
Briones et al. (1987)
Nakayama et al. (1987)
Postigo and Katz (1987)
273.15–363.15
298.15
298.15
308.15–375.15
298.15
298.15
293.15–307.15
292.95–293.95
298.15
273.15–298.15
298.15
293.15–303.15
298.15
298.15
273–373
298.15
323.15–373.15
295.95–298.65
291.15–313.15
273.35–313.15
298.15–348.15
273.15–323.15
291.45–295.05
286.45–347.85
283.15–303.15
278–298
387–592
473.15–603.15
288.15–311.15
284.65–350.15
293.2–311.2
450.15–607.15
323.15–623.15
383.15–623.15
298.15–318.15
423.15–623.15
288.15
293.16–308.16
303.15–353.15
310.15
273.15–298.15
273.35–313.15
273.85–303.15
274.19–308.15
293.74–293.76
323.15–473.15
306.15–486.25
323.15–423.15
323.15
298.2
288.15–308.15
1.013
0.991–1.801
1.007–1.8
29.4–166.7
0.351–1.024
0.363–1.317
1.013
1.036–1.038
1.013
0.082–1.143
1.045
4.903–29.42
1.045
1.045
10.84–94.46
1.045
25.33–709.3
1.041–1.046
25.33–506.6
1.02–1.09
1.013–709.275
1.019–1.137
0.145–0.202
1.029–1.394
1.03–20.31
1–50.7
5.07–164.05
98–490
1.03–1.08
0.128–0.935
1.04–1.08
25.254–198.7
200–3500
100–1500
1.013
100–1400
0.947–4.163
25.33–75.99
9.9–38.9
1.013
10.13–45.6
1.013
1.02–1.056
1.02–1.07
1.038
2.54–53.89
8–58
100–800
68.2–176.8
36.3–109.9
1.013
8
10
12
36
10
18
8
14
2
19
1
9
1
1
60
1
29
6
22
3
39
18
6
19
15
18
36
79
6
54
5
15
109
116
4
33
10
20
13
1
12
3
5
8
5
33
7
14
9
6
5
S. Mao et al. / Chemical Geology 347 (2013) 43–58
Table 2 (continued)
References
T (K)
P (bar)
Nd
D'Souza et al. (1988)
Müller et al. (1988)
Nighswander et al. (1989)
Sako et al. (1991)
King et al. (1992)
Dohrn et al. (1993)
Yuan and Yang (1993)
Teng et al. (1997)
Zheng et al. (1997)
Gu (1998)
Dhima et al. (1999)
Bamberger et al. (2000)
Blencoe et al. (2001)
Servio and Englezos (2001)
Kiepe et al. (2002)
Teng and Yamasaki (2002)
Bando et al. (2003)
Sabirzyanov et al. (2003)
Blencoe (2004)
Chapoy et al. (2004)
Li et al. (2004)
Valtz et al. (2004)
Koschel et al. (2006)
Qin et al. (2008)
Han et al. (2009)
Ferrentino et al. (2010)
Liu et al. (2011)
323.15–348.15
373.15–473.15
352.85–471.25
348.2–421.4
288.15–298.15
323.15
278.15–318.15
278–293
278.15–338.15
304.19–313.15
344.25
323.2–353.1
573.15
277.05–283.15
313.2–393.17
298
303.15–333.15
298.15–423.15
623.15
274.14–351.31
332.15
278.22–318.23
323.1–373.1
323.6–375.8
313.2–343.2
313
308.15–328.15
101.33–152
3.25–81.1
20.4–102.1
101.8–209.4
60.8–243.2
101–301
1.013
64.4–294.9
0.4906–0.8417
17.63–58.25
100–1000
40.5–141.1
276.1–566.9
20–42
0.9509–92.576
75–300
100–200
100–800
209.7–299.5
1.9–93.33
33.4–198.9
4.65–79.63
20.6–194.7
106–499
43.3–183.4
75–150
20.8–159.9
4
49
33
8
27
3
5
24
10
10
7
29
5
9
39
6
12
17
5
27
6
47
8
7
28
4
31
Note: Nd is number of measurements.
increasing runs are not consistent with those measured in the pressure
decreasing runs. The deviation between them is 8–15%. The CO2 solubilities from Tödheide and Franck (1963) are overestimated due to systematic analytical error, as pointed out by Blencoe (2004), and their data are
inconsistent with others. Recently, Ferrentino et al. (2010), Han et al.
(2009), Liu et al. (2011) and Qin et al. (2008) made experimental measurements for the CO2 solubility in water, and their data points are consistent with each other. It can be seen from Table 2 that reliable solubility
data for CO2 in water at high pressures are still lacking, and future
experimental work should focus on pressures above 1500 bar. Most of
45
experimental CO2 solubility data in Table 2 except for those of
Drummond (1981) and Tödheide and Franck (1963) are used in the parameterization. The optimal T-P range of this model for CO2–H2O system
is 273.15–623.15 K and 1–1500 bar.
Experimental solubility data of CO2 in aqueous NaCl solutions are not
as extensive as those in water. Since 1940, about 28 data sets have been
reported for the ternary CO2–H2O–NaCl system (Table 3). Experimental
data of Cramer (1982) are not only internally inconsistent, but also
inconsistent with other data sets. The data of Drummond (1981) for the
CO2–H2O–NaCl system are not as accurate as those for the CO2–H2O system. Part of the experimental data of Gehrig et al. (1986) and Schmidt and
Bodnar (2000) deviates largely from those of Takenouchi and Kennedy
(1965) and the others. Recently, Ferrentino et al. (2010), Liu et al.
(2011) and Yan et al. (2011) measured CO2 solubility in aqueous NaCl
solutions, and their data are consistent with each other. Therefore, all
data points but those of Cramer (1982), Drummond (1981), Gehrig
et al. (1986) and Schmidt and Bodnar (2000) are included in the parameterization, which covers a wide T − P − mNaCl range (273.15–723.15 K,
1–1400 bar and 0–4.5 mol kg−1) for the CO2–H2O–NaCl system.
2.2. Thermodynamic model for CO2 solubility in aqueous NaCl solutions
CO2 solubility in aqueous solutions depends on the balance
between the chemical potential of CO2 in the liquid phase μ lCO2 and
that in the vapor phase μ vCO2 . The potential can be written in terms
of fugacity in the vapor phase and activity in the liquid phase:
vð0Þ
μ vCO2 ðT; P; yÞ ¼ μ CO ðT Þ þ RT lnf CO2 ðT; P; yÞ
2
vð0Þ
¼ μ CO ðT Þ þ RT lnyCO2 P þ RT lnϕCO2 T; P; yCO2
2
vð0Þ
v
¼ μ CO ðT Þ þ RT lnyCO2 P þ RT lnϕpure
CO2 ðT; P Þ þ RT lnγ CO2 ðT; P Þ
2
ð1Þ
lð0Þ
μ lCO2 ðT; P; mNaCl Þ ¼ μ CO ðT; P Þ þ RT lnaCO2 ðT; P; mNaCl Þ
2
lð0Þ
¼ μ CO ðT; P Þ þ RT lnmCO2 þ RT lnγlCO2 ðT; P; mNaCl Þ
ð2Þ
2
Table 3
CO2 solubility measurements in aqueous NaCl solutions since 1940.
References
T (K)
P (bar)
mNaCl(mol kg−1)
Nd
Markham and Kobe (1941)
Harned and Davis (1943)
Gjaldbæk (1953)
Ellis and Golding (1963)
Yeh and Peterson (1964)
Takenouchi and Kennedy (1965)
Onda et al. (1970)
Malinin and Savelyeva (1972)
Malinin and Kurorskaya (1975)
Yasunishi and Yoshida (1979)
Drummond (1981)
Burmakina et al. (1982)
Cramer (1982)
Gehrig et al. (1986)
Nighswander et al. (1989)
He and Morse (1993)
Rumpf et al. (1994)
Vázquez et al. (1994a)
Vázquez et al. (1994b)
Zheng et al. (1997)
Gu (1998)
Schmidt and Bodnar (2000)
Kiepe et al. (2002)
Bando et al. (2003)
Koschel et al. (2006)
Ferrentino et al. (2010)
Liu et al. (2011)
Yan et al. (2011)
273.35–313.15
273.15–323.15
293.15
445–607
298.15–318.15
423.15–723.15
298.15
298.15–358.15
298.15–423.15
288.15–308.15
293.65–673.15
298.15
296.75–511.75
415–783
353.65–473.65
273.15–363.15
313.14–433.12
298.1
293.1–308.1
278.15–338.13
303.15–323.15
548.15–923.15
313.38–353.07
303.15–333.15
323.1–373.1
313
318.15
323.2–413.2
1.02–1.09
1.02–1.14
1.013
25.22–213.37
1.05–1.11
100–1400
1.045
47.95
48
1.03–1.07
34.48–292.78
1
8–62
30–2717
20.4–102.1
0.07–1.0
1.51–96.37
1.045
1.04–1.07
0.69–0.95
17.73–58.96
450–3500
0.98–101
100–200
50–202.4
100–150
21–158.3
50–400
0.2–4
0–3
0–5.67
0–2.822
0–1.155
0–4.28
0–3.209
0–5.09
0–7.081
0–5.733
0–6.48
0–0.201
0–1.95
1.09–4.28
0–0.18
0.1–6.14
0–5.999
0–2.91
0.692–2.903
0.68–3.32
0.5–2.0
1.09–11.41
0.52–4.34
0.18–0.56
1–3
0.18
1.93–1.98
0–5
15
109
4
54
8
123
9
37
36
30
506
9
20
64
67
31
76
5
16
18
60
42
64
36
14
2
8
54
Note: Nd is number of measurements.
46
S. Mao et al. / Chemical Geology 347 (2013) 43–58
lð0Þ
where μ CO , the standard chemical potential of CO2 in liquid, is defined
2
as the chemical potential in a hypothetically ideal solution of unit molalvð0Þ
ity (Denbigh, 1971), and μ CO , the standard chemical potential in vapor,
where λ and ξ are second-order and third-order interaction parameters,
respectively; c and a refer to cation and anion, respectively. Substituting
Eq. (5) into Eq. (3) yields
2
is the hypothetical ideal gas chemical potential when the pressure is set
to 1 bar. ϕpure
CO2 is the fugacity coefficient of pure CO2 calculated from the
ln
equation of state of Duan et al. (1992). yCO2 is the mole fraction of CO2 in
yCO2 P
mCO2
ln
yCO2 P
mCO2
lð0Þ
¼
pure
c
ð6Þ
a
Following Pitzer et al. (1984), we choose the following equation
for the P-T dependence of the parameters (Par) for λ′s, ξ′s and
lð0Þ
μ CO2 =ðRT Þ− lnγvCO2 :
2
2
RT
þ
ParðT; P Þ ¼ c1 þ c2 T þ c3 =T þ c4 T 2 þ c5 =T 2 þ c6 P þ c7 PT
þ C 8 P=T þ c9 PT 2 þ c10 P 2 T þ c11 P 3
v
− ln γ CO2 ðT; P Þ
ð3Þ
l
lnγ CO2 ðT; P; mNaCl Þ
vð0Þ
value μ CO2
can be set to 0 for
In the parameterization, the reference
lð0Þ
vð0Þ
convenience, because only the difference between μ CO2 and μ CO2 is
important. yCO2 is calculated from
ð7Þ
Eqs. (6) and (7) form the basis of our model parameterization. Since
measurements can only be made in electronically neutral solutions, one
of the parameters in Eq. (6) must be assigned arbitrarily (Duan and Sun,
2003; Duan and Mao, 2006). λCO2 Cl is set to zero and then the remaining
parameters are fit to the experimental solubility data selected above,
lð0Þ
P−P H2 O
ð4Þ
P
where the partial pressure of water in vapor phase, P H2 O , is approximated
as the saturated pressure of H2O–NaCl system, which is different from
Duan and Sun (2003) whose P H2 O is approximated as the saturated pressure of pure H2O. This approximation will lead to errors (up to 5%) for
lð0Þ
v
2
− lnγCO2 − lnϕCO2 þ ∑ 2λCO2 c mc
RT
c
þ∑ 2λCO2 a ma þ ∑ ∑ ξCO2 ca mc ma
vð0Þ
μ CO ðT; P Þ−μ CO ðT Þ
pure
− ln ϕCO2 ðT; P Þ
yCO2 ¼
μ CO
a
the vapor phase. γ vCO2 and γ lCO2 are activity coefficients of CO2 in vapor
phase and liquid phase, respectively.
At phase equilibrium μ lCO2 = μ vCO2 , and we obtain
lð0Þ
¼
v
l
μ CO =ðRT Þ− lnγCO2 and lnγCO2 . However, these errors can be canceled
2
to a large extent in the parameterization. In this work, P H2 O is calculated
from the model of Atkinson (2002) whose valid T and xNaCl (mole fraction of NaCl) ranges are 252–1773 K and 0–1.
lnγlCO2 in Eq. (3) is expressed as a virial expansion of excess Gibbs
energy (Pitzer, 1973).
l
ln γ CO2 ¼ ∑ 2λCO2 c mc þ ∑ 2λCO2 a ma þ ∑ ∑ ξCO2 ac mc ma
c
a
c
a
ð5Þ
where μ CO2 =ðRT Þ− lnγvCO2 is evaluated from the solubility data of CO2
in pure water, λCO2 Na and ξCO2 NaCl are then evaluated simultaneously
to the solubility data of CO2 in aqueous NaCl solutions. Eq. (7) is
obviously better than that of Duan and Sun (2003), whose
Par ðT; P Þ ¼ c1 þ c2 T þ c3 =T þ c4 T 2 þ c5 =ð630−T Þ þ c6 P þ c7 P lnT
,
þ C 8 P=T þ c9 P=ð630−T Þ þ c10 P 2 =ð630−T Þ2 þ c11 T lnP
where the Par(T,P) value will be infinite when T approaches 630 K. So
(630 − T) in denominator limits the applicable temperature region of
that model.
In the fitting, we found that one set of parameters for
lð0Þ
μ CO2 =ðRT Þ− lnγvCO2 , from which CO2 solubility in pure water can be
calculated, is sufficient. When a small amount of NaCl is added to
the CO2–H2O system, the immiscible region expands considerably,
compared to the CO2–H2O system, which has a small or nonexistent
vapor–liquid coexistence region at high temperatures. It is very difficult to fit the parameters to a single set of equations. Therefore, the
parameters are divided into two parts in terms of temperatures:
one set of parameters is for 273.15–503 K and the other is for
Table 4
Interaction parameters.
273.15 ≤ T ≤ 503
503 b T b 523
523 ≤ T ≤ 723.15
lð0Þ
T-P coefficient
μ CO2 =ðRT Þ− lnγvCO2
λCO2 Na
ξCO2 NaCl
c1
c2
c3
c4
c5
c6
c7
c8
c9
c10
c11
0.23018254D + 02
−0.36540569D-01
−0.18366895D + 04
0.20330876D-04
−0.39072384D + 06
−0.58269326D-01
0.15061716D-03
0.78086969D + 01
−0.13013307D-06
0.11145375D-08
−0.13073985D-09
−0.31312239D + 00
0.55326470D-03
0.75844401D + 02
0.34096802D-02
−0.27671084D-04
−0.18950519D-03
0.71628762D-06
−0.83847525D-07
−0.14585720D-09
0.34225403D-10
0.38934171D + 00
−0.18177725D-03
−0.65214916D + 02
−0.20867161D-01
AðT; P; mNaCl Þ ¼ Að503; P; mNaCl Þ þ ½Að523; P; mNaCl Þ−Að503; P; mNaCl Þ
ðT−503Þ=20
c1
c2
c3
c4
c5
c6
c7
c8
c9
0.11625439D + 04
−0.13350381D + 01
−0.44522815D + 06
0.57326540D-03
0.64139318D + 08
−0.12735549D + 00
0.21720184D-03
0.25529557D + 02
−0.12542533D-06
lð0Þ
−0.31200574D-03
0.60052047D-06
Note: A(T,P,mNaCl) is defined by Eq. (8). If mNaCl = 0, the parameters of μ CO2 =ðRT Þ− lnγvCO2 below 503 K can be used up to 647 K.
S. Mao et al. / Chemical Geology 347 (2013) 43–58
47
Table 5
Calculated deviations of CO2 solubility in water.
References
T (K)
P (bar)
Nd
AAD (%)
MAD (%)
Findlay and Creighton (1910)
Findlay and Shen (1912)
Findlay and Williams (1913)
Findlay and Howell (1915)
Kunerth (1922)
Buch (1928)
Morgan and Pyne (1930)
Morgan and Maass (1931)
Kobe and Williams (1935)
Kritschewsky et al. (1935)
Shedlovsky and MacInnes (1935)
Orcutt and Seevers (1937)
Zel'venskii (1937)
Curry and Hazelton (1938)
Wiebe and Gaddy (1939)
Van Slyke (1939)
Wiebe and Gaddy (1940)
Markham and Kobe (1941)
Harned and Davis (1943)
Koch et al. (1949)
Morrison and Billett (1952)
Bartholome and Friz (1956)
Malinin (1959)
Bartels and Wrbitzky (1960)
Novák et al. (1961)
Austin et al. (1963)
Ellis and Golding (1963)
Tödheide and Franck (1963)
Takenouchi and Kennedy (1964)
Yeh and Peterson (1964)
Takenouchi and Kennedy (1965)
de Khanikof and Louguinine (1967)
Matous et al. (1969)
Power and Stegall (1970)
Stewart and Munjal (1970)
Barton and Hsu (1971)
Li and Tsui (1971)
Murray and Riley (1971)
Weiss (1974)
Zawisza and Malesińska (1981)
Cramer (1982)
Shagiakhmetov and Tarzimanov (1982)
Briones et al. (1987)
Nakayama et al. (1987)
Postigo and Katz (1987)
D'Souza et al. (1988)
Müller et al. (1988)
Nighswander et al. (1989)
Sako et al. (1991)
King et al. (1992)
Dohrn et al. (1993)
Yuan and Yang, (1993)
Teng et al. (1997)
Zheng et al. (1997)
Gu (1998)
Dhima et al. (1999)
Bamberger et al. (2000)
Blencoe et al. (2001)
Servio and Englezos (2001)
Kiepe et al. (2002)
Teng and Yamasaki (2002)
Bando et al. (2003)
Sabirzyanov et al. (2003)
Blencoe (2004)
Chapoy et al. (2004)
Li et al. (2004)
Valtz et al. (2004)
Koschel et al. (2006)
Qin et al. (2008)
Han et al. (2009)
Ferrentino et al. (2010)
Liu et al. (2011)
298.15
298.15
298.15
298.15
293.15–307.15
292.95–293.95
298.15
273.15–298.15
298.15
293.15–303.15
298.15
298.15
273–373
298.15
323.15–373.15
295.95–298.65
291.15–313.15
273.35–313.15
273.15–323.15
291.45–295.05
286.45–347.85
283.15–303.15
473.15–603.15
288.15–311.15
284.65–350.15
293.2–311.2
450.15–607.15
323.15–540.15
383.15–623.15
298.15–318.15
423.15–623.15
288.15
303.15–353.15
310.15
273.15–298.15
273.35–313.15
273.85–303.15
274.19–308.15
293.74–293.76
323.15–473.15
306.15–486.25
323.15–423.15
323.15
298.2
288.15–308.15
323.15–348.15
373.15–473.15
352.85–471.25
348.2–421.4
288.15–298.15
323.15
278.15–318.15
278–293
278.15–338.15
304.19–313.15
344.25
323.2–353.1
573.15
277.05–283.15
313.2–393.17
298
303.15–333.15
298.15–423.15
623.15
274.14–351.31
332.15
278.22–318.23
323.1–373.1
323.6–375.8
313.2–343.2
313
308.15–328.15
0.991–1.801
1.007–1.8
0.351–1.024
0.363–1.317
1.013
1.036–1.038
1.013
0.082–1.143
1.045
4.903–29.42
1.045
1.045
10.84–94.46
1.045
25.33–709.3
1.041–1.046
25.33–506.6
1.02–1.09
1.019–1.137
0.145–0.202
1.029–1.394
1.03–20.31
98–490
1.03–1.08
0.128–0.935
1.04–1.08
25.254–198.7
200–2000
100–1500
1.013
100–1400
0.947–4.163
9.9–38.9
1.013
10.13–45.6
1.013
1.02–1.056
1.02–1.07
1.038
2.54–53.89
8–58
100–800
68.2–176.8
36.3–109.9
1.013
101.33–152
3.25–81.1
20.4–102.1
101.8–197.2
60.8–243.2
101–301
1.013
64.4–294.9
0.4906–0.8417
17.63–58.25
100–1000
40.5–141.1
276.1–515.2
20–42
0.9509–92.576
75–300
100–200
100–800
209.7–282.4
1.9–93.33
33.4–198.9
4.65–79.63
20.6–194.7
106–499
43.3–183.4
75–150
20.8–159.9
10
12
10
18
8
14
2
19
1
9
1
1
60
1
29
6
22
3
18
6
19
15
79
6
54
5
14
39
116
4
33
10
13
1
12
3
5
8
5
32
7
14
8
6
5
4
49
33
7
27
3
5
24
10
10
7
29
3
9
38
6
12
16
3
27
6
47
8
7
28
4
31
0.28
0.83
2.27
2.64
3.30
5.09
2.10
4.57
1.38
4.42
1.83
1.52
2.31
1.81
2.72
0.65
0.67
3.14
1.26
8.55
2.66
1.07
2.28
3.78
5.10
3.50
3.50
4.51
5.28
6.25
5.18
8.93
1.82
5.52
10.85
2.98
1.26
3.56
0.48
5.09
7.28
2.91
2.30
4.11
2.30
5.07
3.27
6.16
7.31
1.40
1.16
3.52
3.85
1.27
3.20
5.03
1.45
9.31
5.37
7.11
0.89
2.76
9.55
6.55
3.02
7.24
4.65
3.72
3.77
2.10
0.87
1.67
0.93
1.49
6.01
6.51
5.60
18.89
2.11
23.35
1.38
12.50
1.83
1.52
18.65
1.81
6.37
1.41
2.27
5.83
2.89
15.81
5.58
2.11
12.18
4.74
19.98
4.29
13.72
16.47
27.49
14.64
15.22
10.78
4.95
5.52
23.99
4.94
2.46
17.34
0.81
10.92
13.28
10.75
8.81
6.60
3.55
6.48
12.53
26.97
15.53
5.19
2.15
6.61
7.83
2.51
9.33
10.00
7.67
21.18
8.16
16.74
1.39
6.77
20.17
8.30
7.93
10.04
8.51
7.58
6.76
7.47
2.35
4.65
Note: AAD is the average absolute deviation calculated from this model; MAD is the maximal absolute deviation calculated from this model; Nd is number of measurements.
48
S. Mao et al. / Chemical Geology 347 (2013) 43–58
Table 6
Calculated deviations of CO2 solubility in aqueous NaCl solutions.
References
T (K)
P (bar)
mNaCl(mol kg−1)
Nd
AAD (%)
MAD (%)
Markham and Kobe (1941)
Harned and Davis (1943)
Gjaldbæk (1953)
Ellis and Golding (1963)
Yeh and Peterson (1964)
Takenouchi and Kennedy (1965)
Onda et al. (1970)
Malinin and Savelyeva (1972)
Malinin and Kurorskaya (1975)
Yasunishi and Yoshida (1979)
Drummond (1981)
Burmakina et al. (1982)
Gehrig et al. (1986)
Nighswander et al. (1989)
Rumpf et al. (1994)
Vázquez et al. (1994a)
Vázquez et al. (1994b)
Gu (1998)
Schmidt and Bodnar (2000)
Kiepe et al. (2002)
Bando et al. (2003)
Koschel et al. (2006)
Ferrentino et al. (2010)
Liu et al. (2011)
Yan et al. (2011)
273.35–313.15
273.15–323.15
293.15
445–607
298.15–318.15
423.15–723.15
298.15
298.15–358.15
298.15–423.15
288.15–308.15
293.65–673.15
298.15
408–748
353.65–473.65
313.14–433.12
298.1
293.1–308.1
303.15–323.15
548.15–823.15
313.38–353.07
303.15–333.15
323.1–373.1
313
318.15
323.2–413.2
1.02–1.09
1.02–1.14
1.013
25.22–213.37
1.05–1.11
100–1400
1.045
47.95
48
1.03–1.07
34.48–292.78
1
290–2404
20.4–102.1
1.51–96.37
1.045
1.04–1.07
17.73–58.96
560–2470
0.98–101
100–200
50–202.4
100–150
21–158.3
50–400
0.2–4
0–3
0–5.67
0–2.822
0–1.155
0–4.28
0–3.209
0–5.09
0–7.081
0–5.733
0–6.48
0–0.201
1.09–1.90
0–0.18
0–5.999
0–2.91
0.692–2.903
0.5–2.0
1.09–4.28
0.52–4.34
0.18–0.56
1–3
0.18
1.93–1.98
0–5
15
109
4
53
8
123
9
37
36
30
506
9
40
67
70
5
16
60
12
64
36
14
2
8
54
3.66
3.14
1.36
4.48
5.83
5.18
2.90
4.62
7.18
6.63
6.62
1.67
13.21
6.28
2.09
3.07
2.49
4.05
16.97
5.78
2.30
4.50
2.89
4.84
3.73
7.24
8.21
2.19
13.97
9.83
35.06
3.50
14.45
10.23
11.49
50.24
3.32
25.42
26.97
5.23
3.36
3.87
12.06
27.79
17.61
5.63
15.50
4.16
8.36
13.03
Note: AAD is the average absolute deviation calculated from this model; MAD is the maximal absolute deviation calculated from this model; Nd is number of measurements.
523–723.15 K. In order to make the calculated CO2 solubilities and
isopleths continuous during the whole temperature range, a function
A(T,P,mNaCl) is defined as
lð0Þ
AðT; P; mNaCl Þ ¼
μ CO
2
RT
v
− lnγCO2 þ ∑ 2λCO2 Na mNa þ ∑ ∑ ξCO2 ClNa mCl mNa
Na
Cl Na
ð8Þ
When temperatures are between 503 K and 523 K, A(T,P,mNaCl) is
calculated from
AðT; P; mNaCl Þ ¼ Að503; P; mNaCl Þ þ ½Að523; P; mNaCl Þ−Að503; P; mNaCl Þ
ð9Þ
ðT−503Þ=20
Table 4 lists the optimized parameters. Tables 5 and 6 show the average and maximum deviations of our model from each data set for the
CO2 solubility in pure water and aqueous NaCl solutions, respectively.
The average absolute deviation of CO2 solubility in pure water and
aqueous NaCl solutions is 4.62%, which is within experimental uncertainties. With these parameters, the CO2 solubility in pure water
(Table 7) and aqueous NaCl solutions (Tables 8–11) can be calculated.
It should be noted that the blank spaces at low temperatures in
Tables 8–11 represent the P-T region where CO2-clathrate is stable,
and the blank spaces at high temperatures and low pressures represent
the region where the total pressure is below the vapor pressure of H2O–
NaCl, where this model cannot be applied.
Figs. 1 and 2 show the comparisons between the experimental
results and model predictions. As can be seen in Fig. 1, the CO2 solubility
in water increases with pressure at a given temperature, and the experimental CO2 solubility data are accurately reproduced by this model
except in the critical region, where deviations increase to over 10%
(Fig. 1(g–h)). It can be seen from Fig. 2 that the model can also reproduce
the experimental CO2 solubility in aqueous NaCl solutions within or
close to experimental uncertainty, and covers a large T − P − mNaCl
Table 7
Calculated CO2 solubility (mol kg−1) in pure water.
P (bar)
1
50
100
150
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
T (K)
273.15
298.15
323.15
373.15
423.15
473.15
523.15
0.0737
0.0329
1.2052
1.4330
1.4885
1.5355
1.6120
1.6707
1.7151
1.7478
1.7711
1.7870
1.7977
1.8049
1.8106
1.8164
1.8243
1.8359
1.8530
0.0175
0.7734
1.1473
1.2529
1.3239
1.4399
1.5357
1.6168
1.6860
1.7453
1.7966
1.8417
1.8823
1.9204
1.9579
1.9965
2.0385
2.0859
0.4713
0.8087
1.0315
1.1777
1.3731
1.5197
1.6411
1.7442
1.8324
1.9082
1.9736
2.0308
2.0817
2.1285
2.1733
2.2183
2.2657
0.3805
0.7215
0.9916
1.2051
1.5192
1.7483
1.9302
2.0791
2.2018
2.3024
2.3842
2.4501
2.5031
2.5458
2.5812
2.6120
2.6408
0.3085
0.7092
1.0596
1.3661
1.8742
2.2823
2.6226
2.9122
3.1597
3.3707
3.5492
3.6990
3.8237
3.9274
4.0139
4.0875
4.1524
0.1108
0.6280
1.1240
1.5995
2.4949
3.3299
4.1187
4.8700
5.5875
6.2720
6.9235
7.5420
8.1290
8.6868
9.2195
9.7322
10.2319
573.15
598.15
623.15
0.1946
0.9133
1.6807
3.3652
5.2594
7.3783
0.5019
1.4583
3.7649
6.6655
0.8017
3.8777
S. Mao et al. / Chemical Geology 347 (2013) 43–58
49
Table 8
Calculated CO2 solubility (mol kg−1) in 1 mol kg−1 NaCl solutions.
P(bar)
1
50
100
150
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
T (K)
273.15
323.15
373.15
423.15
473.15
523.15
573.15
623.15
673.15
723.15
0.0587
0.0145
0.6346
0.9393
1.0239
1.0804
1.1733
1.2515
1.3200
1.3813
1.4372
1.4896
1.5399
1.5899
1.6413
1.6960
1.7559
1.8231
1.9002
0.0002
0.3892
0.6639
0.8425
0.9574
1.1077
1.2190
1.3114
1.3911
1.4616
1.5250
1.5835
1.6389
1.6931
1.7479
1.8055
1.8680
1.9376
0.3124
0.5859
0.7980
0.9622
1.1960
1.3604
1.4877
1.5909
1.6763
1.7478
1.8087
1.8616
1.9088
1.9529
1.9960
2.0406
2.0889
0.2516
0.5657
0.8335
1.0618
1.4266
1.7063
1.9310
2.1171
2.2737
2.4068
2.5209
2.6199
2.7072
2.7863
2.8606
2.9333
3.0080
0.1080
0.5307
0.9047
1.2370
1.8028
2.2731
2.6806
3.0464
3.3833
3.6993
3.9993
4.2866
4.5636
4.8317
5.0921
5.3456
5.5928
0.1833
0.6967
1.1739
2.0384
2.8117
3.5222
4.1910
4.8329
5.4579
6.0730
6.6830
7.2913
7.9005
8.5124
9.1284
9.7495
0.5295
1.7711
2.9346
4.0453
5.1238
6.1865
7.2459
8.3114
9.3902
10.4879
11.6089
12.7570
13.9351
15.1458
0.4090
2.0375
3.6769
5.3480
7.0696
8.8589
10.7310
12.6997
14.7776
16.9766
19.3082
21.7836
24.4138
1.4533
3.5507
5.8980
8.5361
11.5093
14.8661
18.6590
22.9460
27.7901
33.2613
39.4361
Table 9
Calculated CO2 solubility (mol kg−1) in 2 mol kg−1 NaCl solutions.
P(bar)
1
50
100
150
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
T (K)
273.15
323.15
373.15
423.15
473.15
523.15
573.15
623.15
673.15
723.15
0.0472
0.0121
0.5280
0.7817
0.8526
0.9004
0.9806
1.0502
1.1134
1.1726
1.2296
1.2857
1.3427
1.4021
1.4658
1.5356
1.6139
1.7031
1.8064
0.0004
0.3268
0.5559
0.7038
0.7983
0.9211
1.0124
1.0893
1.1574
1.2196
1.2782
1.3349
1.3916
1.4501
1.5122
1.5801
1.6559
1.7423
0.2614
0.4868
0.6594
0.7911
0.9752
1.1019
1.1991
1.2781
1.3443
1.4016
1.4526
1.4997
1.5450
1.5907
1.6388
1.6914
1.7508
0.2095
0.4632
0.6756
0.8534
1.1301
1.3354
1.4959
1.6264
1.7355
1.8285
1.9098
1.9827
2.0503
2.1156
2.1814
2.2507
2.3264
0.0913
0.4111
0.6940
0.9452
1.3725
1.7274
2.0345
2.3099
2.5633
2.8008
3.0260
3.2415
3.4489
3.6496
3.8443
4.0336
4.2180
0.1616
0.5444
0.8978
1.5314
2.0900
2.5956
3.0645
3.5079
3.9335
4.3465
4.7506
5.1481
5.5410
5.9306
6.3178
6.7034
0.4425
1.3388
2.1572
2.9179
3.6372
4.3276
4.9982
5.6559
6.3055
6.9508
7.5945
8.2386
8.8848
9.5343
0.3932
1.5318
2.6367
3.7225
4.8018
5.8850
6.9804
8.0949
9.2341
10.4028
11.6049
12.8439
14.1229
1.1399
2.5000
3.9550
5.5214
7.2155
9.0544
11.0551
13.2356
15.6142
18.2105
21.0448
Table 10
Calculated CO2 solubility (mol kg−1) in 3 mol kg−1 NaCl solutions.
P(bar)
1
50
100
150
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
T (K)
273.15
323.15
373.15
423.15
473.15
523.15
573.15
623.15
673.15
723.15
0.0382
0.0103
0.4453
0.6611
0.7233
0.7664
0.8406
0.9073
0.9703
1.0316
1.0928
1.1554
1.2209
1.2910
1.3678
1.4533
1.5503
1.6618
1.7919
0.0006
0.2792
0.4748
0.6014
0.6827
0.7895
0.8705
0.9405
1.0044
1.0647
1.1234
1.1824
1.2434
1.3080
1.3784
1.4566
1.5453
1.6474
0.2233
0.4141
0.5596
0.6701
0.8238
0.9294
1.0109
1.0781
1.1358
1.1873
1.2350
1.2811
1.3274
1.3759
1.4285
1.4874
1.5548
0.1786
0.3897
0.5647
0.7097
0.9321
1.0942
1.2194
1.3206
1.4053
1.4783
1.5433
1.6033
1.6612
1.7193
1.7802
1.8466
1.9211
0.0799
0.3322
0.5552
0.7532
1.0897
1.3688
1.6101
1.8263
2.0250
2.2110
2.3873
2.5557
2.7178
2.8743
3.0260
3.1734
3.3169
0.1466
0.4441
0.7168
1.2006
1.6207
1.9952
2.3371
2.6555
2.9565
3.2443
3.5217
3.7906
4.0526
4.3087
4.5597
4.8061
0.3840
1.0574
1.6559
2.1971
2.6947
3.1589
3.5973
4.0154
4.4172
4.8055
5.1825
5.5498
5.9086
6.2599
0.3705
1.1977
1.9705
2.7015
3.4009
4.0766
4.7350
5.3806
6.0171
6.6473
7.2733
7.8969
8.5192
0.8806
1.7940
2.7265
3.6858
4.6790
5.7122
6.7913
7.9216
9.1080
10.3555
11.6687
50
S. Mao et al. / Chemical Geology 347 (2013) 43–58
Table 11
Calculated CO2 solubility (mol kg−1) in 4 mol kg−1 NaCl solutions.
P(bar)
T (K)
1
50
100
150
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
273.15
323.15
373.15
423.15
473.15
523.15
573.15
623.15
673.15
723.15
0.0313
0.0088
0.3808
0.5683
0.6252
0.6662
0.7390
0.8071
0.8737
0.9405
1.0091
1.0809
1.1576
1.2411
1.3336
1.4376
1.5564
1.6938
1.8548
0.0008
0.2425
0.4137
0.5257
0.5988
0.6976
0.7751
0.8442
0.9091
0.9723
1.0355
1.1005
1.1688
1.2425
1.3236
1.4145
1.5181
1.6380
0.1946
0.3606
0.4877
0.5847
0.7209
0.8162
0.8913
0.9548
1.0109
1.0625
1.1117
1.1604
1.2105
1.2637
1.3221
1.3877
1.4630
0.1559
0.3369
0.4866
0.6105
0.8004
0.9388
1.0460
1.1333
1.2071
1.2717
1.3303
1.3855
1.4398
1.4955
1.5547
1.6200
1.6938
0.0720
0.2796
0.4629
0.6256
0.9018
1.1307
1.3285
1.5054
1.6678
1.8197
1.9636
2.1009
2.2328
2.3601
2.4834
2.6030
2.7194
0.1354
0.3763
0.5956
0.9805
1.3097
1.5985
1.8578
2.0955
2.3165
2.5243
2.7215
2.9096
3.0899
3.2634
3.4307
3.5923
0.3420
0.8685
1.3237
1.7236
2.0803
2.4031
2.6985
2.9714
3.2254
3.4631
3.6865
3.8971
4.0962
4.2846
0.3495
0.9736
1.5343
2.0436
2.5111
2.9443
3.3488
3.7290
4.0881
4.4288
4.7531
5.0625
5.3583
0.7090
1.3449
1.9632
2.5693
3.1676
3.7616
4.3542
4.9476
5.5438
6.1443
6.7503
range: 273.15–723.15 K, 1–1400 bar and 0–4.5 mol kg−1 for the CO2–
H2O–NaCl system.
s
The molar heat of CO2 in aqueous NaCl solutions (ΔHm
) can also be
derived from the solubility model:
−
ΔH sm
∂
¼
∂T
RT 2
∂
¼
∂T
ð Þ
μl 0
v
− lnγCO2
RT
!
!
lð0Þ
∂ lnγCO2
þ
μ
v
− lnγ CO2
RT
!
∂λCO2 a
þ Σ 2ma
a
∂T
∂T
P;mNaCl
þ Σ 2mc
P;mNaCl
c
þ Σ Σ mc ma
P;mNaCl
c a
!
P;mNaCl
∂λCO2 c
!
∂T
ð10Þ
P;mNaCl
∂ξCO2 ca
∂T
!
3.1. An iterative method for determining the CO2 contents of CO2–H2O–
NaCl inclusions
P;mNaCl
∂ParðT; P Þ
c
2c
c P
¼ c2 − 32 þ 2c4 T− 35 þ c7 P− 82 þ 2c9 PT
∂T
T
T
T
P;mNaCl
þ c10 P
CO2–H2O–NaCl inclusions with salinities less than 5 wt.% NaCl. However,
how to determine the CO2 contents of CO2–H2O–NaCl inclusions with
higher salinities has not been satisfactorily formulated. The developed
model here for CO2 solubility in aqueous NaCl solutions, together with
the updated PVTx model of Mao et al. (2010), can be used to calculate
CO2 contents, homogenization pressures and volume fractions of
CO2–H2O–NaCl fluid inclusions. A three-phase (liquid CO2 + vapor
CO2 + liquid H2O) CO2–H2O–NaCl inclusion at room temperature
is used here as an example to illustrate how to obtain these thermodynamic parameters by an iterative method.
2
ð11Þ
The predicted molar heat of CO2 in water (Fig. 3) is compared with
some experimental results, which exhibit a good agreement. These,
from another respect, prove the reliability of this model.
3. Application of the model to CO2–H2O–NaCl fluid inclusions
As discussed in the introduction, the main purpose of this study is to
develop a numerical CO2 solubility model that can be applied to CO2–
H2O–NaCl fluid inclusions. For CO2–H2O–NaCl fluid inclusions in either
three phases (liquid CO2 + vapor CO2 + liquid H2O) or two phases
(liquid or vapor CO2 + liquid H2O) at room temperatures, various
methods have been developed and tested to determine the NaCl
contents (salinities) of CO2–H2O–NaCl inclusions. For three-phase (liquid
CO2 + vapor CO2 + liquid H2O) inclusions at room temperatures, equation of Chen (1972) can be used to calculate salinities of CO2–H2O–NaCl
inclusions by the dissociation temperature of CO2 clathrate that melts
in the presence of both liquid and vapor CO2. For two-phase (liquid or
vapor CO2 + liquid H2O) inclusions at room temperatures, two kinds
of methods can be used to estimate the salinities of CO2–H2O–NaCl
inclusions: one is the equations of Diamond (1992), and another method
is combining Raman analysis (Fall et al., 2011 or Wang et al., 2011) with a
thermodynamic clathrate-phase-equilibrium model (Bakker et al., 1996
or Duan and Sun, 2006). Azbej et al. (2007) developed two empirical
equations between inclusion compositions and Raman spectral
parameters, which can be used to calculate the CO2 contents of natural
If a three-phase (liquid CO2 + vapor CO2 + liquid H2O) CO2–H2O–
NaCl inclusion is not a negative-crystal inclusion and the volume fraction of vapor phase at the partial homogenization temperature of CO2
is well measured by the improved approach (Bodnar, 1983; Bodnar
et al., 1985; Bakker and Diamond, 2006), the following equation can
be used to calculate CO2 content of the CO2–H2O–NaCl fluid inclusion:
Bulk
xCO2 ¼
F V yCO2
V Vm
þ
ð1−F V ÞxCO2
V Lm
F
1−F V −1
⋅ VV þ
Vm
V Lm
ð12Þ
where xBulk
CO2 is the bulk mole fraction of CO2 in the total inclusion, FV is
the volume fraction of vapor bubble at the partial homogenization temperature of CO2, yCO2 and xCO2 are the mole fractions of CO2 in the vapor
V
L
phase and aqueous liquid phase, respectively, Vm
and Vm
are the molar
volumes of vapor phase and liquid phase, respectively. Because water
content in vapor phase is very low at the partial homogenization temperature of CO2 (which is generally below 31 °C), Eq. (12) can be
approximated as
Bulk
xCO2 ¼
F V ð1−F V ÞxCO2
F
1−F V −1
þ
⋅ VV þ
V
L
L
Vm
Vm
Vm
Vm
ð13Þ
where xCO2 is a function of temperature, pressure and salinity, which can
be calculated by combining equations of Chen (1972) and Span and
V
Wagner (1996) and the CO2 solubility model. Vm
in Eq. (13) is a function
of temperature and is calculated from the density equations of CO2
along the liquid–vapor (saturation) curve (Span and Wagner, 1996)
L
due to negligible contents of water and NaCl in vapor phase. Vm
in
Eq. (13) is a function of temperature, pressure and composition (mNaCl
and xCO2 ), which can be calculated from the general PVTx model (Mao
S. Mao et al. / Chemical Geology 347 (2013) 43–58
a
b
3.0
Exp. King et al. (1992)
Exp. Teng et al. (1997)
Exp. Valtz et al. (2004)
2.0
T = 298.15 K
.15 K
mCO2(mol kg-1)
This model
2.0
T = 278
mCO2(mol kg-1)
2.5
51
1.5
T = 288.15 K
1.0
1.5
1.0
Exp. Zel'venskii (1937)
Exp. Nakayama et al. (1987)
Exp. King et al. (1992)
Exp. Teng and Yamasaki (2002)
Exp. Valtz et al. (2004)
This model
0.5
0.5
0.0
0.0
0
50
100
150
200
250
0
300
100
200
P (bar)
c 2.5
d
T=
.15
323
1.5
Exp. Zel'venskii (1937)
Exp. Wiebe and Gaddy (1939)
Exp. Todheide and Franck (1963)
Exp. Briones et al. (1987)
Exp. D'Souza et al. (1988)
Exp. Dohrn et al. (1993)
Exp. Bamberger et al. (2000)
Exp. Bando et al. (2003)
Exp. Koschel et al. (2006)
This model
1.0
0.5
400
500
4
3
K
mCO (mol kg-1)
2
mCO2(mol kg-1)
2.0
300
P (bar)
T=
2
.15
423
K
Exp. Todheide and Franck (1963)
Exp. Takenouchi and Kennedy (1964)
Exp. Takenouchi and Kennedy (1965)
Exp. Zawisza and Malesinska (1981)
Exp. Shagiakhmetov and Tarzimanov (1982)
This model
1
0
0.0
0
300
600
900
1200
0
1500
300
600
P (bar)
f
5
mCO2(mol kg-1)
4
3
47
T=
.15
3
2
Exp. Malinin (1959)
Exp. Todheide and Franck (1963)
Exp. Takenouchi and Kennedy (1964)
Exp. Takenouchi and Kennedy (1965)
This model
1
12
8
6
T
.1
23
=5
4
300
600
900
1200
1500
0
300
600
P (bar)
h
Exp. Malinin (1959)
Exp. Takenouchi and Kennedy (1964)
Exp. Takenouchi and Kennedy (1965)
Exp. Blencoe et al. (2001)
This model
mCO2(mol kg-1)
T=
4
57
1200
1500
5
Takenouchi and Kennedy (1964)
Takenouchi and Kennedy (1965)
Blencoe (2004)
This model
4
8
6
900
P (bar)
g
mCO2(mol kg-1)
5K
0
0
10
1500
2
0
12
1200
Exp. Malinin (1959)
Exp. Todheide and Franck (1963)
Exp. Takenouchi and Kennedy (1964)
Exp. Takenouchi and Kennedy (1965)
This model
10
K
mCO2(mol kg-1)
e
900
P (bar)
5K
3.1
2
3
6
T=
2
23
.15
K
1
0
0
100
200
300
P (bar)
400
500
600
200
220
240
260
280
300
P (bar)
Fig. 1. CO2 solubility in water (this model vs. experimental data): (a) 278.15 and 288.15 K, (b) 298.15 K, (c) 323.15 K, (d) 423.15 K, (e) 473.15 K, (f) 523.15 K, (g) 573.15 K, (h) 623.15 K.
52
S. Mao et al. / Chemical Geology 347 (2013) 43–58
b 0.04
a 0.10
Exp. Markham and Kobe (1941)
Exp. Harned and Davis (1943)
Exp. Vá zquez et al. (1994a)
This model
Exp. Harned and Davis (1943)
This model
mCO2(mol kg-1)
mCO2(mol kg-1)
0.08
0.06
0.04
0.02
0.03
0.02
T = 298.15 K; PCO = 1 bar
T = 273.15 K; PCO = 1 bar
2
2
0.01
0.00
0
1
2
3
mNaCl(mol
1
kg-1)
2
3
4
kg-1)
mNaCl(mol
d
0.6
Exp. Rumpf et al. (1994)
This model
T
=
2
3.
K
T=
3
.1
53
mCO2(mol kg-1)
0.4
31
Exp. Takenouchi and Kennedy (1965)
This model
3
2K
T=
2
mCO (mol kg-1)
2
c
0
4
0.2
.9
432
K
T
7
=4
5K
3.1
2
T=4
23.1
5K
1
-1
-1
mNaCl = 4 mol kg
mNaCl = 1.09 mol kg
0.0
0
20
40
60
80
0
100
300
600
P (bar)
e
900
1200
1500
P (bar)
f
16
5
Exp. Takenouchi and Kennedy (1965)
This model
Exp. Takenouchi and Kennedy (1965)
This model
4
-1
mNaCl = 1.09 mol kg
8
T
2
=6
T=
4
T = 52
3.1
mCO2(mol kg-1)
mCO (mol kg-1)
2
12
5K
K
.15
573
3.15 K
3
-1
mNaCl = 4.28 mol kg
2
T=
K
.15
0
0
300
600
900
1200
1500
0
300
600
P (bar)
h
8
mCO2(mol kg-1)
T = 673.15 K
4
2
P (bar)
600
700
3
T = 723.15 K
2
-1
mNaCl = 4.28 mol kg
0
500
1500
4
1
-1
mNaCl = 1.09 mol kg
400
1200
Exp. Takenouchi and Kennedy (1965)
This model
6
300
900
P (bar)
Exp. Takenouchi and Kennedy (1965)
This model
mCO (mol kg-1)
2
52 3
1
0
g
T
K
.15
23
6
=
800
0
400
500
600
700
P (bar)
800
900
1000
S. Mao et al. / Chemical Geology 347 (2013) 43–58
30
ΔHms (kJ mol-1)
Start
Ellis (1959)
Ellis and Golding (1963)
This work
20
53
10
Input Tm(cla), Th (CO2) and Th (tot)
0
-10
Calculate mNaCl , Ph (CO2 ), xCO2 ,
-20
VmV , VmL and FV (max)
PCO = 1 bar
2
-30
300
350
400
450
500
550
600
Initial FV1 =0, FV2 =FV (max),
T (K)
FV =(FV1 +FV2 )/2
Fig. 3. Molar heat of solution of CO2 in water.
Bulk
Calculate xCO
, Vm ,
2
V
L
et al., 2010). Combining FV, Vm
and Vm
, the bulk molar volume Vm of
inclusion can be calculated from the following equation:
Vm ¼
F V 1−F V
þ
V Vm
V Lm
−1
Ph (tot) and Vmcal
ð14Þ
In the above calculation for determining the CO2 contents of CO2–
H2O–NaCl inclusions, FV is an input variable, whose values are obtained
from experimental measurements. The relative accuracy of estimated volume fraction FV is ±4% if the improved method of Bakker and Diamond
(2006) is used. However, this approach is time-consuming, particularly
unfit for the negative-crystal inclusions.
In order to solve this problem, an iterative approach is presented to
calculate the CO2 contents of CO2–H2O–NaCl inclusions on the assumption that the molar volume of the fluid inclusion at the partial homogenization temperature of the CO2 phases equals the molar volume at the
total homogenization temperature. One important advantage of this
method is that the compositions, molar volumes and homogenization
pressures of CO2–H2O–NaCl inclusions can be obtained simultaneously
without using optical volume fractions of the CO2 phase at the partial
homogenization temperatures. The whole calculation is based on a
bisection algorithm, whose main steps are summarized as follows:
Step 1 Input the dissociation temperature of CO2 clathrate Tm(cla), the
partial homogenization temperature of CO2 phases Th(CO2)
including partial homogenization modes (liquid or vapor or critical), and the total homogenization temperature Th(tot), then
use the equation of Chen (1972) to calculate salinity mNaCl
with input Tm(cla).
Step 2 Calculate the partial homogenization pressure Ph(CO2) with
input Th(CO2) by the simple vapor pressure equation of Span
and Wagner (1996).
Step 3 Calculate xCO2 with Th(CO2), Ph(CO2) and mNaCl from the CO2
solubility model (Eq. (6)).
V
Step 4 Calculate Vm
with Th(CO2) from the saturated density equations
of CO2 (Span and Wagner, 1996), where the CO2 phase is approxL
imated as pure CO2; at the same time, calculate Vm
with Th(CO2),
Ph(CO2), mNaCl and xCO2 from the PVTx model of Mao et al. (2010).
Step 5 Calculate the maximal volume fraction FV(max) at Th(CO2) by
Th(tot). Because the maximal applicable pressure of the CO2 solubility model is 1500 bar, FV(max) is calculated from Eq. (13),
where the maximal CO2 content is calculated from the CO2
solubility model with Th(tot), 1500 bar and mNaCl.
Modify FV1 or FV2
by bisection
Vmcal - Vm= 0?
Bulk
Calculated xCO
, Ph (tot),
2
FV and Vm are correct
End
Fig. 4. A bisection algorithm for calculating xBulk
CO2 , Ph(tot), and Vm of the CO2–H2O–NaCl
fluid inclusion at given Tm(cla), Th(CO2) and Th (tot): xBulk
CO2 is mole fraction of CO2 of the
total inclusion, Ph(tot) is the total homogenization pressure, Vm is the bulk molar volume
of the total inclusion, Tm(cla) is the dissociation temperature of CO2 clathrate, Th(CO2) is
the partial homogenization temperature of CO2 phases, and Th(tot) is the total homogenization temperature. For the meanings of other parameters see Section 3.1.
Step 6 Assume initial volume fraction of vapor bubble FV1 = 0, final volume fraction of vapor bubble FV2 = FV(max), FV = (FV1 + FV2)/2,
and calculate xBulk
CO2 and Vm from Eqs. (13) and (14), respectively.
Then, use a bisection method to calculate the total homogenization pressure Ph(tot) with Th(tot), mNaCl andxBulk
CO2 from the CO2 solubility model developed here. Finally, calculate the bulk molar
cal
volume Vm
with Th(tot), Ph(tot), mNaCl and xBulk
CO2 from the PVTx
cal
model of Mao et al. (2010). Generally, Vm
is not equal to Vm.
Therefore, the initial value of FV and the calculated xBulk
CO2 and
Ph(tot) are not right because the molar volume of fluid inclusion
is constant during heating.
Step 7 Go to Step 6 and modify the value of FV1 or FV2 by a bisection
cal
algorithm until the calculated Vm
equals to Vm. Under this condical
tion, FV is right and the calculated xBulk
CO2 , Ph(tot) and Vm represent
the bulk content of CO2, total homogenization pressure and
bulk molar volume, respectively. Fig. 4 shows the flow chart
cal
of the algorithm, whose convergence condition is |Vm
−
−5
3
−1
Vm| b 10
cm mol .
Fig. 2. CO2 solubility in aqueous NaCl solutions (this model vs. experimental data): (a) T = 273.15 K and P CO2 = 1 bar, (b) T = 298.15 K and P CO2 = 1 bar, (c) T = 313.2 K, 353.12 K,
432.9 K with mNaCl = 4 mol kg−1, (d) T = 423.15 K, 473.15 K with mNaCl = 1.09 mol kg−1, (e) T = 523.15 K, 573.15 K, 623.15 K with mNaCl = 1.09 mol kg−1, (f) T = 523.15 K,
623.15 K with mNaCl = 4.28 mol kg−1, (g) T = 673.15 K with mNaCl = 1.09 mol kg−1, (h) T = 723.15 K with mNaCl = 4.28 mol kg−1. P CO2 is the partial pressure of CO2.
54
S. Mao et al. / Chemical Geology 347 (2013) 43–58
700
0
0.08 0
0.07 0
0.06
0
0.0545
0.0
0
0.04
35
0.0
xCO = 0.030
2
600
Ph (tot) (bar)
500
400
CO2+H2O
0.025
300
0.020
200
100
0.015
0.010
0.005
0
0.001
300
350
400
V+L→L
450
500
550
600
Th (tot) (K)
Fig. 5. Isopleths of the CO2–H2O fluid mixtures: Ph(tot) is the total homogenization
pressure, Th(tot) is the total homogenization temperature, and “V + L → L” denotes
homogenization to liquid phase by the bubble disappearance.
If a fluid inclusion is represented by CO2–H2O system and finally
homogenizes to liquid phase, input parameter Tm(cla) in Step 1 can
be omitted owing to the salinity mNaCl = 0, and the maximal pressure
in Step 5 is set as critical pressure at temperature above 538 K, which
is calculated from Blencoe (2004). Other steps are the same as those
of ternary CO2–H2O–NaCl system.
The above iterative algorithm can also be applied to a two-phase
(liquid or vapor CO2 + liquid H2O) CO2–H2O–NaCl inclusion at room
temperatures. If Tm(cla) of the inclusion is above 273.15 K, we first
b
a 1200
Ph (tot) (bar)
0.02
800
600
08
0.
7
0.0
5
600
400
0.06
0.05
0.04
5
0.03
0.03
2
08
0. 7
0.0
0.06
0.05
0.04
5
xCO = 0.025
V+L→L
1000
0.03
0.03
Ph (tot) (bar)
V+L→L
1200
0.02
1000
800
calculate the inclusion salinity by equations of Diamond (1992) or by
the method of combining Raman analysis (Fall et al., 2011 or Wang et
al., 2011) with thermodynamic model (Bakker et al., 1996 or Duan
and Sun, 2006), then calculate the dissociation pressure of CO2 clathrate
by the thermodynamic model (Bakker et al., 1996 or Duan and Sun,
2006), and finally use the above iterative method to calculate the CO2
content of the CO2–H2O–NaCl inclusion based on the approximation
that the molar volume at Tm(cla) equals that at Th(tot). If Tm(cla) of
the inclusion is below 273.15 K, which is beyond the valid temperature
range of the CO2 solubility model developed here, we can first obtain
the inclusion salinity by the same method above, then calculate the
density of CO2 phase at room temperature (e.g., 293 K) by Raman analysis (Fall et al., 2011 or Wang et al., 2011) and further obtain the
pressure at room temperature by the equation of state of Span and
Wagner (1996), and finally use the iterative method to calculate the
CO2 content of the CO2–H2O–NaCl inclusion based on the approximation that the molar volume at room temperature equals that at Th(tot).
From the iterative method, homogenization pressures of CO2–H2O–
NaCl inclusions can also be calculated. Figs. 5 and 6 show the relation of
Ph(tot) and Th(tot) at given compositions. It can be seen that when the
CO2–H2O and CO2–H2O–NaCl fluid inclusions with low CO2 compositions homogenize to liquid phase, Ph(tot) generally increases slowly
with Th(tot) at the beginning, then decreases slowly, and finally
increases rapidly with Th(tot). Some Ph(tot)–Th(tot) curves at given
salinities with low CO2 compositions have a maximal Ph(tot) at low
temperatures. When the CO2–H2O–NaCl fluid inclusions with high
CO2 compositions homogenize to liquid phase, Ph(tot) generally
decreases with the increase of Th(tot) at the beginning and increases
slightly with increasing Th(tot) at last.
xCO = 0.02
2
400
0.015
0.015
0.01
0.005
0.001
200
0
300
200
0.01
0.005
0.001
-1
mNaCl = 1 mol kg
400
500
600
0
300
700
400
500
Th (tot) (K)
700
d
400
08
0.
0.015
6
600
7
2
5
2
800
xCO = 0.02
1000
0.0
xCO = 0.02
0.0
1200
0.0
1400
0.02
1000
V+L→L
4
0.0
35
0.0
0.03
5
0.02
08
0.
7
0.0
0.06
0.05
0.04 35
0.0
0.03
5
1400
1200
Ph (tot) (bar)
600
Th (tot) (K)
Ph (tot) (bar)
c
-1
mNaCl = 2 mol kg
800
0.015
600
400
200
0.01
0.005
0
mNaCl = 3 mol kg
0.001
300
400
0.01
200
500
Th (tot) (K)
600
700
V+L→L
-1
mNaCl = 4 mol kg
0.005
-1
0
0.001
300
400
500
600
700
Th (tot) (K)
Fig. 6. Isopleths of the CO2–H2O–NaCl fluid mixtures: (a) mNaCl = 1 mol kg−1, (b) mNaCl = 2 mol kg−1, (c) mNaCl = 3 mol kg−1, (d) mNaCl = 4 mol kg−1. mNaCl is the molality of
NaCl, Ph(tot) is the total homogenization pressure, Th(tot) is the total homogenization temperature, x(CO2) is the bulk mole fraction of CO2, and “V + L → L” denotes homogenization to liquid phase by the bubble disappearance.
S. Mao et al. / Chemical Geology 347 (2013) 43–58
55
Table 12
Calculated results for natural CO2–H2O–NaCl inclusions in quartz from Moose River, NY, USA (Darling and Bassett, 2002).
Inclusion no.
Tm(CO2) (K)
Tm(cla) (K)
Th(CO2) (K)
Th(tot) (K)
Ph(tot) (bar)
x(CO2)
x(NaCl)
x(H2O)
Vm (cm3 mol−1)
092799
092799
092799
092799
092799
101899
102899
216.25
216.55
–
216.45
216.55
216.45
216.55
282.55
282.65
282.25
282.45
282.45
282.65
282.35
304.05(L)
303.65(L)
303.95(V)
303.35(L)
304.15(C)
304.05(L)
303.95(L)
499.75
486.95
457.95
493.75
489.55
496.95
504.65
979.92
1165.3
772.91
1222.18
867.34
981.45
1041.97
0.07778
0.0722
0.0445
0.07827
0.06403
0.07615
0.08222
0.00351
0.00295
0.00541
0.00408
0.00414
0.00294
0.00463
0.91871
0.92485
0.95009
0.91765
0.93182
0.92092
0.91315
22.18
21.44
20.55
21.68
21.69
22.06
22.33
A-1
A-2
A-3
B-1
B-2
A-1
A-1
Tm(CO2) = CO2 melting temperature; Tm(cla) = final clathrate melting temperature; Th(CO2) = CO2 homogenization temperature; L = liquid; V = vapor; C = critical behavior;
Th(tot) = total homogenization temperature; Ph(tot) = total homogenization pressure; x(CO2) = bulk mole fraction of CO2; x(NaCl) = bulk mole fraction of NaCl; x(H2O) = bulk
mole fraction of H2O; Vm = bulk molar volume of inclusion.
3.2. Application examples and computation program
The solubility model of CO2 in aqueous NaCl solutions together with
the iterative method described above can be applied in analysis of natural CO2–H2O–NaCl fluid inclusions. For example, microthermometric
data from natural CO2–H2O–NaCl inclusions in quartz from Moose
River, NY, USA (Darling and Bassett, 2002) are listed in Table 12. From
Tm(cla), Th(CO2) and Th(tot), the compositions, homogenization
pressure and bulk molar volume can be obtained (Table 12). Another
example is given by natural CO2–H2O–NaCl inclusions in quartz from
Longwangzhuang Pb–Zn deposit, Henan province, China (Xi et al.,
2010). These inclusions approximated by CO2–H2O–NaCl system form
from immiscible boiling fluids and CO2 phases homogenize by critical
behavior (Xi et al., 2010), so the homogenization temperature and pressure represent the temperature and pressure of entrapment. Table 13
lists the microthermometric data and the calculated results, where critical homogenization temperature of CO2 is corrected as 304.1282 K
(Span and Wagner, 1996) in the calculation. It can be seen from
Table 13 that the temperature and pressure of entrapment range from
478.55 to 509.85 K and 898.47–1130.17 bar. The calculated depth of
formation of the Pb–Zn deposit is estimated to be 3.0–3.8 km assuming
a lithostatic pressure gradient of 300 bar/km.
The CO2 solubility model and its application to CO2–H2O–NaCl
fluid inclusions have been programmed in Fortran95 language. The
source code of the program can be obtained from the corresponding
author ([email protected]). Table 14 lists a calculation example
using the CO2 solubility model and the iterative method for a
three-phase CO2–H2O–NaCl inclusion finally homogenizing to liquid
phase, where input variables Tm(cla) = 279.15 K, Th(CO2) =
293.15 K (L + V → V), and Th(tot) = 505 K (L + V → L).
Table 13
Calculated results for natural inclusions in quartz approximated as CO2–H2O–NaCl system from Longwangzhuang rock, Henan province, China (Xi et al., 2010).
Inclusion no.
Tm(CO2) (K)
Tm(cla) (K)
Th(CO2) (K)
Th(tot) (K)
Ph(tot) (bar)
x(CO2)
x(NaCl)
x(H2O)
Vm (cm3 mol−1)
1
2
3
4
5
6
7
8
9
217.05
216.75
216.15
–
216.65
–
–
–
–
281.75
282.05
281.05
280.95
280.45
279.95
280.35
280.35
280.55
304.15(C)
304.15(C)
304.05(C)
304.45(C)
304.15(C)
304.15(C)
304.55(C)
304.45(C)
304.35(C)
499.85
490.05
498.85
509.85
478.55
479.25
483.35
482.95
481.55
898.47
906.99
958.71
961.08
1087.08
1130.17
1080.54
1081.95
1068.56
0.06701
0.06247
0.06270
0.06815
0.04912
0.04752
0.05097
0.05078
0.05092
0.00814
0.00646
0.01213
0.01261
0.01567
0.01846
0.01619
0.01620
0.01508
0.92485
0.93107
0.92517
0.91924
0.93521
0.93402
0.93284
0.93303
0.93400
21.96
21.58
21.70
22.13
20.70
20.62
20.85
20.84
20.83
The meanings of Tm(CO2), Tm(cla), Th(CO2), C, Th(tot), Ph(tot), x(CO2), x(NaCl), x(H2O) and Vm are the same as those in Table 12.
Table 14
Calculated results for a CO2–H2O–NaCl fluid inclusion (V + L → L).
Input variables
Output variables
Tm(cla) = 279.15 K
Th(CO2) = 293.15 K (L + V → V)
Th(tot) = 505 K (L + V → L)
Ph(CO2) = 57.29 bar
mNaCl = 1.360 mol kg−1
FV = 0.1730
x(CO2) = 0.03507
x(NaCl) = 0.02308
x(H2O) = 0.94185
Ph(tot) = 459.19 bar
Vm = 21.76 cm3 mol−1
Isochore (P-T relation)
T (K)
505
525
550
575
600
625
650
675
700
725
750
775
800
825
850
P (bar)
459.19
764.85
1151.66
1543.73
1940.61
2341.77
2746.28
3153.38
3562.16
3971.86
4381.71
4790.95
5199.13
5605.77
6010.44
Note: The meanings of Tm(cla), Th(CO2), Th(tot), Ph(CO2), FV, x(CO2), x(NaCl), x(H2O), Ph(tot) and Vm are the same as those in Table 12; mNaCl is the molality of NaCl; FV is the
volume fraction CO2 phase at Th(CO2); “V + L → V” denotes homogenization to vapor phase by the liquid disappearance; “V + L → L” denotes homogenization to liquid
phase by the bubble disappearance; Isochore is calculated from the PVTx model of Mao et al. (2010).
56
S. Mao et al. / Chemical Geology 347 (2013) 43–58
4. Conclusions
Using updated experimental data available and the electrolyte
solution theory of Pitzer (1973), an improved activity–fugacity phase
equilibrium model is presented to calculate CO2 solubility in pure
water and aqueous NaCl solutions covering a large T − P − mNaCl
range of 273.15–723.15 K, 1–1500 bar and 0–4.5 mol kg−1, with or
close to experimental accuracy. With a bisection algorithm, the CO2 solubility model and the updated volumetric model of the CO2–H2O–NaCl
fluids are finally combined together and applied in the studies of fluid
inclusions, thus compositions, isopleths, homogenization pressures,
homogenization volumes or densities, and isochores can be obtained to
interpret corresponding microthermometric and Raman analysis data
of CO2–H2O–NaCl inclusions. It should be noted that the calculated
results beyond the valid T − P − mNaCl range of the CO2 solubility
model, e.g., Ph(tot) >1500 bar, is not warranted when the model is
applied to CO2–H2O–NaCl fluid inclusions.
Acknowledgments
We thank the two anonymous reviewers for the constructive suggestions. This work is supported by the funds (41173072, 90914010)
awarded by the National Natural Science Foundation of China.
Appendix A. Supplementary data
Supplementary data to this article can be found online at http://
dx.doi.org/10.1016/j.chemgeo.2013.03.010.
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