Densities for Ternary System of CaCl₂–H₂O–CO₂ at Elevated P-T: An Experimental and Modeling Approach

Abstract

Very few thermodynamic models exist for estimation of density alteration due to solution of CO₂ in a pure H₂O and CaCl₂–H₂O system. All of these models require density of CaCl₂ solution to estimate density of CaCl₂–H₂O–CO₂ system. Similarly, models presented to calculate CaCl₂ solution density need pure H₂O density in advance. The main approach to model density of CaCl₂–H₂O–CO₂ system is based on estimation of density alteration of CaCl₂–H₂O system due to the solution of CO₂ mole fraction. Hence, to estimate CO₂–CaCl₂–H₂O system density, density of CaCl₂ solution is necessary, and to estimate density of CaCl₂–H₂O system, density of pure H₂O is required in advance. Firstly in this paper, density of 0, 1.91, and 4.85 mol/kg CaCl₂ solutions saturated with CO₂ at 328.15 to 375.15 °K and 68.9 to 206.8 Bar were measured through laboratory experiments. Then, a new model is developed to estimate the density of CaCl₂ solutions containing CO₂ based on the experiments conducted in this study. The average and maximum absolute deviations of the new model from the experimental data are 0.0047 and 0.0177, respectively. Hence, the new model combined with other existing models to separately calculate density of the CaCl₂ solution can be used to accurately predict density of the CaCl₂–H₂O–CO₂ system in a wide range of P-T applicable for subsurface reservoirs.

1. Introduction

Thermodynamic behavior of CaCl₂–H₂O–CO₂ system is important in different scientific and engineering areas, such as well stimulation (i.e., matrix acidizing and acid fracturing) in both limestone and sandstone reservoirs; geological CO₂ sequestration, especially in sandstone reservoirs; and enhanced oil recovery (EOR) using CO₂ injection. Changes in thermodynamic behavior of pure H₂O and CaCl₂ solution with addition of CO₂ is of great importance for accurate evaluation of capacity of saline aquifer for CO₂ sequestration. In recent years, researchers have been paying attention to environmental issues such as CO₂ capture and geological storage in deep saline aquifers or depleted hydrocarbon reservoirs. However, the target depleted hydrocarbon reservoirs are mostly sandstone reservoirs [1,2,3,4,5,6]. Nevertheless, understanding thermodynamic behavior of this mixture is also important in flow enhancement, and well stimulation in petroleum production operations (i.e., matrix acidizing and acid fracturing) commonly used in carbonate reservoirs (i.e., limestone) are also a subject of great interest.

Different thermodynamic models have been proposed in the literature to simulate parameters affecting subsurface CO₂ storage [7,8,9,10,11]. The CaCl₂–H₂O–CO₂ system also appears in limestone matrix acidizing, where hydrochloric acid (HCl) is the most common acid used for limestone matrix acidizing. A mixture of CaCl₂–H₂O–CO₂ is produced, while HCl and calcite react as follows:

$${2\mathrm{HCl}+\mathrm{CaCO}}_3\leftrightarrow {\mathrm{CaCl}}_2+{\mathrm{H}}_2{\mathrm{O}+\mathrm{CO}}_2$$

Hence, density of CO₂–CaCl₂–H₂O system should be accurately estimated during matrix acidizing and acid fracturing in limestone formations. It should be noted that in spite of extensive studies on matrix acidizing, chemical and physical behavior of spent acid solutions is not yet properly addressed. Different experimental investigations on measurements of density of CO₂–CaCl₂–H₂O at different temperatures and pressures are reported in the literature, as shown in

Table 1. Experimental studies to measure density of the CO₂–CaCl₂–H₂O system.

Reference	Temperature Range (°K)	Pressure Range (Bar)
Nighswander [13]	353.15–471.15	20.4–102.1
King et al. [14]	288.15–298.15	60.8–243.2
Fenghour et al. [15]	415–700	60–300
Teng et al. [16]	278–293	64–295
Yaginuma et al. [17]	304.1	10–100
Tegetmeier et al. [18]	374.15	0–300
Song et al. [19]	273.15–284.15	50–125
Li et al. [20]	332.15	Up to 290
Hebach et al. [21]	284–332	10–300
Song et al. [22]	274–333	100–180

[12,13,14,15,16,17,18,19,20,21,22].

Very few models have been developed to calculate density of saline solutions during CO₂ sequestration at reservoir conditions. As the major salt in aquifers is NaCl, thermodynamic behavior of CO₂–H₂O–NaCl system has been widely investigated to estimate the density. In spite of the attention paid to NaCl solutions, experimental data concerning CO₂–CaCl₂–H₂O system remain scarce.

Teng et al. [16] and Song et al. [19] proposed simple correlations based on available experimental data. Bando et al. [23] further developed the model proposed by Teng et al. [16] for NaCl solutions. Bachu and Adams [24] introduced a thermodynamic model based on apparent molar volume estimation using a correlation presented by Garcia [25]. As mentioned, for all of these models, density of the base fluids should be estimated in advance. Then, by considering the amount of CO₂ dissolved in the solution, the new density is updated.

In this paper, first CaCl₂–H₂O–CO₂ system at different ranges of pressure, temperature, and CaCl₂ molality, similar to reservoir conditions, is investigated in a laboratory to measure density of CaCl₂–H₂O–CO₂. The main approach to model density of CaCl₂–H₂O–CO₂ system is based on the estimation of density alteration of CaCl₂–H₂O system because of the solution of CO₂ mole fraction. Hence, to estimate CO₂–CaCl₂–H₂O system density, density of CaCl₂ solution is necessary and to estimate density of CaCl₂–H₂O system, density of pure water is required in advance. In this paper, experimental studies have been carried out and based on the available data for CO₂–CaCl₂–H₂O system. Then, a new model is developed to calculate the density of CaCl₂–H₂O system. The new model, combined with other existing models to separately calculate density of CaCl₂ solution, can be used to accurately predict density of CaCl₂–H₂O–CO₂ system in a wide range of P-T applicable for subsurface reservoirs.

2. Experimental Work

The purpose of these experiments was to measure density of different CaCl₂ solutions saturated with CO₂. The experiments are similar to the ones previously reported by Bastami et al. [26] to measure CO₂ solubility at different CaCl₂ solutions at temperatures ranging from 328.15 to 375.15 °K and pressures from 68.9 to 206.8 Bar. The temperature and pressure ranges considered for these experiments are similar to typical P-T condition of subsurface reservoirs.

2.1. Materials

The purity of CO₂ is 99.99%. The anhydrous CaCl₂ provided by Merck KGaA (Darmstadt, Germany) is of analytical grade. In addition, distilled water was used throughout the experiments.

2.2. Apparatus and Procedures

Figure 1 shows a schematic of the apparatus used in this study to estimate density of CaCl₂ solutions saturated with CO₂ and solubility of CO₂ in CaCl₂ solutions. The apparatus consisted of a DBR pump for 0.1–2000 cm³/h range with 0.07 Bar precision at pressure constant mode, a high pressure cylinder with a volume of 500 cm³ and maximum bearable pressure of 689 Bar, a pycnometer made of St–Cr alloy, high pressure transfer lines, an oven with maximum temperature of 493.15 °K with 0.1 °K precision, a pressure transducer for 1–620 Bar range with 0.007 Bar precision, a densitometer, and a vacuum pump. An agitating pump system, as shown in Figure 2, was also designed to agitate the high-pressure cylinder in the oven. The system included a motor, a gearbox, a shaft, a pulley, a drive belt, and two bull bearings. The motor and gearbox were out of the oven during the experiments.

First, the high pressure cylinder was opened and washed with distilled water to ensure that no CaCl₂ solution remained in the high pressure cylinder, and evacuated by a vacuum pump. CaCl₂ solution and CO₂ were then pumped into the high pressure cylinder and pressurized up to the final experimental pressure. Then, the high pressure cylinder was agitated at the desired temperature for 3 h. The effects of agitating time were then investigated for 1.91 mol/kg-CaCl₂ solution at 137.9 Bar and 351.65 °K (Figure 3). In this study, the soluble CO₂ mole fraction changed negligibly for data acquired in experiments after 3 h of agitation. Hence, to ensure equilibrium, the samples were agitated for 3 h. Pressure was adjusted to remain constant during the agitating time.

Concentration of the injected CO₂ was set to be more than its saturation concentration to guarantee achieving saturation concentration of CO₂ in the solution. Thus, there was a separate supercritical CO₂ phase in the high pressure cylinder after reaching equilibrium. As the high pressure cylinder had just one exhaust, the cylinder should be reversed for sampling to ensure that only saturated solution was produced. The solution was transferred gradually through lines to a pycnometer. The pycnometer, filled with solution, was weighed at the final temperature and pressure. Knowing the weight of the vacuumed pycnometer and its volume means that we can determine the solution density. The procedure used for determining CO₂ solubility has been published previously by Bastami et al. [26].

3. Model Development

As mentioned earlier in this text, addition of CO₂ to pure water or CaCl₂–H₂O solution causes density alteration. Different models exist for estimation of density of CO₂–CaCl₂–H₂O. In all models, to estimate density of the CO₂ bearing solution, it is necessary to have a good prediction of solution density of either pure H₂O or CaCl₂ solution. Hence, estimation of pure H₂O and CaCl₂ solution density is the first step to compare the ability of different models to estimate density of a complex system such as CaCl₂–H₂O–CO₂. For model development, we investigated density of CO₂–H₂O and CO₂–CaCl₂–H₂O systems in a laboratory and used different available models to find the best model in terms of accuracy in prediction of density of these two systems. Nevertheless, all of these models examined need densities of pure H₂O and CaCl₂ solution in advance.

3.1. Pure Water Density

Density of pure water increases with a decrease in temperature and increase in pressure. However, because of the incompressibility of water, changes in water density with pressure are negligible. The best model to estimate water density is the model presented as IAPWS97. However, because of a large number of parameters and the required considerable computation efforts, simpler equations with insignificant deviations proposed to reduce the number of parameters and consequently increase efficiency of the computations. Batzle and Wang [27] proposed a correlation (Equation (1)) to calculate pure H₂O density as a function of temperature and pressure. They recommended this equation for the P-T range of 293–623 °K and 50–1000 Bar, respectively.

$$\begin{array}{l}{\rho }_w=1+1\times {10}^{-6}(-80T-3.3T^2+0.00175T^3+489P-2TP+0.016T^{2}P\\ -1.3\times {10}^{-5}{T}^{3}P-0.333P^2-0.002T{P}^2)\end{array}$$

(1)

where T is temperature (°C), P is pressure (MPa), and ${\rho }_w$ is pure H₂O density in (g/cm³).

Islam and Carlson [28] proposed another simple correlation (Equation (2)), the deviation of which from IAPWS97 within the P-T range of current study is insignificant.

$${\rho }_w=a_0+{\displaystyle \sum _{i=1}^3{b}_i{10}^{c_{i}T}}+{\displaystyle \sum _{i=0}^2{d}_i{P}^i}$$

(2)

where T is temperature (°K), P is pressure (MPa), and the constants of the equation are listed in Islam and Carlson [28]. These two models are used in this study to achieve an accurate model for estimation of density of CO₂–H₂O and CO₂–CaCl₂–H₂O systems.

3.2. CaCl₂ Solution Density

A number of experimental studies are reported for estimation of density of different electrolyte solutions. Mao and Duan [29] presented a detailed study of available experimental data. Al Ghafri et al. [30] reported density of various electrolyte solutions with molalities up to 6 mol/kg at P-T ranges of 283–472 °K and 10–685 Bar, respectively. Their measurements are of great importance in this study because of the fact that P-T conditions in their study are consistent with typical reservoir P-T.

Kemp et al. [31] proposed a model to calculate density of solutions containing single salts and various salt mixtures. Although they developed a model for estimation of density of brines for drilling fluid systems, it can be used for estimation of density of other electrolyte solutions at high P-T conditions. Their model is based on Equation (3).

$${\rho }_b=\left(1.0+{\displaystyle \sum _i{m}_i{M}_i}\right)/\left[\left(1.0/{\rho }_w\right)+{\displaystyle \sum _i{m}_i{\nu }_i}\right]$$

(3)

where $m_i$ is molality of the ith salt (mol/kg); $M_i$ is the molecular weight of the ith salt; ${\rho }_b$, ${\rho }_w$ are density of brine and pure H₂O, respectively (g/cm³); and ${\nu }_i$ is the apparent molal volume of the ith salt. To find the value of apparent molal volume, they used the “specific interaction model”, which is an extension of the Debye-Hückel model. The complete procedure for calculations can be found in Kemp et al. [31].

Furthermore, Mao and Duan [29] developed a PVTX model for aqueous chloride solutions of binary systems such as CaCl₂–H₂O. They attempted to obtain a universal model for all binary solutions. They showed that their model (Equation (4)) can accurately estimate density of different aqueous chloride solutions.

$$\begin{array}{l}\frac{1000+m{M}_s}{m{\rho }_b}=\frac{V(m_r)}{m_r}+\frac{1000}{{\rho }_w}\left(\frac{1}{m}-\frac{1}{m_r}\right)+\nu \left|z_{+}{z}_{-}\right|A_v\left[h\left(I_m\right)-h\left(I_{m_r}\right)\right]+\\ 2{\nu }_{+}{\nu }_{-}RT\left[B_v\left(m-m_r\right)+{\nu }_{+}{z}_{+}{C}_v\left(m^2-m_r^2\right)\right]\end{array}$$

(4)

where T is temperature (°K); $m$ is molality of brine; $m_r$ is reference molality, which is determined by Mao and Duan [29]; I is ionic strength; $M_s$ is molar mass of calcium chloride (g/mol); $z$ and $\nu$ are ion charges and number of charges, respectively; $A_v$ is the volumetric Debye-Hückel limiting law slope; and $V(m_r)$, $B_v$, and $C_v$ are pressure and temperature dependent parameters, which can be calculated using the correlations introduced by Mao and Duan [29]. As can be observed from these two thermodynamic models, both of them require density of pure H₂O at the desired P-T.

3.3. CO₂–H₂O Density

Teng et al. [16] presented a model to estimate density of water by adding CO₂ (Equation (5)):

$${\rho }_c={\rho }_W+1.96\times {10}^{-1}{x}_{{\mathrm{CO}}_2}+1.54\times 10x_{{\mathrm{CO}}_2}^2$$

(5)

where $x_{{\mathrm{CO}}_2}$ is mole fraction of CO₂ in liquid phase and ${\rho }_c$ is solution density with CO₂ (g/cm³).

Song et al. [19] also proposed a density model for dilute solution of CO₂ (Equation (6)):

$${\rho }_c={\rho }_w\left[1+0.275wt\left({\mathrm{CO}}_2\right)\right]$$

(6)

where ${\rho }_w$ and $wt\left({\mathrm{CO}}_2\right)$ are water density and mass fraction of CO₂, respectively.

Bachu and Adams [24] proposed a thermodynamic model based on apparent partial molar volume (Equation (7)) to estimate density of aqueous solutions with addition of CO₂:

$${\rho }_{aq}=\frac{{\rho }_w}{1-X_{sol}\left(1-\frac{V_{\phi }}{M}{\rho }_w\right)}$$

(7)

where M is the molecular weight of solute (44.01 g/mol for CO₂), $X_{sol}$ is the mass fraction of solute (CO₂), and $V_{\phi }$ is the apparent partial molar volume. Garcia [25] proposed Equation (8) for calculating the apparent molar volume of CO₂:

$$V_{\phi }=37.51-9.585\times {10}^{-2}T+8.740\times {10}^{-4}{T}^2-5.044\times {10}^{-7}{T}^3$$

(8)

where T is temperature (°C) and $V_{\phi }$ is partial molar volume of CO₂ in water (cm³/mol).

Previously, two models for estimating the pure water density were introduced. Using these models, it is possible to estimate the density alteration while CO₂ is added to the system.

3.4. CO₂–CaCl₂–H₂O Density

As was mentioned previously, Bachu and Adams [24] proposed a model based on apparent partial molar volume that can be estimated by Garcia’s correlation [25]. They also studied different methods to consider the effect of salt on the solution density and proposed Equation (9). This makes it possible to use Equation (7) for more general cases, including CO₂ in brines.

$$V_{\phi }^b{\rho }_b=V_{\phi }{\rho }_w$$

(9)

where ${\rho }_b$ is density of brine and $V_{\phi }^b$ is partial molar volume of CO₂ in brine.

Bando et al. [23] extended the model previously proposed by Teng et al. [16] based on experimental measurements of density of electrolyte solution of NaCl–H₂O with addition of CO₂ for CO₂–H₂O system. They proposed Equation (10) to model more general cases of CO₂ in aqueous chloride solutions.

$${\rho }_{aq}={\rho }_b+1.96\times {10}^{-1}{x}_{{\mathrm{CO}}_2}+1.54\times 10x_{{\mathrm{CO}}_2}^2$$

(10)

where ${\rho }_{aq}$ is density of electrolyte solution containing CO₂ (g/cm³) and $x_{{\mathrm{CO}}_2}$ is mole fraction of CO₂ in the solution.

We developed a new correlation based on the experimental data generated in this study (Equation (11)) to calculate density changes of CaCl₂ solution with addition of CO₂:

$${\rho }_{aq}={\rho }_b+2.271\times {10}^{-1}{x}_{{\mathrm{CO}}_2}+1.6129\times {10}^2{x}_{{\mathrm{CO}}_2}^2$$

(11)

The deviation of different model estimates from the experimental measurements of this study is discussed in Section 4.4.

4. Results and Discussion

In this section, we first presented results from experimental investigation of density measurements for 0, 1.91, and 4.85 mol/kg CaCl₂ solutions saturated with CO₂ at 328.15 to 375.15 °K and 68.9 to 206.8 Bar. Then, various models proposed for estimation of density of CaCl₂ solution were examined using experimental data published by Al Ghafari et al. [30] to find the best model. Then, we used the experimental data generated in this study to estimate density of CO₂–H₂O to identify the most appropriate existing model for this purpose. Finally, we used the experimental data from the current study to develop a new model for estimation of CO₂–CaCl₂–H₂O density.

4.1. Experimental Results

The results of experimental work conducted in this study for estimation of density of pure H₂O, saturated with CO₂ at 328.15 °K, are depicted in Figure 4. As shown in Figure 4, experimental measurements in this study are compared with those published by Hebach et al. [21]. Hebach et al. [21] measured density of pure water saturated with CO₂ at 322.85 °K and 331.8 °K. Although the experimental conditions are not the same, this comparison confirms the accuracy of the experimental technique used in this study. Density of CO₂–H₂O system measured at different P-T conditions of this study and molalities of 0, 1.91, and 4.85 mol/kg-CaCl₂ are listed in

Table 2. Results from measurement of density of pure H₂O saturated with CO₂ in P-T conditions of current study and molalities of 0, 1.91, and 4.85 mol/kg CaCl₂.

Temperature(°K)	Pressure (Bar)	Liquid Density (g/cm3), Molality = 0	Liquid Density (g/cm3); Molality = 1.91 mol/kg-CaCl2	Liquid Density (g/cm3); Molality = 4.85 mol/kg-CaCl2
328.15	68.9	0.9975	1.1710	1.3250
	137.9	1.0040		1.3450
	206.8	1.0080	1.1770	1.3650
351.65	68.9	0.9790		1.3130
	137.9		1.1565
	206.8	0.9921		1.3610
368.15	68.9	0.9662		1.3080
	206.8	0.9815		1.3540
375.15	68.9		1.1410
368.15	137.9	0.9600		1.3420
375.15	206.8		1.1450

. These data show the trend of density change for 1.91 and 4.85 mol/kg CaCl₂ solutions saturated with CO₂, with changing P-T conditions.

Addition of CO₂ to pure H₂O increases the solution density [22,24]. Moreover, measured data in the current study indicate that density of CaCl₂ solution saturated with CO₂ increases with an increase in pressure, as shown in Figure 4 and Table 2. Furthermore, density of 1.91 mol/kg CaCl₂ solution saturated with CO₂ measured at 68.9 and 206.8 Bar is plotted versus temperature in Figure 5. As one can see from Figure 5, density of this system decreases with an increase in temperature. This result is consistent with observations made by some previous studies [21,32,33].

The density of aqueous solution saturated with CO₂ increases with an increase in CaCl₂ solution concentration. Figure 6 demonstrates this trend for pure H₂O and different CaCl₂ solutions at 328.15 °K and 68.9–206.8 Bar. CaCl₂ solution concentration is the most effective parameter on density of the solution. This is also in harmony with previously reported research by some previous studies [13,24,25,33,34].

4.2. CaCl₂ Solution Density

To identify the best model for estimation of density of CaCl₂–H₂O under typical reservoir P-T conditions, predictions from models presented by Kemp et al. (1989) [31] and Mao and Duan [29] are compared with experimental measurements reported by Al Ghafri et al. [30]. Al Ghafri et al. [30] measured density of 1–6 mol/kg aqueous CaCl₂ solutions in P-T ranges of 298.15–477 °K and 10.5–689 Bar, respectively. Different models for pure H₂O and CaCl₂ solution density are used here to find the best one with minimum deviation from experimental measurements. Deviations of model estimations from experimental measurements (${\rho }_{calc}-{\rho }_{exp}$) are compared in the desired temperature range of the current study (323.07–372.99 °K), as illustrated in Figure 7.

While the model proposed by Mao and Duan [29] underestimates density of CaCl₂ solution in all of the examined P-T conditions, the model proposed by Kemp et al. [31] overestimates the density of solutions with higher salinities for high P-T ranges. In other words, there are some conditions in which the deviation of estimates by Kemp et al. [31] tend towards zero. Moreover, in all of the cases, estimations of Kemp et al. [31] are more accurate than those of Mao and Duan [29]. It should be noted that Mao and Duan [29] recommended their model to be used with pure H₂O densities from IAPWS97. These deviations can be attributed to the approximations made by either pure H₂O or CaCl₂ solution density models.

To compare model estimates, average absolute deviations (AAD) and maximum absolute deviations (MAD) of model estimates from experimental measurements are defined as Equations (12) and (13), respectively.

$$\mathrm{A}\mathrm{A}\mathrm{D}=\left(\left|{\rho }_{exp}-{\rho }_{calc}\right|/{\rho }_{exp}\right)/\mathrm{N}$$

(12)

$$\mathrm{M}\mathrm{A}\mathrm{D}=\mathrm{M}\mathrm{a}\mathrm{x}\left(\left|{\rho }_{exp}-{\rho }_{calc}\right|/{\rho }_{exp}\right)$$

(13)

where ${\rho }_{exp}$ is density measured experimentally, ${\rho }_{calc}$ is calculated density, and N is number of data.

Table 3. Deviation of various model estimations for the CaCl₂–H₂O system from the experimental data measured by Al Ghafri et al. [30]. AAD is average absolute deviations and MAD is maximum absolute deviations of model (see below) estimates from experimental measurements conducted in current study.

Models for Density Estimation of Pure Water	Deviations	Models for Density Estimation of Brine
		Kemp et al. [31]	Mao and Duan [29]
Batzle and Wang [27]	AAD	0.0089	0.0151
	MAD	0.0304	0.0504
Islam and Carlson [28]	AAD	0.0077	0.0143
	MAD	0.0349	0.0389

shows the average and maximum absolute deviations of density predictions by these two models with the experimental measurements of Al Ghafri et al. [30], which are consistent with reservoir conditions. It should be noted that these models require an estimation of pure H₂O density. Hence, water density is estimated by correlations previously introduced in this article. The AAD and MAD values shown in Table 3 also confirm that the model presented by Kemp et al. [31] performs relatively better than the model proposed by Mao and Duan [29]. In addition, the pure H₂O densities estimated by Islam and Carlson [28] show less deviation from the experimental aqueous CaCl₂ solutions.

4.3. CO₂–H₂O Density

Using experimental data from the current study, the density of pure H₂O saturated with CO₂, three models presented as Equations (5)–(7) are compared at different temperatures, as presented in Figure 8. It should be emphasized that based on the previous results, the pure H₂O density is estimated by the model proposed by Islam and Carlson [28].

The average and maximum absolute deviations of the predictions are listed in

Table 4. Comparison of experimental data from this study with estimations from three different models for pure H₂O saturated with CO₂.

Deviations	Models for Density Estimation of CO2 in Pure Water
	Teng et al. [16]	Bachu and Adams (2003) [24] and Garcia [25]	Song et al. [19]
AAD	0.0016	0.0020	0.0032
MAD	0.0037	0.0051	0.0069

. As can be observed from Figure 8, as well as from the AAD and MAD values presented in Table 4, the correlation proposed by Teng et al. [16] with pure water density estimation from the model proposed by Islam and Carlson [28] results in the best estimation.

4.4. CO₂–CaCl₂–H₂O Density

We used the experimental data generated in this study to develop a new model for estimation of density of the CO₂–CaCl₂–H₂O CaCl₂ system (Equation (11)). Data from experimental measurements from this study and model estimates by Bachu and Adams [24], Bando et al. [23], and also the new model we proposed for prediction of CO₂–CaCl₂–H₂O density for 1.91 and 4.85 mol/kg CaCl₂ solutions saturated with CO₂, are compared in Figure 9 and Figure 10. It should be noted that to predict density of the CO₂–CaCl₂–H₂O system, it is required to estimate density of CaCl₂–H₂O in advance. Moreover, estimation of density of CaCl₂–H₂O requires estimation of density of pure H₂O density. Based on investigations of this study (Section 4.2 and Section 4.3), these estimates are obtained using Kemp et al. [31] and Islam and Carlson [28] to calculate CaCl₂ solution and pure H₂O densities, respectively.

Figure 9 and Figure 10 demonstrate the improvement in the model estimates by the new model developed in this study in comparison with some previous models. Moreover, the estimate deviations for different models, including the new model, are listed in

Table 5. Comparison of various model estimations for CaCl₂ solutions saturated with CO₂ to experimental data from the current study and the new model proposed in this study.

Deviations	Models for Density Estimation of CO2 in CaCl2 Solution
	Bachu and Adams [24] and Garcia [25]	Bando et al. [23]	New Model
AAD	0.0072	0.0080	0.0047
MAD	0.0228	0.0241	0.0177

. Comparing the deviations reported in Table 5 shows that the minimum deviation between predictions and experimental data belongs to the new model developed in this study for estimation of density alteration of CaCl₂ solutions due to addition of CO₂. The average and maximum absolute deviation between experiments and the estimates are 0.0047 and 0.0036, respectively. The average absolute deviation of the new model is 34% and 41% lower than the models previously proposed by Bachu and Adams [24] and Bando et al. [23], respectively.

Based on experimental investigations in this study and examination of different models proposed in the literature using the obtained experimental data, the correlation proposed by Teng et al. [16] and the model proposed by Islam and Carlson [28] are the best available models for estimation of density of pure H₂O saturated with CO₂ and density of pure H₂O, respectively (CO₂–H₂O system). In addition, for the CO₂–CaCl₂–H₂O system, the model proposed by Kemp et al. [31] for estimation of density of CaCl₂ solution and the model proposed by Islam and Carlson [28] for pure H₂O density estimation gives the best estimates with least deviations.

In petroleum engineering applications, CaCl₂ solutions are usually subject to CO₂ addition. The method proposed in this study can be recommended for estimation of density of CaCl₂ solutions in the ternary system of CO₂–CaCl₂–H₂O. The new model can be applied in various areas such as geological CO₂ sequestration to evaluate capacity of saline aquifers for CO₂ storage, as well as for a more accurate simulation of CO₂ sequestration. Moreover, it can be used to further current understanding of physics of matrix acidizing in limestone reservoirs such as interaction between rock and the CO₂–CaCl₂–H₂O system.

5. Conclusions

Accurate estimation of changes in density of CO₂–pure H₂O and CaCl₂–H₂O systems is of paramount importance in areas such as matrix acidizing in limestone reservoirs as a well stimulation technique, as well as geological sequestration of CO₂ in saline aquifers. In this paper, density of 0, 1.91, and 4.85 mol/kg CaCl₂ solutions saturated with CO₂ at 328.15 to 375.15 °K and 68.9 to 206.8 Bar were measured through laboratory experiments. Then, a new model was developed to estimate the density of the CaCl₂ solutions containing CO₂ based on the experiments conducted in this study. Based on the experimental and modeling works conducted in this study the following five main conclusions can be drawn:

• An experimental apparatus was designed and fabricated to measure density of 0, 1.91, and 4.85 mol/kg CaCl₂ solutions saturated with CO₂ at 328.15 to 375.15 °K and 68.9 to 206.8 Bar. The experimental data showed that solution density increases with addition of CO₂. The solution density also increases with increase in CaCl₂ solution concentration, increase in pressure, and decrease in temperature. However, the effect of CaCl₂ solution concentration on density alteration was notable compared with the effect of pressure and temperature.
• CaCl₂ solution density estimated by different models was compared with the experimental data from the current study. The comparison shows that minimum errors belong to the models proposed by Islam and Carlson [28] and Kemp et al. [31] for pure H₂O and CaCl₂ solution, respectively.
• The models proposed by Teng et al. [16] and Islam and Carlson [28] are the best available models for estimation of density of pure H₂O saturated with CO₂ and density of pure H₂O, respectively (CO₂–H₂O system).
• We developed a new model for estimation of density of the CO₂–CaCl₂–H₂O CaCl₂ system (Equation (11)) based on the experimental data generated in this study. Comparing the deviations between the experimental data and the new model and the models proposed by Bachu and Adams [24] and Bando et al. [23] showed that the new model performs 34% and 41% better than these two models, respectively. For the new model, the average and maximum absolute deviation between experiments and the estimates are 0.0047 and 0.0036, respectively.
• The new model can be applied in various areas such as geological CO₂ sequestration to evaluate capacity of saline aquifers for CO₂ storage, as well as for a more accurate simulation of CO₂ sequestration. Moreover, it can be used to further understand physics of matrix acidizing in limestone reservoirs such as interaction between rock and the CO₂–CaCl₂–H₂O system.