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[.]
Accepted Manuscript Applications of the SW96 formulation in the thermodynamic calculation of fluid inclusions and mineral-fluid equilibria Jia Zhang, Shide Mao PII: S1674-9871(17)30026-9 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 This is a PDF file of an unedited manuscript that has been accepted for publication As a service to our customers we are providing this early version of the manuscript The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain AC C EP TE D M AN U SC RI PT ACCEPTED MANUSCRIPT ACCEPTED MANUSCRIPT Applications of the SW96 formulation in the thermodynamic calculation of fluid inclusions and mineral-fluid equilibria RI PT Jia Zhang, Shide Mao* SC School of Earth Sciences and Resources, China University of Geosciences, Beijing 10083, China AC C EP TE D M AN U * Corresponding author E-mail address: maoshide@163.com Abstract ACCEPTED MANUSCRIPT The SW96 formulation explicit in Helmholtz free energy proposed by Span and Wagner (1996) is the most accurate multifunction equation of state of CO2 fluid, from which all thermodynamic properties can be obtained over a wide temperature-pressure RI PT range from 216.592 to 1100 K and from to 8000 bar with or close to experimental accuracy This paper reports the applications of the SW96 formulation in fluid inclusions and mineral-fluid equilibria A reliable and highly efficient algorithm is SC presented for the saturated properties of CO2 so that the formulation can be M AN U conveniently applied in the study of fluid inclusions, such as calculation of homogenization pressures, homogenization densities (or molar volumes), volume fractions of vapor phase and isochores Meanwhile, the univariant curves of some typical decarbonation reactions of minerals are calculated with the SW96 formulation TE D and relevant thermodynamic models of minerals The computer code of the SW96 formulation can be obtained from the corresponding author AC C EP Keywords: Equation of state; CO2; Fluid inclusion; Application Introduction ACCEPTED MANUSCRIPT 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; RI PT 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; SC Leduhovsky et al., 2015) To reduce the amount of CO2 emissions to the atmosphere, M AN U 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., TE D 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, EP phase equilibria, fugacity, enthalpy, and other volumetric properties AC C 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 ACCEPTED MANUSCRIPT the SW96 formulation explicit in Helmholtz free energy, which was developed by Span and Wagner (1996) and has been recommended as the standard equation of CO2 by NIST (National Institute of Standards and Technology) The SW96 formulation RI PT can reproduce all thermodynamic properties of CO2 at a T-P range from 216.592 to 1100 K and from to 8000 bar with or close to experimental accuracy, and also extrapolates well beyond the above range, as will be shown later SC In this work, the SW96 formulation is briefly presented in Section 2, with further M AN U validation by the updated experimental data Then, the application of the SW96 formulation in fluid inclusions is discussed in detail in Section 3, including the algorithm of the saturated properties, isochores, and volume fractions of vapor phase Finally, the SW96 formulation is used to calculate the univariant curves of some TE D decarbonation reactions of minerals The SW96 formulation EP The SW96 formulation, expressed in the form of dimensionless Helmholtz free AC C energy φ (δ ,τ ) = f ( ρ , T ) / RT , is separated into two parts, an ideal-gas part φ (δ ,τ ) and a residual part φ r (δ ,τ ) : f ( ρ , T ) / RT = φ (δ ,τ ) = φ (δ ,τ ) + φ r (δ ,τ ) (1) where ρ is density, T is temperature, δ and τ are reduced parameters which are defined as δ = ρ / ρc ， τ = Tc / T with critical density ρ c = 467.6 kg/m3 and Tc = 304.1282 K, and specific gas constant R= 0.1889241 kJ/(kg K) The ideal-gas part φ (δ ,τ ) is written as ACCEPTED MANUSCRIPT φ (δ ,τ ) = ln (δ ) + a10 + a20τ + a30 ln (τ ) + ∑ ai0 ln 1 − exp ( −τθi0 ) (2) i =1 where ai0 and θi0 are parameters which are listed in Table A1 The residual part φ r (δ ,τ ) can be written as 34 ti i =1 i i RI PT φ = ∑ niδ τ + ∑ niδ d τ t e− c di r i i =8 39 + ∑ niδ diτ ti e − (δ −ε i ) − βi (τ −γ i ) i = 35 42 + ∑ ni ∆bi δ e −Ci (δ −1) SC (3) − Di (τ −1) i = 40 M AN U 1/ ( βi ) ∆ = (1 − τ ) + Ai (δ − 1) + Bi (δ − 1) (4) where ni , d i , ti , ci α i , β i , γ i , ε i , Ci , Di , Ai , and Bi are parameters listed in Table A2 All thermodynamic properties can be derived from above equations by Wagner (1996) TE D thermodynamic relations More details can be found in Tables 29–32 of Span and EP 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 lists AC C 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 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 ACCEPTED MANUSCRIPT agreement with experimental volumetric data up to 1600 bar Application of the SW96 formulation in fluid inclusion 3.1 Calculation method for the saturated properties RI PT From the SW96 formulation, all thermodynamic properties can be obtained including the saturated properties, e.g., saturated pressure Ps, saturated liquid density SC ρ ′ and saturated vapor density ρ′′ At liquid-vapor phase equilibria, the saturated properties are uneasy to calculate, especially when temperature approaches to critical M AN U 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 TE D efficient Newton iteration method: using values of ρ ′ and ρ′′ from auxiliary equations (see Appendix) as initial values and using the density function as iterative EP function The algorithm is given as follows: From the SW96 formulation, molar Gibbs free energy G can be derived: G = RT (1 + φ + φ r + δφδr ) AC C (5) ∂φ r ∂δ τ φδr = (6) At liquid-vapor phase equilibria, Ps = ρ′RT 1 + δ ′φδr (τ ,δ ′) (7) Ps = ρ′′RT 1 + δ ′′φδr (τ ,δ ′′) (8) G ′ = G ′′ (9) ACCEPTED MANUSCRIPT where δ ′ = ρ ′ / ρc , δ ′′ = ρ ′′ / ρ c Based on Eq (5), the phase-equilibrium condition at given P and T can be rewritten as φ (τ , δ ′ ) + φ r (τ , δ ′ ) + δ ′φδr (τ , δ ′ ) = φ (τ , δ ′′ ) + φ r (τ , δ ′′ ) + δ ′′φδr (τ , δ ′′ ) (10) RI PT Considering φ in Eq (2), Eq (10) can be rewritten as ln δ ′ + φ r (τ , δ ′ ) + δ ′φδr (τ , δ ′ ) = ln δ ′′ + φ r (τ , δ ′′ ) + δ ′′φδr (τ , δ ′′ ) (11) two paratactic equations for δ ′ and δ ′′ : δ ′′ 1 + δ ′′φδr (τ , δ ′′) − δ ′ 1 + δ ′φδr (τ , δ ′) = SC Eliminating Ps from Eqs (7) and (8), and rewriting Eq.(10) yield the following M AN U ln δ ′′ + φ r (τ , δ ′′) + δ ′′φδr (τ , δ ′′) − ln δ ′ + φ r (τ , δ ′) + δ ′φδr (τ , δ ′) = (12) (13) In the Newton iteration method for paratactic equations, δ ′ and δ ′′ are calculated with the following equations: {[ K (τ , δ " ) − K (τ , δ ' )] J δ (τ , δ " ) Λ − [ J (τ , δ " ) − J (τ , δ ' )]Kδ (τ , δ " )} (14) {[ K (τ , δ " ) − K (τ , δ ' )] J δ (τ , δ ' ) Λ − [ J (τ , δ " ) − J (τ , δ ' )]K δ (τ , δ ' )} (15) EP δ "( k +1) = δ "( k ) + TE D δ '( k +1) = δ '( k ) + AC C where J, K, J δ , Kδ and Λ are defined as J (τ , δ ) = δ 1 + δφδr (τ , δ ) (16) K (τ , δ ) = δφδr (τ , δ ′) + φ r (τ , δ ) + ln δ (17) ∂J r r J δ (τ , δ ) = = + 2δφδ + δ φδδ ∂δ τ (18) ∂K r r K δ (τ , δ ) = = 2φδ + δφδδ + δ ∂δ τ (19) Λ = J δ (τ , δ ′′) Kδ (τ , δ ′) − J δ (τ , δ ′) Kδ (τ , δ ′′) (20) ACCEPTED MANUSCRIPT The presented method can calculate saturated properties of CO2 from the temperature of triple point (216.592 K) to that of critical point (304.1282 K) Fig shows the flow chart of this method, which adopts the following convergence K (τ , δ ′′) − K (τ , δ ′) + J (τ , δ ′′) − J (τ , δ ′) < 10−9 RI PT condition: (21) Fig and Table show the calculated saturated properties of this method from SC 216.592 to 304.1282 K, where number of iterations until convergence is also given M AN U The reliable experimental data reported by Duschek et al (1990) are plotted for comparison in Fig It can be seen from Table and Fig 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 TE D 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) EP and homogenization density (saturated liquid or vapor density) of CO2 inclusion with AC C the measured homogenization temperature 3.2 Calculation of isochores Isochores, along which fluid inclusion were trapped, can be used to determine the trapping temperatures and pressures Based on the SW96 formulation, isochores of CO2 inclusions can be easily calculated from the following equation: P = ρ RT 1 + δφδr (τ , δ ) (22) The calculated isochores of CO2 fluid are plotted in Fig 4, from which it can be ACCEPTED MANUSCRIPT pressures up to 30 MPa using a new accurate single-sinker densimeter The Journal of Chemical Thermodynamics 33(3), 251-267 Koziol, A.M., Newton, R.C., 1995 Experimental determination of the reactions RI PT magnesite + quartz = enstatite + CO2 and 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dioxide Journal of the American Chemical Society 45, 1167-1184 SC Span, R., Wagner, W., 1996 A new equation of state for carbon dioxide covering the M AN U fluid region from the triple-point temperature to 1100 K at pressures up to 800 MPa Journal of Physical and Chemical Reference Data 25(6), 1509-1596 Stein, W.A., 1972 Die Zustandsgleichung für reine fluide Stoffe Chemical Engineering Science 27(7), 1371-1382 TE D Sterner, S.M., Pitzer, K.S., 1994 An equation of state for carbon dioxide valid from zero to extreme pressures Contributions to Mineralogy & Petrology 117(4), 362-374 EP Sun, R., Dubessy, J., 2010 Prediction of vapor–liquid equilibrium and PVTx AC C properties of geological fluid system with SAFT-LJ EOS including multi-polar contribution Part I: Application to H2O–CO2 system Geochimica et Cosmochimica Acta 74(7), 1982-1998 Tang, X., Zhang, L., Tu, H., Gu, H., Chen, L., 2010 Decarbonization mechanisms of polycarbosilane during pyrolysis in hydrogen for preparation of silicon 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MANUSCRIPT Applications of the SW96 formulation in the thermodynamic calculation of fluid inclusions and mineral -fluid equilibria RI PT Jia Zhang, Shide Mao* SC School of Earth Sciences and Resources,... data Then, the application of the SW96 formulation in fluid inclusions is discussed in detail in Section 3, including the algorithm of the saturated properties, isochores, and volume fractions of. .. 1100 K and from to 8000 bar with or close to experimental accuracy This paper reports the applications of the SW96 formulation in fluid inclusions and mineral -fluid equilibria A reliable and highly