a predictive model for the pvtx properties of co2-h2o-nacl
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Accepted Manuscript
A predictive model for the PVTx properties of CO2-H2O-NaCl fluid mixture up
to high temperature and high pressure
Shide Mao, Jiawen Hu, Yuesha Zhang, Mengxin Lü
PII: S0883-2927(15)00010-4
DOI: http://dx.doi.org/10.1016/j.apgeochem.2015.01.003
Reference: AG 3403
To appear in: Applied Geochemistry
Please cite this article as: Mao, S., Hu, J., Zhang, Y., Lü, M., A predictive model for the PVTx properties of CO2-
H2O-NaCl fluid mixture up to high temperature and high pressure, Applied Geochemistry (2015), doi: http://
dx.doi.org/10.1016/j.apgeochem.2015.01.003
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1
A predictive model for the PVTx properties of
CO2-H2O-NaCl fluid mixture up to high temperature and
high pressure
Shide Mao1*, Jiawen Hu2, Yuesha Zhang1, Mengxin Lü1
1 State Key Laboratory of Geological Processes and Mineral Resources, and School of
Earth Sciences and Resources, China University of Geosciences, Beijing, 100083,
China
2 College of Resources, Shijiazhuang University of Economics, Shijiazhuang, 050031,
China
*The corresponding author: ([email protected])
2
Abstract
A predictive thermodynamic model is constructed to calculate the
pressure-volume-temperature-composition (PVTx) properties of CO2-H2O-NaCl fluid
mixtures by the Helmholtz free energy model of CO2-H2O fluid mixtures. The new
model uses no other mixing parameters but those of the CO2-H2O system, because the
Helmholtz free energy of H2O-NaCl fluid at a given composition is equivalently
converted into that of pure H2O by a scaled temperature redefined at the same
pressure. In addition, the parameters developed by Driesner (2007) in the PVTx model
of H2O-NaCl fluid system are refitted by the IAPWS-95 formulation. Comparisons
with experimental data available show that the model can reproduce the single-phase
PVTx properties of H2O-NaCl and CO2-H2O-NaCl fluid mixtures of all compositions
from 273 to 1273 K and from 0 to 5000 bar, within or close to experimental
uncertainty in most cases (with slightly lower accuracy at 5000-10000 bar). The
isochores of CO2-H2O-NaCl fluid can be obtained from this model by a bisection
algorithm, and an application example is given to analyze some natural
CO2-H2O-NaCl fluid inclusions in quartz from a wolframite deposit. Computer
program code for calculation of molar volume of the CO2-H2O-NaCl fluid as a
function of temperature, pressure and composition can be obtained from the
corresponding author ([email protected]).
Keywords: CO2-H2O-NaCl, fluid mixture, volume, equation of state, PVTx,
isochore
3
1. Introduction
Fluid inclusions that can be approximated by the CO2-H2O-NaCl system are
common in various geologic environments, e.g., hydrothermal ore deposits (Bodnar,
1995; Roedder, 1984; Roedder and Bodnar, 1997; Yoo et al., 2011) and metamorphic
rocks (Crawford, 1981; Cuney et al., 2007; Touret, 1981; Touret, 2001). In the studies
of CO2-H2O-NaCl vapor-liquid inclusions, inclusion compositions (salinities and CO2
contents) and homogenization pressures are often obtained by combining
experimental microthermometric and Raman analysis (Azbej et al., 2007; Becker et
al., 2010; Chen, 1972; Darling, 1991; Diamond, 1992; Fall et al., 2011; Wang et al.,
2011) with theoretical phase-equilibrium models (Bakker et al., 1996; Bakker, 1997;
Barton and Chou, 1993; Diamond, 2003; Duan and Sun, 2006; Dubacq et al., 2013;
Dubessy et al., 2005; Ji and Zhu, 2012; Ji and Zhu, 2013; Mao et al., 2013; Sun and
Dubessy, 2012). In particular, the isochores of CO2-H2O-NaCl fluid inclusions, along
which the fluid inclusions were trapped, are often calculated from the
pressure-volume-temperature-composition (PVTx) models (Bakker, 1999; Brown and
Lamb, 1989; Duan et al., 2008; Mao et al., 2010; Sun and Dubessy, 2012).
In the near decades, there have been some experimental studies for the PVTx
properties of CO2-H2O-NaCl fluid mixture (Gehrig et al., 1986; Johnson, 1992; Li et
al., 2004; Nighswander et al., 1989; Schmidt and Bodnar, 2000; Schmidt et al., 1995;
Song et al., 2005; Song et al., 2013; Teng and Yamasaki, 1998; Yan et al., 2011).
However, these data are very scattered, and most of them cover a limited
temperature-pressure-composition space that is only valid for CO2 sequestration
4
environments, and they are inconvenient and inadequate for fluid inclusion research.
Therefore, theorists devoted extensive efforts to modeling the PVTx properties of
CO2-H2O-NaCl fluid so as to interpolate among experimental data points or
extrapolate beyond the data range (Bachu and Adams, 2003; Bakker, 1999; Bando et
al., 2004; Brown and Lamb, 1989; Duan et al., 2008; Duan et al., 1995; Mao et al.,
2010; Song et al., 2005; Sun and Dubessy, 2012; Teng and Yamasaki, 1998). Simple
empirical volumetric models (Bachu and Adams, 2003; Bando et al., 2004; Song et al.,
2005; Teng and Yamasaki, 1998) are not commented here for their narrowly
applicable temperature-pressure-composition conditions. Brown and Lamb (1989)
proposed that an isochore of CO2-H2O-NaCl fluid can be calculated by a linear
interpolation between the isochore of a pure CO2 and that of the H2O-NaCl system.
This empirical approach is applicable at temperatures above 623 K and pressures of
2000-10000 bar. Duan et al. (1995) developed an equation of state (EOS) on the basis
of perturbation theory to represent the phase-equilibrium and PVTx properties of
CO2-H2O-NaCl fluid, which is valid for 573-1273 K, 0–6000 bar and 0-30 wt% of
NaCl (relative to NaCl-H2O). But Duan’s EOS cannot calculate thermodynamic
properties of CO2-H2O-NaCl fluid below 573 K, where quite a few fluid inclusions
may experience partial and/or total homogenization. Bakker (1999) adapted the EOS
of Bowers and Helgeson (1983) to model the PVTx properties of
H2O-CO2-CH4-N2-NaCl system, which is valid for CO2-H2O-NaCl fluid up to about
30 wt% of NaCl (relative to NaCl-H2O) at pressures above 500 bar and temperatures
between 623 and 973 K. Duan et al. (2008) established an accurate density model for
5
CO2-H2O-NaCl fluid mixture based on their own CO2-H2O density model and the
H2O-NaCl density model of Rogers and Pitzer (1982). This model is only valid at
T≤573 K, P≤1000 bar and salinity below 6 mol kg-1 NaCl, so it has limited
applicability in fluid inclusion studies. Mao et al. (2010) proposed a PVTx model of
gas-H2O-NaCl fluids on the approximation that the apparent molar volume of gas in
water equals to that of gas in aqueous NaCl solution. The model can predict the molar
volume or density of CO2-H2O-NaCl fluid mixture from 273 to 1273 K and from 0 to
5000 bar. Sun and Dubessy (2012) developed an improved SAFT-LJ equation of state
to calculate the vapor–liquid equilibrium and PVTx properties of CO2–H2O-NaCl
system over a temperature-pressure-composition range of 273-573 K, 1-1000 bar and
0-6 mol kg-1 NaCl. Obviously, this EOS cannot predict the PVTx properties above 573
K and 1000 bar. Those models described above can be used to calculate the
volumetric properties of CO2-H2O-NaCl fluid, but most of them have only limited
applicability to fluid inclusions owing to the relatively narrow valid range of
temperature and/or pressure and/or composition.
To overcome deficiencies of previous models in applications to fluid inclusions,
we present a predictive model to calculate the PVTx properties of CO2-H2O-NaCl
fluid (273-1273 K, 0-10000 bar and 0-1 mole fraction of NaCl or CO2) based on
Helmholtz free energy model of CO2-H2O fluid without using additional mixing
parameters. The framework of this paper is as follows: First, a volumetric model
explicit in Helmholtz free energy is presented. Then this model is compared with
experimental PVTx data of H2O-NaCl and CO2-H2O-NaCl fluids. Finally, the isochore
6
calculation and the application of the model to CO2-H2O-NaCl fluid inclusions are
discussed.
2. Volumetric model explicit in Helmholtz free energy
The volumetric model of a multi-component fluid mixture is in terms of
dimensionless Helmholtz free energy α , defined as
A
RTα = (1)
where A is molar Helmholtz free energy, R is molar gas constant
( 1 18.314472 J mol K− −⋅ ⋅ ), and T is temperature in K. So are the same in the
following equations.
The dimensionless Helmholtz free energy α of the mixture is represented by
id Emα α α= + (2)
where idmα is the dimensionless Helmholtz free energy of an ideal mixture and Eα
is the excess dimensionless Helmholtz free energy. idmα directly comes from the
fundamental equations of pure fluids and can be written as
id 0 rm m i i
1
n n0 r
i i i i ii 1 i 1
( , , ) ( , )
( , ) ln( ) ( , )
n
i
x x
x x x
α α δ τ α δ τ
α δ τ α δ τ
=
= =
= +
⎡ ⎤= + +⎣ ⎦
∑
∑ ∑ (3)
where 0mα is the ideal-gas part of dimensionless Helmholtz free energy of the
mixture, 0iα and r
iα are the ideal-gas part and residual part of dimensionless
Helmholtz free energy of component i, respectively, ix is the mole fraction of the
component i. The superscripts “id”, “0” and “r” denote ideal mixing, the ideal-gas part
and residual part of dimensionless Helmholtz free energy, respectively. The subscripts
7
“i" and “m” denote the component and mixture, respectively. δ and τ are reduced
parameters, which are defined by
c
ρδρ
= (4)
cT
Tτ = (5)
where ρ is the density of mixture, and cρ and cT are defined as same as Lemmon
and Jacobsen (1999) by
1n n 1
ic i j ij
i 1 i 1 j i 1ci
nxx xρ ζ
ρ
−−
= = = +
⎡ ⎤= +⎢ ⎥
⎣ ⎦∑ ∑ ∑ (6)
ij
n n 1 n
c i ci i j iji 1 i 1 j i 1
T x T x xβ ς−
= = = +
= +∑ ∑ ∑ (7)
where ciρ and ciT are the critical density and critical temperature of the component
i, respectively, jx denotes mole fraction of component j, and ijζ , ijς , and ijβ are
mixture-dependent binary parameters associated with components i and j.
The Eα in equation of (2) is given by
k k
n 1 n 10E
i j ij ki 1 j=i 1 k 1
d tx x F Nα δ τ−
= + =
= ∑ ∑ ∑ (8)
where kN , kd and kt are general parameters independent of fluids, which can be
found from the model of Lemmon and Jacobsen (1999) (Table 1), ijF is a
mixture-dependent binary parameter of components i and j.
8
Table 1: Coefficients and exponents of mixture equation (8) (Lemmon and Jacobsen, 1999)
k kN kd kt
1 -2.45476271425D-2 1 2
2 -2.41206117483D-1 1 4
3 -5.13801950309D-3 1 -2
4 -2.39824834123D-2 2 1
5 2.59772344008D-1 3 4
6 -1.72014123104D-1 4 4
7 4.29490028551D-2 5 4
8 -2.02108593862D-4 6 0
9 -3.82984234857D-3 6 4
10 2.69923313540 D -6 8 -2
The residual part of dimensionless Helmholtz free energy of a multi-component
fluid mixture rα is defined by
n
r r Ei i
i 1
( , ) ( , , )x xα α δ τ α δ τ=
= +∑ (9)
For CO2-H2O-NaCl fluid, rα can be divided into two components: CO2
component and pseudo H2O-NaCl component. Therefore, Eq. (9) combining with Eq.
(8) can be rewritten as
2 2 2 2
k k
2 2 2 2
r r rCO CO H O-NaCl H O-NaCl
10
CO H O-NaCl CO H O-NaCl kk 1
( , ) ( , )
d t
x x
x x F N
α α δ τ α δ τ
δ τ−=
= + +
∑ (10)
where 2
rCO ( , )α δ τ and
2
rH O-NaCl ( , )α δ τ are the residual parts of dimensionless
Helmholtz free energy of CO2 and H2O-NaCl fluids, respectively, and 2 2CO H O-NaClF − is
the pseudo binary parameter of CO2 and H2O-NaCl. In this work, 2
rCO ( , )α δ τ is
calculated from the EOS of Span and Wagner (1996), and 2
rH O-NaCl ( , )α δ τ can be
calculated from 2
r *H O ( , )α δ τ , the residual part of dimensionless Helmholtz free energy
EOS of pure water (Wagner and Pruß, 2002), where *τ is defined as
9
* c*
V
T
Tτ = (11)
where the scaled temperature *VT in K is refitted with the approach of Driesner
(2007), as will be discussed in the following section. Therefore, Eq. (10) can be
rewritten as
2 2 2 2
k k
2 2 2 2
r r r *CO CO H O-NaCl H O
10
CO H O-NaCl CO H O kk 1
( , ) ( , )
d t
x x
x x F N
α α δ τ α δ τ
δ τ−=
= + +
∑ (12)
Eqns. (9)-(12) are developed in this study. It can be seen that if the residual part
of dimensionless Helmholtz free energy equations of state of CO2, H2O and CO2-H2O
fluids are known, the PVTx properties of CO2-H2O-NaCl fluid mixture can be
obtained. In other words, the Helmholtz free energy of ternary CO2-H2O-NaCl fluid
can be equivalently converted into that of binary CO2-H2O fluid, because the
Helmholtz free energy of H2O-NaCl fluid at a given composition can be equivalently
converted into that of pure H2O at the same pressure by a scaled temperature.
The critical parameters of pure fluids (CO2 and H2O) are listed in Table 2. The
values of ijζ , ijς , ijβ and ijF for the CO2-H2O fluid are taken from Mao et al.
(2010) (Table 3).
10
Table 2: Critical parameters of pure fluids (Mao et al., 2010)
i ci (K)T -3ci (mol dm )ρ ⋅
CO2 304.1282 10.624978698
H2O 647.096 17.87371609
Table 3: Parameters of the CO2-H2O fluid mixture (Mao et al., 2010)
Binary mixture ijF 3 1ij (dm mol )ζ −⋅ ij (K)ς ijβ
CO2-H2O 1.196 0.0108 -223.33 1.0
The density or molar volume of CO2-H2O-NaCl fluid mixture can be calculated
from the following equations with the Newton iterative method.
r
1P RTτ
αρ δδ
⎡ ⎤⎛ ⎞∂= +⎢ ⎥⎜ ⎟∂⎢ ⎥⎝ ⎠⎣ ⎦ (13)
2 2
2 2
*
k k
2 2 2 2
r rrCO H O
CO H O-NaCl
101
CO H O-NaCl CO H O k kk 1
d t
x x
x x F N d
τ τ τ
α ααδ δ δ
δ τ−−
=
⎛ ⎞ ⎛ ⎞∂ ∂⎛ ⎞∂ = +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠
+ ∑ (14)
where the derivatives 2
rCO
τ
αδ
⎛ ⎞∂⎜ ⎟⎜ ⎟∂⎝ ⎠
and 2
*
rH O
τ
αδ
⎛ ⎞∂⎜ ⎟⎜ ⎟∂⎝ ⎠
denote the same meanings as in the
references (Span and Wagner, 1996; Wagner and Pruß, 2002).
In the use of the Newton iterative method, if the CO2-H2O-NaCl fluid mixture is
in vapor or supercritical state, the initial density of mixture can be set equal to that of
ideal gas. If the CO2-H2O-NaCl fluid mixture is in liquid state, the initial density can
be set equal to the saturated liquid density of pure water.
3. Results and discussions
3.1 H2O-NaCl fluid mixture
11
In the last three decades, over ten PVTx models or equations of state have been
proposed for H2O-NaCl fluid mixture (Table 4), each model having its own applicable
range. From the aspect of application of models in fluid inclusion research, the best
PVTx model is the model of Driesner (2007), which covers a large
temperature-pressure-composition range [273-1273 K, 0-5000 bar and 0-1 NaClx (mole
fraction of NaCl)]. However, the equation of state of water used by Driesner (2007) is
the IAPS-84 EOS of Haar et al. (1984). In this work, the more accurate and versatile
formulation of International Association for the Properties of Water and Steam
(IAPWS-95) (Wagner and Pruß, 2002) was used to replace the IAPS-84 EOS.
Therefore, *VT in Eq. (11) was refitted with experimental volumetric data of
H2O-NaCl system based on the IAPWS-95 formulation, and the corresponding
parameters related to *VT ( 11n , 21n , 22n , 300n , 301n , 302n , 310n , 311n and 312n ) are
listed in Table 5. Details on *VT see the Appendix I.
12
Table 4: PVTx models for H2O-NaCl fluid mixture
References T (K) P (bar) NaClx
Rogers and Pitzer (1982) 273-573 Ps-1000 0-0.0901
Bowers and Helgeson (1985) 623-873 500-2000 0-1
Zhang and Frantz (1987) 453-973 <3000 0-1
Brown (1989) >623 2000-10000 0-1
Lvov and Wood (1990) 273-973 1-10000 0-0.2356
Anderko and Pitzer (1993) 573-1200 ≤5000 0-1
Bodnar and Vityk (1994) 323-973 ≤6000 0-0.17
Spivey et al. (2004) 273-548 1-2000 0-0.0975
Driesner (2007) 273-1273 1-5000 0-1
Mao and Duan (2008) 273-573 Ps-1000 0-0.0975
Ji and Zhu (2012) 273-473 ≤600 0-0.0975
Sun and Dubessy (2012) 273-573 0-1000 0-0.0975
Mantegazzi et al. (2013) 293-673 5000-45000 0-0.0513
Note: Ps is the vapor pressure of solutions, and xNaCl is the mole fraction of NaCl in undissociated
state.
13
Table 5: Refitted parameters for the new volumetric model of binary H2O-NaCl fluid based on the IAPWS-95 formulation
11n 22 2 (-0.13786998 10 )-0.45146040 10 -0.29812895 10 e P−×× × ×
21n 1 3-0.26105212 10 -0.20362282 10 P−× ×
22n 1 5 8 20.31998439 10 +0.36137426 10 +0.15608215 10P P− − −× × ×
300n 7 3 20.64988075 10 /( +0.42937670 10 )P× ×
301n 32 2 (-0.59264170 10 )-0.47287373 10 -0.81190283 10 e P−×× ×
302n 23 (-0.56045287 10 )0.28803474 10 e P−××
310n 21 (-0.22339191 10 ) 4-0.68388688 10 e -0.53332903 10P P−− × −× ×
311n 22 2 (-0.10315741 10 )-0.41933849 10 +0.19198040 10 e P−×× ×
312n 3-0.29097042-0.83864808 10 P−×
Once temperature, pressure and composition of binary H2O-NaCl fluid are given,
*VT and the corresponding volumetric properties can be calculated from these
parameters (Table 5). Table 6 shows the average and maximum absolute volume
deviations of the model from each data set. Fig. 1 shows the comparisons between
experimental and predicted results of the H2O-NaCl system. From Table 6 and Fig. 1,
it can be seen that the average absolute volume deviation is within 0.1% at
temperatures and pressures below 573 K and 1000 bar, above which average deviation
is about 1%. The model shows excellent agreement with the 3338 data points of
Hilbert (1979), with an average deviation of 0.08% (including both experimental and
extrapolated data). An average deviation of 1.26% is found between the predicted
results and the experimental data of Mantegazzi et al. (2013) at 5000-9000 bar. As
14
seen from the figure, the molar volumes of H2O-NaCl fluid mixture up to 1273 K and
10000 bar can be reproduced by this improved model within or close to experimental
uncertainties.
15
Table 6: Deviations of this model for the molar volume of H2O-NaCl fluid mixture from experimental data
References T (K) P (bar) NaClx Nd AAD
(%)
MAD
(%)
Ellis (1966) 298.15-473.15 20.27 0.0018-0.0177 32 0.02 0.06
Fabuss et al.
(1966)
298.15-348.15 1.01 0.0018-0.0431 32 0.04 0.18
Vaslow (1966) 298.15 1.01 0-0.0619 21 0.02 0.04
Dunn (1968) 273.2-338.15 1.01 0.0038-0.0178 17 0.10 0.68
Korosi and
Fabuss (1968)
298.15-323.15 1.01 0.0018-0.0609 28 0.03 0.11
Millero and
Drost-Hansen
(1968)
293.15-313.15 1.01 0.0021 11 0.01 0.01
Wirth and
LoSurdo (1968)
273.15-318.21 1.01 0.0672 9 0.06 0.08
Ostroff et al.
(1969)
298.15 1.01 0.0181-0.0917 5 0.09 0.15
Vaslow (1969) 278.15 1.01 0.009-0.0593 19 0.15 0.26
Millero (1970) 273.15-328.15 1.01 0.0002-0.0171 83 0.06 0.59
Perron et al.
(1975)
274.65-318.15 1.01 0.0005-0.0515 69 0.08 0.51
Urusova (1975) 623.15-823.15 107.87-980.68 0.0331-0.3153 376 0.82 8.03
Chen et al. (1977) 273.15-323.15 99.9-1001.2 0.0005-0.0349 179 0.03 0.25
Goncalves and
Kestin (1977)
298.15-323.15 1.01 0.0001-0.0949 44 0.01 0.06
Millero et al.
(1977)
298.15 1.01 0.0002-0.0177 12 0.02 0.03
Hilbert (1979) 293.15-673.15 100-4000 0-0.0932 3338 0.08 1.16
Olofsson (1979) 298.15 1.01 0.0011-0.0482 40 0.15 0.04
Chen et al. (1980) 273.15-308.15 1.01 0.002-0.0263 57 0.06 0.58
Dessauges et al.
(1980)
288.15-318.15 1.01 0.003-0.0964 58 0.02 0.04
Out and Los
(1980)
278.15-368.15 1.01 0.0018-0.0212 60 0.03 0.25
Grant-Taylor
(1981)
446.18-627.38 200 0.0018-0.0672 68 0.99 9.77
Rogers et al.
(1982)
303.15-473.15 20.27 0-0.0733 63 0.03 0.16
Surdo et al.
(1982)
278.15-318.15 1.01 0.0066-0.0971 46 0.05 0.26
Gehrig et al.
(1983)
573.15-873.15 100-3000 0.0193-0.0715 230 0.70 7.57
Romankiw and 298.15-318.15 1.01 0-0.099 45 0.03 0.07
16
Chou (1983)
Albert and Wood
(1984)
298.15-673.1 1.01-385 0.0018-0.0825 18 0.23 2.02
Gates and Wood
(1985)
298.15 1.01-407.8 0.001-0.0825 40 0.03 0.12
Connaughton et
al. (1986)
308.15-368.15 1.01 0.0039-0.1004 141 0.02 0.14
Majer et al.
(1988)
321.57-597.45 1-401.6 0.001-0.0833 442 0.04 0.24
Oakes et al.
(1990)
298.23-308.12 1 0.0015-0.0981
23 0.02 0.09
Bischoff (1991) 573.15-773.15 58-581.8 0-0.3225 416 1.30 5.52
Crovetto et al.
(1993)
623.09-623.19 152.4-167.7 0.0045-0.0513 47 0.19 1.22
Manohar et al.
(1994)
298.15-413.15 1-20 0.0089-0.075 32 0.02 0.07
Simonson et al.
(1994)
298.05-522.98 70.5-415 0.003-0.0899 179 0.04 0.11
Apelblat and
Manzurola (1999)
277.15-343.15 1.01 0.0018-0.0177 201 0.03 0.32
Mironenko et al.
(2001)
273-293 1.01 0.0002-0.0978 90 0.05 0.51
Motin (2004) 308.15-323.15 1.01 0.0011-0.0177 15 0.04 0.18
Mantegazzi et al.
(2013)
293-673 5000-9000 0.0177-0.0513 30 1.26 4.72
Nd: number of data points; cal exp expAAD 100 ( ) /V V V= − , where calV and expV are the calculated
and experimental molar volumes, respectively; MAD: maximal absolute deviations calculated from improved
model.
17
0 100 200 300 400 500-1.0
-0.5
0.0
0.5
1.0
100(
Vca
l-Vex
p)/
Vex
p
P (bar)
Gates and Wood (1985) Number of data points = 40T-P-x
NaCl range: 298.15 K, 1.01-407.8 bar, 0.001-0.0825
a
0 50 100 150 200 250 300 350 400-1.0
-0.5
0.0
0.5
1.0
b
Albert and Wood (1984) Number of data points = 17T-P-x
NaCl range: 298.15-673.1 K, 1.01-385 bar, 0.0018-0.0825
100(
Vca
l-Vex
p)/
Vex
p
P (bar)
0 500 1000 1500 2000 2500 3000 3500-10
-5
0
5
10
c
Gehrig et al. (1983) Number of data points = 230T-P-x
NaCl range: 573.15-873.15 K, 100-3000 bar, 0.0193-0.0715
100(
Vca
l-Vex
p)/
Vex
p
P (bar)
0 200 400 600 800 1000-1.0
-0.5
0.0
0.5
1.0
d
Chen et al. (1977) Number of data points = 179T-P-x
NaCl range: 273.15-323.15 K, 99.9-1001.2 bar, 0.0005-0.0349
100(
Vca
l-Vex
p)/
Vex
p
P (bar)
0 100 200 300 400-1.0
-0.5
0.0
0.5
1.0
e
Majer et al. (1988) Number of data points = 442T-P-x
NaCl range: 321.57-597.45 K, 1-401.6 bar, 0.001-0.0833
100(
Vca
l-Vex
p)/
Vex
p
P (bar)
0 1000 2000 3000 4000-1.0
-0.5
0.0
0.5
1.0
f
Hilbert (1979) Number of data points = 3338T-P-x
NaCl range: 293.15-673.15 K, 100-4000 bar, 0-0.0932
100(
Vca
l-Vex
p)/
Vex
p
P (bar)
Fig. 1: Deviations of this model for the molar volume of H2O-NaCl fluid mixture
from experimental data: T, P, and NaClx are temperature, pressure and mole
fraction of NaCl, respectively. Vcal is the calculated molar volume using the IAPWS-95 formulation, and Vexp is the experimental molar volume.
18
3.2 CO2-H2O-NaCl fluid mixture
From Eq. (13), the molar volume or density of ternary CO2-H2O-NaCl fluid
mixture can be obtained by the Helmholtz free energy model of CO2-H2O fluid. In
order to test the predictive ability of the model, experimental single-phase volume
data available are compared with this model. Table 7 lists the average and maximum
absolute volume deviations, and Fig. 2 shows the volume deviations of Eq. (13) from
experimental data of the CO2-H2O-NaCl mixture. It can be seen from Table 7 and Fig.
2 that the average deviation is generally less than 1% at temperatures and pressures
below 573 K and 1000 bar, above which the average deviation increases to about
1-2%, which is in general agreement with experimental uncertainties. Some big
volume deviations occur in the near-critical or saturated region, where experimental
deviations of molar volume are much more than 2%. Table 8 shows the comparison of
high-pressure experimental data with Eq. (13) and the model of Mao et al. (2010),
which indicates that this model is better than our previous model. However, there are
large deviations from the saturated densities of CO2-H2O-NaCl mixture reported by
Gehrig et al. (1986). Hu et al. (2007) pointed out that the saturated densities of Gehrig
et al. (1986) below 647 K contain big uncertainties, which may be similar to the other
high-temperature saturated density data. Because experimental volumetric data of
CO2-H2O-NaCl mixture are still limited, future experimental work for the system is
needed to further validate Eq. (13).
Table 7: Deviations of this model for the molar volume of CO2-H2O-NaCl fluid mixture from experimental data
19
References T (K) P (bar) 2COx
NaClx Nd
AA
D
(%)
MA
D
(%)
Gehrig et
al. (1986) 673-773 400-3000 0-1 0-0.0331
28
1 1.49 9.76
Nighswand
er et al.
(1989)
353.35-473.65 21.1-100.3 0.0029-0.0154 0.003 34 0.90 2.83
Johnson
(1992)
1197.15-1213.
15 6600-7458 0.187-0.489
0.0244-0.06
17 4 1.98 2.89
Schmidt et
al. (1995) 623.15-973.15 2000-4000 0.0418 0.1633 13 1.73 5.01
Teng and
Yamasaki
(1998)
278-293 64.4-294.9 0.0232-0.0322 0.0106 24 0.15 0.37
Li et al.
(2004) 332.15 2.4-289.3 0-0.018 0.029 37 0.29 0.47
Song et al.
(2005) 276.15-283.15 40-130
0.00422-0.031
52 0.011 90 0.18 0.29
Yan et al.
(2011) 323.2-413.2 50-400 0.0031-0.0274 0-0.0824 54 0.14 0.37
Song et al.
(2013) 332.83-414.19
99.95-180.
17 0-0.014 0.017-0.067
40
0 0.19 0.41
cal exp expAAD 100 ( ) /V V V= − , where calV and expV are the calculated and experimental molar
volumes, respectively; MAD: maximal absolute deviations calculated from this model; Nd: number of data points.
20
Table 8: Comparisons of this model with the experimental PVTx data of CO2-H2O-NaCl fluid mixture at high temperatures and high pressures
References T (K) P (bar) 2COx
NaClx Vexp 1Vcal
2Vcal 1AD 2AD
Johnson (1992)
1211.15 7458 0.187 0.041 27.67 28.35 28.41 2.45 2.66
1203.15 6600 0.280 0.035 30.78 31.67 31.55 2.89 2.50
1197.15 6800 0.489 0.024 35.60 35.54 35.46 -0.16 -0.40
1213.15 7400 0.290 0.062 29.96 30.68 30.81 2.41 2.82
Schmidt et al.
(1995)
623.15 4000 0.042 0.163 21.19 21.00 21.58 -0.91 2.76
673.15 4000 0.042 0.163 21.80 21.64 22.03 -0.76 1.05
723.15 4000 0.042 0.163 22.49 22.31 22.60 -0.81 0.51
773.15 4000 0.042 0.163 23.24 23.02 23.21 -0.96 -0.15
823.15 4000 0.042 0.163 24.03 23.76 23.83 -1.10 -0.83
873.15 4000 0.042 0.163 24.91 24.56 24.48 -1.42 -1.74
923.15 4000 0.042 0.163 25.88 25.39 25.14 -1.90 -2.85
973.15 4000 0.042 0.163 27.07 26.27 25.83 -2.97 -4.58
Note: T, P, 2COx and NaClx denote temperature, pressure, mole fraction of CO2 and mole fraction
of NaCl, respectively. Vexp is experimental molar volume and Vcal is the calculated molar volume.
AD=100(Vcal-Vexp)/Vexp. Superscripts 1 and 2 denote the calculated results from this model and the
model of Mao et al. (2010), respectively.
21
500 1000 1500 2000 2500 3000-10
-5
0
5
10
Gehrig et al. (1986)T-P-x
CO2
-xNaCl
range: 673-773 K, 400-3000 bar, 0-1, 0-0.0331
100(
Vca
l-Vex
p)/
Vex
p
P (bar)a
6000 6500 7000 7500 8000-5.0
-2.5
0.0
2.5
5.0
Johnson (1992) T-P-x
CO2
-xNaCl
range:
1197-1213 K, 6600-7458 bar, 0.187-0.489, 0.0244-0.0617
100(
Vca
l-Vex
p)/
Vex
p
P (bar)b
600 700 800 900 1000-5.0
-2.5
0.0
2.5
5.0
Schmidt et al. (1995)T-P-x
CO2
-xNaCl
range: 623-973 K, 2000-4000 bar, 0.0418, 0.1633
100(
Vca
l-Vex
p)/
Vex
p
T (K)c
0 50 100 150 200 250 300-2
-1
0
1
2
Li et al. (2004)T-P-x
CO2
-xNaCl
range: 332 K, 2.4-289.3 bar, 0-0.018, 0.029
100(
Vca
l-Vex
p)/
Vex
p
P (bar)d
50 100 150 200 250 300-2
-1
0
1
2
Teng and Yamasaki (1998)T-P-x
CO2
-xNaCl
range: 278-293 K, 64.4-294.9 bar,
0.0232-0.0322, 0.0106
100(
Vca
l-Vex
p)/
Vex
p
P (bar)e
Fig. 2: Deviations of this model for the molar volume of CO2-H2O-NaCl fluid mixture from experimental data.
The above discussion suggests that we should be careful of the applicable range
of the model, although the predictive model has no other mixing parameters and
extrapolates well. For the CO2-H2O-NaCl mixture, it can be safely used to calculate
the single-phase molar volume up to 5000 bar, above which it can be used to 10000
0 50 100 150 200 250 300 350 400 450-2
-1
0
1
2
Yan et al. (2011)T-P-x
CO2
-xNaCl
range: 323.2-413.2 K, 50-400 bar
0.0031-0.0274, 0-0.0824
100(
Vca
l-Vex
p)/
Vex
p
P (bar)f
22
bar with slightly lower accuracy.
4. Isochore calculation of CO2-H2O-NaCl fluid inclusions
In the studies of CO2-H2O-NaCl fluid inclusions, isochores
(pressure-temperature relations) used to estimate the trapping temperatures and
pressures of the host minerals can be calculated from the PVTx model of
CO2-H2O-NaCl fluid mixture by a bisection algorithm. As discussed in the
introduction, the total homogenization temperature of CO2-H2O-NaCl inclusion
h (tot)T can be obtained from microthermometric analysis, and compositions, such as
NaClx and 2COx , can be obtained from microthermometric and Raman analysis by
combining with thermodynamic models (Chen, 1972; Fall et al., 2011; Mao et al.,
2013). In the algorithm, first input parameters h (tot)T , NaClx , 2COx , and T, then
calculate the total homogenization pressure h (tot)P from the CO2 solubility model
of Mao et al. (2013) and calculate molar volume mV from Eq. (13). Assume initial
boundary pressure 1 0P = , 2 10000P = , then calculate molar volume calmV at
1 2( ) / 2P P P= + from Eq. (13). Generally, calmV is not equal to mV . Therefore,
modify 1P or 2P by bisection until the calculated calmV equals to mV because the
molar volume of fluid inclusion is almost constant during heating and cooling. Fig. 3
shows the flow chart of the algorithm, whose convergence condition is
cal 5 3 1m m| | 10 cm molV V − −− < .
23
St ar t
h mCalculate (tot) and P V
1 2
1 2
Initial =0, =10000,
=( + )/2
P P
P P P
calmCalculate V
calm m 0?V V− =
1 2Modify or
by bisection
P P
Calculated is correctP
E n d
2h CO
NaCl
Input (tot), ,
and
T x
x T
Fig. 3: A bisection algorithm for calculating the isochores of CO2-H2O-NaCl fluid
inclusions: h (tot)T is the total homogenization temperature, 2COx and NaClx are
the total mole fraction of CO2 and NaCl of inclusion, respectively, h (tot)P is the
total homogenization pressure, mV is the bulk molar volume of the whole inclusion,
and T and P denote the temperature and pressure on the isochores, respectively.
24
The calculated isochores of H2O-NaCl fluid with certain salinities are shown in
Fig. 4, where experimental iso-Th lines (which can be approximated as isochores)
from Bodnar and Vityk (1994) are also used in comparison. It can be seen from Fig. 4
that the calculated isochores agree with the experimental iso-Th lines of H2O-NaCl
fluid. Fig. 5 shows the comparison of the experimental isochores of Gehrig (1980)
with the calculated results of CO2-H2O-NaCl fluid mixtures. It can be seen that the
calculated isochores are almost linear in high temperature-pressure regions, and the
calculated results are also in agreement with the experimental isochores.
Fig. 4: Isochores of H2O-NaCl fluid mixtures: NaClx is mole fraction of NaCl, and
T and P denote temperature and pressure on the isochores, respectively.
400 500 600 700 800
0
1000
2000
3000
4000
5000
6000
P (
bar
)
T (K)
This model Bodnar and Vityk (1994)
xNaCl
= 0.03312
vapor-liquid curve
a
Vm = 19.55 cm3⋅mol-1
20.49
21.75
23.49
400 500 600 700 800 900 1000
0
1000
2000
3000
4000
5000
6000 Vm = 19.59 cm3⋅mol-1
vapor-liquid curve
xNaCl
= 0.07155
This model Bodnar and Vityk (1994)
P (
bar
)
T (K)b
21.40
24.48
30.22
400 500 600 700 800 900 1000 1100 1200
0
1000
2000
3000
4000
5000
6000 Vm = 20.43 cm3⋅mol-1
vapor-liquid curve
xNaCl
= 0.1167
This model Bodnar and Vityk (1994)
P (
bar
)
T (K)c
22.36 25.47
30.83
600 700 800 900 1000 1100 1200 1300
0
1000
2000
3000
4000
5000
6000 Vm = 24.72 cm3⋅mol-1
vapor-liquid curve
xNaCl
= 0.1705
This model Bodnar and Vityk (1994)
P (
bar
)
T (K)d
28.75
35.06
25
Fig. 5: Isochores of CO2-H2O-NaCl fluid mixture: 2COx , NaClx and
2H Ox are the
mole fraction of CO2, NaCl and H2O of fluid mixture, respectively, and mV is the
molar volume of mixture.
5. Application of isochores to estimate the ore-forming conditions of
deposit
The PVTx model of CO2-H2O-NaCl fluid mixture together with the
aforementioned iterative method for calculation of isochores can be applied to the
analysis of natural CO2-H2O-NaCl fluid inclusions. For example, Table 9 lists some
experimental microthermometric data of natural CO2-H2O-NaCl inclusions in quartz
from Piaotang wolframite deposit in Jiangxi province, China. These fluid inclusions
600 700 800 900 1000 11000
1000
2000
3000
4000
5000
40
30
P (
bar)
T (K)
Exp. Gehrig (1980) This model
Vm= 25 cm3⋅mol-1
a
xCO
2
= 0.0018
xNaCl
= 0.0193
xH
2O= 0.9789
600 700 800 900 1000 11000
1000
2000
3000
4000
xCO
2
= 0.0400
xNaCl
= 0.0185
xH
2O= 0.9415
b
72
38
Vm= 28 cm3⋅mol-1
Exp. Gehrig (1980) This model
P (
bar
)
T (K)
600 700 800 900 1000 11000
500
1000
1500
2000
2500
3000
xCO
2
= 0.0969
xNaCl
= 0.0174
xH
2O= 0.8857
c
75
45
Vm= 35 cm3⋅mol-1
Exp. Gehrig (1980) This model
P (
bar
)
T (K)
500 600 700 800 900 1000 11000
1000
2000
3000
4000
5000
6000
xCO
2
= 0.328
xNaCl
= 0.013
xH
2O= 0.659
35
40
d
50
75
Vm= 30 cm3⋅mol-1
Exp. Gehrig (1980) This model
P (
bar
)
T (K)
26
were trapped by quartz at the main mineralization stage. How to calculate
homogenization pressures, compositions and molar volumes of a fluid inclusion
requires an iterative approach, which is same to the method presented by Mao et al.
(2013) where detailed calculation steps were given. The only difference is that the
molar volume of CO2-H2O-NaCl inclusion is calculated from Eq. (13) of this work.
Calculated homogenization pressures, compositions and molar volumes are also listed
in Table 9. Calculated isochores from this model are shown in Fig. 6. The ore-forming
temperature range estimated from oxygen isotope analysis of these inclusions and
quartz is 576.35-635.15 K, and the trapping pressure range is 560.49-1343.77 bar
according to points A and B in Fig. 6. The calculated formation depth of Piaotang
wolframite deposit is estimated to be 2.1-5.0 km assuming a lithostatic pressure
gradient of 270 bar/km.
27
Table 9: Calculation results for some natural CO2-H2O-NaCl inclusions in
quartz from Piaotang wolframite deposit in Jiangxi province, China
Inclusion
no. m (cla)T
(K)
h 2(CO )T
(K)
h (tot)T
(K)
h (tot)P
(bar) 2COx NaClx
2H Ox mV
(cm3
mol-1)
1 279.25 298.55(V) 576.35(L) 560.50 0.05930 0.02197 0.91873 24.84
2 277.15 297.65(V) 598.65(L) 629.79 0.05813 0.03271 0.90916 25.25
3 280.55 301.75(V) 591.15(L) 653.35 0.08365 0.01456 0.90179 26.51
4 277.55 298.45(V) 588.75(L) 623.06 0.05771 0.03075 0.91154 24.87
5 280.75 301.45(V) 573.15(L) 602.64 0.07334 0.01364 0.91303 25.38
6 279.75 302.15(V) 604.45(L) 725.31 0.08939 0.01869 0.89192 26.98
7 280.55 301.25(V) 603.15(L) 655.57 0.08834 0.01449 0.89717 27.43
8 280.45 300.25(V) 601.75(L) 619.72 0.08202 0.01513 0.90285 27.20
m (cla)T = final melting temperature of CO2 clathrate; h 2(CO )T = CO2 homogenization temperature; L =
liquid (homogenization); V = vapor (homogenization); h (tot)T = total homogenization temperature; h (tot)P
= total homogenization pressure; 2COx = bulk mole fraction of CO2; NaClx = bulk mole fraction of NaCl;
2H Ox = bulk mole fraction of H2O; mV = bulk molar volume of inclusion.
28
Fig. 6: Isochores of CO2-H2O-NaCl fluid inclusions in quartz from Piaotang wolframite deposit in Jiangxi province, China: Molar volume of each inclusion is shown in Table 9.
6. Conclusions
A predictive thermodynamic model based on the Helmholtz free energy equation
of state of CO2-H2O fluid is proposed to calculate the molar volume or density of
CO2-H2O-NaCl fluid mixture. The model can reproduce the volume of
CO2-H2O-NaCl fluid mixture of all compositions from 273 to 1273 K and from 0 to
5000 bar, with or close to experimental accuracy (with slightly lower accuracy
between 5000 and 10000 bar). The PVTx model for the CO2-H2O-NaCl fluid mixture
established here not only is fit for calculation of volumetric properties of CO2
geological sequestration, but also is very useful for calculation of isochores of
550 600 650 700 7500
400
800
1200
1600
2000
2400
1343.77
560.49
635.15
B
P (
bar
)
T (K)
Isochore and inclusion no. 1; 2; 3 4; 4; 6 7; 8
A
576.35
29
CO2-H2O-NaCl fluid inclusions, from which the trapping temperatures and pressures
can be obtained. The proposed method should be valid for other gas-H2O-NaCl fluids.
Although the importance of CO2-H2O-NaCl fluid is well known, experimental
volumetric data are insufficient for this fluid system, and future experimental data on
this system is still needed to cover a wide temperature-pressure-composition space.
Acknowledgements:
We thank the two anonymous reviewers for their detailed and helpful comments,
which improved greatly the quality of the manuscript. This work is supported by the
National Natural Science Foundation of China (41173072, 41172118 and 90914010)
and the Fundamental Research Funds for the Central Universities (2652013032).
Appendix I: The equations for calculating *VT of H2O-NaCl fluid
The corresponding equations for calculating *VT of H2O-NaCl fluid are as
follows:
1 2 ( )VT n n T D T∗ = + + (A1)
21 10 11 NaCl 12 NaCl(1 ) (1 )n n n x n x= + − + − (A2)
2 20 21 NaCl 22 23 NaCln n n x n n x= + + + (A3)
NaCl1, 1
8 2 10 3
330.47 0.942876 0.0817193
2.4755 10 3.45052 10
xn P P
P P
=
− −
= + +
− × + × (A4)
NaCl
52, 1
9 2 13 3
0.0370751 0.00237723 5.42049 10
5.94709 10 5.99373 10
xn P P
P P
−=
− −
= − + + ×
+ × − × (A5)
3130( ) n TD T n e= (A6)
30
301 NaCl30 300 302 NaCl( 1)n xn n e n x= − + (A7)
311 NaCl31 310 312 NaCl
n xn n e n x= + (A8)
where T and scaled temperature( VT ∗ ) are in ℃, and P is in bar. Note that if NaCl 0x = ,
then T = VT ∗ , 1 0n = , 2 1n = . This constraint can be used to eliminate 10n and 20n . If
NaCl 1x = , 1n and 2n can be represented by equations (A4) and (A5), and 12n and
23n will vanish from the two equations. The rest parameters of equations (A1)-(A8)
related to *VT (e.g., 11n , 21n , 22n , 300n , 301n , 302n , 310n , 311n and 312n ) are given
in Table 5, which are obtained by refitting to experimental volumetric data of the
H2O-NaCl system listed in Table 6 using the IAPWS-95 formulation.
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Highlights
A predictive PVTx model is developed for the CO2-H2O-NaCl fluid mixtures
The new model uses no other mixing parameters but those of the CO2-H2O system
Parameters of H2O-NaCl fluid model are fitted by the IAPWS-95 formulation
Isochore of CO2-H2O-NaCl fluid inclusion can be calculated by a bisection algorithm