# A New Density Model of Quartz Solubility in H2O-CO2-NaCl Ternary Systems up to High Temperatures and High Pressures

Authors: Weiping Deng; Qing Wei; Xuan Liu
Journal: Geofluids
 (2021)

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## Abstract

A novel density model for computing quartz solubility in H2O-CO2-NaCl hydrothermal fluids applicable to wide ranges of temperature and pressure is proposed. Based on the models of Akinfiev and Diamond (2009) and Wei et al. (2012), the effective partial molar volume of water (VH2O∗) is replaced by the partial molar volume of water (V¯H2O) by implementing an empirical correction, and water molar fraction (xH2O) is modified with water activity (aH2O), in addition to a series of changes to the model coefficient forms. The absolute values of averaged relative deviation of this model compared to the experimental data sets in pure water, H2O-CO2, and H2O-NaCl solutions are 5.74%, 6.69%, and 7.09%, respectively, which are better than existing models in the literature. The model can be reliably used for computing quartz solubilities in pure water from 0°C to 1000°C, from 0 bar to 20,000 bar, and in CO2- and/or NaCl-bearing solutions from 0°C to 1000°C, from 0 bar to 10,000 bar (with slightly lower accuracy at 5000-10,000 bar in H2O-NaCl systems) in the single liquid region. Moreover, the trends and overall ranges of this model may probably be more accurate in the H2O-CO2-NaCl fluid mixtures compared to the limited experimental data. In addition, a bisection algorithm for deriving the isopleths of quartz solubilities based on this new model is first proposed, and application perspectives are discussed for various geologic settings including subduction zone, lower crust-upper mantle, migmatite, pegmatite, porphyry, and orogenic deposits.


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## Body

## 1. Introduction

Quartz is one of the most common silicate minerals in the Earth’s crust. Its solubility behavior in hydrothermal fluids bears fundamental fingerprints on geochemical and petrological processes that govern the formation and evolution of the Earth [1]. For instance, a complete body of knowledge on quartz dissolution and precipitation in crustal fluids is critical for understanding ore-forming processes [2–4]. Particularly, combined with the scanning electron microscope- (SEM-) cathodoluminescence (CL) petrographic technique, formation of quartz veins and veinlets in mineral ore deposits have been intensively studied and better understood due to progressive improvement of solubility models [5–8].In general, there are two means of deriving quartz solubilities in aqueous fluids, i.e., by hydrothermal experiments and by formulating numerical models. Starting with Kennedy [9], early experiments [10–18] were carried out in pure water and reviewed by Walther and Helgeson [19]. Since the electrolyte (e.g., NaCl) and the nonpolar gas CO2 are very common in various hydrothermal environments and may also play important roles in quartz solubilities [20–24], later the experiment were extended to H2O-NaCl [20, 21, 24–28], H2O-CO2 [20, 21, 26, 29], and H2O-CO2-NaCl [22] fluid mixtures. Meanwhile, higher temperature-pressure (T-P) conditions in pure water have been attempted [23, 30–32].In numerical modeling, there are mainly two types of models: one is “electrostatic model” accounting for the electrostatic interactions between aqueous species and surrounding H2O molecules [19, 33, 34]; the other is “density model” which is based on the empirical observations and shown as the linear relationship between the decadic logarithm of quartz solubility and the density of pure water [2, 30, 35–38]. Due to a lack of knowledge on water dielectric constants at high pressures, electrostatic models are commonly limited pressures less than 5000 bar. Absence of such pressure limitation for density models in addition to their simplicity makes them more powerful and attractive [35–37].To some extents, the quartz solubilities in hydrothermal fluids and as such them isopleths control the transport and deposition of silica in association with metals and sulfides. Monecke et al. [39, 40] and Li et al. [41] have constructed the quartz isopleths to illustrate the formation of copper- and gold-bearing quartz vein(let)s in porphyry Cu and orogenic Au deposits by calculating large numbers of quartz solubilities along a closely spaced isochores and importing them into the graphing and data analysis software Origin. For one thing, the previous density models for computing quartz solubility either have narrow applicability in fluid systems (such as the model of Manning [30] which is only suitable for pure water and the models of Fournier [2] and Brooks and Steele-MacInnis [38] which are only valid for saline aqueous fluids), limited accuracies in aqueous NaCl and/or CO2 solutions (such as the models of Akinfiev and Diamond [35] and Wei et al. [36] which overestimate quartz solubilities in NaCl-bearing solutions and underestimate quartz solubilities in CO2-bearing solutions), or yield inconsistent results for H2O-CO2-NaCl solutions (model prediction of Shi et al. [37] shows quartz solubility has enhanced follows the gradually adding of CO2 with the same NaCl contents). For another, a convenient and practical method for quartz solubility isopleths is needful. So, a new accurate density model for the solubilities of quartz in H2O-CO2-NaCl fluid mixtures and a simple generic algorithm to get quartz solubility isopleths are necessitated.The objective of this work is to develop a density model to calculate quartz solubility in H2O-CO2-NaCl hydrothermal solutions cover a wide temperature-pressure-composition (T-P-x) range at higher accuracy. Furthermore, an algorithm for acquiring quartz solubility isopleths based on this new model is designed to facilitate applicability. The model here is based on the models of Akinfiev and Diamond [35] and Wei et al. [36] with three revises: introduction of the partial molar volume of water (V¯H2O) for the effective partial molar volume of water (VH2O∗), replacement of water molar fraction (xH2O) by water activity (aH2O), and coefficient form modifications. Detailed modeling process, algorithm on acquiring isopleths, and an applicable case study on quartz solubility isopleths are proposed in the following text.


## 2. Density Model of Quartz Solubility in H2O-CO2-NaCl Solutions

### 2.1. Existing Models

Compared to the experimental ways of knowing quartz solubilities, the numerical models have broader range of application. Early efforts on density modeling were mostly focused on pure water [9, 12, 14, 18, 19, 23, 30, 42–44]. They were established mainly on empirical observations on the logarithm linear relationship between quartz solubility and the density of pure water at constant temperatures (Figure 1). With the accumulation of experimental data, density models have been expanded to gas- and/or salt-bearing solutions. The first of these density models which could be applied to H2O-CO2-NaCl ternary system was proposed by Shmulovich et al. [22]. Later, Akinfiev and Diamond [35] developed a simple predictive model (thereafter abbreviated as AD 2009 model) based on calculating the effective partial molar volume of water (VH2O∗). With a new form of hydration number, Wei et al. [36] improved the accuracy of AD 2009 model, especially when applied to H2O-NaCl system at high temperatures, pressures, and high NaCl concentrations and proposed a revised model (thereafter abbreviated as WDM 2012 model). Just recently, an accurate density-based model for quartz solubility in aqueous NaCl and/or CO2 solutions was established by Shi and coworkers (thereafter abbreviated as SMH 2019 model [37]). However, the accuracy of SMH 2019 model in H2O-CO2-NaCl fluid mixtures can be further improved as is demonstrated later in Section 4.2.Figure 1
LogmSiO2 versus logρH2O for experimental studies and model predictions of quartz solubility at isothermal conditions in pure water. Square, triangle, star, and diamond represent experimental data, and the asterisk is the critical point of H2O. The curve refers to the saturation vapor pressure of H2O, which is calculated from Wagner and Pruß [45], and the solid lines are logmSiO2 calculated from this model.All the density models for quartz solubility were based on its dissolution and precipitation reaction as the following expression:(1)SiO2quartz+n⋅H2O⇌SiO2⋅H2Onaq.Here,n represents the average number of water molecules complexed with one SiO2quartz molecule in an aqueous solution, and SiO2⋅H2Onaq stands for the aqueous silica abbreviated as SiO2aq. For pure quartz, if adopting standard states of unit activity at any temperatures and pressures, the equilibrium constant (K) for the above reaction is as follows:
(2)logK1=logaSiO2aq−n⋅logaH2O,where aSiO2aq is the activity of aqueous silica and aH2O is the activity of pure water. As an approximation, assuming the ideal behavior of SiO2aq, Eq. (2) could be changed to the following form:
(3)logmSiO2aq=logK1+n⋅logaH2O,where mSiO2aq is the molality of SiO2aq. Mainly on the empirical observations, more than half a century ago, scientists have noticed that the logarithm of equilibrium constant (logK) for the ionization of water and a variety of aqueous species are linear in the logarithm of the density of pure water (logρH2O) as Figure 1 depicts [30, 46–49], and the correlation for the ionization of H2O up to 1000°C and 10,000 bar was first proposed by Marshall and Franck [50] with the following equation:
(4)logK=a+bT+cT2+dT3+e+fT+gT2logρH2O,where a to g are the regressed constants and T is temperature in Kelvin (note: all the temperatures in the following equations are in Kelvins unless a description).Adopting the rapid-quench method, Manning [30] doubled the pressure range which mSiO2aq experimental measured and combined the measurements of Hemley et al. [51] and Walther and Orville [29], and he regressed the constants a through g for SiO2aq and proposed a successful density model (thereafter abbreviated as M 1994 model) for quartz solubility computing in pure water. Because of its simple form and accuracy, Eq. (4) became a successful density model for calculating quartz solubility in pure water over a wide range of temperatures and pressures (from 25°C and 1 bar to 1000°C and 10,000 bar).With the activity of a constituent relative to unit activity in an arbitrary standard state, Fournier [2] proposed a method to calculate quartz solubility in aqueous NaCl solutions by modifying his solubility model in pure water. As quartz dissolved in salt-bearing solutions with the form of reaction (1), Fournier [2] started from Eq. (3) and gave his model (thereafter abbreviated as F 1983 model) to compute quartz solubility in NaCl solutions:
(5)logmSiO2aq=logmSiO2aqo+nlogρeρeo,where mSiO2aqand mSiO2aqo are silica solubility in NaCl solutions and in pure water, respectively. In Eq. (5), ρe and ρeo are defined as the effective density of “free water” which are not tightly bound to hydrations in NaCl solutions and in pure water (Stokes and Robinson [52]).Adopted the same dissolution form of Eq. (1) and built on the models of F 1983 and M 1994, Akinfiev and Diamond [35] first proposed a density model (AD 2009 model) to calculate quartz solubility in water-salt-CO2 systems over a broad T-P conditions with the following form:
(6)logmSiO2aq=AT+BT⋅logMH2OVH2O∗+2logxH2O,where AT and BT are a+b/T+c/T2+d/T3 and e+f/T+g/T2 of the Eq. (4), MH2O=18.0152gmol−1 is the molar mass of water, and xH2O denotes the mole fraction of water in the mixed fluid. The AD 2009 model introduced a new parameter of “effective partial molar volume of water” (VH2O∗), which is obtained from the relationship below.
(7)Vmixmol=xH2OVH2O∗+xNaClVNaCl+xCO2VCO2,where Vmixmol is the molar volume of fluid mixture and xNaCl and xCO2 are the molar fractions of the solute NaCl and CO2. VNaCl and VCO2 are intrinsic volumes of NaCl and CO2, which are the only empirical parameters of AD 2009 model and are derived from experimental data sets of quartz solubilities in H2O-NaCl and H2O-CO2 systems, respectively.Although AD 2009 model was very successful, however across the whole temperature and pressure spectrum of their model, the solvation number was simply fixed to 2, and this might be the reason for their model make overestimations in H2O-NaCl mixed fluids over 600°C and 9000 bar conditions (their Figure 5 of Akinfiev and Diamond [35]).Based on the in situ Raman spectroscopic measurements of aqueous silica in equilibrium with the solid quartz, it was confirmed that the silica speciation depends strongly on temperatures and pressures (Zotov and Keppler [53]). So, the model of Wei et al. [36] was based on the master equation of AD 2009 model but changing the solvation number of aqueous silica complex into a temperature-dependent polynomial CT and obtained quartz solubility in H2O-CO2-NaCl ternary hydrothermal solutions. It expressed as the following equation:
(8)logmSiO2aq=AT+BT⋅logMH2OVH2O∗+CTlogxH2O.Accordingly, WDM 2012 model revised the overestimates of AD 2009 model in low-moderate NaCl-bearing solutions (xNaCl<0.3), but it is subject to underestimate the molality of aqueous silica in high-salinity solutions compared to the experimental data sets (their Figure 5 of Wei et al. [36]).Recently, based on the pressure-volume-temperature-composition (PVTx) model of Mao et al. [54] and a systematically review of experimental data sets of quartz solubilities, an improved density model for quartz solubility was derived by Shi et al. [37] (SMH 2019 model) as the relationship below:
(9)logmSiO2aq=logK+nlogρsolF+ϕSiO2−H2O,where ρsol and F are the density of solution and the mass fraction of water, respectively, and ϕSiO2−H2O is the ratio of the activity coefficient of water corresponding to their concentrations with the power of hydration number and the activity coefficient of aqueous SiO2. Although the SMH 2019 model shows excellent performance in pure water and the binary fluid system, its prediction in aqueous NaCl- and CO2-bearing solutions still lacks versatile evaluation, which will be discussed below.


### 2.2. New Density Model and Its Parameterization

The first step of this modelling work is a systematic and comprehensive review of existing experimental data of quartz solubility. As early as 1935, researchers had carried out experimental studies and those measurements had covered H2O, H2O-CO2, H2O-NaCl, and H2O-CO2-NaCl hydrothermal solutions (Table 1). In constant regression, we use parts of experimental data (data with superscripts in Table 1), and others were either some of what unreliable or used as the testing of model prediction. The select regulation of experimental data for regression is basically like the former researchers discussed, and all the detailed information of data selection is listed in our Supplementary A. (Manning [30]; Shi et al. [37]; Fournier and Potter II [43]).Table 1
Experimental measurements for quartz solubility.
Pure H2OReferencesT (K)P (bar)NdVan Nieuwenburg and Van Zon [55]653.15-698.15289.2-490.218Kennedy [9]●433.15-883.156.2-1750106Morey and Hesselgesser [10]●633.15-873.1533-200020Fournier [56]438.15-513.157-33.53Kitahara [11]473.15-643.1515.5-210.48Van Lier et al. [12]∗333.15-373.150.2-1.05Morey et al. [13]●318.15-573.150.29-1013.351Siever [14]●398.15-454.652.32-10.416Weill and Fyfe [15]673.15-898.151000-400048Anderson and Burnham [16]773.15-1173.151000-985079Anderson and Burnham [17]∗973.1540002Crerar and Anderson [18]●452.15-602.159.8-126.931Novgorodov [26]973.1515001Hemley et al. [51]473.15-773.1515.5-200018Fournier et al. [27]∗623.15165-5005Anderson and Burnham [57]773.15-973.152000-841029Ragnarsdottr and Walther [58]523.15250-10007Walther and Orville [29]●623.15-823.151000-200020Archer [59]473.15-593.1561-100016Xie and Walther [28]685.15-846.152000-20057Manning [30]●773.15-1173.155000-20,00052Rimstidt [60]●294.15-369.150.02-0.8810Newton and Manning [20]●873.15-1123.152000-10,0008Shmulovich et al. [21]●773.15-1173.155000-90007Wang et al. [61]399.15-643.15800-610084Shmulovich et al. [22]●673.15-1073.151000-900019Newton and Manning [31]1023.15-1403.1510,0007Manning et al. [32]573.15-908.1510,0009Hunt and Manning [23] ∗1173.15-1323.1515,000-20,0005H2O-CO2 mixtureReferencesT (K)P (bar)xCO2NdNovgorodov [26]973.1515000.05-0.494Walther and Orville [29]∗873.1520000.023-0.317Newton and Manning [20]∗1073.1510,0000.19-0.685Shmulovich et al. [21]∗1073.1590000.16-0.505Shmulovich et al. [22]∗773.15-1073.152000-90000.06-0.5026Newton and Manning [62]∗1073.1510,0000.08-0.9314H2O-NaCl mixtureReferencesT (K)P (bar)XNaClNdVan Lier et al. [12]∗333.151.00.000211Anderson and Burnham [17]873.1530000.0161Ganeyev [25]∗523.15-773.1510000.03-0.0817Novgorodov [26]973.1515000.02-0.135Hemley et al. [51]∗473.15-673.1510000.009-0.0676Fournier et al. [27]∗623.15180-5250.035-0.06735Von Damm et al. [44]∗472.15-726.55192-10330.009100Xie and Walther [28]∗675.15-868.151725-18000.00189Newton and Manning [20]∗773.15-1173.152000-15,0000.06-0.7650Shmulovich et al. [21]∗773.15-1073.155000-90000.02-0.5514Shmulovich et al. [22]∗673.15-1073.151000-90000.004-0.5544Newton and Manning [63]∗1073.1510,0000.171Foustoukos and Seyfried Jr [64]∗638.15-703.15219-3780.0002-0.0192Cruz and Manning [24]∗1173.15-1373.1515,000-20,0000.05-0.527Scheuermann et al. [65]693.15310-3250.027-0.0464H2O-CO2-NaCl mixtureReferenceT (K)P (bar)xCO2XNaclNdShmulovich et al. [22]1073.1550000.07-0.300.0036-0.2514Note:Nd means the number of measurements. Superscripts ∗ and ● denote whole or parts of the experimental data sets which involved in constants regressions, respectively.Then, this model is also built on the equilibrium reaction of Eq. (1) and assumes ideal behavior for SiO2aq (Eq. (3)), as the empirical expression of the logarithm of the equilibrium constant for aqueous silica is linear with the logarithm of the density of pure water (Eq. (4)) written as follows:
(10)logmSiO2aq=AT,P+BT,P⋅logMH2OV¯H2O⋅ϕT,P,xNaCl,xCO2+nT,P,xNaCl,xCO2⋅loga,where V¯H2O is the partial molar volume of water, MH2O/V¯H2O refers to the density of pure water in H2O-CO2-NaCl ternary fluid mixtures, and ϕT,P,xNaCl,xCO2 is an empirical parameter on T, P, xNaCl, and xCO2 dependent, which corrects the deviation of the density of “pure water” and the effective density of “free water” (water not tightly bound as water of hydration) as introduced by Fournier [2]. AT,P and BT,P are T-P dependent polynomials in Kelvins and bars, respectively (note: all the pressures in the following equations are in bars unless a description), and nT,P,xNaCl,xCO2 is the hydration number and as an empirical polynomial expressed with the parameters of T, P, xNaCl, and xCO2. aH2O is the activity of pure water in H2O-CO2-NaCl solutions which is calculated from the model of Dubacq et al. [66].In this model, the polynomials ofAT,P,BT,P, ϕT,P,xNaCl,xCO2, and nT,P,xNaCl,xCO2 are assumed as follows:
(11)AT,P=A0+A1T+A2T2+A3T3+A4T4+A5P+A6P2+A7P3,(12)BT,P=B0+B1T+B2T2+B3logP+B4logP2,(13)ϕT,P,xNaCl,xCO2=10C0+C1T+C2T2+C3P+C4P2+C5xNaCl+C6xNaCl2xNaCl+D0+D1T+D2P+D3P2+D4xCO2xCO2+F0xNaClxCO22,(14)nT,P,xNaCl,xCO2=2+N00+N01T+N02T2+N03xNaCl2xNaCl+N10+N11P+N12xCO2+N13xCO22xCO2.By adopting the regressed constantsA0 to A7 through B0 to B4 in pure water and by combining the constants N00 to N03, N10 to N13, C0 to C6, D0 to D4, and F0 based on regressing quartz solubility data in H2O-NaCl, H2O-CO2, and H2O-CO2-NaCl solutions, the regressed constants of this model are given in Table 2.Table 2
Regressed constants for Eq. (10).
AT,PValuesBT,PValuesΦT,P,xNaCl,xCO2ValuesnT,P,xNaCl,xCO2ValuesA0-9.52462828E+00B07.92625728E+00C04.83288536E-01N001.76415897E+01A12.91941547E-02B1-1.53823138E-02C1-1.32770018E-04N01-3.05577769E-02A2-4.00569452E-05B29.14289810E-06C21.12520228E-07N021.25299576E-05A32.71067438E-08B34.03311251E-01C34.07784959E-05N03-1.99999983E+00A4-6.73901359E-12B4-6.91374075E-02C4-3.13938041E-09N10-1.15563053E+00A5-1.86478387E-06C5-9.00618419E-01N11-3.69288750E-05A64.10234340E-10C69.50723453E-01N125.56872025E+00A7-1.16951479E-14D0-1.72890706E+00N13-4.03999992E+00D11.41633612E-03D21.43379216E-04D3-7.23903865E-09D42.32574082E+00F06.99791510E+00For one molar homogenous system which is composed of H2O, NaCl, and CO2, the extensive property of the molar volume of H2O-CO2-NaCl mixture solution Vmix could be expressed as the following equation:
(15)Vmix=V¯H2O⋅xH2O+V¯NaCl⋅xNaCl+V¯CO2⋅xCO2,where V¯H2O, V¯NaCl, and V¯CO2 defined as the “partial molar volume” of H2O, NaCl, and CO2, respectively. At the same time, Vmix also can be obtained by the function of Mao et al. [54] as the following form:
(16)Vmix=VmixT,P,xNaCl,xCO2.Based on the definition of the partial molar volume,V¯NaCl and V¯CO2 at constant T and P can be further written as follows:
(17)V¯NaCl=∂Vmix∂xNaClT,P,(18)V¯CO2=∂Vmix∂xCO2T,P.IfVmix is continuous differentiable in a closed interval of [xi, xi+Δh] (xi means the molar fraction of NaCl or CO2 solute at any fixed T and P conditions, and Δh is an incremental change of the molar fraction of NaCl or CO2 solute), according to the Taylor expansion of Vmix, it reduces to
(19)Vmixxi+Δh=Vmixxi+ΔhdVmixxidxi+Δh22d2Vmixcdxi2,where c is a constant and between xi and xi+Δh, and according to the finite difference method (FDM), Eq. (19) can be transformed into
(20)dVmixxidxi=Vmixxi+Δh−VmixxiΔh−oΔh,where oΔh is the equivalence infinitesimal of Δh and equals to Δh/2d2Vmixc/dxi2. In our calculation process when Δh/x values 10-4, the term of dVmixxi/dxi, namely, V¯NaCl or V¯CO2 is solved out.As Shi and coworkers [37] discussed, when the SMH 2019 model applied to H2O-CO2-NaCl solutions, the part of ϕNaCl,CO2r which accounts for the nonadditivity of the contributions of NaCl and CO2 to ϕSiO2−H2O is very small and can be neglected in most cases, so in this model, V¯NaCl and V¯CO2 in Eq. (17) and Eq. (18) could be solved out with FDM in H2O-NaCl and H2O-CO2 systems, respectively, and the value of V¯H2O in Eq. (15) will then be calculated.


### 2.3. Accuracy Test of This Model

The accuracy of this new model is compared with the most competitive three ones, i.e., the AD 2009 model, WDM 2012, model and SMH 2019 model by investigating the absolute values of averaged relative deviations (AARDs) (listed in Table3) from experimental data over a broad T-P-x spectrum. For quartz solubilities in H2O, H2O-CO2, and H2O-NaCl solutions, the overall AARDs of this model are 5.74%, 6.69%, and 7.09%, respectively. Although inferior to the SMH 2019 model in aqueous NaCl solutions, the performance of this model has advantages in aqueous CO2 fluids. In addition, this model has made an obvious progress in pure water and NaCl- and/or CO2-bearing solutions which compared to the AD 2009 model and the WDM 2012 model.Table 3
Deviations of calculated values on quartz solubility.
ReferencesNd1AARD/%2AARD/%3AARD/%4AARD/%Pure H2OKennedy [9]●794.955.127.466.29Morey and Hesselgesser [10]●105.665.462.563.39Kitahara [11]66.376.449.518.92Van Lier et al. [12]∗54.892.8425.8621.66Morey et al. [13]●324.9810.9625.3521.90Siever [14]●106.157.4716.8813.01Weill and Fyfe [15]374.273.674.755.69Anderson and Burnham [16]799.045.7111.3113.46Anderson and Burnham [17]∗24.276.2114.8117.13Crerar and Anderson [18]●157.269.9713.7311.04Hemley et al. [51]184.133.6710.6210.70Fournier et al. [27]∗52.002.488.133.02Anderson and Burnham [57]87.807.339.6611.44Ragnarsdottr and Walther [58]47.755.328.525.09Walther and Orville [29]●154.624.276.415.74Archer [59]138.0830.0735.9133.85Xie and Walther [28]∗610.6712.0713.6813.87Manning [30]●525.326.625.706.80Rimstidt [60]●105.3719.4827.4423.24Newton and Manning [20]●85.295.469.7411.13Shmulovich et al. [21]●56.799.586.648.29Shmulovich et al. [22]●153.964.233.503.91Newton and Manning [31]32.6217.1317.1617.03Hunt and Manning [23]∗45.5524.729.4628.19Overall AARD/%5.749.0113.5312.70H2O-CO2 mixtureWalther and Orville [29]∗52.042.976.537.72Newton and Manning [20]∗47.119.4914.8515.21Shmulovich et al. [21]∗57.196.5216.4117.76Shmulovich et al. [22]∗209.2217.1810.5215.64Newton and Manning [62]∗97.9011.439.3828.62Overall AARD/%6.699.5211.5416.99H2O-NaCl mixtureVan Lier et al. [12]∗78.0114.4833.2929.38Anderson and Burnham [17]13.225.7713.3015.36Ganeyev [25]∗1014.9015.8816.7120.68Novgorodov [26]59.849.1114.7515.19Hemley et al. [51]∗35.542.088.538.34Fournier et al. [27]∗351.381.583.633.66Von Damm et al. [44]∗1002.343.406.775.70Xie and Walther [28]∗910.7510.5013.5215.04Newton and Manning [20]∗456.594.6111.2319.82Shmulovich et al. [21]∗126.866.449.4010.50Shmulovich et al., [22]∗407.735.779.148.28Foustoukos and Seyfried Jr [64]∗754.313.014.673.75Cruz and Manning [24]∗1411.9012.0455.3919.13Scheuermann et al. [65]45.902.843.694.38Overall AARD/%7.096.9714.5712.80Notes:Nd is the number of experimental data points; superscripts ∗ and ● denote whole or parts of the experimental data sets which involved in constant regressions, respectively; AARD is the absolute value of averaged relative deviation calculated from models; superscripts 1 to 4 represent the models of this one, the SMH 2019, the WDM 2012, and the AD 2009, respectively.The AARD of this model in pure water is 5.74%, only <50% of that of WDM 2012 and AD 2009, and is smaller than that of the SMH 2019. As can be seen from Figure2(a), this model can predict the dissolution and precipitation behavior of quartz in pure water over broad T-P conditions (up to 1000°C and 20,000 bar). Two points should be emphasized: first, this model reproduced quartz retrograde phenomenon (quartz increases with decreasing temperature) near the critical point of water at low pressures (200 bar and 500 bar) much more accurate compared to other models discussed here (Figure 2(b)); second, except for the data sets of Manning [30] and parts of Shmulovich et al. [22], the solid curve in Figure 2(c) presented the predictability of this model (for those other data sets were not used in constants regression) and which can be seen that this model had better accuracy compared to other models especially in high pressure conditions (Figures 2(c) and 2(d)).Figure 2
Model predictions of quartz solubility in pure water and the comparisons between different model computes and experimental data sets. Symbols represent experimental data, and curves are the predictions of different models.
(a)(b)(c)(d)Comparisons of different models computing and experimental data sets in H2O-CO2 solutions are shown in Table 3 and Figures 3(a) and 3(b). Generally, the calculated results of this model matched the experimental data quite well, and the overall AARD of this model is 6.69% which is about 50% better than that of WDM 2012 and AD 2009 and about 30% better than the model of SMH 2019. As Figure 3(a) depicts, at 500°C, 2000 bar, both WDM 2012 and AD 2009 models underestimate quartz solubility slightly, while SMH 2019 overestimates. The WDM 2012 and AD 2009 models underestimate the quartz solubility at high temperature and moderate pressure at 800°C and 5000 bar, whereas the SMH 2019 model overestimates (note: the circle points in Figure 3(b) are the experimental data at 10,000 bar condition). Though the data themselves are very scattered at 800°C, 5000 bar condition, compared to other models, this model seems better fit. When xCO2 is higher than 0.5 at 800°C, 10,000 bar condition, the WDM 2012 and AD 2009 models underestimate quartz solubility, but the SMH 2019 model overestimates although the data sets of Newton and Manning [62] are as the extrapolation testing for their model (Figure 3(b)). Overall, our model seems to have the best performance under the conditions as Figures 3(a) and 3(b) depict.Figure 3
Model predictions of quartz solubility in (a, b) aqueous CO2 solutions, in (c–e) aqueous NaCl solutions, and in (f) CO2- and NaCl-bearing solutions. Different curves denote comparisons between four model computes and experimental data sets. Symbols represent experimental data, and curves are the predictions of different models.
(a)(b)(c)(d)(e)(f)Figures3(c)–3(e) show that this model can reproduce quartz solubility well in aqueous H2O-NaCl solutions and is in good agreement with experimental data sets at most temperature and pressure conditions. After excluding some experimental data sets which inconsistent with themselves or having systematic errors (such as some data of Newton and Manning [20] and Cruz and Manning [24]), the overall AARD of this model is 7.09% (Table 3), which has made big improvement compared to the WDM 2012 model and the AD 2009 model, but little inferior than the SMH 2019 model. At 700°C, 2000 bar, compared to the experimental data points of Newton and Manning [20], this model better reproduced the “salting-in” effect (quartz solubility is enhanced with NaCl content) whereas other models all overestimate as Figure 3(c) depicts. At 800°C, 5000 bar high temperature and moderate pressure condition, the accuracy of this model is almost the same as that of the model SMH 2019 and is obviously better than those of the WDM 2012 and the AD 2009 (Figure 3(d)). As Figure 3(e) depicts at 900°C, 10,000 bar condition, this model overestimates quartz solubility in low NaCl-bearing system whereas gives accurate result in higher xNaCl condition when the pressure is much beyond the applicable limitation of the equation of state (EOS) for Vmix calculation from Mao et al. [54]. However, when considering the quartz solubility in pure water at the same T, P condition (as the circle point by Manning [30] in Figure 3(e)), this overestimation may not be that “over,” and the predictions by models of SMH 2019, WDM 2012, and AD 2009 may seem a litter lower.Till now, only 14 experimental data points of quartz solubility have reported in H2O-CO2-NaCl ternary solution by the work of Shmulovich et al. [22] (at xH2O=0.7). As seen from Figure 3(f), although the data sets are fairly scattered covering a wide spectrum of xCO2 and xNaCl condition from 0 to 0.3, however all model computing results are between them (after deleting some data points with obvious discrepancy as the open squares in their Figure 6 of Shi et al. [37]). Moreover, only this model can approach both two end-member systems of H2O-CO2 and H2O-NaCl solutions on quartz solubility reproducing. Certainly, this model invites stricter testing by future experimental data sets.There still some points worth of mentioning for the new model. Firstly, the success of this density model may be derived from the highly accurate EOS forVmix and aH2O computing and more physically meaning parameters the model having. Secondly, only parts of experimental data sets in pure water (the data with superscriptions in Table 1) were used in the constant regression, so the computing result of the solid curve in Figure 2(c) can represent the prediction of this model to some extent. In addition, although the form of the solvation number n in our model is empirical, it may have a physical explanation of assigning it to fixed 2 in pure water (the fixed value of n is for simplistic in model computing for the contribution from aH2O on quartz solubility will always be zero in pure water, and it could be seemed as the hydration number only when temperatures and pressures are below 600°C and 6000-8000 bar conditions) or depending on T, P, xNaCl, and/or xCO2 conditions in NaCl and/or CO2-bearing solutions which were verified from in situ Raman spectroscopic study on silica speciation in aqueous fluids depends strongly on T, P and the study on silica hydration changing with the fluid adding of NaCl or CO2 species [35, 53]. Besides, we think the form of n in our model may be an empirical way to get hydration number on quartz dissolution in NaCl and/or CO2-bearing fluids but should be carefully treated and tested by more studies on silica molecule structures. Lastly, this model can extrapolate to high temperatures and high pressures within or approaching experimental accuracy (for instance, the absolute value of averaged experimental error from the experimental data of Newton and Manning [31] which is carried in H2O at 1223.15 K and 10,000 bar condition is 2.91%, and the absolute value of averaged relative deviation calculated from this model is 2.93%), for those conditions are beyond the T-P validation of the EOS computing on Vmix obtained from Mao et al. [54] (validated from 273 to 1273 K, from 1 to 5000 bar for the H2O-NaCl system, from 273 to 1273 K, and from 1 to 10,000 bar for the H2O-CO2 system) and aH2O obtained from Dubacq et al. [66] (validated from 283 to 653 K and from 1 to 3500 bar).


## 2.1. Existing Models

Compared to the experimental ways of knowing quartz solubilities, the numerical models have broader range of application. Early efforts on density modeling were mostly focused on pure water [9, 12, 14, 18, 19, 23, 30, 42–44]. They were established mainly on empirical observations on the logarithm linear relationship between quartz solubility and the density of pure water at constant temperatures (Figure 1). With the accumulation of experimental data, density models have been expanded to gas- and/or salt-bearing solutions. The first of these density models which could be applied to H2O-CO2-NaCl ternary system was proposed by Shmulovich et al. [22]. Later, Akinfiev and Diamond [35] developed a simple predictive model (thereafter abbreviated as AD 2009 model) based on calculating the effective partial molar volume of water (VH2O∗). With a new form of hydration number, Wei et al. [36] improved the accuracy of AD 2009 model, especially when applied to H2O-NaCl system at high temperatures, pressures, and high NaCl concentrations and proposed a revised model (thereafter abbreviated as WDM 2012 model). Just recently, an accurate density-based model for quartz solubility in aqueous NaCl and/or CO2 solutions was established by Shi and coworkers (thereafter abbreviated as SMH 2019 model [37]). However, the accuracy of SMH 2019 model in H2O-CO2-NaCl fluid mixtures can be further improved as is demonstrated later in Section 4.2.Figure 1
LogmSiO2 versus logρH2O for experimental studies and model predictions of quartz solubility at isothermal conditions in pure water. Square, triangle, star, and diamond represent experimental data, and the asterisk is the critical point of H2O. The curve refers to the saturation vapor pressure of H2O, which is calculated from Wagner and Pruß [45], and the solid lines are logmSiO2 calculated from this model.All the density models for quartz solubility were based on its dissolution and precipitation reaction as the following expression:(1)SiO2quartz+n⋅H2O⇌SiO2⋅H2Onaq.Here,n represents the average number of water molecules complexed with one SiO2quartz molecule in an aqueous solution, and SiO2⋅H2Onaq stands for the aqueous silica abbreviated as SiO2aq. For pure quartz, if adopting standard states of unit activity at any temperatures and pressures, the equilibrium constant (K) for the above reaction is as follows:
(2)logK1=logaSiO2aq−n⋅logaH2O,where aSiO2aq is the activity of aqueous silica and aH2O is the activity of pure water. As an approximation, assuming the ideal behavior of SiO2aq, Eq. (2) could be changed to the following form:
(3)logmSiO2aq=logK1+n⋅logaH2O,where mSiO2aq is the molality of SiO2aq. Mainly on the empirical observations, more than half a century ago, scientists have noticed that the logarithm of equilibrium constant (logK) for the ionization of water and a variety of aqueous species are linear in the logarithm of the density of pure water (logρH2O) as Figure 1 depicts [30, 46–49], and the correlation for the ionization of H2O up to 1000°C and 10,000 bar was first proposed by Marshall and Franck [50] with the following equation:
(4)logK=a+bT+cT2+dT3+e+fT+gT2logρH2O,where a to g are the regressed constants and T is temperature in Kelvin (note: all the temperatures in the following equations are in Kelvins unless a description).Adopting the rapid-quench method, Manning [30] doubled the pressure range which mSiO2aq experimental measured and combined the measurements of Hemley et al. [51] and Walther and Orville [29], and he regressed the constants a through g for SiO2aq and proposed a successful density model (thereafter abbreviated as M 1994 model) for quartz solubility computing in pure water. Because of its simple form and accuracy, Eq. (4) became a successful density model for calculating quartz solubility in pure water over a wide range of temperatures and pressures (from 25°C and 1 bar to 1000°C and 10,000 bar).With the activity of a constituent relative to unit activity in an arbitrary standard state, Fournier [2] proposed a method to calculate quartz solubility in aqueous NaCl solutions by modifying his solubility model in pure water. As quartz dissolved in salt-bearing solutions with the form of reaction (1), Fournier [2] started from Eq. (3) and gave his model (thereafter abbreviated as F 1983 model) to compute quartz solubility in NaCl solutions:
(5)logmSiO2aq=logmSiO2aqo+nlogρeρeo,where mSiO2aqand mSiO2aqo are silica solubility in NaCl solutions and in pure water, respectively. In Eq. (5), ρe and ρeo are defined as the effective density of “free water” which are not tightly bound to hydrations in NaCl solutions and in pure water (Stokes and Robinson [52]).Adopted the same dissolution form of Eq. (1) and built on the models of F 1983 and M 1994, Akinfiev and Diamond [35] first proposed a density model (AD 2009 model) to calculate quartz solubility in water-salt-CO2 systems over a broad T-P conditions with the following form:
(6)logmSiO2aq=AT+BT⋅logMH2OVH2O∗+2logxH2O,where AT and BT are a+b/T+c/T2+d/T3 and e+f/T+g/T2 of the Eq. (4), MH2O=18.0152gmol−1 is the molar mass of water, and xH2O denotes the mole fraction of water in the mixed fluid. The AD 2009 model introduced a new parameter of “effective partial molar volume of water” (VH2O∗), which is obtained from the relationship below.
(7)Vmixmol=xH2OVH2O∗+xNaClVNaCl+xCO2VCO2,where Vmixmol is the molar volume of fluid mixture and xNaCl and xCO2 are the molar fractions of the solute NaCl and CO2. VNaCl and VCO2 are intrinsic volumes of NaCl and CO2, which are the only empirical parameters of AD 2009 model and are derived from experimental data sets of quartz solubilities in H2O-NaCl and H2O-CO2 systems, respectively.Although AD 2009 model was very successful, however across the whole temperature and pressure spectrum of their model, the solvation number was simply fixed to 2, and this might be the reason for their model make overestimations in H2O-NaCl mixed fluids over 600°C and 9000 bar conditions (their Figure 5 of Akinfiev and Diamond [35]).Based on the in situ Raman spectroscopic measurements of aqueous silica in equilibrium with the solid quartz, it was confirmed that the silica speciation depends strongly on temperatures and pressures (Zotov and Keppler [53]). So, the model of Wei et al. [36] was based on the master equation of AD 2009 model but changing the solvation number of aqueous silica complex into a temperature-dependent polynomial CT and obtained quartz solubility in H2O-CO2-NaCl ternary hydrothermal solutions. It expressed as the following equation:
(8)logmSiO2aq=AT+BT⋅logMH2OVH2O∗+CTlogxH2O.Accordingly, WDM 2012 model revised the overestimates of AD 2009 model in low-moderate NaCl-bearing solutions (xNaCl<0.3), but it is subject to underestimate the molality of aqueous silica in high-salinity solutions compared to the experimental data sets (their Figure 5 of Wei et al. [36]).Recently, based on the pressure-volume-temperature-composition (PVTx) model of Mao et al. [54] and a systematically review of experimental data sets of quartz solubilities, an improved density model for quartz solubility was derived by Shi et al. [37] (SMH 2019 model) as the relationship below:
(9)logmSiO2aq=logK+nlogρsolF+ϕSiO2−H2O,where ρsol and F are the density of solution and the mass fraction of water, respectively, and ϕSiO2−H2O is the ratio of the activity coefficient of water corresponding to their concentrations with the power of hydration number and the activity coefficient of aqueous SiO2. Although the SMH 2019 model shows excellent performance in pure water and the binary fluid system, its prediction in aqueous NaCl- and CO2-bearing solutions still lacks versatile evaluation, which will be discussed below.


## 2.2. New Density Model and Its Parameterization

The first step of this modelling work is a systematic and comprehensive review of existing experimental data of quartz solubility. As early as 1935, researchers had carried out experimental studies and those measurements had covered H2O, H2O-CO2, H2O-NaCl, and H2O-CO2-NaCl hydrothermal solutions (Table 1). In constant regression, we use parts of experimental data (data with superscripts in Table 1), and others were either some of what unreliable or used as the testing of model prediction. The select regulation of experimental data for regression is basically like the former researchers discussed, and all the detailed information of data selection is listed in our Supplementary A. (Manning [30]; Shi et al. [37]; Fournier and Potter II [43]).Table 1
Experimental measurements for quartz solubility.
Pure H2OReferencesT (K)P (bar)NdVan Nieuwenburg and Van Zon [55]653.15-698.15289.2-490.218Kennedy [9]●433.15-883.156.2-1750106Morey and Hesselgesser [10]●633.15-873.1533-200020Fournier [56]438.15-513.157-33.53Kitahara [11]473.15-643.1515.5-210.48Van Lier et al. [12]∗333.15-373.150.2-1.05Morey et al. [13]●318.15-573.150.29-1013.351Siever [14]●398.15-454.652.32-10.416Weill and Fyfe [15]673.15-898.151000-400048Anderson and Burnham [16]773.15-1173.151000-985079Anderson and Burnham [17]∗973.1540002Crerar and Anderson [18]●452.15-602.159.8-126.931Novgorodov [26]973.1515001Hemley et al. [51]473.15-773.1515.5-200018Fournier et al. [27]∗623.15165-5005Anderson and Burnham [57]773.15-973.152000-841029Ragnarsdottr and Walther [58]523.15250-10007Walther and Orville [29]●623.15-823.151000-200020Archer [59]473.15-593.1561-100016Xie and Walther [28]685.15-846.152000-20057Manning [30]●773.15-1173.155000-20,00052Rimstidt [60]●294.15-369.150.02-0.8810Newton and Manning [20]●873.15-1123.152000-10,0008Shmulovich et al. [21]●773.15-1173.155000-90007Wang et al. [61]399.15-643.15800-610084Shmulovich et al. [22]●673.15-1073.151000-900019Newton and Manning [31]1023.15-1403.1510,0007Manning et al. [32]573.15-908.1510,0009Hunt and Manning [23] ∗1173.15-1323.1515,000-20,0005H2O-CO2 mixtureReferencesT (K)P (bar)xCO2NdNovgorodov [26]973.1515000.05-0.494Walther and Orville [29]∗873.1520000.023-0.317Newton and Manning [20]∗1073.1510,0000.19-0.685Shmulovich et al. [21]∗1073.1590000.16-0.505Shmulovich et al. [22]∗773.15-1073.152000-90000.06-0.5026Newton and Manning [62]∗1073.1510,0000.08-0.9314H2O-NaCl mixtureReferencesT (K)P (bar)XNaClNdVan Lier et al. [12]∗333.151.00.000211Anderson and Burnham [17]873.1530000.0161Ganeyev [25]∗523.15-773.1510000.03-0.0817Novgorodov [26]973.1515000.02-0.135Hemley et al. [51]∗473.15-673.1510000.009-0.0676Fournier et al. [27]∗623.15180-5250.035-0.06735Von Damm et al. [44]∗472.15-726.55192-10330.009100Xie and Walther [28]∗675.15-868.151725-18000.00189Newton and Manning [20]∗773.15-1173.152000-15,0000.06-0.7650Shmulovich et al. [21]∗773.15-1073.155000-90000.02-0.5514Shmulovich et al. [22]∗673.15-1073.151000-90000.004-0.5544Newton and Manning [63]∗1073.1510,0000.171Foustoukos and Seyfried Jr [64]∗638.15-703.15219-3780.0002-0.0192Cruz and Manning [24]∗1173.15-1373.1515,000-20,0000.05-0.527Scheuermann et al. [65]693.15310-3250.027-0.0464H2O-CO2-NaCl mixtureReferenceT (K)P (bar)xCO2XNaclNdShmulovich et al. [22]1073.1550000.07-0.300.0036-0.2514Note:Nd means the number of measurements. Superscripts ∗ and ● denote whole or parts of the experimental data sets which involved in constants regressions, respectively.Then, this model is also built on the equilibrium reaction of Eq. (1) and assumes ideal behavior for SiO2aq (Eq. (3)), as the empirical expression of the logarithm of the equilibrium constant for aqueous silica is linear with the logarithm of the density of pure water (Eq. (4)) written as follows:
(10)logmSiO2aq=AT,P+BT,P⋅logMH2OV¯H2O⋅ϕT,P,xNaCl,xCO2+nT,P,xNaCl,xCO2⋅loga,where V¯H2O is the partial molar volume of water, MH2O/V¯H2O refers to the density of pure water in H2O-CO2-NaCl ternary fluid mixtures, and ϕT,P,xNaCl,xCO2 is an empirical parameter on T, P, xNaCl, and xCO2 dependent, which corrects the deviation of the density of “pure water” and the effective density of “free water” (water not tightly bound as water of hydration) as introduced by Fournier [2]. AT,P and BT,P are T-P dependent polynomials in Kelvins and bars, respectively (note: all the pressures in the following equations are in bars unless a description), and nT,P,xNaCl,xCO2 is the hydration number and as an empirical polynomial expressed with the parameters of T, P, xNaCl, and xCO2. aH2O is the activity of pure water in H2O-CO2-NaCl solutions which is calculated from the model of Dubacq et al. [66].In this model, the polynomials ofAT,P,BT,P, ϕT,P,xNaCl,xCO2, and nT,P,xNaCl,xCO2 are assumed as follows:
(11)AT,P=A0+A1T+A2T2+A3T3+A4T4+A5P+A6P2+A7P3,(12)BT,P=B0+B1T+B2T2+B3logP+B4logP2,(13)ϕT,P,xNaCl,xCO2=10C0+C1T+C2T2+C3P+C4P2+C5xNaCl+C6xNaCl2xNaCl+D0+D1T+D2P+D3P2+D4xCO2xCO2+F0xNaClxCO22,(14)nT,P,xNaCl,xCO2=2+N00+N01T+N02T2+N03xNaCl2xNaCl+N10+N11P+N12xCO2+N13xCO22xCO2.By adopting the regressed constantsA0 to A7 through B0 to B4 in pure water and by combining the constants N00 to N03, N10 to N13, C0 to C6, D0 to D4, and F0 based on regressing quartz solubility data in H2O-NaCl, H2O-CO2, and H2O-CO2-NaCl solutions, the regressed constants of this model are given in Table 2.Table 2
Regressed constants for Eq. (10).
AT,PValuesBT,PValuesΦT,P,xNaCl,xCO2ValuesnT,P,xNaCl,xCO2ValuesA0-9.52462828E+00B07.92625728E+00C04.83288536E-01N001.76415897E+01A12.91941547E-02B1-1.53823138E-02C1-1.32770018E-04N01-3.05577769E-02A2-4.00569452E-05B29.14289810E-06C21.12520228E-07N021.25299576E-05A32.71067438E-08B34.03311251E-01C34.07784959E-05N03-1.99999983E+00A4-6.73901359E-12B4-6.91374075E-02C4-3.13938041E-09N10-1.15563053E+00A5-1.86478387E-06C5-9.00618419E-01N11-3.69288750E-05A64.10234340E-10C69.50723453E-01N125.56872025E+00A7-1.16951479E-14D0-1.72890706E+00N13-4.03999992E+00D11.41633612E-03D21.43379216E-04D3-7.23903865E-09D42.32574082E+00F06.99791510E+00For one molar homogenous system which is composed of H2O, NaCl, and CO2, the extensive property of the molar volume of H2O-CO2-NaCl mixture solution Vmix could be expressed as the following equation:
(15)Vmix=V¯H2O⋅xH2O+V¯NaCl⋅xNaCl+V¯CO2⋅xCO2,where V¯H2O, V¯NaCl, and V¯CO2 defined as the “partial molar volume” of H2O, NaCl, and CO2, respectively. At the same time, Vmix also can be obtained by the function of Mao et al. [54] as the following form:
(16)Vmix=VmixT,P,xNaCl,xCO2.Based on the definition of the partial molar volume,V¯NaCl and V¯CO2 at constant T and P can be further written as follows:
(17)V¯NaCl=∂Vmix∂xNaClT,P,(18)V¯CO2=∂Vmix∂xCO2T,P.IfVmix is continuous differentiable in a closed interval of [xi, xi+Δh] (xi means the molar fraction of NaCl or CO2 solute at any fixed T and P conditions, and Δh is an incremental change of the molar fraction of NaCl or CO2 solute), according to the Taylor expansion of Vmix, it reduces to
(19)Vmixxi+Δh=Vmixxi+ΔhdVmixxidxi+Δh22d2Vmixcdxi2,where c is a constant and between xi and xi+Δh, and according to the finite difference method (FDM), Eq. (19) can be transformed into
(20)dVmixxidxi=Vmixxi+Δh−VmixxiΔh−oΔh,where oΔh is the equivalence infinitesimal of Δh and equals to Δh/2d2Vmixc/dxi2. In our calculation process when Δh/x values 10-4, the term of dVmixxi/dxi, namely, V¯NaCl or V¯CO2 is solved out.As Shi and coworkers [37] discussed, when the SMH 2019 model applied to H2O-CO2-NaCl solutions, the part of ϕNaCl,CO2r which accounts for the nonadditivity of the contributions of NaCl and CO2 to ϕSiO2−H2O is very small and can be neglected in most cases, so in this model, V¯NaCl and V¯CO2 in Eq. (17) and Eq. (18) could be solved out with FDM in H2O-NaCl and H2O-CO2 systems, respectively, and the value of V¯H2O in Eq. (15) will then be calculated.


## 2.3. Accuracy Test of This Model

The accuracy of this new model is compared with the most competitive three ones, i.e., the AD 2009 model, WDM 2012, model and SMH 2019 model by investigating the absolute values of averaged relative deviations (AARDs) (listed in Table3) from experimental data over a broad T-P-x spectrum. For quartz solubilities in H2O, H2O-CO2, and H2O-NaCl solutions, the overall AARDs of this model are 5.74%, 6.69%, and 7.09%, respectively. Although inferior to the SMH 2019 model in aqueous NaCl solutions, the performance of this model has advantages in aqueous CO2 fluids. In addition, this model has made an obvious progress in pure water and NaCl- and/or CO2-bearing solutions which compared to the AD 2009 model and the WDM 2012 model.Table 3
Deviations of calculated values on quartz solubility.
ReferencesNd1AARD/%2AARD/%3AARD/%4AARD/%Pure H2OKennedy [9]●794.955.127.466.29Morey and Hesselgesser [10]●105.665.462.563.39Kitahara [11]66.376.449.518.92Van Lier et al. [12]∗54.892.8425.8621.66Morey et al. [13]●324.9810.9625.3521.90Siever [14]●106.157.4716.8813.01Weill and Fyfe [15]374.273.674.755.69Anderson and Burnham [16]799.045.7111.3113.46Anderson and Burnham [17]∗24.276.2114.8117.13Crerar and Anderson [18]●157.269.9713.7311.04Hemley et al. [51]184.133.6710.6210.70Fournier et al. [27]∗52.002.488.133.02Anderson and Burnham [57]87.807.339.6611.44Ragnarsdottr and Walther [58]47.755.328.525.09Walther and Orville [29]●154.624.276.415.74Archer [59]138.0830.0735.9133.85Xie and Walther [28]∗610.6712.0713.6813.87Manning [30]●525.326.625.706.80Rimstidt [60]●105.3719.4827.4423.24Newton and Manning [20]●85.295.469.7411.13Shmulovich et al. [21]●56.799.586.648.29Shmulovich et al. [22]●153.964.233.503.91Newton and Manning [31]32.6217.1317.1617.03Hunt and Manning [23]∗45.5524.729.4628.19Overall AARD/%5.749.0113.5312.70H2O-CO2 mixtureWalther and Orville [29]∗52.042.976.537.72Newton and Manning [20]∗47.119.4914.8515.21Shmulovich et al. [21]∗57.196.5216.4117.76Shmulovich et al. [22]∗209.2217.1810.5215.64Newton and Manning [62]∗97.9011.439.3828.62Overall AARD/%6.699.5211.5416.99H2O-NaCl mixtureVan Lier et al. [12]∗78.0114.4833.2929.38Anderson and Burnham [17]13.225.7713.3015.36Ganeyev [25]∗1014.9015.8816.7120.68Novgorodov [26]59.849.1114.7515.19Hemley et al. [51]∗35.542.088.538.34Fournier et al. [27]∗351.381.583.633.66Von Damm et al. [44]∗1002.343.406.775.70Xie and Walther [28]∗910.7510.5013.5215.04Newton and Manning [20]∗456.594.6111.2319.82Shmulovich et al. [21]∗126.866.449.4010.50Shmulovich et al., [22]∗407.735.779.148.28Foustoukos and Seyfried Jr [64]∗754.313.014.673.75Cruz and Manning [24]∗1411.9012.0455.3919.13Scheuermann et al. [65]45.902.843.694.38Overall AARD/%7.096.9714.5712.80Notes:Nd is the number of experimental data points; superscripts ∗ and ● denote whole or parts of the experimental data sets which involved in constant regressions, respectively; AARD is the absolute value of averaged relative deviation calculated from models; superscripts 1 to 4 represent the models of this one, the SMH 2019, the WDM 2012, and the AD 2009, respectively.The AARD of this model in pure water is 5.74%, only <50% of that of WDM 2012 and AD 2009, and is smaller than that of the SMH 2019. As can be seen from Figure2(a), this model can predict the dissolution and precipitation behavior of quartz in pure water over broad T-P conditions (up to 1000°C and 20,000 bar). Two points should be emphasized: first, this model reproduced quartz retrograde phenomenon (quartz increases with decreasing temperature) near the critical point of water at low pressures (200 bar and 500 bar) much more accurate compared to other models discussed here (Figure 2(b)); second, except for the data sets of Manning [30] and parts of Shmulovich et al. [22], the solid curve in Figure 2(c) presented the predictability of this model (for those other data sets were not used in constants regression) and which can be seen that this model had better accuracy compared to other models especially in high pressure conditions (Figures 2(c) and 2(d)).Figure 2
Model predictions of quartz solubility in pure water and the comparisons between different model computes and experimental data sets. Symbols represent experimental data, and curves are the predictions of different models.
(a)(b)(c)(d)Comparisons of different models computing and experimental data sets in H2O-CO2 solutions are shown in Table 3 and Figures 3(a) and 3(b). Generally, the calculated results of this model matched the experimental data quite well, and the overall AARD of this model is 6.69% which is about 50% better than that of WDM 2012 and AD 2009 and about 30% better than the model of SMH 2019. As Figure 3(a) depicts, at 500°C, 2000 bar, both WDM 2012 and AD 2009 models underestimate quartz solubility slightly, while SMH 2019 overestimates. The WDM 2012 and AD 2009 models underestimate the quartz solubility at high temperature and moderate pressure at 800°C and 5000 bar, whereas the SMH 2019 model overestimates (note: the circle points in Figure 3(b) are the experimental data at 10,000 bar condition). Though the data themselves are very scattered at 800°C, 5000 bar condition, compared to other models, this model seems better fit. When xCO2 is higher than 0.5 at 800°C, 10,000 bar condition, the WDM 2012 and AD 2009 models underestimate quartz solubility, but the SMH 2019 model overestimates although the data sets of Newton and Manning [62] are as the extrapolation testing for their model (Figure 3(b)). Overall, our model seems to have the best performance under the conditions as Figures 3(a) and 3(b) depict.Figure 3
Model predictions of quartz solubility in (a, b) aqueous CO2 solutions, in (c–e) aqueous NaCl solutions, and in (f) CO2- and NaCl-bearing solutions. Different curves denote comparisons between four model computes and experimental data sets. Symbols represent experimental data, and curves are the predictions of different models.
(a)(b)(c)(d)(e)(f)Figures3(c)–3(e) show that this model can reproduce quartz solubility well in aqueous H2O-NaCl solutions and is in good agreement with experimental data sets at most temperature and pressure conditions. After excluding some experimental data sets which inconsistent with themselves or having systematic errors (such as some data of Newton and Manning [20] and Cruz and Manning [24]), the overall AARD of this model is 7.09% (Table 3), which has made big improvement compared to the WDM 2012 model and the AD 2009 model, but little inferior than the SMH 2019 model. At 700°C, 2000 bar, compared to the experimental data points of Newton and Manning [20], this model better reproduced the “salting-in” effect (quartz solubility is enhanced with NaCl content) whereas other models all overestimate as Figure 3(c) depicts. At 800°C, 5000 bar high temperature and moderate pressure condition, the accuracy of this model is almost the same as that of the model SMH 2019 and is obviously better than those of the WDM 2012 and the AD 2009 (Figure 3(d)). As Figure 3(e) depicts at 900°C, 10,000 bar condition, this model overestimates quartz solubility in low NaCl-bearing system whereas gives accurate result in higher xNaCl condition when the pressure is much beyond the applicable limitation of the equation of state (EOS) for Vmix calculation from Mao et al. [54]. However, when considering the quartz solubility in pure water at the same T, P condition (as the circle point by Manning [30] in Figure 3(e)), this overestimation may not be that “over,” and the predictions by models of SMH 2019, WDM 2012, and AD 2009 may seem a litter lower.Till now, only 14 experimental data points of quartz solubility have reported in H2O-CO2-NaCl ternary solution by the work of Shmulovich et al. [22] (at xH2O=0.7). As seen from Figure 3(f), although the data sets are fairly scattered covering a wide spectrum of xCO2 and xNaCl condition from 0 to 0.3, however all model computing results are between them (after deleting some data points with obvious discrepancy as the open squares in their Figure 6 of Shi et al. [37]). Moreover, only this model can approach both two end-member systems of H2O-CO2 and H2O-NaCl solutions on quartz solubility reproducing. Certainly, this model invites stricter testing by future experimental data sets.There still some points worth of mentioning for the new model. Firstly, the success of this density model may be derived from the highly accurate EOS forVmix and aH2O computing and more physically meaning parameters the model having. Secondly, only parts of experimental data sets in pure water (the data with superscriptions in Table 1) were used in the constant regression, so the computing result of the solid curve in Figure 2(c) can represent the prediction of this model to some extent. In addition, although the form of the solvation number n in our model is empirical, it may have a physical explanation of assigning it to fixed 2 in pure water (the fixed value of n is for simplistic in model computing for the contribution from aH2O on quartz solubility will always be zero in pure water, and it could be seemed as the hydration number only when temperatures and pressures are below 600°C and 6000-8000 bar conditions) or depending on T, P, xNaCl, and/or xCO2 conditions in NaCl and/or CO2-bearing solutions which were verified from in situ Raman spectroscopic study on silica speciation in aqueous fluids depends strongly on T, P and the study on silica hydration changing with the fluid adding of NaCl or CO2 species [35, 53]. Besides, we think the form of n in our model may be an empirical way to get hydration number on quartz dissolution in NaCl and/or CO2-bearing fluids but should be carefully treated and tested by more studies on silica molecule structures. Lastly, this model can extrapolate to high temperatures and high pressures within or approaching experimental accuracy (for instance, the absolute value of averaged experimental error from the experimental data of Newton and Manning [31] which is carried in H2O at 1223.15 K and 10,000 bar condition is 2.91%, and the absolute value of averaged relative deviation calculated from this model is 2.93%), for those conditions are beyond the T-P validation of the EOS computing on Vmix obtained from Mao et al. [54] (validated from 273 to 1273 K, from 1 to 5000 bar for the H2O-NaCl system, from 273 to 1273 K, and from 1 to 10,000 bar for the H2O-CO2 system) and aH2O obtained from Dubacq et al. [66] (validated from 283 to 653 K and from 1 to 3500 bar).


## 3. Algorithm for Determining Solubility Isopleths

### 3.1. Design of the Numerical Approach

Quartz solubility isopleths have been widely used to quantify the magnitudes and trends of quartz solubility. In some recent studies, those isopleths provided critical knowledges for understanding volcanogenic massive sulfide deposits, the porphyry deposits, and the orogenic gold deposits [39–41, 67, 68]. Compared to the ways of Monecke et al. [39, 40] and Li et al. [41] for plotting their isopleths by repeat calculating numerous quartz solubilities (they were with the same method of calculating quartz solubilities along a closely spaced isochores and imported them into the graphing and data analysis software Origin), here, we provide a method to obtain those isopleths by a bisection algorithm which is based on this new model on quartz solubility in hydrothermal fluid mixtures over a broad T-P-x spectrum.As can be seen from Eqs. (10) and (15) and aH2O calculated from the model of Dubacq et al. [66], the parameters of this density model are only variables of temperatures, pressures, and fluid compositions. So, this model can be transformed as Eq. (21) and simply expressed as follows:
(21)mSiO2aq=MSiO2aqT,P,xNaCl,xCO2,xH2O,where MSiO2aq is the function of quartz solubility with T, P, xCO2, xNaCl, and xH2O as variables.Many experimental studies on quartz solubility have shown that at fixedT, xCO2, xNaCl, and xH2O fluid compositions, the quartz solubility mSiO2for pressures P (below 20,000 bar) is continuous monotonically changed (Figures 2(c) and 2(d)). So, the strategy for computing solubility isopleths is to solve each given mSiO2 quantity with T-P relationships at a fixed composition with a bisection algorithm.


### 3.2. Computing Programs

The above algorithm is embedded in a Fortran program, and the computing procedure is as Figure4 described. For obtaining one point on quartz solubility isopleths (consisting of many points) in the computational steps,
(i)
Input parametersT, xCO2, xNaCl, and mSiO2, in which T is in °C (between 0 and 1000°C) and mSiO2 is in moles per kilogram H2O(ii)
Assume initial boundary pressureP1=0 bar and P2=20,000 bar(iii)
Calculate quartz solubilitymSiO2cal with this density model (Eq. (10)) at P=P1+P2/2(iv)
ModifyP1 or P2 by a bisection method (if mSiO2cal<mSiO2, then P1=P; otherwise, P2=P) and repeat step III until the absolute difference between mSiO2cal and mSiO2 is less than 10-8 mol kg-1(v)
The calculatedP which satisfied the convergence condition is the approximate numerical solution of this programFigure 4
A flow chart on bisection algorithm for calculating quartz solubility isopleths in H2O-CO2-NaCl hydrothermal solutions. Parameters ofT, xCO2, xNaCl, xH2O, mSiO2, mSiO2cal, P1, and P2 are as defined in the text, respectively.Please note that the quartz solubility isopleths are sets ofT-P relations which are based on recomputing the above program with any interval of temperatures T at fixed xCO2, xNaCl, xH2O, and mSiO2 conditions.


### 3.3. Application on Isopleths for Single-Phase Region

Gold mineralization especially for the orogenic intrusion-related gold deposits is dominated by low to intermediate salinity CO2-bearing aqueous solutions and often has embodied themselves on the auriferous quartz veins [4, 39–41, 68]. On the contrary, the T-P-x properties of hydrothermal processes control the type of Au complexing, transport, and deposition in minerals, as well as quartz dissolution and precipitation [41]. Slope patterns of quartz solubility isopleths with different fluid compositions make an insightful way on understanding those veins and even those ores’ formation.An application example on iso-solubility curves in the single liquid region for temperature range of 300 to 500°C, pressure range 0 to 4000 bars, a fixed salinity of 5 wt. %, and CO2 content ranges of 5 mol % and 15 mol% is acquired based on this new model and the algorithms presented above (Figure 5). These solubility diagrams are designed for orogenic gold systems (for the T-P-x condition designed) but are applicable for all other systems satisfying the physical and chemical conditions mentioned above. The phase boundaries between the single phase and the liquid plus vapor phase in the H2O-CO2-NaCl system are calculated using the model of Mao et al. [69]. The P, T ranges of phase boundaries calculating which outside of the applicable range of Mao et al. [69] were extrapolated (model validation is from 273.15 K to 723.15 K and from 1 bar to 1500 bar).Figure 5
Quartz solubility isopleths in H2O-CO2-NaCl hydrothermal fluids containing the same NaCl contents and different CO2 contents in the single phase which are calculated from this density model and the isopleth algorithm, at 300-500°C and 0.01-4000 bar. (a) 5 wt. % NaCl and 5 mol % CO2 containing. (b) 5 wt. % NaCl and 15 mol % CO2 containing. Quartz solubilities are recorded in millimoles per kilogram H2O. F: the single-phase fluid; L+V: liquid plus vapor phase fluid. The thin curves represent quartz solubility isopleths, and the thick curves are phase boundaries.
(a)(b)Generally, the regions of isopleths in the diagrams with parallel and gentle slopes mean quartz solubilities are sensitive to the pressure changing, with steep and drastic slopes mean quartz solubilities are sensitive to the temperature changing, and with crossing higher isopleths when temperature is decreasing mean quartz solubilities reflect retrograde features. In the H2O-CO2-NaCl single-phase field of mesothermal gold mineralization concerned with 5 wt. % NaCl and/or 5-15 mol % CO2 containing (Figure 5), it was revealed from this work that quartz solubilities generally not appear retrograde phenomenon and depend strongly on temperature at pressures higher about 1500 bar whereas little on pressure dependences. Isobaric cooling may be the main reason for quartz precipitating. At lower pressures about less than 1500 bar, quartz solubility isopleths are strongly curved with parallel and gentle slopes which means isothermal decompression and isobaric cooling are act together on quartz precipitations. The above conclusions are similar to the systematical research of Li et al. [41] as their Figures 6–8 depict. A noteworthy feature is that this algorithm can also calculate quartz solubility isopleths in liquid plus vapor phase by adding quartz solubility of both phases only if the phase proportion is known [40, 41].


## 3.1. Design of the Numerical Approach

Quartz solubility isopleths have been widely used to quantify the magnitudes and trends of quartz solubility. In some recent studies, those isopleths provided critical knowledges for understanding volcanogenic massive sulfide deposits, the porphyry deposits, and the orogenic gold deposits [39–41, 67, 68]. Compared to the ways of Monecke et al. [39, 40] and Li et al. [41] for plotting their isopleths by repeat calculating numerous quartz solubilities (they were with the same method of calculating quartz solubilities along a closely spaced isochores and imported them into the graphing and data analysis software Origin), here, we provide a method to obtain those isopleths by a bisection algorithm which is based on this new model on quartz solubility in hydrothermal fluid mixtures over a broad T-P-x spectrum.As can be seen from Eqs. (10) and (15) and aH2O calculated from the model of Dubacq et al. [66], the parameters of this density model are only variables of temperatures, pressures, and fluid compositions. So, this model can be transformed as Eq. (21) and simply expressed as follows:
(21)mSiO2aq=MSiO2aqT,P,xNaCl,xCO2,xH2O,where MSiO2aq is the function of quartz solubility with T, P, xCO2, xNaCl, and xH2O as variables.Many experimental studies on quartz solubility have shown that at fixedT, xCO2, xNaCl, and xH2O fluid compositions, the quartz solubility mSiO2for pressures P (below 20,000 bar) is continuous monotonically changed (Figures 2(c) and 2(d)). So, the strategy for computing solubility isopleths is to solve each given mSiO2 quantity with T-P relationships at a fixed composition with a bisection algorithm.


## 3.2. Computing Programs

The above algorithm is embedded in a Fortran program, and the computing procedure is as Figure4 described. For obtaining one point on quartz solubility isopleths (consisting of many points) in the computational steps,
(i)
Input parametersT, xCO2, xNaCl, and mSiO2, in which T is in °C (between 0 and 1000°C) and mSiO2 is in moles per kilogram H2O(ii)
Assume initial boundary pressureP1=0 bar and P2=20,000 bar(iii)
Calculate quartz solubilitymSiO2cal with this density model (Eq. (10)) at P=P1+P2/2(iv)
ModifyP1 or P2 by a bisection method (if mSiO2cal<mSiO2, then P1=P; otherwise, P2=P) and repeat step III until the absolute difference between mSiO2cal and mSiO2 is less than 10-8 mol kg-1(v)
The calculatedP which satisfied the convergence condition is the approximate numerical solution of this programFigure 4
A flow chart on bisection algorithm for calculating quartz solubility isopleths in H2O-CO2-NaCl hydrothermal solutions. Parameters ofT, xCO2, xNaCl, xH2O, mSiO2, mSiO2cal, P1, and P2 are as defined in the text, respectively.Please note that the quartz solubility isopleths are sets ofT-P relations which are based on recomputing the above program with any interval of temperatures T at fixed xCO2, xNaCl, xH2O, and mSiO2 conditions.


## 3.3. Application on Isopleths for Single-Phase Region

Gold mineralization especially for the orogenic intrusion-related gold deposits is dominated by low to intermediate salinity CO2-bearing aqueous solutions and often has embodied themselves on the auriferous quartz veins [4, 39–41, 68]. On the contrary, the T-P-x properties of hydrothermal processes control the type of Au complexing, transport, and deposition in minerals, as well as quartz dissolution and precipitation [41]. Slope patterns of quartz solubility isopleths with different fluid compositions make an insightful way on understanding those veins and even those ores’ formation.An application example on iso-solubility curves in the single liquid region for temperature range of 300 to 500°C, pressure range 0 to 4000 bars, a fixed salinity of 5 wt. %, and CO2 content ranges of 5 mol % and 15 mol% is acquired based on this new model and the algorithms presented above (Figure 5). These solubility diagrams are designed for orogenic gold systems (for the T-P-x condition designed) but are applicable for all other systems satisfying the physical and chemical conditions mentioned above. The phase boundaries between the single phase and the liquid plus vapor phase in the H2O-CO2-NaCl system are calculated using the model of Mao et al. [69]. The P, T ranges of phase boundaries calculating which outside of the applicable range of Mao et al. [69] were extrapolated (model validation is from 273.15 K to 723.15 K and from 1 bar to 1500 bar).Figure 5
Quartz solubility isopleths in H2O-CO2-NaCl hydrothermal fluids containing the same NaCl contents and different CO2 contents in the single phase which are calculated from this density model and the isopleth algorithm, at 300-500°C and 0.01-4000 bar. (a) 5 wt. % NaCl and 5 mol % CO2 containing. (b) 5 wt. % NaCl and 15 mol % CO2 containing. Quartz solubilities are recorded in millimoles per kilogram H2O. F: the single-phase fluid; L+V: liquid plus vapor phase fluid. The thin curves represent quartz solubility isopleths, and the thick curves are phase boundaries.
(a)(b)Generally, the regions of isopleths in the diagrams with parallel and gentle slopes mean quartz solubilities are sensitive to the pressure changing, with steep and drastic slopes mean quartz solubilities are sensitive to the temperature changing, and with crossing higher isopleths when temperature is decreasing mean quartz solubilities reflect retrograde features. In the H2O-CO2-NaCl single-phase field of mesothermal gold mineralization concerned with 5 wt. % NaCl and/or 5-15 mol % CO2 containing (Figure 5), it was revealed from this work that quartz solubilities generally not appear retrograde phenomenon and depend strongly on temperature at pressures higher about 1500 bar whereas little on pressure dependences. Isobaric cooling may be the main reason for quartz precipitating. At lower pressures about less than 1500 bar, quartz solubility isopleths are strongly curved with parallel and gentle slopes which means isothermal decompression and isobaric cooling are act together on quartz precipitations. The above conclusions are similar to the systematical research of Li et al. [41] as their Figures 6–8 depict. A noteworthy feature is that this algorithm can also calculate quartz solubility isopleths in liquid plus vapor phase by adding quartz solubility of both phases only if the phase proportion is known [40, 41].


## 4. Discussion

### 4.1. Extrapolation to High Temperatures and Pressures

As discussed in Section2.3, the limit on temperatures and pressures for this semiempirical thermodynamic model is mainly set by the PVTx model of Mao et al. [54] in addition to the water activity model of Dubacq et al. [66]. However, this model can be used in higher temperature and pressure hydrothermal conditions. For the upper critical endpoint (T=1353 K and P=9500‐10,000 bar) which Newton and Manning [31] pointed out; although Shi et al. [37] had stated that their model and the models of WDM 2012 and M 1994 are all failure in the H2O-SiO2 system, however, it seems that the present model works better than others on the sharp-enhanced trends of quartz solubility when extrapolated to high temperatures (more than 1000°C) which are shown in Figures 6(a) and 6(b). Summarized in the work of Manning [30], it is convinced that this model can be extrapolated to high pressures (more than 20,000 bar) in pure water which is benefited from the empirical linear relationship between the logarithm of equilibrium constant (logK) for a variety of aqueous and the logarithm of water density (logρH2O), and only parts of experimental data sets were used in constant regression. In H2O-NaCl mixture solutions, the extrapolation ability of this model is obvious enhanced than models of WDM 2012 and AD 2009 as well as is compared to the latest SMH 2019 model at high pressures (Figures 6(c) and 6(d)). We should note that those pressure conditions are far beyond their model application scopes of Mao et al. [54] and Dubacq et al. [66].Figure 6
Comparisons of model extrapolation to high temperature and pressure hydrothermal conditions. (a, b) Computing results of different models in pure water. (c, d) Computing results of different models in H2O-NaCl system. The solid curves, dash curves, dot curves, and dash dot curves present the predictions of this model, the SMH 2019 model, the WDM 2012 model, and the AD 2009 model, respectively.
(a)(b)(c)(d)


### 4.2. Prediction in Complex Fluids

One thing should be mentioned is that this thermodynamic model relies only on few experimental data in the H2O-CO2-NaCl mixtures (Figure 3(f), Shmulovich et al. [22]), which makes it difficult on model predictions in the above ternary system. The SMH 2019 model has adopted the solubility experimental data points of Shmulovich et al. [21] in H2O-CO2 solutions as constant regression of their model as the initial zero of NaCl content. However, the experimental data of Shmulovich et al. [21] are systematically higher than the data of Shmulovich et al. [22] at the same T, P and nearly the identical fluid composition conditions. Because of the higher experimental value SMH 2019 model adopted in constant regression, this may be the reason of their model’s failure when used in H2O-CO2-NaCl fluid mixtures as Figure 7(b) denotes the paradox computing results (please note that when the temperature is below 350°C, the quartz solubility has enhanced follows the gradually adding of CO2 with the same NaCl contents, whereas CO2 as a nonelectrolyte which will always expand the fluid volume and consequently lower quartz solubility).Figure 7
Comparisons of model predictions in H2O-CO2-NaCl hydrothermal fluids with fixed salinity and pressure conditions and with CO2 compositions from 0 to 15 mol %.
(a)(b)(c)(d)As we can see from the Figure7, the simulating results have revealed that the four competitive quartz solubility models here discussed are all displayed “salting-in” effect at low pressure condition (500 bar). Except for the SMH 2019 model, the prediction ability of this model, the WDM 2012 model, and the AD model are all displayed monotonous lowering quartz solubility when gradually enhancing CO2 content in the above ternary fluid systems at the same salinity conditions. In point of fact, we consider that the trends and overall ranges of this present model may probably correct and more accurate when used in the H2O-CO2-NaCl fluid mixtures (Figure 3(e) and 7(a)), whereas it still should be deeper tested by much more experimental data sets on quartz solubilities.


### 4.3. Perspectives for Model Application

A versatile understanding of solubility behaviors of silicate minerals (quartz in particular) in electrolyte and/or volatile-bearing aqueous fluids is vital in deciphering fundamental petrological and ore-forming processes in various geological settings [70]. For instance, in subduction zone, the chemical components in slab (subducting oceanic lithosphere) fluids (essentially metamorphic fluids [71]) may not only affect the chemical and isotopic signatures of arc-derived magmatic rocks [72], but also influence melting behavior of overriding mantle [73]. Similarly, solubility of mineral fluids in deep crust and upper mantle is critical in mass transport and evolution of the crust [20, 74]. Not surprisingly, solubility models applicable to these petrological settings with some extreme pressure and temperature conditions are prerequisite for a better understanding. The novel model of this study is among the best of such models.In ore-forming environment, mostly encountered with middle to upper crust, solubility models with better precision and accuracy may significantly improve our understanding of fluid flow, fluid-rock interaction, and ore deposition [75]. A more accurate model may facilitate to distinguish subtle differences in solubility to fluid P-T-x conditions (e.g., the model of Wei et al. [68]). For instance, it will enable the identification of retrograde solubility in complex CO2-bearing aqueous fluids, which are very important for metal (e.g., Au and Cu) deposition (for porphyry deposit: [39, 40]; for orogenic Au deposit: [41, 68]). In addition, quartz solubility models may also be significant in studying many other settings, for instance, in migmatite [76] and pegmatite [77]. Therefore, there is no doubt that solubility model of this study, with its superior accuracy and wider application range, will provide an improved geochemical tool for geoscience community.


## 4.1. Extrapolation to High Temperatures and Pressures

As discussed in Section2.3, the limit on temperatures and pressures for this semiempirical thermodynamic model is mainly set by the PVTx model of Mao et al. [54] in addition to the water activity model of Dubacq et al. [66]. However, this model can be used in higher temperature and pressure hydrothermal conditions. For the upper critical endpoint (T=1353 K and P=9500‐10,000 bar) which Newton and Manning [31] pointed out; although Shi et al. [37] had stated that their model and the models of WDM 2012 and M 1994 are all failure in the H2O-SiO2 system, however, it seems that the present model works better than others on the sharp-enhanced trends of quartz solubility when extrapolated to high temperatures (more than 1000°C) which are shown in Figures 6(a) and 6(b). Summarized in the work of Manning [30], it is convinced that this model can be extrapolated to high pressures (more than 20,000 bar) in pure water which is benefited from the empirical linear relationship between the logarithm of equilibrium constant (logK) for a variety of aqueous and the logarithm of water density (logρH2O), and only parts of experimental data sets were used in constant regression. In H2O-NaCl mixture solutions, the extrapolation ability of this model is obvious enhanced than models of WDM 2012 and AD 2009 as well as is compared to the latest SMH 2019 model at high pressures (Figures 6(c) and 6(d)). We should note that those pressure conditions are far beyond their model application scopes of Mao et al. [54] and Dubacq et al. [66].Figure 6
Comparisons of model extrapolation to high temperature and pressure hydrothermal conditions. (a, b) Computing results of different models in pure water. (c, d) Computing results of different models in H2O-NaCl system. The solid curves, dash curves, dot curves, and dash dot curves present the predictions of this model, the SMH 2019 model, the WDM 2012 model, and the AD 2009 model, respectively.
(a)(b)(c)(d)


## 4.2. Prediction in Complex Fluids

One thing should be mentioned is that this thermodynamic model relies only on few experimental data in the H2O-CO2-NaCl mixtures (Figure 3(f), Shmulovich et al. [22]), which makes it difficult on model predictions in the above ternary system. The SMH 2019 model has adopted the solubility experimental data points of Shmulovich et al. [21] in H2O-CO2 solutions as constant regression of their model as the initial zero of NaCl content. However, the experimental data of Shmulovich et al. [21] are systematically higher than the data of Shmulovich et al. [22] at the same T, P and nearly the identical fluid composition conditions. Because of the higher experimental value SMH 2019 model adopted in constant regression, this may be the reason of their model’s failure when used in H2O-CO2-NaCl fluid mixtures as Figure 7(b) denotes the paradox computing results (please note that when the temperature is below 350°C, the quartz solubility has enhanced follows the gradually adding of CO2 with the same NaCl contents, whereas CO2 as a nonelectrolyte which will always expand the fluid volume and consequently lower quartz solubility).Figure 7
Comparisons of model predictions in H2O-CO2-NaCl hydrothermal fluids with fixed salinity and pressure conditions and with CO2 compositions from 0 to 15 mol %.
(a)(b)(c)(d)As we can see from the Figure7, the simulating results have revealed that the four competitive quartz solubility models here discussed are all displayed “salting-in” effect at low pressure condition (500 bar). Except for the SMH 2019 model, the prediction ability of this model, the WDM 2012 model, and the AD model are all displayed monotonous lowering quartz solubility when gradually enhancing CO2 content in the above ternary fluid systems at the same salinity conditions. In point of fact, we consider that the trends and overall ranges of this present model may probably correct and more accurate when used in the H2O-CO2-NaCl fluid mixtures (Figure 3(e) and 7(a)), whereas it still should be deeper tested by much more experimental data sets on quartz solubilities.


## 4.3. Perspectives for Model Application

A versatile understanding of solubility behaviors of silicate minerals (quartz in particular) in electrolyte and/or volatile-bearing aqueous fluids is vital in deciphering fundamental petrological and ore-forming processes in various geological settings [70]. For instance, in subduction zone, the chemical components in slab (subducting oceanic lithosphere) fluids (essentially metamorphic fluids [71]) may not only affect the chemical and isotopic signatures of arc-derived magmatic rocks [72], but also influence melting behavior of overriding mantle [73]. Similarly, solubility of mineral fluids in deep crust and upper mantle is critical in mass transport and evolution of the crust [20, 74]. Not surprisingly, solubility models applicable to these petrological settings with some extreme pressure and temperature conditions are prerequisite for a better understanding. The novel model of this study is among the best of such models.In ore-forming environment, mostly encountered with middle to upper crust, solubility models with better precision and accuracy may significantly improve our understanding of fluid flow, fluid-rock interaction, and ore deposition [75]. A more accurate model may facilitate to distinguish subtle differences in solubility to fluid P-T-x conditions (e.g., the model of Wei et al. [68]). For instance, it will enable the identification of retrograde solubility in complex CO2-bearing aqueous fluids, which are very important for metal (e.g., Au and Cu) deposition (for porphyry deposit: [39, 40]; for orogenic Au deposit: [41, 68]). In addition, quartz solubility models may also be significant in studying many other settings, for instance, in migmatite [76] and pegmatite [77]. Therefore, there is no doubt that solubility model of this study, with its superior accuracy and wider application range, will provide an improved geochemical tool for geoscience community.


## 5. Conclusions

A density-based model on quartz solubility in pure water and NaCl and/or CO2-bearing solutions has been developed in this work. This model is semiempirical and is based on the pervious results of the AD 2009 model and the WDM 2012 model. The success of this model is benefitted from two aspects: one is the logarithm linear relationship between quartz solubility and the density of pure water, and the other is the accurate thermodynamic model of Dubacq et al. [66] and Mao et al. [54] for water activity (aH2O) and molar volume of the fluid mixture (Vmix) calculating in H2O-CO2-NaCl systems. Compared to the experimental data sets and the existing most competitive models, the accuracy of this model has been greatly enhanced. Particularly, this model may be extrapolated to high temperature and pressure conditions, provided new experimental data sets are available. It seems that the trends and overall ranges of this model may be probably correct and more accurate when used in the H2O-CO2-NaCl fluid mixtures. Moreover, this model can reproduce the retrograde and “salting-in” phenomenon on quartz solubility in pure water and NaCl-bearing solutions clearly and more accurate. At the same time, an algorithm on quartz solubility isopleths in the single-phase region based on this new model has first proposed.



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*Source: 6631152-2021-02-13.xml*