3D Plotting of Gold Solubility and Gold Fineness: Quantitative Analysis of Ore-Forming Conditions in Hydrothermal Gold Deposits

Au-Ag minerals, which are mostly composed of gold and silver, include three minerals defined by the gold fineness scale [GF = 1000 × Au/(Au + Ag), by weight] as native gold (GF > 800), electrum (GF = 200–800), and native silver (GF < 200) [1-3]. As the major ore minerals of hydrothermal gold deposits, Au-Ag minerals (Au-Ag solid solutions) are commonly associated with Fe-S and Fe-O minerals, such as pyrite, arsenopyrite, pyrrhotite, magnetite, and hematite, which play important roles in studies of deposits [4-11]. GF is a key mineral characteristic of Au-Ag minerals that can be directly precipitated by a decrease in gold solubility in hydrothermal fluids. Accordingly, gold solubility and GF are both crucial for the study of hydrothermal gold deposits.

Gold solubility is a key factor in determining gold mineralization, and it is also an important topic in mineralogy [1-3]. Early studies of gold solubility were focused on identifying the types and thermodynamic data of gold complexes [e.g., AuCl₂⁻, AuHS⁰, and Au(HS)₂⁻] dominant in hydrothermal fluids under varied P‒T conditions [12-17]. Recent studies have mainly measured gold solubility in simulation experiments of hydrothermal fluids and plotted well-known pH-logfO₂ diagrams that show variations in gold solubility in hydrothermal fluids [18-24]. However, it is difficult to summarize the effective factors controlling gold solubility since there are too many variables that show different effects on gold solubility, such as temperature [25], pressure [26], pH [27], redox state [28, 29], and chloride and sulfur concentrations [30, 31]. Therefore, quantifying the correlations between ore-forming conditions and gold solubility is a problem worth studying.

As the standard for identifying Au-Ag minerals, GF is the fundamental parameter of gold deposits. GF is commonly determined by electron probe microanalysis and widely reported for various gold deposits. Earlier than 30 years ago, some researchers collected experimental data on GF and summarized the relationship between the GF range and gold deposit type [2, 32]. However, since increasingly more GF data have been published, recent studies [3, 20, 33] have revealed that the range of GF for most types of gold deposits is wider than previously thought. Hence, the indication of GF for gold deposits and ore-forming conditions is questioned and requires illustration from theoretical studies. Theoretical studies on GF were started by White et al. [34], who proposed a mixing model for Au-Ag solid solutions. Pal’yanova [3] successfully examined the detailed physicochemical conditions for equilibria of Au-Ag solid solutions and fluids by accounting for recently determined thermodynamic data for mono-ligands of gold and silver with other known species. Liang and Hoshino [20] proposed an ideal-mixing model for the thermodynamic calculation of Au_xAg_1-x-fluid equilibria and investigated suitable ore-forming conditions for high GF. The above studies suggest that the role of GF can be an indicator of ore-forming conditions. Although theoretical studies have been gradually improved in the literature, how to quantify the correlation between ore-forming conditions and GF and more precisely explain the experimental data of GF are still under debate for the following reasons. Many variables, including temperature, pressure, pH, oxygen fugacity (fO₂), and the concentrations of chloride, sulfide, gold, and silver complexes, control GF [3, 10, 33, 35, 36]. Moreover, the correlations and their rates of change (ROCs) between the conditions and GF are distinct and variable, respectively. Therefore, quantifying the correlation between the ore-forming conditions and GF plays a decisive role in understanding GF.

This contribution aims to quantify and evaluate the abovementioned correlations, specifically their ROCs. For instance, how can the pressure change of fluid cause a decrease in gold solubility by one order of magnitude or GF by 100 units? Although theoretical methods and thermodynamic data for calculating gold solubility and GF have gradually improved in previous studies, there are still limitations for quantification. This is because 2D results from thermodynamic calculations, such as isopleths of gold solubility in the pH-logfO₂ diagram, are not ideally suitable for multiple-dimensional quantification for various ore-forming conditions. Recently, computer technology has been widely used in geology, and 3D plotting has been introduced into geological research. For example, the spatial characteristics involved in the exploration of concealed ore bodies, underground geophysical prospecting, and groundwater evaluation cannot be directly shown by 2D graphs but can be fully displayed by 3D plotting [37-40]. In addition, the software development has improved the efficiency of thermodynamic calculations and the building of 3D plots. For instance, Mathematica software [41] has promoted intelligent and automatic thermodynamic calculations, and MATLAB can interpolate 2D data [42] to set up a 3D dataset that is a basis for plotting 3D plots for quantitative analyses. Here, we use thermodynamic calculations and 3D plotting of gold solubility and GF to quantify their correlations with ore-forming conditions, including temperature (200℃, 400℃), pressure (0.1, 5 kbar), salinity (1, 40 wt% NaCl eq.), and sulfur concentration (0.01~0.5 mol/kg); the ranges of which are commonly reported for hydrothermal gold deposits [4, 5, 26, 35, 43-47] and Au-Ag mineral experiments [3, 15, 16, 19, 22, 48, 49].

Therefore, the goals of this study are (1) to plot 3D plots of gold solubility and GF with varied physicochemical conditions by using thermodynamic calculations in Mathematica and 3D plotting in MATLAB, (2) to quantify the correlations between the conditions, gold solubility, and GF in 3D plots, and (3) to analyze gold precipitation and the GF in different types of hydrothermal gold deposits. According to the results of this study, we expect to provide a new method to apply the experimental data of ore-forming fluids and GF in gold deposits and to be used practically in the exploration industry.

Based on the thermodynamic properties of species published in previous studies [3, 16, 17, 22, 45, 50-52], the log K and thermodynamic data of related reactions are listed in Appendix 1 under varied temperatures (200℃, 400℃) and pressures (0.1, 5 kbar). Although Au(OH)S₃⁻ in hydrothermal fluids has recently been reported [53], it can be a dominant gold species under high temperatures (350℃) and high sulfur concentrations (0.5 m K₂S₂O₃). Unfortunately, the thermodynamic data of Au(OH)S₃⁻, such as the data presented in Appendix 1, are not available in the present literature. Hence, this gold species [Au(OH)S₃⁻] is excluded in our calculations. We calculated the equilibria of hydrosulfide and hydrosulfate ions in the pH-log fO₂ diagram (Figure 1) and plotted the following four fields: (I) HSO₄⁻, (II) SO₄²⁻, (III) H₂S, and (IV) HS⁻. Fields I and II located at higher log fO₂ values are called oxidized fields (or oxidized fluids) in this study, while fields III and IV located at lower log fO₂ values are called reduced fields (or reduced fluids). The stability fields of Fe-O and Fe-S minerals commonly associated with Au-Ag minerals, including pyrite, pyrrhotite, magnetite, and hematite, are also plotted in Figure 1.

The method of thermodynamic calculation adopted in this study is the ideal mixing model of Au-Ag solid solutions published by Liang and Hoshino [20]. The reaction of Au_XAg_1-X -fluid equilibria proposed by the model is: Au_XAg_1-X + H⁺ + 1/4O_2(aq) = XAu⁺ + (1 X) Ag⁺ + 1/2H₂O, where X is the mole fraction of gold, and Au⁺ and Ag⁺ are all dissolved species of gold and silver. Simultaneous equations for the equilibria of all dissolved species with the Au-Ag solid solution of a fixed composition (fixed GF) under certain pressure, temperature, pH, and log fO₂ conditions are solved by using Mathematica (Appendix 2). The mixing model of Au-Ag solid solutions proposed by White et al. [34] is also calculated in Appendix 2 to show the difference in the activity coefficients of gold and silver calculated by the two models.

Based on a given log K value for each reaction (Appendix 1), Mathematica can calculate the activity coefficients of gold and silver under fixed pH and log fO₂ conditions (Appendix 2). Mathematica can also calculate the total concentrations of all dissolved gold (m_ΣAu) and silver (m_ΣAg) species after inputting salinity (m_ΣCl) and the concentration of total dissolved sulfur (m_ΣS) into the calculation. In that case, the result of m_ΣAu and m_ΣAg can be shown at a certain pH-log fO₂ point in Figure 1, and log (m_ΣAu/m_ΣAg) (R) and dominant species of gold and silver can also be shown at the point. In addition, by using the calculation program in Mathematica (Appendix 2), we can directly calculate all 2D data (pH and log fO₂) for the isopleth of m_ΣAu or m_ΣAg and rapidly plot red dotted lines (m_ΣAu), blue dotted lines (m_ΣAg), and black dotted lines (R) in Figure 1.

The plotting of 3D diagrams is completed by the following three steps. First, pH-logfO₂ coordinate values for the isopleth are calculated under different conditions by using Mathematica, which provides the point data for the 3D plotting. The second step is focused on the interpolation method (Appendix 3) of point data. We use the Xlsread function in MATLAB to read the point data (x = pH, y= logfO₂) of the isopleth and the Linspace (x_min, x_max, and n) function to generate equidistant vectors between x_min and x_max, where n is the same as the number of vectors. Then, the Interp1 (x, y, x_i, Spline) function in MATLAB is used to complete the 2D interpolation, where x_i is the vector generated by the Linspace function, and “Spline” is the interpolation method adopted in this function. By using the above functions, a 2D dataset can be created including most pH-logfO₂ coordinate values on the isopleth. The data randomly selected in the 2D dataset can be verified by calculations in Mathematica, indicating no obvious error. Based on the 2D dataset, the Plot function in MATLAB can plot the pH-logfO₂ diagram (Figure 2(b)) that is compared with the diagram plotted before the interpolation (Figure 2(a)), illustrating that the interpolation method is helpful for the precision of isopleths.

The last step of 3D plotting is building 3D plots. The specific condition, such as GF or temperature, is chosen as the Z-axis of the 3D plots. The 2D data with varied Z values can be calculated by using Mathematica and interpolated by using MATLAB, forming a vector group (x, y, and z). By reading the data from vector groups, the 3D dataset can be created by the spatial interpolation function in MATLAB. Based on the 3D dataset, we use the Surf (x, y, and z) function in MATLAB (Appendix 3) to build the 3D plot of the m_ΣAu isopleth (Figure 2(c)). The 3D plotting is achieved by the combination of different isopleth plots in MATLAB. The logical connection between pH and logfO₂ is established by the 2D plotting of thermodynamic calculations, while the connection between the 2D plots and GF is established by 3D plotting in MATLAB. According to the 3D plotting, the variation in the isopleth plots of gold solubility and GF can be displayed in 3D to quantify their correlations with different ore-forming conditions.

Based on the methodology of 3D plotting, we built 3D plots of gold solubility at four pH - log fO₂ fields (I, II, III, and IV) to quantify the correlation and its ROCs between gold solubilities (m_∑Au) and the conditions of GF, pressure (P), temperature (T), salinity (m_ΣCl), and concentration of total dissolved sulfur (m_ΣS).

We chose pH, log fO₂, and GF as the X-, Y-, and Z-axes for the 3D plots, respectively, and plotted isopleth plots of varied m_ΣAu values (Figure 3(a)). The shapes of the plots are similar to inverted cones since the plots gradually shrink with decreasing GF. For example, the plots of m_ΣAu at 10⁻⁶ mol/kg and 10⁻⁷ mol/kg in Figure 3(a) present pinch-out sites at GF equal to 392 and 46, respectively. This result thus indicates a positive correlation between m_ΣAu and GF, suggesting that a higher m_ΣAu is more likely to be dissolved in the fluid forming a higher GF. The inner structure of the plots is basically a parallel arrangement, as shown in Figure 3(a), indicating that the variation in GF - log m_∑Au may be consistent in the four fields. We plotted the varying curves of GF - log m_∑Au (Figure 3(b)) to quantify the positive correlation in the four fields and found that the varying curves are parallel with each other under varied pH - log fO₂ conditions. This result means that the positive correlation is independent of pH and fO₂, and the Mathematica calculations also confirm that it is independent of other conditions, including T, P, m_ΣCl, and m_ΣS. By using the function analysis of the trend line in Figure 3(b), the varying curves of GF - log m_∑Au present the function of log m_ΣAu = 0.495 * ln (GF) – 3.5 – a, where a is the log m_ΣAu at GF = 999. The precision of the function is verified by comparing log m_ΣAu calculated by the function with log m_ΣAu calculated by Mathematica, showing that the errors of log m_ΣAu are less than 0.1 (at GF = 10–999) and 0.24 (at GF < 10).

Quantitative analyses of the plots show that the ROC values (log m_ΣAu /100 GF) of the positive correlation (the slope of the curves in Figure 3(b)) range from 0.11 to 3.49. The GF - log m_∑Au curves at GF = 200–1000 have an ROC = 0.11, which is consistent with the average rates of changes (ARCs), since they are straight lines. However, the curves at GF = 0–200 have ROC values ranging from 0.14 to 3.49 that rapidly increase with decreasing GF. The quantification results indicate that a decrease in gold solubility by one order of magnitude is possibly caused by a decrease in GF of ≥ 885 units.

We chose pH, log fO₂, and pressure as the X-, Y-, and Z-axes for the 3D plots, respectively, and plotted isopleth plots of varied m_ΣAu values at GF = 999 (Figure 4(a)). The shapes of the plots resemble shield cones since the plots gradually enlarge with decreasing pressure. For example, the plot of m_ΣAu at 10⁻⁶ mol/kg has an area at P = 0.1 kbar that is ~5 times as large as that at P = 5 kbar. This result thus indicates a negative correlation between gold solubility and pressure, suggesting that a higher m_ΣAu is more likely to be dissolved in the fluid at a lower pressure. The inner structure of the plots is basically parallel, as shown in Figure 4(a), indicating that the variation in log m_ΣAu - P is consistent in the four fields. The varying curves of log m_ΣAu - P are plotted in Figure 4(b), showing a vertical line in field II and three curves with small curvatures in fields I, III, and IV. This result suggests that the negative correlation is probably absent in field II, and the ARC (log m_ΣAu/kb) can approximately quantify the correlation since the three curves are all close to the straight line. The calculations for the ARC identify that the ARC is independent of fO₂ but controlled by pH. Hence, varying curves of pH-ARC are plotted in Figure 4(c) to show changes in the ARC in the oxidized fields and the reduced fields. The red line in Figure 4(c) shows that the ARC values increase from −0.34 to 0 with increasing pH in field I and are equal to 0 in field II. The blue line in Figure 4(c) shows that the pH-ARC curve is V-shaped, the lowest value (−0.33) appears in field IV, and the highest value (0) appears in field III. The range of the ARC in Figure 4(c) is thus from −0.34 to 0, indicating that a decrease in gold solubility by one order of magnitude can be caused by an increase in pressure of ≥ 3 kbar.

We chose pH, log fO₂, and T as the X-, Y-, and Z-axes for the 3D plots (Figure 5(a)), respectively. The shapes of the plots in the reduced fields are similar to cylindrical folds with a dip direction of log fO₂ = −55 since a decrease in temperature can shift the stability fields of pyrite and hydrosulfide ions to the lower fO₂ fields. Most plots generally shrink from 400℃ to 200℃ in the reduced fields, indicating a positive correlation between gold solubility and temperature. For example, the plot of m_ΣAu at 10^-6 mol/kg has an area at T = 400℃ that is ~4 times as large as that at T = 200℃. In addition, the shapes of the plots in the oxidized fields are similar to that of inclined layers, with a dip direction of log fO₂ = −20. This result thus indicates a positive correlation between gold solubility and temperature in the two fields. As shown in Figure 5(a), the inner structure of the plots is complicated due to different shapes in the two fields. Hence, a quantitative analysis of the plots is made in the “vertical direction” (changing only T) for pH - logfO₂ sections and the “inclined direction” (changing both T and pH-fO₂) for the buffer conditions of minerals.

The quantitative analysis of the “vertical direction” reveals that the correlation of m_ΣAu - T has complex changes in the plots. As shown by the red and blue solid curves in Figure 5(b), the m_ΣAu values under two specific pH-logfO₂ conditions (red and blue solid lines in Figure 5(a)) both present rapid decreases when the temperature decreases from 400℃ to 360℃, and the ARC values (=log m_ΣAu/10 ℃) are 0.8 and 0.6, respectively. In contrast, the two m_ΣAu values both show a slow increase when the temperature decreases from 360℃ to 200℃, and the ARCs are −0.05 and −0.06, respectively. Moreover, the quantification of other vertical directions in Figure 5(a) confirms that the ARC can be more variable when the temperature decreases.

The quantitative analysis of the “inclined direction” identifies a positive correlation of m_ΣAu - T. As shown by the red and blue dotted lines in Figure 5(b), the m_ΣAu values under two specific pH-logfO₂ conditions (red and blue dotted lines in Figure 5(a)) both slowly decrease with decreasing temperature, showing that the average of changes (AOC) is approximately consistent with the ARC. Hence, the varying curves of ARC - pH are calculated under Py - Ht (- Mt) buffer conditions (red line in Figure 5(c)) and Py - Po (-Mt) buffer conditions (blue line in Figure 5(c)). The red line in Figure 5(c) shows that the ARC-pH curve is shaped as a recumbent “S,” and ARC values range from 0.20 (at pH = 4.1) to 0.25 (at pH = 6.4) in the oxidized fields. The blue line in Figure 5(c) shows that the ARC-pH curve is V-shaped, and the ARC values range from 0.04 (at pH = 6.4) to 0.24 (at pH = 2) in the reduced fields. Figure 5(c) indicates that the ARC in the oxidized fields is generally higher than that in the reduced fields, and a decrease in gold solubility by one order of magnitude is possibly caused by a decrease of 40℃–50℃ in the oxidized fields and 42℃–250℃ in the reduced fields.

We chose pH, log fO₂, and salinity as the X-, Y-, and Z-axes for the 3D plots (Figure 6(a)), respectively. The plots in (and near) field I are shaped as inclined layers, indicating a positive correlation between m_ΣAu and m_ΣCl. This is because the concentration of AuCl₂⁻ that is dominant in field I is positively correlated with m_ΣCl. The plots in other fields in Figure 6(a) are shaped as upright folds, suggesting that the correlation might be absent. The XZ sections in the plots display varying curves of m_ΣAu and salinity as straight lines, and the ARC (log m_ΣAu /10 wt% NaCl eq.) of the curves is mainly controlled by pH conditions. Hence, the variation between ARC and pH is quantified in Figure 6(b). The red line in Figure 6(b) shows that the ARC values decrease from 1.04 to 0 with increasing pH in the oxidized field. The blue line in Figure 6(b) shows that the ARC values decrease from 0.43 to 0 with increasing pH from 2 to 4.8 in field III and are consistent with 0 in field IV. Figure 6(b) shows that the ARC in the oxidized fields (0–1.04) is generally higher than that in the reduced fields (0–0.43), indicating that a decrease in gold solubility by one order of magnitude is possibly caused by a decrease in salinity of ≥9.6 wt% NaCl eq. in the oxidized fields and ≥23.2 wt% NaCl eq. in the reduced fields.

We chose pH, log fO₂, and m_ΣS as the X-, Y-, and Z-axes for the 3D plots (Figure 7(a)), respectively. The shapes of the plots are similar to inverted semicones since m_ΣAu generally decreases with decreasing m_ΣS in the reduced fields. For example, the plots of 10⁻³ mol/kg and 10⁻⁴ mol/kg in Figure 7(a) display pinch-out sites at m_ΣS equal to 0.16 mol/kg and 0.05 mol/kg, respectively. This result thus indicates a positive correlation between m_ΣAu and m_ΣS in the reduced fields, suggesting that a higher m_ΣAu is more likely to be dissolved in the reduced fluid with a higher m_ΣS. The plots present varying curves of m_ΣAu and salinity that are similar to straight lines in the XZ sections in Figure 7(a), and the ARC (log m_ΣAu / 0.1 mol. kg^-1) of m_ΣAu - m_ΣS is mainly controlled by the pH in the reduced fields. Hence, the varying curves of ARC - pH are plotted in Figure 7(b). The red line in Figure 7(b) shows that the ARC decreases from 0.14 to 0 with increasing pH in field I and is consistent with 0 in field II. The blue line in Figure 7(b) shows that the ARC increases from 0.4 to 0.7 with increasing pH in field III and is consistent with 0.7 in field IV. Figure 7(b) shows that the ARC values are higher (0.4, 0.7) in the reduced fields and lower (0, 0.14) in the oxidized fields, indicating that a decrease in gold solubility by one order of magnitude is possibly caused by a decrease in m_ΣS of ≥0.71 mol/kg in the oxidized fields and 0.14–0.25 mol/kg in the reduced fields.

The 3D plots are built to quantify the correlation and its ROCs between GF and ratios of the total concentrations of dissolved gold and silver [R = log (m_ΣAu/m_ΣAg)], pressure (P), temperature (T), salinity (m_ΣCl), and concentration of total dissolved sulfur (m_ΣS).

We chose pH, log fO₂, and GF as the X-, Y-, and Z-axes for the 3D plots, respectively, and plotted isopleth plots of varied R (Figure 8(a)). The plots are shaped as horizontal layers parallel to the XY sections in fields I and IV and as laminated lenses parallel to the Y-axis that dip in the direction of pH = 10 in fields II and III. The plots in the four fields are unrelated to fO₂ since the plots are all parallel to the Y-axis. In addition, both the XZ and YZ sections in Figure 8(a) show that GF has a positive correlation with R; for example, native gold (GF > 800) can be formed at R > −0.6. A comparison between the oxidized field and the reduced fields reveals that the plots have a similar shape in the two fields and thus can mostly coincide, showing only a difference in pH coordinate values. The boundary of the two fields resembles a strike-slip fault that shifts the plots along the boundary. Thus, the varying curves of pH - R are plotted, showing a positive correlation between R and pH in field II and field III (Figure 8(b)), and the ROC (R/100 GF) values are both close to 1 (1.041 and 0.995, respectively). Moreover, the pH - R curves at various GFs are parallel to each other in Figure 8(b), explaining why the plots have a similar shape in the two fields.

Quantitative analyses of the 3D plots reveal that the positive correlation between R and GF is independent of other conditions, such as T and P. As shown in Figure 8(b), the interval of parallel curves can represent the ROC of the correlation between R and GF, and it is quite similar in the two fields. For example, the ROC of the R–GF correlation in the two fields is close to 0.2 at GF = 200–800 and sharply increases at GF = 800–1000 and GF = 0–200. Hence, we plotted the varying curves of R–GF in the reduced fields in Figure 8(c). The results show that the varying curves in field IV are unvaried at pH ≥ 6.4, while the curves are parallel to each other in field III and gradually shift to the curve in field IV when the pH increases. By using a function analysis of the trend line (Figure 8(c)), the varying curves of GF–R at GF = 1–500 and GF = 500–1000 present two functions as R = 0.476 * ln (GF) + b − 0.07 and R = −0.476 * ln (1000 − GF) + c + 0.07, where b and c are the R values at GF = 1 and GF = 999, respectively. The precision of the two functions is verified by comparing the R values calculated by the functions with the R values calculated by Mathematica, showing that the error of the R values is less than 0.11.

Because the 3D plots in the two fields both show a similar shape in Figure 8(a) and the ROC of the R–GF correlation is consistent in the two fields, the 3D plots in the reduced fields are plotted in the following sections to quantify the correlation between GF and the conditions that include P, T, m_ΣCland m_ΣS.

Since R is independent of fO₂ in the reduced fields, we chose pH, GF, and P as the X-, Y-, and Z-axes for the 3D plots, respectively, and plotted isopleth plots of varied R values in Figure 9(a). The plots in field III are shaped as nearly upright lenses, while the plots in field IV resemble wedge-shaped blocks with a dip direction of GF=0. Figure 9(a) shows that GF generally decreases with decreasing pressure, and the positive correlation is more obvious in field IV. In addition, the plots in Figure 9(a) show that the varying curves of GF–P are similar to straight lines in the YZ sections (such as pH = 10), and the ARC (GF/kb) of the correlation is controlled by pH and R. Hence, the varying curves of ARC - R and ARC - pH are plotted in Figures 9(b) and 9(c), respectively. Figure 9 shows that ARC values are higher (the highest ARC value is 72 GF/kb) in (or near) field IV when R ranges from −1 to −2, and Figure 9(c) shows that ARC values are less than 20 in field III or when R does not range from −1 to −2. This illustrates that a decrease in GF by 100 units is possibly caused by a decrease in pressure of 1.4 kb in field IV, while it is difficult to decrease GF by 100 units when the pressure decreases by 5 kbar in field III.

We chose pH, GF, and T as the X-, Y-, and Z-axes for the 3D plots, respectively (Figure 10(a)). The plots in field III are shaped as parallel lenses with a dip direction of GF = 0, while the plots in field IV are mostly shaped as upright layers parallel to the X-axis. The varying curves of GF–T are plotted in Figure 10(b), showing that a decrease in temperature can result in a decrease in GF when pH = 4 (field III), indicating a positive correlation between T and GF. However, a decrease in temperature can result in both a decrease and an increase in GF when pH = 8 (field IV), suggesting that the positive correlation at T < 300℃ can be changed into a negative correlation at T > 300℃. Thus, the ARC (GF/10℃) of the GF–T correlation at 200℃–300℃ and 300℃–400℃ is quantified in Figure 10(c). The results show that the ARC values are distinct in field III and field IV. For example, the ARC values in field III (colored lines in Figure 10(c)) are all greater than 0 and vary with increasing pH, and higher values appear at R < −1 (the highest ARC value is 73 GF/10℃ at pH = 6 and R = −3). In contrast, the ARC values in field IV (black lines in Figure 10(c)) are smaller than 0 when T = 200℃–300℃, and the values are unrelated to pH but controlled by R (the highest ARC value is 7.8 GF/10℃ at R = −1). This illustrates that a decrease in GF by 100 units is possibly caused by a decrease in temperature of ≥14℃ in field III and a decrease from 300℃ to ≤172℃ or an increase from 300℃ to ≥428℃ in field III.

We chose pH, GF, and salinity as the X-, Y-, and Z-axes for the 3D plots (Figure 11(a)), respectively. The shape of the plots in Figure 11(a) is similar to that in Figure 10(a), except that the dip angle of the plots in Figure 11(a) is lower. Figure 11(a) shows that GF generally decreases with decreasing salinity, indicating a positive correlation between GF and salinity that is more obvious in field III. Figure 11(a) can directly display the varying curves of GF–salinity, similar to straight lines in YZ sections (such as pH = 10); ARC (GF/wt% NaCl eq.) values of the correlation are thus plotted in Figure 11(b) to show the difference in the ARC values in field III and field IV. The ARC values in field IV are generally low (the highest ARC is 9 GF/wt% NaCl eq.), while the ARC values in field III vary by pH and are high at pH = 4–6 and R > −2 (the highest ARC value is 25 GF/wt% NaCl eq.). This illustrates that a decrease in GF by 100 units is possibly caused by a decrease in salinity of ≥4 wt% NaCl eq. in field III and ≥11.2 wt% NaCl eq. in field IV, respectively.

We chose pH, GF, and m_ΣS as the X-, Y-, and Z-axes for the 3D plots (Figure 12(a)), respectively. The plots in field III are shaped as parallel lenses with a dip direction of GF = 1000, while the plots in field IV are mostly shaped as upright layers parallel to the X-axis. Figure 12(a) shows that GF generally increases with decreasing m_Σs, indicating a negative correlation between GF and m_Σs that is more obvious in field III. Because the YZ sections (such as pH =10) can directly display varying curves of GF–m_Σs, similar to straight lines, ARC (GF/0.1 mol. kg⁻¹) values of the correlation are quantified in Figure 12(b) to show the difference in the ARC values in field III and field IV. The results show that the ARC values in field IV are all close to 0, while the ARC values in field III are lower when R ≤ −1 (the lowest ARC value is −155 GF/0.1 mol.kg⁻¹). This illustrates that a decrease in GF by 100 units is possibly caused by an increase in m_ΣS of ≥0.07 mol. kg⁻¹ in field III.

The 3D plotting of gold solubility can provide important implications and feasible methods for studies of gold precipitation. For instance, the plots in Figure 3(a) indicate that GF has a positive correlation with gold solubility. Moreover, the varying curves of GF - log m_ΣAu in Figure 3(b) present a function that is unaffected by other conditions. Hence, the pH - log fO₂ - GF plot can be rapidly built by the function when knowing the 2D data (pH - logfO₂) of the m_ΣAu isopleth, and 3D plots display the variation in gold solubility based on the determined range of GF.

The 3D plots plotted in this study illustrate that pressure and temperature have a distinct correlation with gold solubility. The 3D plots in Figure 4(a) generally show a negative correlation between gold solubility and pressure, except in field II, where AuOH⁰ is dominant (Figure 4(b)). This was explained by previous studies [50] that AuOH⁰ and its log K have a very low dependence from pressure. The lowest value of ARC (−0.34 log m_ΣAu/kb) in Figure 4(c) indicates that a decrease in gold solubility by one order of magnitude is possibly caused by an increase in pressure of ≥3 kbar. A similar result has been reported by other researchers [48, 50, 54], showing that the variation in pressure within 10 kbar can result in a change in the gold solubility of only less than one order of magnitude. This is because of the small variation in P - log K for most gold complexes; for instance, Benning and Seward [54] and Tagirov et al. [50] both found that although the log K of AuCl₂⁻, AuHS⁰, and Au(HS)₂⁻ can change with pressure, the variation in log K is small when the pressure varies within 5 kbar. According to the pressure gradient (0.27 kb/km) in the lithosphere [55], 3 kbar in pressure is equal to a depth of ~11 km. Most previous studies [56-58] on geofluids and hydrothermal gold deposits reported pressures ranging within 3 kbar. A variation in pressure greater than 3 kbar is generally considered to be caused by specific geological activities and events, such as high-pressure metamorphic belt activity [59, 60] and meteorite impacts [61]. Thus, the quantification suggests that pressure is unlikely to be an effective factor in controlling gold solubility.

Although the 3D plots in Figure 5(a) generally show a positive correlation between gold solubility and temperature, the correlation quantified by the “vertical direction” is variable and complex in Figure 5(b). This result is consistent with previous results; for example, Benning and Seward [54] and Stefánsson and Seward [62] both reported that gold solubility appears to decrease and increase in turn when the temperature decreases from 500℃ to 150℃, and Hu et al. [23] published that a variable correlation between temperatures (600℃, 800℃) and gold solubilities was found in an experimental study of synthetic fluid inclusions. This variable correlation can be explained by the variation in T - log K for the gold complexes; for example, Pal'yanova [3] collected the thermodynamic data of gold complexes to show that the log K of AuHS_(aq) and Au(HS)₂⁻ are positively correlated with temperatures from 0℃ to 350℃ but negatively correlated with temperatures higher than 350℃, and the variation in T - log K is much greater than that in P- log K. Figure 5(c) shows a positive correlation between gold solubility and temperature, as quantified by the “inclined direction,” and the ARC is distinct in varied pH-logfO₂ fields and presents high values in the oxidized fluid that can precipitate hematite and pyrite. The highest ARC value (0.24 log m_ΣAu / 10℃) of the oxidized fluid in Figure 5(c) indicates that the gold solubility may decrease by one order of magnitude when temperatures decrease by ≥40℃. The temperature of 40℃ is equal to a depth of ~2 km, which is calculated by the geothermal gradient as 20℃/km [63]. However, a decrease of 40℃ is usually reported in hydrothermal deposits [64-66], geothermal fluids [67, 68], and underground waters [69]. In addition, most previous studies [70-72] have stressed that a sharp decrease in temperature can have an obvious effect on gold solubility, indicating an effective mechanism of gold precipitation. Thus, the quantification suggests that temperature is an effective factor in controlling gold solubility, and a sharp decrease in temperature may lead to large-scale gold mineralization in the orogenic gold deposit [64-66].

The quantification of 3D plots illustrates that both m_ΣS and m_ΣCl may have a positive correlation with gold solubility, but the ARC of correlation is distinct in varied pH-logfO₂ fields. The 3D plots in Figure 6(a) show that gold solubility can have a positive correlation with salinity in acidic fluid dominated by AuCl₂⁻ since salinity can have a great effect on the concentration of AuCl₂⁻. Figure 6(b) shows that ARC values range from 0 to 1.04 in oxidized fluids and from 0 to 0.43 in reduced fields, indicating that a decrease in gold solubility by one order of magnitude is possibly caused by a decrease in salinity of ≥9.6 wt% in oxidized fluids and ≥23.2 wt% in reduced fluids. Previous studies [73-75] have reported that salinities in geofluids and gold deposits commonly change within 10 wt% NaCl eq. A variation in salinity (≥10 wt% NaCl eq.) is more likely to occur in the underground brine or the ore-forming fluid of porphyry gold deposits [6, 76, 77], and such mixing with surface water is required for dilution. Thus, a positive correlation between salinity and gold solubility is more likely to occur in a specific fluid with a high salinity that is dominated by AuCl₂⁻, and large-scale gold mineralization may be caused by a sharp decrease in salinity in a porphyry deposit that particularly precipitates hematite and pyrite.

The plots in Figure 7(a) show that gold solubility has a positive correlation with m_ΣS in the reduced fields since m_ΣS can have a great effect on the concentrations of AuHS_(aq) and Au(HS)₂⁻. Figure 7(b) shows that the highest ARC is 0.7 log m_ΣAu /0.1 mol. kg⁻¹ in the reduced field, indicating that a decrease in gold solubility by one order of magnitude is possibly caused by a decrease in m_ΣS of ≥ 0.14 mol. kg⁻¹. According to previous studies on m_ΣS in most geofluids and gold deposits [46, 67, 78-81], the above variation in m_ΣS is more likely to occur in high-sulfidation fluids with a magma source, and mixing with surface water is required to dilute the sulfur concentration. For example, Hu et al. [82] published that gold precipitation around a submarine “hydrothermal chimney” is caused by a rapid decrease in m_ΣS when magmatic fluid is mixed with seawater. Thus, a positive correlation between m_ΣS and gold solubility is more likely to occur in a specific fluid with a high m_ΣS that is dominated by AuHS⁰ and Au(HS)₂⁻, and large-scale gold mineralization may be caused by a sharp decrease in m_ΣS in a high-sulfide deposit that can also precipitate pyrite, pyrrhotite, and magnetite.

Based on the above discussion, 3D plots of gold solubility can be rapidly built to quantify the correlation between gold solubility and ore-forming conditions, indicating that a sharp decrease in temperature may be the most effective mechanism for gold precipitation. Although the results of this study are illustrated under fixed conditions, it is expected that the 3D plotting of gold solubility can be applicable to other studies on gold precipitation.

The 3D plots with GF as a variable can provide an effective method to illustrate GF as a geochemical indicator for ore-forming fluids in hydrothermal gold deposits. The 3D plots in Figure 8(a) indicate a positive correlation between GF and R, and the correlation can be checked by statistical research for different types of gold deposits [2, 3, 20, 33]. For example, orogenic-type gold deposits commonly produce native gold with high GF (Figure 13), which is probably due to the high R values (mostly >0) identified in those gold deposits [3, 6, 43, 44, 83]. In contrast, volcanogenic massive sulfide (VMS)-, skarn-, and porphyry-type gold deposits (Figure 13) related to magmatic intrusions often contain electrum with low GF [2, 3, 20, 47], which may be because of the low R values (mostly ≤−1) in magmatic fluids. Thus, R plays a decisive role in the ore-forming fluids of gold deposits, and it has an essential effect on GF. The varying curves of GF–R in Figure 8(d) present a function that is unaffected by other conditions. By knowing the 2D data (pH-log fO₂) of R isopleths, the pH - log fO₂ - GF plots can thus be rapidly plotted by the function, and 3D plots display the variation in R based on the determined range of GF.

The quantification of 3D plots illustrates that both pressure and temperature show a positive correlation with GF, but the ARC is varied at different pH - log fO₂ conditions and R values. The 3D plots in Figure 9(a) indicate a positive correlation of P - GF that is more obvious in field IV because the log K of Au(HS)₂⁻ has a higher dependence on pressure than that of AuHS⁰, resulting in an increase in R values. The quantifications in Figures 9(b) and 9(c) both show that the ARC is higher (the highest ARC is 72 GF/kb) in field IV when R values range from −1 to −2, indicating that a decrease in GF by 100 units is possibly caused by a decrease in pressure of ≥1.4 kbar. Thus, gold with higher fineness is more likely to be formed at higher pressure, and GF can vary over a large range (>500) if the pressure changes by >7 kbar. According to the pressure gradient in the lithosphere (0.27 kbar/km), 7 kbar in pressure is equal to ~25 km in depth. By comparing the variation in pressures in geofluids and gold deposits mentioned above, pressure is unlikely to be an effective factor resulting in a large range of GF in hydrothermal gold deposits.

The 3D plots in Figure 9(a) indicate a positive correlation between GF and temperature that is more pronounced in field III, where AuHS⁰ and AgCl₂⁻ are dominant. This result is because AuOH⁰ and AgCl₂⁻ have different dependencies on the temperature in field III, resulting in a higher increase in R than in field IV, where Ag(HS)₂⁻ is dominant. The quantifications in Figures 10(b) and 10(c) show that the ARC is high (the highest ARC is 73 GF/10℃) at pH values ranging from 4 to 6 and R values ranging from −2 to −4, indicating that a decrease in GF by 100 units is possibly caused by a decrease in temperature of ≥14℃. Thus, a higher GF is more likely to be formed in fluids with higher temperatures, and GF can vary over a large range (>500) if the temperature changes by ≥68.5℃. By comparing the variation in temperatures in the geofluids and gold deposits mentioned above, temperature might be an effective factor controlling the variation in GF in the Ag-rich gold deposits associated with pyrite or pyrrhotite. For example, a range of GF greater than 500 is often reported in epithermal- and skarn-type gold deposits (Figure 13) that commonly contain silver minerals, sericite, and pyrite, which may be due to a large variation in temperatures and lower R values in those gold deposits. Thus, the indication of GF for temperature is more effective than that for pressure, and 3D plotting can provide numerical analysis to explain the indication.

The quantification of the 3D plots illustrates that m_ΣS and m_ΣCl may have distinct correlations with gold solubility, and the ARC of the correlation is distinct in varied pH-fO₂ fields. The reason for the above result is similar to that for gold solubility, that is, m_ΣCl and m_ΣS can have different effects on m_ΣAu and m_ΣAg. The 3D plots in Figure 11(a) indicate a positive correlation between GF and salinity that is more obvious in field III. The quantification in Figure 11(b) shows that the ARC in field III is high at R > −2 and pH = 4–6 (the highest ARC is 25 GF/wt% NaCl eq.), indicating that a decrease in GF by 100 units is possibly caused by a decrease in the salinity of ≥4 wt% NaCl eq. in field III. Thus, a higher GF is more likely to be formed in fluid with a higher salinity, and GF can vary in a large range (>500) if the salinity changes greatly (such as 20 wt% NaCl eq.). In comparison with the variations in salinities in the geofluids and gold deposits mentioned above, salinity is more likely to have a greater effect on the GF in specific fluid with a high salinity. For example, the native gold produced in porphyry-type gold deposits and skarn-type gold deposits (Figure 13) commonly shows high GF with a smaller GF range than other types of gold deposits of magmatic origin since the salinity of ore-forming fluids is higher in porphyry-type deposits and skarn-type gold deposits than in other types.

The 3D plots in Figure 12(a) indicate a negative correlation between GF and m_ΣS that is more obvious in field III. The quantification in Figure 12(b) shows that the ARC in field III is low at R < −1 (the lowest ARC is −155 GF/0.1 mol. kg⁻¹), indicating that a decrease in GF by 100 units is possibly caused by an increase in m_ΣS ≥ 0.07 mol. kg⁻¹ in field III. Thus, a higher GF is more likely to be formed in fluids with a lower m_ΣS, and the GF can vary over a large range (>500) if the m_ΣS changes by more than 0.33 mol/kg. By comparing the variation in m_ΣS in the geofluids and gold deposits mentioned above, m_ΣS is more likely to have a greater effect on the GF in specific fluid with a high m_ΣS. For example, VMS-type gold deposits usually contain electrum with lower GF (Figure 13), which may be formed by ore-forming fluids with high m_ΣS values. Conversely, the orogenic gold deposits contain native gold with higher GF (Figure 13), which may be formed by some geologic parameters, such as fluid boiling triggered by fault-valve processes. Li et al. [84] studied the fluid boiling triggered by fault-valve processes in orogenic gold deposits and concluded that a decrease in logfS₂ during fluid boiling can increase the fluid pH, resulting in a decrease in the solubilities of base metal chloride complexes and thus explaining the coprecipitation of Au and base metal sulfides. In this case, a decrease in logfS₂ and an increase in fluid pH can both result in an increase in GF according to our results.

Thus, indications of GF for salinity and sulfur concentration are both effective in the specific fluid under the restrictive condition of high m_ΣCl and m_ΣS, respectively, and 3D plotting is thus useful to quantify the indication of GF in those specific deposits.

Based on the above discussion, the 3D plots in this study can be rapidly built to quantify the correlation between GF and ore-forming conditions, suggesting that temperature might be an effective factor to change GF in most hydrothermal gold deposits, and other conditions can be effective in specific deposits. Although the results of this study are for fixed conditions, it can be expected that the 3D plotting of gold solubility and GF is applicable to other studies of gold deposits. The role of gold remobilization in changes in GF is determined by the physicochemical conditions during remobilization, which can also be analyzed by this study.

The conclusions of this study are summarized as follows:

1. The correlation between gold solubility and GF can be expressed by the function of log m_ΣAu = 0.495 * ln (GF) – 3.5 – a, where a is the log m_ΣAu at GF = 999, thereby providing an efficient method to build 3D plots of gold solubility.

2. Gold solubility may have a positive correlation with GF, temperature, salinity and sulfur concentration, and a negative correlation with pressure. The ARC of correlation can be varied at different pH-logfO₂ conditions.

3. The quantification of 3D plots in this study illustrates that a decrease in gold solubility by one order of magnitude is possibly caused by a decrease in temperature of ≥40℃, salinity of ≥9.6 wt% NaCl eq. or sulfur concentration of ≥0.14 mol/kg, or by an increase in pressure of ≥3 kbar, indicating that temperature might be an effective factor for gold precipitation, while the other conditions can be effective in specific fluids under restricted conditions, such as high salinity or a high sulfur concentration.

4. The correlation between GF and the ratio of all dissolved gold and silver species can be expressed by the two functions of R = 0.476 * ln (GF) + b − 0.07 and R = −0.476 * ln (1000 − GF) + c + 0.07, where b and c are the R values at GF = 1 and GF = 999, respectively, thereby providing an efficient method to build 3D plots of GF.

5. GF may have a positive correlation with R, temperature, salinity, and pressure and a negative correlation with the sulfur concentration, and the ARC of correlation is distinct at varied pH-logfO₂ fields since dominant gold and silver complexes have different dependencies on the conditions, resulting in variations in R.

6. The quantification of 3D plots in this study illustrates that a decrease in GF by 100 units is possibly caused by a decrease in temperature of ≥14℃, pressure of ≥1.4 kbar or salinity of ≥4 wt% NaCl eq. or by an increase in sulfur concentration of ≥0.07 mol/kg, indicating that temperature might be an effective factor to change GF, while the other conditions can be effective in the specific deposits.

This study was financially supported by the Natural Science Foundation of China (41702077) and the Fundamental Research Funds for the Central Universities (Grant no. 2682016CX090). We are grateful to Dr. Junyan Yu (Kyushu University) for his constructive suggestions. Great thanks to Prof. Yi He (Southwest Jiaotong University) for his insightful discussion.

The authors declare that they have no conflicts of interest.

The data are provided in supplementary files.

Yi Liang: Conceptualization, Methodology, Thermodynamic calculation, 3D plotting, Validation, Writing-original draft, Writing - Review and Editing, and Supervision;

Hangfei Ge: Methodology, Thermodynamic calculation, 3D plotting, Software, Data Curation, Validation, Visualization, Writing-original draft, and Co-supervision;

Qiuming Pei: Conceptualization, Methodology, and Writing - Review and Editing;

Haonan Huang: Thermodynamic calculation, 3D plotting, and Data Curation;

Kenichi Hoshino: Methodology, Software, and Thermodynamic calculation.