SAFT2 equation of state for the CH 4 – CO 2 – H 2 O – NaCl quaternary system with applications to CO 2 storage in depleted gas reservoirs

Understanding the phase equilibria and physical-chemical characteristics of the CH 4 – CO 2 – H 2 O – NaCl quaternary system is important for evaluating costs and risks for the storage of CO 2 in depleted natural gas reservoirs as well as fluid inclusion studies. In this study, phase equilibria and thermodynamic properties of this system were investigated through the utilization of a statistical association fluid theory-based (SAFT) equation of state (EOS) at temperatures from 298 to 513 K (25 – 240 ◦ C), pressures up to 600 bar (60 MPa) and concentration of NaCl up to 6 mol/kgH 2 O. The model parameters were obtained from the fitting of available experimental data of sub-systems (i.e., CH 4 – H 2 O, CH 4 – CO 2 , and CH 4 – H 2 O – NaCl) that were judged to be reliable and incorporation of available parameters for the subsystems (i.e., pure component, CO 2 – H 2 O, and CO 2 – H 2 O – NaCl). Using the SAFT EOS developed in this study, we predicted the solubility of (CH 4 + CO 2 ) gas mixtures in pure H 2 O and compared it with the available experimental data and the predicted values from four popular numerical simulators. The results indicate that our model can provide reliable predictions for the CH 4 – CO 2 – H 2 O ternary system. Subse-quently, we further predicted the phase equilibria and density of the CH 4 – CO 2 – H 2 O – NaCl system with NaCl varying from 0 to 6 mol/kgH 2 O. We also employed the SAFT EOS to predict the solubility of CO 2 and CH 4 in the water-alternating-gas process for CO 2 -enhanced oil recovery, demonstrating good agreement with the simulation results obtained through the Peng-Robinson EOS for predicting the CO 2 and CH 4 solubility. These predicted thermodynamic properties and phase behaviors in the CH 4 – CO 2 – H 2 O – NaCl system provide quantitative insights into the implications of CO 2 storage in depleted oil and gas reservoirs.


Introduction
Geologic carbon sequestration (GCS) is considered a promising method for mitigating CO 2 emissions (Metz et al., 2005).CO 2 injection into depleted gas/oil reservoirs for enhanced gas/oil recovery (EOR/ EGR) has emerged as a synergetic strategy for both CO 2 storage and energy production because the produced hydrocarbons can provide additional economic benefits (Dai et al., 2016;Oldenburg, 2003).Wateralternating-gas (WAG) injection is a widely used EOR technology in the petroleum industry.WAG combines the advantages of enhanced macroscopic sweep efficiency in water flooding and the high displacement efficiency of gas injection to improve oil production operations (Kulkarni and Rao, 2005).Practically, WAG is conducted by re-injecting the produced gas into water injection wells in mature petroleum fields (Hustad and Holt, 1992).CH 4 is a common component in the produced gas, and a mixture of CO 2 and CH 4 is usually injected together.Storing greenhouse gases (CO 2 and CH 4 ) in the reservoir during the WAG process has a synergistic effect on both revenue generation and environmental protection.In addition to the EOR technology, natural gas storage (primarily composed of CH 4 , with CO 2 as an impurity) in salt caverns has been proposed to facilitate underground energy storage to meet load variations, as well as support a variety of secondary purposes (Wei et al., 2023).In all these applications, the phase equilibria and thermodynamic properties of injected CO 2 and CH 4 can impact the gas storage processes.For instance, as demonstrated in a numerical simulation study for a large-scale GCS pilot project near Cranfield, Mississippi (Soltanian et al., 2018), there exists a solubility competition between CO 2 and CH 4 .CH 4 tends to exsolve from the aqueous to gaseous phase, imposing a substantial risk of CH 4 leakage.Therefore, understanding the phase equilibria and thermodynamic properties of CH 4 -CO 2 -H 2 O-brine fluids and their subsystems is important for advancing CO 2 storage and utilization technologies, particularly in applications such as GCS (temperatures up to 473 K and pressures up to 600 bar), WAG injection process (temperatures to 363 K and pressures to 150 bar), and salt cavern gas storage (temperatures to 333 K and pressures to 250 bar).
The phase equilibria and volumetric properties of pure CO 2 or CH 4 in water or aqueous salt solutions have been extensively measured, and the relevant experimental data have been summarized in the previous publications (Duan and Mao, 2006;Duan and Sun, 2003).However, to the best of our knowledge, experimental data for the (CH 4 + CO 2 ) gas mixture in water or aqueous salt solutions are limited.Only a few publications have reported experimental vapor-liquid equilibrium (VLE) data of the CH 4 -CO 2 -H 2 O ternary system at various temperatures and pressures (Al Ghafri et al., 2014;Dhima et al., 1999;Hayashi, 2014;Kastanidis et al., 2018;Qin et al., 2008), and no reliable experimental VLE data for the CH 4 -CO 2 -H 2 O-NaCl quaternary system are available.Because experimental measurements are time-consuming and costintensive, especially under severe conditions, developing theoretical models for predicting phase equilibria, densities, and other properties of this system is necessary to provide guides for future experiments.
Theoretically, there are two widely employed approaches for modeling the VLE of mixed gases: the γ-ϕ approach and the ϕ-ϕ approach.In this context, γ and ϕ denote the activity coefficient and fugacity coefficient, respectively.In the γ-ϕ approach, the non-ideality of the liquid phase is characterized using an activity model, while the non-ideality of the vapor phase is described by an equation of state (EOS).The γ-ϕ approach has been applied in the development of models for CO 2 -H 2 O (Spycher et al., 2003), CH 4 -H 2 O (Li et al., 1997;Li et al., 2015b), CO 2 -H 2 O-NaCl (Duan and Li, 2008;Duan and Sun, 2003;Spycher and Pruess, 2005;Zhao et al., 2015), CH 4 -H 2 O-NaCl (Duan and Mao, 2006;Li et al., 2001) and CH 4 -CO 2 -H 2 O-NaCl (Shabani and Vilcáez, 2017;Zirrahi et al., 2012) systems.However, it should be mentioned that this approach cannot estimate the densities of the liquid phase.
In the ϕ-ϕ approach, the non-idealities of both phases are described by an EOS.This methodology has found widespread application in describing the CH 4 -CO 2 -H 2 O-NaCl quaternary system and its subsystems (e.g., CO 2 -H 2 O-NaCl and CH 4 -H 2 O-NaCl).For example, Duan et al. (1995Duan et al. ( , 2003) ) developed an EOS based on perturbation theory to describe phase equilibria and volumetric properties of the quaternary system as well as its constituent subsystems.Although the model proposed by Duan et al. (1995Duan et al. ( , 2003) ) has good accuracy, such models are not predictive because they require many adjustable parameters and a large number of experimental data to get the model parameters.Li et al. (2015a) improved the Peng-Robinson (PR) EOS to model the phase equilibria of CO 2 , CH 4 , and their mixture in brine.However, their model primarily focuses on solubility, and its predictive performance for volumetric properties is unknown.Therefore, to improve the predictive capability of the model and decrease the number of adjustable parameters, extensive research endeavors have been dedicated to developing the statistical association fluid theory (SAFT) family EOS (Emborsky et al., 2011;Ji et al., 2005;Miri et al., 2014;Novak et al., 2021;Patel et al., 2003;Sun andDubessy, 2010, 2012;Sun et al., 2014) and the cubic plus association (CPA) EOS (Bian et al., 2019;Courtial et al., 2014;Li and Firoozabadi, 2009;Oliveira et al., 2007;Perfetti et al., 2008;Wu and Prausnitz, 1998;Xiong et al., 2021;Yan et al., 2009;Yang et al., 2024) for describing phase behavior and the densities in binary, ternary and quaternary systems containing CO 2 , CH 4 , H 2 O and NaCl.Notably, Courtial et al. (2014) utilized an electrolyte version of the CPA EOS to describe the phase behavior of systems containing CO 2 , CH 4 , H 2 O, and NaCl at a wide range of temperatures (up to 773 K for salt-free systems and 573 K for salt systems) and pressures (up to 200 MPa).However, this study did not include predictions regarding the phase equilibria of the CH 4 -CO 2 -H 2 O-NaCl quaternary system.On the other hand, Miri et al. (2014) applied SAFT1-RPM EOS to predict phase equilibria and density of the CH 4 -CO 2 -H 2 O-NaCl quaternary system-but limited to 298 K and at pressures up to 100 bar.The predictions at other temperatures and pressures, which are widely needed in GCS, are unavailable.
In this study, we used the SAFT2 EOS to represent the phase equilibria and densities of the CH 4 -CO 2 -H 2 O-NaCl system covering a wide range of temperatures (from 298 to 513 K), pressures (up to 600 bar), and NaCl concentrations (up to 6 mol/kgH 2 O).Here, the SAFT2 EOS is an improved version of the SAFT1-RPM EOS, offering advancements in representing multiple-salt solutions compared to other electrolyte SAFTbased EOS (Tan et al., 2006).In the research, at first, the model parameters were obtained from the fitting of available and reliable experimental data of binary and ternary mixtures (i.e., CH 4 -CO 2 , CH 4 -H 2 O, and CH 4 -H 2 O-NaCl) and the incorporation of the existing parameters of some subsystems (i.e., pure components, CO 2 -H 2 O, and CO 2 -H 2 O-NaCl) reported in previous works (Ji and Adidharma, 2010;Tan et al., 2006).Secondly, with the obtained parameters, the available VLE data of the CH 4 -CO 2 -H 2 O system were compared with the model results to verify the model performance, and the phase equilibria and densities of the CH 4 -CO 2 -H 2 O-NaCl quaternary system were further predicted.Lastly, the solubilities of CO 2 and CH 4 in the WAG injection process were predicted and then compared with the simulation results obtained from the PR EOS.

Theoretical background
The ion-based SAFT2 is expressed by the dimensionless residual Helmholtz energy (ã res ): where the superscripts refer to the terms representing residual, hardsphere repulsive, dispersive, chain, associative, and ionic (Coulomb) interactions, respectively.The description of each term on the right side of Eq. ( 1) has been provided in previous studies (Ji and Adidharma, 2006, 2007, 2008;Ji et al., 2005Ji et al., , 2006;;Tan et al., 2006).
For uncharged and simple components (i.e., CH 4 , CO 2 , and H 2 O), each substance was represented using four parameters: segment number m, segment volume v oo , segment energy u/k, and the reduced range of the potential well λ.However, for the components with association interactions, two additional parameters are required: the well depth of the association site-site potential ε and the parameter concerning the volume available for bonding κ.The parameters for each substance were derived through the fitting process of liquid density and vapor-pressure data for pure components (Ji and Adidharma, 2010;Tan et al., 2006).In this study, the salt (NaCl) was treated as the fully dissociated cation (Na + ) and anion (Cl − ), and each ion was modeled as charged, but nonassociating, spherical segments, and one more parameter, the effective diameter d is required.
For the multi-component system, the mixing rules in our previous work were followed.To account for the special interaction within binary systems, i.e., the cross-association (i.e., CO 2 -H 2 O), two additional parameters ε and κ were assigned.Furthermore, the cross parameter k ij was employed to regulate the cross-dispersion energy between disparate segments of the molecule, that is,

Results and discussions
To access the phase equilibria of the CH 4 -CO 2 -H 2 O-NaCl quaternary system, it is imperative to obtain the parameters of SAFT2 EOS for individual pure substances (CH 4 , CO 2 , and H 2 O) and aqueous NaCl solution, together with the cross parameters for binary and ternary subsystems (CH 4 -CO 2 , CO 2 -H 2 O, CH 4 -H 2 O, CH 4 -H 2 O-NaCl, and CO 2 -H 2 O-NaCl).
In our previous work (Ji and Adidharma, 2010;Tan et al., 2006), the parameters of pure components CH 4 , CO 2 , and H 2 O were obtained by Z. Zuo et al. fitting the statured liquid density and vapor pressure data.The ion parameters of Na + and Cl − were derived by adjusting them to fit the activity coefficients and liquid densities of a selection of individual electrolytes dissolved in water (Ji and Adidharma, 2007).The phase equilibrium data of binary mixture CO 2 -H 2 O and ternary mixture CO 2 -H 2 O-NaCl were used to obtain the cross parameters k ij between CO 2 and H 2 O (Ji and Zhu, 2012).The specifics regarding the model development were elaborated upon in our previous publications (Ji and Zhu, 2012;Tan et al., 2006).Therefore, in this study, the properties and phase equilibria of the remaining subsystems (i.e., CH 4 -CO 2 , CH 4 -H 2 O, and CH 4 -H 2 O-NaCl) were utilized to determine the cross parameters of SAFT2 EOS.

CH 4 (1)-CO 2 (2)
For CH 4 (1)-CO 2 (2), we modeled methane (CH 4 ) as a nonassociating, single spherical segment, and CO 2 was represented as a molecule possessing three association sites, i.e., two O-type sites and one C-type site.Thus, there was no cross-association between CH 4 and CO 2 , and only the self-association in pure CO 2 was considered.Additionally, the cross-dispersion energy of this binary system is described by a temperature-dependent binary interaction parameter k 12 , where Cs are the adjustable parameters.
For the CH 4 -CO 2 system, the VLE data have been measured by Davales et al. (1976) at 230 K, 250 K, and 270 K and pressure from 15 to 85 bar.Similarly, Wei et al. (1995) measured the VLE of the CH 4 -CO 2 system under the same conditions.Webster and Kidnay (2001) measured the VLE at 230 K and 270 K and pressure from 8.9 to 84 bar.Moreover, Liu et al. (2018) systematically compared their experimentally determined densities of the CH 4 -CO 2 mixture with the existing data, showing high accuracy.Their measurements span a wide range of temperatures (313-353 K), pressures (30-180 bar), and CO 2 mole fractions (0.0998-0.8988).Consequently, only the data from Liu et al. (2018) was selected for the subsequent analysis of density calculations.
The experimental VLE data were utilized to determine the parameters of C 1 and C 2 in Eq. 3. The fitted results are summarized in Table 1, showing an average relative deviation of 2.4% between the calculated equilibrium compositions of the CO 2 -rich (y CH4 ) and CH 4 -rich phases (x CO2 ) and experiment data.Fig. 1 illustrates the comparison between the model results and the experimental data at different temperatures.
As it shows, the model results are in good agreement with the experimental composition data in both phases, while the accuracy of the calculation near the critical region is reduced as the temperature increases.The inaccurate description near the critical region is common for all EOSs due to their intrinsic defect (Bymaster et al., 2008).
With the obtained parameters, the densities of the CH 4 -CO 2 mixture across a range of CO 2 mole fractions (0.0998 to 0.8988) at various temperatures and pressures were predicted.Fig. 2 compares the predicted results and experimental data at temperatures of 313.15 and 353.15 K, showing that over the full range of mole fractions and pressures, the model prediction is in good agreement with the experimental data.At 313.15 K, the densities of the binary system at CO 2 mole fractions of x CO2 = 0.0998, 0.2017, 0.3997, and 0.6015 are linearly increased with increasing pressures.However, as the CO 2 mole fractions further increase (especially, x CO2 = 0.8988 and 0.7985), the relationship between density and pressure becomes non-linear.At 353.15 K, the variation of the density for the CH 4 -CO 2 mixture with the pressure is almost linear at various CO 2 mole fractions.The comparison between Figs. 2(a) and 2(b) reveals that the deviations between the model predictions and experimental values increase as the temperature rises.

CH 4 (1)-H 2 O(3)
For CH 4 (1)-H 2 O(3), we assumed the absence of cross-association between CH 4 -H 2 O, and only the self-association in pure H 2 O was considered.H 2 O was represented as a molecule possessing four association sites, i.e., two O-type sites and two H-type sites.Meanwhile, the cross-dispersive energy for this binary system was adjusted using a temperature-dependent binary interaction parameter k 13 , represented as: For the CH 4 -H 2 O system, experimental phase equilibrium data have been summarized in previous publications (Duan and Mao, 2006;Ou et al., 2015;Xiong et al., 2020).Ou et al. (2015) systematically measured the solubility of CH 4 in the H 2 O-rich phase, covering temperatures from 273.15 to 603.15 K and pressures from 50 to 1400 bar.It should be noted that the measurements by Ou et al. (2015) at low temperatures and high pressures were conducted under metastable equilibrium, indicating that no methane hydrate was formed under these conditions due to the absence of the specific conditions required for nucleation.Additionally, Culberson and McKetta (1951) measured the solubility of CH 4 in pure H 2 O at temperatures from 298.15 to 444.15 K and pressures up to 600 bar.Price (1979) measured the solubility of CH 4 in pure H 2 O at temperatures from 424.15 to 589.15 K and pressures up to 2000 bar.Thus, the experimental data measured by Ou et al. (2015), Culberson and McKetta (1951), and Price (1979) were used to obtain the coefficients in Eq. 4.During parameterization, it was observed that the linear relation described by Eq. 3 has limitations in representing the solubility of CH 4 in the H 2 O-rich phase across a broad temperature range from 273.15 to 603.15 K. Consequently, a non-linear

Parameter
Value 2.351 × 10 − 3  Guillot and Guissani (1993) employed computer simulations, the non-monotonic temperature dependence of the solubility for the non-polar gases, including CH 4 , in water is originated from the entropic contribution of the structural change.

P x
Apart from the experimental data on the solubility of pure CH 4 in H 2 O, the molar volume data of the CH 4 -H 2 O binary mixture and the phase equilibrium composition of H 2 O in the CH 4 -rich (y) gas phase were reported in the literature.Shmonov et al. (1993) determined the molar volumes experimentally at temperatures of 653, 673, and 723 K and pressures from 100 to 2000 bar.The solubility of H 2 O in pure CH 4 was measured by Olds et al. (1942) at temperatures from 310 to 511 K and pressures up to 689 bar.Yarrison et al. (2006) 4 shows the comparison between the predicted density and experimental data at varying temperatures, pressures, and molar fractions of CH 4 .The results indicate good agreements between predicted densities and experimental data.Furthermore, it is observed that the densities of the CH 4 -H 2 O binary mixture rise with increasing pressure, while they decline with an increase in the mole fractions of CH 4 .The comparison of the predicted solubility of H 2 O in pure CH 4 with experimental data is illustrated in Fig. 5, showing that the prediction performance of the model is relatively poor at low temperatures and high pressures.However, with rising temperatures, the disparities between the predicted outcomes and experimental observations diminish.
To further analyze the reason, we re-optimized the binary parameter k 13 to improve the model accuracy for the liquid-phase (i.e., x CH4 ) and vapor-phase (i.e., y H2O ) compositions, while capturing the pressuredependent y H2O at low temperatures.However, k 13 determined from either x CH4 or y H2O data exhibits limited predictability for the respective mole fractions due to discrepancies in the obtained values.Similar findings were reported by Perfetti et al. (2008), who used the CPA EOS to model the VLE of the CH 4 -H 2 O mixture.Given the extensive experimental data of x CH4 , it is reason to use this dataset to determine k 13 , while it is worth mentioning that determining one set of parameters that accurately predicts the compositions in both phases and describes the variations of y H2O with pressure at T < 324 K remains challenging.2015), Culberson and McKetta (1951), and Price (1979); Lines stand for the calculated results in this study.

CH 4 (1)-H 2 O(3)-NaCl(4)
Based on the parameters of CH 4 -H 2 O obtained in this study and the parameters of H 2 O-NaCl determined in the previous work (Ji and Adidharma, 2007), the phase equilibrium of this ternary system was investigated, where one additional binary interaction parameter k 14 was used to describe the short-range interactions between the segments of CH 4 and Na + /Cl − , (5) where the subscripts 4+ and 4-denote the Na + and Cl − , respectively.The temperature-dependent binary parameter k 14 was expressed as follows: For CH 4 (1)-H 2 O(3)-NaCl( 4), the available experimental data have also been summarized previously (Duan and Mao, 2006;Duan et al., 1992;Xiong et al., 2020).Among the available experimental data, the data from Blount and Price (1982) have been extensively utilized in parametrization due to their measurements covering a broad range of temperature, pressure, and NaCl molality (373-513 K, 100-1600 bar, and 0-6 mol/kg).Therefore, the experimental data from Blount and Price (1982) were utilized to determine the coefficients in Eq. 7 in this study.The fitted coefficients, along with an average relative deviation of 7.90%, are provided in Table 1.Fig. 6 showcases the comparison between the model results and the experimental data, indicating agreement throughout wide ranges of temperature, pressure, and molality of NaCl.Furthermore, Fig. 6 shows that the solubility of CH 4 in the aqueous NaCl solution increases with increasing pressure and temperature, particularly above 373.15K. Notably, below 373.15 K, the solubility of CH 4 in aqueous NaCl decreases with increasing temperature, similar to the solubility of CH 4 observed in the pure water.However, the solubility of CH 4 decreases with increasing molality of NaCl.Besides the experimental data from Blount and Price (1982), the CH 4 solubility in aqueous NaCl solutions was determined by Stoessell and To further evaluate the model performance in predicting the vaporphase compositions, we compared the predicted results with the correction factors measured by Katz et al. (1959).These correction factors represent the ratio of the water content in the vapor phase of the CH 4 -H 2 O system to that in the vapor phase of the CH 4 -H 2 O-NaCl system.The correction factor is solely dependent on the concentration of NaCl and is independent of temperature, pressure, and the composition of the gas mixture (Zirrahi et al., 2012).Consequently, the correction factors were calculated at various NaCl concentrations, at a temperature of 298.15 K, and a pressure of 1 bar, and then compared with the available experimental data.The comparison is illustrated in Fig. 8, demonstrating that our developed model can reliably predict the correction factors for the vapor phase of the CH 4 -H 2 O-NaCl system up  Stoessell and Byrne (1982), Michels et al. (1936), andO'Sullivan andSmith (1970); Lines stand for the predicted results in this study.

CH 4 (1)-CO 2 (2)-H 2 O(3)-NaCl(4)
Thus far, the parameters for the four components (CH 4 , CO 2 , H 2 O, and NaCl) required to predict the phase equilibria properties of the CH 4 -CO 2 -H 2 O-NaCl quaternary system have been collected.Among them, the parameters of pure components (i.e., CH 4 , CO 2 , and H 2 O) were determined from their saturated liquid density and vapor pressures.Cross parameters between each pair of components were fitted from their respective subsystems: CH 4 -CO 2 , CH 4 -H 2 O, CH 4 -H 2 O-NaCl, CO 2 -H 2 O, and CO 2 -H 2 O-NaCl.Specifically, they are cross parameters of CH 4 -CO 2 from CH 4 -CO 2 system; CH 4 -H 2 O from CH 4 -H 2 O system; CH 4 -NaCl from CH 4 -H 2 O-NaCl system; CO 2 -H 2 O from CO 2 -H 2 O system; and CO 2 -NaCl from CO 2 -H 2 O-NaCl system.Therefore, using these obtained parameters, the properties of the CH 4 -CO 2 -H 2 O-NaCl quaternary system can be predicted.

Discussion and applications predictions from the new EOS
As explained earlier, this study aimed to develop a SAFT-based EOS for predicting the phase equilibria and densities of the CH 4 -CO 2 -H 2 O-NaCl quaternary system, for which experimental data are lacking.Consequently, the comparison between EOS predictions with the available experimental data for the quaternary system is limited.Nevertheless, the phase equilibria of the CH 4 -CO 2 -H 2 O ternary system has been reported in the literature, so we could predict the phase behavior of the ternary system and compare the predicted results with the available experimental data.The solubility data of the (CH 4 + CO 2 ) gas mixture in pure water were determined by Dhima et al. (1999) at 344.25 K and pressures from 100 to 1000 bar.Qin et al. (2008) measured the VLE data of the CH 4 -CO 2 -H 2 O ternary system at temperatures from 324 to 375 K and pressures from 100 to 500 bar.Kastanidis et al. (2018) conducted solubility measurements for various ratios of (CH 4 + CO 2 ) gas mixture in water at 323 K and 100 bar.Al Ghafri et al. (2014) measured the VLE data of the CH 4 -CO 2 -H 2 O system at temperatures from 323.15 to 423.15 K and pressures up to 200 bar.Hayashi (2014) measured the solubility of CH 4 and CO 2 in the CH 4 -CO 2 -H 2 O system at 333.15 K and pressures from 22.1 to 180.3 bar.According to the Gibbs phase law, by keeping the temperature (T), pressure (P), and mole fractions of CH 4 to CO 2 in liquid phase or vapor phase consistent with the experimental conditions, the liquid phase composition (x i ) and vapor phase composition (y i ) were calculated.The comparison of the predicted solubility of the CH 4 -CO 2 gas mixture in water and the predicted CH 4 concentration in the vapor phase with experiment data is illustrated in Fig. 9, with the average relative deviations of 7.20% and 4.30%, respectively.Thus, the phase behavior of the CH 4 -CO 2 -H 2 O ternary system is well predicted by our model.Also, our model demonstrates comparable predictive performance to other thermodynamic models reported in the literature, such as eCPA (Courtial et al., 2014), SAFT-LJ (Sun et al., 2014), modified CPA equations (Xiong et al., 2021;Yang et al., 2024), and modified PR equation (Li et al., 2015a).Notably, the model performs better in predicting the vapor-phase compositions for the CH 4 -CO 2 -H 2 O system than for the CH 4 -H 2 O system, suggesting that the model is not limited to liquid phase calculations.These results indicate the potential for extending the model to multi-component mixtures with the obtained binary parameters.However, assessing the model performance in the CH 4 -CO 2 -H 2 O-NaCl system, particularly in the vapor phase calculations, is limited by the lack of experimental data.
Moreover, Oldenburg et al. (2003) found that minor differences between estimated physical properties led to substantial differences in simulation results when modeling the mixing of supercritical CO 2 and CH 4 in reservoir processes.For example, four popular numerical simulation codes (i.e., CHEMTOUGH, GEM, SIMUSCOPP, and TOUGH2) predict aqueous solubility of the (CH 4 + CO 2 ) gas mixtures with different degrees of deviation from the reference values.In these four simulators, the Redlich-Kwong equation was utilized in CHEMTOUGH (White, 1995), while the PR EOS or its modifications were employed in GEM (Nghiem et al., 2004), SIMUSCOPP (Le Thiez et al., 1996), and TOUGH2 (Oldenburg et al., 2004) to model the phase equilibria of the CH 4 -CO 2 -H 2 O system and its subsystems.To evaluate the accuracy of our model, we predicted the aqueous solubility of the (CH 4 + CO 2 ) gas mixtures under the same conditions (at the temperature of 313.15K and pressures of 40 and 100 bars) as those in the report of Oldenburg et al. (2003) and compared the predicted results with the values calculated using the four numerical simulation codes.The results are presented in Table 2, showing that the deviations between the values of CH 4 and CO 2 solubility predicted in this study and reference values are the smallest compared to other models.This observation suggests that the model developed in this study can provide more accurate prediction results, enhancing the applicability of the subsequent numerical simulations for gas mixtures within reservoirs.
With the obtained parameters, we calculated properties and phase equilibria for the CH 4 -CO 2 -H 2 O-NaCl system.Based on the mole fraction in the H 2 O-rich phase at the temperature of interest, the equilibrium pressures were predicted.In our calculations, the molar ratios of CH 4 to CO 2 in the H 2 O-rich phase were set as 1:1, 2:1, 4:1, 9:1, and 19:1, and the concentration of NaCl varied from 0, 1, 2 to 6 mol/kgH 2 O, and the temperature was set as 298.15, 323.15, 348.15, 373.15, and 423.15 K. Additionally, the equilibrium compositions, fugacity coefficients of components, and aqueous solution density for the CH 4 -H 2 O-NaCl system and the CH 4 -CO 2 -H 2 O-NaCl systems with various molar ratios of CH 4 to CO 2 in the H 2 O-rich phase were also predicted.These predictions are listed in Table S2-S7 (Supporting Material S1).Moreover, based on the EOS, a user-friendly software with a graphical user interface (GUI) was developed, employing the interpolation approach to calculate gas solubility and aqueous solution in the CH 4 -CO 2 -H 2 O-NaCl system and its subsystems, with details available in Supporting Material S2.
Equilibrium pressures for the CH 4 -CO 2 -H 2 O system at 298.15 K were compared with those for CH 4 -H 2 O and CO 2 -H 2 O (Fig. 10a).For a fixed mole fraction in the H 2 O-rich phase, the equilibrium pressure of CH 4 surpasses that of CO 2 by a considered margin, indicating that CO 2 is more soluble than CH 4 .For the mixture of CH 4 -CO 2 -H 2 O at the same concentration of (CH 4 + CO 2 ), the equilibrium pressure for the mixture falls between those observed for CH 4 -H 2 O and CO 2 -H 2 O, which means that the (CH 4 + CO 2 ) gas mixture is more soluble than pure CH 4 but less soluble than pure CO 2 .Additionally, with the decrease of the mole ratio of CO 2 to CH 4 , the equilibrium pressure for the mixture is monotonically m NaCl Fig. 8.Comparison between the calculated correction factors and the experimental values of Katz et al. (1959) for the CH 4 -H 2 O-NaCl system at atmospheric pressure.increased, indicating that the solubility of the gas mixture is expected to be a linear combination of the solubilities of the two pure gases.
Predictions were made for the equilibrium pressures of the CH 4 -CO 2 -H 2 O-NaCl system at 298.15, 348.15, and 423.15 K, with a molar ratio of CH 4 to CO 2 of 2:1 in the liquid phase, and various molalities (m NaCl ), as illustrated in Fig. 10b-d.At a specific temperature, the equilibrium pressure increases as the concentration of the (CH 4 + CO 2 ) in the liquid phase increases and as the concentration of NaCl increases.The results suggest that the inclusion of NaCl has adverse effects on the solubility of the gas mixture in the H 2 O-rich phase.Moreover, the variation of solubility of the (CH 4 + CO 2 ) gas mixture in aqueous NaCl solutions with temperature exhibits the same tendency as that of the mixture in pure water.Specifically, the variation in solubility of the (CH 4 + CO 2 ) gas mixture within the H 2 O-rich phase exhibits a nonmonotonic behavior as temperature varies.
As discussed in our previous work (Ji and Zhu (2012), the dissolution of CO 2 increases the aqueous solution density, which can cause natural convection to occur, thereby helping to dissolve the CO 2 in the reservoir.Therefore, the investigation of the injection of (CH 4 + CO 2 ) gas mixture on the solution density is necessary.Using the model in this study, we predicted the solution density at 334.15 K, 135 bar with a NaCl molality (m NaCl ) of 2.05 mol/kgH 2 O when the mole fraction of gases changes from zero to saturated concentration for CH 4 -CO 2 -H 2 O-NaCl solutions.Specifically, we considered x CO2 * = x CO2 /(x CH4 + x CO2 ) = 0, 0.5, 0.75, 0.9, and 1.The results are shown in Fig. 11.At x CO2 * = 0 (i.e., solutions of CH 4 -H 2 O-NaCl), the solution density decreases with the increasing CH 4 concentration.Conversely, when x CO2 * = 1, i.e., solutions of CO 2 -H 2 O-NaCl, the solution density increases with the increasing CO 2 concentration.For the gas mixture, the density of the solution increases as the mole fraction of CO 2 increases.

Application for CO 2 and CH 4 solubility calculation in WAG injection operations
In the area of WAG injection, the majority of previous studies have examined the impacts of impurities in the CO 2 stream on the minimum miscibility pressure.However, the effects of CH 4 hysteresis and solubility for residual and solubility trapping mechanisms have received less attention.Specifically for WAG process simulation, Cho et al. (2020) conducted a compositional simulation of the co-injection of CO 2 and CH 4 during the WAG process to assess the efficiency of carbon capture and storage in combination with EOR.The Weyburn Field in Saskatchewan, Canada, was used as an example by Cho et al. (2020) and the simulation period is 13 years (2007)(2008)(2009)(2010)(2011)(2012)(2013)(2014)(2015)(2016)(2017)(2018)(2019)(2020).The WAG process was designed as three years of waterflooding followed by ten years of WAG (with the gas injection for one year followed by water injection for one year and the alternation continued for five gas-water cycles).Four cases were simulated: 100% CO 2 + 0% CH 4 (Case 1), 90% CO 2 + 10% CH 4 (Case 2), 80% CO 2 + 20% CH 4 (Case 3), and 70% CO 2 + 30% CH 4 (Case 4).The gas injection rate was constant at 2265 m 3 /day (surface condition) for each case.The reservoir temperature is 63 • C (336.15 K) with varying average reservoir pressure ranging from 14,200 to 17,000 kPa (Cho et al., 2020).Gas fugacity was calculated using the PR EOS (Cho et al., 2020).
x x y Fig. 9. Comparison of (a) the predicted solubility of (CH 4 + CO 2 ) gas mixture in the liquid phase and (b) the predicted mole fraction of CH 4 in the vapor phase for the (CH 4 -CO 2 -H 2 O) ternary system with experimental data.Symbols, experimental data from Qin et al. (2008), Dhima et al. (1999), Kastanidis et al. (2018), Al Ghafri et al. (2014) and Hayashi (2014).

Table 2
Comparison of the solubility of the (CH 4 + CO 2 ) gas mixture in water from our model prediction with values estimated by four numerical simulation codes.We conducted the solubility calculations using the SAFT2 model in this study and compared them with the results from Cho et al. (2020).The calculated solubility values for CO 2 and CH 4 (in molality) across four CO 2 -CH 4 WAG cases are detailed in Table 3.
Because the required information is not available to calculate the total amount of dissolved CO 2 and CH 4 in the reservoir using SAFT2, we focus our comparison on the ratios of solubility-trapped CO 2 for each case with those from Cho et al. (2020).The ratios of solubility-trapped CO 2 for each case were first calculated for Cho et al.'s simulation results (Table 4).These ratios were compared with those from the SAFT2 model across different cases, as depicted in Fig. 12.It can be seen that ratios of solubility-trapping CO 2 from Cho et al. (2020) across four cases exhibit good alignment with those of calculated CO 2 solubility using the SAFT2 model.relative to the emissions of 1 ton of CO 2 .The GWPs of the trapped CO 2 in four cases are 3.94 × 10 7 , 3.04 × 10 7 , 2.48 × 10 7 and 2.02 × 10 7 mol, respectively (Fig. 13 of Cho et al. (2020)).The trapped CO 2 consists of residual and solubility trapping CO 2 , and the GWP of solubility trapping CO 2 is assumed to be the GWP of trapped CO 2 multiplied by the proportion of solubility trapping CO 2 .Based on the results of the proportion of CO 2 remaining in the reservoir (Fig. 12 of Cho et al. (2020)), the proportion of solubility trapping CO 2 to the sum of proportions of residual trapping, solubility trapping, and movable CO 2 in four cases are 0.115, 0.133, 0.146 and 0.159, respectively.Consequently, the GWPs of solubility trapping CO 2 are estimated to be 4.53 × 10 6 , 4.04 × 10 6 , 3.63 × 10 6 and 3.22 × 10 6 mol, respectively.If the GWP of solubility trapping CO 2 for Case 1 was considered as the reference, the ratios of GWP between Case 2 and Case 1, Case 3 and Case 1, and Case 4 and Case 1 would be 0.8918, 0.8013, and 0.7108, respectively.They are closely aligned with the ratios of calculated CO 2 solubility for various cases using the SAFT2 model; the ratio of CO 2 solubility in the liquid phase between Case 2 and Case 1 is 0.8923, that of Case 3 and Case 2 is 0.8038, and that of Case 3 is 0.7108.However, for CH 4 , the proportion of solubility trapping CH 4 in total trapped CH 4 is unknown.In this case, the comparison between the ratios of solubility trapping CH 4 across various cases and those of CH 4 solubility in the liquid phase was not conducted.
From this example, we found that the SAFT2 model compared well with the simulation results using PR EOS for the solubility prediction of CO 2 and CH 4 .Reservoir simulation is time-consuming and computationally expensive.The advantage of using the SAFT2 model is that it can reduce the number of scenarios in the sensitivity analysis of reservoir simulations.With the results of one simulation scenario (e.g., Case 1), we can predict the amounts of solubility trapped CO 2 for the scenarios with variable CH 4 percentages (e.g., Cases 2, 3, and 4).Given the information on the proportion of solubility trapping CH 4 in total trapped CH 4 , the amount of solubility trapped CH 4 can also be predicted.This supports the quantification of the carbon storage effects of greenhouse gases in CO 2 -EOR operations.Besides the application in CO 2 EOR, the SAFT2 model is also useful in measuring CO 2 storage in depleted natural gas fields, where CH 4 is presented as the primary component of remaining hydrocarbon gas.

Conclusions
We used a SAFT-based EOS for predicting phase equilibria and densities of the CH 4 -CO 2 -H 2 O-NaCl quaternary system, covering wide ranges of temperatures, pressures, and NaCl concentrations, where the parameters for the pure components, CO 2 -H 2 O, and CO 2 -H 2 O-NaCl were taken from previous work (Ji and Adidharma, 2010;Tan et al., 2006).The cross parameters of SAFT2 were fitted based on the experimental phase equilibrium data for CH 4 -CO 2 , CH 4 -H 2 O, and CH 4 -H 2 O-NaCl.The new model exhibits good performance in representing the properties and phase equilibria when compared to the experimental data.
Using the EOS in this study, we predicted the phase equilibria of the CH 4 -CO 2 -H 2 O ternary system.Comparing our model predictions with those predicted using four popular numerical simulators shows that our model predictions are closer to experimental data.Subsequently, we further predicted the phase equilibria and properties of the CH 4 -CO 2 -H 2 O-NaCl quaternary system.Our predictions reveal (CH 4 + CO 2 ) gas mixture is more soluble than pure CH 4 but less soluble than pure CO 2 .The dissolved concentration of the gas mixture increases with increasing pressure and decreasing concentration of NaCl, while its solubility exhibits a non-monotonic trend as temperature increases.Moreover, dissolving CH 4 will reduce the solution density as opposite to dissolving CO 2 .These predictions of phase equilibria and thermodynamic properties are beneficial for evaluating costs and risks for the transportation and storage of CO 2 and natural gas as well as the WAG process.
Furthermore, we employed the SAFT2 model to predict the solubilities of CO 2 and CH 4 in four WAG processes for CO 2 EOR.The predicted results exhibited good alignment with reservoir simulation results using the PR EOS for the solubility prediction of CO 2 and CH 4 .This underscores the capability of the SAFT2 model to provide quantitative insights into the carbon storage implications of greenhouse gases within CO 2 -EOR operations.(Cho et al., 2020), lines: CO 2 solubility calculated by SAFT2.

Fig. 2 .Fig. 3 .
Fig. 2.Comparison of the predicted densities with the experimental data of the CH 4 -CO 2 mixture at various mole fractions of CO 2 (0.0998, 0.2017, 0.3997, 0.6015, 0.7985, and 0.8988) and temperatures of (a) 313.15K and (b) 353.15 K. Symbols stand for experimental data fromLiu et al. (2018).Lines stand for the predicted results in this study.

Fig. 4 .PFig. 5 .
Fig. 4.Comparison of the predicted densities with the experimental data of the CH 4 -H 2 O mixture at various mole fractions of CH 4 (0, 0.2, 0.4, 0.6, 0.8, and 1) and temperatures of (a) 653.15K and (b) 723.15 K. Symbols represent experimental data fromShmonov et al. (1993); Lines stand for the calculated results in this study.

Fig. 6 .Fig. 7 .
Fig. 6.Comparison of the calculated CH 4 solubility in aqueous NaCl solutions with experimental data at temperatures of (a) 373.15,(b) 443.15,(c) 478.15, and (d) 513.15K at varying concentrations of NaCl.Symbols represent experimental data fromBlount and Price (1982); Lines stand for the calculated results in this study.

Fig. 10 .
Fig. 10.Solubility variation with pressure for the (CH 4 + CO 2 ) gas mixture in pure H 2 O (a), across different CH 4 to CO 2 ratios.Solubility variation for the (CH 4 + CO 2 ) gas mixture with a constant CH 4 to CO 2 ratio of 2:1 in the NaCl solution with various molalities, at temperatures of 298.15K (b), 348.15K (c), and 423.15K (d).
given in Eq. 4 was utilized to capture the temperature dependence of k 13 .The fitted results of C 3 -C 6 are shown in Table1, with an average relative deviation of 5.47% between the calculated CH 4 solubilities in the H 2 O-rich phase and experiment data.Fig.3depicts the comparison of the calculated results with experimental data, demonstrating good agreement between the model results and the experimental data in the wide ranges of temperatures and pressures.The solubility of CH 4 in pure H Fig. 1.Comparison of calculated equilibrium compositions with experimental data at different temperatures for the CH 4 -CO 2 mixture.Symbols (experimental data), □, Davales et al. (1976), ○, Wei et al. (1995), △, Webster and Kidnay (2001).Lines (calculated results), black, 230 K, red, 250 K, and blue, 270 K. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)Z.Zuo et al.relation 2 O demonstrates an upward trend with increasing pressure, whereas its behavior with temperature is notably non-monotonic.For instance, the solubility of CH 4 in pure H 2 O decreases with increasing temperature in the range of 273 to 345 K. Conversely, within the temperature range of 373 to 603 K, the solubility of CH 4 increases with increasing temperatures.As studied by measured the solubility of H 2 O in supercritical CH 4 under temperatures from 310 to 477 K and pressure up to 1100 bar.Tabasinejad et al. (2011) measured the H 2 O content in the CH 4 vapor phase at temperatures from 422 to 483 K and pressures up to 733 bar.With the obtained parameters (i.e., C 3 ~C6 ) shown in Table 1, the density of the CH 4 -H 2 O binary mixture and the solubility of H 2 O in pure CH 4 were predicted.Fig.

Table 3
The solubility of CO 2 and CH 4 calculated with SAFT2 for four CO 2 -CH 4 WAG cases.

Table 4
The solubility-trapped CO 2 values of Cho et al.'s simulations and the corresponding calculated ratios across four cases.Comparison between the solubility-trapping CO 2 and calculated CO 2 solubility for four CO 2 -CH 4 cases.Symbols: Data of solubility trapping CO 2