Benoît Dubacq
email: benoit.dubacq@upmc.fr

Mikej Bickle

Katya Evans

An activity model for phase equilibria in the H 2 O-CO 2 -NaCl system

Keywords: Fluid-rock interactions, activity-composition model, H 2 O-CO 2 -NaCl, CO 2 solubility, minerals solubility, speciation, thermodynamics, salting-out effect, Carbon Capture and Storage, Enhanced Oil Recovery, water activity

We present a semi-empirical thermodynamic model with uncertainties that encompasses the full range of compositions in H 2 O-CO 2 -NaCl mixtures in the range of 10-380°C and 1-3500 bars. For binary H 2 O-CO 2 mixtures, the activitycomposition model is built from solubility experiments. Theparametersdescribing interactions between H 2 OandCO 2 are independent of the absolute thermodynamic properties of the end-members and vary strongly non-linearly with pressure and temperature. The activity of water remains higher than 0.88 in CO 2 -saturated solutions across the entire pressure-temperature range. IntheH 2 O-NaCl system, it is shown that the speciation of aqueous components can be accounted for by athermodynamicformalismwhereactivitiesaredescribedbyinteractionparameters varying with intensive properties such as pressure andt e m p e r a t u r eb u tn o t with concentration or ionic strength, ensuring consistencywiththeGibbs-Duhem relation. The thermodynamic model reproduces solubility experiments of halite up to 650°C and 10 kbar, and accounts for ion pairing of aqueouss o d i u ma n d chloride ions with the use of associated and dissociated aqueous sodium chloride end-members whose relative proportions vary with salinity.IntheH 2 O-CO 2 -NaCl system, an activity-composition model reproduces the salting-out effect with in-

INTRODUCTION

Fluids play a key role in the evolution of the lithosphere, at the surface (e.g. [START_REF] Nesbitt | Weathering of granodioritic crust, longterm storage of elements in weathering profiles, and petrogenesis of siliciclastic sediments[END_REF]Tipper et al., 2006), where crust is created at subduction zones (e.g. [START_REF] Tatsumi | Migration of fluid phases and genesis of basalt magmas in subduction zones[END_REF][START_REF] Hacker | Subduction factory -1 .T h e o r e t i c a l mineralogy, densities, seismic wave speeds, and H 2 Ocontents[END_REF] as well as in the deep crust and upper mantle [START_REF] Newton | Carbonic metamorphism, granulites and crustal growth[END_REF][START_REF] Thompson | Water in the Earth's upper mantle[END_REF], in seawater composition [START_REF] Edmond | Ridge crest hydrothermal activity and the balances of the major and minor elements in the ocean: The Galapagos data[END_REF], seismicity [START_REF] Chester | Internal Structure and Weakening Mechanisms Of The San-Andreas Fault[END_REF], mantle dynamics [START_REF] Molnar | Mantle dynamics,upliftoftheT ibetan plateau, and the Indian monsoon[END_REF], exhumation of subducted material [START_REF] Angiboust | Effect of fluid circulation on subduction interface tectonic processes: Insights from thermomechanical numerical modelling[END_REF], ore deposits (e.g. [START_REF] Wilkinson | Pressure fluctuations, phase separation, and gold precipitation during seismic fracture propagation[END_REF], for H 2 O-CO 2 -NaCl) and melting [START_REF] White | Calculation of partial melting equilibria in the system Na 2 O-CaO-K 2 O-FeO-MgO-Al 2 O 3 -SiO 2 -H 2 O(NCKF-MASH)[END_REF]. Aqueous solutions transform their hostrockbydissolution -p r e c i p i t a t i o nr e a c t i o n sa n di o ne x c h a n g e ,t r a n s p o r t i n gg eochemical fluxes and changing rock properties.

Concern about the environmental impacts of greenhouse gasesemissionshas created an interest in geological carbon storage where safe,long-termstoragewill require prediction of reactions between CO 2 ,aqueousformationfluidsandreservoir minerals [START_REF] Bickle | Geological carbon storage[END_REF][START_REF] Wigley | Fluid-mineral reactions and trace metal mobilization in an exhumed natural CO 2 reservoir, Green River, Utah[END_REF]. Understanding the behavior of mixed H 2 O-CO 2 fluids is also important for modeling the global carbon cycle and estimating metamorphic CO 2 fluxes [START_REF] Kerrick | Metamorphic CO 2 degassing from orogenic belts[END_REF][START_REF] Becker | Himalayan metamorphic CO 2 fluxes: Quantitative constraints from hydrothermal springs[END_REF].

Description, quantitative modeling and prediction of thesephenomenaisbased on knowledge of the thermodynamic properties of end-member minerals, fluids, and solutes, as well as activity-composition relationships, which describe the thermodynamic properties of mixtures as a function of their composition. Reaction rates have also been shown to depend on the approach to equilibrium.

Although the thermodynamic properties of mineral and fluid end-members are relatively well known, Oelkers et al. (2009) highlightedtheimportanceofusing internally consistent thermodynamic databases in geochemical modeling. Research in metamorphic petrology has produced reliable internally consistent thermodynamic databases, amongst which the regularly updated databases of THER-MOCALC [START_REF] Holland | An internally consistentthermodynamicdata set for phases of petrological interest[END_REF][START_REF] Powell | Calculating phase diagrams involving solid solutions via non-linear equations, with examples using THERMOCALC[END_REF]Holland andPowell, 2003, 2011) and TWEEQ [START_REF] Berman | Internally-consistent thermodynamic dataf o rm i n e r a l si nt h e system Na 2 O-K 2 O-CaO-MgO-FeO-Fe 2 O 3 -Al 2 O 3 -SiO 2 -TiO 2 -H 2 O-CO 2[END_REF][START_REF] Berman | Optimized standard state and solution properties of minerals[END_REF][START_REF] Aranovich | Optimized standard state and solution properties of minerals[END_REF] provide thermodynamics propertiesformorethan150 minerals of petrological interest each, where enthalpies off o r m a t i o no fv a r i o u s end-members at standard state and their uncertainties have been estimated from calorimetric measurements and phase equilibria. THERMOCALC can be used with the database of [START_REF] Helgeson | Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: IV. Calculation of activity coefficients, osmotic coeffcients, and apparent molal and standard and relative partial molal properties to 600C and 5kb[END_REF] for aqueous fluids and incorporates the facility for error propagation and for complex activity-composition calculations in fluids and mineral phases. However, replication of observed fluid properties by activity-composition expressions, particularly for salt-rich, mixed solvent fluids at pressure-temperature conditions where liquid and gas phases coexist, in the regions of the critical points of CO 2 and of water, and along the critical mixing line of their mixtures, has proved more challenging.

Anumberofworkershasdevelopedactivity-compositionrelationships for fluids that are salt-rich and/or mixed solvent and/or mixed phase or close to the critical point (e.g [START_REF] Helgeson | Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: IV. Calculation of activity coefficients, osmotic coeffcients, and apparent molal and standard and relative partial molal properties to 600C and 5kb[END_REF][START_REF] Pitzer | Thermodynamics of electrolytes.1. Theoretical basis and general equations[END_REF]Pitzer andMayorga,1973;[START_REF] Pitzer | Thermodynamics of multicomponent, miscible, ionic systems: theory and equations[END_REF][START_REF] Chapman | SAFT: Equation-ofstate solution model for associating fluids[END_REF]Clegg and Pitzer, 1992;Clegg et al., 1992;[START_REF] Duan | An improved model calculating CO 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF][START_REF] Tan | SAFT1-RPM Approximation Extended to Phase Equilibria and Densities of CO 2 -H 2 Oa n dC O 2 -H 2 O-NaCl Systems[END_REF]Duan et al., 2006;[START_REF] García | Prediction of mineral scale formation in geothermal and oilfield operations using the Extended UNIQUAC model: Part II. Carbonate-scaling minerals[END_REF][START_REF] Zhu | A SAFT equation of state for the quaternary H 2 S-CO 2 -H 2 O-NaCl system[END_REF]. However, there is still a need for a model that:

1. replicates available data for mixed phase, salt-rich, mixed-solvent fluids over a large range of pressures and temperatures; 2. allows realistic propagation of uncertainties; 3. allows dissociation of ionic solutes such as NaCl; 4. can be extended readily to more complex systems; 5. is compatible with existing thermodynamic databases and software 1 . 6. is based on a relatively small number of fitted thermodynamic quantities, which facilitates fits for systems where data are sparse; 7. is based on physically realistic expressions with a minimum reliance on empirical expressions such as power-law series. Such an approach increases the ability of a model to extrapolate beyond the limits of experimental data.

In this paper, we use the the Debye-Huckel ASymmetric Formalism (DH-ASF) model developed by [START_REF] Evans | A method for activity calculations in saline and mixed solvent solutions at elevated temperature and pressure: A framework for geological phase equilibria calculations[END_REF] to describe activity-composition relations between H 2 O-CO 2 -NaCl in mixed solvent fluids up to 10 kbars and 650°C. The model is compatible with the computer program THERMOCALC.

First, phase diagrams are constructed with the ASF model applied to the binary mixture of water and carbon dioxide. Then the DH-ASF model is parameterized to reproduce experimental results of halite solubility in water, taking the pairing of aqueous Na + and Cl -into account. Finally, the effect of aqueous NaCl on CO 2 solubility is fitted in the H 2 O-CO 2 -NaCl system. The choice of the H 2 O-CO 2 -NaCl chemical system is dictated by its geological importance and the large number of experimental results available. The approach is designed to be readily extended to additional end-members.

APPROACH FOR THERMODYNAMIC MODELING

There are a number of published approaches to modeling water-gas-minerals interactions and the dissolved species they involve. The fundamental challenges in modeling the thermodynamics of fluids are the choices of components and the parameterizations used to describe activity-composition relationships.

Terminology

In the following, we use end-members for components, defined as the smallest set of chemical formulae needed to describe the composition of all the phases in the system [START_REF] Anderson | Thermodynamics in geochemistry: The equilibrium model[END_REF][START_REF] Spear | CO 2 -H 2 Omixturesinthegeological sequestration of CO 2 .I.Assessmentandcalculationofmutualsolubilities from 12 to 100°C and up to 600 bar[END_REF]. We modelthermodynamic equilibrium and the phases considered here are assumed to be chemically and physically homogeneous substances bounded by distinct interfaces with adjacent phases and may be minerals exhibiting solid-solutions, melts, aqueous liquids, gases or supercritical fluids. AkeytoallsymbolsisprovidedinT able1.

Thermodynamic background

The Debye-Hückel limiting law (Debye and Hückel, 1923a,b)a n di t se x t e nsions (e.g. [START_REF] Helgeson | Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: IV. Calculation of activity coefficients, osmotic coeffcients, and apparent molal and standard and relative partial molal properties to 600C and 5kb[END_REF] have been extensively usedtocalculatethethermodynamic properties of solutions for geological applications but are restricted to ionic strengths below 0.1 molal, whereas solutions of interest such as sedimentary brines and metamorphic fluids have often higher ionic strengths (e.g. [START_REF] Houston | Therelativeimportance of buffering and brine inputs in controlling the abundance of Na and Ca in sedimentary formation waters[END_REF].

The Pitzer model [START_REF] Pitzer | Thermodynamics of electrolytes.1. Theoretical basis and general equations[END_REF][START_REF] Pitzer | Thermodynamics of electrolytes .2. Activity and osmotic coefficients for strong electrolytes with one or both ions univalent[END_REF][START_REF] Pitzer | Thermodynamics of multicomponent, miscible, ionic systems: theory and equations[END_REF]Clegg and Pitzer, 1992;Clegg et al., 1992) has provided a framework to express the activity coefficient, γ,ofaqueousspeciesathigherionicstrengths and can be applied to salt-bearing solutions from infinite dilution to fused salt mixtures. The large range of concentrations is accounted forbyanactivitymodel describing a short-range force term for highly-concentrated solutions, with interaction parameters fitted to a convenient expression such as a Margules expansion (e.g., [START_REF] Pitzer | Thermodynamics of multicomponent, miscible, ionic systems: theory and equations[END_REF], to which is added a Debye-Hückel term resulting from long-range ionic forces which dominate at low concentrations.

AdifferentapproachhasbeensuggestedbyDuanetal. (2006)who proposed an activity model to calculate the solubility of CO 2 in aqueous fluids by mapping variations of the fugacity coefficient φ CO 2 in the two-phase mixture. φ CO 2 is linked to the activity of CO 2 in the mixture,

a CO 2 ,suchasa CO 2 = φ CO 2 φ 0 CO 2 X CO 2 ,whereφ 0 CO 2
is the fugacity coefficient of CO 2 in its pure phase and X CO 2 the mole fraction of CO 2 in the gas phase [START_REF] Flowers | Correction of Holloway's (1977) adaptation of the modified Redlich-Kwong equation of state for calculation of the fugacities of molecular species in supercritical fluids of geologic interest[END_REF]. The ratio

φ CO 2 φ 0 CO 2
is the activity coefficient γ CO 2 (e.g., [START_REF] Holland | Activity-composition relations for phases in petrological calculations: an asymmetric multicomponent formulation[END_REF]. [START_REF] Duan | An improved model calculating CO 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF] note that φ CO 2 differs very little from φ 0 CO 2 at temperatures between 0 and 260°C for all pressures lower than 2kbar, and subsequently assume them to be equal so that γ CO 2 =1and Powell, 1998), and also has the disadvantage that little information is provided on the activity of water in the mixture.

a CO 2 = X CO
SAFT (Statistical Associating Fluid Theory, [START_REF] Chapman | SAFT: Equation-ofstate solution model for associating fluids[END_REF]equations of state (EOS) can be altered to describe variations of φ CO 2 and have been shown

by [START_REF] Tan | SAFT1-RPM Approximation Extended to Phase Equilibria and Densities of CO 2 -H 2 Oa n dC O 2 -H 2 O-NaCl Systems[END_REF] and Ji and [START_REF] Zhu | A SAFT equation of state for the quaternary H 2 S-CO 2 -H 2 O-NaCl system[END_REF] to reproduce well the experimentallyderived solubility and density of H 2 O-CO 2 mixtures to which their short-range parameters are fitted over the range 20-200°C and 1-600 bars. However their predictive power is not greater than other semi-empirical, simpler models.

The ASF and DH-ASF models

In this study, we use both the ASF model developed by [START_REF] Holland | Activity-composition relations for phases in petrological calculations: an asymmetric multicomponent formulation[END_REF] and its extension for aqueous species (DH-ASF, from [START_REF] Evans | A method for activity calculations in saline and mixed solvent solutions at elevated temperature and pressure: A framework for geological phase equilibria calculations[END_REF]. ASF and DH-ASF are frameworks for activity-composition models used in the THERMOCALC software together with its internally consistent database.

ASF

For a binary mixture, the ASF model is essentially equivalentt ot h es u bregular model (e.g. [START_REF] Thompson | Thermodynamic properties of simple solutions[END_REF] where the shape of the Gibbs free energy function along the mixture is a function of two Margules parameters determining the activity coefficients needed to describe the non-ideal mixing (see [START_REF] Capitani | Corresponding states in binary solutions and graphical determination of Margules parameters[END_REF]. ASF differs from the sub-regular model in that it links the shape of the Gibbs free energy function along a mixture to the relative molar volumes of the end-members, in a mixing scheme derived from the van Laar equation, and there are no ternary interaction parameters in ASF.

The chemical potential of an end-member l in a solution can be expressed as:

µ l (x l )=µ 0 l + RT log(x l )+RT log(γ l ) (1)
where R is the ideal gas constant, T is temperature, µ l is the chemical potential of l, µ 0 l the chemical potential of l at standard pressure and temperature, x l is the mole fraction of l and γ l the activity coefficient of l.W i t ht h eA S Ff o r m a l i s m of [START_REF] Holland | Activity-composition relations for phases in petrological calculations: an asymmetric multicomponent formulation[END_REF], a single activity-compositionmodelconstrainsthe activities of the two end-members in the two phases along a binary mixture. The activity coefficient γ of l in a system with n end-members is expressed as:

RT log(γ l )=- n-1 i=1 n j>i q i q j W * ij ( 2
)
where i and j are end-members of the mixture, q i =1-Φ i when i = l,a n d q i = -Φ i when i = l where Φ i and W * ij are the size parameter-adjusted proportion and interaction energy defined respectively as:

Φ i = x i α i n j=1 x j α j (3) W * ij = W ij 2α l α i + α j (4) W ij
G xs m (x)= n l x l RT log(γ l )= n-1 i=1 n j>i Φ i Φ j 2 n l=1 α l x l α i + α j W ij (5)
The asymmetry of the excess Gibbs free energy function is controlled by the ratio of the different α values, which describe the properties of the end-members in the mixture and have been primarily related to their relative volumes in the mixture, although these parameters may be adjusted to fit experimental constraints [START_REF] Holland | Activity-composition relations for phases in petrological calculations: an asymmetric multicomponent formulation[END_REF]. There are therefore n -1 independent α parameters and one of them may be set to unity [START_REF] Holland | Activity-composition relations for phases in petrological calculations: an asymmetric multicomponent formulation[END_REF]. In a binary ij solution where α i = α j ,t h em o d e li ss y m m e t r i ca n dt h ee x p r e s s i o no fG xs m (x)

reduces to the sub-regular symmetric model where G xs m (x)=W ij x i x j .T h eA S F model assumes that neither α i nor W ij vary with the composition of the solution although they may vary with pressure and temperature to satisfy the Gibbs-Duhem equation (see [START_REF] Spear | CO 2 -H 2 Omixturesinthegeological sequestration of CO 2 .I.Assessmentandcalculationofmutualsolubilities from 12 to 100°C and up to 600 bar[END_REF].

In the case of a binary mixture with a solvus, α i and W ij can be evaluated solely from the compositions at binodal equilibrium and/or from the conditions of critical mixing, a separation of a two-phase system from a single-phase system.

Details of the derivation of the parameters are given in Appendix A where the equations have been obtained in a manner similar to the derivation of [START_REF] Capitani | Corresponding states in binary solutions and graphical determination of Margules parameters[END_REF] for the subregular model. It is noteworthy that in this case the chemical potentials of the end-members at standard state arenotr e quir e dtoc a lculate compositions along solution binaries; it is thus possible to construct phase diagrams of binary solutions from the activity model alone, which can therefore be used with any database. Uncertainties on the activity-composition model may then be estimated independently from the uncertainties on the thermodynamic properties of the end-members.

DH-ASF

The DH-ASF model of [START_REF] Evans | A method for activity calculations in saline and mixed solvent solutions at elevated temperature and pressure: A framework for geological phase equilibria calculations[END_REF] temperature and the properties of the solvent as shown later.

Selected equations of state

For aqueous species, we use the EOS of [START_REF] Holland | An internally consistentthermodynamicdata set for phases of petrological interest[END_REF], modified from [START_REF] Holland | An enlarged and updated internally consistent thermodynamic dataset with uncertainties and correlations -the system K 2 O-Na 2 O-CaO-MgO-MnO-FeO-FE 2 O 3 -Al 2 O 3 -TiO 2 -SiO 2 -C-H 2 -O 2[END_REF] to incorporate the density model of [START_REF] Anderson | The density model for estimation of thermodynamic parameters of reactions at high temperatures and pressures[END_REF]. This EOS is selected as a simple tool to estimatethethermodynamic properties of aqueous species at standard state over a large range of pressures and temperatures. We use the EOS for water given in the 2009 revised release of the International Association for the Properties of Water and Steam (IAPWS, avail-able at http://www.iapws.org) and the [START_REF] Sterner | An Equation Of State For Carbon-Dioxide Valid From Zero To Extreme Pressures[END_REF] EOS for CO 2 to calculate their respective densities, volumes and fugacities. The other routines for calculation of fugacities investigated during our study arenotappropriateforthe pressure and temperature range of interest: the CORK EOS [START_REF] Holland | A compensated Redlich-Kwong (CORK) equation for volumes and fugacities of CO 2 and H 2 Ointherange1barto50kbar and 100-1600 degrees[END_REF] have not been constrained at temperatures lower than 100°C for either water or carbon dioxide, and the EOS derived by [START_REF] Pitzer | Equations of state valid continuously from zero to extreme pressures for H 2 OandCO 2[END_REF], currently used in THERMOCALC, fails to reproduce the density of water accurately at temperatures less than 130°C with a maximum density at about 45°C at all pressures less than 1kb rather than at 4°C at 1 bar. We also use the equations given by the IAPWS in their 1997 release to calculate the dielectric constant of water. The thermodynamic properties of the end-members used in this study are taken from the THERMOCALC database [START_REF] Powell | Calculating phase diagrams involving solid solutions via non-linear equations, with examples using THERMOCALC[END_REF][START_REF] Holland | An internally consistentthermodynamicdata set for phases of petrological interest[END_REF], 2003, 2011).

BINARY MIXTURES OF H 2 OANDCO 2

The phase diagram controlling the solubilities of CO 2 and H 2 Ointheaqueous and CO 2 -rich phases can be modeled with two components, for which we use the end-members H 2 Oa n dC O 2 .S u p p l e m e n t a r yc o m p o n e n t ss u c ha sc a r b o n a t e species in the aqueous solutions are not required to calculate solubilities.

Derivation and parameterization of the activity model

Although the H 2 O-CO 2 system has been extensively studied and different parameterizations are available for calculating CO 2 solubility in water or brines, there is currently no activity-composition model which encompasses both low pressures and temperatures and high-grade metamorphic conditions. The very simple model of [START_REF] Holland | Activity-composition relations for phases in petrological calculations: an asymmetric multicomponent formulation[END_REF] where

W H 2 O-CO 2 =1 0 .5 V CO 2 +V H 2 O V CO 2 .V H 2 O , α CO 2 = V CO 2 and α H 2 O = V H 2 O
is based on the high pressure and temperature experiments by [START_REF] Aranovich | Experimental determination of CO 2 -H 2 O activity-composition relations at 600-1000 degrees C and 6-14 kbar by reversed decarbonation and dehydration reactions[END_REF] and gives a satisfactory fit to the experimental data at pressures greater than 5 kbars but deviates from other experimental constraints at lower pressures (Fig. 1). and co-workers [START_REF] Duan | An equation of state for the CH 4 -CO 2 -H 2 Osystem:II.Mixturesfrom50to1000°Cand0to1000bar[END_REF][START_REF] Duan | Equation of state for the NaCl-H 2 O-CO 2 system: prediction of phase equilibria and volumetric properties[END_REF][START_REF] Duan | A general equation of state for supercritical fluids and molecular dynamics simulation of mixturePVTXproperties[END_REF][START_REF] Duan | Accurate prediction of thermodynamic properties of fluids in the system H 2 O-CO 2 -CH 4 -N2[END_REF](Duan et al., , 2003(Duan et al., , 2006(Duan et al., , 2008;;[START_REF] Duan | An improved model calculating CO 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF]Duan and Li, 2008;[START_REF] Duan | Equation of state of the H 2 O-CO 2 system up to 10 GPa and 2573K: Molecular dynamics simulations with ab initio potential surface[END_REF][START_REF] Hu | PVTx properties of the CO 2 -H 2 Oa n dC O 2 -H 2 O-NaCl systems below 647 K: Assessment of experimental data and thermodynamic models[END_REF][START_REF] Li | The speciation equilibrium coupling with phase equilibrium in the H 2 O-CO 2 -NaCl system from 0 to 250 °C, from 0 to 1000 bar, and from 0 to 5 molality of NaCl[END_REF]Mao and Duan, 2009) allow calculation of the solubility of CO 2 and densities of mixtures at various pressures and temperatures. [START_REF] Akinfiev | Thermodynamic model of aqueous CO 2 -H 2 O-NaCl solutions from -22 to 100 °C and from 0.1 to 100 MPa[END_REF] have proposed a thermodynamic model from a compilation and critical analysis of experimental results in the system H 2 O-CO 2 -NaCl reproducing the solubility of CO 2 and NaCl. However, their model is restricted to a small pressure-temperature range (less than 100°C and 1 kbar). As shown earlier, the models of [START_REF] Duan | An improved model calculating CO 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF] and Duan et al. (2006) and the results of [START_REF] Todheide | Das Zweiphasengebiet die kritische Kurve im System Kohlendioxid-Wasser bis zu Drucken von 3500 bar[END_REF] ranging from 50°C to 350°C and from 200 to 3500 bar have been used assuming binodal equilibrium. We have also used the results of [START_REF] Sterner | Synthetic fluid inclusions.X:E x p e r i me n t a ld etermination of P-V-T-X properties in the CO 2 -H 2 0systemto6kband700°C[END_REF] and [START_REF] Mather | Phase-equilibria in the systemcarbondioxidewater at elevated pressures[END_REF] who investigated the discrepancy between the measurements of [START_REF] Todheide | Das Zweiphasengebiet die kritische Kurve im System Kohlendioxid-Wasser bis zu Drucken von 3500 bar[END_REF] and that of [START_REF] Takenouchi | The bynary system H 2 O-CO 2 at high temperatures and pressures[END_REF], who report asolubilityofwater in the CO 2 -rich phase about 20% higher at 1kb and 200°C, as noted by [START_REF] Joyce | An experimental determination of the thermodynamic properties of H 2 O-CO 2 -NaCl fluids at high pressures and temperatures[END_REF]. [START_REF] Sterner | Synthetic fluid inclusions.X:E x p e r i me n t a ld etermination of P-V-T-X properties in the CO 2 -H 2 0systemto6kband700°C[END_REF] and Mather andFranck(1992) found agreement with [START_REF] Todheide | Das Zweiphasengebiet die kritische Kurve im System Kohlendioxid-Wasser bis zu Drucken von 3500 bar[END_REF] and therefore, we discarded the results of [START_REF] Takenouchi | The bynary system H 2 O-CO 2 at high temperatures and pressures[END_REF] obtained at temperatures greater than 110°C. The CO 2 solubilities measured by [START_REF] Takenouchi | The bynary system H 2 O-CO 2 at high temperatures and pressures[END_REF] and [START_REF] Takenouchi | Solubility of carbon dioxide in NaCl solutions at high temperatures and pressures[END_REF] also differ from [START_REF] Todheide | Das Zweiphasengebiet die kritische Kurve im System Kohlendioxid-Wasser bis zu Drucken von 3500 bar[END_REF], especially at pressures below 600 bars and temperatures above 200°C, whereas the results of [START_REF] Todheide | Das Zweiphasengebiet die kritische Kurve im System Kohlendioxid-Wasser bis zu Drucken von 3500 bar[END_REF] [START_REF] Takenouchi | The bynary system H 2 O-CO 2 at high temperatures and pressures[END_REF] on the pressure and temperature conditions ofc r i t i c a lm i x i n g .

To the previously described experimental results, we have added the measurements of CO 2 solubility reviewed and selected by [START_REF] Diamond | Solubility of CO 2 in water from -1.5 to 100 °C and from 0.1 to 100 MPa: evaluation of literature data and thermodynamic modelling[END_REF] in the range 0-100°C and1 [START_REF] Bartholomé | Solubility of CO 2 in water[END_REF][START_REF] Vilcu | L öslichkeit Der Gase Unter Druck In Flüssigkeiten .I. Das System Kohlendioxid-Wasser[END_REF][START_REF] Matous | Solubility of carbon dioxide in water at pressures up to 40 atm[END_REF][START_REF] Stewart | Solubility of carbon dioxide in pure water, synthetic sea water, and synthetic sea water concentrates at-5deg.to25deg.and 10-to 45-atm. pressure[END_REF][START_REF] Malinin | Experimental investigations of CO 2 solubility in NaCl and CaCl 2 solutions at temperatures of 25, 50 and 75 degrees and elevated CO 2 pressure[END_REF][START_REF] Malinin | Investigation of CO 2 solubility in solutions of chlorides at elevated-temperatures and pressures of CO 2[END_REF][START_REF] Zawisza | Solubility of carbon-dioxide in liquid water and of water in gaseous carbon-dioxide in the range 0.2-5 MPa and at temperatures up to 473-K[END_REF][START_REF] Gillespie | Vaporliquid and liquidliquid equilibria: watermethane, watercarbon dioxide, waterhydrogen sulfide,w a t e r n -p e n t a n e , watermethanen-pentane[END_REF][START_REF] Müller | Vapor-liquid-equilibrium in the ternary-system ammonia-carbon dioxide-water at high waterc o n t e n t si nt h e range 373-K to 473-K[END_REF][START_REF] Namiot | Solubility of Gases in Water[END_REF][START_REF] King | The mutualsolubilities of water with supercritical and liquid carbon dioxides[END_REF][START_REF] Teng | Solubility ofl i q u i dC O 2 in water at temperatures from 278 K to 293 K and pressures from 6.44 MPa to 29.49 MPa and densities of the corresponding aqueous solutions[END_REF][START_REF] Bamberger | High-pressure (vapor+liquid) equilibrium in binary mixtures of (carbon dioxide+water or acetic acid) at temperatures from 313 to 353 K[END_REF]Yang et al., 2000;[START_REF] Servio | Effect of temperature and pressure on the solubility of carbon dioxide in water in the presence of gas hydrate[END_REF][START_REF] Anderson | Solubility of carbon dioxide in water under incipient clathrate formation conditions[END_REF]. When these measurements do not include the composition of the gas phase, it has been evaluated for each experi- Interestingly, the quantity W H 2 O-CO 2 /RT is remarkably linear over the range of pressure and temperature where CO 2 is liquid or supercritical. α CO 2 is similarly nearly linear with pressure and temperature at pressures greater than 500 bars.

The data scatter close to the following least squares fits to linear relationships:

W H 2 O-CO 2 /RT =5.41 -0.276P -1.07T * 10 -2 (6) α CO 2 =0.742 -0.0974P +3.32T * 10 -3 (7)
with correlation coefficients (r

N i =1+k i,1 .a + k i,2 .a 2 + k i,3 .a 3 + k i,4 .a 4 + k i,5 .b + k i,6 .b 2 +k i,7 .b 3 + k i,8 .b 4 + k i,9 .a.b + k i,10 .a 2 .b 2 (8)
D i = k i,11 + k i,12 .a + k i,13 .a 2 + k i,14 .a 3 + k i,15 .a 4 + k i,16 .b +k i,17 .b 2 + k i,18 .b 3 + k i,19 .b 4 + k i,20 .a.b + k i,21 .a 2 .b 2 (9)
where i represents either Simpler expressions than equations 8 and 9 would be preferable but the approach used here gives accurate results for both critical mixing and reciprocal solubilities of CO 2 and H 2 Ooverthewholepressure-temperaturerange.

W H 2 O-CO 2 /RT or α CO 2 , a is the density of pure H 2 Oand b is that of CO 2 .T h eu
The adjustable parameters were calibrated by minimization of the reduced χ 2 ν (defined as

χ 2 ν = 1 ν n j=1 (v Mj -v Cj ) 2 σ 2 j
where v Mj and v Cj are measured and calculated values for experiment j and ν the degree of freedom) using the Levenberg-Marquardt algorithm [START_REF] Levenberg | A Method for the Solution of Certain Non-Linear Problems in Least Squares[END_REF][START_REF] Marquardt | An Algorithm for Least-Squares Estimation of Nonlinear Parameters[END_REF]. Toreducethenumber of adjustable parameters, an F test (as defined by [START_REF] Bevington | Data Reduction and Error Analysis for The Physical Sciences[END_REF] has been run where parameters were successively zeroed untilthequalityofthe fit was significantly lowered.

The fits to the experimental results and calculated phase diagrams are illustrated in figures 3 and 4. At pressures lower than the critical point of water, the solvus closes on the boiling curve of water with a CO 2 -free composition (i.e. on the y axis of Fig. 3a, 3b and 3c). At pressures higher than the critical point of water, the temperature of critical mixing decreases with increasing pressure (Fig. 1 and 3d). CO 2 solubility is also well calculated at low pressures and temperatures, where the sharp difference of increase in CO 2 solubility with pressure along the boiling curve of CO 2 is well reproduced (figure 4).

Uncertainties on the adjustable parameters of equations 8 to9h a v eb e e ne s -

Activities of water and carbon dioxide in saturated mixtures

Phase assemblages may be very sensitive to variations of the activity of water, in both low-and high-grade metamorphic rocks [START_REF] Greenwood | Wollastonite: Stability in H 2 O-CO 2 mixtures and occurence in a contact-metamorphic aureole near Salmo, British Columbia, Canada[END_REF][START_REF] Ferry | A Biotite Isograd in South-Central Maine, U.S.A.: Mineral Reactions, Fluid Transfer, and Heat Transfer[END_REF][START_REF] Nicollet | Two contrasted P-T-time paths of coronitic metanorites of the French Massif Central: are reaction textures reliable guides to metamorphic histories[END_REF][START_REF] Bayon | Modelling phase-assemblage diagrams for magnesian metapelites in the system K 2 O-FeO-MgO-Al 2 O 3 -SiO 2 -H 2 O: geodynamic consequences for the Monte Rosa nappe[END_REF]; Vidal andD u b a c q ,2 0 0 9 ) .

The dilution of water by addition of CO 2 in solution decreases the activity of water (e.g., [START_REF] Santosh | CO 2 flushing: A plate tectonic perspective[END_REF] and may consequently decrease the temperatures of dehydration reactions. decreases with increasing pressure roughly parallel to the water vapor curve. At pressures higher than 1kb, a H 2 O is more sensitive to temperature. a H 2 O remains elevated even in the vicinity of the critical curve (dashed line on Fig. 6), because the composition of the mixture tends to that of water along theb o i l i n gc u r v eo f water; at pressures higher than the critical point of water, the positive deviation from ideal gas behavior (γ H Consequently, even if errors on the activity model are small,errorsonthecalculated densities are large at very low densities. Similarly, the experimental results at 59°C of [START_REF] Li | Densities and Solubilities for Binary Systems of Carbon Dioxide + Water and Carbon Dioxide + Brine at5 9 °C a n d Pressures to 29 MPa[END_REF] are well reproduced above 100 bars after recalibration for systematic deviations (see Duan et al., 2008). A map of densities calculated over the range 0.1-3kb and 50-370°C is presented in figure 7.

ACTIVITY MODEL IN H 2 O-NaCl

The H 2 O-NaCl system has been extensively studied for decades (see [START_REF] Driesner | The system H 2 O-NaCl. Part I: Correlation formulae for phase relations in temperature-pressure-composition space from 0to1 0 0 0 °C ,0to5 0 0 0b a r ,a n d0to1X N a C l[END_REF], for a review). The solubility of halite isk n o w nt ob em o r e sensitive to temperature than to pressure (e.g, [START_REF] Bodnar | Synthetic fluid inclusions: XII. The system H2ONaCl. Experimental determination of the halite liquidus and isochores for a 40 wt% NaCl solution[END_REF], and NaCl dissociates into several species in solution (see Oelkers and Helgeson, 1993b;[START_REF] Sharygin | Multiple Ion Association versus Redissociation in Aqueous NaCl and KCl at High Temperatures[END_REF] because of the pairing of Na + and Cl -ions. The neutral aqueous sodium chloride species NaCl 0 is thought to dominate the associated NaCl species in solution over a large range of pressure, temperature and composition [START_REF] Sharygin | Multiple Ion Association versus Redissociation in Aqueous NaCl and KCl at High Temperatures[END_REF], but it has been proposed that polynuclear species (Na 2 Cl + ,N a C l - 2 ,e t c ) also occur (e.g. Oelkers and Helgeson, 1993a,b;[START_REF] Sherman | Ion association in concentrated NaCI brines from ambient to supercritical conditions: results from classical molecular dynamics simulations[END_REF].

The derivation of thermodynamic properties of such polynuclear species is beyond the scope of this study and all associated species are here considered as NaCl 0 .

DH-ASF formalism and the Anderson et al. density model

With the DH-ASF model, Na + and Cl -ions are represented by a mean ionic end-member (NaCl±) 1/2 as defined in section 2.3, and the reaction describing halite NaCl (cr) dissolution is:

NaCl (cr) ↔ 2(NaCl±) 1/2 (10)
The pairing of Na + and Cl -into NaCl 0 (or its dissociation) is calculated with the reaction: The thermodynamic properties of the end-member NaCl 0 were based on those of molten halite [START_REF] Evans | A method for activity calculations in saline and mixed solvent solutions at elevated temperature and pressure: A framework for geological phase equilibria calculations[END_REF].

2(NaCl±) 1/2 ↔ NaCl 0 (
The uncertainties on the formation enthalpy ∆H 0 f ,asreportedbyHollandand [START_REF] Holland | An improved and extended internally consistent thermodynamic dataset for phases of petrological interest, involving a new equation of state for solids[END_REF] for halite, NaCl± and NaCl 0 ,a r eo fs i m i l a rr e l a t i v em a g n i t u d ea t ∼ 0.5 of their ∆H 0 f .T h i sc o r r e s p o n d st oap r e c i s i o no f∼ ±0.1 molal for the calculation of the solubility of halite at STP.

Parameterization of the activity model

The relative amounts of NaCl± and NaCl 0 vary non-linearly as a function of the concentration of aqueous sodium chloride and may be described by five parameters between the end-members in equations 10 and 11:

W NaCl±-H 2 O , W NaCl 0 -H 2 O ,
W NaCl±-NaCl 0 , α NaCl± and α NaCl 0 .I ti sa s s u m e dt h a tW NaCl±-NaCl 0 =0as this term has little effect on the calculated equilibria.

Solubility, density and conductivity measurements were used to parameterize the activity model. Selected solubility experiments range from 15 to 650°C and from 1 bar to 10 kbar (table 2). The data are mostly consistent,withtheexception of the high pressure measurements of [START_REF] Sawamura | Solubility of sodium chloride in water under high pressure[END_REF] who report solubility values for halite higher by up to 2% than the measured values at2-3kband25°C

of Adams (1931, figure 9).

Measurements of the conductance of NaCl solutions are used toestimatethe degree of dissociation of NaCl 0 into Na + and Cl -(eq. 11) via:

x NaCl± = Λ /Λe (12)
where x NaCl± is the fraction of NaCl dissolved as NaCl±, Λ is the experimentally determined equivalent conductance and Λe is the equivalent conductance of a hypothetical completely dissociated NaCl solution of the same effective ionic strength (Oelkers and Helgeson, 1988). The procedure selected to calculate Λ /Λe is described in Appendix C, and we used the measurements of [START_REF] Bianchi | The conductivity of concentrated aqueous mixtures of NaCl and MgCl 2 at 25°C˙Journal of Solution[END_REF], [START_REF] Chambers | Conductances of Concentrated Aqueous Sodium and Potassium Chloride Solutions at 25°C˙The[END_REF] and [START_REF] Quist | Electrical conductancesofaqueoussodium chloride solutions from 0 to 800°C and at pressures to 4000 bars[END_REF] measurements [START_REF] Surdo | The (p, V, T) properties of concentrated aqueous electrolytes I. Densities and apparent molar volumes of NaCl, Na 2 SO 4 ,M g C l 2 ,a n dM g S O 4 solutions from 0.1 molkg-1 to saturation and from 273[END_REF]. The temperature dependency of these parameters has been estimated together with that of W NaCl 0 -H 2 O and α NaCl± by fitting high-temperature solubility experiments (Fig. 9a) and the calculated degree of association (Fig. 10a, equation C1).

The following expressions have been found to provide a good description of the system in the pressure-temperature range 1bar-10kbar and 20-650°C, as shown in figures 9, 10 and 11:

W NaCl 0 -H 2 O = C 1 + C 2 .P + C 3 .T + C 4 .P 2 (13) α NaCl 0 = C 5 + C 6 .P + C 7 .T (

Results and error propagation

The correlation between the calculated and measured solubilities is good (Fig. 9). This model gives a better fit to high-temperatures data than the model of Mao and Duan (2008, dotted line on Fig. 9). The obtained parameters agree well with the results of [START_REF] Aranovich | H 2 OactivityinconcentratedNaClsolutions at high pressures and temperatures measured by the brucite-periclase equilibrium[END_REF] who found that the system is very close to ideality around 2 kbar and 500°C (W NaCl 0 -H 2 O =0.56 kJ.mol -1 ). Density measurements in the aqueous phase are very well reproduced below the critical point of water and are in close agreement with the model of [START_REF] Driesner | The system H2O-NaCl. Part II: Correlations for molar volume, enthalpy, and isobaric heat capacity from 0 to 1000°C, 1 to 5000 bar, and 0 to 1 XNaCl[END_REF]. At higher temperatures, our model diverts from the model of Driesner (2007) (Fig. 11b andc) for pressures around 1 kbar and below. The calculated dissociation of NaCl 0 presented in figure 10a is consistent with the measurements ofQ u i s ta n d Marshall (1968) as they show the same variations with pressure and temperature but appear offset from a few to about 15 percents, which could not be reproduced without allowing unrealistically low values for α NaCl 0 .T h eu n c e r t a i n t i e sa s s o c iated with the original values of Λ and Λ 0 reported by [START_REF] Quist | Electrical conductancesofaqueoussodium chloride solutions from 0 to 800°C and at pressures to 4000 bars[END_REF] cumulate to a minimal uncertainty of about 10% of the given x NaCl± value at water densities below 0.5 and between 4 and 10% at higher densities.Because1)uncertainties associated with the use of equations C2 and C3 are unknown, 2) the possible role of large polynuclear species, although unclear [START_REF] Sharygin | Multiple Ion Association versus Redissociation in Aqueous NaCl and KCl at High Temperatures[END_REF],

can not be ruled out and 3) the low ratio of NaCl± to NaCl 0 makes W NaCl±-H 2 O poorly constrained at temperatures greater than 500°C, we have adopted the simplest expression that could be derived for the activity modelinwhichW NaCl±-H 2 O is independent of temperature above 400K and α NaCl± varies linearly with pressure but with a temperature-dependent slope. Overall, the model gives a better fit to the data and is in better agreement with other models for high density solutions than for low density solutions, partially because of the use of the density model of [START_REF] Anderson | The density model for estimation of thermodynamic parameters of reactions at high temperatures and pressures[END_REF], but also because the lack of theory together with the scarcity of precise high-temperature high-concentration experimental data made assumptions on the pressure-temperature dependency of the thermodynamic parameters necessary.

Error propagation has been performed by a Monte-Carlo simulation where the system has been refitted 100 times, by varying the original experimental results within the misfit of the model to the data and the enthalpy of formation of the aqueous species within their uncertainties. To simplify thepropagationoferrors on the activity model, the correlation of uncertainties associated with the enthalpy of formation of the aqueous species given in the THERMOCALC database has been neglected. The covariance matrix associated with C 1 to C 16 is given in Table D3.

THE H 2 O-CO 2 -NaCl SYSTEM

The solubility of CO 2 in the water phase decreases with increasing salinity of the aqueous phase (termed salting-out, e.g.: [START_REF] Markham | The Solubility of Carbon Dioxide and Nitrous Oxide in Aqueous Salt Solutions[END_REF][START_REF] Drummond | Boiling and mixing of hydrothermal fluids : chemical effects on mineral precipitation[END_REF]Nighswander et al., 1989;[START_REF] Rumpf | Solubility of carbon dioxide in aqueous solutions of sodium chloride: Experimental results and Correlation[END_REF] in the CO 2 phase at 670 bars and 400°C. In H 2 O-CO 2 -NaCl, taking the CO 2 -rich phase to be free of NaCl is an acceptable approximation up to atleast300°C(e.g. Hu et al., 2007, and references herein). With this assumption, the solubility of CO 2 in the aqueous NaCl solution can be calculated by minimizing the Gibbs free energy of the system, composed of a water-saturated CO 2 -rich phase and of the aqueous phase at fixed NaCl content. The relative proportions of NaCl 0 and NaCl± are calculated by solving equation 11 with the appropriate interaction parameters (eq. 8,9,13,14,15,16).

As pointed out by [START_REF] Hu | PVTx properties of the CO 2 -H 2 Oa n dC O 2 -H 2 O-NaCl systems below 647 K: Assessment of experimental data and thermodynamic models[END_REF], there is currently no accurate model to predict the solubility of CO 2 in NaCl brines at temperatures greater than about 60°C, mainly due to the scarcity of experimental results at these pressures and temperatures. In particular, the measurements of [START_REF] Takenouchi | Solubility of carbon dioxide in NaCl solutions at high temperatures and pressures[END_REF] (ranging up to 1400 bars and 450°C) are not in agreement with other experiments at pressures below 300 bars, some of which we used here [START_REF] Ellis | Solubility of carbon dioxide above 100°C in water and in sodium chloride solutions[END_REF][START_REF] Drummond | Boiling and mixing of hydrothermal fluids : chemical effects on mineral precipitation[END_REF]Nighswander et al., 1989;[START_REF] Rumpf | Solubility of carbon dioxide in aqueous solutions of sodium chloride: Experimental results and Correlation[END_REF]

DISCUSSION

Accuracy and advantages of the method

The method used here to parameterize the H and Sun ( 2003) on which the model of [START_REF] Duan | Equation of state of the H 2 O-CO 2 system up to 10 GPa and 2573K: Molecular dynamics simulations with ab initio potential surface[END_REF] is based. At pressures and temperatures under 100°C and 100 bars, our model gives results very similar to [START_REF] Duan | An improved model calculating CO 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF]. At higher pressures and temperatures, the model of [START_REF] Duan | An improved model calculating CO 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF] gives a better fit to the data of [START_REF] Takenouchi | Solubility of carbon dioxide in NaCl solutions at high temperatures and pressures[END_REF], which are often lower than expected from the measurements of [START_REF] Todheide | Das Zweiphasengebiet die kritische Kurve im System Kohlendioxid-Wasser bis zu Drucken von 3500 bar[END_REF], well reproduced by our model. For this reason, our model gives solubilities greater than the data of [START_REF] Takenouchi | Solubility of carbon dioxide in NaCl solutions at high temperatures and pressures[END_REF] and the model of [START_REF] Duan | An improved model calculating CO 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF] at pressures and temperatures where [START_REF] Todheide | Das Zweiphasengebiet die kritische Kurve im System Kohlendioxid-Wasser bis zu Drucken von 3500 bar[END_REF] and [START_REF] Takenouchi | Solubility of carbon dioxide in NaCl solutions at high temperatures and pressures[END_REF] are not in agreement (Fig. 12). However, the curvature of the CO 2 solubility is quite similar for both mod-

Range of applicability and limitations

The pressure-temperature range of the activity model proposed for H [START_REF] Johnson | Experimentally determined limits for H 2 O-CO 2 -NaCl immiscibility in granulites[END_REF]Holland and Powell, 2003, Fig. 1) suggest that the critical temperature increases significantly at pressure greater than 3.5 kb, implying that W H 2 O-CO 2 increases with increasing pressure, opposite to the trend observed below 3500 bars. As shown in figure 13, thedeparturesat higher temperatures from the model of [START_REF] Holland | Activity-composition relations for phases in petrological calculations: an asymmetric multicomponent formulation[END_REF], calibrated in the one-phase domain, imply that the decrease of W H 2 O-CO 2 /RT with temperature (Fig. 2) is likely to be much smaller or reversed in the one-phase domain. In the two-phase domain, differences between the model of [START_REF] Holland | Activity-composition relations for phases in petrological calculations: an asymmetric multicomponent formulation[END_REF] and this model, of the order of 1.5 kJ/mol for W H 2 O-CO 2 around 3kbar and 200°C (Fig. 13), are very significant for the solvus as illustrated on figure 3d. For the same conditions, the precision on the activity model in this work is that obtained from the experimental data, of the order of 0.1 kJ/mol for W H 2 O-CO 2 and 0.01 for

α CO 2 /α H 2 O .
There are therefore abrupt changes in the parameters describing the thermodynamic properties of the mixture both along the critical mixing curve (Fig. 13) and along along the phase transitions of the end-members (e.g. steps in

W H 2 O-CO 2
along the boiling curve of CO 2 ,F i g . 2 d ;α CO 2 →∞along the boiling curve of water, Fig. 2c).

For these reasons the model (equations 8-9) should not be extrapolated to the one-phase region (Fig. 1) where the model of [START_REF] Holland | Activity-composition relations for phases in petrological calculations: an asymmetric multicomponent formulation[END_REF] is appropriate (and conversely the model of [START_REF] Holland | Activity-composition relations for phases in petrological calculations: an asymmetric multicomponent formulation[END_REF] should not be used in the two-phase domain). The critical curve which delimits the domain of applicability can be closely approximated from the criticalp o i n to fw a t e ru pt o 3500 bars by: T C = q 1 + q 2 .P + q 3 .P 2 + q 4 . log(P ) (18)

where q 1 to q 4 are constants obtained by regression analysis (R 2 =0 .993)o f the data of [START_REF] Todheide | Das Zweiphasengebiet die kritische Kurve im System Kohlendioxid-Wasser bis zu Drucken von 3500 bar[END_REF] and [START_REF] Takenouchi | The bynary system H 2 O-CO 2 at high temperatures and pressures[END_REF]:

q 1 =195.3,q 2 =90
.36,q 3 = -8.945,q 4 = -107.9.

Another limitation of our approach (derived from the phase rule and relying on a minimum number of end-members used to describe the system) is that speciation (and its effects) is hard to predict for high concentrations although easy to reproduce. In other words, experimental results are necessary to parameterize the model. For instance, the formation of Na carbonate complexes might result in increasing CO 2 solubility with increasing Na concentration, when our model predicts a continuous decrease of CO 2 solubility with increasing salinity. For the same reason, CO 2 solubility in rock-buffered systems may be different from what is estimated with this model.

CONCLUSIONS

The model presented here is unique in that it replicates available data for mixed phase, salt-rich, mixed-solvent fluids close to the criticale n dp o i n t ,a l l o w sr e a listic propagation of uncertainties, simulates dissociation of ionic solutes such as NaCl, can be extended readily to more complex systems as it uses transferable parameters, is compatible with existing thermodynamic databases and software, and is based, as far as possible, on physically realistic expressions with a minimum reliance on empirical expressions such as power-law series.T h el a r g en u m b e ro f empirically fitted constants is however a drawback to a well-constrained activity model.

It is possible to calculate accurate phase diagrams for CO 2 -H 2 Om i x t u r e si n the two-phase domain with macroscopic interaction parameters and associated uncertainties independently of the standard state chemicalpotentialsofwaterand carbon dioxide. Such calculations are straightforward, fast, lead to a unique result and do not depend on the quality of the chosen thermodynamic database. They can be implemented in Gibbs energy minimization routines (e.g. [START_REF] Connolly | The geodynamic equation of state: What and how[END_REF][START_REF] Capitani | The computation of equilibrium assemblage diagrams with Theriak/Domino software[END_REF].

The interaction parameters W H 2 O-CO 2 and a CO 2 calculated from experimental results show non-linear variations with pressure and temperature, reflecting significant changes of solubilities and volumes for example along the vapor-liquid transition curves of water and carbon dioxide. This contrasts with the description of the higher temperature-pressure conditions applicable to metamorphic petrol-ogy where the linear relation W G = W H -T * W S + P * W V holds for the many systems described by the subregular model or the ASF formalism. As a consequence, it is not possible to extrapolate mixing parameters in the one-phase domain from compositions at equilibrium in the two-phase domain in the H 2 O-

CO 2 system.
An activity model in the H 2 O-NaCl system was derived with a similar thermodynamic formalism, DH-ASF [START_REF] Evans | A method for activity calculations in saline and mixed solvent solutions at elevated temperature and pressure: A framework for geological phase equilibria calculations[END_REF], whichuseshypothetical end-members with unit activity at standard state for aqueous species. This model requires a small number of concentration-independentinteractionparameters, which we parameterized up to 10kbars and 650°C as a function of pressure and temperature using density, conductivity and solubilitymeasurements.

The activity models in H 2 O-CO 2 and H 2 O-NaCl were combined to derive an activity model in the H 2 O-CO 2 -NaCl system where the decrease of CO 2 solubility with salinity is reproduced by positive values for W parameters between dissolved NaCl and CO 2 .I ti ss h o w nt h a tt h i sa p p r o a c he x t r a p o l a t e st oh i g hs a l i n i t ies with a continuous decrease in CO 2 solubility, in contrast with the model of [START_REF] Duan | An improved model calculating CO 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF]. The fit to the experimental data is good and uncertainties on all parameters of the activity model are estimated.

Addition of activity models for aqueous species and gases to the THERMO-CALC database with the DH-ASF formalism is in progress (e.g. [START_REF] Evans | Internally consistent data for sulphur-bearing phases and application to the construction of pseudosections for mafic greenschist facies rocks in Na 2 O-CaO-K 2 O-FeO-MgO-Al 2 O 3 -SiO 2 -CO 2 -O-S-H 2 O[END_REF].

ACKNOWLEDGMENTS

This work was supported by the EU Marie Curie GRASP RTN and the NERC through the CRIUS consortium. We thank Éric Lewin for help in regression anal-ysis and Roger Powell, Tim Holland, Niko Kampman, Olivier Namur, Oli Shorttle and Ed Tipper for helpful discussions. Jacques Schott (A.E.)andthreeanonymous reviewers are thanked for their numerous comments and suggestions on previous versions of the manuscript.

APPENDIX A

It is demonstrated here that if unmixing occurs in a binary ij solution, it is possible to calculate the values of the parameters W ij and α i /α j from the compositions of the coexisting phases x 1 and x 2 at equilibrium, i.e. from the binodal solvus.

In a binary ij solution, where x is the proportion of the j end-member so that x i =1-x,chemicalpotentialscanbeexpressedas(e.g. Guggenheim,1977):

µ i (x)=G m (x) -x δG m (x) δx (A1) µ j (x)=G m (x)+(1-x) δG m (x) δx (A2)
where G m (x) is the molar Gibbs free energy of the mixture:

G m (x)=(1-x)µ 0 i + xµ 0 j + RT (1 -x) log(1 -x) +RTxlog(x)+G xs m (x) (A3)
The Gibbs free energy of mixing G mix m (x) is the energy added to the mechanical mixture of the pure end-members:

G mix m (x)=RT (1 -x) log(1 -x)+RTxlog(x)+G xs m (x) (A4)
Expressed with the ASF formalism, G xs m (x) reduces to:

G xs m (x)= 2α i α j W ij (1 -x)x (α i + α j )(α i (1 -x)+α j x) (A5)
Note that in an ideal mixing scheme, G Equilibrium implies that the chemical potential of each end-member must be the same in each phase:

µ i (x) x=x 1 = µ i (x) x=x 2 (A6) µ j (x) x=x 1 = µ j (x) x=x 2 (A7)
Using equation A1 and equation A2, we obtain the two conditions A8 and A9:

δG m (x 1 ) δx = δG m (x 2 ) δx (A8)
and

G m (x 1 ) -G m (x 2 )=(x 1 -x 2 ) δG m (x 1 ) δx (A9)
where

δG m (x) δx = µ 0 j -µ 0 i + RT log( x 1 -x ) (A10) + 2α i α j W ij (α i (x -1) 2 -α j x 2 ) (α i + α j )(α i (x -1) -α j x) 2
In the special case of a symmetric solvus, the gradient of Gibbs energy with composition δGm(x) δx =0at binodal compositions.

Combining equations A8 and A9 with equation A5 gives

W ij = RT ((1 -x 2 ) log( 1 -x 2 1 -x 1 )+x 2 log( x 2 x 1 )) * D (A11)
where D is given by:

D = - (α i + α j )(α i (x 1 -1) -α j x 1 ) 2 (α i (x 2 -1) -α j x 2 ) 2α 2 i α 2 j (x 1 -x 2 ) 2 (A12)
and

α j = α i (2(x 1 -1)(x 2 -1) log( x 1 -1 x 2 -1 )+(x 1 + x 2 -2x 1 x 2 ) log( x 1 x 2 )) (-x 2 + x 1 (2x 2 -1)) log( x 1 -1 x 2 -1 ) -2x 1 x 2 log( x 1 x 2 ) (A13)
Defining α i =1 ,o n l yo n eo fα i and α j varies independently and D can be expressed as D = E/F,whereE and F depend only on x 1 and x 2 :

E =2((x 1 (2x 2 -1) -x 2 ) log( x 1 -1 x 2 -1 ) -2x 1 x 2 log( x 1 x 2 )) 2 (2(x 1 -1)(x 2 -1) log( x 1 -1 x 2 -1 ) (A14) +(-2x 2 x 1 + x 1 + x 2 ) log( x 1 x 2 )) 2 F =(x 2 -x 1 )((x 1 -1) log( x 1 -1 x 2 -1 ) -x 1 log( x 1 x 2 )) 2 ((x 2 -1) log( x 1 -1 x 2 -1 ) -x 2 log( x 1 x 2 ))(A15) ((-3x 2 + x 1 (4x 2 -3) + 2) log( x 1 -1 x 2 -1 )+(-4x 2 x 1 + x 1 + x 2 ) log( x 1 x 2
))

The conditions of critical mixing (consolute point along isotherms or isobars on phase diagrams such as presented in figure 3) are reached when:

δ 2 G m (x c ) δx 2 = RT C x C -x 2 C - 4α 2 i α 2 j W ij (α i + α j )(α i + x C (α j -α i )) 3 =0 (A16)
and

δ 3 G m (x c ) δx 3 = RT C ( 1 (x C -1) 2 - 1 x 2 C ) (A17) + 12α 2 i α 2 j W ij (α j -α i ) (α i + α j )(α i + x C (α j -α i )) 4 =0
where T C is the critical temperature. This defines W ij as:

W ij = RT C (α i + α j )(α i (x C -1) -α j x C ) 3 4α 2 i α 2 j (x 2 C -x C ) (A18)
and α j as

α j = α i (x 2 C -1) x C (x C -2) (A19)
The two coexisting compositions can be calculated by minimizing the Gibbs free energy of the system (see [START_REF] Connolly | The geodynamic equation of state: What and how[END_REF][START_REF] Capitani | The computation of equilibrium assemblage diagrams with Theriak/Domino software[END_REF] or by solving equations A8 and A9 for x 1 and x 2 .E q u a t i o nA 8g i v e s :

2α 2 i α 2 j W ij (x 1 -x 2 )(α i (x 1 + x 2 -2) -α j (x 1 + x 2 )) (α i + α j )(α i (x 1 -1) + α j x 1 ) 2 (α i (x 2 -1) -α j x 2 ) +RT (log( 1 -x 2 1 -x 1 ) + log( x 1 x 2 )) = 0 (A20)
Equation A9 gives: [START_REF] Evans | A method for activity calculations in saline and mixed solvent solutions at elevated temperature and pressure: A framework for geological phase equilibria calculations[END_REF].

-2α 2 i α 2 j W ij (x 1 -x 2 ) 2 (α i + α j )(α i (x 1 -1) + α j x 1 ) 2 (α i (x 2 -1) -α j x 2 ) +RT ((x 2 -1) log( 1 -x 2 1 -x 1 )+x 2 log( x 1 x 2 )) = 0 (A21)

APPENDIX C

Measurements of the conductance of NaCl solutions are used toestimatethe degree of dissociation of NaCl 0 into Na + and Cl -(eq. 11) via:

x NaCl± = Λ /Λe (C1)
where x NaCl± is the fraction of NaCl dissolved as NaCl±, Λ is the experimentally determined equivalent conductance and Λe is the equivalent conductance of ah y p o t h e t i c a lc o m p l e t e l yd i s s o c i a t e dN a C ls o l u t i o no ft h esame effective ionic strength (Oelkers and Helgeson, 1988). Λe is calculated from the limiting equivalent conductance of the electrolyte (Λe 0 ). Λe for NaCl solutions has been obtained

with the equation of [START_REF] Monica | A conductivitye q u a t i o nf o r concentrated aqueous solutions[END_REF], which is valid up to at least 6 molal:

Λe.η/η 0 =(Λe 0 - B 2 √ c 1+Bå √ c )(1 - B 1 √ c 1+Bå √ c e 0.2929κα -1 0.2929κα ) (C2)
where η is the viscosity of the solution, η 0 the viscosity of the pure solvent, c the concentration of NaCl and å the ion size parameter taken from [START_REF] Helgeson | Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: IV. Calculation of activity coefficients, osmotic coeffcients, and apparent molal and standard and relative partial molal properties to 600C and 5kb[END_REF]. B 1 and B 2 are coefficients originating from the formula of Onsager (1927) and modified by [START_REF] Monica | A conductivitye q u a t i o nf o r concentrated aqueous solutions[END_REF]

κα = Bå √ c =50.294å √ c/ √ T( C 3 )
Equation C2 is a refinement of the [START_REF] Wishaw | The Diffusion Coefficientsa n dC o n d u ctances of Some Concentrated Electrolyte Solutions at 25°C[END_REF] equation which allows correction of the ionic mobility for changing viscosity and is based on the original equations proposed for dilute solutions by [START_REF] Falkenhagen | Zur Theorie der Leitfhigkeit starker nicht assoziierender Elektrolyte bei hheren Konzentrationen[END_REF].

Oelkers and Helgeson (1988) have used the Shedlovsky equation [START_REF] Shedlovsky | The computation of ionization constants and limiting conductance values from conductivity measurements[END_REF] to estimate the degree of association in solutions up to0.1molalforvarious electrolytes. However, the Shedlovsky equation is not validforthecalculationof Λe for solutions containing more than about 0.1m. For some high-temperature conditions, the calculated values of Λe are less than the observed Λ ,l e a d i n gt o

x NaCl± > 1.

Λe has been calculated for the data of Bianchi et al. (1989, 25°C,1bar,0.5-3.6 molal), [START_REF] Chambers | Conductances of Concentrated Aqueous Sodium and Potassium Chloride Solutions at 25°C˙The[END_REF] (25°C, 1 bar, 0.1-5.35 molal) and from [START_REF] Quist | Electrical conductancesofaqueoussodium chloride solutions from 0 to 800°C and at pressures to 4000 bars[END_REF] for solutions of 0.1molal NaCl in the range 100°C-600°C. For each experimental result, η 0 has been calculated with the IAPWS EOS of water.

When not measured, η was calculated with the equation of Mao et al. (2009). The increase in viscosity for solutions of 0.1 molal NaCl or less has little impact on the calculated Λe and has been neglected for the data of [START_REF] Quist | Electrical conductancesofaqueoussodium chloride solutions from 0 to 800°C and at pressures to 4000 bars[END_REF].

The densities reported along with temperatures by [START_REF] Quist | Electrical conductancesofaqueoussodium chloride solutions from 0 to 800°C and at pressures to 4000 bars[END_REF] were individually converted to pressures with the IAPWS EOS.

Va l u e k 1 [START_REF] Hnedkovský | Volumes of aqueous solutions of CH 4 ,CO 2 ,H 2 SandNH 3 at temperatures from 298.15 K to 705 K and pressures to 35 MPa[END_REF], and measurements above 1M CO 2 are taken from [START_REF] Teng | Solubility ofl i q u i dC O 2 in water at temperatures from 278 K to 293 K and pressures from 6.44 MPa to 29.49 MPa and densities of the corresponding aqueous solutions[END_REF]. T&K1964: Measurements of [START_REF] Takenouchi | The bynary system H 2 O-CO 2 at high temperatures and pressures[END_REF]. H&P2 0 0 3 : Model of [START_REF] Holland | Activity-composition relations for phases in petrological calculations: an asymmetric multicomponent formulation[END_REF]. High pressure -low temperature solid phases (ices and clathrates) are not represented.

k 2 k 3 k 4 k 5 k 6 k 7 k 8 k 9 k 10 k 11 k 12 k 13 k 14 k 15 k 16 k 17 k 18 k 19 k 20 k 21 k - - - - - - - - - -- - - - - - - - - - - - k 3.5237E-05 -3.14E-
- - - - - - - - - -- - - - - - - - - - - - k 10 - - - - - - - - - -- - - - - - - - - - - - k 11 2.
- - - - - - - - - -- - - - - - - - - - - - k 17 - - - - - - - - - -- - - - - - - - - - - - k 18 -9.

  describes the magnitude of the excess Gibbs free energy function in terms of the parameters α l attributed to each end-member. This model yields the following expression for the excess Gibbs energy G xs m (x) of a mixture comprised of n endmembers:

  rely on erroneous assumptions. The thermodynamic models of[START_REF] Duan | An equation of state for the CH 4 -CO 2 -H 2 Osystem:II.Mixturesfrom50to1000°Cand0to1000bar[END_REF];[START_REF] Spear | CO 2 -H 2 Omixturesinthegeological sequestration of CO 2 .I.Assessmentandcalculationofmutualsolubilities from 12 to 100°C and up to 600 bar[END_REF][START_REF] Tan | SAFT1-RPM Approximation Extended to Phase Equilibria and Densities of CO 2 -H 2 Oa n dC O 2 -H 2 O-NaCl Systems[END_REF] and Ji and Zhu (2012) account for the compositionofthegasphase, but these models are either poorly constrained at low pressures and temperatures (e.g.[START_REF] Duan | An equation of state for the CH 4 -CO 2 -H 2 Osystem:II.Mixturesfrom50to1000°Cand0to1000bar[END_REF] valid at conditions > 100°C-200 bars) or restricted to low temperatures (<100°C, Spycher et al., 2003) or low pressures (<600 bars, Ji et al., 2005; Ji and Zhu, 2012). None of the above models give uncertainties on their calculated parameters. We use the ASF formalism, with α H 2 O set to 1 and W H 2 O-CO 2 and α CO 2 functions of pressure and temperature, to model the mutual solubilities of H 2 Oa n d CO 2 up to temperatures of 370°C and pressures of 3500 bars. Figure2a -csho w the values of W H 2 O-CO 2 and α CO 2 as a function of pressure and temperature calculated for experimental results where both the amount of CO 2 in the aqueous phase and the amount of water in the CO 2 -rich phase are measured. To do this, the experimental data gathered by Spycher et al. (2003) at temperatures lower than 110°C

  are generally higher. The results of[START_REF] Takenouchi | Solubility of carbon dioxide in NaCl solutions at high temperatures and pressures[END_REF] have therefore not been used top a r a m e t e r i z e the model in H 2 O-CO 2 ,b utthe irme a s ur e me ntsinH 2 O-CO 2 -NaCl were used as described below.When not given in the original publication, experimental uncertainties have been estimated as ±3% of the measured CO 2 content in the water-rich phase and ±1% of the measured CO 2 content in the CO 2 -rich phase, in line with the commonly reported uncertainties. Although reported compositions vary, there is ag e n e r a la g r e e m e n tb e t w e e nT o d h e i d ea n dF r a n c k( 1 9 6 3 )a n dT akenouchi and

  -1000 bars (namely: Sander, 1912; Hä h n e l , 1 9 2 0 ; Kritschewsky et al., 1935; ?;W i e b ea n dG a d d y ,1 9 3 9 ,1 9 4 0 ;B a r t h o l o m éa n d

  mental point from a version of the model with W H 2 O-CO 2 and α CO 2 derived from the dataset without these measurements. The estimated gas composition was given a very large uncertainty (100% of the water content) to ensure that the uncertainty on W H 2 O-CO 2 and α CO 2 is effectively constrained by the measured aqueous composition, giving negligible weight to the estimated gas compositions.As shown on figure2, neither W H 2 O-CO 2 nor α CO 2 are linear functions of pressure and temperature, and they vary rapidly close to the phaset r a n s i t i o n so ft h e end-members. At conditions below the critical point of water, the solvus closes on the composition of water. In this case the value of α CO 2 tends towards +∞ and it is not possible to use ASF at the exact conditions of boilingwater.Thisfeature is nonetheless a strong constraint on the shape of the α CO 2 = f (P, T ) surface (Fig.2c). The scale of the α CO 2 diagram on figure2chas been chosen to exclude the very high values from experimental results close to the boiling curve of water.

  s eo fr a t i o so fp o l y n o m i a le q u a t i o n sh a sb e e nc h o s e nt o accommodate trends towards infinity. Values for k W H 2 O-CO 2 /RT and k α CO 2 are given in tables D1 and D2 of Appendix D.

  timated with the help of a Monte-Carlo simulation. The systemw a sc a l c u l a t e d 1000 times by allowing the compositions of the experimental results used to calculate W H 2 O-CO 2 and α CO 2 to vary within the limits of their uncertainty (or defined by the misfit of the model to the data when this was greater than experimental uncertainty). The resulting covariance matrices are given in tables D1 and D2 of appendix D and allow the uncertainty on the compositions to ber e c o v e r e df r om the uncertainty on G xs m (x) with the usual error propagation equation (e.g. Bevington and Robinson, 2002, their equation 3.13). The calculated parameters are strongly correlated due to thenatureofthefunctions and the covariance terms can consequently not be neglected. An indicative map of uncertainties in calculated CO 2 solubility is provided in figure 5. At pressures greater than 500 bars, experiments are generally reproduced within their uncertainties (of the order of 10% of the measured value at low temperatures). At low pressure and high temperature, themodelcandiverge from experimental values, especially in the vicinity of the boiling curve of water where CO 2 solubility is very small. Below the critical point of CO 2 ,t h em o d e l reproduces the experiments less well at very low pressures where the CO 2 concentrations are very small.

Figure 6

 6 Figure 6 illustrates the calculated activity of water (a H 2 O ,F i g . 6 a )a n da ctivity coefficient of water (γ H 2 O ,F i g . 6 b )i naC O 2 -saturated aqueous phase in the pressure-temperature range where two phases coexist. Atlo wpressure,a H 2 O

  11) The total number of moles of dissolved NaCl is thus equal to (1/2)(NaCl±) 1/2 + NaCl 0 .T h e s u b s c r i p t 1 / 2 w i l l b e d r o p p e d f r o m t h i s p o i n t o n w a r d s f o rc o n v enience.The thermodynamic properties of NaCl± have been calculated from that of Na + and Cl -from[START_REF] Helgeson | Theoretical prediction of the thermodynamic behavior of aqueous electrolytes at high pressures and temperatures: IV. Calculation of activity coefficients, osmotic coeffcients, and apparent molal and standard and relative partial molal properties to 600C and 5kb[END_REF] as used in the THERMOCALC database[START_REF] Holland | An improved and extended internally consistent thermodynamic dataset for phases of petrological interest, involving a new equation of state for solids[END_REF]. It has been found necessary to recalculate the heat capacity terms (Cp 0 and b Cp ,seeHollandandPowell,1998)ofNaCl± by fitting them to the measurements reported in[START_REF] Pitzer | Thermodynamic properties of aqueous sodium-chloride solutions[END_REF] as shown in figure8a.The regressed values are Cp 0 = -0.0417156 kJ/K and b Cp = -20.8763 * 10 5 kJ.K 2 .T h ev o l u m e sc a l c u l a t e da ti n fi n i t ed i l u t i o no fN a C l ± using the calculated heat capacity via the modified[START_REF] Anderson | The density model for estimation of thermodynamic parameters of reactions at high temperatures and pressures[END_REF] density model EOS fit the volumes inferred from experimental results (Fig.8b). However, volumes obtained above the critical temperature of water show an unreasonablepressuredependency at pressures lower than about 1 kbar, as illustrated in figure8c. This has been identified as originating from the use of the ratio ρ H 2 O /ρ 0 H 2 O in the derivation of the chemical potential ofAnderson et al. (1991, Fig. 8d), where ρ 0 H 2 O is the density of water at standard pressure and temperature and ρ H 2 O is the density of water at the considered pressure-temperature. It is therefore not possible to calculate accurate densities for aqueous fluids in the range of approximately 370-550°C at pressures below about 1 kbar with this EOS, which nevertheless shows good accuracy in many geologically relevant thermal gradients.

  .A t2 5 °Ca n d1 bar, the calculated Λ /Λe indicate that, within uncertainties of the data, corrections and equations used, at least 95% of the aqueous NaCl is dissociated up to 5.35 molal, in agreement with[START_REF] Monica | A conductivitye q u a t i o nf o r concentrated aqueous solutions[END_REF] but strikingly different from the results of[START_REF] Sherman | Ion association in concentrated NaCI brines from ambient to supercritical conditions: results from classical molecular dynamics simulations[END_REF] whose molecular dynamic simulations predict about 50% of aqueous NaCl to be as NaCl 0 or larger polynuclear species at 6 molal. Only values of x NaCl± estimated from conductance measurements have been used in this work. At constant molality, Λ /Λe decreases with temperature and increases with pressure, as noted by Oelkers andHelgeson(1988)andshown in figure10.The calculated volume of the NaCl± end-member at its hypothetical pure standard state depends on both 1) the thermodynamic properties oftheusualhypothetical Na + and Cl -1molalaqueousspeciesderivedfrominfinitedilutionand2)on the interaction parameters W NaCl±-H 2 O and α NaCl± because they affect the chemical potential of the NaCl± end-member within the DH-ASF framework. NaCl being largely dissociated under 100°C, NaCl 0 has little effect on the Gibbs energy of the mixture and on its pressure dependency. Consequently, the pressure dependency of W NaCl±-H 2 O and α NaCl± are constrained mainly by the chosen density

Figure 12

 12 Figure 12 illustrates the calculated salting-out effect andp r e s e n tsth eq u a lity of the fit of the model to some of the experimental data. Error propagation has been carried out as in the previous systems by a Monte-Carlo simulation where experimental uncertainties are estimated from the misfit of the model with the experimental results. The covariance matrix for D 1 to D 5 is given together with their values in table D4. The correlation of uncertainties betweenparametersregressed

  2 O-CO 2 activity model is independent of the thermodynamic properties of the co-solvents and successfully reproduces the experimentally-derived phase diagram over thewholetwo-phasedomain. The solubility of CO 2 is also calculated well by other models, in particular Duan et al. (2006, figure 3), although it has been shown that this model does not provide a correct derivation of the chemical potential of CO 2 .T h e m o d e l s o f Spycher et al. (2003) and[START_REF] Akinfiev | Thermodynamic model of aqueous CO 2 -H 2 O-NaCl solutions from -22 to 100 °C and from 0.1 to 100 MPa[END_REF] give excellent fits to the aqueous phase experimental results at temperatures below 100°C. The principal advantages of the model presented here are 1) the facility to model the CO 2 -rich phase, 2) the wider pressure-temperature range of applicability, and 3) the ease of error propagation provided by separate uncertainties fortheactivitymodeland the end-members.In the H 2 O-NaCl system, the DH-ASF activity model reproduces the experimental data on the solubility of halite over a wide range of pressures and temperatures. Here, the form of the pressure and temperature dependency of the activity parameters have been fitted to solubilities, densities and conductivity measurements, giving reasonable values for α NaCl± and α NaCl 0 whose variations over 10 kb and 600 °C do not exceed 0.4 and 0.8 α unit, respectively. Their pressure derivatives, implied by density measurements, are particularly small (at a max-imum magnitude of 5.10 -3 kbar -1 )a n do fo p p o s i t es i g n s ,m a k i n gi td i f fi c u l tt o link them to variations of the properties of the solvent. The pressure dependencies of W NaCl 0 -H 2 O and W NaCl±-H 2 O are similarly small, which is imposed by the insensitivity of halite solubility to pressure.Results in H 2 O-CO 2 -NaClarecomparedinfigure12tothemodelofDuan

  els at NaCl concentrations below 3 molal. The model presentedi nt h i ss t u d y has been fitted with essentially the same dataset as in Duan andS u n( 2 0 0 3 )f o r the system H 2 O-C O 2 -N a C l ,w h e r eh i g h -p r e s s u r e -t e m p e r a t u r em e a s u r e m e n t s are restricted below about 4 molal NaCl. However our model gives different extrapolation to higher concentrations thanDuan and Sun (2003, e.g. Fig.12b), where their approach predicts an increase of the CO 2 solubility with increasing NaCl molality, which might happen if for instance the formation of Na carbonate complexes increases the solubility of CO 2 ,b u ti sn o ts u p p o r t e db ya n yd a t at o our knowledge. Our model, whose parameters do not depend on concentration, predicts a continuous decrease in CO 2 solubility with increasing salinity over the entire pressure-temperature range and up to 10 molal NaCl.

  p e r a t u r e -d e p e n d e n tm i n i m u mv a l u ea tx =0 .5 reflecting the entropy of mixing.

Figure B1 :

 B1 Figure B1: Schematic illustration of the relationship between chemical potential of the aqueous species A and its concentration x A in the DH-ASF formalism. The solid line indicates the measured chemical potential. The broken line shows the chemical potential of an ideal solution referenced to infinite dilution for A. The usual 1M standard state is indicated with a gray square, and its equivalent projected at x=1 along the ideal mixing line shown with an open square. The dotted line shows the chemical potential for an ideal solution referenced to a hypothetical ideal solution at x =1. µ 0A ,thestandardstatechemicalpotentialofA used in the DH-ASF formalism, is shown with a closed square at x A =1 ,itdiffersfromthe ideal mixing line by the sum of long-and short-range interactions to the excess free energy. Modified after[START_REF] Evans | A method for activity calculations in saline and mixed solvent solutions at elevated temperature and pressure: A framework for geological phase equilibria calculations[END_REF].

Figure 1 :

 1 Figure 1: Phase diagram for CO 2 -H 2 Om i x t u r e s . N o t et h a tp r e s s u r ei so na logarithmic scale. T&F 1963: Measurements of Todheide and Franck (1963).

Figure 2 :

 2 Figure 2: Representative values of W H 2 O-CO 2 and a CO 2 from selected experimental results. a) and b) respectively show isobaric values of W H 2 O-CO 2 and W H 2 O-CO 2 /RT as a function of temperature. c) Isobaric values of a CO 2 as a function of temperature. Note the log scale to represent low pressure value of a CO 2 .d )V a r i a t i o n s of W H 2 O-CO 2 (open symbols and dashed lines) and a CO 2 (closed symbols and plain lines) with pressure at 25°C. A step is seen at the boiling curve of CO 2 .L i n e sa r e calculated wih eq. 8 and 9. Uncertainties are smaller than thes i z eo ft h ep o i n t when not indicated.

Figure 3 :

 3 Figure 3: Phase equilibrium in the H 2 O-CO 2 system at various pressures and temperatures (cross-sections through figure 1). Lines are calculated and points are experimental results at the pressure -temperature corresponding to the lines closest to which they plot. a) Pressure-composition phase diagram at different temperatures. The solvus closes at low pressure against the boiling curve of water (X CO 2 =0 ). Dashed lines illustrate the dependency of the phase diagram to temperatures around 270°C. Corresponding experimental measurements are not indicated for clarity. b), c) and d) show temperature-composition phase diagrams at different pressures. c) is a close-up view of b) for the water-rich phase. Dotted lines are calculated with the model of Duan and Sun (2003). Thethickgrayline in d) is calculated with the parameters of Holland and Powell (2003) plotted in figure 13.

Figure 4 :

 4 Figure 4: a) Isopleths of calculated CO 2 solubility (mol. %) in pure water at low pressures and temperatures. The dark circle indicates the critical point of CO 2 and the dashed line is the liquid-gas transition of CO 2 .

Figure 5 :

 5 Figure 5: Map of uncertainties on CO 2 solubility, in percent of the calculated value. This map takes into account experimental uncertainties and the misfit of the model to the data.

Figure 6 :

 6 Figure 6: Isopleths of calculated a) water activity (a H 2 O )a n db )w a t e ra c t i vity coefficient (γ H 2 O )i nC O 2 -saturated solutions, c) CO 2 activity (a CO 2 )a n dd ) CO 2 activity coefficient (γ CO 2 )i nt h ew a t e r -s a t u r a t e dC O 2 -rich phase, between 25 and 380°C and 1 bar to 3 kbar. Dashed curves illustrate the two critical mixing lines. Points in a) represent pressure-temperature conditions of experimental results used in this study. The square indicates the criticalpointofwater .

Figure 7 :

 7 Figure 7: Calculated density isopleths of the CO 2 -saturated aqueous phase over the range 0.1-3kb and 50-370°C, in kg.m -3 .

Figure 8 :

 8 Figure 8: Heat capacity (Cp) and volume of NaCl± at infinite dilution (V ∞ )calculated with the modified density model (Holland and Powell, 1998). a) Heat capacity at 200 bars compared to measurements from Pitzer et al. (1984). b) Comparison with the experimentally-derived volumes of Grant-Taylor (1981) and L'Vov et al. (1981). c) Isotherms of calculated V ∞ (NaCl±)u pt o2k b a r s . d ) Isotherms of the pressure derivative of ln(ρ H 2 O /ρ 0 H 2 O ),plotte donalogs c a lea nd used in the calculation of V ∞ (NaCl±).

Figure 9

 9 Figure 9: a) Solubility of halite calculated as a function of temperature with the model of Mao and Duan (2008, dotted line) and with the present model (bold line). The solubility is calculated at water vapor pressure below the critical point of water and along a linear pressure gradient linking experimental pressures above the critical point of water. b) Solubility of halite calculated at 25°C as a function of pressure. Points are experimental measurements indicated in table 2.

Figure 10

 10 Figure 10: a) Fraction of dissociated aqueous NaCl (x NaCl± )asafunctionofpressure, calculated at different temperatures and compared to the degree of association calculated from the conductivity measurements of QuistandMarshall(1968) at 0.1 molal, as described in the text. b) x NaCl± calculated as a function of the total concentration of NaCl for various pressures and temperatures. The fraction of NaCl 0 is x NaCl 0 =1-x NaCl± .S ta r sindic a teha lites a tur a tion.

Figure 11

 11 Figure 11: a) Density of NaCl solutions calculated at 1 bar (dark line) compared to the measurements of Surdo et al. (1982) at 25°C (open symbols)and45°C(closed symbols). b) and c): density of NaCl solutions calculated at 400 and 500 °C for various pressures with this model (dark lines) and with the model of Driesner (2007) (thick gray lines).

  CO 2 is only a function of pressure and temperature. However it is incorrect at conditions close to the closure of the H 2 O-CO 2 solvus because experimental results imply γ CO 2 =1(e.g.[START_REF] Todheide | Das Zweiphasengebiet die kritische Kurve im System Kohlendioxid-Wasser bis zu Drucken von 3500 bar[END_REF]. We show below that γ CO 2 =1.07 at 100°C -1bar and γ CO 2 =1.35 at 260°C -2kbar. Furthermore, assumptions in[START_REF] Duan | An improved model calculating CO 2 solubility in pure water and aqueous NaCl solutions from 273 to 533 K and from 0 to 2000 bar[END_REF] reduce the chemical potential of CO 2 in the gas phase (µ v CO 2 )t oµ v CO

2 .D u a n e t a l . ( 2 0 0 6 ) h a v e e x t e n d e d t h i s a s s u m p t i o n t o a m i x i n g scheme where φ 2 (P, T, y)=RT ln(P -

P H 2 O )+ RT ln(φ 0 CO 2 ),w h e

r ey is amount of CO 2 in the gas phase, T is temperature, P is pressure, and P H 2 O is boiling pressure for pure water at T .T h i sa s s u m e st h a tC O 2 has no enthalpy of formation and describes µ v CO 2 as independent of the composition of the mixture but sensitive to P H 2 O ,evenwhenthecompositiontendstowards pure CO 2 .T hismode listhe r e f or einc ons is te ntw ithothe rw or k( e . g. ,Hollandand

  2 O and W H 2 O-CO 2 > 0)increasesa H 2 O and compensatesfor the dilution of water by CO 2 :f o re x a mp l e ,a t2 7 0 °Ca n d1k b a r ,t h ew a t e r -r i c h phase is a mixture of 18 mol. % of CO 2 and 82 mol. % of water but the water activity is as high as 0.91.Figure6c and dpresent a CO 2 and γ CO 2 in the water-saturated CO 2 -rich phase as afunctionofpressureandtemperature.γ CO 2 is significantly greater than one over alargerangeofpressuresandtemperatures,especiallyclose to the boiling curve of water. At 1 bar, γ CO 2 =1 .01 at 50°C and γ CO 2 =1.07 close to 100°C. Generally,

	are reproduced well away from the boiling curve of water but there is a significant
	deviation with the calculated values (>10 )atlowwaterdensity .Inthegaseous
	field of CO 2 ,d e n s i t yc a l c u l a t i o n sd on o tg i v es a t i s f a c t o r yr e s u l t se i t her. This is
	attributed to the use of H

the calculated a CO 2 decreases with temperature but increases with pressure. Very low a CO 2 is obtained at low pressure in the vicinity of the boiling curve of water, and a CO 2 becomes gradually less sensitive to pressure with increasing pressure.

3.3. Density calculations for H 2 O-CO 2 mixtures

Densities calculated with the present model show a good accuracy at conditions greater than the boiling curves of CO 2 and water, at both low and high CO 2 concentrations. As shown in table 3, the densities of the water phase saturated in CO 2 measured by

[START_REF] Teng | Solubility ofl i q u i dC O 2 in water at temperatures from 278 K to 293 K and pressures from 6.44 MPa to 29.49 MPa and densities of the corresponding aqueous solutions[END_REF]

at temperatures lower than 20°Ca r er eproduced within 4 at pressures greater than 150bars. The measurements of

[START_REF] Hnedkovský | Volumes of aqueous solutions of CH 4 ,CO 2 ,H 2 SandNH 3 at temperatures from 298.15 K to 705 K and pressures to 35 MPa[END_REF]

, carried out below CO 2 saturation (∼0.15mol/kgH 2 O), 2 OandCO 2 as end-members in the calculations. In the gaseous field of CO 2 ,t h ev o l u m eo ft h eC O 2 end-member is very sensitive to pressure with the result that the calculated density shows a comparable sensitivity.

  14) W NaCl±-H 2 O = C 8 + C 9 .T 400 + C 10 .T 2 400 + C 11 .P + C 12 .P 2 (15) α NaCl± = C 13 + C 14 .T +(C 15 . + C 16 .T )P (16) where T 400 is temperature below 400K such as W NaCl±-H 2 O is independent of temperature above 400K. Values for the C constants are given in Table D3 of Ap-

	pendix D.

  . Thec o m b i n a t i o n of the subsystems H 2 O-CO 2 and H 2 O-NaCl needs two additional interaction parameters to describe H 2 O-CO 2 -NaCl: W CO 2 -NaCl 0 and W CO 2 -NaCl± .S a l t i n g -o u t implies positive values for these parameters. The magnitudes of W CO 2 -NaCl 0 and W CO 2 -NaCl± are expected to be high given that NaCl is barely incorporatedinthe CO

2 -rich phase in the absence of water, even at high pressure and temperature: the measurements of Zakirov et al. (2007) indicate a NaCl mole fraction of 30 * 10 -7

  CO 2 -NaCl 0 = W CO 2 -NaCl± because the quality of the fit to the data is not statistically sensitive to the ratioW CO 2 -NaCl 0 /W CO 2 -NaCl± . W CO 2 -NaCl 0and W CO 2 -NaCl± may then be estimated from solubility experiments independently of the thermodynamic properties of pure H 2 Oa n dC O 2 and of their uncertain-

	et al., 2002). The measurements of Takenouchi and Kennedy (1965) at zero NaCl
	content are also inconsistent (by up to 15%, see Fig. 12b) withthemeasurements
	of Todheide and Franck (1963) which we used to constrain the H 2 O-CO 2 activity
	model. However, all the data of Takenouchi and Kennedy (1965)a tx NaCl > 0
	have been kept in our regression as they are the major source ofm e a s u r e m e n t s
	available above 500 bars.
	It was assumed that W
	al.,1 9 9 4 ;K i e p e

ties. However the values depend on the activity models in H 2 O-CO 2 and H 2 O-NaCl. As observed in the H 2 O-CO 2 system, the quantity W CO 2 -NaCl± /(RT ) shows smoother variations in pressure-temperature space than W CO 2 -NaCl± ,althoughboth values increase strongly with pressure especially under 100bars.Forthisreason, W CO 2 -NaCl± has been parameterized as a function of the natural logarithmofpressure:

W CO 2 -NaCl± /(RT )=D 1 + D 2 ln P + D 3 .T + D 4 T ln P + D 5 (ln P ) 2 (17)

where pressure is in bar and temperature in Kelvin. Values forD 1 to D 5 are given in table

D4

.

  2 O-CO 2is limited to the two-phase domain. Equations A11, A13, A18 and A19 provide a simple and accurate parameterization of the system H 2 O-CO 2 but only over pressure-temperature conditions where two phases coexist. The parameterization can not be extrapolated to the one-phase domain belowthec r itic a lpoint of water (Fig.1): although a CO 2 /α H 2O tends towards infinity near the boiling curve of water, there is no simple relation between a CO 2 and V CO 2 /V H 2

O .E xtrapolation of the model towards low temperatures or high pressures in the liquid field of CO 2 is speculative as there are few experimental results to constrain its pressure-temperature dependency. It has been noted in section 2.2 and figure

2b

that away from the boiling curve of water and critical point ofc a r b o nd i o x i d e , both W H 2 O-CO 2 /RT and a CO 2 approximate a linear behavior. However, it is unlikely that the observed trends can be extrapolated over a larger range of pressures and temperatures. Models (e.g.

  such as B 1 =8 .2053 * 10 5 √ ρ/( T) 3/2 and

	B 2 =8 2 .48	√ ρ/(η	√	T),w i t hρ density of the solvent and dielectric constant
	of water. κα is a dimensionless quantity from the Debye-Huckel formula and is
	calculated as:

Table D1 :

 D1 Values of parameters and associated covariance matrix for equations 8 and 9 for W H 2 O-CO 2 /RT .

		Va l u e	C 1	C 2	C 3	C 4	C 5	C 6	C 7	C 8	C 9	C 10		C 11	C 12	C 13	C 14	C 15	C 16
	C	-3.67E+00	1.12E+00	-2.26E-04	-2.84E-03	-8.47E-05	5.16E-04	1.22E-04	-1.38E-06	1.81E-01	1.74E-03	2.37E-05	-4.56E-02	-6.69E-04	-3.57E-03	9.14E-06	-8.39E-07	3.04E-08
	C	-1.38E-03	-2.26E-04	1.63E-07	5.74E-07	7.06E-08	9.23E-07	-1.24E-07	-2.23E-09	-6.06E-05	-3.14E-07	-5.16E-09	8.12E-06	1.21E-07	2.44E-06	-6.30E-09	-8.19E-11	1.54E-11
	C	8.92E-03	-2.84E-03	5.74E-07	7.22E-06	2.15E-07	-1.31E-06	-3.11E-07	3.50E-09	-4.60E-04	-4.42E-06	-6.03E-08	1.16E-04	1.70E-06	9.09E-06	-2.33E-08	2.13E-09	-7.74E-11
	C	7.20E-04	-8.47E-05	7.06E-08	2.15E-07	3.11E-08	5.34E-07	-5.49E-08	-1.30E-09	-2.38E-05	-1.09E-07	-1.98E-09	1.28E-06	5.20E-08	1.00E-06	-2.60E-09	-4.12E-11	7.38E-12
	C	1.00E+00	5.16E-04	9.23E-07	-1.31E-06	5.34E-07	2.33E-04	-7.21E-07	-5.84E-07	-8.56E-04	4.81E-05	-6.87E-07	-1.60E-03	-1.62E-06	5.38E-05	-1.44E-07	-5.52E-09	-2.69E-10
	C	-4.23E-04	1.22E-04	-1.24E-07	-3.11E-07	-5.49E-08	-7.21E-07	1.01E-07	1.73E-09	3.67E-05	2.00E-07	1.98E-09	-3.20E-06	-9.09E-08	-1.66E-06	4.30E-09	8.70E-11	-2.32E-11
	C	-1.53E-03	-1.38E-06	-2.23E-09	3.50E-09	-1.30E-09	-5.84E-07	1.73E-09	1.48E-09	2.59E-06	-1.22E-07	1.75E-09	4.06E-06	1.80E-09	-1.60E-07	4.16E-10	1.43E-11	7.22E-13
	C	-9.85E+00	1.81E-01	-6.06E-05	-4.60E-04	-2.38E-05	-8.56E-04	3.67E-05	2.59E-06	1.17E-01	3.45E-04	5.12E-06	2.89E-04	-3.97E-04	-5.56E-03	1.38E-05	1.03E-07	-5.21E-10
	C	-4.18E-02	1.74E-03	-3.14E-07	-4.42E-06	-1.09E-07	4.81E-05	2.00E-07	-1.22E-07	3.45E-04	4.94E-05	-6.20E-07	-1.94E-04	-5.10E-06	-7.13E-06	1.80E-08	-2.44E-09	-1.90E-12
	C 10	1.99E-04	2.37E-05	-5.16E-09	-6.03E-08	-1.98E-09	-6.87E-07	1.98E-09	1.75E-09	5.12E-06	-6.20E-07	9.84E-09	1.11E-06	3.80E-08	-1.72E-07	4.35E-10	5.02E-12	1.73E-12
	C 11	-6.14E-01	-4.56E-02	8.12E-06	1.16E-04	1.28E-06	-1.60E-03	-3.20E-06	4.06E-06	2.89E-04	-1.94E-04	1.11E-06	6.57E-02	2.24E-06	-2.68E-04	6.88E-07	5.53E-08	1.80E-09
	C 12	3.16E-02	-6.69E-04	1.21E-07	1.70E-06	5.20E-08	-1.62E-06	-9.09E-08	1.80E-09	-3.97E-04	-5.10E-06	3.80E-08	2.24E-06	3.37E-06	1.81E-05	-4.43E-08	5.62E-10	2.15E-11
	C 13	8.70E-01	-3.57E-03	2.44E-06	9.09E-06	1.00E-06	5.38E-05	-1.66E-06	-1.60E-07	-5.56E-03	-7.13E-06	-1.72E-07	-2.68E-04	1.81E-05	2.95E-04	-7.34E-07	-1.13E-08	2.90E-10
	C 14	-5.60E-04	9.14E-06	-6.30E-09	-2.33E-08	-2.60E-09	-1.44E-07	4.30E-09	4.16E-10	1.38E-05	1.80E-08	4.35E-10	6.88E-07	-4.43E-08	-7.34E-07	1.83E-09	2.86E-11	-7.65E-13
	C 15	7539E-08 --4.58E-14 1.77E-16 -2.78E-19 9.59E-12 -1.48E-13 9.29E-16 -1.13E-18 --6.34E-11 -8.33E-13-1.42E-14 5.49E-17 -6.85E-20 -2.80E-06 -8.39E-07 -8.19E-11 2.13E-09 -4.12E-11 -5.52E-09 8.70E-11 1.43E-11 1.03E-07 -2.44E-09 5.02E-12 -8.83E-17 -1.47E-19 6.89E-15 -1.98E-20 5.53E-08 5.62E-10 -1.13E-08	2.86E-11	2.01E-12	-5.29E-14
	C 16	k 19 1.8277E-10 -6.92E-17 -2.57E-19 4.01E-22 -1.66E-14 2.56E-16 -1.60E-18 2.01E-21 ---1.12E-13 1.10E-15 2.16E-17 -8.19E-20 1.01E-22 -6.48E-08 3.04E-08 1.54E-11 -7.74E-11 7.38E-12 -2.69E-10 -2.32E-11 7.22E-13 -5.21E-10 -1.90E-12 1.73E-12 --1.47E-19 2.55E-22 -1.02E-17 3.13E-23 1.80E-09 2.15E-11 2.90E-10	-7.65E-13	-5.29E-14	2.66E-14
		k 20 -5.7627E-06 --4.63E-12 1.93E-14 -3.03E-17 6.64E-10 -1.05E-11 6.62E-14 -7.05E-17 --4.30E-09 -1.12E-10-1.40E-12 5.65E-15 -7.18E-18 -	-6.89E-15 -1.02E-17 7.39E-13 -1.86E-18
		k 21 2.2035E-11 -1.15E-17 -4.54E-20 6.90E-23 -2.01E-15 3.19E-17 -2.00E-19 2.31E-22 ---1.23E-14 2.66E-16 3.49E-18 -1.38E-20 1.67E-23 -	--1.98E-20 3.13E-23 -1.86E-18 5.32E-24

Table D3 :

 D3 Values and covariance matrices for C 1 to C 16 (equations 13 to 16).

	46
	Va l u e	D 1	D 2	D 3	D 4	D 5
	D 5.046E01	8.15E00	-9.42E-01 -2.89E-02 5.76E-03 -1.41E-01
	D -3.533E00 -9.42E-01 2.69E-01 1.64E-03 -3.91E-04 -9.28E-03
	D -1.525E-01 -2.89E-02 1.64E-03 1.22E-04 -2.38E-05 7.87E-04
	D 2.307E-02 5.76E-03 -3.91E-04 -2.38E-05 4.73E-06 -1.52E-04
	D -4.801E-01 -1.41E-01 -9.28E-03 7.87E-04 -1.52E-04 7.02E-03

Table D4 :

 D4 Values and covariance matrices for D 1 to D 5 (equation 17).

	Nighswander J., Kalogerakis N. and Mehrotra A. (1989) Solubilities of carbon-Symbol Quantity
	dioxide in water and 1 wt-percent NaCl solution at pressures up to 10-mpa and R Ideal gas constant (=8.314472 J.mol -1 .K -1 )
	temperatures from 80 degrees C to 200 degrees C. Journal of Chemical and T Temperature
	Engineering Data 34,355-360. P Pressure
	Oelkers E. and Helgeson H. (1993a) Multiple ion association in supercritical aque-STP Standard temperature and pressure, 298.15K and 1 bar
	Critical temperature and composition ous solutions of single electrolytes. Science 261,8 8 8 -8 9 1 . C i t e dB y( s i n c e T C , x C Chemical potential of end-member i µ i 1996) 61. µ 0 i Standard state chemical potential of i
	x i	Mole fraction of i
	γ i	Activity coefficient of i
	a i	Activity of i:a i = x i .γ i
	σ 2 i	Va r i a n c e o f i
	G mix m (x)	Gibbs free energy of mixing
	G m (x)	Gibbs free energy of the mixture
	Oelkers E.H. and Helgeson H.C. (1993b) Calculation of dissociation constants and G xs m (x) Excess Gibbs free energy of the mixture
	the relative stabilities of polynuclear clusters of 1:1 electrolytes in hydrother-Φ i Proportion of i adjusted to a size parameter
	mal solutions at supercritical pressures and temperatures. Geochimica et Cos-W * ij Interaction energy between i and j adjusted to a size parameter
	mochimica Acta 57,2673-2697. doi:10.1016/0016-7037(93)90383-8. W j Interaction energy between i and j
	Oelkers E.H., Benezeth P. and Pokrovski G.S. (2009) Thermodynamic Databases α i Size parameter for i
		Interaction parameters on the Gibbs free energy,
	for Water-Rock Interaction. In THERMODYNAMICS AND KINETICS OF W G , W H , W S , W V enthalpy, entropy and volume of mixing, respectively.
	WATER-ROCK INTERACTION (eds. E. Oelkers and J. Schott), Mineralogi-cal Society of America, vol. 70 of Reviews in Mineralogy & Geochemistry,pp. Equivalent conductance, respectively experimentally-determined and Λ , Λe hypothetical from a completely dissociated electrolyte
	1-46. doi:10.2138/rmg.2009.70.1. Λe 0 Limiting equivalent conductance
	Onsager L. (1927) On the theory of electrolytes. II. Physikalische Zeitschrift 28, η, η 0 Viscosity, respectively of the solution and of the pure solvent
	277-298. ρ	Density of the solution
		Dielectric constant of water
	å	Ion size parameter

Oelkers E.H. and Helgeson H.C. (1988) Calculation of the thermodynamic and transport properties of aqueous species at high pressures and temperatures: dissociation constants for supercritical alkali metal halidesa tt e m p e r a t u r e sf r o m 400 to 800°C and pressures from 500 to 4000 bar. The Journal of Physical Chemistry 92,1631-1639.doi:10.1021/j100317a049.

Table 1 :

 1 Key to symbols used in the text.

	Pbars T°C molality CO 2 ρ Meas	ρ Calc	Deviation
	10	0.155	996.9	970.7	26.2
	200	0.155	1005.3 1005.0 0.3
	200	0.155	996.0	996.5	-0.5
	200	0.155	966.5	965.8	0.7
	200	0.155	926.4	926.7	-0.4
	200	0.155	875.9	879.1	-3.7
	200	0.185	813.4	821.0	-9.4
	200	0.155	731.1	740.2	-12.4
	280	0.155	631.7	642.1	-16.4
	350	0.155	1011.7 1011.4 0.3
	350	0.155	1002.2 1002.4 -0.2
	350	0.155	973.1	973.9	-0.8
	350	0.155	934.1	934.7	-0.6
	350	0.155	885.9	887.2	-1.4
	350	0.185	827.7	831.3	-4.4
	350	0.155	754.7	760.6	-7.8
	197	1.741	1020.8 1016.4 -4.2
	246	1.796	1021.7 1020.5 -1.2
	295	1.840	1024.0 1023.0 -1.0
	197	1.702	1019.7 1015.8 -3.8
	246	1.757	1021.0 1019.6 -1.3
	295	1.801	1022.9 1022.0 -0.9
	197	1.614	1017.7 1015.0 -2.7
	246	1.675	1019.1 1019.0 -0.1
	295	1.719	1019.8 1021.4 1.6

Table 3 :

 3 Comparison of measured (ρ Meas )andcalculated(ρ Calc )densitiesatvarious pressures, temperatures and CO 2 concentrations. Measurements at CO 2 concentrations around 0.15 M originate from

Mineral phases in fluid-rock systems in CO

sequestration and geothermal environments have astronginfluenceonfluidcompositionsviafluid-rockreaction, and almost always involve phases with complex activity-composition relationships such as ternary carbonates and feldspars. At this time, there is no software capable of combining the most recent and sophisticated activitycomposition models for fluids and mineral phases.

APPENDIX B

The DH-ASF formalism allows description of both the Debye-Hü c k e l e f f e c t arising from long-range interactions at low concentrationsandshort-rangeforces at high concentrations. The evolution of the chemical potential of the aqueous species A is illustrated in figure B1. The chemical potential of A at standard state µ 0

A is defined at x A =1 . µ 0 A is augmented from the ideal line at x =1 by the sum of all non-ideal contributions: long-range electrostatic forces at low concentrations where µ is below the ideal mixing line and short-range interactions at higher concentrations. At high dilutions of A, µ A approaches ideal solution (dashed line on Fig. B1). With increased concentration of A, µ A evolves to µ A < mu ideal (e.g. at x = x 1 )d u et ot h eD e b y e -Hü c k e le f f e c t . T h ee f f e c to fl o n grange interactions has been limited in the modeling to a maximum of 0.1 molal by [START_REF] Evans | A method for activity calculations in saline and mixed solvent solutions at elevated temperature and pressure: A framework for geological phase equilibria calculations[END_REF]. For solutions more concentrated than 0.1 molal, the activities of the end-members in solution are described by the short-range interaction parameters of the ASF model. On figure B1, these constrain the part of the diagram where µ A is greater than the line of ideal mixing (e.g. at x = x 2 ).

APPENDIX D

The following tables provide values and covariance matricesf o rc o e f fi c i e n t s used in the modeling and described in the text. The diagonal ofac o v a r i a n c e matrix is the variance.

Va l u e j 1 j 2 j 3 j 4 j 5 j 6 j 7 j 8 j 9 j 10 j 11 j 12 j 13 j 14 j 15 j 16 j 17 j 18 j 19 j 20 j 21

31E-15 6.51E-08 6.31E-12 -2.03E-03 1.30E-04 -1.57E-06 --1.39E-05 -2.59E-08 -2.03E-14 -5.53E-07j 13 -6.7895E-04 1.08E-06 2.23E-08 1.69E-11 -1.61E-14 -1.25E-10 -3.08E-13 1.45E-16 -9.27E-10-7.93E-14 2.11E-05 -1.57E-06 1.96E-08 ---1.41E-07 2.54E-10 --1.64E-16 6.75E-09 -

Table D2: Values of parameters and associated covariance matrix for equations 8 and 9 for α CO 2 .