An improved phase-partitioning model is proposed for the prediction of the mutual solubility in the CO2-brine system containing Na+, K+, Ca2+, Mg2+, Cl-, and SO42-. The correlations are computationally efficient and reliable, and they are primarily designed for incorporation into a multiphase flow simulator for geology- and energy-related applications including CO2 sequestration, CO2-enhanced geothermal systems, and CO2-enhanced oil recovery. The model relies on the fugacity coefficient in the CO2-rich phase and the activity coefficient in the aqueous phase to estimate the phase-partitioning properties. In the model, (i) the fugacity coefficients are simulated by a modified Peng-Robinson equation of state which incorporates a new alpha function and binary interaction parameter (BIP) correlation; (ii) the activity coefficient is estimated by a unified equilibrium constant model and a modified Margules expression; and (iii) the simultaneous effects of salting-out on the compositions of the CO2-rich phase and the aqueous phase are corrected by a Pizter interaction model. Validation of the model calculations against literature experimental data and traditional models indicates that the proposed model is capable of predicting the phase-partitioning behaviors in the CO2-brine system with a higher accuracy at temperatures of up to 623.15 K and pressures of up to 350 MPa. Using the proposed model, the phase diagram of the CO2+H2O system is generated. An abrupt change in phase compositions is revealed during the transfer of the CO2-rich phase from vapor to liquid or supercritical. Furthermore, the preliminary simulation shows that the salting-out effect can considerably decrease the water content in the CO2-rich phase, which has not been well experimentally studied so far.
CO2-water/brine is one of the most important and commonly encountered systems [1–3] in CO2 sequestration [4–11], CO2-enhanced oil recovery (EOR) [2, 12, 13], CO2-enhanced geothermal systems (EGS) [14, 15], global CO2tracing [16–19], and so on. In these chemical-, petroleum-, and environment-related technological applications, accurate prediction of the phase-partitioning properties over a wide pressure-temperature-salt composition (--) range is essential for the understanding of the CO2 flow and trapping mechanism in subsurface formations [7, 11, 20, 21] and the potential rock-fluid chemical interactions [4, 14].
By now, CO2 solubility in water/brine has attracted great interest. The injected CO2 can dissolve into formation water, form carbonic acid, and react with reservoir rocks, altering the porosity and permeability of the porous media [22–24]. This complicated geochemical process related to CO2 dissolution has a long-term positive or adverse influence on the performance of subsurface systems. Firstly, the preliminary simulation revealed that the heat exploitation efficiency was decreased by 27% in a CO2 geothermal system due to CO2 dissolution and mineral precipitation . Borgia et al.  indicated that the geothermal reservoir could even be dead in 1 year. Secondly, Enick and Klara  and Chang et al.  demonstrated that the ultimate CO2 recovery was significantly decreased owing to a large proportion of CO2 trapping in the formation water in the CO2 EOR process. Thirdly, CO2 dissolution is a controlling factor for long-term environmental safe storage [11, 20, 21, 29, 30], given the fact that the solubility trapping accounts for 90% of the total storage capacity in CO2 sequestration in saline aquifers [7, 20, 21]. Compared to CO2 solubility, the water content in the CO2-rich phase has been largely ignored [4, 31]. However, it is of same importance because the amount of water in the CO2-rich phase determines the capability of injected CO2 to dry subsurface rocks [14, 32, 33] and affects the type of chemical fluid-rock interactions [14, 34, 35]. Furthermore, water vaporization can lead to salinity concentrating and then decrease the CO2 solubility in the aqueous phase. Therefore, it is of fundamental and practical importance to build an accurate model of the mutual solubility in the CO2-brine system.
Regarding the CO2+H2O system, a large abundance of experimental studies have been carried out to obtain straightforward knowledge of the phase-partitioning behaviors [1, 36–42], which can facilitate the development of an accurate phase equilibrium model. The modelling approaches can be generally divided into two categories: - models and - models. The - model relies on a homogeneous equation of state (EOS) to estimate the fugacity coefficient of different components in the CO2-rich phase and the aqueous phase. However, the classic cubic EOS is not capable of accurately characterizing the phase behaviors in a strongly nonideal system [2, 43], in which the water and CO2 molecules can form a hydrogen bond and can associate . A feasible approach is the incorporation of the excess Gibbs energy model into a cubic EOS or statistical associated fluid theory (SAFT) [21, 43]. The main advantages of - models are their capacities for reproducing volumetric properties and estimating the phase properties near the critical point. However, they are commonly much more computationally complicated compared to the - models . Furthermore, the microscopic knowledge of molecular structures is necessary but not applicable in some industry applications . The - model relies on the activity coefficient in the aqueous phase and the fugacity coefficient in the CO2-rich phase. Although these type of models are not physically rigorous and accurate enough near the critical point , they could be much more amenable to integration with chemical equilibrium simulation [4, 45]. Considering its good extensibility and computational efficiency, the - approach was most commonly used in the large-scale multiphase flow simulations [8, 14, 46]. From this concern, this study focuses on developing a new - type model.
Based on Peng-Robinson’s EOS and Henry’s law, Li and Ngheim  developed a model to predict the CO2 solubility below 473 K, in which a scaled-particle theory was incorporated to account for the effect of NaCl concentration. However, it was indicated that the model calculations were generally not accurate enough . Hu et al.  demonstrated that the cubic and virial EOS were capable of predicting CO2 solubility up to 50 MPa, but the simulation error at high pressure was found to be unacceptable. Assembled from 21 literature experimental studies, a databank of CO2 solubility containing 508 pieces of data was developed by Akinfiev and Diamond . They proposed an accurate - model with a valid range of 0-100 MPa and below 100°C, but the model is not accurate enough for estimating the water content in the CO2-rich phase. Sorensen et al.  developed a model of CO2 solubility in pure water (348-623 K and 1.6-140 MPa) and in NaCl solutions (298-523 K and 0.1-138.2 MPa). However, the corresponding simulation errors can reach 37% and 20.3%, respectively. Mao et al.  built an accurate model for CO2 solubility in NaCl solution, but it cannot be used for estimating CO2 solubility in other brines or water content in the CO2-rich phase. Dubessy et al.  proposed an unsymmetric - model to simulate the compositions of different phases in 40°C-270°C. However, it is only valid below 30 MPa. Similarly, Portier and Rochelle  developed a model of CO2 solubility in pure water and brine with a valid thermodynamic range of 0-300°C and 0-30 MPa.
Using the unsymmetric - approach, Duan and Sun  and Spycher and Pruess  developed the most commonly used and cited models in the geological scientific community [3, 11, 30]. Combining a virial EOS for pure CO2 and a semiempirical Pizter interaction equation, the model of Duan and Sun  can be used to simulate the CO2 solubility in pure water and brines up to 533 K and 200 MPa. It is in general computationally accurate and efficient. However, Hou et al.  claimed that the model calculations of Duan and Sun  disagreed significantly with their experimental measurements at 448.15 K. Similarly, Guo et al.  demonstrated that the model calculation substantially deviates from the measurements above 523.15 K. Affected by the scope of the experimental database used in model development, the model calculations in other brines were not as accurate as those in NaCl solutions . Furthermore, this model cannot be used for accurately estimating the water content in the CO2-rich phase. The models of Spycher et al.  and Spycher and Pruess  rely on the Redlich–Kwong EOS for calculating the fugacity coefficient in the CO2-rich phase, whereas the activity coefficient in the aqueous phase is treated by the correlations of an equilibrium constant for pure water and a Pizter interaction expression for brines. Compared to the model of Duan and Sun , it includes two more experimental studies for parameter determination  and is suitable for both CO2 solubility in the aqueous phase and water content in the CO2-rich phase. However, the valid pressure range was limited to 0-60 MPa [1, 11]. In order to accurately estimate the phase compositions, different model parameter sets were utilized according to the temperature ranges and phase transition of CO2. This discontinuity may affect the smoothness of derivatives and damage the Jacobian-based numerical formulation in large-scale multiphase flow simulation. Furthermore, the effect of salting out on the water content in the CO2-rich phase was neglected in their model. Owing to the improvement of experimental approaches, more available literature data used as fitting constraints can help to increase the accuracy of new thermodynamic formulations [2, 14, 42, 45, 53].
Based on the pioneering experimental and modelling studies, a unified model is developed in this study to characterize the phase-partitioning behaviors in the CO2-water/brine system. There are three major improvements compared to the traditional models. Firstly, both the CO2 solubility in the aqueous phase and water content in the CO2-rich phase can be accurately estimated at temperatures up to 623.15 K and pressures up to 350 MPa. Secondly, an updated phase equilibrium databank assembled from 114 literature experimental studies was developed for model calibration. Thirdly, the salting-out effect of Na+, K+, Ca2+, Mg2+, Cl-, and SO42- on the composition of both the aqueous phase and the CO2-rich phase is considered.
2. Thermodynamic Modelling of CO2-H2O System
2.1. Thermodynamic Framework for Vapor-Liquid Phase Equilibrium
When the gas solubility in the aqueous phase is low without salts, the proposed model can be simplified as Raoult’s law, i.e., the water activity is equal to its mole fraction in the aqueous phase ().
2.2. Fugacity Modelling of the CO2-Rich Phase
The fugacity and thermodynamic properties of the CO2-rich phase are commonly estimated by EOS. Owing to its advantages of computational efficiency and accuracy, cubic EOS is the most commonly used type of EOS models in numerical simulations of multiphase flow .
2.2.1. Alpha Model
The accuracy of the phase equilibrium of a pure material mainly relies on the cohesion factor, which is the basis for multiphase equilibrium simulation [2, 59]. The cohesion factor popularly known as the alpha function represents the effect of mutual attraction between molecules. It is in general a function of temperature and acentric factor, as indicated by previous models [58, 59]. An alpha function should fulfill the following criteria: (1) As the temperature increases and tends to infinity, it should approach zero. (2) Embodying the attraction forces between molecules, it must always be positive. (3) It should be equal to unity at the critical point.
2.2.2. Mixing Rule
2.3. Fugacity Modelling of the Aqueous Phase
It can be found that the fugacity coefficient of gas in the aqueous phase is controlled by three parameters: (1) the equilibrium constant of CO2 in the aqueous phase (), which can be simplified as Henry’s constant at low pressure; (2) the relative activity coefficient including the effect of temperature and pressure for pure water, and the effect of electrolytes; and (3) specific volume which accounts for the effect of pressure.
2.3.1. Equilibrium Constant
The CO2-rich phase has significant variations in thermodynamic properties, as it transfers from vapor to liquid or supercritical. In order to accurately describe the effect of CO2 phase transition, a common approach is to select different equilibrium constant models according to the CO2 phase states [14, 60, 61]. This piecewise parameter group may lead to an unsmooth functional form and a discontinuity of its derivative, which is crucial for the Jacobian-based numerical formulation in a multiphase flow simulator . Therefore, a unified and continuous model of the gas equilibrium constant is derived in this study.
2.3.2. Specific Volume and Relative Activity Coefficient
2.4. Simulation Method
The flow chart for model simulation includes the following steps:
Based on prediction of the water saturation pressure, the water content in the CO2-rich phase can be roughly estimated using the law of partial pressure, . With the estimated CO2 phase composition, the fugacity of water in the CO2-rich phase can be calculated using the modified Peng-Robinson EOS. Then, the water content () can be corrected by equation (25) assuming
Using a Newton-Raphson algorithm, can be estimated by equation (27). Then, the mole fraction of water in aqueous phase can be calculated based on mass conservation
Repeat steps (2) and (3) until the convergence criterion of maximum permissible error is satisfied
3.1. Experimental Database in CO2+H2O System
An extensive experimental databank is developed for the CO2+H2O and CO2+brine systems, which includes the data of compositions of the CO2-rich phase and the aqueous phase assembled from 114 literature studies, as shown in Tables 1–3 of Appendix B. The measurements in the databank are obtained at temperatures up to 623.15 K and pressures up to 350 MPa, which cover the potential thermodynamic conditions in common geological applications, such as petroleum engineering, geothermal development, and CO2 sequestration.
3.2. Parameter Determination
The proposed model contains a variety of parameters. Overall, the sensitivity of model parameters to different types of experimental data is different. For example, the water content in the CO2-rich phase is mainly controlled by the binary interaction parameters between different components in the fugacity model of the gas phase , although it may be not very accurate at high temperature (higher than 150°C). In this study, the different types of model parameters are firstly determined by their closely related experimental data. Then, using the determined parameters as initial values, all the model parameters are corrected simultaneously based on the developed experimental databank. In detail, the Newton-Rapson iterative algorithm is used to determine the model parameters as follows :
The parameters in alpha model are determined by the experimental data of the saturation pressure of CO2 and water
The binary interaction parameters between CO2 and water in the gas phase is determined by the experimental data of water content in the CO2-rich phase. While in this step of parameter determination, the necessary data of CO2 solubility is referred to measured data of mutual solubility or the simulated results of the traditional model
The model parameters in the fugacity model of the aqueous phase are determined by the experimental data of CO2 solubility in the aqueous phase. Similarly, the other phase composition data that are necessary in this parameter determination procedure can be referred to measured data or simulated results of the traditional model
The parameters in the Pizter interaction models are determined by the measurements of CO2 solubility in brines
Using the determined results in steps (1)-(4) as initial values, all the model parameters are corrected simultaneously by the developed experimental databank. The determined model parameters in this study are listed in Table 4
4. Model Performance in CO2-Water System
4.1. Model Verification
4.1.1. CO2 Solubility
Figure 1 represents the comparison between the simulated and measured data of CO2 solubility below 100°C. As seen, the solubility of CO2 generally increases with temperature increasing. There is an abrupt change in the variation trend of the curves at around 10 MPa, which is mainly caused by the variations of thermodynamic properties during the CO2-rich phase transferring from vapor to liquid or supercritical.
A large quality of studies have been carried out to measure the CO2 solubility below 100°C. Overall, the literature experimental data is in close agreement with the calculations of the proposed model, Spycher and Pruess , and Duan and Sun . By comparison, a significant deviation is found using the model of Li and Yang .
However, there is considerable consistency between the experimental datasets of different literature studies. At , the experimental data of Hou et al.  and Nakayama et al.  deviates obviously from those of other studies. Its deviation approaches to 15% at . When the pressure is larger than 20 MPa, the experimental data of Greenwood and Barnes  agrees well with those of Wiebe , but it has a significant deviation from those of Teng et al.  and Gillepsie and Wilson . At , 75°C, or 100°C, the experimental data adopted from different literature studies are generally consistent, except for the data of Qin et al.  at 50°C, and the data of Sako et al.  and Kiepe et al.  at 100°C.
It can be seen that the simulated results of Spycher and Pruess  deviate significantly from the experimental data at and . Similarly, the simulation error of Duan and Sun  tends to significantly increase at pressures larger than 150 MPa. By comparison, the proposed model is in good agreement with the experimental datasets at 0-350 MPa.
Figure 2 represents the comparison between the simulated and measured data of CO2 solubility between 100°C and 300°C. As seen, the experimental data in different studies are generally consistent at same temperature, except for those of Shagiakhmetov and Tarzimanov  at 150°C.
However, significant deviations exist between the predicted results of different models at temperatures from 100°C to 300°C: (i) The model of Li and Yang  is accurate below 200°C; however, it significantly deviates from the experimental measurements above 200°C. (ii) The model of Spycher and Pruess  agrees well with the literature data below 60 MPa; however, the simulation error increases rapidly with pressure above 60 MPa. (iii) Regarding the previous models, the model of Duan and Sun  is more accurate than those of Spycher and Pruess  and Li and Yang . (iv) Compared to other models, the proposed model has a better accuracy over a wide temperature and pressure range.
As shown in Figure 3, the CO2 solubility in the aqueous phase and H2O content in the CO2-rich phase significantly increase with temperature above 300°C. The inconsistency is obvious as the fluid system approaches to be miscible.
The experimental measurements of different studies agree well at and . At , the data of Blencoe  is consistent with that of Takenouchi and Kennedy , but shows a significant deviation with that of Todheide and Franck . As already indicated by Spycher and Pruess , a complete mixing is necessary to reach a fully miscible state. However, this appears to be not sufficient in experiments of Takenouchi and Kennedy . By comparison, the model appears in much better agreement with the data determined by Todheide and Franck .
4.1.2. H2O Content
A comparison between the simulated and measured data of H2O content in the CO2-rich phase is shown in Figure 4. As seen, the H2O content decreases rapidly with pressure increasing, and approaches to constant at high pressure.
Similar to that of CO2 solubility, there is an abrupt change in H2O content as the CO2-rich phase transfers from vapor to liquid or supercritical. This variation appears to be obvious at low temperature and tends to decrease with temperature increasing. Although the identical model parameters are used for different CO2 phases, the proposed model can accurately reproduce the phase-partitioning behaviors over a wide temperature and pressure range. This approach is the derivative, continuous and smooth, which facilitates its incorporation into Jacobian-based numerical formulation in a multiphase flow simulator.
There exists a considerable inconsistency between different experimental datasets. At 50°C, the data of Todheide and Franck  differs slightly from those of other studies. For example, the simulation error between Todheide and Franck  and Wiebe and Gaddy  approaches 28% at 20 MPa. At 200°C, the measured data of Takenouchi and Kennedy  and Malinin  obviously deviate from those of Todheide and Franck . The possible reason for inconsistency between these experimental studies has been widely analyzed, which is beyond the scope of this work. The simulated results of the proposed model appear to be closer to the measurements of Todheide and Franck .
There are obvious differences in the calculation results of water content by different models: (i) In the model of Duan and Sun , the water content in the gas phase is equal to the ratio of water vapor pressure to the total pressure. Although this simplification is valid at low pressure, it can generate a significant deviation at high pressure. (ii) The model of Li and Yang  can accurately reproduce the composition of the CO2-rich phase. However, successful convergence is difficult in many thermodynamic conditions, dependent on the valid temperature and pressure range of this model and limitations of the - type approach. (iii) Although both the proposed model and Spycher and Pruess  can accurately reproduce the water content, our model shows a better accuracy at temperatures up to 300°C and pressures larger than 200 MPa.
4.1.3. Error Analysis
Figure 5 shows the comparison in calculation errors of the proposed model and the traditional models. Both the models of Duan and Sun  and Spycher and Pruess  rely on the fugacity coefficient in the CO2-rich phase and the activity coefficient in the aqueous phase, which are the two most commonly used models in the geological field [1–3]. By comparison, the overall simulation error of Spycher and Pruess  is slightly larger than that of Duan and Sun , owing to a valid pressure range of 0-60 MPa. However, the model of Duan and Sun  can only produce a rough estimate of water content in the CO2-rich phase with an overall simulation error larger than 50%.
Compared to Duan and Sun  and Spycher and Pruess , the simulation error of Li and Yang  is considerably larger. The possible reasons are analyzed as follows: (i) The cubic EOS is not perfectly suitable for characterizing the fugacity of the aqueous phase in a strongly nonideal system containing water, although it can be improved via the incorporation of a modified alpha equation and a binary interaction model. (ii) The quality of experimental data determines the calculation accuracy of the model to a large extent. It may be the main factor that affects the model accuracy given the fact that a limited experimental databank containing 109 pieces of data is used by Li and Yang .
Regarding the proposed model, the average simulation errors for CO2 solubility and H2O content are 4.765% and 8.182%, respectively, which are better than other models over a wide temperature and pressure range. We cannot find a further improvement of model performance by increasing the number of parameters and altering the forms of equations. The increase of calculation accuracy needs more high-precision experimental data at high temperature and pressure conditions.
The aforementioned analysis indicates that all the models have good accuracy for experimental data at low temperatures and pressures, which accounts for the majority of the databank. This suggested that the difference in average simulation errors of different models is mainly generated by the simulation error at high temperatures and pressures.
4.2. Phase Diagram
4.2.1. CO2 Solubility
Figure 6 shows the phase diagram of CO2 solubility at different temperatures and pressures. The color region represents the gas-liquid state. The gray area on the bottom represents the single CO2-rich phase, in which the system pressure is lower than the water saturation pressure. The gray area on the top is the single aqueous phase. The fluid system tends to be completely miscible as temperature and pressure increases .
The distribution of contours and variations of colors indicate that the CO2 solubility increases with pressure increasing. Furthermore, the CO2 solubility firstly increases and then decreases with temperature . A more rapid change in CO2 solubility can be found above 150°C compared to that at a low temperature region .
4.2.2. Water Content
Figure 7 shows the phase diagram of water content at different temperatures and pressures. Similar to Figure 6, the color area represents the immiscible region, while the gray area represents the miscible region.
As shown, the water content in the CO2-rich phase increases with temperature increasing. It is more sensitive to changes in temperature above 100°C. Furthermore, the water content rapidly decreases with pressure increasing, showing a complicated variation trend.
5. Extension of the Model to Brines
One advantage of the - model is the good extensibility to complicated systems containing multiple ions [4, 45]. Commonly, the salting-out effect is widely indicated to considerably decrease gas solubility . However, its effect on composition of the CO2-rich phase is generally neglected in previous studies [4, 14]. In this study, a modified Pizter interaction model is incorporated into the developed model to demonstrate the effect of multiple solutes on the phase-partitioning behaviors of the CO2-brine system.
5.2. Comparison to the Experimental Data
The most commonly encountered ions in the geological field include Na+, K+, Ca2+, Mg2+, Cl-, and SO42-. An extensive databank of experimental measurements in such a brine of single salt or mixed salts are developed, covering a wide temperature and pressure range. Using the collected data, the binary and ternary interaction parameters between CO2 and salt are determined and listed in Table 3.
5.3. Phase-Partitioning Behaviors in CO2-Brine System
Figure 8 shows the CO2 solubility in a mixed brine of NaCl, KCl, and CaCl2. The comparison indicates that the proposed model can accurately reproduce the CO2 solubility in a complicated brine system. Similar to that in a single salt solution, the CO2 solubility in a mixed brine decreases with ion concentration.
In traditional models, the salting-out effect on water content in the CO2-rich phase is commonly neglected. However, it has been widely indicated that the binary interaction between electrolytes and water can decrease the water activity in the aqueous phase. This suggested that the composition of the gas phase could be altered by the dissolved solutes in water.
A comparison of water content in the CO2-rich phase considering the salting-out effect or not is shown in Figure 9. As seen, the salting-out effect can decrease the water content in the gas phase which has been preliminary verified by the experimental data of Mousavi . The inset in Figure 9 shows that the water fraction in the gas phase with salt is 21% lower than that without salt. Furthermore, it has been found that the increase of temperature can enhance the effect of salting out on water content.
To the best of our knowledge, the experimental studies on water content in the CO2-rich phase for the CO2-brine system are still very few. However, the solubility of water in CO2 is crucial to the capacity of injected CO2 to dry the formations [14, 32, 33] and the potential fluid-rock reactivity [14, 34, 35]. More experimental data on this topic is still necessary to help obtain a comprehensive understanding of the phase-partitioning behaviors in the CO2-brine system.
In this study, a unified --type thermodynamic model is developed to estimate the phase-partitioning behaviors of the CO2-brine system containing Na+, K+, Ca2+, Mg2+, Cl-, or SO42- at temperatures up to 623.15 K and pressures up to 350 MPa. In the model, the fugacity coefficient in the CO2-rich phase is treated using a modified Peng-Robinson EOS which incorporates a new alpha equation and binary interaction parameter correlation. The activity coefficient in the aqueous phase relies on a unified model of a gas equilibrium constant, the Margules expression, and a Pizter interaction model.
An extensive experimental databank is developed to calibrate the proposed model. The simulation errors for CO2 solubility and water content in the CO2+H2O system are 4.765% and 8.182%, respectively. Regarding the brine containing multiple ions, the simulation deviation of CO2 solubility is less than 5.1%. A detailed comparison indicates that the proposed model has a better accuracy and a wider valid temperature-pressure range compared to the traditional models. Furthermore, the effect of salting out on the composition of the CO2-rich phase can be accurately evaluated.
Using the proposed model, the phase diagrams of mutual solubility in the CO2+H2O system are generated. More sensitivity of phase compositions to temperature is revealed above 100°C. There exist the abrupt changes in CO2 solubility and water content as the CO2-rich phase transfers from vapor to liquid or supercritical.
A. Fugacity Model of the Aqueous Phase
B. Experimental Database
All data, models, and code generated or used during the study appear in the submitted article.
Conflicts of Interest
The authors declare that they have no conflicts of interest. Parts of this work were carried out in 2018-2019 by the senior author (Xiaohui Sun) while he was a visiting scholar at Lawrence Berkeley National Laboratory.
This work was supported by the Postdoctoral Innovative Talents Support Program in Shandong Province (SDBX2020005) and the Postdoctoral Applied Research Project of Qingdao (qdyy20200086). We thank Curtis M. Oldenburg and Lehua Pan (LBNL) for their hospitality, assistance, and weekly group meetings where some of these methods were first presented.