Abstract

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.

1. Introduction

CO2-water/brine is one of the most important and commonly encountered systems [13] in CO2 sequestration [411], CO2-enhanced oil recovery (EOR) [2, 12, 13], CO2-enhanced geothermal systems (EGS) [14, 15], global CO2tracing [1619], 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 (P-T-x) 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 [2224]. 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 [25]. Borgia et al. [26] indicated that the geothermal reservoir could even be dead in 1 year. Secondly, Enick and Klara [27] and Chang et al. [28] 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, 3642], which can facilitate the development of an accurate phase equilibrium model. The modelling approaches can be generally divided into two categories: φ-φ models and φ-γ models. The φ-φ 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 [44]. 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 φ-φ 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 φ-γ models [4]. Furthermore, the microscopic knowledge of molecular structures is necessary but not applicable in some industry applications [2]. The φ-γ 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 [43], they could be much more amenable to integration with chemical equilibrium simulation [4, 45]. Considering its good extensibility and computational efficiency, the φ-γ 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 φ-γ type model.

Based on Peng-Robinson’s EOS and Henry’s law, Li and Ngheim [47] 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 [17]. Hu et al. [48] 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 [42]. They proposed an accurate φ-γ 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. [49] 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. [50] 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. [51] proposed an unsymmetric φ-γ 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 [52] 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 φ-γ approach, Duan and Sun [17] and Spycher and Pruess [14] 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 [17] 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. [45] claimed that the model calculations of Duan and Sun [17] disagreed significantly with their experimental measurements at 448.15 K. Similarly, Guo et al. [9] 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 [45]. Furthermore, this model cannot be used for accurately estimating the water content in the CO2-rich phase. The models of Spycher et al. [8] and Spycher and Pruess [14] 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 [17], it includes two more experimental studies for parameter determination [30] 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

According to the thermodynamic principle, the necessary criterion for multiphase equilibrium is equality of the fugacity of each fluid component in the aqueous phase and the CO2-rich phase.
(1)fiG=fiAq,
where f is the fugacity (in Pa), i is the component, G and Aq represent the CO2-rich phase and the aqueous phase, respectively.
The fugacity of component i in the CO2-rich phase is as follows:
(2)fiG=pφiyi,
where p is the pressure (in Pa); yi is the mole fraction of component i in the CO2-rich phase, yH2O+yCO2=1; φi is the fugacity coefficient of component i, dimensionless. Regarding the strongly nonideal system composed of CO2 and H2O, an accurate binary interaction correlation is necessary for estimation of the effect of asymmetric interaction between molecules.
In equation (1), the fugacity of component i in the aqueous phase can be estimated by Jager et al. [54]:
(3)fiAq=fioexpμiAgioRT,
where fio is the fugacity of component i in the ideal gas state (in Pa); gio is the Gibbs free energy of component i in the ideal gas state (in Pa); μiA is the chemical potential of component i in the aqueous phase (in Pa); R is the gas constant, R=8.314472Pam3mol1K1; and T is temperature (in K).
Substituting the expressions of Gibbs free energy and chemical potential (as shown in Appendix A) into equation (3), the following equation can be obtained:
(4)fiAq=KiaiexpP0PviARTdp,
where Ki is the equilibrium constant of component i, which is a function of temperature and pressure; ai is the activity of component i in the aqueous phase; viA is the partial molar volume of component i(in m3/mol); P0 is the reference pressure (in Pa), which is set as the saturated vapor pressure of water.

2.1.1. Water

In equation (4), the activity of component i in the aqueous phase is as follows:
(5)aH2O=xH2OγH2O,(6)γH2O=γH2ORefxH2O=1γH2O,γH2O=γH2O0γH2Osolute,
where xi is the mole fraction of component i in the aqueous phase (in mol/mol); γH2ORefxH2O=1 is the activity coefficient of water at the reference state, dimensionless. Following the symmetric convention, it is equal to one for pure water. γH2O is the relative activity coefficient of water that includes two parts: (i) γH2O0 represents the relative activity coefficient of water in the CO2+H2O system. It approaches one when the CO2 solubility is negligible at low pressure and temperature. (ii) γH2Osolute represents the effect of electrolytes on the activity of water, which is generated by the long-range and short-range interactions between the cations, anions, and water molecules. It is equal to one for pure water and generally decreases with increasing salt concentration.

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 (aH2OxH2O).

2.1.2. CO2

In equation (4), the activity coefficient of CO2 can be estimated by
(7)aCO2=mCO2γCO2,γCO2=γCO2RefmCO2=0γCO2,
where mCO2 is the molality of CO2 in the aqueous phase (in mol/kg), and γCO2RefmCO2=1 is the activity coefficient of CO2 at the reference state:
(8)γCO2RefmCO2=0=11+mCO2MHO2=1xCO2,
where MH2O is the molality of water, MH2O=0.01802kg/mol. The calculation of CO2 activity in the aqueous phase corresponds to the unsymmetric convention. Namely, the CO2 activity coefficient is equal to unity as the concentration of dissolved CO2 is 0 mol/kg. Equation (8) is used to generate a molality to mole fraction correction [55].
With the substitution of equation (8) into equation (7), the following expression can be obtained:
(9)aCO2=xCO2γCO2MHO2,γCO2=γCO20γCO2solute,
where γCO2 is the relative activity coefficient of CO2 in the aqueous phase that contains two parts: (i) γCO20 represents the relative activity coefficient in the CO2+H2O system. It approaches one if the solubility of CO2 is negligible. It was revealed that the simulation error of CO2 solubility approaches 7% if the activity coefficient γg0 is simplified as unity [6]. (ii) γCO2solute represents the effect of electrolytes on the relative activity coefficient of CO2, which is equal to unity for pure water and increases with salt content increasing in brine.

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 [56].

It has been indicated that the Peng-Robinson EOS is suitable for predicting thermodynamic properties and phase equilibrium of the fluid system containing sour gas and water [57]. In this study, we present a modified Peng-Robinson EOS, which incorporates a new alpha model and binary interaction correlation. The expression of the Peng-Robinson EOS is as follows [58]:
(10)p=RTvbaTvv+b+bvb,
where v is specific volume (in m3/mol), a is intermolecular attraction parameter, and b is the intermolecular repulsion parameter.
Using the Peng-Robinson EOS, the fugacity coefficient of component i can be written as follows:
(11)Lnϕi=1RTVpniT,V,njiRTVdVlnZ=bibZ1lnZBAδ1δ2B2j=1NxjaijabiblnZ+δ1BZ+δ2B,
where ni is the mole of component i(in mol); V is volume (in m3); Z is the gas compression factor; δ1 and δ2 are 1+2 and 12, respectively; A=ap/RT2; and B=bp/RT.

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.

In this study, a modified alpha equation is proposed to increase the accuracy of simulated saturation pressure of CO2 and water:
(12)αi=1+κi1Tr1/22expai1Trbi,
where α is the cohesion factor; Tr is the reduced temperature; κi, ai, and bi are the model parameters, which can be estimated by the experimental data of saturation pressure of CO2 and H2O. If b1 and b2 are set as zero, equation (12) can be simplified as the original expression in the Peng-Robinson EOS.

2.2.2. Mixing Rule

The mixing rule is primarily intended for characterizing the interaction force between different molecules. Using the Van der Waals mixing rule, the expression for a and b in the Peng-Robinson EOS can be written as follows [58]:
(13)a=ijxixjaij,(14)b=ixibi,
where
(15)aij=1kijaiaj,
where kij is the parameter of binary interaction between the components i and j, which has a significant influence on the accuracy of phase equilibrium simulation.
The previous analysis indicated that the traditional Van der Waals mixing rule has a good applicability for hydrocarbon mixtures. However, it is not accurate enough for the strongly nonideal system containing CO2 and water, owing to the existence of asymmetric intermolecular forces. Here, a correlation of temperature is proposed to describe the binary interaction parameter between CO2 and water.
(16)k=k0+k1TTC,CO2+k2TTC,CO22,
where k0-k2 are constants, TC,CO2 is the critical temperature of CO2(in K).

2.3. Fugacity Modelling of the Aqueous Phase

Substituting equation (5), equation (6), and equation (9) into equation (2) and equation (4), the fugacity model of CO2 and water in the aqueous phase can be obtained:
(17)fH2OAq=KH2OxH2OγH2O0γH2OsoluteexpVH2ORT,(18)fCO2Aq=1MHO2KCO2xCO2γCO20γCO2soluteexpVCO2RT.

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 (KCO2), 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 [46]. Therefore, a unified and continuous model of the gas equilibrium constant is derived in this study.

Based on the principle of thermodynamics, the expression of a gas equilibrium constant is as follows:
(19)LnKi=ΔμAG0RT=lnfio+giA0gio0RT0T0ThiAhioRT2dT,
where gio0 is the molar Gibbs energy of component i in the ideal gas state (in J/mol), giA0 is the molar Gibbs energy of component i in a hypothetical 1 molal solution at the reference temperature and pressure (in J/mol), hiA is the molar enthalpy of component iin the aqueous phase (in J/mol), fioandhiorepresent the fugacity and the gas specific enthalpy at the reference pressure (in J/mol), and T0 is the reference temperature (in K).
In this study, a unified model of the gas equilibrium constant that varies with temperature is developed:
(20)LnKi=f0+f2Tf12+f3T2+f4T+f5T+f6T2+f7T3+f8lnT+f9expd1TT,
where d1 is a constant, which is equal to 4.60128×103. f0-f9 are the fitted model parameters. It should be noted that in theory equation (20) can reproduce the effect of temperature over a wide range and is not limited by the gas types.

2.3.2. Specific Volume and Relative Activity Coefficient

In equation (17) and equation (18), VH2O and VCO2 represent the effect of specific volume on gas solubility. Spycher and Pruess [14] indicated that the CO2 specific volume varied with temperature in a linear manner while the effect of pressure was neglected. However, a significant simulation error can be generated in the traditional models, especially at a high pressure, since a constant value of CO2-specific volume is employed. This is because pressure may have a considerable influence on the CO2-specific volume [1]. In this study, a correlation of the specific volume item as a function of temperature and pressure is proposed:
(21)VCO2=P0PvCO2dp=c1+c2T+c3pT+c4plnTpp0,
where c1-c4 are fitted parameters and p_0 is the reference pressure (in Pa).
The relative coefficients γCO20 and γH2O0 at different temperature and pressure conditions can be estimated using the modified Margules model developed by Carlson and Colburn [62]:
(22)LnγCO20=2AMxCO2xH2O2,(23)LnγH2O0=AM2AMxH2OxCO22.
Incorporating equation (22) and equation (23) into the developed mutual solubility model will not influence the symmetric and unsymmetrical conventions for the fugacity models of CO2 and water in the aqueous phase. Namely, the activity coefficients of CO2 and water are equal to one in infinite dilution solution.
(24)AM=AMaT2+AMbT+AMc,
where AMa, AMb, and AMc are fitted parameters.

2.4. Simulation Method

Substituting equation (17) and equation (18) into equation (2), the equations for water content in the CO2-rich phase and CO2 solubility in the aqueous phase can be obtained:
(25)yH2O=KH2OγH2O0γH2OsoluteexpVH2O/RTpφH2OxH2O,(26)yCO2=KCO2γCO20γCO2soluteMHO2pφCO2expVCO2RTxCO2.
The sum of mole fractions of CO2 and water in the CO2-rich phase is equal to one. Therefore, the following expression can be obtained combining equation (25) and equation (26):
(27)KH2OγH2O0γH2OsolutepφH2OexpVH2ORTxH2O+KCO2γCO20γCO2soluteMHO2pφCO2expVCO2RTxCO21=0.

The flow chart for model simulation includes the following steps:

  • (1)

    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, yH2O,0=PH2O/P. 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 (yH2O,1) can be corrected by equation (25) assuming xH2O=1

  • (2)

    Using a Newton-Raphson algorithm, xCO2 can be estimated by equation (27). Then, the mole fraction of water xH2O in aqueous phase can be calculated based on mass conservation

  • (3)

    According to the estimated composition of the aqueous phase, the mole fractions of CO2 and water in the CO2-rich phase can be recalculated using equation (25) and equation (26).

  • (4)

    Repeat steps (2) and (3) until the convergence criterion of maximum permissible error is satisfied

3. Parameterization

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 13 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 [1], 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 [14]:

  • (1)

    The parameters in alpha model are determined by the experimental data of the saturation pressure of CO2 and water

  • (2)

    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

  • (3)

    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

  • (4)

    The parameters in the Pizter interaction models are determined by the measurements of CO2 solubility in brines

  • (5)

    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 [14], and Duan and Sun [17]. By comparison, a significant deviation is found using the model of Li and Yang [2].

However, there is considerable consistency between the experimental datasets of different literature studies. At T=25°C, the experimental data of Hou et al. [45] and Nakayama et al. [63] deviates obviously from those of other studies. Its deviation approaches to 15% at P=18MPa. When the pressure is larger than 20 MPa, the experimental data of Greenwood and Barnes [64] agrees well with those of Wiebe [65], but it has a significant deviation from those of Teng et al. [66] and Gillepsie and Wilson [67]. At T=50°C, 75°C, or 100°C, the experimental data adopted from different literature studies are generally consistent, except for the data of Qin et al. [68] at 50°C, and the data of Sako et al. [69] and Kiepe et al. [38] at 100°C.

It can be seen that the simulated results of Spycher and Pruess [14] deviate significantly from the experimental data at T=100°C and P>60MPa. Similarly, the simulation error of Duan and Sun [17] 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 [70] 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 [2] is accurate below 200°C; however, it significantly deviates from the experimental measurements above 200°C. (ii) The model of Spycher and Pruess [14] 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 [17] is more accurate than those of Spycher and Pruess [14] and Li and Yang [2]. (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 T=300°C and P<50MPa. At T=300°C, the data of Blencoe [71] is consistent with that of Takenouchi and Kennedy [72], but shows a significant deviation with that of Todheide and Franck [73]. As already indicated by Spycher and Pruess [14], a complete mixing is necessary to reach a fully miscible state. However, this appears to be not sufficient in experiments of Takenouchi and Kennedy [72]. By comparison, the model appears in much better agreement with the data determined by Todheide and Franck [73].

Above 300°C, the proposed model and Spycher and Pruess [14] have a better agreement with the scattered literature data, but the model of Duan and Sun [17] tends to be invalid.

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 [73] differs slightly from those of other studies. For example, the simulation error between Todheide and Franck [73] and Wiebe and Gaddy [74] approaches 28% at 20 MPa. At 200°C, the measured data of Takenouchi and Kennedy [72] and Malinin [75] obviously deviate from those of Todheide and Franck [73]. 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 [73].

There are obvious differences in the calculation results of water content by different models: (i) In the model of Duan and Sun [17], 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 [2] 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 φ-φ type approach. (iii) Although both the proposed model and Spycher and Pruess [14] 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 [17] and Spycher and Pruess [14] 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 [13]. By comparison, the overall simulation error of Spycher and Pruess [14] is slightly larger than that of Duan and Sun [17], owing to a valid pressure range of 0-60 MPa. However, the model of Duan and Sun [17] 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 [17] and Spycher and Pruess [14], the simulation error of Li and Yang [2] 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 [2].

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 [14].

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 [20]. A more rapid change in CO2 solubility can be found above 150°C compared to that at a low temperature region [53].

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

5.1. Modelling

One advantage of the φ-γ 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 [76]. 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.

Considering the interaction between different components in the aqueous phase, the excess Gibbs free energy of an aqueous phase can be estimated as follows [76]:
(28)GERTnH2OMH2O=f1I+ijmimjλijI+ijkmimjmkμijk,
where GE is the excess Gibbs free energy of an aqueous phase (in J/mol); mi is the molality in component iin the aqueous phase (in mol/kg); λij is the binary interaction parameter; μijk is the ternary interaction parameter; I is the ion strength, where I=0.5mizi2; zi is the charge number of ion i; and f1I is the Debye-Huckel item.
Using the expansion form of equation (28), the activity coefficient of CO2 and water can be obtained:
(29)Lnγisolute=2smsλGs+6nmnsmsμGns+3camcmaχGca,(30)LnaH2Osolute=MH2O2AΦI1.51+bI0.52camcmaλca+i=a,cmiziCcaϕ2zcza2nmnsmsλns6nmncamcmaχncac,ami,
where λGs is the parameter of binary interaction between gas and salt; ms is the molality of salt (in mol/kg); χGca and μGns are the parameters of ternary interaction between ions and CO2. μGns is set to zero in this study. The subscript a represents anion, while c represents cation; Aϕ is Debye-Huckel parameter for the osmotic coefficient; and Ccaϕ is the third osmotic virial coefficient in Pizter’s equation.
Considering the effect of temperature variation, the parameter of binary interaction between CO2 and salt is estimated using the following correlation:
(31)λGs=B0+B1T+B2T2,
where B0, B1, and B2 are the fitted parameters; and S represents salt. Using the literature experimental data of phase-partitioning properties in the CO2-brine system, the ternary and binary interaction parameters in the developed model are determined, as shown in Table 4.

Substituting equation (29) and equation (30) into equation (25), equation (26), and equation (27), the mutual solubility in the CO2-brine system can be estimated.

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.

Validation of the model simulation against the literature experimental data and other models is shown in Table 5. The average simulation error of CO2 solubility in different brines is 5.15%, which has a better accuracy over the models of Spycher and Pruess [14] and Duan and Sun [17].

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 [77]. 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.

6. Conclusions

In this study, a unified φ-γ-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.

Appendix

A. Fugacity Model of the Aqueous Phase

The fugacity equation of a fluid component in the aqueous phase is as follows [54]:
(A.1)fiAq=fioexpμiAgioRT,
where gio is the Gibbs free energy at the ideal gas state, which can be estimated as follows:
(A.2)gio=μik0+RTlnfio=Tgio0T0TT0ThioT2dT,
where μik0 is the chemical potential of component iat the ideal gas state (in J/mol).
The chemical potential of component i in the aqueous phase is as follows:
(A.3)μiA=μiA0+RTlnai,(A.4)μiA0=TgiA0T0TT0ThiAT2dT+P0PviAdp,
where μiA0 is the chemical potential of component i in the aqueous phase (in J/mol).
Substituting equation (A.2) and equation (A.3) into equation (A.1), we obtain
(A.5)fiAq=expμiA0μik0RT+lnai.
The difference between chemical potentials of component i in the gas and aqueous phases at the reference state is as follows:
(A.6)μiA0μik0=ΔμAG0+P0PviAdp,(A.7)ΔμAG0=TgiA0T0TT0ThiAT2dT+RTlnfioTgio0T0+TT0ThioT2dT.
Substituting equation (A.6) into equation (A.5), the expression for the fugacity of component i in the aqueous phase is as follows:
(A.8)fiAq=expΔμAG0RT+lnaiexpP0PviARTdp.
Equation (A.8) can be rewritten as a function of the equilibrium constant:
(A.9)fiAq=KiaiexpViRT,
where
(A.10)Ki=expΔμAG0RT,Vi=P0PviAdp,

B. Experimental Database

The experimental databases for CO2 solubility in the aqueous phase and brines are presented in Table 1 and Table 3, respectively, while those for the H2O content in the CO2-rich phase are listed in Table 2.

Data Availability

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.

Acknowledgments

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.

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