Applications of the SW96 formulation in the thermodynamic calculation of fluid inclusions and mineral-fluid equilibria

Applications of the SW96 formulation in the thermodynamic calculation of fluid inclusions and mineral fluid equilibria Accepted Manuscript Applications of the SW96 formulation in the thermodynamic cal[.]

Applications of the SW96 formulation in the thermodynamic calculation of fluid

inclusions and mineral-fluid equilibria

Jia Zhang, Shide Mao

DOI: 10.1016/j.gsf.2017.01.007

Reference: GSF 533

To appear in: Geoscience Frontiers

Received Date: 11 November 2016

Revised Date: 16 January 2017

Accepted Date: 29 January 2017

Please cite this article as: Zhang, J., Mao, S., Applications of the SW96 formulation in the

thermodynamic calculation of fluid inclusions and mineral-fluid equilibria, Geoscience Frontiers (2017),

doi: 10.1016/j.gsf.2017.01.007.

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Jia Zhang, Shide Mao*

School of Earth Sciences and Resources, China University of Geosciences, Beijing

10083, China

* Corresponding author E-mail address: maoshide@163.com

Abstract

Keywords: Equation of state; CO2; Fluid inclusion; Application

1 Introduction

It is well-known that CO2 and CO2-bearing fluid inclusions are frequently found

in hydrothermal ore deposits, whose isochores are often used to estimate the trapping temperatures and trapping pressures of ore-forming fluids (Yamamoto et al., 2011; Lamadrid et al., 2014; Hudgins et al., 2015) In decarbonation reactions of minerals, fugacity of CO2 at given temperature and pressure must be known to calculate the univariant curves of reaction equilibria (Omori and Santosh, 2008; Tang et al., 2010; Leduhovsky et al., 2015) To reduce the amount of CO2 emissions to the atmosphere,

CO2 capture and sequestration (CCS) has become a technologically feasible method, but the thermodynamic properties of CO2, especially the PVT and vapor-liquid phase

equilibrium properties, must be known for studying the volumetric changes and dynamic mechanism after CO2 is injected into deep saline formations (Kelemen et al., 2011; Mathias et al., 2015) Equation of state developed on the basis of thermodynamic theories and reliable experimental data is a powerful tool for quantitative calculation of various thermodynamic properties of CO2, e.g., density, phase equilibria, fugacity, enthalpy, and other volumetric properties

Over the last several decades, a lot of equations of state have been proposed for

CO2 (Table 1) Each equation has its strength and weakness Some of them can be used in a large temperature-pressure space with less accuracy, and some others are only valid in a small temperature-pressure region but with high accuracy Some good equations of state of CO2 can be found from these literatures (Kerrick and Jacobs, 1981; Sterner and Pitzer, 1994; Span and Wagner, 1996; Duan and Zhang, 2006; Sun and Dubessy, 2010) However, among these equations of state, the best one is likely

As stated in the Section 1, the SW96 formulation is the best equation of state for

CO2 up to now, which can reproduce most of existed experimental data Table 2 lists the calculated density deviations from the latest experimental data, which are not used

in the development of the SW96 formulation It can be seen from Table 2 that most of deviations are within experimental uncertainties except for some of low-pressure data (Pečar and Doleček, 2007; Mazzoccoli et al., 2012; Deering et al., 2016) The

comparisons between the SW96 formulation and experimental PVT data (Klimeck et

al., 2001; Tsuji et al., 2004; Pensado et al., 2008; Mantilla et al., 2010;Yang et al., 2015) are plotted in Fig 1, which indicates that the calculated densities are in good

i i

agreement with experimental volumetric data up to 1600 bar

3 Application of the SW96 formulation in fluid inclusion

3.1 Calculation method for the saturated properties

From the SW96 formulation, all thermodynamic properties can be obtained

including the saturated properties, e.g., saturated pressure Ps, saturated liquid density and saturated vapor density At liquid-vapor phase equilibria, the saturated properties are uneasy to calculate, especially when temperature approaches to critical temperature of CO2 Under these conditions, an iterative method must be used to obtain the saturated properties of CO2, which involves two aspects: one is how to choose initial values of variables, and another is how to choose iterative functions

In the calculation of saturated properties, we found that a reliable and highly efficient Newton iteration method: using values of and from auxiliary

equations (see Appendix) as initial values and using the density function as iterative

function The algorithm is given as follows:

From the SW96 formulation, molar Gibbs free energy G can be derived:

G=RT + + +φ φ δφδ (5)

r r

where δ′=ρ ρ′/ c, δ′′=ρ ρ′′/ c Based on Eq (5), the phase-equilibrium condition

at given P and T can be rewritten as

δ δ

δ δ

K τ δ′′ −Kτ δ′ + J τ δ′′ −J τ δ′ < − (21) Fig 3 and Table 3 show the calculated saturated properties of this method from 216.592 to 304.1282 K, where number of iterations until convergence is also given The reliable experimental data reported by Duschek et al (1990) are plotted for comparison in Fig 3 It can be seen from Table 3 and Fig 3 that five iterations are enough to meet the requirement, that is, the method here is a reliable and stable method for calculating the saturated properties of CO2 with the SW96 formulation In experimental microthermometric analysis of fluid inclusions, homogenization temperature (phase-transition temperature) can be directly measured Therefore, the new method can be used to calculate homogenization pressure (saturated pressure) and homogenization density (saturated liquid or vapor density) of CO2 inclusion with the measured homogenization temperature

3.3 Calculation of volume fraction of vapor phase

Volume fraction of vapor phase (FV) before homogenization can be obtained from either experimental or calculation method Traditional experimental method for

FV requires measuring area-fractions of the phase projected in the microscope and then making rough corrections for the third dimension, so the resulting uncertainties may be very large for the inclusions of irregular shape On the contrary, the calculated

FV based on equation of state is more accurate than traditional experimental method

Generally, FV at a given temperature is defined as

V V/ ( V L)

F =V V +V (23) where VV is the volume of vapor phase, and VL is the volume of liquid phase Because the total density of fluid inclusions is almost unchanged during the heating and cooling process, Eq (23) can be rewritten as

H FV V (1 FV) L

ρ = ρ + − ρ (24)

where ρH is the density at the homogenization temperature, and ρL and ρv denote saturated liquid density, and vapor density at a given temperature, respectively Rewriting Eq (24) yields

V ( H L) / ( V L)

F = ρ −ρ ρ −ρ (25)According to Eq (25) and the algorithm for the saturated properties of CO2

aforementioned, FV of CO2 inclusion can be accurately calculated from the SW96

formulation For a vapor-liquid CO2 inclusion finally homogenizing to liquid phase at

300 K, some calculated FV below 300 K are listed in Table 4

4 Applications of the SW96 formulation in the calculation of mineral-fluid equilibria

When a reaction reaches equilibrium at a given P and T, the changes of Gibbs

free energy ∆rG T P is zero, so

is the change of standard Gibbs free energy of the reaction, ∆Vs is the changes of solid molar volume, ∆Vfluid is the changes of fluid molar volume, and

P$is the standard pressure defined as 1 bar ∆rG T P$

f = f P

(29)where R is the gas constant,

2

C O

f is the fugacity of CO2, v is the stoichiometric

number of CO2 in the reactions, and fc is the fugacity coefficient which is a function

of temperature and pressure and can be calculated from

c

ln( f ) = − − Z 1 ln( ) Z + φr (30)

where Z is the compressibility factor (P / ( ρ RT )) and ϕr is the residual part of

= 2Forsterite + CO2; (b) Calcite = Lime + CO2; (c) Calcite + Quartz = Wollastonite +

CO2; (d) Magnesite = Periclase + CO2 All thermodynamic data of CO2 and minerals are from Holland and Powell (1998) From Fig 5, it can be seen that the calculated equilibrium curves with the SW96 formulation are in good agreement with experimental results (Smyth and Adams, 1923; Harker and Tuttle, 1955; Baker, 1962; Johannes, 1969; Zhu et al., 1994; Koziol and Newton, 1995; Koziol and Newton, 1998; Aranovich and Newton, 1999)

5 Conclusion

The SW96 formulation developed by Span and Wagner (1996) can reproduce all thermodynamic properties of CO2 from 216.592 to 1100 K and from 0 to 8000 bar within experimental uncertainties A reliable and highly efficient method is presented for the calculation of saturated properties of CO2 so that the equation of state can be conveniently applied in the studies of fluid inclusions This method should be valid to all other pure fluids if their equations of state are in form of Helmholtz free energy The univariant curves of some decarbonation reactions at high pressures further

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