Prediction of vapor–liquid equilibrium and PVTx properties of geological fluid system with SAFT-LJ EOS including multi-polar contribution. Part II: Application to H2O–NaCl and CO2–H2O–NaCl System
Introduction
The compositions of fluids in various geological environments, such as metamorphic rocks, pegmatites, carbonatites, geothermal systems, and a wide variety of hydrothermal deposits (e.g. Roedder, 1984, Dubessy et al., 1989, Labotka, 1991, Schmidt and Bodnar, 2000, Anovitz et al., 2004, Heinrich, 2007), can be reasonably approximated by the H2O–NaCl system or the CO2–H2O–NaCl system. Understanding the thermodynamic properties (e.g. PVTx properties, vapor–liquid equilibrium) of the H2O–NaCl and the CO2–H2O–NaCl system over a wide P–T range is important for the use of fluid inclusions, for modeling heat and mass transfer in various geological environments, and for many process designs in chemical engineering and environmental engineering, such as enhanced oil recovery using H2O or CO2, CO2 storage in saline aquifers (e.g. Spycher et al., 2003, Hu et al., 2007).
There are two kinds of methods for modeling the thermodynamic properties of aqueous electrolyte solutions, the activity coefficient model and equation of state approach. Most of activity coefficient models for electrolyte solutions are based on the Debye–Hückel theory. The Pitzer model (Pitzer, 1973, Pitzer and Mayorga, 1973), a famous one of these models, can calculate the activity coefficients of many single and mixed aqueous electrolyte solutions and has gained wide application in the field of geochemistry and chemical engineering. However, the activity coefficient models are in terms of the excess Gibbs energy so that they are only useful in predicting activity coefficients. The vapor–liquid equilibrium and PVTx properties of electrolyte solutions cannot be derived from the activity coefficient model itself. Pitzer et al. (1984) combined their model with an empirical EOS for pure water to calculate the density of aqueous NaCl solution. However, a large number of experimental data are required to evaluate the parameters of the Pitzer model and it is difficult to apply the Pitzer model to vapor–liquid equilibria of gas–water–salt systems.
Applying an activity coefficient model or Henry’s law to describe the non-ideality of the aqueous solution, many solubility models, such as the model of Li and Ngheim, 1986, Duan and Sun, 2003, Papaiconomou et al., 2003, Dubessy et al., 2005, Spycher and Pruess, 2005, Akinfiev and Diamond, 2010, have been proposed to calculate CO2 solubilities in aqueous NaCl solution. Although they can give accurate representation of CO2 solubility, solubility models does not allow for the calculation of the composition of vapor phase and PVTx properties of the CO2–H2O–NaCl system.
Equations of state (EOS) are the second type of methods to model the PVTx properties and fluid phase equilibria, which also provide enthalpies and other thermodynamic properties of fluids. There are numerous types of EOS. In the family of EOS, the classical cubic EOS and the virial-type EOS can model the thermodynamic properties of the fluid systems composed of uncharged non-polar or weakly-polar molecules (e.g. N2, CO2, and hydrocarbons) between which the dominant intermolecular forces are the repulsion force and the dispersion force. However, cubic EOS and virial-type EOS are not suitable for modeling highly non-ideal aqueous electrolyte solutions because the long-range electrostatic forces (Coulombic forces) between ions and the hydrogen-bonding forces between H2O molecules are different from the repulsion forces and dispersion forces. In contrast, the molecular-based EOS, which take into account the energetic contribution of various types of intermolecular forces in terms of statistical mechanical approaches, provide the possibility to model the thermodynamic behavior of aqueous electrolyte solutions.
The analytical equations accounting for the energetic contribution of different types of intermolecular forces (e.g. the Coulombic force, the multi-polar force, the repulsion force between hard spheres) have been derived from statistical mechanical approaches (such as perturbation theory, integral equation theory and molecular simulation). These equations form the theoretical basis of molecular-based EOS. Considerable advances have been made in molecular-based EOS for non-electrolyte systems. The SAFT (Statistical Association Fluid Theory) EOS, one popular example of the molecular-based EOS, is suitable for description of the thermodynamic properties of long-chain molecules (e.g. hydrocarbon, alkanol) and molecules with hydrogen bonding. A review on the state of the art of SAFT EOS has been given in our previous paper (Sun and Dubessy, 2010) and is omitted here. Many studies (Jin and Donohue, 1988, Wu and Prausnitz, 1998, Galindo et al., 1999, Liu et al., 1999, Tan et al., 2005) tried to model electrolyte solutions with SAFT EOS or other molecular-based EOS in the last two decades. However, there still remains important work to improve thermodynamic models.
Before modeling the H2O–NaCl system with molecular-based EOS, we should note that the speciation of NaCl (and other strong electrolytes) in aqueous solution changes with temperature (and, to a lesser extent, with pressure). At ambient conditions, NaCl is fully dissociated so that only single ions exist in aqueous solutions. The dominant interaction between ions is the long-range electrostatic force. As temperature increases, ions in aqueous environment tend to associate due to the decrease of the dielectric constant of the solvent resulting from the decrease of hydrogen bonding. At temperatures above 573 K, ion pairs (NaCl° and polyatomic clusters) rather than single ions are the dominant species of NaCl in aqueous solution (Pitzer and Simonson, 1984, Oelkers and Helgeson, 1993). Using an empirical method to treat with salt effect, Bowers and Helgeson (1983) applied a modified cubic EOS to calculate the PVTx properties of the CO2–H2O–NaCl system at temperatures higher than 623 K. Taking account of the repulsion force, dispersion force and dipole–dipole force between H2O and NaCl° based on perturbation theory, Anderko and Pitzer (1993) developed an EOS to calculate the phase equilibrium and PVTx properties of H2O–NaCl system at temperatures above 573 K. Duan et al., 1995, Duan et al., 2003 extended the Anderko–Pitzer EOS to the H2O–NaCl–CO2 and H2O–NaCl–CH4 systems.
Since the long-range electrostatic force is the major difference between strong electrolyte solutions and non-electrolyte systems, one way to develop an EOS for electrolyte solutions is superimposing an equation for the energetic contribution of long-range electrostatic forces on the formula of an existing EOS for non-electrolyte systems. Some models (such as Aasberg-Petersen et al., 1991, Zuo and Guo, 1991, Kiepe et al., 2004) combined the traditional Debye–Hückel term or an electrostatic term derived from a semi-empirical activity coefficient model with a cubic EOS. Although the formulas of these EOS are relatively simple, a satisfactory quantitative description of the thermodynamic properties of aqueous electrolyte solutions is difficult to achieve. Poor agreement with experimental data is due to the fact that the models used to represent the intermolecular interactions in systems they investigated are not exact from the molecular point of view. For instance, the Debye–Hückel model considers the ions as point charges without size, which is only valid at the infinite dilution limit.
In contrast to the Debye–Hückel theory, the mean spherical approximation (MSA) of integral equation approach and the perturbation theory can take account of the ion sizes, and not only the thermodynamic properties but also the detailed structural information can be obtained (Thiery and Dubessy, 1998, Thiery et al., 1998, Galindo et al., 1999). Jin and Donohue (1988) combined the long-range term from Henderson’s perturbation theory (Henderson, 1983) with perturbed anisotropic chain theory (PACT) EOS proposed by Vimalchand and Donohue (1985). Fürst and Renon (1993) tried to combine the long-range term from MSA with a modified Redlich–Kwong–Soave EOS. Considering the poor representation of the cubic EOS for the liquid density, Myers et al. (2002) added the long-range term from MSA to a volume-translated Peng–Robinson EOS. The model of Myers et al. (2002) can represent the activity coefficient, osmotic coefficient and density of aqueous NaCl solution from 273 to 573 K and from 1 to 86 bar. Wu and Prausnitz, 1998, Lin et al., 2007, Inchekel et al., 2008 added the long-range term from MSA to a CPA EOS (cubic EOS plus an association term). In recent years, many studies (Galindo et al., 1999, Liu et al., 1999, Behzadi et al., 2005, Liu et al., 2005, Liu et al., 2008, Tan et al., 2005, Tan et al., 2006, Zhao et al., 2007, Seyfi et al., 2009, Herzog et al., 2010) superimposed the long-range term from MSA on different SAFT-type EOS to model the thermodynamic properties of the H2O–NaCl, H2O–CaCl2 and other water–salt binary systems. The SAFT1E and SAFT2E EOS (Tan et al., 2005, Tan et al., 2006) were extended to some multi-salt–water systems by Ji et al., 2005a, Ji et al., 2006.
Although there are many EOS adopting the long-range term from MSA or perturbation theory as mentioned above, most of them are limited to room temperature and 1 bar or a very narrow P–T range except the SAFT2E EOS and Myers’s EOS. The model of Myers et al. (2002) is suitable for the H2O–NaCl system from 273 to 573 K and from 0 to 86 bar, and the SAFT2E EOS improved by Ji and Adidharma (2007) can model the H2O–NaCl system from 273 to 473 K and from 0 to 1000 bar.
Some EOS for water–salts systems below 573 K have been extended to some gas–water–salt ternary systems (Jin and Donohue, 1988, Wu and Prausnitz, 1998, Patel et al., 2003, Ji et al., 2005b). Among them, only the model of Ji et al. (2005b) was designed for the CO2–H2O–NaCl system. Its calculation is accurate but the P–T range is small (from 298 to 373 K, up to 200 bar). Jin and Donohue (1988) just calculated the vapor–liquid equilibrium of the CH4–H2O–NaCl system at 375 K. The model of Patel et al. (2003) can represent CH4 solubilities in water () from 398 to 603 K but overestimate the content of H2O in vapor () by more than 30%. The absolute average deviations of the model of Wu and Prausnitz (1998) for both and are more than 20%. The inaccuracy of these models for ternary systems mainly arises from the inaccuracy for binary systems (water–gas binary system and H2O–salt binary system). In general, there is no model allowing the calculation of the thermodynamic properties of the H2O–NaCl and CO2–H2O–NaCl systems over 273–573 K and 0–1000 bar.
It’s important to develop an EOS suitable for the CO2–H2O–NaCl system over a wide P–T range for the studies of fluid inclusion, metamorphic rocks, and CO2 sequestration so on. In the first companion paper (Sun and Dubessy, 2010), we improved the SAFT-LJ EOS proposed by Kraska and Gubbins (1996) and modeled the vapor–liquid equilibrium and PVTx properties of the H2O–CO2 system over a wide P–T range. The purpose of this paper is to extend this EOS to the H2O–NaCl and CO2–H2O–NaCl systems below 573 K. Considering that the dissociation constant of NaCl in aqueous solution is greater than 1 at temperatures below 573 K (according to Mesmer et al., 1988) and the lower temperature limit of the EOS of Duan et al. (1995) for the CO2–H2O–NaCl system is 573 K, we chose 573 K as the upper temperature limit of the new EOS. Assuming that NaCl is fully dissociated in aqueous solution at temperatures below 573 K, the restricted primitive model of mean spherical approximation was included in SAFT-LJ EOS to account for the energetic contribution due to long-range electrostatic forces between ions. The theory of our model will be explained in the next section in detail. Section 3 describes the evaluation procedure of the parameters of the model. Section 4 compares the prediction of this model for the phase equilibrium and volumetric properties of the H2O–NaCl and CO2–H2O–NaCl systems with previous studies.
Section snippets
Models and theory
The SAFT-LJ EOS improved by Sun and Dubessy (2010) takes account of the energetic contribution of the main types of molecular interactions in the CO2–H2O system, such as the repulsion force, the dispersion force, the hydrogen-bonding force and the multi-polar forces. It can be extended to the CO2–H2O–NaCl system by adding a new term to account for the energetic contribution due to the long-range electrostatic force between ions considering that the Columbic force is the characteristic of
Evaluation of parameters
There are four species in the CO2–H2O–NaCl system: CO2, H2O, Na+ and Cl− because NaCl is considered fully dissociated in aqueous solution below 573 K. The SAFT-LJ parameters for CO2 have been determined in the previous study (Sun and Dubessy, 2010). Because this study made a simplification for the calculation of parameter I in association term, the parameters for H2O and the binary parameters for CO2–H2O were re-evaluated, which are listed in Table 1, Table 2. Besides the energetic contribution
H2O–NaCl system
Fig. 1(a) compares the representation of the mean ionic activity coefficient of aqueous NaCl solution by our model with the experimental data reported by Liu and Lindsay, 1972, Gibbard et al., 1974 at low pressure (1 bar at temperature below 373 K or saturation pressure of pure water at temperature above 373 K). This model can represent the experimental data over a wide range of temperature from 273 to 573 K and a wide range of salinity from 0 to 6 mol/kg with high accuracy. The average
Conclusion
The SAFT-LJ EOS improved by Sun and Dubessy (2010) recently is extended to the H2O–NaCl and CO2–H2O–NaCl systems below 573 K. Assuming that NaCl is fully dissociated in aqueous solution, this study adopted the restricted primitive model of mean spherical approximation to account for the energetic contribution due to long-range electrostatic forces between ions. The association between Na+ and Cl− and the association between ion and H2O are neglected. The SAFT-LJ parameters describing the
Acknowledgments
The authors acknowledge TOTAL Company for their financial support. The work is further financially supported by National Natural Science Foundation of China (Project 41073049), by MOST Special Fund from the State Key Laboratory of Continental Dynamics, Northwest University, and by the Science Foundation of Northwest University (No.09NW01). The authors acknowledge Matthew Steel-MacInnis, Denis Zezin and one anonymous reviewer for their detailed, helpful and pertinent comments which improved
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