A Helmholtz Free Energy Equation of State of CO₂-CH₄-N₂ Fluid Mixtures (ZMS EOS) and Its Applications

Abstract

An equation of state (EOS) of CH₄-N₂ fluid mixtures in terms of Helmholtz free energy has been developed by using four mixing parameters, which can reproduce the pressure-volume-temperature-composition (PVTx) and vapor-liquid equilibrium (VLE) properties of CH₄-N₂ fluid mixtures. The average absolute deviation of all the PVTx data available up to 673.15 K and 1380 bar from this EOS is 0.38%. Combining this EOS of CH₄-N₂ fluid mixtures and the EOS of CH₄-CO₂ and CO₂-N₂ fluid mixtures in our previous work, an EOS of CO₂-CH₄-N₂ fluid mixtures has been developed, which is named ZMS EOS. The ZMS EOS can calculate all thermodynamic properties of ternary CO₂-CH₄-N₂ fluid mixtures and the average absolute deviation of the PVTx data from the ZMS EOS is 0.40% for the CO₂-CH₄-N₂ system. The ZMS EOS can be applied to calculate excess enthalpies of CO₂-CH₄-N₂ fluid mixtures, predict the solubility of CO₂-CH₄-N₂ fluid mixtures in brine and water, and quantitatively estimate the impact of the impurities (CH₄ and N₂) on the CO₂ storage capacity in the CO₂ capture and storage (CCS) processes. The ZMS EOS can also be applied to calculate the isochores of CO₂-CH₄-N₂ system in the studies of fluid inclusions. All Fortran computer codes and Origin drawing projects in this paper can be obtained freely from the corresponding author.

1. Introduction

The CO₂, CH₄ and N₂ are important natural fluids, and non-aqueous CO₂–CH₄–N₂ mixtures are often reported in the studies of fluid inclusions [1,2,3]. In recent decades, the amount of greenhouse gas CO₂ in the atmosphere has gradually increased because of the development of industry. Numerous studies have shown that CO₂ capture and storage (CCS) is an effective method to reduce the amount of CO₂ in the atmosphere [4,5,6]. However, CO₂ in industrial exhaust gases is usually impure, mixing with other components, such as CH₄ and N₂ [7,8]. The impurities can affect the design of the CCS processes [9,10,11]. Therefore, predicting thermodynamic properties of the CO₂-CH₄-N₂ fluid mixtures, especially the pressure-volume-temperature-composition (PVTx) and vapor-liquid equilibrium (VLE) properties at different temperatures and pressures, is of great significance for the related CCS and fluid inclusion studies [12,13,14,15].

Although experimental thermodynamic data are the most reliable in the corresponding applications, they are costly and time-consuming to obtain. On the other hand, the predictive EOS is a better choice for us. In the past century, the cubic EOSs (e.g., Peng-Robinson (PR) EOS [16], Soave–Redlich–Kwong (SRK) EOS [17]) originated from the van der Waals EOS [18] are highly preferred for the CCS researchers because of their simplicity. Based on the standard PR EOS, some more accurate EOSs have been presented for the CCS mixtures, such as a model that integrates the classical PR EOS with advanced mixing rules, called the “Peng-Robinson + residual part of excess Helmholtz energy model” [19], and the Enhanced-Predictive-PR78 (E-PPR78) EOS [20]. However, the cubic EOSs cannot well reproduce the PVTx properties of fluids under high pressure-temperature conditions, as can be seen in Section 2. Based on Heyen EOS [21], Heyen et al. [22] simulated the phase equilibria of the CH₄-CO₂ system below 50 °C and 100 bar; Darimont and Heyen [23] predicted the phase equilibria of the CO₂-N₂ system below 20 °C and 90 bar. Because cubic EOSs cannot well reproduce the PVTx properties, Thiery et al. [24,25] and Thiery and Dubessy [26] modeled the phase equilibria of the CO₂-CH₄-N₂ system including liquid, vapor and solid by the cubic EOSs and the PVTx properties by the Lee–Kesler correlation [27]. However, these models have the disadvantage of using two EOSs that may cause incoherency of various fluid parameters [28].

In recent years, the Statistical Associating Fluid Theory (SAFT) EOS [29] from Wertheim’s thermodynamic perturbation theory (TPT) has become popular for CCS fluid mixtures. Because it is based on statistical mechanics and has an explicit physical meaning, it has more reliable extrapolation and more accurate predictive capability than traditional empirical models. Many successful modifications of the SAFT-based EOSs have been presented, such as the EOS of perturbed-chain SAFT (PC-SAFT) [30,31], variable range SAFT (SAFT-VR) [32], and Lennard-Jones SAFT (LJ-SAFT) [33]. However, the SAFT EOSs cannot well reproduce experimental VLE data in the near-critical region of the CO₂-N₂ fluid mixture, as can be seen in the previous work [34]. Up to now, the EOS in terms of Helmholtz free energy is also commonly used to calculate the thermodynamic properties of CCS fluid mixtures [35,36,37,38,39], because the EOS in the form of Helmholtz free energy has high precision, wide temperature-pressure range, and can calculate all thermodynamic properties of the fluids by the derivation from the Helmholtz free energy EOS, such as enthalpy, heat capacity, entropy, etc. The derivation from the Helmholtz free energy EOS is not complex, as can be seen later.

In our previous work [34,40], the EOS in terms of dimensionless Helmholtz free energy for the CH₄-CO₂ and CO₂-N₂ fluid mixtures have been established by using four mixing parameters, which have been applied to the studies of the CH₄-CO₂ and CO₂-N₂ fluid inclusions. In this work, firstly, the EOS in terms of dimensionless Helmholtz free energy of the CH₄-N₂ fluid mixture has been developed by the same approach. The ZMS EOS is obtained by combining the binary interaction parameters of CH₄-CO₂, CO₂-N₂ and CH₄-N₂ systems. Experimental data available for the binary CH₄-N₂ and ternary CO₂-CH₄-N₂ fluid mixtures are used to verify the accuracy of the ZMS EOS. Then the ZMS EOS is applied to calculate the excess enthalpies, the solubility of CO₂-CH₄-N₂ gas mixtures in brine and water, the impact of impurities (CH₄ and N₂) on the CO₂ storage capacity, and the isochores of the CO₂-CH₄-N₂ fluid inclusions.

2. The ZMS EOS

The EOS of the CO₂-CH₄-N₂ mixtures is in terms of dimensionless Helmholtz free energy $\alpha$, which is represented by

$$\alpha ={\alpha }^0+{\alpha }^r$$

(1)

where ${\alpha }^0$ and ${\alpha }^r$ are the ideal-gas part and the residual part of dimensionless Helmholtz free energy. ${\alpha }^0$ and ${\alpha }^r$ are defined by

$${\alpha }^0={\displaystyle \sum }_{i=1}^n{x}_i\left[{\alpha }_i^0+\mathit{ln}{x}_i\right]$$

(2)

$${\alpha }^r={\displaystyle \sum }_{i=1}^n{x}_i{\alpha }_i^r\left(\delta ,\tau \right)+{\alpha }^E\left(\delta ,\tau ,x\right)$$

(3)

where n is the number of components in the mixture, x_i is the mole fraction of component i in the mixture, the superscripts ‘‘0”, “r” and “E” denote the ideal-gas part, the residual part, and the excess of dimensionless Helmholtz free energy, respectively. The subscript i denotes the component i. ${\mathsf{\alpha }}_i^{\mathrm{r}}$ can be calculated by the EOSs of pure CO₂, CH₄ and N₂ fluids [41,42,43], which are in terms of dimensionless Helmholtz free energy and are recommended as the standard EOSs of pure CO₂, CH₄ and N₂ fluids by the National Institute of Standards and Technology (NIST).

$\delta$ and $\tau$ of Equation (3) are the reduced mixture density ($\delta =\rho /{\rho }_c$) and the inverse reduced mixture temperature ($\tau =T_c/T$), where $\rho$ and T are the density and temperature of the mixture, and ${\rho }_c$ and $T_c$ are the pseudo-critical density and temperature of the mixture. ${\rho }_c$ and $T_c$ are defined by

$${\rho }_c={\left[{\displaystyle \sum }_{i=1}^n\frac{x_i}{{\rho }_{ci}}+{\displaystyle \sum }_{i=1}^{n-1}{\displaystyle \sum }_{j=i+1}^n{x}_i{x}_j{\zeta }_{ij}\right]}^{-1}$$

(4)

$$T_c={{\displaystyle \sum }}_{i=1}^n{x}_i{T}_c{}_i+{{\displaystyle \sum }}_{i=1}^{n-1}{{\displaystyle \sum }}_{j=i+1}^n{x}_i^{{\beta }_{ij}}{x}_j{\varsigma }_{ij}$$

(5)

where x_j is the mole fraction of component j in the mixture, ${\rho }_{ci}$ and $T_c{}_i$ are the critical density and temperature of component i, respectively. The critical temperatures and densities of the three components considered in this study are listed in

Table 1. Critical parameters of pure fluids.

Component i	${\mathit{T}}_{\mathit{c}}{}_{\mathit{i}} \left(\mathbf{K}\right)$	${\mathit{\rho }}_{\mathit{c}\mathit{i}} (\mathbf{mol}/{\mathbf{dm}}^3)$	References
CH4	90.6941	10.139	Setzmann and Wagner [42]
CO2	216.592	10.625	Span and Wagner [41]
N2	63.151	11.184	Span et al. [43]

. ${\zeta }_{ij}$, ${\beta }_{ij}$ and ${\varsigma }_{ij}$ are binary parameters associated with components i and j.

${\alpha }^E$ of Equation (3) takes the general form developed by Lemmon and Jacobsen (LJ-1999 EOS) [35].

$${\alpha }^E\left(\delta ,\tau ,x\right)={\displaystyle \sum }_{i=1}^{n-1}{\displaystyle \sum }_{j=i+1}^n{x}_i{x}_j{F}_{ij}{\displaystyle \sum }_{k=1}^{10}{N}_k{\delta }^{d_k}{\tau }^{t_k}$$

(6)

where F_ij is a binary parameter associated with components i and j, N_k, d_k and t_k are the general parameters independent of fluids, which can be obtained from Lemmon and Jacobsen [35]. There are four binary parameters (${\zeta }_{ij}$, ${\beta }_{ij}$, ${\varsigma }_{ij}$ and F_ij) in the above equations. Binary interaction parameters of CH₄-CO₂ and CO₂-N₂ have been determined in previous work [34,40]. In this work, the binary interaction parameters of CH₄-N₂ systems are obtained by the nonlinear regression of selected experimental data. Since experimental PVTx data available are much more than experimental VLE data, only part of new and accurate experimental PVTx data are selected to fit the parameters to make the PVTx and VLE data keep similar weights in the fitting. In this work, the Levenberg–Marquardt algorithm of the nonlinear least squares method is used for fitting, which is an efficient and widely used mathematical optimization technique. The fitting condition is to minimize the weighted sum of squares of the calculation errors of the equation on the selected experimental PVTx and VLE data. Compared to Newton’s method (e.g., The Newton–Raphson method used in REFPROP [44]), it combines the advantages of two numerical minimization algorithms: the gradient descent method and the Gauss–Newton method, and it has good convergence and robustness. Regressed parameters of the CH₄-N₂ system are listed in

Table 2. Binary interaction parameters of the CO₂-CH₄-N₂ system.

References	Binary Mixtures	${\mathit{F}}_{\mathit{i}\mathit{j}}$	${\mathit{\zeta }}_{\mathit{i}\mathit{j}} ({\mathbf{dm}}^3/\mathbf{mol})$	${\mathit{\varsigma }}_{\mathit{i}\mathit{j}} \left(\mathbf{K}\right)$	${\mathit{\beta }}_{\mathit{i}\mathit{j}}$
Mao et al. [40]	CH4-CO2	0.12844025 × 10	0.35751245 × 10−2	−0.43720344 × 102	0.10358865 × 10
Zhang et al. [34]	CO2-N2	0.16671494 × 10	0.58411078 × 10−2	−0.22952094 × 102	0.16878787 × 10
This work	CH4-N2	0.63739997	0.3812517 × 10−2	−0.17790001 × 102	0.1009001 × 10

Note: Subscripts i and j refer to the first component and the second component, respectively.

, which also includes the parameters of CH₄-CO₂ and CO₂-N₂ systems from previous studies.

2.1. The Binary CH₄-N₂ Mixture

The calculated deviations of the PVTx and VLE properties from each experimental data set [45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66] by this EOS for the CH₄-N₂ fluid mixture are listed in

Table 3. Calculated PVTx and VLE deviations for the CH₄-N₂ mixture.

References	Number of Data Points		T-P-x Range for xCH4-(1-x) N2			AAD%
	Styles	Total (Used)	T (K)	P (bar)	x
Liu and Miller [45]	PVTx	7 (0)	91–115	3–12	0.5	0.57
Rodosevich and Miller [46]	PVTx	8 (0)	91–115	43–454	0.8–0.95	3.83
Pan et al. [47]	PVTx	7 (0)	91–115	1–11	0.5–0.86	0.11
Hiza et al. [48]	PVTx	21 (0)	95–140	1–21	0.5–0.95	0.10
Da Ponte et al. [49]	PVTx	182 (50)	110–120	15–1380	0.3–0.68	0.12
Straty and Diller [65]	PVTx	578 (0)	82–320	6–356	0.32–0.7	0.24
Haynes and McCarty [56]	PVTx	85 (85)	140–320	10–164	0.3–0.71	0.09
Seitz et al. [64]	PVTx	190 (90)	323–573	99–999	0.1–0.9	0.28
Seitz and Blencoe [63]	PVTx	43 (0)	673.15	199–999	0.1–0.9	5.36
Ababio et al. [50]	PVTx	83 (83)	308–333	9–120	0.5–0.78	0.12
Chamorro et al. [52]	PVTx	242 (56)	230–400	9–192	0.8–0.9	0.23
Janisch et al. [57]	PVTx	17 (0)	129–180	15–50	0.4–0.9	2.10
Li et al. [60]	PVTx	27 (27)	17–270	1–16	0.9	0.06
Gomez-Osorio et al. [54]	PVTx	142 (42)	304–470	100–1379	0.25–0.75	0.07
Brandt and Stroud [51]	VLE	23 (0)	128–179	34	0.05–0.98	1.48
Cheung and Wang [53]	VLE	20 (0)	92–124	0.2–6	0.85–1.0	5.02
Pan et al. [47]	VLE	60 (60)	95–120	0.2–25	0.05–1	5.05
Miller et al. [62]	VLE	11 (0)	112	2–13	0.2–0.97	3.30
Kidnay et al. [59]	VLE	83 (83)	112–180	2–49	0.1–0.99	1.31
McClure et al. [61]	VLE	8 (8)	91	1–3	0.1–0.8	5.60
Jin et al. [58]	VLE	10 (10)	123	4–26	0.1–0.95	2.96
Parrish and Hiza [66]	VLE	48 (43)	95–120	2–20	0.1–0.9	3.14
Janisch et al. [57]	VLE	16 (6)	130–180	0.5–5	0.4–0.96	1.14
Han et al. [55]	VLE	77 (60)	110–123	4–13	0.7–1.0	2.49

Note: Deviations of the PVTx data are the density deviations, and the deviations of VLE data are the equilibrium composition deviations. AAD is the abbreviation of average absolute deviation.

. Detail calculation method for the PVTx and VLE properties can be found in Mao et al. [40]. The deviations of the PVTx data are the density deviations, and the temperature and pressure of the experimental PVTx data are up to 673.15 K and 1380 bar, respectively. The deviations of VLE property are the deviations of the vapor-liquid phase equilibrium composition, and the experimental VLE composition covers the full range. The average absolute deviation of all PVTx data from this EOS is 0.38% for the CH₄-N₂ fluid mixture.

Figure 1 shows the deviations between this EOS and the experimental density data of the CH₄-N₂ mixture [49,52,54,64] at different pressures. It can be seen in Figure 1 that most of the density deviations are within ±1%, with or close to experimental accuracy. The comparisons between experimental density and the calculated densities by this EOS are shown in Figure 2, where the calculated densities from the PR and SRK EOSs [67] are also included. The errors of three EOSs are also plotted in Figure 2. Figure 2 shows the PR and SRK EOSs cannot well reproduce the PVTx properties, especially under high-pressure conditions. In contrast, the calculated densities from this EOS are in good agreement with experiment data [68].

Figure 3 compares the vapor-liquid phase equilibrium curves calculated from this EOS with experimental data [55,57,59,66] at three temperatures of 170 K, 115 K and 110 K. It can be seen from Figure 3 that the calculated vapor-liquid phase equilibrium curves from this EOS are consistent with experimental data, indicating that this EOS can accurately calculate the vapor-liquid phase equilibrium of CH₄-N₂ mixture.

2.2. The Ternary CO₂-CH₄-N₂ Mixture

Combining binary interaction parameters of the CH₄-CO₂, CO₂-N₂ and CH₄-N₂ systems, the EOS can also predict the thermodynamic properties of the CO₂-CH₄-N₂ mixture. Here, the experimental PVTx and vapor-liquid equilibrium data are used to verify the accuracy of this EOS for the ternary mixture.

The calculated PVTx deviations for the ternary CO₂-CH₄-N₂ mixture from each experimental data set [1,64,69,70,71] are given in

Table 4. Calculated PVTx deviations for the ternary CO₂-CH₄-N₂ mixture.

References	N	T-P-x Range for xCO2-yCH4-(1-x-y) N2				AAD%
		T (K)	P (bar)	x	y
McElroy et al. [59]	242	303–333	6–126	0–0.9998	0–0.999	0.45
Seitz et al. [60]	42	474.15	1000	0.0–1.0	0.0–1.0	0.28
Seitz et al. [55]	271	323–573	199–999	0.1–0.8	0.1–0.8	0.28
Zhang et al. [61]	200	293.15–353.25	5–180	0.098–0.9949	0.02–0.6525	0.52
Le et al. [1]	84	305.15	5–600	0.499–0.899	0.0505–0.331	0.82

Note: Deviations of the PVTx data are the density deviations. N is the Number of data points, AAD is the abbreviation of average absolute deviation.

, where the temperature and pressure of the PVTx data are up to 573.15 K and 1000 bar, and the composition almost covers the full range. The average absolute deviation of all PVTx data from this EOS is 0.40% for the CO₂-CH₄-N₂ fluid mixture. Figure 4 shows the comparison between experimental data [71] and calculated densities for the CO₂-CH₄-N₂ mixture at different temperatures. It can be seen from Figure 4 that calculated densities from this EOS are in good agreement with experiment data, indicating that this EOS has a good predictive ability for the CO₂-CH₄-N₂ mixture.

The vapor-liquid phase equilibrium properties of the ternary CO₂-CH₄-N₂ system are shown in Figure 5, where the curves are calculated from this EOS, and the experimental data are from the literature of [72,73,74]. As can be seen from Figure 5, all the experimental data points in the non-critical region agree well with this EOS, but in the near-critical region (Figure 5c), the calculated values deviate largely from experimental data [72,73,74].

3. The Applications of the ZMS EOS

3.1. Calculating Excess Enthalpies

The excess enthalpy ($H^E$) is an important thermodynamic property of the mixture for the mixing and separation processes, which is defined by

$$H^E={\left\{H_m-{{\displaystyle \sum }}_i{x}_i{H}_i\right\}}_{P,T}$$

(7)

where $H_m$ is the enthalpy of the mixture, $H_i$ is the enthalpy of pure component i, and x_i is the mole fraction of component i in the mixture. According to the thermodynamic relationship between Helmholtz free energy and enthalpy, $H_m$ and $H_i$ are given by

$$H_m=RT\left(1+\tau \left({\alpha }_{\tau }^0+{\alpha }_{\tau }^r\right)+\delta {\alpha }_{\delta }^r\right)$$

(8)

$$H_i=RT\left(1+{\tau }_i\left({\alpha }_{\tau i}^0+{\alpha }_{\tau i}^r\right)+{\delta }_i{\alpha }_{\delta i}^r\right)$$

(9)

where ${\alpha }_{\tau }^0={\left(\frac{\partial {\alpha }^0}{\partial \tau }\right)}_{\delta }$, ${\alpha }_{\tau }^r={\left(\frac{\partial {\alpha }^r}{\partial \tau }\right)}_{\delta }$, ${\alpha }_{\delta }^r={\left(\frac{\partial {\alpha }^r}{\partial \delta }\right)}_{\tau }$, ${\alpha }_{\tau i}^0={\left(\frac{\partial {\alpha }_i^0}{\partial {\tau }_i}\right)}_{\delta i}$, ${\alpha }_{\tau i}^r={\left(\frac{\partial {\alpha }_i^r}{\partial {\tau }_i}\right)}_{\delta i}$ and ${\alpha }_{\delta i}^r={\left(\frac{\partial {\alpha }_i^r}{\partial {\delta }_i}\right)}_{\tau i}$. ${\delta }_i$ and ${\tau }_i$ are the reduced density and inverse reduced temperature of pure component i, which are defined by

$${\delta }_i={\rho }_i/{\rho }_{ci}$$

(10)

$${\tau }_i=T_{ci}/T$$

(11)

where ${\rho }_i$ is the density of component i. Based on Equations (8)–(11), $H_m$ can be calculated by this EOS of the CO₂-CH₄-N₂ fluid mixtures and $H_i$ can be calculated by the above-mentioned equations of pure CO₂, CH₄ and N₂ fluids.

The calculated excess enthalpy curves of the CH₄-CO₂ and CO₂-N₂ mixtures are, respectively, shown in Figure 6 and Figure 7, where the experimental data [75,76] are included for comparison. The following EOSs are also included for comparison: the standard Peng–Robinson (PR) EOS, either optimized (optimal k_ij) or not (k_ij = 0); the PR EOS with the residual part of the Wilson excess Helmholtz energy model (PR + EOS/$a_{res}^{E,Wilson}$) [19]. It can be seen from Figure 6 and Figure 7 that this EOS is more accurate than the standard PR EOS, either optimized (optimal k_ij) or not (k_ij = 0) at most cases. In general, the PR + EOS/$a_{res}^{E,Wilson}$ and this EOS shows a similar predictive capability for the excess enthalpies of the CH₄-CO₂ mixture. However, this EOS is slightly more accurate than the PR + EOS/$a_{res}^{E,Wilson}$ for the CO₂-N₂ mixture at low pressures.

Figure 8 compares the excess enthalpy curves of the CH₄-N₂ mixture calculated from the ZMS EOS with experimental data [77], and enthalpies calculated from the Enhanced-Predictive-PR78 (E-PPR78) EOS [20] are also added for comparison. Figure 8a shows the ZMSEOS are significantly more accurate than the (E-PPR78) EOS at high and middle pressures. Figure 8b shows the enthalpies of mixing calculated from the E-PPR78 EOS are slightly better than those of the ZMS EOS at low and middle pressures, but the enthalpies of mixing calculated from the ZMS EOS are better than those from the E-PPR78 EOS at P = 101.33 bar.

3.2. Calculating the Solubility of CO₂-CH₄-N₂ Mixtures in Aqueous Electrolyte Solution

The solubility of the CO₂-CH₄-N₂ gas mixtures in the electrolyte solutions can provide the quantitative assessment for the storage of CO₂ in deep saline aquifers. According to the thermodynamic principle, when the CO₂-CH₄-N₂ gas mixtures reach dissolution equilibrium in the electrolyte solution, the chemical potential of component i in the liquid phase (${\mu }_i^L$) and its chemical potential in the vapor phase (${\mu }_i^V$) are equal. The chemical potential can be written in terms of fugacity in the vapor phase and activity in the liquid phase

$${\mu }_i^L={\mu }_i^{L\left(0\right)}+RT\mathit{ln}{m}_i{\gamma }_i$$

(12)

$${\mu }_i^V={\mu }_i^{V\left(0\right)}+RT\mathit{ln}{\phi }_i{y}_{i}P$$

(13)

where i is the gas component in vapor mixtures, and P is the pressure in bar. y_i is the molar fraction of i in the vapor phase, ${\phi }_i$ is the fugacity coefficient of i in the vapor phase and m_i is the solubility of i in the liquid phase in mol/kg. ${\gamma }_i$ is the activity coefficient of component i in the liquid phase. ${\mu }_i^{L\left(0\right)}$ and ${\mu }_i^{V\left(0\right)}$ are the standard chemical potential of i in the liquid and vapor phase, respectively.

At the dissolution phase equilibrium, ${\mu }_i^V={\mu }_i^L$. From Equations (12) and (13), we can obtain

$$\mathit{ln}{m}_i=\mathit{ln}{y}_{i}P+\mathit{ln}{\phi }_i-\frac{{\mu }_i^{L\left(0\right)}-{\mu }_i^{V\left(0\right)}}{RT}-\mathit{ln}{\gamma }_i$$

(14)

Since the water content in the vapor phase at equilibrium is generally small, it has a negligible effect on the fugacity coefficient. When calculating the fugacity coefficient, the water content in the vapor phase can be ignored, the vapor phase can be approximated as the CO₂-CH₄-N₂ system and the mole fraction of gas component i in the vapor phase is expressed as $y_i$. Consequently, ${\phi }_i$ can be calculated from the ZMS EOS developed in this work.

Pitzer activity coefficient model [78] is chosen to calculate ${\gamma }_i$

$$\mathit{ln}{\gamma }_i={\displaystyle \sum }_{c}2{\lambda }_{i-c}{m}_c+{\displaystyle \sum }_{a}2{\lambda }_{i-a}{m}_a+{\displaystyle \sum }_c{\displaystyle \sum }_a{\zeta }_{i-c-a}{m}_c{m}_a$$

(15)

where λ and $\zeta$ are second-order and third-order interaction parameters, respectively; c and $a$ refer to cation and anion, respectively. Substituting Equation (15) into Equation (14) yields

$$\begin{array}{cc}\mathit{ln}{m}_i=& \mathit{ln}{y}_{i}P+\mathit{ln}{\phi }_i-\frac{{\mu }_i^{L\left(0\right)}-{\mu }_i^{V\left(0\right)}}{RT}-{\displaystyle \sum _c}2{\lambda }_{i-c}{m}_c\\ & -{\displaystyle \sum _a}2{\lambda }_{i-a}{m}_a-{\displaystyle \sum _c}{\displaystyle \sum _a}{\zeta }_{i-c-a}{m}_c{m}_{a}a\end{array}$$

(16)

As can be seen from Equation (17), m_i is related to the difference between ${\mu }_i^{L\left(0\right)}$ and ${\mu }_i^{V\left(0\right)}$, and is not related to the specific value of ${\mu }_i^{L\left(0\right)}$ or ${\mu }_i^{V\left(0\right)}$. Therefore, to simplify the model, ${\mu }_i^{V\left(0\right)}$ is assumed to be zero. Equation (16) can be simplified as

$$\begin{array}{cc}\mathit{ln}{m}_i=& \mathit{ln}{y}_{i}P+\mathit{ln}{\phi }_i-\frac{{\mu }_i^{L\left(0\right)}}{RT}-{\displaystyle \sum _c}2{\lambda }_{i-c}{m}_c\\ & -{\displaystyle \sum _a}2{\lambda }_{i-a}{m}_a-{\displaystyle \sum _c}{\displaystyle \sum _a}{\zeta }_{i-c-a}{m}_c{m}_a\end{array}$$

(17)

where $\frac{{\mu }_i^{L\left(0\right)}}{RT}$, $\lambda$ and $\zeta$ are the function of temperature and pressure, and the parameters can be determined by the solubility model of pure gases (CO₂, CH₄, N₂). Since the 1980s, many solubility models for CO₂, CH₄ and N₂ gases in pure water and brine have been developed [79,80,81,82,83,84,85,86,87,88,89,90,91], most of which do not exceed 473 K and 1000 bar. Among them, the solubility models of CH₄, CO₂ and N₂ established by Mao and co-workers [83,86,91] are applicable to a wider range of temperature, pressure, and salinity with higher accuracy. They have been chosen as the solubility models of pure CH₄, CO₂ and N₂ gases in this work.

Table 5. Ranges of pure gas solubility models.

Pure Gas	T	P	$\mathbf{Salinity} \left({\mathit{m}}_{\mathbf{NaCl}}\right)$	References
CH4	273–523 K	1–2000 bar	0–6 mol/kg	Duan and Mao [83]
CO2	273.15–723.15 K	1–1500 bar	0–4.5 mol/kg	Mao et al. [91]
N2	273–590 K	1–600 bar	0–6 mol/kg	Mao and Duan [86]

lists the applicable temperature, pressure, and salinity ranges of these solubility models for pure gases.

To verify the accuracy of the predictions, the experimental data for the solubility of the CO₂-CH₄-N₂ gas mixtures are compared with the calculated results. Figure 9 compares the experimental data on the solubility of the CH₄-CO₂ mixture in pure water [92,93] for the temperature range of 324.5–375.5 K and the pressure range of 100–750 bar, and it can be seen that the agreement is very good.

The solubility experimental data of the CO₂-N₂ gas mixture in pure water [94] and the calculated results are compared in Figure 10, where the temperature range is 308.15–318.15 K and the pressure range is 80–160 bar. As can be seen from Figure 10 the predictive results are in good agreement with the experimental data. The average absolute deviations between the calculated N₂ and CO₂ solubility of this model and the experimental data are 6.04% and 1.52%, respectively. Figure 11 compares the solubility experimental data of the CO₂-N₂ gas mixture in saline water. The average absolute deviations between the calculated N₂ and CO₂ solubility and the experimental data are 2.04% and 2.49%, respectively.

3.3. The Impact of Impurities (CH₄ and N₂) on the CO₂ Storage Capacity

CH₄ and N₂ are non-condensable impurities, which reduce the CO₂ storage capacity in geological formations. Based on the EOS of the CO₂-CH₄-N₂ fluid mixtures, the impact of CH₄ and N₂ on CO₂ storage capacity can be calculated quantitatively by the normalized storage capacity proposed by Wang et al. [9]. For a certain reservoir with a certain volume, the normalized storage capacity of the CO₂-CH₄-N₂ gas mixtures is shown as

$$\frac{A}{A_{{\mathrm{CO}}_2}}=\frac{\rho }{{\rho }_{{\mathrm{CO}}_2}(1+\frac{M_{{\mathrm{CH}}_4}}{M_{{\mathrm{CO}}_2}}+\frac{M_{N_2}}{M_{{\mathrm{CO}}_2}})}$$

(18)

where $A$ and $A_{{\mathrm{CO}}_2}$ are the mass of CO₂ in the mixtures and the mass of pure CO₂ under the same volume, respectively. ${\rho }_{{\mathrm{CO}}_2}$ and $\rho$ are the density of the pure CO₂ and CO₂-CH₄-N₂ gas mixtures, respectively; $M_{{\mathrm{CH}}_4}$,$M_{N_2}$ and $M_{{\mathrm{CO}}_2}$ are the mass of CH₄, N₂, and CO₂ in the mixture, respectively. $A$/$A_{{\mathrm{CO}}_2}$ is the ratio of the mass of CO₂ per unit volume in the mixture to that in the pure state. The value of $A$/$A_{{\mathrm{CO}}_2}$ can be viewed as the normalized storage capacity for CO₂ (i.e., the storage capacity for structural trapping of CO₂). For pure CO₂, the normalized storage capacity ($A$/$A_{{\mathrm{CO}}_2}$) is 1. For the CO₂-CH₄-N₂ gas mixtures of a given composition, $M_{{\mathrm{CH}}_4}$, $M_{N_2}$ and $M_{{\mathrm{CO}}_2}$ are also known. ${\rho }_{{\mathrm{CO}}_2}$ can be calculated by the above-mentioned equation of pure CO₂ fluid and $\rho$ can be calculated by the ZMS EOS.

The normalized storage capacity of the CO₂-CH₄-N₂ fluid mixture calculated by the ZMS EOS is plotted in Figure 12. Figure 12a shows the normalized storage capacity of a given composition at different temperatures. With the increase in pressure, the normalized storage capacity decreases at first and then increases, and there is a minimum point. When the temperature changes, the pressure corresponding to the lowest point of the normalized storage also changes. Figure 12b shows the normalized storage capacity of different compositions at the same temperature, which indicates that the content of impurities is much larger and the normalized storage capacity is much smaller.

3.4. Isochores of the CO₂-CH₄-N₂ fluid Inclusions

In the studies of fluid inclusions, the isochores (pressure-temperature relation at constant density and composition) are frequently used to estimate the trapping temperatures and pressures.

Based on the EOS of the CO₂-CH₄-N₂ fluid mixtures, isochores of the CO₂-CH₄-N₂ inclusions can be calculated by the following equation:

$$P\left(\delta ,\tau ,x\right)=\rho RT\left[1+\delta {\alpha }_{\delta }^r\right]$$

(19)

The calculated isochores of the CO₂-CH₄-N₂ inclusions at two different compositions are plotted in Figure 13, from which it can be seen that the isochores of the CO₂-CH₄-N₂ inclusions are a bit curved.

This section may be divided into subheadings. It should provide a concise and precise description of the experimental results, their interpretation, as well as the experimental conclusions that can be drawn.

4. Conclusions

A fundamental EOS for the Helmholtz free energy of the CH₄-N₂ mixture has been developed by using four binary interaction parameters. Comparisons with experimental PVTx and VLE data available showed that the EOS can satisfactorily reproduce the experimental volumetric and vapor-liquid phase equilibria data of binary CH₄-N₂ mixtures up to 673.15 K and 1380 bar, with or close to experimental accuracy. Combining this EOS of the CH₄-N₂ fluid mixtures and the EOS of the CH₄-CO₂ and CO₂-N₂ fluid mixtures developed in our previous work, an EOS of ternary CO₂-CH₄-N₂ fluid mixtures has been presented, which is named ZMS EOS. The ZMS EOS can be applied to calculate excess enthalpies, predict the solubility of the CO₂-CH₄-N₂ gas mixtures in water and brine, estimate the impurities CH₄ and N₂ on the CO₂ storage capacity, and calculate the isochores of the CO₂-CH₄-N₂ fluid mixtures.