A model to predict the thermodynamic stability of abiotic methane-hydrogen binary hydrates in a marine serpentinization environment

Abiotic methane (CH₄) and hydrogen (H₂), which are produced during marine serpentinization, provide abundant gas source for hydrate formation on ocean floor. However, previous models of CH₄–H₂ hydrate formation have generally focused on pure water environments and have not considered the effects of salinity. In this study, the van der Waals–Platteeuw model, which considered the effects of salinity on the chemical potentials of CH₄, H₂, and H₂O, was applied in a marine serpentinization environment. The model uses an empirical formula and the Peng–Robinson equation of state to calculate the Langmuir constants and fugacity values, respectively, of CH₄ and H₂, and it uses the Pitzer model to calculate the activity coefficients of H₂O in the CH₄–H₂–seawater system. The three-phase equilibrium temperature and pressure predicted by the model for CH₄–H₂ hydrates in pure water demonstrated good agreement with experimental data. The model was then used to predict the three-phase equilibrium temperature and pressure for CH₄–H₂ hydrates in a NaCl solutions, for which relevant experimental data are lacking. Thus, this study provides a theoretical basis for gas hydrate research and investigation in areas with marine serpentinization.

1 Introduction

Natural gas hydrates are crystalline solids formed from a mixture of water and gases. Natural gas hydrates are not only a new clean energy resource, but also have an important role in environmental effects and marine hazard assessment (Wan et al., 2022). The microstructure comprises cavities (hosts) formed by water molecules through hydrogen bonding and gas molecules (guests) trapped inside (Dendy Sloan and Koh, 2007). Typical gas molecules include methane (CH₄), ethane, propane, and carbon dioxide, which can be biotic or abiotic in origin. Serpentinization, which primarily occurs at mid-ocean ridges and fore-arc systems, plays an important role in producing abiotic gases (Holm et al., 2015). Serpentinization is the hydration of olivine and orthopyroxene minerals, the main constituents of ultramafic rocks, creating a reducing chemical environment characterized by high H₂ concentrations. The general reaction equation is: $\mathrm{6[}M{g}_{1.8}F{e}_{0.2}Si{O}_4]+7H_{2}O\to \mathrm{3[}M{g}_{3}S{i}_2{O}_5{(OH)}_4]+F{e}_3{O}_4+H_2$. Excess H₂ reduces CO₂ dissolved in water to CH₄ and low-molecular-weight hydrocarbons through Fischer-Tropsch type (FTT) reactions, The general reaction equation is: $C{O}_{2aq}+[2+(m/2n)]H_2\to (1/n)C_n{H}_m+2H_{2}O$ (Proskurowski et al., 2008). The ultraslow-spreading ridges in the Fram Strait between the North Atlantic Ocean and Arctic Ocean are a typical serpentinization area with bottom simulating reflectors (BSRs) in their seismic profiles, which is characteristic of CH₄ hydrate development (Rajan et al., 2012; Johnson et al., 2015). In addition to CH₄, serpentinization produces substantial amounts of H₂ (Coveney et al., 1987; McCollom and Bach, 2009). The formation of hydrates from H₂ has been a well-explored research topic in recent years. Experimental studies have shown that the pressure required to form stable H₂ hydrates is hundreds of times higher than that of CH₄ hydrates under the same low-temperature conditions (Dyadin et al., 1999; Mao et al., 2002). However, the pressure required to form H₂ hydrates can be effectively reduced by mixing in a small amount of a second guest molecule such as tetrahydrofuran (THF) (Hashimoto et al., 2006). Similar to THF, CH₄ can be used as a thermodynamic promoter to stabilize H₂ hydrate formation, and it is small enough to afford H₂ a higher occupancy in hydrate cages compared to THF (Matsumoto et al., 2014). This effect of the second guest molecule may allow CH₄–H₂ hydrates to form from the abundance of abiotic CH₄ and H₂ produced by serpentinization. To confirm this possibility, the three-phase (hydrate, liquid, and vapor) equilibrium conditions require to be determined for the formation of CH₄–H₂ hydrates in a serpentinization environment.

Researchers have experimentally measured the phase equilibrium conditions for forming CH₄–H₂ hydrates at various molar fraction ratios. Holder et al. (1983) measured the phase equilibrium conditions for the formation of H₂-rich gas hydrates at temperatures of<282.3 K and concluded that hydrate formation is strongly dependent upon the gas composition. Zhang et al. (2000) used the pressure search method to measure the phase equilibrium conditions for hydrate formation from H₂ and hydrocarbon gas mixtures in pure water. They considered temperature and pressure ranges of 274.3–278.2 K and 3.72–6.63 MPa, respectively, and H₂ molar fractions of 22.13 and 36.18 mol%. Their results suggested that increasing H₂ molar fractions was not conducive to the formation of CH₄–H₂ hydrates. Measurements by Chen et al. (2002) demonstrated that the pressure range for CH₄–H₂ hydrate formation at H₂ molar fractions of 22–70 mol% and a temperature of 274.15 K was 3.72–9.67 MPa. Skiba et al. (2007) used differential thermal analysis to investigate the phase equilibrium of CH₄–H₂ hydrates at H₂ molar fractions of 0–70 mol% and a pressure of up to 250 MPa. Their results suggested that the decomposition temperature of the formed hydrate decreased as the H₂ concentration in the initial gas mixture increased. Pang et al. (2012) injected gas or gas mixtures into an equilibrium cell containing an appropriate amount of water and at a constant pressure. They nucleated and decomposed hydrates by adjusting the temperature, where the equilibrium temperature was defined as the point at which hydrates appeared to melt for the second time. The phase equilibrium conditions for CH₄–H₂ hydrates were measured for H₂ molar fractions of 5–66 mol%. Li et al. (2022) used the isochoric pressure-search method to measure the phase equilibrium conditions for CH₄–H₂ hydrates in the temperature range of 274.24–287.43 K at H₂ molar fractions of 22 and 80 mol%. Additionally, Researchers have established several models to obtain a wider range of phase equilibrium data. Skiba et al. (2007) obtained the coefficients of the equation $T(°C)=A+B\times P+C\times P^2+D\times P^3+E\times \ln P(P,MPa)$ by fitting the experimental data using the method of least squares. The equation demonstrated good agreement with the experimental results of pure CH₄ hydrates and initial gas mixtures with H₂ molar fractions of<40 mol%. However, the difference between the predicted values and experimental results considerably widened when the H₂ molar fractions was >40 mol%. Pang et al. (2012) used the Ng–Robinson model (Ng and Robinson, 1976) to establish a model that can predict the three-phase equilibrium conditions for the formation of CH₄–H₂ hydrates at different gas molar fractions. Li et al. (2022) used the Chen–Guo model (Chen and Guo, 1998) to establish a model that can predict the three-phase equilibrium conditions for the formation of multi-component mixtures from different molar fractions of H₂ with one or more hydrocarbons.

The abovementioned models can accurately predict the three-phase equilibrium conditions for the formation of multi-component gas hydrates from H₂ in a pure water system. However, salinity can inhibit hydrate formation, and currently available models cannot accurately predict the three-phase equilibrium conditions for CH₄–H₂ hydrates in a marine serpentinization environment. Although various abiotic and microbially mediated reactions affect the chemical compositions of pore waters in serpentinization surface sediments, resulting in some differences in major and trace elements, the pore water ions are still dominated by NaCl and are generally similar in species to those found in the upper layers of ocean water (Hulme et al., 2010). For example, The Ocean Drilling Program Site 1200, located on the South Chamorro Seamount, has highly permeable and strongly alkaline (pH 12.5) in the deep pore water compared to other serpentinization areas. The contents of chloride, magnesium, and calcium ions are lower while the alkalinity and contents of sodium, potassium, sulfate, and light hydrocarbon ions are significantly higher compared with the upper layers of ocean water. The content of sodium and chloride ions in this region are above 500mmol/kg, potassium and sulfate ions are 10-20mmol/kg, while magnesium and calcium ions are less than 5mmol/kg. Therefore, sodium and chloride ions are still the leading components, and Na/Cl can reach a maximum of 1.2 (Salisbury et al., 2002). The effects of alkalinity and other ions on hydrate formation are essentially negligible compared with the effects of sodium and chloride ions. Thus, a thermodynamic model that can predict the three-phase equilibrium conditions for the formation of CH₄–H₂ hydrates in both pure water and a sodium chloride (NaCl) solution is required. In this study, the van der Waals–Platteeuw thermodynamic model of classical adsorption theory was used to establish a thermodynamic model that can accurately predict the temperature and pressure conditions for the three-phase equilibrium of CH₄–H₂ hydrates in a marine serpentinization environment. The model incorporates the molecular potential energy model to consider the effects of temperature, pressure, and salinity. The performance of the model was evaluated by comparison with experimental data in the literature.

2 Thermodynamic model of gas hydrates

At phase equilibrium, the chemical potential or fugacity of each component in the system is identical in various phases. Water has low volatility and is not compatible with hydrocarbons, and therefore it generally accounts for a low proportion of the vapor and liquid phases of hydrocarbons. Therefore, the chemical potentials of water in the hydrate and liquid phases can be selected as the constraints. If water is selected as the reference component, the phase equilibrium constraint is given by

$\begin{array}{ll}{\mu }_{\text{w}}^H={\mu }_{\text{w}}^L& (1)\end{array}$

where ${\mu }_{\text{w}}^H$ is the chemical potential of water in the hydrate phase (J/mol) and ${\mu }_{\text{w}}^L$ is the chemical potential of water in the liquid phase (J/mol).

If the chemical potential of water in the hypothetical empty hydrate lattice is selected as the intermediate state, the constraints are given by

$\begin{array}{ll}\Delta {\mu }_{\text{w}}^H={\mu }_{\text{w}}^{\beta }-{\mu }_{\text{w}}^H={\mu }_{\text{w}}^{\beta }-{\mu }_{\text{w}}^L=\Delta {\mu }_{\text{w}}^L& (2)\end{array}$

where ${\mu }_{\text{w}}^{\beta }$ is the chemical potential of water in the hypothetical empty hydrate lattice (J/mol). $\Delta {\mu }_{\text{w}}^H$ is the difference between the chemical potentials of water in the empty hydrate and hydrate phases (J/mol). $\Delta {\mu }_{\text{w}}^L$ is the difference between the chemical potentials of water in the empty hydrate and liquid phases (J/mol).

Determining the structural type for CH₄–H₂ hydrates is crucial in this approach, but the thermodynamic properties of CH₄–H₂ hydrates are currently not well understood. From the available studies, pure H₂ hydrates tend to naturally form Structure II (Mao et al., 2002). However, as the initial molar fractions of CH₄ increases, there is a higher likelihood that Structure I will become more favorable (Grim et al., 2012). Based on the dependence of hydrate structure on thermodynamic and kinetic conditions, the hydrate structure of the CH₄ + H₂ mixed system is dependent on several factors: structure induction, driving force of hydrate independent nucleation, composition of gas mixture, pressure, and formation period (Matsumoto et al., 2014; Gao et al., 2022). Considering the pressure and gas composition in the serpentinization area, Structure I was finally chosen as the calculation standard. $\Delta {\mu }_{\text{w}}^H$ can be calculated as per the statistical mechanics model proposed by van der Waals and Platteeuw (2007):

$\begin{array}{ll}\Delta {\mu }_{\text{w}}^H=RT{\displaystyle \sum _i{\nu }_{\text{i}}\ln }(\mathrm{1+}{\displaystyle \sum _{j=1}{C}_{ij}{f}_j})& (3)\end{array}$

where R is the universal gas constant (8.314 J/mol/K), T is the temperature (K), ${\nu }_{\text{i}}$ is the number of type i cages in each water molecule (Table 1), $C_{ij}$ is the temperature-dependent Langmuir constant of the gas component j in type i cavities, and $f_j$ is the fugacity of the gas component j in the hydrate phase (MPa).

TABLE 1

Table 1 Values of v_i in two types of hydrate cages.

The Langmuir constant is a critical parameter of the van der Waals–Platteeuw model and depends on the chemical potential for the interaction between guest and water molecules. For Structure I hydrates, the Langmuir constants of CH₄ and H₂ are calculated from the empirical equations fitted by Sun and Duan (2007) and Klauda and Sandler (2003), respectively.

$\begin{array}{ll}{C}_{ij}(T)=e^{A_{ij}+\frac{B_{ij}}{T}}& (4)\end{array}$

$\begin{array}{ll}{C}_{ij}(T)=e^{A_{ij}+\frac{B_{ij}}{T}+\frac{D_{ij}}{T^2}}& (5)\end{array}$

where the values of A, B, and D are given in Table 2.

TABLE 2

Table 2 Calculated parameters of Langmuir constant.

Another important aspect of an accurate thermodynamic model is calculating the fugacity of the gas component. When three phases coexist in equilibrium, the fugacity values of gas j in the hydrate, liquid water, and vapor phases are identical:

$\begin{array}{ll}{f}_j^H=f_j^L=f_j^V& (6)\end{array}$

where the superscripts H, L, and V indicate the hydrate, liquid, and vapor phases, respectively. In this study, the fugacity values of CH₄ and H₂ in the vapor phase were calculated using the equation of state for gas mixtures proposed by Peng and Robinson (1970). $\Delta {\mu }_{\text{w}}^L$ can be calculated by using the equations proposed by Holder et al. (1980):

$\begin{array}{ll}\frac{\Delta {\mu }_w^L}{RT}=\frac{\Delta {\mu }_w^0}{R{T}_0}-{\displaystyle {\int }_{T_0}^T\frac{\Delta h_w}{R{T}^2}}dT+{\displaystyle {\int }_0^P\frac{\Delta V_w}{RT}}dp-\ln ({\alpha }_w)& (7)\end{array}$

$\begin{array}{ll}\Delta h_w=\Delta h_w^0+{\displaystyle {\int }_{T_0}^T\Delta C_{pw}}dT& (8)\end{array}$

$\begin{array}{ll}\Delta C_{pw}=\Delta C_{pw}^0+b(T-T_0)& (9)\end{array}$

where $\Delta {\mu }_{\text{w}}^0$ is the difference between the reference chemical potentials of water in the empty hydrate and ice phases at the reference temperature $T_0$ (generally 273.15 K) and zero pressure (J/mol). $\Delta h_{\text{w}}$ is the difference between the molar enthalpies of the empty hydrate lattice and liquid or ice phase of pure water (J/mol). $\Delta V_{\text{w}}$ is the difference between the molar volumes of the empty hydrate lattice and the liquid or ice phase of pure water (m³/mol). ${\alpha }_w$ is the water activity. $\Delta C_{pw}$ is the difference between the molar heat capacities of the empty hydrate lattice and liquid or ice phase of pure water (J/mol/K). Table 3 presents the above parameters.

TABLE 3

Table 3 Thermodynamically relevant parameters of Structure I hydrate at T_{0 =}273.15 K.

To calculate the three-phase equilibrium pressure at a given temperature T, the molar fractions of H₂ (mol%), and salinity (mol/kg), the initial pressure P₁ is estimated first. Then, $\Delta {\mu }_{\text{w}}^H$ and $\Delta {\mu }_{\text{w}}^L$ are calculated and compared at P₁. If the absolute difference between $\Delta {\mu }_{\text{w}}^H$ and $\Delta {\mu }_{\text{w}}^L$ is sufficiently small, then the pressure can be considered the equilibrium pressure P under the above conditions. Otherwise, the pressure is modified, then $\Delta {\mu }_{\text{w}}^H$ and $\Delta {\mu }_{\text{w}}^L$ are calculated at the modified pressure. The abovementioned processes are repeated by using Newton’s method or dichotomy until the equilibrium pressure is determined. If the absolute difference between $\Delta {\mu }_{\text{w}}^H$ and $\Delta {\mu }_{\text{w}}^L$ is less than 1 × 10⁻², then the iteration is terminated, and an equilibrium pressure is obtained with a deviation of<0.1%.

2.1 Fugacity calculation

As described above, the fugacity values of CH₄ and H₂ can be calculated by using the equation of state for gas mixtures proposed by Peng and Robinson (1970):

$\begin{array}{ll}P=\frac{RT}{\nu -\text{b}}-\frac{a(T)}{\nu (\nu +\text{b)+b(}\nu -\text{b)}}& (10)\end{array}$

Equation 11 presents rules by which Equation 10 can be rearranged in the form of Equation 12:

$\begin{array}{ll}A=\frac{aP}{R^2{T}^2};B=\frac{bP}{RT};Z=\frac{Pv}{RT}& (11)\end{array}$

$\begin{array}{ll}{Z}^3-(1-B)Z^2+(A-3B^2-2B)Z-(AB-B^2-B^3)=0& (12)\end{array}$

where R is the universal gas constant (8.314 J/mol/K), $v$ is the gas molar volume, a is a measure of the intermolecular attraction, and b is a constant related to the gas molecule size. The values of a and b at the critical point can be obtained as per the critical properties of gases:

$\begin{array}{ll}{T}_r=\frac{T}{T_c};a(T_c)=0.45724\frac{R^2{T}_c^2}{P_c};b(T_c)=0.07780\frac{R{T}_c}{P_c};Z_{\text{c}}=0.307& (13)\end{array}$

$\begin{array}{ll}a(T)=a(T_c)·\alpha (T_r,w);b(T)=b(T_c)& (14)\end{array}$

$\begin{array}{ll}\sqrt{\alpha (T_r,w)}=1+\kappa (1-\sqrt{T_r})& (15)\end{array}$

$\begin{array}{ll}\kappa =0.37464+1.54226\omega -0.26992{\omega }^2& (16)\end{array}$

where $\kappa$ is a constant characteristic of each substance. where the critical temperature T_c of H₂ is 33.2 K, the critical pressure P_c is 1.3 MPa, and the acentric factor $\omega$ is −0.216. The critical temperature T_c of CH₄ is 190.4 K, the critical pressure P_c is 4.6 MPa, and the acentric factor $\omega$ is 0.012. The above method is followed when calculating pure gases, but certain rules should be followed when calculating gas mixtures. The mixing rule is as follows:

$\begin{array}{ll}\begin{array}{l}a={\displaystyle \sum _i{\displaystyle \sum _j{x}_i{x}_j{a}_{ij}}}\\ b={\displaystyle \sum _i{x}_i{b}_i}\\ a_{ij}=(1-{\delta }_{ij})\sqrt{a_i}\sqrt{a_j}\end{array}& (17)\end{array}$

In eq 17, $x$ is the initial molar fractions of component i and component j in the gas mixture. ${\delta }_{ij}$ is an empirically determined binary interaction coefficient characterizing the binary formed by component i and component j. The value ${\delta }_{ij}$ is 0.9035 between component H₂ and component CH₄ (Matsumoto et al., 2014). As per the state parameters and mixing rule of CH₄ and H₂, the critical values of a and b can be calculated by Equation 17. Then, the fugacity coefficients $f_k$ of H₂ and CH₄ can be calculated from the following equation:

$\begin{array}{ll}\ln \frac{f_k}{x_{k}P}=\frac{b_k}{b}(Z-1)-\ln (Z-B)-\frac{A}{2\sqrt{2}B}\times (\frac{2{\displaystyle \sum _i{x}_i{a}_{ik}}}{a}-\frac{b_k}{b})\ln (\frac{Z+2.414B}{Z-0.414B})& (18)\end{array}$

2.2 Water activity calculation

The water activity a_w in Equation 7 is calculated using the Pitzer model (Pitzer, 1975). The relationship between a_w and the permeability coefficient $\varphi$ is denoted by the following equation:

$\begin{array}{ll}\ln a_w=-\frac{M_W}{1000}({\displaystyle \sum _i{m}_i})\varphi & (19)\end{array}$

where $M_W$ is the molecular weight of water. $m_i$ is the molality of solute i, which can be cations, anions, or neutral substances. $\varphi$ is the permeability coefficient, which was first proposed by Pitzer and Silvester (Pitzer and Silvester, 1976) and was eventually rearranged by Harvie et al. (1984) and Felmy and Weare (1986) to obtain:

$\begin{array}{ll}\begin{array}{l}({\displaystyle \sum _i{m}_i})(\varphi -1)=2\{-\frac{A^{\varphi }{I}^{1.5}}{1+1.2I^{1.5}}+{\displaystyle \sum _c{\displaystyle \sum _a{m}_c{m}_a(B_{ca}^{\varphi }+Z{C}_{ca}}})\\ +{\displaystyle \sum {\displaystyle \sum _{c<c\text{'}}{m}_c{m}_{c\text{'}}({\Phi }_{cc\text{'}}^{\varphi }+{\displaystyle \sum _a{m}_a{\psi }_{cc\text{'}a}})}}+{\displaystyle \sum {\displaystyle \sum _{a<a\text{'}}{m}_a{m}_{a\text{'}}({\Phi }_{aa\text{'}}^{\varphi }+{\displaystyle \sum _c{m}_c{\psi }_{aa\text{'}c}})},}\\ +{\displaystyle \sum _n{\displaystyle \sum _c{m}_n{m}_c{\lambda }_{nc}+{\displaystyle \sum _n{\displaystyle \sum _a{m}_n{m}_a{\lambda }_{na}+{\displaystyle \sum _n{\displaystyle \sum _c{\displaystyle \sum _a{m}_n{m}_c{m}_a}}}}}}}{\zeta }_{nca}\}\end{array}& (20)\end{array}$

where I is the ionic strength. The subscripts c, a, and n are cations, anions, and neutral substances, respectively. The summation index, c, denotes the sum over all cations in the system. The double summation index, c< c’, denotes the sum over all distinguishable pairs of dissimilar cations. Analogous definitions apply to the summation indices for anions. $A^{\varphi }$ is one-third of the Debye–Hückel limiting slope. $B^{\varphi }$, ${\Phi }^{\varphi }$, and $\lambda$ are measurable combinations of the second virial coefficient. $C$, $\psi$, and $\zeta$ are measurable combinations of the third virial coefficient.

The second virial coefficients $B_{ca}^{\varphi }$ and ${\Phi }_{cc\text{'}}^{\varphi }$ are functions of the ionic strength, and the third virial coefficients $C_{ca}$ and ${\psi }_{cc\text{'}a}$ are assumed independent of the ionic strength. Duan and Sun (2006) described the equations for calculating the above parameters in detail. Because gas hydrates in an aqueous solution of electrolytes exist in equilibrium at low temperatures of -25 to 25°C, the relevant parameters determined by Spencer et al. (1990) were selected. The temperature-dependent ion interaction parameters in the Pitzer model are expressed as follows (The values of c₁-c₆ can be found in the paper of Spencer et al. (1990)):

$\begin{array}{ll}Par(T)=c_1+c_{2}T+\frac{c_3}{T}+c_4\ln T+c_5{T}^2+c_6{T}^3.& (21)\end{array}$

The effect of pressure on activity coefficients at a specific temperature should be included in the theoretical calculation. However, Monnin (1990) reported that the effect of pressure on the activity of water is small and can be ignored. Duan and Sun (2006) also confirmed this result from temperature- and pressure-dependent parameters for aqueous NaCl solutions. Thus, this model will neglect the effect of pressure on water activity.

The second virial coefficient ${\lambda }_{ni}$ and third virial coefficient ${\zeta }_{nij}$ are the interactions between ions and neutral substances (Duan and Sun (2006)). ${\lambda }_{C{H}_4-i}$ and ${\zeta }_{C{H}_4-ij}$ have been determined by the CH₄ solubility model established by Duan and Mao (2006), and ${\lambda }_{H_2-i}$ and ${\zeta }_{H_2-ij}$ have been determined by the H₂ solubility model established by Zhu et al. (2022). Duan and Sun (2006) set ${\lambda }_{C{H}_4-Cl}$ to zero and fitted ${\lambda }_{C{H}_4-Na}$ and ${\zeta }_{C{H}_4-Na-Cl}$ based on the solubility of CH₄ in an aqueous NaCl solutions. All interaction parameters between CH₄ and monovalent and divalent cations can be approximated as ${\lambda }_{C{H}_4-Na}$ and $2{\lambda }_{C{H}_4-Na}$, respectively. ${\lambda }_{H_2-Cl}$, ${\lambda }_{H_2-Na}$, and ${\zeta }_{H_2-Na-Cl}$ are treated by the same method, where all interaction parameters between H₂ and monovalent and divalent cations approximated as ${\lambda }_{H_2-Na}$ and $2{\lambda }_{H_2-Na}$, respectively.

$\begin{array}{ll}\begin{array}{c}{\lambda }_{C{H}_4-Na}=-0.81222036+0.10635172\times {10}^{-2}T+0.18894036\times {10}^3\frac{1}{T}\\ +0.44105635\times {10}^{-4}P-0.46797718{10}^{-10}{P}^{2}T\end{array}& (22)\end{array}$

$\begin{array}{ll}\begin{array}{l}{\lambda }_{H_2-Na}=-7.68559552+1.91233146\times {10}^{-2}T+1.04890475\times {10}^3\frac{1}{T}-1.52746819\times {10}^{-5}{T}^2\\ +1.59803686\times {10}^{-4}P-19.2667249\frac{P}{T^2}-47.5822792\frac{1}{P}+0.472712503\frac{T}{P}\\ -1.56750050\times {10}^{-3}\frac{T^2}{P}+1.73272315\times {10}^{-6}\frac{T^3}{P}\end{array}& (23)\end{array}$

$\begin{array}{ll}{\zeta }_{C{H}_4-Na-Cl}=-0.29903571\times {10}^{-2}& (24)\end{array}$

$\begin{array}{ll}{\zeta }_{H_2-Na-Cl}=-1.44839161\times {10}^{-2}& (25)\end{array}$

In summary, Equation 19 and 20 form the fundamental equations of the model for predicting the stability of gas hydrates in an aqueous solution of electrolytes. All parameters in these equations have been assessed by researchers. The phase equilibrium data for gas hydrate formation in an aqueous solution of electrolytes should not be adjusted before use.

3 Results and discussion

At present, only experimental data of CH₄–H₂ hydrates in pure water are available. In this study, 60 data points were collected for H₂ molar fractions of 4.55%–80.00%. The model was used to predict the pressure necessary to form CH₄–H₂ hydrates at a specific temperature. The model predictions in pure water were confirmed against experimental data from the literature, and deviations of the predicted results from the experimental data were tabulated (Table 4). The minimum and maximum mean absolute percentage errors (MAPEs) between the predicted results and experimental data were 0.86% and 18.03%, respectively. MAPE was >10.00% only at H₂ molar fractions of 36.18% and 65.90%.

TABLE 4

Table 4 Errors occurring when model predicts conditions for hydrate formation in pure water.

To confirm the prediction accuracy of the model, the pressures for forming CH₄–H₂ hydrates at temperatures of 274.15–293.15 K were calculated at different H₂ molar fractions (Figure 1). The three-phase equilibrium pressure increased with the temperature regardless of the initial H₂ molar fractions. For example, when the H₂ molar fractions was 33.85%, the three-phase equilibrium pressures were 4.59, 7.71, 13.56, 25.40, and 43.37 MPa at temperatures of 274.15, 279.15, 284.15, 289.15, and 293.15 K, respectively. The calculated pressure increments were 3.13, 5.85, 11.84, and 17.97 MPa, respectively. These results suggest that the pressure does not increase linearly with temperature. The experimental data obtained by Zhang et al. (2000) corresponded to H₂ molar fractions of 22.13% and 36.18%; these were lower than those obtained by Li et al. (2022) and Pang et al. (2012) of 20.00% and 33.85%, respectively. The model predictions were greater than the above experimental data when the H₂ molar fractions were 22.13% and 36.18%. Notably, the error increased with the H₂ molar fractions, which is consistent with the results of other researchers (Pang et al., 2012; Ma et al., 2013; Wang et al., 2015). In general, the conventional equation of state cannot accurately calculate the phase equilibrium properties because of the quantum properties of H₂ molecules, and the parameters of the equation of state of H₂ molecules may be specifically regressed (Deiters, 2013; Privat and Jaubert, 2013).

FIGURE 1

Figure 1 Comparison between model-predicted results and experimental data in pure water.

The model can predict the three-phase equilibrium conditions for CH₄–H₂ hydrates in NaCl solutions of different concentrations, for which relevant experimental data are lacking. To consider the effects of solution ions on the three-phase equilibrium of CH₄–H₂ hydrates, four initial H₂ molar fractions were selected: 4.55%, 20.00%, 28.03%, and 33.85%. Then, the three-phase equilibrium conditions were calculated for the formation of CH₄–H₂ hydrates from H₂ in pure water and in 0.1, 0.55, and 1 mol/kg NaCl solutions at temperatures of 274.15–293.15 K. Figure 2 plots the results with experimental data. Figure 2A shows that the three-phase equilibrium pressures for the formation of CH₄–H₂ hydrates at temperatures of 274.15, 283.15, and 293.15 K were 3.06, 7.73, and 27.3 MPa, respectively, in pure water; 3.15, 7.98, and 28.45 MPa, respectively, in the 0.1 mol/kg NaCl solutions; 3.49, 9.04, and 33.62 MPa, respectively, in the 0.55 mol/kg NaCl solutions; and 3.88, 10.33, and 39.94 MPa, respectively, in the 1 mol/kg NaCl solutions. Compared with pure water, the 1 mol/kg NaCl solutions increased the three-phase equilibrium pressures at 274.15, 283.15, and 293.15 K by 0.82, 2.60, and 12.67 MPa, respectively. This suggests that the three-phase equilibrium pressure increases with the temperature as well as the NaCl concentration. Note that experimental data are still required to verify the accuracy of the model in NaCl solutions, but the trend of the effect of NaCl solutions on the phase equilibrium of CH₄-H₂ hydrates is the same as that of pure methane hydrate.

FIGURE 2

Figure 2 The model predictions at H₂ molar fractions of 4.55 (A), 20.00 (B), 28.03 (C), and 33.85 mol% (D).

At the same temperature, the equilibrium pressure of CH₄–H₂ hydrates is associated with the initial molar fractions of CH₄ and H₂. Four temperatures were selected to calculate the three-phase equilibrium pressures for hydrate formation with initial H₂ molar fractions of 10–80mol%: 274.15, 280.15, 286.15, and 293.15 K. Figure 3 plots the results with experimental data obtained by Chen et al. (2002). Figure 3A shows that the three-phase equilibrium pressures at initial H₂ molar fractions of 10%, 35%, 60%, and 80% were 3.28, 4.67, 7.72, and 15.23 MPa, respectively. Based on these results, the model was used to calculate the three-phase equilibrium pressures of CH₄–H₂ hydrates in 0.1, 0.55, and 1 mol/kg NaCl solutions. At the above H₂ molar fractions, the three-phase equilibrium pressures were 3.36, 4.79, 7.92, and 15.60 MPa, respectively, in the 0.1 mol/kg NaCl solutions; 3.73, 5.33, 8.83, and 17.34 MPa, respectively, in the 0.55 mol/kg NaCl solutions; and 4.15, 5.96, 9.87, and 19.32 MPa, respectively, in the 1 mol/kg NaCl solutions. Figure 3A shows that the model predictions were consistent with experimental data in pure water. The three-phase equilibrium pressure for the formation of CH₄–H₂ hydrates was low when the initial molar fractions of CH₄ was high and increased with the H₂ content until H₂ was dominant. Then, the pressure increased to a value that cannot be reached under natural conditions. Figures 3B–D show similar trends. A higher H₂ content resulted in more demanding conditions for hydrate formation in submarine sediments, particularly in serpentinization areas.

FIGURE 3

Figure 3 The model predictions when temperatures are 274.15 (A), 280.15 (B), 286.15 (C), and 293.15 K (D).

4 Conclusions

A thermodynamic model was established based on classical adsorption theory, the van der Waals theory, and the molecular potential model to predict the three-phase (hydrate, liquid, and vapor) equilibrium temperature and pressure for the formation of CH₄–H₂ hydrates at different molar fractions of CH₄ and H₂. The model predictions were consistent with recent experimental data of CH₄–H₂ hydrates, and the proposed model demonstrated its superiority to previously established models in terms of temperature and pressure ranges as well as prediction accuracy. Moreover, the model can consider the effects of salinity on the hydrate stability. The model predictions indicated that the pressure necessary for the formation of CH₄–H₂ hydrates in saltwater increases with the temperature, NaCl concentration, and H₂ molar fractions when other conditions remain unchanged. In the sedimentary layers of serpentinization areas, the combined effects of the temperature, NaCl concentration, and H₂ molar fractions on the formation of CH₄–H₂ hydrates should be comprehensively considered. This study thus provides a theoretical basis for identifying CH₄–H₂ hydrates on ocean floor in marine areas.

Data availability statement

Publicly available datasets were analyzed in this study. This data can be found here: https://gashydrates.nist.gov/.

Author contributions

ZWZ: Conceptualization, investigation, methodology, software and writing—original draft. YC: Conceptualization, software, funding acquisition, writing-review and editing. ZHZ: methodology, software. NW: writing-review. DC: writing-review and funding acquisition. All authors contributed to the article and approved the submitted version.

Funding

This study is financially supported by the National Natural Science Foundation of China (No. 41776050, No. 41776080, and No. 91858208).

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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