Modelling of the Phase-Partitioning Behaviors for CO2-Brine System at Geological Conditions

Xiaohui Sun; Zhiyuan Wang; Yingjie Li; Hangyu Li; Haikang He; Baojiang Sun; Xixin Wang

doi:10.2113/2021/3474828

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 SO₄^2-. 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 [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 (⁠ $P$ - $T$ - $x$ ⁠) 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 [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 [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, 36–42], 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 SO₄^2- 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.

\begin{matrix} (1) & f_{i}^{G} = f_{i}^{Aq}, \end{matrix}

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:

\begin{matrix} (2) & f_{i}^{G} = p φ_{i} y_{i}, \end{matrix}

where

p

is the pressure (in Pa);

y_{i}

is the mole fraction of component

i

in the CO₂-rich phase,

y_{H 2 O} + y_{CO 2} = 1

⁠;

φ_{i}

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 (), the fugacity of component

i

in the aqueous phase can be estimated by Jager et al. [54]:

\begin{matrix} (3) & f_{i}^{Aq} = f_{i o} \exp [\frac{μ_{i A} - g_{i o}}{R T}], \end{matrix}

where

f_{i o}

is the fugacity of component

i

in the ideal gas state (in Pa);

g_{i o}

is the Gibbs free energy of component

i

in the ideal gas state (in Pa);

μ_{i A}

is the chemical potential of component

i

in the aqueous phase (in Pa);

R

is the gas constant,

R = 8.314472 Pa ∙ m^{3} ∙ mo l^{- 1} ∙ K^{- 1}

⁠; and

T

is temperature (in K).

Substituting the expressions of Gibbs free energy and chemical potential (as shown in Appendix A) into equation (), the following equation can be obtained:

\begin{matrix} (4) & f_{i}^{Aq} = K_{i} a_{i} \exp [\int_{P_{0}}^{P} \frac{v_{i A}}{R T} d p], \end{matrix}

where

K_{i}

is the equilibrium constant of component

i

⁠, which is a function of temperature and pressure;

a_{i}

is the activity of component

i

in the aqueous phase;

v_{i A}

is the partial molar volume of component

i

(in m³/mol);

P_{0}

is the reference pressure (in Pa), which is set as the saturated vapor pressure of water.

2.1.1. Water

In equation (), the activity of component

i

in the aqueous phase is as follows:

\begin{matrix} (5) & a_{H_{2} O} = x_{H_{2} O} γ_{H 2 O}, \\ (6) & γ_{H 2 O} = {γ_{H_{2} O}^{Re f}|}_{x_{H_{2} O} = 1} γ_{H_{2} O}^{*}, \\ γ_{H_{2} O}^{*} = γ_{H_{2} O}^{0} γ_{H_{2} O}^{solute}, \end{matrix}

where

x_{i}

is the mole fraction of component

i

in the aqueous phase (in mol/mol);

{γ_{H 2 O}^{Ref}|}_{x_{H 2 O} = 1}

is the activity coefficient of water at the reference state, dimensionless. Following the symmetric convention, it is equal to one for pure water.

γ_{H 2 O}^{*}

is the relative activity coefficient of water that includes two parts: (i)

γ_{H 2 O}^{0}

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)

γ_{H 2 O}^{solute}

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 (⁠ $a_{H 2 O} \approx x_{H 2 O}$ ⁠).

2.1.2. CO₂

In equation (), the activity coefficient of CO₂ can be estimated by

\begin{matrix} (7) & a_{{CO}_{2}} = m_{{CO}_{2}} γ_{{CO}_{2}}, \\ γ_{{CO}_{2}} = {γ_{{CO}_{2}}^{Re f}|}_{m_{{CO}_{2}} = 0} γ_{{CO}_{2}}^{*}, \end{matrix}

where

m_{CO 2}

is the molality of CO₂ in the aqueous phase (in mol/kg), and

{γ_{CO 2}^{Ref}|}_{m_{CO 2} = 1}

is the activity coefficient of CO₂ at the reference state:

\begin{matrix} (8) & {γ_{{CO}_{2}}^{Re f}|}_{m_{{CO}_{2}} = 0} = \frac{1}{1 + m_{{CO}_{2}} M_{H {}_{2}O}} = 1 - x_{{CO}_{2}}, \end{matrix}

where

M_{H 2 O}

is the molality of water,

M_{H 2 O} = 0.01802 kg / 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 () is used to generate a molality to mole fraction correction [55].

With the substitution of equation () into equation (), the following expression can be obtained:

\begin{matrix} (9) & a_{{CO}_{2}} = \frac{x_{{CO}_{2}} γ_{{CO}_{2}}^{*}}{M_{H {}_{2}O}}, γ_{{CO}_{2}}^{*} = γ_{{CO}_{2}}^{0} γ_{{CO}_{2}}^{solute}, \end{matrix}

where

γ_{CO 2}^{*}

is the relative activity coefficient of CO₂ in the aqueous phase that contains two parts: (i)

γ_{CO 2}^{0}

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

γ_{g 0}

is simplified as unity [6]. (ii)

γ_{CO 2}^{solute}

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]:

\begin{matrix} (10) & p = \frac{R T}{v - b} - \frac{a (T)}{v (v + b) + b (v - b)}, \end{matrix}

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:

\begin{matrix} (11) & Ln ϕ_{i} = \frac{1}{R T} \int_{V}^{\infty} [{(\frac{\partial p}{\partial n_{i}})}_{T, V, n_{j \neq i}} - \frac{R T}{V}] d V - \ln Z = \frac{b_{i}}{b} (Z - 1) - \ln [Z - B] - \frac{A}{(δ_{1} - δ_{2}) B} [\frac{2 \sum_{j = 1}^{N} x_{j} a_{i j}}{a} - \frac{b_{i}}{b}] \ln [\frac{Z + δ_{1} B}{Z + δ_{2} B}], \end{matrix}

where

n_{i}

is the mole of component

i

(in mol);

V

is volume (in m³);

Z

is the gas compression factor;

δ_{1}

and

δ_{2}

are

1 + \sqrt{2}

and

1 - \sqrt{2}

⁠, respectively;

A = a p / (R T) 2

⁠; and

B = b p / (R T)

⁠.

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:

\begin{matrix} (12) & α_{i} = {[1 + κ_{i} (1 - T_{r}^{1 / 2})]}^{2} \exp [a_{i} (1 - T_{r}^{b_{i}})], \end{matrix}

where

α

is the cohesion factor;

T_{r}

is the reduced temperature;

κ_{i}

⁠,

a_{i}

⁠, and

b_{i}

are the model parameters, which can be estimated by the experimental data of saturation pressure of CO₂ and H₂O. If

b_{1}

and

b_{2}

are set as zero, equation () 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]:

\begin{matrix} (13) & a = \sum_{i} \sum_{j} x_{i} x_{j} a_{i j}, \\ (14) & b = \sum_{i} x_{i} b_{i}, \end{matrix}

where

\begin{matrix} (15) & a_{i j} = (1 - k_{i j}) \sqrt{a_{i} a_{j}}, \end{matrix}

where

k_{i j}

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.

\begin{matrix} (16) & k = k_{0} + k_{1} (\frac{T}{T_{C, CO 2}}) + k_{2} {(\frac{T}{T_{C, CO 2}})}^{2}, \end{matrix}

where

k_{0}

-

k_{2}

are constants,

T_{C, CO 2}

is the critical temperature of CO₂(in K).

2.3. Fugacity Modelling of the Aqueous Phase

Substituting equation (), equation (), and equation () into equation () and equation (), the fugacity model of CO₂ and water in the aqueous phase can be obtained:

\begin{matrix} (17) & f_{H_{2} O}^{Aq} = K_{H_{2} O} x_{H_{2} O} γ_{H_{2} O}^{0} γ_{H_{2} O}^{solute} \exp (\frac{V_{H_{2} O}}{R T}), \\ (18) & f_{{CO}_{2}}^{Aq} = \frac{1}{M_{H {}_{2}O}} K_{{CO}_{2}} x_{{CO}_{2}} γ_{{CO}_{2}}^{0} γ_{{CO}_{2}}^{solute} \exp (\frac{V_{{CO}_{2}}}{R T}) . \end{matrix}

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 (⁠ $K_{CO 2}$ ⁠), 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:

\begin{matrix} (19) & Ln (K_{i}) = \frac{Δ μ_{A G}^{(0)}}{R T} = \ln (f_{i o}) + \frac{g_{i A 0} - g_{i o 0}}{{R T}_{0}} - \int_{T_{0}}^{T} \frac{h_{i A} - h_{i o}}{{R T}^{2}} d T, \end{matrix}

where

g_{i o 0}

is the molar Gibbs energy of component

i

in the ideal gas state (in J/mol),

g_{i A 0}

is the molar Gibbs energy of component

i

in a hypothetical 1 molal solution at the reference temperature and pressure (in J/mol),

h_{i A}

is the molar enthalpy of component

i

in the aqueous phase (in J/mol),

f_{i o}

and

h_{i o}

represent the fugacity and the gas specific enthalpy at the reference pressure (in J/mol), and

T_{0}

is the reference temperature (in K).

In this study, a unified model of the gas equilibrium constant that varies with temperature is developed:

\begin{matrix} (20) & Ln (K_{i}) = f_{0} + \frac{f_{2}}{{(T - f_{1})}^{2}} + \frac{f_{3}}{T^{2}} + \frac{f_{4}}{T} + f_{5} T + f_{6} T^{2} + f_{7} T^{3} + f_{8} \ln (T) + f_{9} \frac{\exp (d_{1} T)}{T}, \end{matrix}

where

d_{1}

is a constant, which is equal to

- 4.60128 \times 10^{- 3}

⁠.

f_{0}

-

f_{9}

are the fitted model parameters. It should be noted that in theory equation () 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 () and equation (),

V_{H 2 O}

and

V_{CO 2}

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:

\begin{matrix} (21) & V_{{CO}_{2}} = \int_{P_{0}}^{P} v_{{CO}_{2}} d p = \frac{c_{1} + c_{2} T + c_{3} p}{T + c_{4} p \ln T} (p - p_{0}), \end{matrix}

where

c_{1}

-

c_{4}

are fitted parameters and

p_0

is the reference pressure (in Pa).

The relative coefficients

γ_{CO 2}^{0}

and

γ_{H 2 O}^{0}

at different temperature and pressure conditions can be estimated using the modified Margules model developed by Carlson and Colburn [62]:

\begin{matrix} (22) & Ln (γ_{{CO}_{2}}^{0}) = 2 A_{M} x_{CO 2} x_{H 2 O}^{2}, \\ (23) & Ln (γ_{H_{2} O}^{0}) = (A_{M} - 2 A_{M} x_{H_{2} O}) x_{{CO}_{2}}^{2} . \end{matrix}

Incorporating equation () and equation () 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.

\begin{matrix} (24) & A_{M} = A_{Ma} T^{2} + A_{Mb} T + A_{Mc}, \end{matrix}

where

A_{M a}

⁠,

A_{M b}

⁠, and

A_{M c}

are fitted parameters.

2.4. Simulation Method

Substituting equation () and equation () into equation (), the equations for water content in the CO₂-rich phase and CO₂ solubility in the aqueous phase can be obtained:

\begin{matrix} (25) & y_{H_{2} O} = \frac{K_{H_{2} O} γ_{H_{2} O}^{0} γ_{H_{2} O}^{solute} \exp (V_{H_{2} O} / R T)}{p φ_{H_{2} O}} x_{H_{2} O}, \\ (26) & y_{{CO}_{2}} = \frac{K_{{CO}_{2}} γ_{{CO}_{2}}^{0} γ_{{CO}_{2}}^{solute}}{M_{H {}_{2}O} p φ_{{CO}_{2}}} \exp (\frac{V_{{CO}_{2}}}{R T}) x_{{CO}_{2}} . \end{matrix}

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 () and equation ():

\begin{matrix} (27) & \frac{K_{H_{2} O} γ_{H_{2} O}^{0} γ_{H_{2} O}^{solute}}{p φ_{H_{2} O}} \exp (\frac{V_{H_{2} O}}{R T}) x_{H_{2} O} + \frac{K_{{CO}_{2}} γ_{{CO}_{2}}^{0} γ_{{CO}_{2}}^{solute}}{M_{H {}_{2}O} p φ_{{CO}_{2}}} \exp (\frac{V_{{CO}_{2}}}{R T}) x_{{CO}_{2}} - 1 = 0 . \end{matrix}

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, $y_{H 2 O, 0} = P_{H 2 O / P}$ ⁠. 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 (⁠ $y_{H 2 O, 1}$ ⁠) can be corrected by equation (25) assuming $x_{H 2 O = 1}$
(2)
Using a Newton-Raphson algorithm, $x_{CO 2}$ can be estimated by equation (27). Then, the mole fraction of water $x_{H 2 O}$ 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 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 [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 $P = 18 MPa$ ⁠. 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 $T = 10 0^{°} C$ and $P > 60 MPa$ ⁠. 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 $T = 30 0^{°} C$ and $P < 50 MPa$ ⁠. At $T = 30 0^{°} C$ ⁠, 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 [1–3]. 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]:

\begin{matrix} (28) & \frac{G^{E}}{{R T n}_{H_{2} O} M_{H_{2} O}} = f_{1} (I) + \sum_{i} \sum_{j} m_{i} m_{j} λ_{i j} (I) + \sum_{i} \sum_{j} \sum_{k} m_{i} m_{j} m_{k} μ_{i j k}, \end{matrix}

where

G^{E}

is the excess Gibbs free energy of an aqueous phase (in J/mol);

m_{i}

is the molality in component

i

in the aqueous phase (in mol/kg);

λ_{i j}

is the binary interaction parameter;

μ_{i j k}

is the ternary interaction parameter;

I

is the ion strength, where

I = 0.5 \sum m_{i} z_{i}^{2}

⁠;

z_{i}

is the charge number of ion

i

⁠; and

f_{1} (I)

is the Debye-Huckel item.

Using the expansion form of equation (), the activity coefficient of CO₂ and water can be obtained:

\begin{matrix} (29) & Ln γ_{i}^{solute} = 2 \sum_{s} m_{s} λ_{G s} + 6 \sum_{n} m_{n} (\sum_{s} m_{s} μ_{G n s}) + 3 \sum_{c} \sum_{a} m_{c} m_{a} χ_{G c a}, \\ (30) & Ln a_{H_{2} O}^{solute} = M_{H_{2} O} [2 A^{Φ} \frac{I^{1.5}}{1 + {b I}^{0.5}} - 2 \sum_{c} \sum_{a} m_{c} m_{a} (λ_{c a} + \sum_{i = a, c} m_{i} |z_{i}| \frac{C_{c a}^{ϕ}}{2 |z_{c} z_{a}|}) - 2 \sum_{n} m_{n} \sum_{s} m_{s} λ_{n s} - 6 \sum_{n} m_{n} \sum_{c} \sum_{a} m_{c} m_{a} χ_{n c a} - \sum_{c, a} m_{i}], \end{matrix}

where

λ_{G s}

is the parameter of binary interaction between gas and salt;

m_{s}

is the molality of salt (in mol/kg);

χ_{G c a}

and

μ_{G n s}

are the parameters of ternary interaction between ions and CO₂.

μ_{G n s}

is set to zero in this study. The subscript

a

represents anion, while

c

represents cation;

A_{ϕ}

is Debye-Huckel parameter for the osmotic coefficient; and

C_{c a}^{ϕ}

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:

\begin{matrix} (31) & λ_{G s} = B_{0} + \frac{B_{1}}{T} + \frac{B_{2}}{T^{2}}, \end{matrix}

where

B_{0}

⁠,

B_{1}

⁠, and

B_{2}

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 SO₄^2-. 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 SO₄^2- 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]:

\begin{matrix} (A.1) & f_{i}^{Aq} = f_{i o} \exp [\frac{μ_{i A} - g_{i o}}{R T}], \end{matrix}

where

g_{i o}

is the Gibbs free energy at the ideal gas state, which can be estimated as follows:

\begin{matrix} (A.2) & g_{i o} = μ_{i k}^{(0)} + R T \ln (f_{i o}) = T \frac{g_{i o 0}}{T_{0}} - T \int_{T_{0}}^{T} \frac{h_{i o}}{T^{2}} d T, \end{matrix}

where

μ_{i k}^{(0)}

is the chemical potential of component

i

at the ideal gas state (in J/mol).

The chemical potential of component

i

in the aqueous phase is as follows:

\begin{matrix} (A.3) & μ_{i A} = μ_{i A}^{(0)} + R T \ln a_{i}, \\ (A.4) & μ_{i A}^{(0)} = T \frac{g_{i A 0}}{T_{0}} - T \int_{T_{0}}^{T} \frac{h_{i A}}{T^{2}} d T + \int_{P_{0}}^{P} v_{i A} d p, \end{matrix}

where

μ_{i A}^{(0)}

is the chemical potential of component

i

in the aqueous phase (in J/mol).

Substituting equation () and equation () into equation (), we obtain

\begin{matrix} (A.5) & f_{i}^{Aq} = \exp [\frac{μ_{i A}^{(0)} - μ_{i k}^{(0)}}{R T} + \ln (a_{i})] . \end{matrix}

The difference between chemical potentials of component

i

in the gas and aqueous phases at the reference state is as follows:

\begin{matrix} (A.6) & μ_{i A}^{(0)} - μ_{i k}^{(0)} = Δ μ_{A G}^{(0)} + \int_{P_{0}}^{P} v_{i A} d p, \\ (A.7) & Δ μ_{A G}^{(0)} = T \frac{g_{i A 0}}{T_{0}} - T \int_{T_{0}}^{T} \frac{h_{i A}}{T^{2}} d T + R T \ln (f_{i o}) - T \frac{g_{i o 0}}{T_{0}} + T \int_{T_{0}}^{T} \frac{h_{i o}}{T^{2}} d T . \end{matrix}

Substituting equation () into equation (), the expression for the fugacity of component

i

in the aqueous phase is as follows:

\begin{matrix} (A.8) & f_{i}^{Aq} = \exp [\frac{Δ μ_{A G}^{(0)}}{R T} + \ln (a_{i})] \exp [\int_{P_{0}}^{P} \frac{v_{i A}}{R T} d p] . \end{matrix}

Equation () can be rewritten as a function of the equilibrium constant:

\begin{matrix} (A.9) & f_{i}^{Aq} = K_{i} a_{i} \exp (\frac{V_{i}}{R T}), \end{matrix}

where

\begin{matrix} (A.10) & K_{i} = \exp [\frac{Δ μ_{A G}^{(0)}}{R T}], \\ V_{i} = \int_{P_{0}}^{P} v_{i A} d p, \end{matrix}

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.

Exclusive Licensee GeoScienceWorld. Distributed under a Creative Commons Attribution License (CC BY 4.0).

No.	Acronym	Author	Time	Exp.	Temperature (°C)	Pressure (MPa)
1	W1	Wiebe and Gaddy [39]	1939	29	50-100	2.5-70.9
2	W2	Wiebe [65]	1941	73	0-100	2.5-70.9
3	P1	Prutton and Savage [78]	1945	26	101-120	2.3-70.3
4	H1	Houghton et al. [79]	1957	171	0-100	0.1-3.6
5	E1	Ellis [80]	1959	36	114-348	0.5-16.4
6	M1	Malinin [75]	1959	36	200-300	9.7-49.0
7	T1	Todheide and Franck [73]	1963	104	50-350	20.0-350.0
8	T2	Takenouchi and Kennedy [72]	1964	116	110-350	10.0-150.0
9	T3	Takenouchi and Kennedy [81]	1965	21	150-300	10.0-60.0
10	G1	Greenwood et al. [82]	1966	16	11.85-24.85	5.0-40.0
11	S1	Stewart and Munjal [83]	1970	12	0-25	1.0-4.6
12	M2	Malinin and Kurovskaya [84]	1975	2	100-150	4.8-4.9
13	D1	Drummond [85]	1981	2	300-300	19.1-19.6
14	S2	Shagiakhmetov and Tarzimanov [70]	1981	6	100-150	10.1-60.1
15	Z1	Zawisza and Malesinska [86]	1981	33	50-200	0.2-5.4
16	G2	Gillepsie and Wilson [67]	1982	33	15.6-260	0.7-20.3
17	B1	Briones et al. [87]	1987	8	50-50	6.8-17.7
18	N1	Nakayama et al. [63]	1987	6	25.05-25.05	3.6-11.0
19	D2	D’Souza et al. [56]	1988	4	50-75	10.1-15.2
20	M3	Müller et al. [88]	1988	20	100-200	0.3-8.0
21	N2	Nighswander et al. [40]	1989	33	79.7-198.1	2.0-10.2
22	S3	Sako et al. [69]	1991	7	75.15-148.25	10.2-19.7
23	K1	King et al. [89]	1992	27	15-25	6.1-24.3
24	D3	Dohrn et al. [90]	1993	5	50-50	10.1-30.4
25	R1	Rumpf et al. [36]	1994	7	50.01-50.03	1.1-5.8
26	T4	Teng et al. [66]	1997	40	4.85-24.85	5.0-40.0
27	D4	Dhima et al. [91]	1999	7	71-71	10.0-100.0
28	B2	Bamberger et al. [92]	2000	29	50.05-79.95	4.1-14.1
29	B3	Blencoe et al. [71]	2001	8	300-300	27.6-56.7
30	K2	Kiepe et al. [38]	2002	43	40.05-120.02	0.0-9.3
31	B4	Bando et al. [93]	2003	12	30-60	10.0-20.0
32	C1	Chapoy et al. [94]	2004	27	0.99-78.16	0.2-9.3
33	V1	Valtz et al. [95]	2004	47	5.07-45.08	0.5-8.0
34	B5	Bermejo et al. [37]	2005	26	23.58-96.5	1.6-8.3
35	K3	Koschel et al. [96]	2006	8	49.95-99.95	2.1-20.2
36	Q1	Qin et al. [68]	2008	7	50.45-102.65	10.6-49.9
37	H2	Han et al. [97]	2009	28	40.05-70.05	4.3-18.3
38	F1	Ferrentino et al. [98]	2010	18	35-50	7.6-13.1
39	L1	Liu et al. [41]	2011	31	35-55	2.1-16.0
40	Y1	Yan et al. [30]	2011	18	50.05-140.05	5.0-40.0
41	L2	Lucile et al. [99]	2012	30	25-120	0.5-5.1
42	S4	Savary et al. [100]	2012	3	150-150	11.3-33.7
43	H3	Hou et al. [45]	2013	42	25-175	1.1-17.6
44	T5	Tong et al. [20]	2013	18	101-101.84	7.2-27.3
45	A1	Al Ghafri et al. [101]	2014	8	50-50	2.1-18.7
46	B6	Bastami et al. [102]	2014	10	55-102	6.9-20.7
47	G3	Guo et al. [9]	2014	131	0-300	10.0-120.0
48	C2	Carvalho et al. [10]	2015	66	10.09-90.27	0.3-12.1
49	M4	Mohammadian et al. [29]	2015	20	60-80	0.1-21.3
50	C3	Caumon et al. [18]	2016	22	65-65	0.3-19.5
51	J1	Jacob and Saylor [11]	2016	50	23.85-23.85	1.4-12.4
52	T6	Truche et al. [19]	2016	7	200-280	3.1-13.2
53	M5	Mousavi [77]	2017	8	50-100	7.9-52.4
Sum				1597	0-350	0.1-350.0

No.	Acronym	Author	Time	Exp.	Temperature (°C)	Pressure (MPa)
1	W1	Wiebe and Gaddy [39]	1939	29	50-100	2.5-70.9
2	W2	Wiebe [65]	1941	73	0-100	2.5-70.9
3	P1	Prutton and Savage [78]	1945	26	101-120	2.3-70.3
4	H1	Houghton et al. [79]	1957	171	0-100	0.1-3.6
5	E1	Ellis [80]	1959	36	114-348	0.5-16.4
6	M1	Malinin [75]	1959	36	200-300	9.7-49.0
7	T1	Todheide and Franck [73]	1963	104	50-350	20.0-350.0
8	T2	Takenouchi and Kennedy [72]	1964	116	110-350	10.0-150.0
9	T3	Takenouchi and Kennedy [81]	1965	21	150-300	10.0-60.0
10	G1	Greenwood et al. [82]	1966	16	11.85-24.85	5.0-40.0
11	S1	Stewart and Munjal [83]	1970	12	0-25	1.0-4.6
12	M2	Malinin and Kurovskaya [84]	1975	2	100-150	4.8-4.9
13	D1	Drummond [85]	1981	2	300-300	19.1-19.6
14	S2	Shagiakhmetov and Tarzimanov [70]	1981	6	100-150	10.1-60.1
15	Z1	Zawisza and Malesinska [86]	1981	33	50-200	0.2-5.4
16	G2	Gillepsie and Wilson [67]	1982	33	15.6-260	0.7-20.3
17	B1	Briones et al. [87]	1987	8	50-50	6.8-17.7
18	N1	Nakayama et al. [63]	1987	6	25.05-25.05	3.6-11.0
19	D2	D’Souza et al. [56]	1988	4	50-75	10.1-15.2
20	M3	Müller et al. [88]	1988	20	100-200	0.3-8.0
21	N2	Nighswander et al. [40]	1989	33	79.7-198.1	2.0-10.2
22	S3	Sako et al. [69]	1991	7	75.15-148.25	10.2-19.7
23	K1	King et al. [89]	1992	27	15-25	6.1-24.3
24	D3	Dohrn et al. [90]	1993	5	50-50	10.1-30.4
25	R1	Rumpf et al. [36]	1994	7	50.01-50.03	1.1-5.8
26	T4	Teng et al. [66]	1997	40	4.85-24.85	5.0-40.0
27	D4	Dhima et al. [91]	1999	7	71-71	10.0-100.0
28	B2	Bamberger et al. [92]	2000	29	50.05-79.95	4.1-14.1
29	B3	Blencoe et al. [71]	2001	8	300-300	27.6-56.7
30	K2	Kiepe et al. [38]	2002	43	40.05-120.02	0.0-9.3
31	B4	Bando et al. [93]	2003	12	30-60	10.0-20.0
32	C1	Chapoy et al. [94]	2004	27	0.99-78.16	0.2-9.3
33	V1	Valtz et al. [95]	2004	47	5.07-45.08	0.5-8.0
34	B5	Bermejo et al. [37]	2005	26	23.58-96.5	1.6-8.3
35	K3	Koschel et al. [96]	2006	8	49.95-99.95	2.1-20.2
36	Q1	Qin et al. [68]	2008	7	50.45-102.65	10.6-49.9
37	H2	Han et al. [97]	2009	28	40.05-70.05	4.3-18.3
38	F1	Ferrentino et al. [98]	2010	18	35-50	7.6-13.1
39	L1	Liu et al. [41]	2011	31	35-55	2.1-16.0
40	Y1	Yan et al. [30]	2011	18	50.05-140.05	5.0-40.0
41	L2	Lucile et al. [99]	2012	30	25-120	0.5-5.1
42	S4	Savary et al. [100]	2012	3	150-150	11.3-33.7
43	H3	Hou et al. [45]	2013	42	25-175	1.1-17.6
44	T5	Tong et al. [20]	2013	18	101-101.84	7.2-27.3
45	A1	Al Ghafri et al. [101]	2014	8	50-50	2.1-18.7
46	B6	Bastami et al. [102]	2014	10	55-102	6.9-20.7
47	G3	Guo et al. [9]	2014	131	0-300	10.0-120.0
48	C2	Carvalho et al. [10]	2015	66	10.09-90.27	0.3-12.1
49	M4	Mohammadian et al. [29]	2015	20	60-80	0.1-21.3
50	C3	Caumon et al. [18]	2016	22	65-65	0.3-19.5
51	J1	Jacob and Saylor [11]	2016	50	23.85-23.85	1.4-12.4
52	T6	Truche et al. [19]	2016	7	200-280	3.1-13.2
53	M5	Mousavi [77]	2017	8	50-100	7.9-52.4
Sum				1597	0-350	0.1-350.0

No.	Acronym	Author	Time	Exp.	Temperature (°C)	Pressure (MPa)
1	W1	Wiebe and Gaddy [74]	1941	39	25-75	0.1-70.9
2	M1	Malinin [75]	1959	26	200-300	19.5-58.8
3	T1	Todheide and Franck [73]	1963	104	50-350	20.0-350.0
4	T2	Takenouchi and Kennedy [72]	1964	116	110-350	10.0-150.0
5	K1	King Jr and Coan [103]	1971	22	25-100	1.7-5.1
6	G1	Gillepsie and Wilson [67]	1982	33	15.6-260	0.7-20.3
7	B1	Briones et al. [87]	1987	8	50-50	6.8-17.7
8	N1	Nakayama et al. [63]	1987	1	25.05-25.05	3.6-3.6
9	S1	Song and Kobayashi [104]	1987	19	12-31.06	0.7-13.8
10	D1	D’Souza et al. [56]	1988	4	50-75	10.1-15.2
11	M2	Müller et al. [88]	1988	20	100-200	0.3-8.1
12	S2	Sako et al. [69]	1991	8	75.15-148.25	10.2-20.9
13	K2	King et al. [89]	1992	41	15-40	5.2-20.3
14	D2	Dohrn et al. [90]	1993	5	50-50	10.1-30.4
15	J1	Jackson et al. [105]	1995	2	50-75	34.5-34.5
16	B2	Bamberger et al. [92]	2000	29	50.05-79.95	4.1-14.1
17	V1	Valtz et al. [95]	2004	30	5.07-45.07	0.5-8.0
18	Q1	Qin et al. [68]	2008	7	50.45-102.65	10.6-49.9
19	K3	Kim et al. [44]	2012	26	10.05-38.85	8.1-20.1
20	H1	Hou et al. [45]	2013	39	25-175	1.1-17.5
21	M3	Meyer and Harvey [106]	2015	58	10-80	0.5-5.0
22	C1	Caumon et al. [18]	2016	21	100-100	0.5-20.1
23	M4	Mousavi [77]	2017	14	100-150	1.6-55.2
Sum				672	5.07-350	0.1-350.0

No.	Acronym	Author	Time	Exp.	Temperature (°C)	Pressure (MPa)
1	W1	Wiebe and Gaddy [74]	1941	39	25-75	0.1-70.9
2	M1	Malinin [75]	1959	26	200-300	19.5-58.8
3	T1	Todheide and Franck [73]	1963	104	50-350	20.0-350.0
4	T2	Takenouchi and Kennedy [72]	1964	116	110-350	10.0-150.0
5	K1	King Jr and Coan [103]	1971	22	25-100	1.7-5.1
6	G1	Gillepsie and Wilson [67]	1982	33	15.6-260	0.7-20.3
7	B1	Briones et al. [87]	1987	8	50-50	6.8-17.7
8	N1	Nakayama et al. [63]	1987	1	25.05-25.05	3.6-3.6
9	S1	Song and Kobayashi [104]	1987	19	12-31.06	0.7-13.8
10	D1	D’Souza et al. [56]	1988	4	50-75	10.1-15.2
11	M2	Müller et al. [88]	1988	20	100-200	0.3-8.1
12	S2	Sako et al. [69]	1991	8	75.15-148.25	10.2-20.9
13	K2	King et al. [89]	1992	41	15-40	5.2-20.3
14	D2	Dohrn et al. [90]	1993	5	50-50	10.1-30.4
15	J1	Jackson et al. [105]	1995	2	50-75	34.5-34.5
16	B2	Bamberger et al. [92]	2000	29	50.05-79.95	4.1-14.1
17	V1	Valtz et al. [95]	2004	30	5.07-45.07	0.5-8.0
18	Q1	Qin et al. [68]	2008	7	50.45-102.65	10.6-49.9
19	K3	Kim et al. [44]	2012	26	10.05-38.85	8.1-20.1
20	H1	Hou et al. [45]	2013	39	25-175	1.1-17.5
21	M3	Meyer and Harvey [106]	2015	58	10-80	0.5-5.0
22	C1	Caumon et al. [18]	2016	21	100-100	0.5-20.1
23	M4	Mousavi [77]	2017	14	100-150	1.6-55.2
Sum				672	5.07-350	0.1-350.0

No.	Author	Temperature (°C)	Pressure (MPa)	Exe.	Salt
1	Bando et al. [93]	30-60	10-20	36	NaCl
2	Carvalho et al. [10]	19.93-80.08	1.02-14.29	44	NaCl
3	Guo et al. [3]	5-200	10-40	142	NaCl
4	Hou et al. [107]	50-150	2.61-18.21	36	NaCl
5	Jacob and Saylor [11]	23.85-23.85	1.72-10.69	28	NaCl
6	Kiepe et al. [38]	40.23-79.93	0.19-10.1	63	NaCl
7	Koschel et al. [96]	49.95-99.95	5-20.24	14	NaCl
8	Liu et al. [41]	45-45	2.1-15.83	8	NaCl
9	Messabeb et al. [108]	50-150	4.99-20.23	40	NaCl
10	Mohammadian et al. [29]	60-80	2.1-21.3	42	NaCl
11	Nighswander et al. [40]	80-200.5	2.11-10.03	32	NaCl
12	Rumpf et al. [36]	39.99-159.93	0.47-9.64	63	NaCl
13	Savary et al. [100]	120-120	12.5-30.5	6	NaCl
14	Yan et al. [30]	50.05-140.05	5-40	35	NaCl
15	Mousavi et al. [77]	5-150	0.90-57.24	85	NaCl
16	Takenouchi and Kennedy [81]	150-300	10-60	29	NaCl
17	Hou et al. [107]	50-100	2.92-18.22	24	KCl
18	Jacob and Saylor [11]	23.85-23.85	2.1-9.65	20	KCl
19	Perez-Salado Kamps et al. [109]	39.95-159.95	0.41-9.40	93	KCl
20	Kiepe et al. [38]	40.01-80.25	0.09-10.51	88	KCl
21	Liu et al. [41]	45-45	2.09-15.81	8	KCl
22	Mousavi et al. [77]	50-150	8.21-54.83	19	KCl
23	Bastami et al. [102]	55-102	6.89-20.68	22	CaCl₂
24	Liu et al. [41]	45-45	2.09-15.86	8	CaCl₂
25	Prutton and Savage [78]	75.5-121	1.52-71.23	116	CaCl₂
26	Tong et al. [20]	34.85-151.49	1.53-37.99	36	CaCl₂
27	Mousavi et al. [77]	50-150	6.52-59.59	43	CaCl₂
28	Tong et al. [20]	36.37-151.53	1.25-34.93	39	MgCl₂
29	Mousavi et al. [77]	50-150	8.23-54.48	35	MgCl₂
30	Mousavi et al. [77]	50-150	10.02-57.10	18	NaKCaMg
31	Bermejo et al. [37]	13.82-95.66	1.98-13.11	113	Na₂SO₄
32	Rumpf and Maurer [110]	39.96-160.01	0.42-9.71	102	Na₂SO₄
33	Liu et al. [41]	45-45	2.48-15.99	8	NaK
34	Tong et al. [20]	35.75-151.52	1.07-17.16	14	NaK
35	Liu et al. [41]	45-45	2.46-16.02	8	NaCa
36	Poulain et al. [53]	49.85-149.85	1.01-19.93	24	NaCa
37	Liu et al. [41]	35-55	1.34-15.87	83	NaKCa
38	Poulain et al. [53]	49.85-149.85	1-19.97	24	NaKCa
	Sum			1648

Equations	Parameter	Parameter units	Regression coefficients
Equation (12)	$α_{CO 2}$	Pa∙m⁶∙Mol^-2	$κ_{CO 2} : 0.6903117$	$a_{CO 2} : 2.0441667 E ‐ 02$	$b_{CO 2} : 1.227268$
Equation (12)	$α_{H 2 O}$	Pa∙m⁶∙Mol^-2	$κ_{H 2 O} : 1.517423$	$a_{H 2 O} : - 0.5593541$	$b_{H 2 O} : 1.060686$

Equation (16)	$k$	NA	$k_{0} : 0.2851749$	$k_{1} : - 3.4677936 E ‐ 02$	$k_{3} : - 6.8869593 E ‐ 02$

Equation (20)	$K_{CO 2}$	Bar	$f_{0} : - 17.92880$	$f_{1} : 228.0000$	$f_{2} : - 253.8662$
			$f_{3} : 0.0$	$f_{4} : - 3.464000$	$f_{5} : 0.1356416$
			$f_{6} : - 2.6472550 E ‐ 04$	$f_{7} : 1.6690060 E ‐ 07$	$f_{8} : - 2.2076023 E ‐ 04$
			$f_{9} : - 1.7600000 E ‐ 04$	$d_{1} : - 4.6012800 E ‐ 03$
	$K_{H 2 O}$	Bar	$f_{0} : 91.74931$	$f_{3} : 1.29014 E 6$	$f_{4} : - 19131.8984$
	$K_{H 2 O}$	Bar	$f_{5} : - 0.21106$	$f_{6} : 2.53575 E ‐ 4$	$f_{7} : - 1.21111 E ‐ 07$

Equation (24)	$A_{M}$	NA	$A_{M a} : 2.4489199 E ‐ 06$	$A_{M b} : - 2.9597802 E ‐ 02$	$A_{M c} : 10.7031064$

Equation (21)	$V_{CO 2}$	Pa∙cm³∙Mol^-1	$c_{1} = - 6.5618993$	$c_{2} = 0.1020634$	$c_{3} = 1.629998$
Equation (21)	$V_{CO 2}$	Pa∙cm³∙Mol^-1	$c_{4} = - 7.8243337 E ‐ 04$

Equation (21)	$V_{H 2 O}$	Pa∙cm³∙Mol^-1	$c_{1} = 6.993863$	$c_{2} = 3.1268786 E ‐ 02$	$c_{3} = 0; c_{4} = 0$

Equation (29) Equation (30)	$γ_{CO 2}^{}$ $γ_{H 2 O}^{}$	NaCl	$B_{0} = 0.1428448$	$B_{1} = - 1.247759$	$χ_{G c a} = - 7.1575899 E ‐ 03$
		KCl	$B_{0} = 4.0705356 E ‐ 02$	$B_{1} = 10.58909$	$χ_{G c a} = - 3.2059969 E ‐ 03$
		CaCl₂	$B_{0} = 0.1418454$	$B_{1} = 31.15499$	$χ_{G c a} = - 4.6038739 E ‐ 03$
		MgCl₂	$B_{0} = 0.1845848$	$B_{1} = 11.49574$	$χ_{G c a} = - 4.1764954 E ‐ 03$
		Na₂SO₄	$B_{0} = 8.6264725 E ‐ 02$	$B_{1} = 100.1299$	$χ_{G c a} = - 2.2485980 E ‐ 02$

Salt	Temperature (°C)	Pressure (MPa)	Exe.	DS2003	SP2010	This model
NaCl	5-300	0.19-60	703	5.20%	9.37%	5.78%
KCl	23.85-159.95	0.09-54.83	252	14.88%	9.84%	5.98%
CaCl₂	34.85-151.49	1.52-71.23	225	4.90%	11.43%	4.37%
MgCl₂	36.37-151.53	1.25-54.48	74	6.92%	9.22%	4.11%
NaKCaMg	50-150	10.02-57.10	18	7.87%	12.82%	9.99%
Na₂SO₄	13.82-160.01	0.42-13.11	215	27.33%	33.58%	4.33%
NaK	35.75-151.52	1.07-17.16	22	4.39%	6.28%	3.48%
NaCa	49.85-149.85	1.01-19.93	32	3.73%	6.28%	5.06%
NaKCa	35-149.85	1-19.97	107	3.73%	3.31%	2.56%
Sum				9.56%	12.42%	5.15%

Modelling of the Phase-Partitioning Behaviors for CO2-Brine System at Geological Conditions

Abstract

1. Introduction

2. Thermodynamic Modelling of CO2-H2O System

2.1. Thermodynamic Framework for Vapor-Liquid Phase Equilibrium

2.1.1. Water

2.1.2. CO2

2.2. Fugacity Modelling of the CO2-Rich Phase

2.2.1. Alpha Model

2.2.2. Mixing Rule

2.3. Fugacity Modelling of the Aqueous Phase

2.3.1. Equilibrium Constant

2.3.2. Specific Volume and Relative Activity Coefficient

2.4. Simulation Method

3. Parameterization

3.1. Experimental Database in CO2+H2O System

3.2. Parameter Determination

4. Model Performance in CO2-Water System

4.1. Model Verification

4.1.1. CO2 Solubility

4.1.2. H2O Content

4.1.3. Error Analysis

4.2. Phase Diagram

4.2.1. CO2 Solubility

4.2.2. Water Content

5. Extension of the Model to Brines

5.1. Modelling

5.2. Comparison to the Experimental Data

5.3. Phase-Partitioning Behaviors in CO2-Brine System

6. Conclusions

Appendix

A. Fugacity Model of the Aqueous Phase

B. Experimental Database

Data Availability

Conflicts of Interest

Acknowledgments

Data & Figures

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Modelling of the Phase-Partitioning Behaviors for CO₂-Brine System at Geological Conditions

2. Thermodynamic Modelling of CO₂-H₂O System

2.1.2. CO₂

2.2. Fugacity Modelling of the CO₂-Rich Phase

3.1. Experimental Database in CO₂+H₂O System

4. Model Performance in CO₂-Water System

4.1.1. CO₂ Solubility

4.1.2. H₂O Content

4.2.1. CO₂ Solubility

5.3. Phase-Partitioning Behaviors in CO₂-Brine System