INTRODUCTION

Currently available systems of compositometer equations (i.e., equations for calculating the composition of minerals in equilibrium with melt under specified parameters) involve variables that are experimental data: temperature, pressure, chemical composition of the system, and oxygen fugacity (Ar’yaeva et al., 2016; Koptev-Dvornikov et al., 2012, 2019, 2020; Romanova et al., 2020). However, these compositometers were derived by means of statistical processing results of anhydrous experiments, whereas the great majority of natural magmas and lavas contain variable concentrations of water.

Numerous experiments have been conducted with the involvement of pure H2O fluids and those of complex composition. In this publication, we constrained ourselves to analysis of interaction between silicate melt and pure H2O fluid.

The following two problems should be resolved to be able to account for the effect of water on melt differentiation in a chamber. When compiling a representative dataset, we became aware that data of only 20% of the experiments referred to as water-saturated by the authors involve information on water concentrations in the melts, which leads to that much of the data should be rejected from statistical processing. At the same time, water is an incompatible component at the crystallization of rock-forming minerals in mafic systems and, hence, enriches the residual melts. It is necessary to know the highest water concentration in melt above which water makes up an individualized vapor phase.

This problem can be resolved by developing an equation that would enable the calculation of water saturated concentration in melt. Such an equation would make it possible to also resolve the problem of the limited volume of the dataset on hydrous experiments. A.A. Ariskin and G.S. Barmina were the first to use such an equation to calculate water concentrations in hydrous experiments in which water contents had not been unmeasured, and this made it possible to expand the potential size of the dataset for deriving melt–mineral thermobarometers (Ariskin and Barmina, 2000). Moreover, dissolved volatile gases significantly modify the density and viscosity of the magmas and thus play a decisive role in magma ascent. The density and viscosity of ascending magmas can be evaluated based on models for the solubility of volatile components that make it possible to calculate changes in the compositions of the melt and vapor as functions of pressure and temperature (Newman and Lowenstern, 2002). Such models are also useful for planning experiments with hydrous systems and for testing experimental data on water solubility in melts.

Several equations of the type have already been proposed, but they yield different values of water contents at the same parameters. Our study was aimed at deriving an equation that would best fit all currently available experimental data. Such an equation could be obtained by either selecting from all preexisting models for water solubility or by developing an original equation.

COMPILING A SET OF SELECTED DATA OF WATER-SATURATED EXPERIMENTS

Previously, most of the experimentalists derived their equations based on a limited amounts of data on original experiments in hydrous systems and more rarely used data sets compiled from several publications (these datasets comprised results of approximately 20 to 100 experiments with pure H2O fluid). In analyzing the literature, we have found 33 publications presenting data on dissolved water in hydrous melt according to results of 412 experiments.

A principal data source from which out dataset was compiled was the INFOREX database (Ariskin et al., 1996), and we also compiled experimental data from publications whose data were utilized to derive earlier equations (Shishkina et al., 2010; Berndt et al., 2002; Botcharnikov et al., 2004; Moore at al., 1995, 1998; Carrol and Blank., 1997; Silver et al., 1990; Shaw et al., 1963; Liu et al., 2005; Yamashita, 1999; Schmidt and Behrens, 2008).

The criteria used to include experimental data in the set were the availability of information on equilibrium between melt of known composition and a fluid phase that contained no components other than water.

REVIEW OF PREEXISTING EQUATIONS FOR CALCULATING WATER MOLE FRACTION IN MELT

Several equations have been proposed to describe the saturated solubility of water in silicate melts. When selecting the equations, we proceeded from the simplest (in terms of both their derivation and the calibration, which involve a single variable) to more complicated ones.

The simplest equations that are often applied describe saturated water concentration in silicate melt as a function of pressure alone (Carroll and Blank., 1997; Moore et al., 1995, 1998; Zhang et al., 2007). Of the equations of this type, we selected Eq. (1) in (Shishkina et al., 2010), because this equations has been derived from data of experiments within a broad pressure range.

Equation (Shishkina et al., 2010)

Equation (1) proposed in (Shishkina et al., 2010) is

$${\text{ln}}~{{C}_{{{{{\text{H}}}_{2}}{\text{O}}}}} = 0.2351{{P}^{{0.5758}}},$$
(1)

where \({{C}_{{{{{\text{H}}}_{2}}{\text{O}}}}}\) is the saturated H2O concentration (in wt %) in melt, and P is the pressure in MPa.

The ranges of the intensive parameters in the dataset from which the equation has been derived are a temperature of 1250oC, pressure of 1 bar to 5 kbar, and water concentrations of 1 to 9 wt %; the size of the set is 27 data points, the composition is tholeiitic basalt.

Comparison of calculation results yielded by this model with experimental data from our dataset (Fig. 1) shows that a single calculated concentration corresponds to a wide range of experimentally determined water concentrations, which led us to the conclusion that other intensive parameters have also be taken into account.

Fig. 1.
figure 1

Comparison of water solubility calculated by the model (Shishkina et al., 2010) with experimental data (data of 412 experiments in the dataset): solid circles are experimental data used by the authors to calibrate the equation, open symbols are other data from our dataset.

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Equation (Liu et al., 2005)

Another equation for the pressure dependence of the saturated water concentration was presented in (Liu et al., 2005), and unlike the previous equation, this one involves temperature

$$\begin{gathered} {{C}_{{\text{w}}}} = \frac{{354.94\sqrt {{{P}_{w}}} + 9.623{{P}_{w}} - 1.5223P_{w}^{{\frac{3}{2}}}}}{T} \\ + \,\,0.0012439P_{w}^{{\frac{3}{2}}} + {{P}_{{{\text{C}}{{{\text{O}}}_{2}}}}}\, \times \,{{10}^{{ - 4}}}( - 1.084\sqrt {{{P}_{w}}} - 0.1362{{P}_{w}}), \\ \end{gathered} $$
(2)

where Cw is saturated water concentration (in wt %), T is the temperature in K, Pw is the partial H2O pressure in MPa, and \({{P}_{{{\text{C}}{{{\text{O}}}_{2}}}}}\)is the partial CO2 pressure in MPa.

The ranges of the intense parameters in the dataset is a temperature of 700–1200°C, pressure of 1 bar to 5 kbar, and saturated water concentration of 0.5 to 11 wt %. The equation was derived by fitting data of 299 experimental runs, with about 80 of them conducted with natural (mostly felsic) compositions and a pure H2O fluid phase. These data points are shown in Fig. 2.

Fig. 2.
figure 2

Comparison of water solubility calculated by the model (Liu et al., 2005) with experimental data (data of 412 experiments in the dataset): solid circles are experimental data used by the authors to calibrate the equation, open symbols are other data from our dataset.

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Comparison of the calculation results of this model with experimental data from our dataset (Fig. 2) shows that, on the one hand, the experimental data are better fitted at low concentrations, and on the other hand, the values remain scattered at high water concentrations (>7 wt %), with the calculated results systematically overestimated relative the experimental data.

Obviously, the quality of fitting of the experimental data can be improved by taking into account the effect of still another parameters: the composition of the melt.

Equation (Zhang et al., 2007)

The authors of (Zhang et al., 2007) have further advanced efforts (Liu et al., 2005), but in contrast to the above two equations, they introduced one additional parameters into their equation: the agpaitic coefficient of the melt, which depends on its composition

$$\begin{gathered} {{C}_{{\text{w}}}} = \left( { - 0.231 + \frac{{651.1}}{T}} \right)\sqrt P \\ + \,\,\left( {0.03424 - \frac{{32.57}}{T} + 0.02447AI} \right)P, \\ \end{gathered} $$
(3)

where Cw is the saturated water concentration (in wt %), T is the temperature (in K), P is the total pressure (in MPa), and AI = (Na + K – Al), where Na, K, and Al are the mole fractions of the respective components, and AI is the agpaitic coefficient.

Zhang et al., (2007) have derived this equation using a dataset of 93 selected experimental data points on felsic, intermediate, and mafic melts.

Comparison of the calculation results obtained with this model with experimental data from our dataset (Fig. 3) shows that the model better than the previous ones reproduces experimental values at low and intermediate water concentrations, but the calculated values at high water contents are systematically overestimated compared to the experimental ones.

Fig. 3.
figure 3

Comparison of water solubility calculated by the model (Zhang et al., 2007) with experimental data (data of 412 experiments in the dataset): solid circles are experimental data used by the authors to calibrate the equation, open symbols are other data from our dataset.

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Equation (Al’meev and Ariskin, 2007)

Another equation that accounts, in a simplified form, for the effect of melt composition on the solubility of pure H2O in this melt is the equation in (Al’meev and Ariskin, 1996). The effect of melt composition on water solubility is accounted for by using the Si/O and Al/Si ratios

$$\begin{gathered} ~\ln ~{{C}_{{{{{\text{H}}}_{2}}{\text{O}}}}} = 4.39 + \frac{{\left[ {38483{{{\left( {\frac{{{\text{Si}}}}{{\text{O}}}} \right)}}^{l}} - 14710} \right]}}{T} \\ + \,\,0.59\ln P - 21.45{{\left( {\frac{{{\text{Si}}}}{{\text{O}}}} \right)}^{l}} + 3.89{{\left( {\frac{{{\text{Al}}}}{{{\text{Si}}}}} \right)}^{l}}, \\ \end{gathered} $$
(4)

where T is the temperature (in K), P is the pressure (in bar), melt parameters are in atomic units, and l is melt.

The ranges of intensive parameters of the dataset are as follows: temperature 800–1200oC, pressure is 200 bar to 9 kbar, dissolved water concentration is 0.5 to 12 wt %; the size of the dataset is 79 data points, and the compositions of the experimental systems cover a range from basalt to rhyolite.

Comparison of the calculated results and experimental data from our dataset is illustrated in Fig. 4. Comparison of Figs. 3 and 4 shows that the use of the Si/O and Al/Si ratios has not improved the reproducibility of the experimental data by the calculation results.

Fig. 4.
figure 4

Comparison of water solubility calculated by the model (Al’meev and Ariskin, 1996) with experimental data (data of 412 experiments in the dataset): solid circles are experimental data used by the authors to calibrate the equation, open symbols are other data from our dataset.

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Equation (Moore et al., 1998)

The dependence of water solubility on melt composition is expressed in this equation through the mole fractions of Al2O3, FeO, and Na2O normalized to the anhydrous silicate matrix (Moore et al., 1998):

$$2\ln X_{{{{{\text{H}}}_{2}}{\text{O}}}}^{{melt}} = \frac{a}{T} + \sum {{b}_{i}}{{X}_{i}}\left( {\frac{P}{T}} \right) + c\ln {\text{f}}_{{{{{\text{H}}}_{2}}{\text{O}}}}^{{fluid}} + d,$$
(5)

where \(X_{{{{{\text{H}}}_{2}}{\text{O}}}}^{{melt}}\) is the saturated mole fraction of water in melt, T is the temperature (in K), Xi is the oxide mole fraction in melt, P is the pressure (in bar), \({\text{f}}_{{{{{\text{H}}}_{2}}{\text{O}}}}^{{fluid}}\) is the water fugacity in the fluid (in bar), which is calculated by a modified Redlich–Kwong equation, a, bi, and c are coefficients at respective variables, and d is a constant.

Inasmuch as water fugacity of pure water is close to the total pressure, we attempted to check how much the water concentrations calculated using its fugacity and total pressure are different. When the same coefficients were used, the maximal difference between the results calculated with these two approaches was no greater than 0.08 wt % H2O (for 41 experiments by Moore et al., 1998). To simplify the Moore equation, we transformed it into

$$2\ln X_{{{{{\text{H}}}_{2}}{\text{O}}}}^{{melt}} = \frac{a}{T} + \sum {{b}_{i}}{{X}_{i}}\left( {\frac{P}{T}} \right) + c\ln P + d.$$
(6)

The range of the intensive parameters of the dataset used in (Moore et al., 1998) is a temperature of 800–1200oC, pressure of 190 bar to 6 kbar, dissolved water content of 1.5 to 10 wt %; the size of the set is 41 experiments (with pure water fluid), samples of phonolite, andesite, and rhyolite composition.

Comparison of the results calculated by this model with experimental data from our dataset is illustrated in Fig. 5.

Fig. 5.
figure 5

Comparison of water solubility calculated by Eq. (6) with experimental data (data of 412 experiments in the dataset): solid circles are experimental data used in (Moore et al., 1998) to calibrate the equation, open symbols are other data from our dataset.

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Analysis of Figs. 1–5 shows that each of the equations generally well reproduces experimental data that have been used by the authors to derive the coefficients of the respective equations, but experimental data from our greater dataset are best reproduced by the equation (Moore et al., 1998). This follows from the highest value of the coefficient of determination, the closeness of the slope of the regression line to unity, and the closeness of the constant term to zero.

However, this equation was derived based on a small set of experimental data (41 experiments), and hence, 25 years later we decided to recalibrate it on a greater amount of experimental data from our dataset.

In addition to models discussed above, the literature presents numerous other, more complicated models (e.g., Newman and Lowenstern, 2002; Papale et al., 2006 and references therein), but we decided to use the equation in (Moore et al., 1998) and consider more complicated models only if the relalibrated equation from (Moore et al., 1998) did not satisfy our requirements.

RECALIBRATED EQUATION (MOORE ET AL., 1998)

The recalibration procedure involved the selection of such values of the constants of Eq. (6) that optimally reproduced the experimental data. Equation (6) is linear with respect to the variables at the coefficients and the logarithm of water mole fraction. This seems to call for preferring multivariate linear regression in finding the coefficients and constant. This method is advantageous thanks to its good theoretical fundamentals and the ability to find the optimal values of the coefficients and constant with the assessment of their confidence (Vorob’ev, 2016).

However, the least-squares method attaches the greatest relative weight to data of the maximal value. When utilizing Eq. (6), we optimized the logarithms of water concentration expressed as a mole fraction. By definition, mole fractions are smaller than unity, and hence, the smaller the water concentration, greater the absolute values of the logarithms of this concentration. Thus, if the deviation of the logarithms are minimized, overestimated weights are acquired by low water contents. Analytical practice indicates that low concentrations are inevitably measured with greater inaccuracies, and hence, the risk is high that the derived dependence would predict disturbed concentrations. Indeed, Fig. 5 obviously demonstrates that the scatter of the data points is the greater, the higher water concentrations in melt. We have earlier faced this problem when deriving an equation for sulfur concentration in sulfide-saturated mafic melts (Koptev-Dvornikov et al., 2012), and this problem have been resolved by modifying the equation linear with respect to concentration logarithm to a form exponential with respect to the concentration itself and the direct optimization of the differences between the experimental and calculated concentrations (but not their logarithms). This transformation does not allow one to apply the apparatus of multivariate linear regression, which led us to apply the option SOLUTION SEARCH in MS Excel and the option SEARCH FOR SOLUTIONS OF NONLINEAR PROBLEMS BY MEANS OF GRG as the solution method.

Optimization of Exponential Equations

Equation (5) provides an expression for the saturated water mole fraction in melt

$$X_{{{{{\text{H}}}_{2}}{\text{O}}}}^{{melt}} = {\text{ex}}{{{\text{p}}}^{{\left( {\frac{a}{T} + \sum {{b}_{i}}{{X}_{i}}\left( {\frac{P}{T}} \right) + cLnP + d} \right)}}},$$
(7)

where Xi are expressed as the mole fractions of the selected oxides in the single-cation form normalized to 100 wt % anhydrous basis. Note that we eliminated multiplier 2 when transforming Eq. (6).

Our experience in fitting experimental data by equations in form (7) demonstrates that the distributions of the excesses (differences between the calculated and experimental values of the optimized parameter) are close to normal (e.g., Koptev-Dvornikov et al., 2012; Koptev-Dvornikov and Bychkov, 2019). This also follows from the shapes of the histograms and the concordance criterion. The normality of the excess distributions makes it possible to assess the quality of the thermobarometers using the well-developed apparatus of statistics.

The quality of optimization is commonly assessed by the standard deviation (Herzberg and O’Hara, 2002 and many others). However, this approach is inaccurate because standard deviations characterize the mean deviations of an experimental value from the value calculated with a model but provides no information on the deviation of the value calculated with the model from the unknown real value.

A more accurate approach to assessing the approximation quality of experimental data is the use of confidence levels. An advantage of this approach is that confidence levels are targeted (in contrast to the standard deviation) not on assessing the quality of any single measurement but at determining (with a prespecified probability) the deviation limits of the calculated value from the real one. A useful property of confidence levels is that they can be determined for dependences (if the distribution is normal) and can be narrowed by increasing the number of measurements. Thanks to the large size of the dataset, the width of the confidence corridor is much lower than ±2σ throughout the whole range of experimental values.

Based on our experience in optimizing equations that approximate experimental data, we decided to assess the quality of the prediction of this concentration by a corridor width at 95% confidence level.

Preparatorily to the recalibration of Eq. (7), we tested the optimality of the set of oxides proposed in (Moore et al., 1998). The best results were achieved when using mole fractions of FeO, CaO, and NaO0.5 as the arguments. According to some literature data (Papale et al., 2006), water solubility may also notably depend on KO0.5 concentration in the melt, but testing datasets that included concentrations of KO0.5 or (KO0.5 + NaO0.5) led us to conclude that the sums of the squared excesses decreases therewith only very insignificantly (by 0.07%), in spite of the fact that KO0.5 concentrations in our dataset broadly varied from 0 to 12.5 wt %, and the sums of (K2O + Na2O) ranged from 1.7 to 17.5 wt %.

In optimizing equations of form (7), we rejected data of experiments in which excesses were greater than 3σ, and this resulted in a reduction of the dataset to 394 experiments. Characteristics of the final dataset are presented in Table 1 and Fig. 6.

Table 1.   Characteristics of the dataset used to calibrate models for water solubility (394 experiments)
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Fig. 6.
figure 6

Variations in the experimental melt compositions (394 experiments).

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The polyhedron of 394 experimental melt compositions in coordinates of oxide concentrations for the final dataset is characterized by the following values (wt %): SiO2 from 45.8 to 77.5, TiO2 from 0 to 2.92, Al2O3 from 8 to 20.4, FeO* from 0.1 to 13.74 (FeO* is all Fe recalculated to FeO), MgO from 0 to 9.59, CaO from 0 to 12.6, Na2O from 1.2 to 9.72, K2O from 0 to 12.25, and P2O5 from 0 to 2.14. The dataset thus presents melt compositions ranging from basalt to rhyolite.

The ranges of the intensive parameters were as follows: temperature from 550 to 1300oC, pressure from 1 bar to 14.9 kbar, and oxygen fugacity log\(f{\text{O}}2\) from −14.2 to −6.8.

Results of optimization with arguments of FeO, CaO, and NaO0.5 mole fractions are displayed in Fig. 7a, and the coefficients of this equation are listed in Table 2.

Fig. 7.
figure 7

(a) Optimization of Eq. (7); (b) optimization of exponential Eq. (8); dashed lines are the confidence intervals and 95% confidence. The dataset comprises information on 394 experiments.

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Table 2.   Parameters of Eq. (7) obtained by optimizing data on a dataset of 394 experiments
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Equations of form (7) are better to use in numerical simulations of magmatic evolution processes. At the same time, many researchers working with experimental or natural materials prefer to express water concentrations in wt % and the arguments of the equations as concentrations (in wt %) of oxides in the melt normalized to anhydrous basis

$$C_{{{{{\text{H}}}_{2}}{\text{O}}}}^{{melt}} = {\text{ex}}{{{\text{p}}}^{{\left( {\frac{a}{T} + \sum {{b}_{i}}{{C}_{i}}\left( {\frac{P}{T}} \right) + c\ln P + d} \right)}}},$$
(8)

where Ci is the wt % of oxides in the melts, \(C_{{{{{\text{H}}}_{2}}{\text{O}}}}^{{melt}}\) is the saturated water concentration (in wt %), and other symbols as in Eq. (7).

Optimization results of Eq. (8) with arguments in the form of FeO, CaO, and NaO0.5 concentrations (in wt %) are shown in Fig. 8, and the coefficients of this equation are listed in Table 3.

Fig. 8.
figure 8

Comparison of solubility calculated by Eq. (8) and coefficients in Table 3 with experimental data (412 experiments in the dataset).

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Table 3.   Parameters of Eq. (8) obtained by optimizing data on a dataset of 394 experiments
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As follows from Fig. 7, the optimization of the exponential equation yields good results in both cases.

The experimental and calculation data displayed in Figs. 7a and 7b are obviously well consistent, as also follows from the closeness of the slope coefficients in the regression equation to unity and the absolute terms to zero, the closeness of the coefficients of determination, and the fairly small width of the confidence corridors. In Fig. 7a, the maximum width of the confidence corridor at high water concentrations is no greater than ±0.01 mole fraction (and is much smaller in the rest of the range). In Fig. 7b, the maximum width of the confidence corridor is ±0.2 wt %. The configurations of the histograms of the excesses in Fig. 7 indicate that the distributions are close to normal, and the estimated water solubility values are nonbiased. The calculated mean deviations of the calculated concentrations from experimental values (in mole fractions) are −0.000081, and those for the concentrations in wt % are 0.0042. The standard deviations are 0.017 and 0.45, and the deviation ranges of the calculated values from experimental ones are –0.051 to 0.059 in mole fractions and –1.29 to 1.66 in wt %.

Thus, optimization of equations for the calculation of water concentrations (in both mole fractions and wt %) show similar and fairly good results, in spite of that water concentrations in the experiments were analyzed by various techniques: FTIR, SIMS, tritium autoradiography, EPMA, Karl Fischer titration (KFT), and some other techniques (Table 1). These methods have been compared in (Devine et al., 1995; Shishkina et al., 2010; Schmidt and Behrens, 2008; Silver et al., 1990; and many others), and it has been proved that their results are highly consistent, with the greatest dispersion typical of the EPMA results, which are strongly dependent on the quality of the microprobe analysis. In view of this, the authors of the reviews suggest that this method should be selectively control by other techniques, which are more accurate but more time- and labor-consuming. Water concentrations in EPMA are calculated as the differences between analytical totals of 100 wt % and the total oxide concentrations determined by microprobe. The good consistency of results yielded by various analytical techniques for water concentrations is illustrated in Fig. 8 by the example of our dataset (412 experiments).

As obviously seen in Fig. 8, the EPMA values show a broader dispersion, but nevertheless most of them does not disturb the general totality of our dataset.

AN EXAMPLE OF THE APPLICATION OF THE MODEL

The recently published paper (Crisp and Berry, 2022) stimulated us to demonstrate the usefulness of the model for water solubility in silicate melt. In this experimental study, conducted within broad ranges of temperature (800–1500°C), pressure (1 bar–40 kbar), and compositions of water-bearing melts, water concentrations were analyzed by EPMA. Using Eq. (8), we have calculated the saturated water concentration in melts for the experiments conducted under pressures no higher than 15 kbar (constraints on the pressure in our dataset) to make sure that the experimentally determined water concentration were no higher than the calculated values (in this set of 151 experiments). The results are presented in Fig. 9.

Fig. 9.
figure 9

Comparison of saturated water concentrations calculated by Eq. (8) and experimental data in (Crisp and Berry, 2022); 151 experiments in the dataset.

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Figure 9 clearly demonstrates that the overwhelming majority of the data points plot within the field of melts unsaturated with water or close to the saturation line. Because the authors did not discuss water saturation in the melts, water concentrations in these experiments do not contradict the evaluations by Eq. (8). At the same time, doubt is provoked by the unrealistically high water concentrations in the products of 18 experiments conducted at 1 bar. The solution of Eq. (8) with respect to pressure leads to the conclusion that such dissolved water concentrations can be reached only under pressures of 3 to 7 kbar.

Inasmuch as the authors evaluated water concentrations by EPMA, it follows from the text of the paper that these evaluations have not been confirmed by analyses by other methods (in spite of the methodological recommendations), and hence, this approach puts in doubt the realism of the results pertaining to the water-bearing samples.

CONCLUSIONS

We have compiled a set of experimental data from the literature that comprises results of 394 quenching experiments characterizing water concentrations in melts within broad ranges of intensive parameters of silicate systems.

Analysis of the most popular types of models for water solubility in silicate melts indicate that the equation (Moore et al., 1998) best fits the experimental results.

The Moore equation recalibrated on the basis of an extended dataset enables predicting water concentrations in silicate melt within a compositional range from basalt to rhyolite, pressures up to 15 kbar, and temperatures of 550 to 1300oC with a maximum uncertainty of no greater than ±0.01 mole fraction or ±0.2 wt %.