# HyPiT: A thermodynamic database for modeling geochemical systems up to high temperatures, pressures and salinities

Arnault Lassin, Christophe Chiaberge, Sylvain Guignot, Johann Tuduri, Laurent André

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Arnault Lassin, Christophe Chiaberge, Sylvain Guignot, Johann Tuduri, Laurent André. HyPiT: A thermodynamic database for modeling geochemical systems up to high temperatures, pressures and salinities. Applied Geochemistry, 2026, 199, pp.106727. 10.1016/j.apgeochem.2026.106727. hal-05523848

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# HyPiT: A thermodynamic database for modeling geochemical systems up to high temperatures, pressures and salinities

Arnault Lassin<sup>a,b,*</sup>, Christophe Chiaberge<sup>a</sup>, Sylvain Guignot<sup>a</sup>, Johann Tuduri<sup>a,b</sup>,
Laurent André<sup>a,b</sup>

## ARTICLE INFO

Editorial handling by Dr Dmitrii A. Kulik

### Keywords:

Thermodynamic database

Geochemical modeling

Pitzer model

Saline groundwater

Parameter estimation

Ion-pairing

Hydrothermal multicomponent systems

## ABSTRACT

Modeling water-rock interactions in underground environments requires the use of thermodynamic databases that account for the complexity of groundwater chemical compositions under relevant pressure and temperature conditions. A key issue concerning saline groundwaters is their high ionic strength, which are often not reliably calculated and calls for the use of specific thermodynamic approaches. Among such approaches, the Pitzer model appears to be one of the most promising. However, its applicability to underground fluids is limited by the fact that the values of its numerous parameters are only known for a subset of chemical systems and, more particularly, are often missing for trace elements and at temperatures other than 25 °C. This paper describes a new way to develop a relevant thermodynamic database to account for the trace elements in these types of deep environments. It combines two in-house databases: Thermodem, which relies on the B-dot Extended Debye-Hückel activity model, and PhreeSCALE, which uses the Pitzer formalism. The resulting Hybrid Pitzer-Thermodem (HyPiT) thermodynamic database (TDB) integrates a simplified Pitzer model to account for major and trace elements. Based on a series of recent works (Simoes et al., 2016, 2017a, 2017b), the Pitzer binary interaction parameters, $β^{(0)}$ and $β^{(1)}$ and their first derivative with respect to temperature ($dβ^{(0)}/dT$ and $dβ^{(1)}/dT$) are determined for simulating the first order interactions between trace elements and major species. The HyPiT database is successfully applied to different geochemical systems relevant to underground environments. They include (i) the solubility of gypsum in NaCl and sea-type brines of varying ionic strength at temperatures between 0 and 25 °C, (ii) the solubility of barite in NaCl brines at 25, 60 and 80 °C, (iii) the solubility of calcite in 0.1 and 4 M NaCl brines for temperatures and pressures up 250 °C and 1450 bars, respectively, (iv) the solubility of amorphous silica up to 150 °C in various single electrolyte solutions, namely HCl, NaCl, MgCl₂, HNO₃, NaNO₃, Na₂SO₄, and MgSO₄. Particular attention is also given to the modeling of iron-bearing systems such as the speciation of Fe<sup>III</sup> in LiCl solutions up to 20 mol/kg.

## 1. Introduction

Climate change is a major issue of the 21st century. One way of contributing to its long-term mitigation is to consider the subsurface of the Earth as a source of carbon-free renewable energy (geothermal production of heat and electricity), as a vast geological space to permanently store gases like CO₂ or temporarily store energy vectors (methane, compressed air, hydrogen), and as a source of valuable elements contained in geothermal waters or rock formations. This enhanced use of the subsurface must be implemented safely and efficiently (Shukla et al., 2019). In addition, increased exploitation of the

underground is also encouraged by new technologies, which use more and more chemical elements covering a larger proportion of the periodic table (Girtan et al., 2021). Despite the development of recycling chains, such materials are mostly supplied by mineral extraction, which means that new ore deposits, potentially hidden in the deep underground, must be discovered and the processes responsible for their development better understood. This includes a variety of geochemical processes like serpentinization of mantle rocks and generation of molecular hydrogen, lithium enrichment in geothermal brines, scaling in geothermal plants and oilfield facilities, or mineral deposition in hydrothermal systems.

The expected growth in the use of the underground environment

<footnote><sup>a</sup> BRGM, 3 Av. Claude Guillemin, 45060, Orléans, France</footnote>

<footnote><sup>b</sup> ISTO, UMR 7327, CNRS, BRGM, Université d'Orléans, 45071, Orléans, France</footnote>

<footnote>* Corresponding author. BRGM, 3 Av. Claude Guillemin, 45060, Orléans, France.</footnote>

<footnote>E-mail address: a.lassin@brgm.fr (A. Lassin).</footnote>
means an increase in fluid exchanges between the surface and the sub-surface, leading to a rise in chemical, thermal and mechanical disturbances at depth. This will affect reservoir rock properties, fluid properties (viscosity, density, and enthalpy), and their mutual interactions, including dissolution and precipitation of minerals. These changes further modify rock surface and petrophysical properties such as pore connectivity, porosity and permeability. Fluid-rock interactions also play a key role in various deep geological environments (Miron et al., 2016), including the production of hot brines (Debure et al., 2019; Jackson et al., 2000; Warren, 2010) and the enrichment of deep waters in valuable substances and the development of metal ore deposits (Bock et al., 2025; Cui et al., 2023, 2023b; Jackson and Helgeson, 1985a,b; Lacroix et al., 2020; Lebedev et al., 2019; Moroni et al., 2019; Scharrer et al., 2019, 2021; Staude et al., 2023).

One approach to investigate such systems relies on reactive transport modeling (Miron et al., 2016, and references therein) since the environments of interest are not easily accessible in space and time. In addition to the nature of the processes implemented in the reactive transport simulation tools, the thermodynamic database is a key component in the reliability of the simulation. The development of such databases has a long history (Helgeson et al., 1978, 1981; Hummel et al., 2019; May and Filella, 2023; Miron et al., 2016; Semerjian and Burgess, 2022; Tanger and Helgeson, 1988). Their domains of validity depend on the target applications and strong efforts have been made to describe high temperature – high pressure systems for more or less extended chemical systems but at low salinity (Awolayo and Tutolo, 2022; Huang and Sverjensky, 2019; Miron et al., 2016; Zhang et al., 2020), or highly saline systems over limited ranges of T-P conditions or reduced chemical systems (Christov and Møller, 2004a, 2004b; Lach et al., 2015; Lassin et al., 2015; Lassin and André, 2023). To date, a thermodynamic database that applies to wide ranges of T-P conditions and salinities, and to extended chemical systems, is still lacking.

Modeling fluid-rock interactions are mathematically described by a combination of different sets of equations that depend on the nature of the compounds: solids, gases, water, and solutes. The goal is to compute the distribution of every chemical element of a given system among the various phases at thermodynamic equilibrium. This can be solved by calculating the chemical potential of every species of interest, which relies on their thermodynamic properties. The calculation of the thermodynamic properties of solids is generally based on standard states and mostly uses a simple equation of state, except in the case of solid solutions. Equations of state for gases are more complex, ranging from the simple van der Waals equation to the more sophisticated GERG (Kunz and Wagner, 2012) or extended cubic Peng-Robinson equations (Jaubert et al., 2022; Paes et al., 2023). Water is the main solvent in natural environments, and its thermodynamic properties are calculated with an equation of state for pure water. The International Association for the Properties of Water and Steam recommends the equation of state developed by Wagner and Pruβ (2002). The calculation of the thermodynamic properties of chemical species dissolved in water distinguishes the standard state contribution from the non-ideal contribution, which is due to interactions between the coexisting aqueous chemical compounds. In a wide range of T-P conditions, the revised-Helgeson-Kirkham-Flowers (HKF) equations of state can describe the standard thermodynamic properties of many solutes up to 1000 °C and 5000 bars (Johnson et al., 1992; Tanger and Helgeson, 1988; Zimmer et al., 2016). These standard state properties are used to calculate reaction properties, like equilibrium constants of aqueous complexation and of dissolution/precipitation reactions, in the same conditions.

The non-ideal contribution to the chemical potential of a solute is expressed by its activity. This helps to calculate the aqueous speciation (i.e. the distribution of matter within the aqueous solution) and the saturation state of the solution with respect to minerals, to quantitatively predict the mass transfers among minerals, gases, water and solutes. Numerous more or less empirical models have been developed to

describe the activity coefficients of solutes, but the application of these models is limited by their range of validity, particularly with respect to salinity. The latter can be related to the ionic strength I (mol·kg⁻¹) of the solution:

$$I = \frac{1}{2} \sum_{i} z_{i}^{2} m_{i} \quad (1)$$

where m is the concentration (mol·kg⁻¹) and z is the charge (–) of the dissolved species i.

This characteristic of an aqueous solution had been introduced by Lewis and Randall (1921) who found that the activity coefficient of any electrolyte is the same in all dilute solutions sharing the same ionic strength. The general trend is the simpler the model, the more limited the domain of application. For example, activity models of the “extended Debye-Hückel” (EDH) type are generally limited to ionic strengths below that of seawater, but they apply to extended chemical systems. They are associated with very comprehensive thermodynamic databases such as the Thermoddem database (Blanc et al., 2012, https://thermoddem.brgm.fr) that includes the standard state thermodynamic properties of a large number of solutes (1684), solids (922) and gases (23).

Among EDH-type models, the activity coefficient of an aqueous species i (γi) (–) can depend solely on the electric charge of the species, the ionic strength, and the temperature and pressure. This is the case, for example, with the Davies equation:

$$\log \gamma_i = -Az_i^2 \left( \frac{\sqrt{I}}{1 + \sqrt{I}} - 0.3I \right) \quad (2)$$

where A is the Debye-Hückel coefficient (kg⁰.⁵ mol⁻⁰.⁵), which depends on the temperature (K), density (kg·m⁻³) and dielectric permittivity of water (F·m⁻¹), these latter two properties also depend on T and P (Pa).

Other EDH models are valid above the ionic strength of seawater. They take into account the size of the aqueous species by including the notion of ionic radius, such as the B-dot model (Helgeson, 1969):

$$\log \gamma_i = - \frac{A z_i^2 \sqrt{I}}{1 + B a_{0,i} \sqrt{I}} + \dot{B}I \quad (3)$$

where B is another Debye-Hückel coefficient (kg⁰.⁵ mol⁻⁰.⁵ Å⁻¹), which depends on the temperature, density and dielectric permittivity of water. a₀,i is the ionic radius of species i (Å), B is the empirical parameter “B-dot” (kg·mol⁻¹) defined by (Helgeson, 1969) and which depends only on temperature. The ionic strength limit of these activity models can vary from few tenths to few units of eq·kg⁻¹, depending on the chemical system of interest (see section 2.2.2). Either one of these EDH-type models can be used with the same thermodynamic database because the equations of aqueous complexation reactions used to calculate speciation in the aqueous phase are the same.

Alternately, more complex models, like the Pitzer equations (Pitzer, 1973, 1991), are able to describe solution properties and solution-mineral equilibria up to very high salinities, well above 10 mol kg⁻¹ (or up to several hundred grams per liter). The Pitzer equations are based on a different interpretation of the behavior of dissolved species. They assume full dissociation of strong electrolytes and consider specific short-range interactions between species. However, they do not exclude the possibility that aqueous complexes may form and account for part of the short-range interactions. Pitzer's equations explicitly include terms to describe these interactions, in addition to long-range interactions. These specific interaction terms between two or three distinct aqueous species become more important the higher the concentration. In this way, the Pitzer equations are able to describe brine properties and mineral-gas-solution equilibria in hyper-saline systems. The major drawback of this approach is the large number of interaction parameters that need to be estimated, which increases drastically with the size of the chemical system to be described. The number of parameters is further
increased by the need to describe their temperature (and/or pressure) dependence.

Pitzer's equations (Pitzer, 1991; see also Supplementary materials 1, S1) are semi-empirical and require more or less complex optimization procedures (André et al., 2009; Voigt, 2001). Considering these solute-specific interaction parameters makes it possible to avoid explicit consideration of aqueous complexes in many cases, up to salinities of several moles per kg of water. From a mathematical point of view, however, this approach does not preclude consideration of aqueous complexes. It is then possible to extend the range of application of the model to very high concentrations (Lach et al., 2015; Lassin et al., 2015).

The numerous works based on Pitzer's equations generally focus on restricted chemical systems, usually with less than 10 chemical elements (Christov and Møller, 2004a, 2004b; Greenberg and Møller, 1989; Harvie et al., 1984), and without necessarily seeking methodological consistency between the systems. As a result, very little work has been done on possible systematics that could be expressed for each type of interaction parameter. Consequently, Pitzer-compatible databases available for multicomponent geochemical calculations contain 42 chemical elements at best (André et al., 2020; Appelo, 2015; Moog et al., 2015).

To date, the use of an EDH-type model in a geochemical calculation code excludes the use of the Pitzer equations while it appears that their respective capacity could be advantageously combined to describe many natural and industrial fluids of high salinity and complex chemical compositions. A generalized approach is therefore needed to combine and take benefit from these two types of models, and thus to be able to describe complexities of saline chemical systems. In addition, by considering the HKF theory to describe the standard state partial molar thermodynamic properties of the solutes, the approach can be used under high temperature and pressure conditions.

In the present work, the ability of existing thermodynamic approaches to calculate the activities of major and trace elements in brines was first assessed. Efforts were focused on the description of the non-ideal contribution to thermodynamic properties of aqueous species and its impacts on the chemical reactivity of multicomponent systems at elevated temperatures, pressures and salinities. Because of the various limitations of existing thermodynamic approaches, the objective was to develop a thermodynamic database that could be used to calculate thermal and volumetric properties of aqueous solutions up to high salinities, in addition to solid-water-gas equilibria. Such database combines the in-house Thermoddem database (Blanc et al., 2012; Lassin et al., 2024) with the Pitzer formalism (Pitzer, 1973, 1991) implemented in the in-house PhreeSCALE.dat database (André et al., 2020). The resulting Hybrid Pitzer-Thermoddem database, or HyPiT, is then applied to different geochemical systems in relation with natural and industrial environments like geothermal energy, oil and gas exploitation, seawater desalination, and more specifically on heat exchanger surfaces, membrane surfaces of reverse osmosis processes, and oilfield facilities (Ezuber et al., 2023; Hassani and Zheng, 2025; Todd and Bluemle, 2022).

## 2. Theoretical background and applications of existing tools

### 2.1. Estimation of Pitzer's interaction parameters

The Pitzer model is based on several parameters that account for binary and ternary interactions between the dissolved species composing the electrolytic solution. As such, its use is restricted to the specific chemical systems for which the values of the parameters are known. In a series of recent works, Simoes et al. (2017a, 2017b, 2016) developed new equations to estimate the Pitzer binary interaction parameters, $β^{(0)}$ and $β^{(1)}$ and their first derivative with respect to temperature ($dβ^{(0)}/dT$ and $dβ^{(1)}/dT$), from the ionic radii and the electrical charges of the ions concerned. The aim was to automate the Pitzer parametrization of binary electrolytic systems over ranges of

concentrations that vary in extent according to the nature of the electrolyte. The authors first verified that they could use a degraded form of the Pitzer equations to describe the osmotic coefficient of numerous binary electrolyte solutions without excessively reducing the range of applicable concentrations compared with the original Pitzer equations.

The set of parameters $β^{(0)}$, $β^{(1)}$, $dβ^{(0)}/dT$, and $dβ^{(1)}/dT$ thus obtained was then used to establish complex correlations with the charges and ionic radii of the electrolytes:

$$β_{MX}^{(0)}(T_r) = \kappa_1 z_M^{\epsilon_1} z_X^{\epsilon_2} |r_M - \kappa_2 r_X|^{\epsilon_3} + \kappa_3 \quad (4)$$

$$\beta_{MX}^{(1)}(T_r) = \kappa_1 z_X^{\epsilon_1} (z_M^{\epsilon_2} z_X^{\epsilon_3} (1 + |r_M - \kappa_2 r_X|^{\epsilon_4}))^{\epsilon_5} + \kappa_3 z_M^{\epsilon_6} z_X^{\epsilon_7} (1 + |r_M - \kappa_2 r_X|^{\epsilon_4}) + \kappa_4 z_X^{\epsilon_3} \quad (5)$$

$$β_{MX}^{(0)}(T) = β_{MX}^{(0)}(T_r) + [\kappa_1 z_M^{\epsilon_1} + \kappa_2 (r_M^{\epsilon_2} + r_X^{\epsilon_3}) z_M^{\epsilon_1}] (T - T_r) \quad (6)$$

$$β_{MX}^{(1)}(T) = β_{MX}^{(1)}(T_r) + [\kappa_1 z_M^{\epsilon_1} + \kappa_2 (r_M^{\epsilon_2} + r_X^{\epsilon_3}) z_M^{\epsilon_1}] (T - T_r) \quad (7)$$

where $T_r$ is the reference temperature (298.15 K), $z_M$ and $z_X$ are the electric charges (–) of cation M and anion X, respectively, and $r_M$ and $r_X$ are their respective ionic radii (Å). For each interaction parameter, the $\kappa_i$ and $\epsilon_i$ coefficients (with $i = 1$ to 7) are reported in Table 1. Note that, following the notation for electrolytes used by Pitzer (1991), MX designates an electrolyte made of $\nu_M$ cations of charge $z_M$ and $\nu_X$ anions X of charge $z_X$. When two cations (alternately, two anions) are present, the M and M' (alternately X and X') notation is used. Based on the charges of the ions, an electrolyte belongs to a type of electrolytes noted $z_M : |z_X|$. A ternary mixture containing two different cations and one anion is noted $z_M : z_M : |z_X|$, and so on. In some instances, a cation $M^{2+}$ and an anion $X^{2-}$ can form aqueous complexes MX of charge $z_{MX} \pm$. These notations apply throughout this document where needed.

The application of equations (4) to (7) comes with several limitations/conditions:

i. Simoes et al. (2016) indicate that equations (4) and (5) apply more specifically to strong electrolytes that dissociate completely over the range of concentrations considered.

In fact, the authors compared their correlations to a classic Pitzer formulation applied under the assumption that dissociation was complete even for weak electrolytes. However, to make a more meaningful comparison, the speciation of weak electrolytes should have been considered. It has not been demonstrated that equations (4) and (5) apply preferably to strong electrolytes and it is therefore possible that

<caption>Table 1</caption>

– Coefficients for equations (4) to (9) to compute binary interaction parameters and their temperature dependence (Simoes et al., 2016, 2017b), as well as radii of complex ions (Simoes et al., 2017a).

<table><thead><tr><th rowspan="2">Coefficient</th><th colspan="6">Parameter</th></tr><tr><th>β<sub>MX</sub><sup>(0)</sup>(T<sub>r</sub>)</th><th>β<sub>MX</sub><sup>(1)</sup>(T<sub>r</sub>)</th><th>dβ<sub>MX</sub><sup>(0)</sup>/dT</th><th>dβ<sub>MX</sub><sup>(1)</sup>/dT</th><th>r<sub>MX</sub> (anion)</th><th>r<sub>MX</sub> (cation)</th></tr></thead><tbody><tr><td>κ<sub>1</sub></td><td>0.04432</td><td>0.01001</td><td>-1.61e-3</td><td>1.94e-5</td><td>-</td><td>1.249</td></tr><tr><td>κ<sub>2</sub></td><td>1.5</td><td>-1.2</td><td>3.73e-4</td><td>2.76e-5</td><td>-</td><td>-</td></tr><tr><td>κ<sub>3</sub></td><td>0.05758</td><td>0.12017</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>κ<sub>4</sub></td><td>-</td><td>0.05226</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>ε<sub>1</sub></td><td>1.62</td><td>-0.4</td><td>1.03</td><td>2.712</td><td>0.057</td><td>-0.264</td></tr><tr><td>ε<sub>2</sub></td><td>-1.35</td><td>2</td><td>1.485</td><td>5.092</td><td>-</td><td>1.5</td></tr><tr><td>ε<sub>3</sub></td><td>1.2</td><td>0.6</td><td>1.39</td><td>3.561</td><td>-</td><td>-</td></tr><tr><td>ε<sub>4</sub></td><td>-</td><td>0.2</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>ε<sub>5</sub></td><td>-</td><td>2</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>ε<sub>6</sub></td><td>-</td><td>2</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>ε<sub>7</sub></td><td>-</td><td>0.2</td><td>-</td><td>-</td><td>-</td><td>-</td></tr></tbody></table>
they may apply to weak electrolytes as long as speciation is explicitly taken into account.

ii. When the constitutive ions of an electrolyte exhibit chaotropic character, Simoes et al. (2017b) consider hydrated rather than ionic radii in equations (6) and (7).

The chaotropic/kosmotropic character of a solvated ion, i.e. its propensity to destructure (respectively organize) the solvent hydrogen bond network, is defined according to a criterion introduced by Marcus (1994). This criterion includes the entropy value and the orientation heat capacity of water molecules in the vicinity of the ion. In general, anions are more chaotropic while cations are more kosmotropic.

iii. According to Simoes et al. (2017b), their equations are valid up to 150 °C.

iv. To extend the use of their formulae for estimating Pitzer parameters to a larger number of aqueous species, Simoes et al. (2017a) have also developed equations for estimating the so-called “thermochemical” radii of anionic, $r_{MX^{z-}}$ (8), and cationic, $r_{MX^{z+}}$ (9), complex species:

$$r_{MX^{z-}} = |z_{MX}|^{\epsilon_1} \left( \sum_{j} \nu_j r_{M_j}^{\epsilon_3} + \sum_{k} \nu_k r_{X_k}^{\epsilon_3} \right)^{1/3} \quad (8)$$

$$r_{MX^{z+}} = z_{MX}^{\epsilon_1} \kappa_1 \left( \sum_{j} \nu_j r_{M_j}^{\epsilon_2} + \sum_{k} \nu_k r_{X_k}^{\epsilon_2} \right)^{1/3} \quad (9)$$

Simoes et al. (2017a) use the ionic radii of the constituent ions of these complex species, considering the covalent nature of the bonds in the case of complex cations. This covalent (as opposed to ionic) nature of the bond results in an apparent ionic radius, $r'(\AA)$, that is smaller than the real ionic radius, $r(\AA)$. The difficulty lies mainly in determining the degree of covalency to be assigned to a bond and the value to be given to the apparent ionic radius. The apparent radii of a number of ions making up complex cationic species (with a high degree of covalency) are given by the authors. However, their database is not exhaustive, which makes it difficult to apply equations (8) and (9) to the large number of known species and complexes.

Once calculated, these “thermochemical” radii can be used to apply equations (4) to (7) to estimate the binary interaction parameters $\beta^{(0)}$, $\beta^{(1)}$, $d\beta^{(0)}/dT$, and $d\beta^{(1)}/dT$ between the complex species and any other aqueous species of opposite charge. At this stage, it is expected that the chemical behavior of numerous relatively concentrated binary aqueous solutions can be described. Of course, such an approach should only be considered if it is not possible to optimize all the required interaction parameters with experimental data on osmotic coefficient, salt solubility, partial pressure of water vapor ($P_{H_2O}$), etc. This is precisely the case when chemical elements are present in trace amounts in salt solutions: traces concentrations have no impact on the osmotic coefficient or $P_{H_2O}$, and the analysis of traces contained in brines remains a challenge.

It should also be noted, that there is no equation for estimating binary interaction parameters between a neutral, generally complex, species and any other aqueous species, whether electrically charged or not.

## 2.2. Preliminary test cases and comparisons between existing thermodynamic approaches

As a first approximation, it can be considered an ion present in trace amounts in a saline solution will preferentially only interact in a binary manner with the major ion(s) of opposite charge. The equations proposed by Simoes et al. (2017a, 2017b, 2016) give these binary interactions a specific character with respect to the trace species and the major ion(s) of opposite charge. This leads to the hypothesis that

applying such an approach should make it possible to extend the use of the Pitzer equations to complex chemical systems, clearly distinguishing the major elements from the trace elements to which the degraded form of the Pitzer equations applies.

To test this hypothesis, three different preliminary test cases were performed:

i. Test case 1 assesses the maximum concentration of the trace compound above which its activity coefficient no longer depends solely on the concentration of the major salt, but also on its own concentration. The degraded form of the Pitzer equations may no longer apply beyond this point.

ii. Test case 2 involves a comparison between the activity coefficients of a trace compound calculated according to the complete Pitzer equations, the degraded form of these equations, and the Davies and B-dot EDH models. This is a major test of the hypothesis.

iii. Test case 3 considers aqueous speciation explicitly and, thus, can be envisaged as soon as aqueous phase speciation data exist for the compound present in trace amounts in brine.

For the comparisons to be meaningful, they must be carried out on well-known ternary systems, with the full Pitzer parametrization established to serve as a reference. For each ternary system, the test must be carried out by considering each electrolyte alternately as a trace and as a major component.

Many electrolytic systems have been studied through the prism of Pitzer's equations. Among them, the most extensively studied are chloride systems, particularly in terms of internal coherence in the presence of different cations e.g. André et al. (2018, 2013), Lassin et al. (2018, 2015), Lassin and André (2023). Up to 29 M-M'-Cl ternary systems can be described, where M and M' represent two cations among $H^+$, $Li^+$, $Na^+$, $K^+$, $Mg^{2+}$, $Ca^{2+}$, $Sr^{2+}$, $Ba^{2+}$, $Fe^{3+}$, or $Al^{3+}$. Alternately, nitrate aqueous systems are also interesting candidates because of their high solubility (Guignot et al., 2019; Lach et al., 2018; Lassin et al., 2020).

### 2.2.1. Test case 1: effect of trace element concentration on its activity coefficient

The test consisted in calculating the activity coefficient of the trace $M^{zM}$ ion in an aqueous $MCl_{zM}$ solution with a concentration varying between 0 and 7 mol kg$^{-1}$. Calculations were made using PhreeSCALE software (Lach et al., 2016) and the associated BRGM in-house PhreeSCALE database (André et al., 2020). The PhreeSCALE code was developed based on PhreeqC-v3 software (Appelo et al., 2014; Parkhurst and Appelo, 2013). For the chemical calculation tests, the $M^{zM}$ concentration was fixed at various values between $10^{-7}$ and $10^{-1}$ mol kg$^{-1}$. The same operation was carried out, this time considering trace $M^{zM}$ in an aqueous solution of $MCl_{zM}$ over the same concentration range. The goal was to determine a concentration level of the trace element above which its activity coefficient no longer depends solely on the concentration of the major electrolyte. This aims at defining a limit for the proposed approach, namely the use of the degraded Pitzer equations for trace solutes. As an illustration, the LiCl-KCl-H$_2$O system was selected. The results are shown in Fig. 1. They show that the activity coefficient of the trace ion is independent of its concentration as long as it remains below $10^{-2}$ mol kg$^{-1}$. Furthermore, the activity coefficient of the trace element is highly dependent on the major salt concentration and can behave very differently depending on the nature of the trace element. To confirm this trend, a second series of activity coefficient calculations was carried out in the Ca(NO$_3$)$_2$-Mg(NO$_3$)$_2$-H$_2$O system to consider another anion and divalent cations (Fig. 2). The results lead to the same conclusions. This, together with other tests (not shown for the sake of clarity), makes it possible to generalize the conclusion drawn above and to choose to continue the following tests only for a single trace element concentration, set at $10^{-5}$ mol kg$^{-1}$.
<figure><img src="image_3.png" /><figcaption>Fig. 1. – (A) Activity coefficient of the Li⁺ ion (trace element) as a function of KCl molality, at various Li⁺ concentrations; (B) activity coefficient of the K⁺ ion (trace element) as a function of LiCl molality, at various K⁺ concentrations. Lines for Li⁺ (A) and K⁺ (B) concentrations from 1e⁻⁷ to 1e⁻² mol kg⁻¹ plot on top of each other.</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>Fig. 2. – (A) Activity coefficient of the Ca²⁺ ion (trace element) as a function of Mg(NO₃)₂ molality, at various Ca²⁺ concentrations; (B) activity coefficient of the Mg²⁺ ion (trace element) as a function of Ca(NO₃)₂ molality, at various Mg²⁺ concentrations. Lines for Ca²⁺ (A) and Mg²⁺ (B) concentrations from 1e⁻⁷ to 1e⁻² mol kg⁻¹ plot on top of each other.</figcaption></figure>

### 2.2.2. Test case 2: comparison between different calculation models for activity coefficients of trace elements

The goal of this test is to assess the accuracy of the approach compared with the conventional EDH activity models, and thus its utility, with respect to classic EDH activity models. The test is performed on the ternary aqueous chloride systems for which the complete sets of Pitzer parameters are known for reference.

For each ternary system, the concentration of one cation is fixed at 10⁻⁵ mol kg⁻¹ while the concentration of the other cation and chloride varies between 0 and 5 mol kg⁻¹. The situation is then reversed. Under these conditions, the activity coefficient of the trace cation is calculated, at 25 °C, in five different ways:

(i) The full Pitzer model, thus providing the reference value. The PhreeSCALE.dat database was used (André et al., 2020), to which interaction parameters from additional studies were added, as for the FeCl₃–H₂O system (André et al., 2013, 2018);

(ii) The Pitzer degraded mode using the parameters $\beta^{(0)}$ and $\beta^{(1)}$ estimated from the equations of Simoes et al. (2016);

(iii) The Pitzer equations, neglecting all the short-range interactions relating to the trace element (only the Debye-Hückel term is retained, with all the specific interaction parameters set to 0);

(iv) The Davies model presented in eq. (2). The ionic strength is calculated according to eq. (1) and the Debye-Hückel coefficient A is 0.5095 mol⁻¹/²dm³/² at 25 °C;

(v) The B-dot model, presented in eq. (3). The Debye-Hückel coefficient B is 3.284 Å⁻¹ mol⁻¹/²dm³/² and, B̄ is 0.0410 kg mol⁻¹ at 25 °C, and $\alpha_0$ (Å), the ionic radius of species i, is from the Thermodiem database.

Up to 22 of the ternary chloride systems were considered. Fig. 3 presents the results for a few of them for illustration of the various possible configurations of $z_M : z_M : 1$-type aqueous mixtures that can be encountered, with $z_M, z_M = 1, 2$ or $3$ (Fig. 3). As a general observation, the degraded Pitzer equations using specific interaction parameters $\beta^{(0)}$ and $\beta^{(1)}$ calculated after Simoes's equations predict activity coefficients of trace cations in better agreement with those computed with the full Pitzer equations than those calculated with the EDH-type models. More specifically, the Pitzer-degraded model is clearly system-specific like the full Pitzer model. The activity coefficients calculated with EDH-type models depend mostly on the charge of the cations and not on the nature of the cations themselves. For monovalent trace cations in monovalent electrolytes, the EDH-type models frame the results obtained with the Pitzer equations (Fig. 3-A,B). The Davies equation generally overestimates while the B-dot equation underestimates the activity coefficient. Disregarding specific interaction parameters for the trace cation leads to meaningless activity coefficient values. For divalent trace cations in 1:1 or 2:1 electrolyte solutions, the match between the B-dot and Pitzer equations is reasonably good (Fig. 3-C to E) but poorer than with the degraded mode of the Pitzer equations. For a higher electric charge of the trace cation or of the electrolyte (Fig. 3-F to H), EDH equations cannot compete with the degraded mode of the Pitzer equations. In summary, the latter is generally more precise than the EDH equations, which is encouraging for its use in modeling complex and saline aqueous systems.
<figure><img src="image_5.png" /><figcaption>Fig. 3. – Calculated activity coefficient of various trace cations in fully dissociated chloride aqueous electrolytes for z<sub>M</sub> : z<sub>M</sub> : 1-type mixtures, with z<sub>M</sub>, z<sub>M</sub> = 1, 2 or 3 as shown and according to different activity models: Davies (EDH), B-dot (EDH), PhreeSCALE (full Pitzer), Simoes (degraded Pitzer), no-interaction (Pitzer's DH term with no specific interactions).</figcaption></figure>

The degraded model of the Pitzer equations deviates from the full Pitzer equations for trace cations that have a strong chaotropic character, i.e. with a large size and a low charge (like Na⁺ and K⁺). At this stage, it is not clear whether this is due to the correlations developed by Simoes et al. (2016) or because of their parameterization.

This type of comparison cannot be taken further because EDH models assume that part of the activity coefficient that is not handled by the model is actually described by the reactions of aqueous complexation. Thus, a different activity coefficient value can be compensated for by the concentration value for the aqueous species to give the same value for the ion activity, aᵢ. Indeed, one can write: aᵢ = γᵢmᵢ/m₀ = γ'ᵢm'ᵢ/m₀, where γᵢ ≠ γ'ᵢ are activity coefficients of species i calculated with two different models, and mᵢ ≠ m'ᵢ are the respective molalities for the same activity value (m₀ = 1 mol kg⁻¹ is the reference molality in both cases). By extension, the same value for the ion activity product involved in the mass action law of a given reaction can be calculated with both EDH and Pitzer approaches. This is precisely what is expected by combining aqueous complexation reactions and the degraded form of the Pitzer equations through which the activity coefficients of aqueous complexes can be calculated.

### 2.2.3. Test case 3: considering the constraint of aqueous speciation

The objective of this test case is to predict the speciation of ferric iron in the FeCl₃–LiCl–H₂O system at 25 °C based on the work of Liu et al. (2006). These authors used UV-vis absorption spectrophotometry to “measure” the aqueous speciation of ferric iron in aqueous electrolyte solutions of LiCl between 25 and 90 °C, for ionic strengths up to 15 M, and pH lower than 1. The concentration of dissolved iron ranged between 0.2 and 0.5 mmol kg⁻¹. The experimental aqueous speciation at 25 °C was obtained by deconvolution of the measured absorbance spectra and led to the identification of five different ferric iron aqueous species, namely Fe³⁺, FeCl²⁺, FeCl₃⁻, FeCl₃⁺, and FeCl₄ (Fig. 4).

For the thermodynamic modeling of this system, two approaches are compared. Firstly, Thermoddem is used with PhreeqC-v3 software. The database is adapted to rather dilute aqueous solutions and so the only aqueous species considered in the FeCl₃–H₂O system are Fe³⁺, FeCl²⁺ and FeCl₃⁻ (leaving aside the hydroxo complexes). The aqueous speciation calculated with the Thermoddem database that uses the B-dot activity model is plotted in Fig. 4-A, where it can be compared to the experimental speciation obtained by Liu et al. (2006). The correspondence is reasonable only for LiCl concentrations below 1 mol kg⁻¹. Beyond this value, iron is mostly present in the form of FeCl²⁺, while FeCl₃⁻ remains limited. Secondly, the PhreeSCALE.dat database is used
<figure><img src="image_6.png" /><figcaption>Fig. 4. – Calculated aqueous speciation of Fe(III) in LiCl aqueous solutions at 25 °C (dashed black lines) overlaid with experimental values from the literature (grey lines, modified after Liu et al., 2006), A) using the Thermoddem database, B) using the log K of the iron species from Thermoddem, the degraded Pitzer equations for their activity coefficients and PhreeSCALE for LiCl (see text).</figcaption></figure>

since it contains Pitzer's interaction parameters to describe the chemical properties of the LiCl–H₂O system for concentrations up to 20 M at 25 °C, and beyond for higher temperatures (Lassin et al., 2015). However, the PhreeSCALE database does not include iron. Thus, the Fe³⁺, FeCl²⁺, FeCl₂⁻ species and their corresponding formation constants were first included in the PhreeSCALE database to consider the same species with the two approaches.

The β⁽⁰⁾, β⁽¹⁾, dβ⁽⁰⁾/dT, and dβ⁽¹⁾/dT parameters were calculated for the Fe³⁺/Cl⁻, FeCl²⁺/Cl⁻ and FeCl₂⁻/Cl pairs using the equations of Simoes et al. (2017a, 2017b, 2016). However, the calculation is not straightforward since several criteria that are necessary for applying these equations are not known for every aqueous species. In particular, the ionic radii of the aqueous complex cations are not known and have to be calculated according to equation (9), prior to the calculation of interaction parameters. Because of the covalent nature of the Fe–Cl chemical bonding due to the strong electronegativity of chlorine, the apparent ionic radii of Fe³⁺ and Cl⁻ ions should be used. Unfortunately, Simoes et al. (2017a) only give the value for Cl⁻ (1.02 Å, while the ionic radius is 1.81 Å). For Fe³⁺, the value of the actual ionic radius, namely 0.65 Å, had to be used.

According to Simoes et al. (2017b), the type of ionic radii to use for computing the dβ⁽⁰⁾/dT and dβ⁽¹⁾/dT parameters depends on the chaotropic nature of the electrolyte (see section 2). If assumed to be fully dissociated, the FeCl₃ electrolyte is not chaotropic because of the kosmotropic nature of the Fe³⁺ ion, and the true ionic radii values of Fe³⁺ and Cl⁻ can be used. Furthermore, the kosmotropic nature of the FeCl²⁺ and FeCl₂⁻ complexes is less clear, but equation (9) only helps to compute thermochemical radii of aqueous complexes, the meaning of which is not straightforward in reference to true and apparent ionic radii. Despite this, these calculated thermochemical radii of the FeCl²⁺ and FeCl₂⁺ complexes were used in equations (6) and (7) to compute the dβ⁽⁰⁾/dT and dβ⁽¹⁾/dT parameters. The results are reported in Table 2. The aqueous speciation calculated using the iron-implemented PhreeSCALE database is plotted in Fig. 4-B, clearly showing a better agreement with the experimental data of Liu et al. (2006) compared to Thermoddem at high LiCl concentrations. Indeed, the FeCl₂⁻ species becomes the majority at the expense of FeCl²⁺ for LiCl concentrations above 6 mol kg⁻¹. However, despite the fact that the trend is in accordance with the experimental data, the deviation remains significant. This is partly due to the absence of values for higher order aqueous ferric chloro-complexes, like FeCl₃⁻ and FeCl₄⁻ in the database. At this stage, one can nevertheless conclude that the degraded mode of the Pitzer equations can apply to aqueous complexes of trace elements in strong electrolyte solutions. Therefore, it has the potential to significantly improve the description of the aqueous speciation in complex brines. This will be further illustrated in the discussion section (5).

According to the results obtained in test cases 1 to 3, it was decided to

<caption>**Table 2**</caption>

– Size characteristics and chaotropic/kosmotropic nature of solutes, and specific interaction parameters calculated for trace ferric iron in LiCl aqueous brines. Numbers in italics represent values calculated in this work. For charged aqueous complexes, the ionic radius (rᵢ) is assimilated to the thermochemical radius (rᵣₘ) estimated according to eq. (8) for anions and (9) for cations. Question marks indicate that the value of the corresponding ionic (rᵢ), hydrated (rₕ) or apparent (rₑ) radius is not known.

<table><thead><tr><th rowspan="2">Parameter</th><th colspan="6">Interaction</th></tr><tr><th>Fe³⁺/Cl⁻</th><th>FeCl²⁺/Cl⁻</th><th>FeCl₂⁻/Cl⁻</th><th>FeCl₃⁻/Cl⁻</th><th>FeCl₄⁻/Li⁺</th><th>FeCl₄⁻/Li⁺</th></tr></thead><tbody><tr><td>r<sub>t,cat</sub>, r<sub>t,an</sub><sup>a</sup></td><td>0.65, 1.81</td><td>1.20, 1.81</td><td>1.71, 1.81</td><td>?, 1.81</td><td>?, 0.69</td><td>2.88, 0.69</td></tr><tr><td>r<sub>h,cat</sub>, r<sub>h,an</sub><sup>b</sup></td><td>?, 2.24</td><td>?, 2.24</td><td>?, 2.24</td><td>?, 2.24</td><td>?, ?</td><td>2.3, ?</td></tr><tr><td>r<sub>c,cat</sub>, r<sub>c,an</sub><sup>c</sup></td><td>?, 1.02</td><td>?, 1.02</td><td>?, 1.02</td><td>?, 1.02</td><td>?, ?</td><td>?, ?</td></tr><tr><td>chao/<br/>kosmo<sup>d</sup></td><td>k, c</td><td>?, c</td><td>?, c</td><td>?, c</td><td>?, k</td><td>?, k</td></tr><tr><td>β⁽⁰⁾</td><td>0.6848</td><td>0.2810</td><td>0.1020</td><td>—</td><td>—</td><td>0.130<sup>f</sup></td></tr><tr><td>β⁽¹⁾</td><td>5.8438</td><td>1.6468</td><td>0.3097</td><td>—</td><td>—</td><td>0.3694</td></tr><tr><td>dβ⁽⁰⁾/dT</td><td>-8.338e-4</td><td>5.359e-5</td><td>3.646e-4</td><td>—</td><td>—</td><td>-2.079e-4</td></tr><tr><td>dβ⁽¹⁾/dT</td><td>1.003e-2</td><td>3.790e-3</td><td>9.361e-3</td><td>—</td><td>—</td><td>5.594e-4</td></tr><tr><td>λ<sup>e</sup></td><td>—</td><td>—</td><td>3</td><td>4</td><td>0</td><td>0.1</td></tr></tbody></table>

a : rᵢ, cation (cat) or anion (an) ionic radius (Å) —

b : rₕ, hydrated radius (Å) —

c : rₑ, apparent ionic radius for covalent bonding —

d : chaotropic or kosmotropic nature of interacting ions —

e : binary specific interaction parameter involving a neutral species and a charged species. The 0 value for λ<sub>FeCl₃-Cl</sub> was arbitrarily chosen in this study (see text) —

f : the Simoes equations gave a value of 0.2662 that had to be refined in order to match the experimental speciation by Liu et al. (2006).

extend the application of the Simoes et al. (2017a, 2017b, 2016) equations to the whole Thermoddem database, excluding the chemical systems already parameterized for the full Pitzer equations. The next step is to build up the hybrid thermodynamic database, the HyPiT database, and to test it against experimental applications or the natural environment, as described below.

### 3. Building of the hybrid Pitzer-Thermoddem database

The building of the HyPiT database relies on the Thermoddem database, which includes aqueous species, minerals and gases for 90 chemical elements, and on the PhreeSCALE database, which includes aqueous species, minerals and gases for 42 chemical elements. Thermoddem is consistent with the B-dot EDH model for calculating activity
coefficients while PhreeSCALE is consistent with the Pitzer equations. Combining the two databases involved the following stages (Fig. 5):

i. Literature compilation of the ionic radii of simple aqueous species (dissociated ions or small ionic molecules) and of their hydrated radii when possible;

ii. Collating of the kosmotropic or chaotropic nature of the ions from the literature;

iii. Computation of the unknown thermochemical radii for all aqueous complexes referenced in the Thermoddem database (using equations (8) and (9));

iv. Identification of all possible pairs of oppositely charged species;

v. Computation of the $β^{(0)}$, $β^{(1)}$, $dβ^{(0)}/dT$, and $dβ^{(1)}/dT$ parameters for the above ion pairs (equations (4) to (7));

vi. Identification of all possible pairs of ion/neutral species, except water (H₂O), with an associated λ interaction parameter to which a value is given as detailed below;

vii. When a mineral is initially present in the two databases, the properties sourced from PhreeSCALE are kept at the expense of those from Thermoddem to preserve consistency between the solubility product and the Pitzer-based activity model;

viii. Consistent combination of this modified Thermoddem database with the PhreeSCALE database, to be used with the PhreeSCALE software (Lach et al., 2016).

The latter action included the careful tracking of duplicates, with priority given to the PhreeSCALE database: the full Pitzer parameterization is preserved at the expense of the degraded version. When

necessary, the aqueous complexes associated with a given electrolyte in the Thermoddem database are eliminated if the corresponding parameterization in the PhreeSCALE database does not include them (ghosted species in Fig. 5). As an example, the species $KSO_4$ and $MgSO_4^{2-}$ are excluded from HyPiT since the Pitzer parameterization implemented in PhreeSCALE after Greenberg and Møller (1989) and Lach et al. (2016) for the $K_2SO_4-H_2O$ and $MgSO_4-H_2O$ binary systems, respectively, rely on the full dissociation hypothesis. Implementing $KSO_4^-$ and $MgSO_4^{2-}$ would thus be redundant.

Ternary systems described in the PhreeSCALE database according to the full Pitzer equations are kept unchanged during the combination process with the Thermoddem database.

While the first five bullet points above have been previously described, point vi) deserves some clarification. Indeed, the Simoes equations do not consider the $λ$ parameters for characterizing binary neutral-ion specific interactions. Yet, the example of the aqueous speciation of ferric iron (Case 3 in the previous section) highlights the need to consider such interactions.

The case of silica, one of the most abundant elements in the earth's crust, is considered below. Silica is present in any groundwater, especially in geothermal waters and it can cause severe trouble during their exploitation because of its prograde solubility. Indeed, when geothermal waters are pumped and treated at the surface for heat extraction, the decrease of the fluid temperature may cause the precipitation of silicate minerals (including amorphous silica) in the surface facilities (pipes, heat exchangers, etc.). Many studies have been conducted at lab scale to measure the solubility of amorphous silica in a wide variety of brine compositions. This study focuses on the works of Marshall and Warakomski (1980) who published solubility data for amorphous silica in chloride solutions, especially in binary systems like NaCl-H₂O, KCl-H₂O, CaCl₂-H₂O, MgCl₂-H₂O, and LiCl-H₂O systems. They also studied the solubility of amorphous silica in nitrate solutions with the same cations (Fig. 6). These binary systems of main interest have been extensively studied in geochemical modeling and Pitzer specific interaction parameters have been determined and already implemented in the PhreeSCALE database (Lach et al., 2017, 2018; Lassin et al., 2015; Lassin and André, 2023).

The pH of all these binary solutions is close to neutrality and, therefore, aqueous silica is mainly in the $H_4SiO_4$ form. The description of silica solubility in the various binary electrolytes thus requires the determination of the $λ_{H_4SiO_4-M}$ and $λ_{H_4SiO_4-X}$ interaction parameters, where $M = H^+$, $Li^+$, $Na^+$, $K^+$, $Mg^{2+}$, or $Ca^{2+}$ and $X = Cl^-$, $NO_3^-$ or $SO_4^{2-}$. According to Felmy and Weare (1986) and Harvie et al. (1984), one of the specific neutral-ion interaction parameters $λ$ must be assigned arbitrarily as a reference, since aqueous solutions are electrically neutral. This means that measurements describe the effect of the electrolyte, rather than of its constitutive ions, on the solubility of amorphous silica. Felmy et al. (1994) and Felmy and Weare (1986) used the convention to consider the neutral-proton $λ$ parameters equal to zero. However, this can only be done when experimental data exist for strong acids. Fortunately, Marshall and Warakomski (1980) did measure the solubility of amorphous silica in HCl and $HNO_3$ solutions. By setting $λ_{H_4SiO_4-H} = 0$, it was possible to determine $λ_{H_4SiO_4-Cl} = λ_{H_4SiO_4-NO_3} = 0.14$ and reproduce amorphous silica solubility trends in these acidic brines (Fig. 6-A,B). For comparison, the silica solubility calculated with the Thermoddem database is plotted. Then, a step-by-step adjustment process led to the determination of the $λ_{H_4SiO_4-M}$ interaction parameters ($M = Li^+$, $Na^+$, $K^+$, $Mg^{2+}$, or $Ca^{2+}$).

The results are plotted in Fig. 6-C to J showing that the solubility of silica can be reproduced very satisfactorily up to very high concentrations (up to 20 mol kg⁻¹ of LiCl). Marshall and Warakomski (1980) also measured amorphous silica solubility in $Na_2SO_4$ and $MgSO_4$ brines. By adjusting the only remaining parameter $λ_{H_4SiO_4-So_4}$ (with a value of 0.18), it is possible to obtain a fairly good match with experimental solubility data, contrary to the original Thermoddem database (Fig. 6-K, L). $λ$ parameters for the $H_4SiO_4-M-H_2O$ and $H_4SiO_4-X-H_2O$ systems are

<figure><img src="image_7.png" /><figcaption>Fig. 5. – Sketch of the process for building the HyPiT database.</figcaption></figure>
<figure><img src="image_8.png" /><figcaption>**Fig. 6.** – Solubility of amorphous silica represented by the molality of dissolved silicium, mSi, at 25 °C in different aqueous electrolyte solutions of 1:1 type (A to H), 1:2 type (I-J), 2:2 type (K) and 2:1 type (L). Symbols represent data from Marshall (1980) or Marshall and Warakomski (1980), grey lines represent calculations using the Thermoddem database, and black lines represent calculations using the HyPiT database.</figcaption></figure>

listed in Table 3.

However, setting values for the λ interaction parameters remains challenging. Indeed, the values given in the literature seem case specific. For instance, λ<sub>B(OH)<sub>3</sub></sub>-Na = -0.097 and λ<sub>B(OH)<sub>3</sub></sub>-K = -0.14 in the borate system (Felmy and Weare, 1986), which are very close to the values

<table><caption>**Table 3**

– Values of λ<sub>H<sub>4</sub>SiO<sub>4</sub>-M</sub> and λ<sub>H<sub>4</sub>SiO<sub>4</sub>-X</sub> parameters defined in this study.</caption><thead><tr><th>Cations M or anions X</th><th>Value</th></tr></thead><tbody><tr><td>NO<sub>3</sub><sup>-</sup></td><td>0.14</td></tr><tr><td>Cl<sup>-</sup></td><td>0.14</td></tr><tr><td>SO<sub>4</sub><sup>2-</sup></td><td>0.18</td></tr><tr><td>H<sup>+</sup></td><td>0</td></tr><tr><td>Li<sup>+</sup></td><td>-0.06</td></tr><tr><td>Na<sup>+</sup></td><td>-0.11</td></tr><tr><td>K<sup>+</sup></td><td>-0.15</td></tr><tr><td>Mg<sup>2+</sup></td><td>-0.065</td></tr><tr><td>Ca<sup>2+</sup></td><td>-0.10</td></tr></tbody></table>

obtained in the present work for the silica system, but with λ<sub>B(OH)<sub>3</sub></sub>-Cl = 0.091. In dos Santos et al. (2022, 2021), λ<sub>CO<sub>2</sub>-M</sub> = 0.094, 0.16, 0.20, 0.063 for M = Na<sup>+</sup>, Ca<sup>2+</sup>, Mg<sup>2+</sup>, and CaCl<sup>+</sup>, respectively, all these parameters having a positive sign, with λ<sub>CO<sub>2</sub>-Cl</sub> = λ<sub>CO<sub>2</sub>-H</sub> = 0. As there is no clear correlation between λ values and some characteristics of both the neutral and the anionic species, it is preferable to set a default λ value of zero initially. Non-zero values should be set only for specific chemical systems for which experimental data allow the parameters to be estimated.

In other respects, the PhreeSCALE database includes the HKF coefficients (Johnson et al., 1992) for aqueous species so that the density, the heat capacity and the enthalpy of the solutions can be calculated using the PhreeSCALE software (Lach et al., 2016). As this information is recorded for all the aqueous species in the core of the Thermoddem database, it was also used for implementation in the HyPiT database. This makes it possible to consider pressure effects on the equilibrium constants and, thus, on geochemical calculations.

In the end, 134,772 pairs of oppositely charged species and 45 pairs of ion/neutral species are included in HyPiT, with as many β<sup>(0)</sup>, β<sup>(1)</sup>, or non-zero λ parameters. The number of dβ<sup>(0)</sup>/dT and dβ<sup>(1)</sup>/dT that could be estimated using equations (6), (7) and (9) is much lower (c.a. 2950 estimated values each) because the chaotropic/kosmotropic character of most solutes and the degree of covalency of the bonds within complex solutes are not known. As a result, the size of the HyPiT database is about 8 Mb (Supplementary Materials 2). Consequently, the PhreeSCALE software was modified to optimize database loading and shorten reading time to less than a minute before running a simulation.

## 4. Application tests

A series of tests was carried out to ensure the capability and consistency of the HyPiT database. The latter was first applied to simple chemical solutions containing few chemical elements (quaternary systems). Because the tests could not cover the whole database, the selection of the chemical systems of contrasted solubility was rather arbitrary, based on the availability of relevant data.

The HyPiT database was then applied to simulate the cooling and heating of a complex geothermal fluid. The results were compared with those obtained using the Thermoddem database.

### 4.1. The CaSO<sub>4</sub>-NaCl-H<sub>2</sub>O system

In the CaSO<sub>4</sub>-NaCl-H<sub>2</sub>O chemical system, the mineral of interest is gypsum. Gypsum is widespread in sedimentary environments, and usually precipitates out from highly saline waters (Herrero et al., 2009). It is a sparingly soluble salt that controls calcium/sulfate concentrations around a few tens of mmol·kg<sup>-1</sup>. The CaSO<sub>4</sub>-H<sub>2</sub>O system is not present in the PhreeSCALE database. The corresponding specific interaction parameters were therefore automatically calculated during the generation of the HyPiT database, while those corresponding to the complementary aqueous electrolyte systems CaCl<sub>2</sub>-H<sub>2</sub>O, Na<sub>2</sub>SO<sub>4</sub>-H<sub>2</sub>O and
NaCl–H₂O are inherited from PhreeSCALE. The solubility of gypsum is calculated as a function of NaCl concentration at various temperatures between 0 and 25 °C, using either the Thermoddem or the HyPiT database. Results are plotted with experimental data for comparison in Fig. 7.

Clearly, the range of chlorine concentrations for which the Thermoddem database can describe experimental data is limited to values below 1 mol kg⁻¹. Conversely, the HyPiT database (Fig. 7-A) correctly records the variable solubility behavior of gypsum with respect to chlorine concentration up to 6 mol kg⁻¹. However, there is a slight systematic overestimation of experimental data at temperatures below 25 °C, although it remains very satisfactory. The match with experimental data is perfect at 25 °C. The slight discrepancy could be due to the temperature function of the solubility product rather than to the activity model.

As some experimental data of gypsum solubility in “seawater-like” solutions are also available at 0 and 25 °C (Furby et al., 1968; Hara et al., 1934; Tanaka et al., 1931), additional calculations were performed using the experimental seawater-like composition of Na = 43.8 mol%, Mg = 1.63 mol%, Cl = 50% and S = 4.89 mol% (Furby et al., 1968). The results are plotted in Fig. 7-A and D with experimental data for comparison.

Even though Thermoddem records the influence of magnesium and sulfate on the solubility of gypsum (dashed grey lines), its ability to reproduce experimental data remains limited to chlorine concentrations below 1-1.2 mol kg⁻¹. Here again, the HyPiT database gives results much closer to the experimental data, especially when the CaSO₄ species is discarded (Fig. 7-A). This has been a subject of debate for some time, a debate which remains unresolved (Christov and Møller, 2004b; Harvie et al., 1984; Li et al., 2017; Shen et al., 2019).

This test clearly demonstrates the gain brought by the HyPiT database and the use of the degraded Pitzer activity model compared to the Thermoddem database and its associated EDH-type B-dot activity model. The gypsum solubility case also shows that the HyPiT database benefits from the ability of the Thermoddem database to consider

complex chemical systems that compensate for the use of the degraded Pitzer activity model.

## 4.2. The BaSO₄-NaCl-H₂O system

Barium sulfate (barite) is known to deposit on the filters at the head of production wells of geothermal plants, for example in the Rhine Graben, near the France-Germany border, Europe, and more specifically at the Soultz-sous-Forêts site (Scheiber et al., 2012, 2015). These deposits are controlled by the addition of precipitation inhibitors to the fluid during production, but geochemical modeling should help improve the anticipation of these deposits.

The PhreeSCALE database contains Pitzer interaction parameters for the BaSO₄-H₂O system. These parameters were naturally implemented in the HyPiT database. Two series of simulations were carried out to verify the solubility of barite in NaCl solutions when the salinity varies between 0 and 5 mol kg⁻¹, using either the Thermoddem or the HyPiT database (Fig. 8).

<figure><img src="image_9.png" /><figcaption>**Fig. 8.** – Solubility of barite represented by the molality of dissolved barium, mBa, as a function of NaCl concentration at 25, 50 and 80 °C. Symbols: experimental data from Appelo et al. (2014); grey lines: solubility calculated with the Thermoddem database; solid black lines: solubility calculated with the HyPiT database.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>**Fig. 7.** – Solubility of gypsum represented by the molality of dissolved calcium, mCa, as a function of Cl concentration at A) 0-0.5 °C, B) 5, C) 10 and D) 25 °C in NaCl and seawater-like solutions. Symbols: experimental data; solid lines: calculated gypsum solubility in NaCl; dashed lines: calculated gypsum solubility in seawater; grey lines: calculations with Thermoddem; black lines: calculations with HyPiT.</figcaption></figure>
Using the Thermoddem database, barite solubility is consistent with experimental data up to 5 mol·kg⁻¹ NaCl at 25 °C. This agreement remains limited to salinities below 1.5 mol·kg⁻¹ as temperature increases. Indeed, for higher salinities, the model tends to overestimate the solubility of this mineral. In contrast, the Pitzer parameters implemented in the HyPiT database enable barite solubility to be calculated with very good consistency with experimental data even at higher temperatures. It should be stressed here that with the HyPiT database, the BaSO₄ species is assumed to be totally dissociated and the aqueous BaCl⁺ and NaSO₄⁻ complexes were removed automatically from the HyPiT database during its generation process, in accordance with the model parameterized in the original PhreeSCALE database (Lach et al., 2016; Monnin, 1995).

### 4.3. Solubility of calcite upon temperature, pressure and salinity

In subsurface systems, calcite is an important mineral, whether primary or secondary. It mainly controls the calco-carbonic equilibrium, and it is therefore very interesting to be able to predict its chemical behavior when in contact with hot brines under pressure, as in deep reservoirs. An interesting set of calcite solubility data was measured by Shi et al. (2013) in 0.1 M NaCl aqueous solutions and 4 M NaCl brines, up to 250 °C and 1450 bars, with an initial CO₂ pressure of 1 bar. These data were used previously in Appelo (2015) where the author highlighted the difficulty to reproduce the solubility measurements using the PHREEQC database adapted to the Pitzer equations. These calculations are repeated in the present work using the HyPiT database (Fig. 9). In 0.1 M NaCl solutions, calculated solubilities are consistent with the measured values although significant discrepancies can be observed at 0 and 100 °C. The increasing solubility of calcite with pressure is correctly reproduced at the different temperatures. In 4 M NaCl brines, a similar global accordance can be observed between calculated and experimental calcite solubility.

Aqueous speciation calculations provide a better understanding of these results in consistency with calculated pH values. The latter are relatively stable with temperature, pressure and salinity (Fig. 10), ranging between 5.4, in 4 M NaCl brines at 250 °C and 1450 bars, and 6.1, in 0.1 M NaCl solutions at 0 °C, 1 bar.

Fig. 11 shows the distribution of aqueous carbonate species at 0, 100 and 250 °C, in 0.1 and 4 M NaCl solutions saturated with respect to calcite. As can be expected from the initial equilibrium with 1 bar of

CO₂(g), one of the main dissolved carbon species is CO₂. Interestingly, the other main species depends on the NaCl content. In 0.1 M NaCl solutions, the second main carbon species is HCO₃⁻. In 4 M NaCl solutions, it is NaHCO₃. The calculated stability increases with pressure, at the expense of the CO₂ species, especially at temperatures up to 100 °C. The NaHCO₃ mobilizes part of the bicarbonates that do not participate in stabilizing calcite. On the contrary, it contributes to enhancing calcite solubility, in varying degrees depending on the temperature. At 0 °C the NaHCO₃ concentration increases up to 48 mmol·kg⁻¹ at 1450 bars, which explains the overestimation of the solubility of calcite in these conditions. At 100 °C, the concentration of NaHCO₃ dominates the carbon speciation, with a maximum value of 21 mmol·kg⁻¹ at 1450 bars. In this case, it helps increase the solubility of calcite in accordance with the experimental data. At 250 °C, the speciation of dissolved carbon is dominated by CO₂, but a non-negligible amount of NaHCO₃ in the 4 M NaCl brine mobilizes part of the bicarbonates. This favors further calcite dissolution, in comparison with the 0.1 M NaCl solution, but not in sufficient quantity to match the measured solubility. Note that the a₁-a₄ HKF coefficients used in Thermoddem for calculating the standard partial molar volume of the NaHCO₃ species are simply approximated by the additional stoichiometric contributions of the HKF coefficients of Na⁺ and HCO₃⁻ ions. As shown in Fig. 11, this provides consistent estimates of calcite solubility up to high temperatures and pressures in NaCl brines.

Comparatively, the aqueous speciation of calcium (not shown) does not vary importantly from one condition to another and is marked by the predominance of the Ca²⁺ species.

In these calculations, the standard partial molar volume of the aqueous species is explicitly considered to calculate the effect of pressure on the equilibrium constants of the reactions (Appelo et al., 2014). This contribution can thus be considered in addition to the activity model for calculating the pressure-dependent chemical behavior of aqueous solutions of different ionic strengths.

In the end, the solubility of calcite calculated with HyPiT upon temperature, pressure, and NaCl concentration is comparable to the one calculated with the Pitzer.dat database developed by Appelo (2015) (see Supplementary Materials 1, S2). However, it should be noted that HyPiT includes the description of aqueous speciation, namely NaHCO₃ and NaCO₃ species consistent with the works of Bychkov et al. (2020) and Stefánsson et al. (2013). In addition, HyPiT is also consistent with the

<figure><img src="image_11.png" /><figcaption>Fig. 9. – Solubility of calcite represented by the molality of dissolved calcium, mCa, as a function of pressure, at 0, 25, 100, 200 and 250 °C, A) in 0.1 M NaCl aqueous solutions and B) in 4 M NaCl aqueous solutions. <sup>a</sup> Symbols: experimental data (Shi et al., 2013); <sup>b</sup> lines: solubility calculated with the HyPiT database.</figcaption></figure>
<figure><img src="image_12.png" /><figcaption>Fig. 10. – Calculated pH of solutions at equilibrium with calcite as a function of pressure, at 0, 25, 100, 200 and 250 °C, A) in 0.1 M NaCl aqueous solutions and B) in 4 M NaCl aqueous solutions.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>Fig. 11. – Calculated distribution of aqueous carbonate species in solutions saturated with respect to calcite, in 0.1 M NaCl (left) and 4 M NaCl (right) solutions, at 0 °C (A and B), 100 °C (C and D) and 250 °C (E and F).</figcaption></figure>

constant b value (=1.2 kg$^{1/2}$ mol$^{-1/2}$) of the coefficient defined by Pitzer (1991) in the Debye-Hückel term $f_γ$ (–) of the activity coefficient equation:

$$f_γ = -A_φ \left[ \frac{I^{1/2}}{1 + bI^{1/2}} + \frac{2}{b} \ln (1 + bI^{1/2}) \right] \quad (10)$$

where $A_φ$ is related to the A Debye-Hückel parameter of equation (3) according to $A_φ = A \frac{ln(10)}{3}$ (Wolery, 1992). On the contrary, Appelo (2015) introduced a dependency of the b coefficient with respect to both temperature and pressure.

Additional tests have been done for further evaluation of the HyPiT database and are summarized in the Supplementary Materials 1, S2.

# 5. Discussion

## 5.1. On the extended application of the Simoes equations

To apply the Simoes equations (4) to (7), the ionic radii or the hydrated radii of the cations and anions must be known. For complex ionic species, a thermochemical ionic radius can be estimated using equations (8) and (9) that account for some degree of covalency between constitutive ions. Although Simoes et al. (2017a, 2017b, 2016) covered a large number of chemical systems, their study could not be exhaustive due to lack of experimental data of solution properties for all chemical systems, and venturing beyond its scope should be done cautiously.

As an example, the Simoes equations were not successfully applied to the $Eu_2(SO_4)_3$–$Na_2SO_4$–$H_2O$ and $Eu_2(SO_4)_3$–$MgSO_4$–$H_2O$ chemical
systems recently studied by dos Santos et al. (2024). These authors measured experimental solubility data and time-resolved laser-induced fluorescence spectra, which they modeled by developing a partial dissociation model based on Pitzer equations. Initially, they attempted to use the degraded Pitzer approach with parameters obtained with the Simoes equations. This approach failed so a full set of aqueous complexation constants and specific interaction parameters were optimized to reproduce the whole set of data. The resulting parameters differed significantly from those estimated with the Simoes equations (Table 4). In addition to the discrepancies observed between the parameter values, it should be noted that, in their model, dos Santos et al. (2024) used the additional C<sup>φ</sup> binary interaction parameter for the Eu<sup>3+</sup>/SO<sub>4</sub><sup>2-</sup>, Mg<sup>2+</sup>/Eu(SO<sub>4</sub>)<sub>2</sub> and Mg<sup>2+</sup>/Eu(SO<sub>4</sub>)<sub>3</sub><sup>-</sup> cation/anion pairs and a θ parameter for the Mg<sup>2+</sup>/EuSO<sub>4</sub><sup>2-</sup> cation/cation interaction. Furthermore, the automatized estimation of binary interaction parameters across all possible combinations of cation/anion pairs using the Simoes equations, as applied in the present work, provides parameters that may not be meaningful in terms of active interactions in the chemical system. In the present case, these parameters are summarized in the last 4 lines of Table 4. They concern interactions between ions in the series of aqueous complexes of Europium and sulfate: Eu(SO<sub>4</sub>)<sub>3</sub><sup>(3-2x)</sup>, with x = 0 to 3. In this series, two non-succcessive species may not both coexist at significant concentrations. The only possible meaningful cation/anion binary interaction could concern the EuSO<sub>4</sub>/Eu(SO<sub>4</sub>)<sub>2</sub> ion pair. Such an analysis of the physically meaningful interactions would be necessary for any chemical system to eliminate the unnecessary interaction parameters. However, it is expected that, even if these interaction parameters are not zero, the contribution of such interactions would be minimized by the very low concentration of at least one of the two species involved.

## 5.2. On the need for a consistent revision of the formation constants of aqueous complexes

In section 2.2.3, it was shown that using the Simoes equations to apply the degraded Pitzer equation in the Fe(III)-LiCl-H<sub>2</sub>O system could improve the description of the aqueous speciation of Fe(III) in LiCl solutions. However, there was still room for improvement. Starting from the situation detailed in section 2.2.3 (and shown again in Fig. 12-A), two main steps are necessary to build a consistent set of parameters. The first step includes:

<caption>**Table 4**</caption>

Specific binary interaction parameters β(0) and β(1) for the Eu<sub>2</sub>(SO<sub>4</sub>)<sub>3</sub>-Na<sub>2</sub>SO<sub>4</sub>-H<sub>2</sub>O and Eu<sub>2</sub>(SO<sub>4</sub>)<sub>3</sub>-MgSO<sub>4</sub>-H<sub>2</sub>O chemical systems according to literature (normal font) and estimated with the Simoes equations (italic font).

<table><thead><tr><th>Species1</th><th>Species2</th><th>β<sup>(0)</sup></th><th>β<sup>(1)</sup></th></tr></thead><tbody><tr><td rowspan="2">Eu<sup>3+</sup></td><td rowspan="2">SO<sub>4</sub><sup>2-</sup></td><td>1.792</td><td>15.040</td></tr><tr><td>0.390</td><td>9.167</td></tr><tr><td rowspan="2">EuSO<sub>4</sub><sup>+</sup></td><td rowspan="2">SO<sub>4</sub><sup>2-</sup></td><td>-0.281</td><td>1.560</td></tr><tr><td>0.082</td><td>0.366</td></tr><tr><td rowspan="2">Na<sup>+</sup></td><td rowspan="2">Eu(SO<sub>4</sub>)<sub>2</sub></td><td>-0.056</td><td>0.340</td></tr><tr><td>0.261</td><td>0.368</td></tr><tr><td rowspan="2">Na<sup>+</sup></td><td rowspan="2">Eu(SO<sub>4</sub>)<sub>3</sub><sup>-</sup></td><td>0.137</td><td>5.788</td></tr><tr><td>0.119</td><td>0.500</td></tr><tr><td rowspan="2">Mg<sup>2+</sup></td><td rowspan="2">Eu(SO<sub>4</sub>)<sub>2</sub><sup>-</sup></td><td>0.990</td><td>1.843</td></tr><tr><td>0.747</td><td>1.934</td></tr><tr><td rowspan="2">Mg<sup>2+</sup></td><td rowspan="2">Eu(SO<sub>4</sub>)<sub>3</sub><sup>-</sup></td><td>1.755</td><td>8.744</td></tr><tr><td>0.263</td><td>3.454</td></tr><tr><td rowspan="4">Eu<sup>3+</sup><br>EuSO<sub>4</sub><sup>-</sup><br>Eu<sup>3+</sup><br>EuSO<sub>4</sub><sup>+</sup></td><td>Eu(SO<sub>4</sub>)<sub>2</sub><sup>-</sup></td><td>1.293</td><td>6.455</td></tr><tr><td>Eu(SO<sub>4</sub>)<sub>2</sub><sup>-</sup></td><td>0.196</td><td>0.350</td></tr><tr><td>Eu(SO<sub>4</sub>)<sub>3</sub><sup>-</sup></td><td>0.431</td><td>13.306</td></tr><tr><td>Eu(SO<sub>4</sub>)<sub>3</sub><sup>-</sup></td><td>0.104</td><td>0.479</td></tr></tbody></table>

i. Introducing the aqueous complexes that were observed experimentally by Liu et al. (2006) but were missing in Thermoddem, namely FeCl<sub>3</sub><sup>0</sup> and FeCl<sub>4</sub><sup>-</sup>.

ii. Giving a default value of 0 to the λ<sub>FeCl<sub>3</sub>-Cl</sub> and λ<sub>FeCl<sub>3</sub>-Li</sub> interaction parameters, following the option taken in section 3.

iii. Estimating the β<sup>(0)</sup>, β<sup>(1)</sup>, dβ<sup>(0)</sup>/dT, and dβ<sup>(1)</sup>/dT parameters for the FeCl<sub>4</sub><sup>-</sup>/Li<sup>+</sup> pair with the Simoes equations. As mentioned previously, these calculations are not straightforward since the ionic radius of FeCl<sub>4</sub><sup>-</sup> is not known and has to be calculated according to equation (8) prior to interaction parameters. Contrary to complex cations, the covalent nature of the Fe-Cl chemical bonding does not affect the calculation of the thermochemical radius of complex anions (Simoes et al., 2017a). Thus, using the ionic radii of Fe<sup>3+</sup> and Cl<sup>-</sup> ions (0.65 Å and 1.81 Å, respectively), equation (8) gave a value of 2.88 Å for the thermochemical radius of the FeCl<sub>4</sub><sup>-</sup> aqueous complex. The resulting β<sup>(0)</sup>, β<sup>(1)</sup>, dβ<sup>(0)</sup>/dT, and dβ<sup>(1)</sup>/dT parameters for the FeCl<sub>4</sub><sup>-</sup>/Li<sup>+</sup> pair, using equations (4) to (7), are reported in Table 2.

iv. Giving a value to the formation constants of the FeCl<sub>3</sub><sup>0</sup> and FeCl<sub>4</sub><sup>-</sup> species since, at this point, they are the only missing information needed to calculate the aqueous speciation of Fe(III) in LiCl solutions. The first option is to use the formation constants proposed by Liu et al. (2006) including that of FeCl<sub>2</sub><sup>+</sup> (namely 1.87) for consistency, instead of the value that is selected in Thermoddem and that was used in the calculations shown in section 2.2.3.

This resulted in the aqueous speciation drawn in Fig. 12-B, which shows a significant destabilization of species FeCl<sup>2+</sup> and FeCl<sub>2</sub><sup>+</sup>, and the appearance of large amounts of FeCl<sub>3</sub><sup>0</sup>. This latter species becomes the majority when the LiCl concentration is above 6 mol kg<sup>-1</sup>. The maximum concentration of FeCl<sup>2+</sup> is now underestimated compared to the experimental speciation. The maximum for the FeCl<sub>3</sub><sup>0</sup> species becomes comparable to the maximum experimental value but shifted towards lower LiCl contents. In addition, the FeCl<sub>4</sub><sup>-</sup> species remains unstable in the model.

Based on these results, the second improvement step includes:

i. Refinement of the formation constants of the aqueous complexes. This is justified by the fact that the formation constants estimated by Liu et al. (2006) are linked to the EDH approach they used for calculating activity coefficients.

ii. Optimization of the λ<sub>FeCl<sub>3</sub>-Li</sub> parameter, assuming λ<sub>FeCl<sub>3</sub>-Cl</sub> = 0, like in dos Santos et al. (2022, 2021). As a first guess, λ<sub>FeCl<sub>3</sub>-Li</sub> = 0.1 was proposed in analogy with the Setchénow coefficient for FeCl<sub>3</sub><sup>0</sup> used by Liu et al. (2006), following the recommendation of Brugger et al. (2001) for the CuCl<sub>2</sub><sup>0</sup> aqueous species. The Setchénow coefficient represents the linear relationship between the logarithm of the activity coefficient of the FeCl<sub>3</sub><sup>0</sup> species and the ionic strength of the solution.

iii. Refinement of the β<sup>(0)</sup> binary interaction parameter of the FeCl<sub>4</sub><sup>-</sup>/Li<sup>+</sup> ion couple.

The results obtained at the end of this second step are plotted in Fig. 12-C, which shows a very good match with the experimental speciation from Liu et al. (2006). The corresponding formation constants of the aqueous complexes are reported in Table 5, which also contains formation constants taken from literature for comparison. The revised interaction parameters are reported in Table 2. Using λ<sub>FeCl<sub>3</sub>-Li</sub> = 0.1 and β<sup>(0)</sup> = 0.13 for the FeCl<sub>4</sub><sup>-</sup>/Li<sup>+</sup> ion couple instead of 0.2662 at 25 °C was found to be satisfactory. In Fig. 12-D, the Fe(III) aqueous speciation calculated by Liu et al. (2006) is overlaid with calculations done in this work for comparison. This chart shows that the model proposed by these authors disagrees with their experimental data. Indeed, it shifts the predominance domains of Fe(III) aqueous species towards the lower LiCl molalities, lowers the height of the maxima for the FeCl<sub>2</sub><sup>+</sup> and FeCl<sub>3</sub><sup>0</sup>.
<figure><img src="image_14.png" /><figcaption>**Fig. 12.** – Aqueous speciation of Fe(III) in LiCl aqueous solutions at 25 °C (black lines) overlaid with experimental literature values (grey lines in A to C, modified after Liu et al., 2006), according to various hypotheses (see text): A) initial situation, B) step 1, C) step 2. D) comparison of the model of this study (back lines) with the model of Liu et al. (2006) (grey lines).</figcaption></figure>

<caption>**Table 5**</caption>

– Formation constants at 25 °C and 1 bar, and coefficients of the corresponding temperature functions of the $FeCl_n^{3-n}$ aqueous complexes (n = 1 to 4), with $log K(T) = A + BT + C/T + D logT + ET^{-2}$, and T in K.

<table><thead><tr><th>Species<br>Ref.</th><th>FeCl<sup>2+</sup></th><th>FeCl<sub>2</sub><sup>+</sup></th><th>FeCl<sub>3</sub><sup>0</sup></th><th>FeCl<sub>4</sub></th><th>FeCl<sub>6</sub><sup>3-</sup></th><th>FeOHCl<sub>2</sub><sup>0</sup></th></tr></thead><tbody><tr><td>[1]</td><td>1.52</td><td>2.4</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>[2]</td><td>1.48</td><td>2.13</td><td>1.13</td><td>-0.79</td><td>-</td><td>-</td></tr><tr><td>[3]</td><td>1.52</td><td>1.87</td><td>0.77</td><td>-1.26</td><td>-</td><td>-</td></tr><tr><td>[4]</td><td>1.52</td><td>0.700</td><td>-</td><td>-</td><td>-</td><td>-</td></tr><tr><td>[5]</td><td>-</td><td>2.22</td><td>1.02</td><td>-0.82</td><td>-</td><td>-</td></tr><tr><td>[6]</td><td>1.49</td><td>2.13</td><td>1.11</td><td>-1.26</td><td>-</td><td>-0.8</td></tr><tr><td>[7]</td><td>1.42</td><td>1.98</td><td>0.9*</td><td>-1.18*</td><td>-</td><td>-</td></tr><tr><td>[8]</td><td>-</td><td>2.28</td><td>2.02</td><td>0.25</td><td>-5.5</td><td>-</td></tr><tr><td>[9]</td><td>1.40</td><td>1.10</td><td>-0.20</td><td>-3.10</td><td>-</td><td>-</td></tr><tr><td>A</td><td>1441.71</td><td>6028.82</td><td>9408.77</td><td>10514.87</td><td>-</td><td>-</td></tr><tr><td>B</td><td>0.21403</td><td>0.89761</td><td>1.40144</td><td>1.56661</td><td>-</td><td>-</td></tr><tr><td>C</td><td>-83894.8</td><td>-348037.4</td><td>-542774.5</td><td>-606761.3</td><td>-</td><td>-</td></tr><tr><td>D</td><td>-517.72</td><td>-2171.251</td><td>-3389.957</td><td>-3789.503</td><td>-</td><td>-</td></tr><tr><td>E</td><td>5184603</td><td>21743559</td><td>33948038</td><td>37949218</td><td>-</td><td>-</td></tr></tbody></table>

[1]: Tagirov et al. (2000) — [2]: Wolery and Jarek (2003) — [3]: Liu et al. (2006) — [4]: Blanc et al. (2012) — [5]: Lemire et al. (2013) — [6]: Jamett et al. (2018) —

[7]: Stefánsson et al. (2019) — [8]: Uchikoshi et al. (2022) — [9]: this work — *: extrapolated values.

species, and overestimates the concentration of $FeCl_4$ at the highest LiCl molalities.

The temperature effects on the aqueous speciation of Fe(III) in LiCl solutions were also considered by Liu et al. (2006), at 60 and 90 °C, in addition to 25 °C. The authors did not provide the experimental aqueous speciation of Fe(III) at 60 and 90 °C, but gave their modeling results instead, which cannot be used to refine our model calculations at these temperatures. Therefore, the temperature variation functions proposed by Liu et al. (2006) were simply kept to recalculate the formation constants of the aqueous complexes at 60 and 90 °C (Fig. 13-C). Coefficients for these temperature functions are reported in Table 5. This resulted in the aqueous speciation at these temperatures reported in Fig. 13-A and Fig. 13-B, respectively. They show that the appearance of the successive aqueous complexes shifts towards the lower LiCl concentrations as temperature increases.

In addition, Table 5 also gathers formation constants of Fe–Cl aqueous complexes collected from the literature. It highlights the variability of these properties, depending on authors and implicitly on the activity model they used to derive the formation constants. At 25 °C, the $Fe^{3+}$ is the dominant species if the LiCl concentration is below 0.3 mol kg$^{-1}$ (Fig. 12-C). Above this concentration, $FeCl^{2+}$ becomes the

dominant species up to a LiCl concentration of 4.11 mol kg$^{-1}$ at which the limits of the extended Debye-Hückel equation may be exceeded. The discrepancy is exacerbated for the higher order Fe–Cl complexes, which can explain the large variability of their formation constants amongst authors, including the values proposed in the present work. It should be noted that some authors have introduced additional aqueous complexes like $FeCl_6^3-$ (Uchikoshi et al., 2022) or $FeOHCl_2^0$ (Jamett et al., 2018), which may also affect the estimated log K values of the other aqueous complexes.

In summary, the degraded Pitzer activity model parameterized after the equations of Simoes et al. (2017a, 2017b, 2016) can be used to efficiently describe the chemical behavior of the trace $FeCl_n^{3-n}$ aqueous complexes in LiCl brines up to about 10 M. The aqueous speciation could be improved in the specific case of the $FeCl_4$ species, for LiCl concentrations above 10 M, by modifying both the $\beta^{(0)}$ binary interaction parameter of the $FeCl_4/Li^+$ ion couple and the formation constant of $FeCl_4$. While reconsidering the Simoes equations is beyond the scope of the work presented here, it is clear that the application of these equations to the behavior of trace elements in highly concentrated aqueous electrolyte solutions should be handled with caution. Nevertheless, the approach is adapted to such systems. In addition, numerous case-specific
<figure><img src="image_15.png" /><figcaption>Fig. 13. – Aqueous speciation of Fe(III) in LiCl aqueous solutions (A) at 60 and (B) at 90 °C. (C) Formation constants of Fe–Cl aqueous complexes as a function of temperature (solid lines: this work, dashed lines and symbols: model of Liu et al. (2006), for comparison).</figcaption></figure>

situations can be expected in which the a priori calculation of specific interaction parameters should be refined as soon as the relevant experimental data allows it.

### 5.3. Considerations about the specific neutral-ion interaction parameters

As mentioned in section 3, a default value of 0 is set for the λ interaction parameter if not known. However, many examples show that this parameter is generally not 0 (see sections 3 and 5.2 for H₄SiO₄ and FeCl₃, respectively). The literature also suggests that λ is most often not 0, like for oxygen, carbon dioxide, or borate (Bok et al., 2023; Clegg and Brimblecombe, 1990, 1989; dos Santos et al., 2022, 2021; Felmy and Weare, 1986; Marion et al., 2003). However, no clear correlation could be made between the value of the λ parameter, the nature of the neutral species and some properties of the ions with which it interacts. This is partly explained by the fact that not all authors considered the same reference, namely the null value for the interaction with the proton. For instance, Clegg and Brimblecombe (1990) considered λ<sub>O₂⁻H</sub> = 0.0353. Moreover, the λ parameter is most often used in combination with ζ parameters, which accounts for interactions of the neutral species with one cation and one anion.

Regarding metal-chloro complexes, different Setchénow coefficients can be found in the literature. As an example, for FeCl₃, Liu et al. (2006) proposed the value of 0.1 used in the present study, but Uchikoshi et al. (2022) suggested a value of 0.156. Similarly, Brugger et al. (2001) proposed a value of 0.1 for CuCl²⁻ while Uchikoshi and Shinoda (2019) suggested a value of 0.188. The latter authors considered the Setchénow coefficient as an adjustable parameter to improve the fit of their experimentally determined aqueous speciation of Cu(II) in HCl solution. However, their extended Debye-Hückel activity model (Uchikoshi, 2017) was different from the one used by Brugger et al. (2001). This may partly explain the difference between the resulting Setchénow coefficients for CuCl²⁻.

In the end, using a null value for the λ parameters is expected to generate errors, but setting a default non-zero value would also generate errors of unknown magnitude. The ideal situation for using a non-zero

value for λ is to determine the correct value during a specific preliminary study, in consistency with the (possibly) degraded Pitzer activity model implemented in the HyPiT database.

### 5.4. On the temperature dependence of the estimated binary interaction parameters

There are several points to bear in mind when taking temperature into account. First, the Pitzer parameters implemented in the current state of development of the HyPiT database were determined at 25 °C only. This is the case for nitrate-bearing systems, which need to be extended to variable temperature conditions.

The second point concerns the correlations of Simoes et al. (2017b) to estimate the first derivative of the β⁽⁰⁾ and β⁽¹⁾ parameters with respect to temperature. By construction, these derivatives are assumed to be independent of temperature, which means that β⁽⁰⁾ and β⁽¹⁾ are expected to vary linearly with temperature (see equations (6) and (7)). However, linearity does not hold in general when considering temperature variations (Christov and Møller, 2004a, 2004b; Greenberg and Møller, 1989; Lach et al., 2017; Lassin and André, 2023). This applies to major elements and may also concern minor and trace elements. However, depending on the effect of temperature on solubility and/or on aqueous complexation reactions, the importance of such considerations may vary from one chemical species to another. For instance, carbonates and alkaline-earth sulfates (above 100 °C) are characterized by retrograde solubility behavior (De Visscher et al., 2012; Gamsjäger et al., 2011; Königsberger et al., 1999; Krumgalz, 2018), thus reducing the concentrations of the chemical elements involved and, finally, the specific interaction term in the calculation of their activity coefficient. In addition, effect of temperature on the chemical behavior of solutes is largely controlled by their formation constant, which also results most often from the model optimization process. In other words, the empirical character of the thermodynamic models applied to high temperature and salinity provides a certain degree of freedom to balance the ideal and non-ideal contributions to the overall thermodynamic properties of a solute. Furthermore, the chemical context, like the domain of pH, can
also strongly affect the aqueous speciation and thus the mineral solubility. Therefore, such considerations are case-specific and should be dealt with individually.

A third point is related to the interaction parameter λ between neutral species and ions. Those presented in this paper have been used to describe the solubility of minerals (such as amorphous silica - see section 3) in aqueous solutions at 25 °C. These parameters can also be used to

describe the speciation of a complex solution at 25° and up to 90 °C without adding any temperature dependence (see section 5.2).

Here, the temperature limit of the constant λ parameter assumption, i.e. whether it can reproduce the experimental data for solubility as a function of temperature, is investigated. To do this, focus is made on the solubility of amorphous silica in different electrolytes between 25 and 200 °C. Due to the pH conditions chosen in the following tests, only the

<figure><img src="image_16.png" /><figcaption>**Fig. 14.** – Solubility of amorphous silica in different aqueous electrolyte solutions of 1:1 type (A to D), 1:2 type (E), 2:2 type (F), 2:1 type (G) and according to temperature. Symbols represent experimental data from Felmy et al. (1994) (A-B), Chen and Marshall (1982) (D to G) and Marshall (1980) (C) and solid black lines represent calculations using the HyPiT database.</figcaption></figure>
$H_4SiO_4$ form is considered in solution. The aim here is not to introduce a temperature dependence for the interaction parameters $\lambda_{H_4SiO_4-M}$ and $\lambda_{H_4SiO_4-X}$ that were defined at 25 °C in section 3 (Table 3), where $M = H^+$, $Li^+$, $Na^+$, $K^+$, $Mg^{2+}$, or $Ca^{2+}$ and $X = Cl^-$, $NO_3^-$ or $SO_4^{2-}$, but rather to test whether these two parameters can describe the solubility of the mineral as the solution temperature increases.

Elmer and Nordberg (1958) and Felmy et al. (1994) published experimental data on the solubility of amorphous silica in $HNO_3$ and HCl solutions up to 95 and 47 °C, respectively (Fig. 14-A to B). Even if some discrepancies can be observed between modeled and experimental data, $\lambda_{H_4SiO_4-H}$ and $\lambda_{H_4SiO_4-NO_3}$ are able to represent the solubility of amorphous silica by respecting the overall tendency up to 90 °C in $HNO_3$ solutions and 47 °C in HCl solutions. It is worth noting here that the binary interaction parameters of the $HNO_3-H_2O$ system have been only determined at 25 °C (Lach et al., 2018) and a new parameterization including a dependency on temperature could also improve the simulation results. Marshall (1980) measured the effect of temperature on the solubility of amorphous silica in $NaNO_3$ brines. Fig. 14-D shows that the proposed approach can reproduce this effect of temperature with temperature-independent $\lambda_{H_4SiO_4-Na}$ and $\lambda_{H_4SiO_4-NO_3}$ values. The modeling of such systems could be improved by adjusting the $\lambda$ parameter but also, and as a priority, the binary parameters of the $NaNO_3-H_2O$ system since they were only determined at 25 °C (Lach et al., 2018). Finally, Chen and Marshall (1982) investigated solubilities of amorphous silica in separate aqueous solutions of sodium chloride, sodium sulfate, magnesium chloride, and magnesium sulfate at temperatures up to 350 °C (Fig. 14-C,E,F,G). Calculations performed using the HyPiT database show that the solubility of amorphous silica in these 4 binary systems are well reproduced up to 150 °C. Above this temperature, the non-variation of $\lambda$ parameter with temperature leads to erratic solubility values (Fig. 14-C,G) meaning that a temperature-regression of $\lambda$ must be carried out. However, this optimization procedure is beyond the scope of this paper.

These results show that the $\lambda$ parameter can be considered as independent of temperature up to around 150 °C. This assumption is in line with the overall tendency of amorphous silica solubility in various electrolytes. This hypothesis is also consistent with previous works dealing with this topic (Azaroual et al., 1997; Felmy et al., 1994). These authors gave temperature regressions of $\lambda$ parameters for different chemical systems up to 350 °C. They show that except for Mg and Ca, $\lambda_{SiO_2-M}$ varies in small proportions up to 150 °C. It is therefore not unreasonable to consider these parameters to be constant over these temperature ranges. It is however a perspective to extend the temperature range of validity by introducing a simple temperature-variation function to the $\lambda$ interaction parameters (Miron, 2025).

## 6. Conclusions

This article described the building of the HyPiT thermodynamic database aimed at being applied to a variety of natural and industrial environments covering wide ranges of temperature, pressure, salinity, and chemical complexity. The main principle relies on a two-step building process for the database. The first step consists in developing a robust activity model for strong aqueous electrolytes, here based on the Pitzer equations. As far as possible, this step should include partial dissociation processes, with standard partial thermodynamic properties of solutes described at any temperature and pressure by the HKF equations of state. Then, on top of this solid foundation, the second step consists in combining a detailed aqueous speciation approach for minor and trace elements/species implemented with a degraded Pitzer interaction model and, here again, the HKF equations of state. The degraded Pitzer model uses only the $\beta^{(0)}$ and $\beta^{(1)}$ parameters for anion-cation specific binary interactions and the $\lambda$ parameters for the neutral-ion specific binary interactions. Indeed, this approach relies on the hypothesis that minor and trace element activity coefficients are primarily affected by the specific binary interactions with the major ions that are

in the aqueous phase.

The development of HyPiT involved two in-house databases: PhreeSCALE (André et al., 2020), which is dedicated to strongly saline systems, and Thermoddem (Blanc et al., 2012; Lassin et al., 2024), which is dedicated to dilute but complex aqueous solutions up to high temperatures and pressures. The resulting HyPiT database contains a degraded Pitzer model for minor and trace elements whose specific interaction parameters $\beta^{(0)}$ and $\beta^{(1)}$ and their first derivative with respect to temperature ($d\beta^{(0)}/dT$ and $d\beta^{(1)}/dT$) are determined a priori, according to a series of recent works (Simoes et al., 2016, 2017a, 2017b). It can be used with the PhreeSCALE software (Lach et al., 2016), which is accessible on demand.

Test cases and application tests showed that the HyPiT database is promising for the modeling of water-rock-gas interactions in hot, pressurized and saline underground environments, as illustrated by the successful modeling of the solubility of calcite in NaCl brines up to 250 °C and 1450 bars (Fig. 9). The parameterized degraded Pitzer activity model can be used to efficiently describe both the speciation of trace elements in strong electrolyte solutions and the solubility of various minerals in brines.

However, various points for consideration can be highlighted.

- Preliminary tests on calculated activity coefficients have underlined that the correlations used to estimate the $\beta^{(0)}$ and $\beta^{(1)}$ interaction parameters and their first derivative with respect to temperature may not work in all cases (Fig. 3), and that the Simoes equations may need to be refined (case of the activity coefficient of trace Mg in KCl brines, or of trace Al in NaCl brines above 3 M, section 2.2.2).

- In relation to the previous item, some adjustments and optimizations of interaction parameters can be needed to really fit experimental data or the results obtained using the full Pitzer equation (e.g., case of the $Eu_2(SO_4)_3-Na_2SO_4-H_2O$ and $Eu_2(SO_4)_3-MgSO_4-H_2O$ chemical systems).

- Neutral-ion specific interaction parameter $\lambda$ is not included in the Simoes equations despite it plays a significant role as soon as neutral species are present in non-negligible amounts in the aqueous solution. In the case of the solubility of amorphous silica in mono-electrolytic brines at temperatures up to 200 °C, the binary $H_4SiO_4^0$-ion specific $\lambda$ parameter had to be estimated. It could be assumed independent of temperature for 12 electrolytes up to 150 °C (Figs. 6 and 14). Beyond, a temperature dependence must be implemented. Such considerations need to be extended to other neutral-ion interactions, in consistency with HyPiT.

- Conversely, by fixing the specific interaction parameters (estimated with the Simoes equations), the formation constants of aqueous complexes of trace elements that only form in concentrated brines can be optimized, like in the case of $FeCl_3-n$ species ($n = 0-4$) in LiCl aqueous brines (Fig. 12).

- From the computational viewpoint, the size of the HyPiT database significantly increases the CPU time of a geochemical simulation. Numerical techniques for solving geochemical calculations need to be developed to mitigate this drawback. Also, tracking non-physical interactions, like between ions stable in acidic pH solutions and ions stable in basic pH solutions, may help reduce the size of the database since there is no need for interaction parameters for such ion pairs.

The various examples presented here also highlight the fact that, despite the encouraging results of the geochemical simulations obtained with the HyPiT database, the users must keep in mind the need to check the modeled results against relevant experimental data when dealing with new specific chemical systems.

## CRediT authorship contribution statement

**Arnault Lassin:** Writing – original draft, Methodology, Investigation, Conceptualization. **Christophe Chiaberge:** Software, Data
curation. **Sylvain Guignot:** Formal analysis, Data curation. **Johann Tuduri:** Writing – review & editing, Supervision, Funding acquisition. **Laurent André:** Writing – review & editing, Validation, Methodology.

## Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

## Acknowledgements

This project has received funding from BRGM (CODES project), from the European Union's Horizon 2020 research and innovation programme under grant agreement no. 850626 (REFLECT project), and from the French National Agency for Research PRC programme under the grant agreement no. ANR-24-CE49-6543 (MagBrines project).

## Appendix A. Supplementary data

Supplementary data to this article can be found online at https://doi.org/10.1016/j.apgeochem.2026.106727.

## Data availability

The HyPiT database can be downloaded as supplementary material and the associated PhreeSCALE software can be obtained on demand

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