Predicting CO₂ and H₂ Solubility in Pure Water and Various Aqueous Systems: Implication for CO₂–EOR, Carbon Capture and Sequestration, Natural Hydrogen Production and Underground Hydrogen Storage

Abstract

The growing energy demand and the need for climate mitigation strategies have spurred interest in the application of CO₂–enhanced oil recovery (CO₂–EOR) and carbon capture, utilization, and storage (CCUS). Furthermore, natural hydrogen (H₂) production and underground hydrogen storage (UHS) in geological media have emerged as promising technologies for cleaner energy and achieving net–zero emissions. However, selecting a suitable geological storage medium is complex, as it depends on the physicochemical and petrophysical characteristics of the host rock. Solubility is a key factor affecting the above–mentioned processes, and it is critical to understand phase distribution and estimating trapping capacities. This paper conducts a succinct review of predictive techniques and present novel simple and non–iterative predictive models for swift and reliable prediction of solubility behaviors in CO₂–brine and H₂–brine systems under varying conditions of pressure, temperature, and salinity (T–P–m salts), which are crucial for many geological and energy–related applications. The proposed models predict CO₂ solubility in CO₂ + H₂O and CO₂ + brine systems containing mixed salts and various single salt systems (Na⁺, K⁺, Ca²⁺, Mg²⁺, Cl⁻, SO₄²⁻) under typical geological conditions (273.15–523.15 K, 0–71 MPa), as well as H₂ solubility in H₂ + H₂O and H₂ + brine systems containing NaCl (273.15–630 K, 0–101 MPa). The proposed models are validated against experimental data, with average absolute errors for CO₂ solubility in pure water and brine ranging between 8.19 and 8.80% and for H₂ solubility in pure water and brine between 4.03 and 9.91%, respectively. These results demonstrate that the models can accurately predict solubility over a wide range of conditions while remaining computationally efficient compared to traditional models. Importantly, the proposed models can reproduce abrupt variations in phase composition during phase transitions and account for the influence of different ions on CO₂ solubility. The solubility models accurately capture the salting–out (SO) characteristics of CO₂ and H₂ gas in various types of salt systems which are consistent with previous studies. The simplified solubility models for CO₂ and H₂ presented in this study offer significant advantages over conventional approaches, including computational efficiency and accuracy across a wide range of geological conditions. The explicit, derivative–continuous nature of these models eliminates the need for iterative algorithms, making them suitable for integration into large–scale multiphase flow simulations. This work contributes to the field by offering reliable tools for modeling solubility in various subsurface energy and environmental–related applications, facilitating their application in energy transition strategies aimed at reducing carbon emissions.

1. Introduction

Mitigating CO₂ emissions through carbon capture and storage (CCS) and carbon capture, utilization, and storage (CCUS) is an increasingly promising approach to curbing the rise of temperature on earth [1,2,3]. Geologic formations such as deep saline aquifers [4], depleted oil and gas reservoirs [5,6,7] and unmineable coal seams are examples of structures viable for CO₂ storage [8]. Among these, deep saline aquifers have the most significant storage potential. Carbon capture and storage (CCS) in saline aquifers depends on four fundamental mechanisms: structural/stratigraphic, dissolution/solubility, capillary/residual, and mineral trapping [9]. Dissolution trapping involves CO₂ dissolving into the brine, which decreases its buoyancy, whereas capillary trapping holds CO₂ in rock pores [10]. Mineral trapping occurs when CO₂ reacts with minerals to form stable carbonates that ensure long–term storage [11]. When CO₂ is injected into a geological formation, it dissolves into the reservoir brine, increasing the brine density and creating a density gradient [12]. This gradient induces convective mixing, which speeds up the dissolution of CO₂ and enhances solubility trapping. The effectiveness of dissolution trapping is crucial for successful geological carbon storage. Furthermore, the interaction between CO₂ and brine solubility significantly impacts long–term storage security and the reservoir’s complex physical and chemical interactions among minerals and fluids. To accurately evaluate these effects and optimize carbon storage, a reliable model for predicting CO₂ solubility in reservoir brines under geological conditions is essential [13].

Predictive solubility models are essential not only for CO₂ storage but also for CO₂ utilization in enhanced oil recovery (EOR) projects [5,14,15,16]. These models are crucial in determining the amount of CO₂ that can be stored during EOR and the volume of CO₂ that is miscible with the oil front for improved recovery—for instance, Pi et al. [17] observed a decrease in the pH of formation water during CO₂ flooding in rock cores. They also recorded a reduction in the minimum miscibility pressure as the salinity of connate water in rock cores increased. This indicates changes in CO₂ concentration within the porous media at lower salinity levels of connate water. When modeling CO₂–EOR, it is essential to have reliable CO₂ dissolution models to accurately represent the amounts of CO₂ available for compositional interaction with the oil front. Similarly, during water–alternating–gas (WAG) injection, the salinity of injected water has been noted as a critical factor in creating optimal miscible flood fronts [18]. For example, saline–injected water with 1–2 wt.% showed good synergy and improved oil recovery. These solubility predictive models can also provide insight into the concentration or volume of CO₂ available for miscibility with oil front. Another scenario in which predictive models are essential is during the drilling of acid gas reservoirs. Acid gas reservoirs contain high amounts of CO₂, which have the potential to dissolve in water–based or oil–based drilling fluid [19,20]. The dissolution degree depends on the drilling fluid downhole salinity, pressure, and temperature. Near the wellhead, the dissolved gas will separate from the aqueous phase and expand rapidly due to a decrease in gas solubility. Therefore, the gas kick commonly exhibits the characteristics of early latency and late burst, which brings significant challenges to the safety of well control [21]. To improve safety, CO₂ predictive models can help understand the degree of solubility and well control choking required for the wellhead’s safety.

In UHS, hydrogen is stored within geological formations, with its capacity and security maintained by specific trapping mechanisms akin to those used for CO₂ storage. These mechanisms include (a) structural or stratigraphic trapping [22,23], (b) residual or capillary trapping [6,24], (c) mineral trapping, (d) dissolution trapping [25], and (e) the more recently studied adsorption trapping potential of coal seams [26,27] or shale [28]. A graphical illustration of these mechanisms is shown in Figure 1. Among these trapping phenomena, trapping by solubility in aqueous media offers the highest efficiency because it requires minimal monitoring and maintenance compared to capillary trapping, which requires close monitoring. Unfortunately, storing hydrogen at surface facilities is challenging because of its low density, thereby requiring high pressures to adequately compress it due to its high diffusivity [29]. Furthermore, H₂ has a low ignition temperature that makes it highly inflammable. Hence, the safest place to store H₂ is likely underground [30,31]. The storage of H₂ in geologic structures is regarded as the most feasible option for large–scale energy storage on a global basis, mimicking the carbon capture and utilization of the storage industry. In particular, salt caverns previously used for hydrocarbon storage are promising geologic candidates for UHS [32]. The interest in salt caverns is primarily due to the high capacity, rapid operation, and minimal contamination risk. However, cavern storage may be constrained by geography and transportation costs. Alternatively, depleted hydrocarbon reservoirs provide established infrastructure and proven containment capabilities for hydrogen storage. However, there are concerns about H₂ mixing with the residual hydrocarbons or reacting with underground minerals. Saline aquifers trap CO₂ effectively in CCUS but need more study for H₂ storage due to potential interactions with brine and rock minerals. Further, hard rock caverns (e.g., granite) offer stability but are costly to develop and may leak if not sealed properly.

Raza et al. [33] conducted a review on underground hydrogen storage, and they concluded that the accurate prediction of H₂ solubility across a wide range of geographical depth–dependent temperatures, pressures, and salinities (T–P–S) is vital for assessing the feasibility of storing hydrogen in geological formations such as aquifers depleted oil and gas fields, and salt caverns [30,32,34]. This understanding is essential not only for determining the storage capacity and retention potential but also for evaluating the risks of hydrogen loss due to dissolution into formation waters or reactions with minerals [35].

Secondly, evaluating the natural production of hydrogen requires knowledge of the solubility of H₂ in the subterranean environment, particularly from serpentinization processes. Serpentinization is a metamorphic process in which mainly ultramafic rocks are oxidized by water into serpentine, producing hydrogen [36]. A typical ultramafic rock is a peridotite, which contains olivine and fayalite minerals that undergo oxidation to release H₂ into brines at temperatures between 200 to 300 °C. Pressure has also been reported to increase the rate of peridotite serpentinization [37]. The salinity of the water is a crucial factor in the serpentinization process, as reported by [38]. The authors reported a decrease in serpentinization rates as salinity and concentration of dissolved Mg increase. Hence, predictive models that can estimate H₂ solubility may be vital in evaluating the H₂–producing potential of target ultramafic rocks. These factors influence the solubility and subsequent migration of hydrogen in subsurface environments [39]. The solubility of hydrogen in these settings can control the concentration of hydrogen that accumulates in fractures and pores, ultimately affecting the efficiency of hydrogen extraction [40].

Critical factors influencing geological carbon storage (GCS) and underground hydrogen storage (UHS) can be broadly categorized into solid, fluid, and solid–fluid interaction factors [30]. Solid factors include absolute permeability and effective porosity, which dictate the capacity and efficiency of the storage formations. Fluid factors such as fluid density, viscosity, solubility, and diffusivity govern the behavior and movement of the gases within the storage medium. Meanwhile, solid–fluid interaction factors like wettability, solid–fluid interfacial tension, capillary pressure, relative permeability, mobility ratio, and adsorption–desorption processes play crucial roles in determining the ease with which gases can be injected, stored, and retained within the geological formations. Figure 2 presents the classifications of the parameters influencing the storage of hydrogen gas in subsurface formation. Understanding these parameters is essential for optimizing storage strategies and ensuring the long–term stability and safety of CO₂ and H₂ storage. Solubility is a key parameter that impacts several of the other parameters significantly.

Given the complexity and the plethora of parameters influencing GCS, UHS, and natural hydrogen production, this paper focuses on the solubility factors of CO₂ and H₂ in water and saline systems, which are pivotal for various subsurface and industrial applications. The solubility of CO₂ and H₂ in solvent directly affects the capacity and stability of the stored gases, influencing how CO₂ and H₂ interact with the storage medium at a molecular and pore–scale level. The critical review of techniques for predicting CO₂ solubility and H₂ solubility, concludes that currently available predictive models employed in predicting CO₂ solubility and H₂ solubility from equation of state (EoS) to machine learning models (ML) have the following limitations: EoS modelling [41,42,43,44,45,46,47,48,49,50,51,52,53] involve iterative approach that might have convergence problems, most of published empirical models [54,55,56,57,58,59] also involve iterative method and many parameters, Molecular dynamic simulation [44,60,61,62,63,64,65] require additional parameter and high computation costs, and ML models [66,67,68,69,70,71,72] lack interpretability. Therefore, we proposed simple, non–iterative and reliable models to accurately predict CO₂ solubility and H₂ solubility in pure water and various salt systems. This paper contributes to the body of knowledge by enhancing the understanding of CO₂ solubility in natural formation brines, aiding in the assessment of CO₂ and H₂ storage capacity and the long–term behavior of CO₂ and H₂ at storage sites in the context of GCS, UHS, and natural hydrogen production.

The rest of the paper’s sections following the introduction are outlined as follows: Section 2 provides a concise review of the progress of experimental work and different techniques of modeling the solubility of CO₂ and H₂ in water and saline systems. Section 3 focuses on analyzing the collected CO₂ and H₂ solubility data and developing CO₂ and H₂ correlations. Section 4 presents the validation and evaluation of the predicted performance of the developed CO₂ and H₂ solubility correlations. Also, discusses the salting out characteristics of CO₂ and H₂ in various brine solutions. Lastly, Section 5 provides major findings and limitations of the study.

2. Literature Review

This section briefly reviews the state–of–the–art techniques on CO₂ and H₂ solubility determination in various pure water and aqueous systems. The techniques include experimental works, equation of state (EoS) models, empirical correlations, molecular dynamics simulations (MDS), and machine learning (ML) techniques. Each approach offers unique insights and advantages that contribute to a deeper understanding of gas solubility under diverse conditions with implications for both large–scale industrial applications and scientific research.

2.1. Carbon Dioxide (CO₂)

Knowledge of CO₂ solubility in geological fluids is essential for studying fluid inclusions [73], carbonate precipitation [74,75,76], and the global carbon cycle. Despite extensive research, accurately predicting CO₂ solubility across different temperatures, pressures, and ionic strengths remains a challenge. Researchers have primarily focused on CO₂ solubility in water and NaCl solutions, which are common in geological fluids [73]. Among potential long–term CO₂ storage sites, deep saline aquifers stand out, with an estimated storage capacity of 2400 to 21,600 billion metric tons [77]. To estimate this storage potential accurately, it is crucial to evaluate the pore volume available for supercritical CO₂ storage and the capacity for CO₂ to dissolve into the brine. Understanding how CO₂ dissolves in brine and groundwater helps predict possible migration pathways and environmental impacts, especially in the event of a leak. CO₂ solubility is influenced by factors like temperature, pressure, and solute concentrations, which can vary both between and within different formations [13,78].

2.1.1. CO₂ Solubility–Experimental Works

Carbon dioxide (CO₂) solubility in water has been extensively studied over the years due to its relevance in various industrial and environmental applications, including carbon sequestration, enhanced oil recovery, and understanding natural carbon cycles. Various experimental techniques have been employed to explore CO₂ solubility in water under different conditions (see

Table 1. Published experimental studies of CO₂ solubility in water and aqueous systems in the compiled database.

References	Data Points	Tmin (K)	Tmax (K)	Pmin (MPa)	Pmax (MPa)	Smin (mol/kg)	Smax (mol/kg)	Systems
Hou et al. (b) [46] *	71	323.15	423.15	2.6	18.2	2.50	4.00	KCl, NaCl
Todheide and Franck [80] *	104	323.15	623.15	10.0	350.0	0.00	0.00	H2O
Takenouchi and Kennedy (b) [81] *	21	423.15	723.15	10.0	140.0	0.00	4.28	H2O, NaCl
Takenouchi and Kennedy (a) [82] *	116	383.15	573.15	10.0	150.0	0.00	0.00	H2O
King et al. [83]*	28	288.15	298.15	6.1	24.3	0.00	0.00	H2O
Ahmandi and Chapoy [84] *	29	300.95	423.48	1.3	42.1	0.00	0.00	H2O
Al Ghafri, S.Z.S. [85] *	8	323.15	323.15	2.1	18.7	0.00	0.00	H2O
Anderson, G.K. [86] *	54	274.15	288.15	0.1	2.2	0.00	0.00	H2O
Bamberger et al. [87] *	29	323.20	353.10	4.1	14.1	0.00	0.00	H2O
Bando et al. [88] *	45	303.15	333.15	10.0	20.0	0.00	0.55	H2O, NaCl
Bastami et al. [89] *	32	328.15	375.15	6.9	20.7	0.00	4.80	H2O, CaCl2
Bermejo et al. [90] *	92	286.97	368.81	2.0	13.1	0.25	0.99	Na2SO4
Bo liu et al. [91] *	538	298.00	373.00	0.1	20.3	0.00	0.26	CaCl2, H2O, MgCl2, (MgCl2 + CaCl2)
Campos et al. [92]	50	298.20	323.20	0.1	0.5	0.00	0.00	H2O
Carvalho et al. [93]	107	283.24	363.42	0.3	40.0	0.00	2.00	H2O, NaCl
Chapoy et al. [94]	27	274.14	351.31	0.2	9.3	0.00	0.00	H2O
Corti et al. [95]	10	323.15	348.15	3.8	14.5	0.96	2.72	Na2SO4
Dalmolin et al. [96]	49	288.00	323.00	0.1	0.5	0.00	0.00	H2O
Dell’Era, C. et al. [97]	7	298.48	298.63	0.3	0.7	0.00	0.00	H2O
dos Santos et al. [98]	62	303.15	423.15	1.5	20.3	1.00	6.00	MgCl2, NaCl, (NaCl + Na2SO4)
Ellis and Golding [99]	54	445.15	607.15	1.6	9.3	0.00	2.00	H2O, NaCl
Gilbert et al. [100]	35	308.15	413.15	1.9	35.8	0.00	3.40	CaCl2, H2O, Na2SO4, NaCl, NaHCO3
Guo et al. [101] *	168	273.15	453.15	10.0	40.0	1.00	5.00	NaCl
Han et al. [102] *	75	313.00	333.00	0.3	2.0	0.00	1.00	NaHCO3
Han Ji et al. [103] *	28	313.20	343.20	4.3	18.3	0.00	0.00	H2O
He and Morse [104]	157	298.15	363.18	0.0	0.1	0.02	6.14	CaCl2, K2SO4, KCl, MgCl2, MgSO4, Na2SO4, NaCl
He et al. [105]	35	293.15	348.15	5.2	38.3	0.00	2.70	H2O, NaCl
Hoballah [106] *	12	348.15	398.15	5.0	50.0	0.80	0.80	NaHCO3
Hou et al. (a) [107] *	41	298.15	448.15	1.1	17.6	0.00	0.00	H2O
Kamps et al. [108] *	138	313.10	433.10	0.3	9.4	0.43	4.05	K2CO3, KCl
Kiepe et al. [109] *	190	313.16	393.17	0.1	10.5	0.00	4.34	H2O, KCl, NaCl
Koschel et al. [110] *	49	323.00	423.00	5.0	20.2	0.33	4.50	KCl, MgCl2, NaCl
Malinin and Kurovskaya [111]	25	298.15	348.15	4.8	4.8	0.36	5.21	CaCl2
Malinin and Kurovskaya [111]	27	298.15	423.15	4.8	4.8	0.32	1.91	NaCl
Martin et al. [112] *	5	353.00	393.00	10.0	30.0	0.00	0.00	H2O
Messabeb et al. [113] *	40	323.15	423.15	5.0	20.2	0.00	6.00	H2O, NaCl
Mohammadian et al. [114] *	68	333.15	353.15	0.1	21.3	0.00	0.26	H2O, NaCl
Muromachi et al. [115] *	17	286.15	298.15	0.2	4.0	0.00	0.00	H2O
Nighswander et al. [116]	33	353.00	473.65	4.1	100.3	0.17	0.17	NaCl
Portier and Rochelle [117] *	35	291.15	353.15	8.0	12.0	0.38	0.38	NaCl, KCl, MgCl2, CaCl2, NaHCO3
Poulain et al. [118] *	48	323.00	423.00	1.0	20.0	1.40	1.50	NaCl, CaCl2, KCl
Prutton and Savage [119]	139	348.65	394.15	1.5	71.2	0.00	3.90	CaCl2, H2O
Qin et al. [120] *	7	323.60	375.80	10.6	49.9	0.00	0.00	H2O
Ruffine and Trusler [121] *	3	333.00	333.00	4.9	11.5	0.00	0.00	H2O
Rumpf and Maurer [122]	111	313.11	433.16	0.0	9.7	0.99	2.01	Na2SO4
Rumpf et al. [123]	63	313.14	433.08	0.5	9.6	4.00	6.00	NaCl
Sako et al. [124]	7	348.30	421.40	10.2	19.7	0.00	0.00	H2O
Serpa et al. [125] *	9	298.00	323.00	0.1	0.4	0.00	0.00	H2O
Servio and Englezos [126] *	6	278.05	283.15	2.0	3.7	0.00	0.00	H2O
Tang et al. [127] *	70	308.15	408.15	8.0	40.0	0.00	1.41	H2O, NaCl, CaCl2, MgCl2, NaHCO3
Tong et al. [128] *	94	308.00	424.68	1.1	38.0	0.00	5.00	CaCl2, H2O, MgCl2, NaCl, KCl
Valtz et al. [129] *	47	278.22	318.23	0.5	8.0	0.00	0.00	H2O
Wiebe [130] *	73	273.15	373.15	2.5	70.9	0.00	0.00	H2O
Wiebe and Gaddy [131]*	29	323.15	373.15	2.5	70.9	0.00	0.00	H2O
Y. Liu et al. (a) [132] *	154	308.15	328.15	1.3	16.0	0.00	1.90	CaCl2, KCl, NaCl,
Y. Liu et al. (b) [133] *	6	308.15	318.15	8.0	16.0	0.00	0.00	H2O
Yan et al. [134] *	54	323.20	413.20	5.0	40.0	0.00	5.00	H2O, NaCl
Zhao et al. (b) [135] *	70	323.00	423.00	15.0	15.0	0.33	4.50	CaCl2, KCl, MgCl2, Na2SO4
Zhao et al. (a) [136] *	21	323.15	423.15	15.0	15.0	0.00	6.00	H2O, NaCl

* Experimental CO₂ solubility data were used in the regression of this study model.

). Among the earliest and most influential studies were those by Wiebe and Gaddy [79], who investigated the CO₂ solubility in water at temperatures ranging from 323 to 373 K and pressures approaching 71 MPa. Their work has provided foundational data for the study of CO₂ behavior under sequestration conditions. In the literature, the highest pressure at which CO₂ solubility in water has been experimentally measured is around 350 MPa, according to studies by Todheide and Franck [80], and later by Takenouchi and Kennedy [81,82]. These works significantly expanded the understanding of CO₂ solubility across a wide range of temperatures (323 to 623 K) and pressures (up to 350 MPa), offering critical insights into CO₂ behavior in deep geological formations. Finally, the study of CO₂ solubility in water and aqueous salt solutions has made considerable progress, with a rich database of experimental data now available for modeling efforts. However, the field still faces challenges, particularly in obtaining high–quality data for mixed–salt solutions and under extreme conditions. Continued experimental and theoretical work is necessary to build refined models and enhance the understanding of CO₂ behavior in these complex systems.

2.1.2. CO₂ Solubility–Equation of State

Generally, Peng Robinson Equation of State, PR (EoS) [137,138] have been commonly used to describe the phase behavior of pure components and mixtures in the gas, liquid, and supercritical fluid states. The traditional PR (EoS) model often results in significant errors, particularly in predicting gas solubility in aqueous phases. It struggles to account for the effects of salts like NaCl, leading to underestimation of the salting–out effect and inaccurate phase behavior predictions in subsurface engineering simulations. To enhance the prediction of CO₂ solubility in brine and hydrocarbon systems, Søreide and Whitson [139] introduced a tailored α–term to the PR (EoS) [137,138]. Other authors have expanded on this work, Duan and Sun [52] extended the pressure (0 to 200 MPa), temperature (273 to 533 K), and ionic strength up to 4.3 M using specific particle interaction theory for the liquid phase. Duan et al. [53] introduced Pitzer’s specific interaction model to calculate the chemical potential of CO₂ in the vapor phase. Advancing on earlier works, various authors [46,47,48,49,50,51,140] have contributed to the modeling of CO₂ solubility in water and saline systems.

2.1.3. CO₂ Solubility–Empirical Correlations

Despite the advantage of EoS modeling being fast and requiring less computing power compared to molecular dynamics, the application of this method to estimate CO₂ solubility in brine, particularly under extreme subsurface injection conditions, remains challenging. These EoS models are usually accurate within a given constraint of pressure and temperature [139]. In addition, EoS can sometimes be computationally extensive, hence the need for accurate, fast, and cheap empirical models. In an earlier study, Darwish [55] developed an extension of the Setschenov model to predict CO₂ solubility in the H₂O–NaCl system under geological sequestration conditions, spanning temperatures from 300 to 500 K, pressures from 5 to 200 MPa, and salt concentrations of 1 to 4 mol/kg. Barta and Bradley [56] applied an interaction model to study gas solubility in high ionic strength and high–temperature aqueous salt solutions, focusing on CO₂, H₂O, and CH₄ in aqueous NaCl up to 6 molar, 623.15 K and 15.19 MPa. Furthermore, to accurately correlate CO₂ solubility in water by analyzing 110 data points from literature covering a temperature range of 298 to 523 K and a pressure range of 3.4 to 72.41 MPa, Enick and Klara [58] employed Henry’s law to model CO₂ solubility in pure water, utilizing an equation of state for hydrocarbon phase equilibria. Diamond and Akinfiev [141] present a detailed evaluation of the solubility of carbon dioxide in pure water over a temperature range of 271.65 to 373.15 K and pressures between 0.1 and 100 MPa. Sun et al. [57] developed a simple model for the prediction of mutual solubility in CO₂ + H₂O and CO₂ + Brine systems that cover 0–250 °C, 0–200 MPa and brines containing Na⁺, K⁺, Ca²⁺, Mg²⁺, Cl⁻, and SO₄²⁻.

2.1.4. CO₂ Solubility–Molecular Dynamics Studies

MDS and other atomistic techniques like Monte Carlo (MC) simulations have become essential for exploring the molecular foundations of various phase behavior and geochemical processes. These simulations offer a unique capability to study the behavior of individual atoms, providing insights that are often challenging to obtain through experimental approaches, which typically capture the average behavior of large molecular groups. The versatility of MDS, which can examine a wide range of length and time scales —from angstroms to nanometers and from femtoseconds to microseconds—makes them particularly effective in understanding how small–scale molecular properties lead to more pr collective behaviors in geochemical systems. One significant application of these molecular simulations has been in investigating the vapor–liquid equilibrium (VLE) in H₂O–CO₂ and CO₂ –NaCl brine systems.

Liu et al. [63] examined the phase behavior of H₂O–CO₂ mixtures using histogram–reweighting Grand–Canonical Monte Carlo (GCMC) simulations over a temperature range of 323.15 to 723.15 K and pressures from 0 to 100 MPa. The study by Yasaman et al. [65] leverages comprehensive MDS with a focus on force field parameters using Large–scale Atomic and Molecular Massively Parallel Simulator (LAMMPS) [142] to predict the equilibrium, interfacial, and transport properties of CO₂ –brine systems under realistic CO₂ storage conditions. In another study, Vorholz et al. [64] utilized Gibbs Ensemble Monte Carlo (GEMC) [143] simulations to study the VLE of water (H₂O) at temperatures between 323 and 573 K, CO₂ between 230 and 290 K, and H₂O–CO₂ mixtures between 348 and 393 K, employing both Number of Particles Volume and Temperature (NVT) and Number of Particle Pressure Temperature (NPT) ensembles. Furthering this research, Liu et al. [62] also investigated the phase behavior and interfacial tension of H₂O–CO₂ –NaCl mixtures using MDS across a temperature range of 323.15 to 523.15 K, pressures from 0 to 60 MPa, and NaCl concentrations between 1 and 4 molal. Moreover, Orozco et al. [144] applied NPT GEMC simulations to optimize intermolecular potential parameters for describing the phase behavior of H₂O–CO₂ mixtures. Similarly, Lobanova et al. [145] modeled the VLE of CO₂ –H₂O systems at 423 and 548 K using SAFT–CG Mie force fields for CO₂ and H₂O, achieving good agreement with experimental solubility data. In another article, Jiang et al. [146] studied the phase equilibria of H₂O–CO₂ and H₂O/n–alkane mixtures using GEMC simulations, covering a temperature range of 323 to 523 K and pressures from 20 to 80 MPa. Hulikal et al. [147] tackles the challenge of predicting VLE in H₂O–CO₂ and CO₂ –NaCl brine systems over a temperature range of 230 to 723.15 K and pressures up to 100 MPa using advanced methods like Gibbs Ensemble Monte Carlo (GEMC) and Grand–Canonical Monte Carlo (GCMC). Their work revealed discrepancies between simulation results and experimental data, highlighting the need for better force fields. Moreover, Adam et al. [148] utilized equilibrium MD using the LAMMPS software (23Jun22 version) to explore the solubilities of H₂ and CO₂ in brine under conditions relevant to CCUS.

2.1.5. CO₂ Solubility–Machine Learning Studies

Machine learning (ML) provides alternative and reliable techniques for predicting the solubility of CO₂ in aqueous solutions. In contrast to conventional models, which frequently rely on simplified equations with restricted variables, ML is a data–driven approach that uses readily available parameters such as temperature, pressure, and electrolyte concentration to predict solubility. Here, we present a succinct review of ML studies to estimate CO₂ solubility in water and brine systems, particularly under subsurface processes.

Menad et al. [67] applied machine learning to model CO₂ solubility in brine, utilizing Multilayer Perceptron (MLP) optimized by the Levenberg–Marquardt algorithm, and Radial Basis Function Neural Network (RBFNN) optimized using Genetic Algorithm (GA), Particle Swarm Optimization (PSO), and Artificial Bee Colony (ABC). Mohammadian et al. [68] used the Group Modeling Data Handling (GMDH) in predicting CO₂ solubility in aqueous solutions at pressures up to 400 atm and temperatures between 283 and 298 K. Continuing this research, Mohammadian et al. [149] applied four data–driven techniques—extreme gradient boosting (XGB), multilayer perceptron (MLP), K–nearest neighbor (KNN), and an in–house genetic algorithm (GA)—to estimate CO₂ solubility across a broader range of conditions (salinity 0–15,000 ppm, temperatures 298–373 K, and pressures up to 200 atm). Ratnakar et al. [150] developed a machine learning–based workflow for accurately estimating CO₂ solubility in brine for CCUS applications. In another study, Jeon and Lee [69] used an artificial neural network (ANN) to predict CO₂ solubility, utilizing 2406 experimental data points in salt–dissolved solutions across a broad range of pressures (0.92–712.31 bar), temperatures (273.15–473.65 K), concentrations of water molecules (0–90.12 mol/kg), and overall mole fractions of dissolved salts (0–25.39 mol%), including supercritical CO₂ conditions. Recently, Mahmoudzadeh et al. [70] developed two tree–based models, LightGBM and GBoost, to predict CO₂ solubility in pure H₂O using a comprehensive dataset of 785 experimental data points from various sources. Pressure and temperature were the input parameters, and solubility was the output. The GBoost model outperformed LightGBM, achieving an R² of 0.9976 and an RMSE of 0.137 mol/kg. Similarly, Zou et al. [151] used machine learning to predict CO₂ solubility in water and brine–based on 1278 experimental data points covering temperatures from 273.15 to 453.15 K and pressures ranging from 0.06 to 100 MPa. They employed two ANNs—cascade forward neural network (CFNN) and generalized regression neural network (GRNN)—along with three optimization algorithms.

The advancement of machine learning techniques in these studies has significantly boosted the precision and reliability of CO₂ solubility predictions. Initial models encountered difficulties in formulating practical equations and handling high–pressure conditions. However, later research improved accuracy by utilizing advanced optimization algorithms, hybrid models, and thorough feature analysis. The use of extensive and diverse datasets, along with careful validation against experimental data, has been key in overcoming the shortcomings of earlier models. This has positioned data–driven approaches as a reliable alternative for estimating CO₂ solubility in brine.

2.2. Hydrogen (H₂)

The study of hydrogen (H₂) solubility is crucial for understanding various natural and engineered processes, particularly in the context of UHS and the natural production of hydrogen. There is a pressing need for more experimental and theoretical work to accurately model hydrogen solubility across the full spectrum of relevant T–P–S conditions, particularly in complex saline aquifer systems and under the high–pressure conditions found in deep geological formations [59,152]. This review of hydrogen solubility is not only fundamental for the safe and efficient design of hydrogen storage systems but also for understanding the broader implications of hydrogen in natural geochemical processes, including its role in the global hydrogen cycle and its potential as a renewable energy resource.

2.2.1. H₂ Solubility–Experimental Works

In the literature, there are numerous data on the solubility of hydrogen in pure water, the majority of which comes from very old studies. All these data are also limited to low temperature and atmospheric pressure. On the other hand, the solubility of hydrogen in various brines have been studied less than in pure water. A substantial amount of experimental data on hydrogen solubility in pure water is available in the literature. This data mainly includes volume ratio measurements, such as Ostwald, Knudsen, and Bunsen coefficients. Additionally, some data are provided as direct solubility measurements in mole fraction using PVT cells (“TPxy” data) or as Henry’s constants. Some experimental data on hydrogen solubility in seawater solutions and NaCl_(aq) are available in the literature; however, these data cover only a narrow range of temperatures and salinities. According to IUPAC’s [153] evaluation of experimental data, Wiebe and Gaddy [154] data are the most reliable and largely cover the pressure and temperature range of geological storage conditions. The most significant contribution comes from Crozier and Yamamoto [155], who collected extensive data on hydrogen solubility in seawater and NaCl aqueous solutions, albeit experiments were conducted at atmospheric pressure. Conversely, the data from Braun [156] were excluded because they were found to be inconsistent with all other available data under similar conditions. Recently, Chabab et al. [59,152] carried out measurements of hydrogen solubility of the H₂ + H₂O + NaCl system, as well as data from the literature. Therefore, it is necessary to overcome data gaps, especially for the H₂ + H₂O + NaCl system, which is the most important since Na⁺ and Cl^— are the predominant species in natural saline water. These experimental data on hydrogen solubility in water and salt were collected and listed in

Table 2. Published experimental studies of H₂ solubility in aqueous systems in the compiled database.

Authors	Year	Min. Temp (K)	Max. Temp (K)	Min. Press (MPa)	Max. Press (MPa)	Max. Molality (mol/kg)	Systems
Chabab et al. [59] *	2020	323.00	372.00	2.00	20.00	5.0	NaCl
Chabab et al. [152]. *	2023	298.15	373.15	10.00	20.00	4.0	NaCl
Wiebe and Gaddy [154]	1934	273.15	373.15	2.53	101.33	0.0	H2O
Crozier and Yamamoto [155] *	1974	274.60	302.47	0.10	0.10	0.0	H2O
Crozier and Yamamoto [155] *	1974	274.03	301.51	0.10	0.10	0.5	NaCl
Braun [156]	1900	278.15	298.15	1.022	1.045	0.0	H2O
Braun [156]	1900	278.15	298.15	1.022	1.045	1.1	NaCl
Torin–Oilarves and Trusler [157] *	2021	323.00	323.00	12.00	39.40	2.5	NaCl
Bunsen [158]	1855	277.15	296.75	0.10	0.10	0.0	H2O
Bohr and Bock [159]	1891	273.20	373.15	0.10	0.20	0.0	H2O
Winkler [160]	1891	273.65	323.25	0.10	0.11	0.0	H2O
Steiner [161] *	1894	288.20	288.20	0.10	0.10	0.0	H2O
Steiner [161] *	1894	286.32	286.95	0.10	0.10	5.3	NaCl
Steiner [161]	1894	291.77	292.38	0.10	0.10	4.0	KCl
Steiner [161]	1894	290.83	291.67	0.10	0.10	3.2	CaCl2
Steiner [161]	1894	291.56	291.72	0.10	0.10	1.4	Na2SO4
Steiner [161]	1894	290.25	291.41	0.10	0.10	2.6	MgSO4
Geffcken [162] *	1904	288.15	298.15	0.10	0.10	0.0	H2O
Knopp [163] *	1904	293.15	293.15	0.10	0.10	0.0	H2O
Knopp [163]	1904	293.15	293.15	0.10	0.10	2.1	KCl
Huefner [164] *	1907	293.15	293.34	0.10	0.10	0.0	H2O
Findlay and Shen [165]	1912	298.15	298.15	0.10	0.18	0.0	H2O
Muller [166]	1913	289.35	290.35	0.10	0.10	0.0	H2O
Ipatiew et al. [167]	1932	273.65	318.15	2.03	14.19	0.0	H2O
Morrison and Billett [168] *	1952	285.65	345.65	0.10	0.14	0.0	H2O
Pray et al. [169] *	1952	324.82	588.71	0.69	2.42	0.0	H2O
Zoss [170] *	1952	273.15	606.48	3.45	20.70	0.0	H2O
Stephan et al. [171]	1953	373.15	435.93	1.41	10.03	0.0	H2O
Wet [172] *	1964	291.65	304.55	0.10	0.11	0.0	H2O
Ruetschi and Amlie [173] *	1966	303.15	303.15	0.11	0.11	0.0	H2O
Shoor et al. [174]	1969	298.15	333.15	0.10	0.12	0.0	H2O
Longo et al. [175] *	1970	310.15	310.15	0.11	0.11	0.0	H2O
Power and Stegall [176]	1970	310.15	310.15	0.11	0.11	0.0	H2O
Gerecke and Bittrich [177] *	1971	298.15	298.15	0.10	0.10	0.0	H2O
Gerecke and Bittrich [177] *	1971	288.15	298.15	0.10	0.10	4.3	NaCl
Gerecke and Bittrich [177] *	1971	288.15	288.15	0.10	0.10	1.0	KCl
Jung et al. [178]*	1971	373.15	423.15	1.00	8.58	0.0	H2O
Schroder [179]	1973	298.15	373.15	10.13	10.13	0.0	H2O
Gordon et al. [180] *	1977	273.29	302.40	0.10	0.10	0.0	H2O
Devaney [181] *	1978	366.48	588.7	1.38	11.03	0.0	H2O
Cargill [182]	1978	277.70	344.83	0.10	0.14	0.0	H2O
Meyer et al. [183] *	1980	310.15	310.15	0.10	0.10	0.0	H2O
Gillespie and Wilson [184] *	1980	310.93	588.71	0.35	13.80	0.0	H2O
Choudhary et al. [185]	1982	323.15	373.15	2.53	10.13	0.0	H2O
Dohrn and Brunner [186]	1986	473.15	623.15	10.00	30.00	0.0	H2O
Alvarez et al. [187] *	1988	318.90	497.50	0.44	4.59	0.0	H2O
Kling and Maurer [188] *	1991	323.15	423.15	3.18	15.37	0.0	H2O
Jauregui–Hazaetal. [189]	2004	353.00	373.00	0.15	0.20	0.0	H2O

* Experimental H₂ solubility data used in the regression of this study model.

. Figure 3 and Figure 4 show experimental data as a function of temperature and pressure.

2.2.2. H₂ Solubility–Equation of State

In the last few decades, several researchers have relied on the EoS to estimate hydrogen solubility in a wide variety of brine solutions [41,42,43,44,45]. A reliable and accurate model for hydrogen solubility was developed by Li et al. [43]. This model calculated hydrogen solubility in formation fluids and tracked changes in fluid density. It accounted for factors like system pressure, temperature, fluid salinity, molar fraction, fugacity coefficient, Henry’s constant, Poynting factor, and the activity coefficient of hydrogen. The model performed well within typical geological storage conditions (273–373 K, 1–50 MPa, and 0–5 mol/kg NaCl), accurately reproducing experimental data and predicting hydrogen solubility and fluid density. Additionally, the model handled H₂–N₂ or H₂–CH₄ gas mixtures and mixed electrolyte solutions containing Na, K, Ca, Mg, Cl, or SO₄. Rahbari et al. [41] conducted thermodynamic modeling to investigate the phase coexistence of the H₂O–H₂ system at temperatures ranging from 283 K to 423 K and pressures between 10 bar and 1000 bar. The authors found that both the Peng–Robinson and Soave Redlich–Kwong equations of state, even with van der Waals mixing rules and fitted binary interaction parameters were unable to accurately predict the equilibrium compositions of the liquid and gas phases. In another study, Lopez–Lazaro [44] used Monte Carlo simulations to generate new data on hydrogen solubility in aqueous NaCl solutions across temperature and salinity ranges relevant to geological applications where experimental data are currently lacking. They fitted a binary interaction parameter for the Soreide and Whiston equation of state using the simulated data generated through molecular simulations. This model enables fast and reliable phase equilibrium calculations and was applied to scenarios pertinent to hydrogen geological storage and natural hydrogen emissions.

Sun et al. [45] applied the SAFT–LJ EoS, which incorporates a multipolar term to explicitly account for dipole–dipole, dipole–quadrupole and quadrupole–quadrupole interactions, to model the VLE of H₂O–H₂ systems. In their model, H₂ was treated as chain molecules with evenly distributed quadrupole moments. The van der Waals one–fluid mixing rule was used to calculate mixture parameters and evaluated two binary parameters for H₂O–H₂ interactions based on mutual solubility data from binary water–gas systems. When compared with experimental data, this molecular–based EoS accurately represented the VLE of H₂O–N₂ and H₂O–H₂ systems across a wide P–T range (273–623 K and 0–1000 bar). The authors concluded that this ability to account for multipolar interactions is why the SAFT–LJ EoS can quantitatively represent the VLE of highly non–ideal water–gas systems. Ratnakar et al. [42] developed a PVT model for the H₂–H₂O system using the traditional PR EoS along with a non–classical Huron–Vidal (HV) mixing rule. Their model effectively captures molecular interactions and aligns well with available experimental data. They also established a robust workflow for determining HV parameters from binary solubility data, which was validated against experimental observations.

2.2.3. H₂ Solubility–Empirical Correlations

Various researchers developed correlations to predict the hydrogen Henry constants in pure water, comparing experimental data. The pioneered work by Harvey [190] was later refined by Fernández–Prini [191]. Trinh et al.’s developed model for estimating hydrogen Henry’s constant in oxygenated solvents [192] and adjusted by Lopez–Lazaro et al. [44] for water. Akinfiev and Diamond’s utilize a virial–like equation of state to describe the properties of aqueous nonelectrolytes at infinite dilution [193]. Li et al. [43] and Torín–Ollarves and Trusler [157] empirical models were calibrated using high–pressure H₂ solubility data in water. Chabab et al. [59] presented an empirical correlation to predict H₂ solubility in pure H₂O and brine. The correlation considers the effects of temperature, pressure, and molal concentration (mol/kg of NaCl). Although Chabab et al.’s correlation provides a useful tool for predicting hydrogen solubility in pure water and NaCl–brine systems, its limitations make it less applicable to more complex or mixed brine environments commonly encountered in natural hydrogen production and UHS scenarios. The correlation does not account for these interactions, which can lead to inaccuracies when applied to systems that are not dominated solely by NaCl. Zhu et al. [54] used a semi–empirical approach to model H₂ solubility in purified water and aqueous NaCl solutions by utilizing a high–precision equation of states and a specific particle interaction theory for the liquid and vapor phases of H₂, respectively. Although the model effectively predicts H₂ solubility in pure water and NaCl solutions, its application is limited by its calibration range, which has a maximum temperature of 373.15 K (100 °C). This limitation restricts its use in high–temperature environments like natural hydrogen production, where temperatures often exceed this range.

2.2.4. H₂ Solubility–Molecular Dynamics Studies

Molecular dynamics studies are increasingly being used to model the solubility behavior of fluid systems, particularly for H₂ in brine, due to the complexities associated with its very low solubility, as well as the high costs and hazards posed by experimental measurements under extreme conditions (e.g., high pressure, flammability of H₂, and the corrosive nature of salt). While these simulations are invaluable, experimental measurements remain essential for validating and refining models used in classical and statistical thermodynamics. Additionally, molecular simulations complement experimental work by offering deeper insights into the underlying physical mechanisms, generating vapor–liquid equilibrium data under severe conditions (such as high temperature, pressure, and salinity), and aiding in the development of consistent and reliable models for predicting solubility.

MDS application to H₂ solubility in pure water and brine is relatively recent and limited, likely due to the low solubility of H₂, which poses significant statistical challenges [44,61]. Rahbari et al. [41] used molecular simulations to study the phase coexistence of the H₂O–H₂ system across temperatures from 283 K to 423 K and pressures between 10 and 1000 bar. They found that force–field–based molecular simulations could predict the solubility of water in compressed hydrogen. Lopez–Lazaro et al. [44] applied this indirect method to calculate Henry’s constant for H₂ in brine and used it to calibrate the Søreide and Whitson equation of state for predicting solubilities at high pressures. Recently, Kerkache et al. [60] employed both direct and indirect molecular dynamic approaches to calculate H₂ solubility in water and brine, demonstrating the applicability of these approaches under various conditions. Similarly, van Rooijen et al. [61] used this method to estimate H₂ solubility at pressures up to 10 MPa, noting that Henry’s law requires correction beyond this pressure. In NaCl brine, a significant lack of experimental data has resulted in few recent molecular dynamic studies [44,60,61] on high–pressure H₂ solubility. These sources show notable discrepancies, particularly in the salt–free molar fraction (x’H₂). This inconsistency highlights the variations between the experimental and simulation results, underscoring the need for further investigation to reconcile these differences. Although molecular dynamic techniques are powerful tools, they are computationally intensive and necessitate additional parameters that are often challenging to obtain. This complexity increases significantly when transitioning from simpler systems like pure water to more complex ones involving mixed salts, making the modeling process even more resource demanding and less accessible for practical applications.

2.2.5. H₂ Solubility–Machine Learning Studies

Machine learning offers a robust approach to predicting hydrogen solubility in aqueous solutions, especially underground hydrogen storage conditions. Unlike traditional models that rely on simplified equations with limited variables like temperature and pressure [194], machine learning can incorporate a wider range of input variables that includes the chemical properties of other solutes [195]. This broader analysis significantly improves the accuracy and reliability of predictions. Recent studies have successfully applied machine learning techniques to predict hydrogen solubility in various mediums [196,197,198], demonstrating their effectiveness in complex systems where conventional models may fall short.

Various machine learning algorithms have been widely used to predict hydrogen solubility. Lv et al. [71] employed advanced models including adaptive boosting decision tree (AdaBoost–DT), adaptive boosting support vector regression (AdaBoost–SVR), gradient boosting decision tree (GB–DT), gradient boosting support vector regression (GB–SVR), and k–nearest neighbors (KNN) to estimate H₂ solubility in both pure and saline water. Similarly, Zhu et al. [66] used a wavelet neural network that combines ANN with wavelet transform to model hydrogen storage in saline water under real–world conditions. Vo Thanh et al. [72] also investigated hydrogen solubility in aqueous systems using various machine learning models such as adaptive gradient boosting (AdaBoost), gradient boosting, random forest, and extreme gradient boosting, confirming their effectiveness. Recently, various authors [199,200,201,202] have contributed to the machine learning modeling of H₂ solubility in water and saline systems.

The application of machine learning to predict hydrogen solubility in aqueous solutions marks a significant advancement over traditional methods; however, the current approaches are constrained by the major limitations of the lack of an extensive experimental database. To improve the accuracy and reliability of these predictions, future research should focus on expanding the datasets to cover a wider range of conditions relevant to both underground hydrogen storage and natural hydrogen production. Additionally, integrating more comprehensive data could help mitigate the risks of overfitting and enhance the generalizability of these models to diverse real–world scenarios.

3. Methodology

Here, we present a detailed analysis of the CO₂ and H₂ solubility data, encompassing various stages from data collection and characterization to model development and performance evaluation. We begin by discussing the methodologies employed in gathering and processing the data, which form the foundation for subsequent empirical model development. The focus then shifts to the formulation and development of predictive correlations fitted from the experimental data, specifically tailored for CO₂ and H₂ solubility under varying conditions. Finally, we assess the accuracy and reliability of these models through statistical evaluation, providing insights into their predictive capabilities and limitations.

3.1. Data Collection and Characterization

To develop a robust and quick predictive model that can accurately predict CO₂ and H₂ solubility in aqueous systems for subsurface applications, reliable and consistent database is essential. In light of this, we collected experimental data points for CO₂ and H₂ solubility from published studies (see Table 1 and Table 2). Because most experimental data are published in different measurement units, a conversion was conducted for all measured data to establish a consistent data unit system. All pressure data was converted to megapascals (MPa), temperature to Kelvin (K), salinity to molality (mol/kg) and solubility to moles per kilogram of water (mol/kg). These datasets cover diverse aqueous systems with three main input variables, including temperature, pressure and salinity (mol/kg). The objective of the section was to gather consistent experimental data and develop swift and accurate correlations to predict CO₂ and H₂ solubility as a function of pressure, temperature and salinity in terms of ionic strength. They are formulated as follows:

$$Solubility=f(P, T, and IS)$$

(1)

Furthermore, the salinity of all experimental data was converted into ionic strength (IS), expressed in mol/kg, using the Debye–Hückel [203] model expressed by Equation (2). The Debye–Hückel model presents an easy approach to determining ionic strength. However, its limitation is that it does not consider the impact of ion interaction.

$$IS={\displaystyle \frac{1}{2}}\sum m_i{z}_i^2$$

(2)

where $m_i$ and $z_i$ represent the molality (mol/kg) and charge of the ion, respectively.

In the context of the experimental measurement unit, Equation (3) can be used to convert the molality and weight fraction of ions to total dissolved salt (TDS). Alternatively, TDS can be calculated from the weight fraction of the ion using Equation (4).

$$TDS=m_{ion}\times M_{ion}\times 1000$$

(3)

$$TDS={\displaystyle \frac{\left(\frac{{wt\%}_{ion}}{M{M}_{ion}}\times 100\right)}{\left(100-{wt\%}_{ion}\right)\times {10}^3\times M_{ion}}}$$

(4)

where $wt{\%}_{ion}$ = weight fraction of ion, $M_{ion}=$ molar mass of ion, and $m_{ion}=$ mole concentration of ion.

Because most industrial applications measure brine salinity in terms of TDS, mostly in parts per million (ppm), it is essential to capture this rationale in our empirical correlation. Electrical conductivity, salinity, and TDS measurements are often used to quantify the ionic strength of water. These measurements typically show good correlation with the concentration of ions. Therefore, we introduce Langelier et al. [204] correlation to estimate ionic strength based on TDS. The empirical correlation (Equation (5)) with the assumption that the relative composition of natural waters is fairly constant. The correlation is expressed as:

$$IS=2.5\times {10}^{-5}\times TDS$$

(5)

Furthermore, these equations were employed to ensure accurate conversions between salinity, ionic strength, and TDS, facilitating a comprehensive analysis of the experimental data.

Next, we collected and compiled CO₂ solubility data. The dataset includes solubility measurements under a wide range of conditions, with maximum values reaching 71 MPa for pressure, 523 K for temperature, and 6.14 mol/kg for salinity. Table 1 outlines the specific experimental conditions under which the CO₂ solubility data was collected. In this study, the oriented temperature and pressure range is 273.15–523.15 °K and 0–71 MPa, respectively and across all salinity levels in which the CO₂–rich and water–rich phases are commonly immiscible. The chosen pressure range covers the domain of applicability for CO₂–EOR and geologic carbon storage. From Figure 3 it is evident that most data points are concentrated between temperatures of 280 K to 350 K and pressures ranging from 0 to 30 MPa. There is a notable concentration of data within the lower temperature (below 350 K) and low–to–medium pressure ranges (0 to 20 MPa), particularly for the CO₂–H₂O–salt system. These temperatures and pressure conditions are typical of CO₂–EOR studies. However, as temperatures rise above 350 K and pressures exceed 30 MPa, the data points become sparse, indicating less experimental coverage in these regions. This suggests that the region between 350 K to 500 K and 20 MPa to 70 MPa lacks sufficient data, which may be critical for understanding CO₂ solubility under conditions of injection into deep saline aquifer. This gap highlights the need for additional experimental studies to extend the coverage to more extreme conditions, thereby improving the reliability of CO₂ solubility predictions for these important applications.

Similarly, for H₂ solubility data, available published data were collected and compiled into a databank. H₂ solubility data were collected. The dataset includes solubility measurements under a wide range of conditions, with maximum values reaching 101.3 MPa for pressure, 636.1 K for temperature, and 5.0 mol/kg for salinity. Notably, most of the available solubility data is from pure water systems. For more detailed information, please refer to Table 2, which outlines the specific experimental conditions under which the H₂ solubility data was collected. Finally, a total number of 438 experimental data records were filtered and selected from our preliminary database. The final dataset comprises solubility data in water and NaCl brine. From Figure 4, it is evident that most data points are concentrated within the temperature range of 250 K to 400 K and pressure range of 0 to 40 MPa. There is a noticeable clustering of data in the lower to medium temperature range, particularly between 250 K and 350 K, and pressures between 0 and 20 MPa. This suggests that the data coverage is quite robust in these regions, which may be relevant for standard conditions encountered in underground hydrogen storage applications. However, there is a significant lack of data at higher temperatures (above 400 K) and higher pressures (above 40 MPa), with very few data points extending up to 640 K and 100 MPa. This indicates that the data coverage is sparse in these high–temperature and high–pressure regions, which are critical for understanding H₂ solubility under extreme conditions like those encountered in natural hydrogen production. The limited data in these regions suggests that more experimental studies are needed to explore and accurately model H₂ solubility under such conditions. Details of the final datasets for CO₂ and H₂ solubility that were used for CO₂ and H₂ correlation development are provided in Table 1 and Table 2.

Although other contaminants do not significantly affect solubility, Na and Cl are often the primary species taken into account when analyzing brines in aquifers [59,152] Commonly for thermodynamic characteristics, most subsurface brines are treated as NaCl brine, and the other species are either ignored or converted to NaCl equivalents [205]. Chemical reactivity, however, requires considering actual existent species rather than assimilating them to NaCl.

3.2. Development of CO₂ Solubility Correlations

Various experimental methods were used to obtain these measurements, including water-trapping [81,87,111], gas chromatography [107,124,129], volumetric expansion [116,135], synthetic methods. [94,108], Raman spectroscopy [206,207], calorimetric [110], and potentiometric techniques [91]. Overall, most of the data is consistent across a broad range of temperatures and pressures. The reliability and consistency of these measurements have been extensively analyzed in previous studies [49,53,141] which is beyond the scope of this work. Table 1 provides details of the databank for the CO₂ + water system.

An accurate model of CO₂ solubility should not only reflect the overall trends across a wide range of temperatures and pressures but also account for abrupt local changes under specific thermodynamic conditions. Traditional models often address this by using different parameters for various CO₂ phases or temperature–pressure ranges [49]. However, this approach can disrupt the smoothness and continuity of the function’s derivative, potentially causing issues like an ill–conditioned Jacobian matrix in large–scale multiphase flow simulations [208,209]. A simple model is proposed to maintain continuity in both the function and its gradient. Herein, the focus is on a temperature range of 273.15–523.15 °K and a pressure range of 0–71 MPa, where the CO₂–rich and water–rich phases are generally immiscible. Due to the strongly non–ideal nature of the CO₂ and water mixture, the composition of different phases varies in a complex manner with changes in pressure and temperature [49,118,128].

In this section, we use our compiled and filtered data to develop mathematical correlations to predict CO₂ solubility in water and various brine systems. These empirical correlations consider temperature and pressure for a CO₂–water system and temperature, pressure, and salinity in terms of ionic strength for CO₂+water+salt systems. The developed correlations were adapted and formulated from the work of Chen et al. [210]. Due to the complex behavior of CO₂ solubility in different aqueous systems, we further separated the entire data into various subsets, namely pure water, mixed salt, monovalent, and divalent single salt systems.

Table 3. Description of the experimental data used for developing the CO₂ and H₂ solubility models.

Correlation	Systems	Data Points	T Min (K)	T Max (K)	P Min (MPa)	P Max (MPa)	IS Min mol/kg)	IS Max (mol/kg)
CO2–Water	—	926	274.14	523.15	0.10	71	–	–
CO2–Water– Salt	Mixed Salts (K+, Na+, Ca2+, Mg2+)	391	291.15	424.67	0.10	40.0	0.024	6.00
	KCl	260	313.1	433.1	0.13	18.22	0.427	4.50
	NaCl	766	273.15	523.15	0.10	40.0	0.017	6.00
	Na2SO4	226	286.97	433.16	0.42	15.0	0.300	8.16
	NaHCO3	75	313	398.15	0.31	50.0	0.050	1.00
	CaCl2	355	298	424.64	0.10	67.4	0.027	15.63
	MgCl2	223	298	424.68	0.10	34.9	0.031	15
H2–Water	—	360	273.15	636.1	0.629	101.35	–	–
H2–Water– Salt	NaCl	78	298.05	423.155	1.9884	45.81	1.00	5.00

provides a summary of the dataset used to develop correlations between CO₂–water and CO₂–water–salt systems.

The final mathematical model that describes the quantitative effects of temperature, pressure, and salinity on the solubility of CO₂ for water and different saline systems is expressed in the following equation forms:

For CO₂ − H₂O system:

$$mC{O}_2=\left(1-\epsilon \right) {\displaystyle \frac{A. {P_r}^{a1}{ T }^{a2}}{B.{{ P}_r}^{a3}{ T }^{a4}+C.{P_r}^{a5}{ T }^{a6}+D}}+\epsilon {\displaystyle \frac{E. {P_r}^{a7}{ T }^{a8}}{F.{{ P}_r}^{a9}{ T }^{a10}+G.{P_r}^{a11}{ T }^{a12}+H}}$$

(6)

$$\epsilon ={\displaystyle \frac{T-T_{min} }{T_{max}-T_{min}}}$$

(7)

$\mathsf{\epsilon }$ = weighting factor, $T_{min}=273.15 \mathrm{K}$ and $T_{max}=523.15 \mathrm{K}$

$$P_r={\displaystyle \frac{P }{P_0}}$$

(8)

$P_r$ = relative pressure, P₀ is the pressure at which the variation trend of CO₂ solubility undergoes an abrupt change, MPa. In this study, the expression of P₀ was derived from Sun et al. [57] and it is expressed as:

$$P_0=16.2086-{\displaystyle \frac{12.1147 }{1+exp(0.049635\times (T-273.15)-2.8034)}}$$

(9)

Regarding the aqueous phase containing either one or a combination of Na⁺, K⁺, Ca²⁺, Mg²⁺, Cl⁻ HCO₃⁻ and SO₄²⁻, a simple model for CO₂ solubility in salt solution is proposed:

For CO₂ − H₂O salts system:

$$mC{O}_{2, salt}={mC{O}_2}^0\mathit{exp}(a_{1}IS+a_{2}I{S}^{a3})$$

(10)

where $P, T, IS, {m{CO}_2}^0 \mathrm{a}\mathrm{n}\mathrm{d} {mCO}_{2, salt}$ represent the pressure (MPa), temperature (K), ionic strength (mol/kg), solubility of CO₂ in pure water (mol/kg) and solubility of CO₂ in salt solution (mol/kg), respectively. Also, A, B, C, D, E, F, G, H and a₁–a₁₂ are constant fitted by experimental measurements of CO₂ solubility, respectively.

3.3. Development of H₂ Solubility Correlations

Next, similar to the previous section, we utilized processed hydrogen solubility data to develop mathematical models to predict H₂ solubility in water and various brine systems. The empirical correlations consider temperature and pressure for an H₂–water system and temperature, pressure, and salinity in terms of ionic strength for H₂—water—salt systems. The developed correlations are formulated as multivariable nonlinear regression models. Table 3 provides a summary of the dataset used to develop correlations for H₂–water and H₂–water–salt systems.

The final mathematical model that describes the quantitative effects of temperature, pressure, and salinity on the solubility of H₂ for water and different saline systems is expressed in the following equation forms:

For H₂ − H₂O system:

$$m{H}_2=\left(1-\epsilon \right) {\displaystyle \frac{A. {P_r}^{a1}{ T }^{a2}}{B.{{ P}_r}^{a3}{ T }^{a4}+C.{P_r}^{a5}{ T }^{a6}+D}}+\epsilon {\displaystyle \frac{E. {P_r}^{a7}{ T }^{a8}}{F.{{ P}_r}^{a9}{ T }^{a10}+G.{P_r}^{a11}{ T }^{a12}+H}}$$

(11)

$$\epsilon ={\displaystyle \frac{T-T_{min}}{T_{max}-T_{min}}}$$

(12)

where $T_{min}=273.15 \mathrm{K}$ and $T_{max}=636.1 \mathrm{K}$

$$P_r={\displaystyle \frac{P }{P_0}}$$

(13)

Regarding the aqueous phase containing Na⁺ and Cl⁻, a simplified model for H₂ solubility in brine is proposed:

For H₂ − H₂O − salts system:

$$m{H}_{2, salt}={m{H}_2}^0\mathit{exp}(a_{1}IS+a_{2}I{S}^{a3})$$

(14)

where $P, T, IS, {m{H}_2}^0 \mathrm{a}\mathrm{n}\mathrm{d} {mH}_{2, salt}$ represent the pressure (MPa), temperature (K), ionic strength (mol/kg), solubility of H₂ in pure water (mol/kg) and solubility of H₂ in salt solution (mol/kg), where A, B, C, D, E, F, G, H and a₁–a₁₂ are constants, fitted by experimental measurements of H₂ solubility, respectively.

3.4. Model Parameter Determination

Experimental measurements are crucial for determining model parameters in phase equilibria models, as the quality of this data directly impacts the model’s accuracy [57,118]. For CO₂ systems, an updated databank has been developed, which includes 923 measurements for the CO₂–water system and 1565 measurements for the CO₂+water+salts system. In the case of the H₂ system, the databank contains 438 measurements of H₂ solubility, 360 for the H₂–water system and 78 for the H₂ + water + salt systems. The Generalized Reduced Gradient algorithm was employed to determine the model parameters using the developed experimental database [211]. Due to the uneven distribution of experimental data across the targeted temperature and pressure range, the parameters were fitted exclusively with the available data. The resulting model parameters for CO₂ and H₂ solubility models are presented in

Table 4. Empirical constants of developed CO₂ and H₂ solubility models.

Constants	CO2 Solubility Correlations								H2 Solubility Correlations
	CO2–H2O	Mixed Salts	NaCl	KCl	CaCl2	MgCl2	Na2SO4	NaHCO3	H2–Water	H2–Water–Salts (NaCl)
A	0.284888	–	–	–	–	–	–	–	0.101466	–
B	−5.02511	–	–	–	–	–	–	–	−5.632826	–
C	4.094051	–	–	–	–	–	–	–	3.732906	–
D	0.507286	–	–	–	–	–	–	–	−0.113223	–
E	0.006187	–	–	–	–	–	–	–	0.543337	–
F	−4.164112	–	–	–	–	–	–	–	−4.379279	–
G	4.939346	–	–	–	–	–	–	–	4.570177	–
H	0.340918	–	–	–	–	–	–	–	0.136001	–
a1	0.756798	0.52944	0.26827	0.287342	1.008286	1.801932	−0.11701	1.565179	1.036691	−0.180909
a2	−0.328316	−0.72297	−0.49775	−0.43852	−1.16212	−1.94698	−0.20067	−1.69733	−0.731073	−0.066281
a3	0.144697	0.998793	0.922111	0.926434	0.987298	0.99382	0.283081	0.961564	−0.003084	−7.126735
a4	−0.182119	–	–	–	–	–	–	–	−0.069526	–
a5	0.208901	–	–	–	–	–	–	–	−0.001675	–
a6	−0.200669	–	–	–	–	–	–	–	0.010505	–
a7	0.573537	–	–	–	–	–	–	–	0.60633	–
a8	−0.097774	–	–	–	–	–	–	–	−0.429898	–
a9	0.043382	–	–	–	–	–	–	–	−0.187142	–
a10	−0.205101	–	–	–	–	–	–	–	0.110585	–
a11	0.059729	–	–	–	–	–	–	–	−0.192603	–
a12	−0.287825	–	–	–	–	–	–	–	0.102827	–

showing the empirical constants for the CO₂ solubility models and H₂ solubility models.

4. Results and Discussions

4.1. Validation of the Developed CO₂ Solubility Correlations

The prediction performance of the developed correlations is validated against experimentally measured data using statistical analysis and graphical plots (isotherms curves and response surface plots). The different CO₂ solubility systems studied namely are pure water, mixed salts, and seawater with ions (Na⁺, K⁺, Ca²⁺, Mg²⁺, Cl⁻, HCO₃⁻, SO₄²⁻), monovalent salt (KCl, NaCl, NaHCO₃ and Na₂SO₄) and divalent salts (CaCl₂ and MgCl₂). The NaCl and KCl salts were separated because, at the same concentration, the solubility of CO₂ is generally greater in KCl solutions than in NaCl solutions. This is a result of the ionic interactions between the CO₂ molecules and the ions in the solutions. In contrast to Na⁺ ions in NaCl, the K⁺ ions in KCl have a larger ionic radius, which leads to a less structured water network and reduced hydration. This less structured water network enables a greater amount of CO₂ to dissolve in the solution [102,212]. Conversely, Na⁺ ions exert a significant influence on the structure and hydration of the water molecules in their vicinity, thereby diminishing the amount of space available for the dissolution of CO₂ molecules [102,212,213].

4.1.1. CO₂ + H₂O Systems

The developed model shows strong agreement with experimental data across various isotherms. The model captures the general trend of increasing solubility, as seen in Figure 5, with rising pressure and saturation behavior at higher pressures, which is consistent with observations in the Duan and Sun [52]. Figure 5 shows the comparison of the simulated results and measured data on CO₂ solubility in the pure water system. Overall, there is a good agreement between the published experimental data and our computed data. As can be seen, the solubility of CO₂ rises gradually in a non–linear manner. A clear turning point in the upward trend occurs when the phase with a high concentration of CO₂ transitions from the gaseous phase to the liquid or supercritical phase. The suggested model can precisely quantify the impact of phase transition on the solubility of CO₂. Nevertheless, there is still some ambiguity, particularly at some temperatures such as 298 and 353 °K, due to substantial inconsistencies across the available literature sources. The inconsistencies mostly arose from variations in experimental principles, techniques, and measurement precision [193,214]. Given that other scholars [50,52,141] have extensively examined the dependability of phase partitioning data, and doing a rigorous review of that topic is not within the scope of this article.

In Figure 6, the 3D response surface highlights the non–linear relationship between temperature, pressure, and CO₂ solubility. Notably, the surface plot also serves as a phase diagram. The plot demonstrates the obvious effect of pressure on solubility, with solubility increasing steeply at lower temperatures. This behavior aligns with the work of Duan and Sun [52], who reported similar trends in CO₂ solubility in pure water. There exists a high CO₂ solubility region around pressure greater than 5 MPa and temperature less than 300 °K. This high–value region corresponds to the liquid phase of the non–supercritical region based on the temperature and pressure condition of the CO₂ phase diagram. Wiebe and Gaddy [79] conducted early experiments that demonstrated the inverse relationship between temperature and CO₂ solubility. This inverse relationship has been supported by more recent studies, such as those by Sander [215], who reviewed gas solubility data and confirmed that colder temperatures enhance, as seen in Figure 6, solubility due to reduced kinetic energy and enhanced molecular interactions. The surface also suggests that at higher temperatures, the effect of pressure on solubility diminishes slightly, a phenomenon also observed in previous studies by Spycher et al. [49]. Carroll et al. [216] discussed how the exothermic nature of dissolution results in greater solubility at lower temperatures. This is because the lower kinetic energy of molecules in cold water allows for more CO₂ to be absorbed before temperature changes shift the equilibrium.

4.1.2. CO₂–Water–Salts

The predictive capability of the developed model to predict CO₂ solubility in water (CO₂–H₂O– salt systems) as a function of temperature, pressure, and ionic strength. The developed model strongly agrees with experimental data across various isotherms presented in the following subsections. The subsections present the isotherm curves of mixed salts (Na⁺, K⁺, Ca²⁺, Mg²⁺, Cl⁻, HCO₃⁻, SO₄²⁻), monovalent salt (KCl), monovalent salt (NaCl), monovalent salts (NaCl, NaHCO_3, and Na₂SO₄) and divalent salts (CaCl₂ and MgCl₂) system respectively. The isotherm curves presented the predictions of the developed model against experimental data at minimum and maximum ionic strength values for the various systems.

The developed models capture the general trend of increasing CO₂ solubility with rising pressure and the saturation behavior at higher pressures, consistent with observations in the Duan et al. [53]. Similarly, the developed models follow the inverse relationship between temperature and CO₂ solubility. The three–dimensional (3D) surface response was used to illustrate the impact of salinity variation on the solubility of CO₂ in water. The 3D plots depicted in the sections are presented as a function of two factors. In contrast, the third factor was kept at three distinct salinities corresponding to different ionic strength levels. The amount of dissolved CO₂ in water is depicted as a function of the aqueous systems’ pressure (MPa) and temperature (K).

Mixed Salts

Mixed salt systems are the commonly encountered salinity in geological engineering applications. The CO₂ solubility in the aqueous phase of mixed salt systems is analyzed in Figure 7, Figure 8, Figure 9 and Figure 10. The CO₂ solubility is generally less than 1.5 mol/kg at low ionic strength value, with a good agreement between published experimental and model data. Compared to pressure, CO₂ solubility is less sensitive to temperature in mixed salt systems. As shown in Figure 7, CO₂ solubility decreases from about 1.1 mol/kg to 0.8 mol/kg when the temperature increases from 308.15 °K 328.15 °K at 10 MPa. The CO₂ solubility increases gradually as pressure increases, but the variation rate decreases. This same trend of an inverse relationship between temperature and CO₂ solubility was observed in Figure 8, Figure 9 and Figure 10 with ionic strengths of 1.80, 2.743 and 6.0 mol/kg, respectively. Furthermore, Figure 11 illustrates the salting–out effect on a 3D response surface plot. This effect results from the interaction between the CO₂ molecules, water, and ions, responsible for decreased CO₂ solubility in a saline system as salinity increases.

Finally, as seen in Figure 5 through to Figure 11, the developed model’s ability to accurately predict CO₂ solubility under various conditions validates its application for sequestration and enhanced oil recovery (EOR) scenarios, where such solubility data is crucial. The consistency of the model with empirical data supports its robustness and reliability for simulating behavior in subsurface environments, which is essential for predicting the performance injection in geological formations. For brevity, details of the CO₂ solubility validation results for the singles salts are presented in the Supplementary Material.

4.2. Validation of the Developed H₂ Solubility Correlations

Here, we present the outcome of the developed correlations for H₂ solubility in water and saline systems composed of NaCl and seawater. The performance of developed correlations is validated against experimentally measured data and presented as isotherms curves and response surface plots.

4.2.1. H₂–H₂O

Figure 12 shows the isotherms of H₂ solubility at varying pressures. The model exhibits excellent prediction performance of the experimentally measured data from low and higher temperatures. The developed model shows strong agreement with experimentally measured data across various isotherms. The model captures the typical relationship of increasing H₂ solubility with rising pressure and the saturation behavior at higher pressures, as seen in Figure 12, which is consistent with observations from Zhu et al. [54]. Further, the developed correlation is reliable for modeling the quadratic relationship between temperature and H₂ solubility. As observed in Figure 12, at lower temperatures (273.15 K), H₂ solubility is higher compared to medium temperatures (323.15 K and 373.15 K) at equivalent pressures. However, at 323.15 K, the solubility is lower than at 373.15 K, affirming the quadratic relationship between H₂ solubility and temperature. Thus, H₂ solubility initially decreases as the temperature rises from low to medium levels, then begins to increase as the temperature shifts from medium to high values. According to Lopez–Lazaro [44], the Henry characteristic curve peaks at a temperature of approximately 330 K, which is about 57 °C. This means that hydrogen dissolution is lowest at this temperature. This outcome generally explains how small molecule dissolves in pure water. Also, the temperature at which the minimal solubility occurs depends on the presence and properties of the solute. This pattern has also been noted in the research by Zhu et al. [54]

Figure 13 shows the 3D response surface that highlights the non–linear relationship between temperature, pressure, and H₂ solubility. Notably, the surface plot demonstrates the effects of pressure and temperature on solubility beyond the physical condition of available experimental data. There exists a high H₂ solubility region around pressure greater than 50 MPa and temperature less than 500 °K. This high–value region corresponds to the fluid phase region based on the temperature and pressure condition of the H₂ phase diagram. For instance, the 3D response surface plot reveals high H₂ solubility values up to 5 mol/kg at high temperature and pressure regions. The solubility results of this high pressure and temperature region are essential to natural hydrogen production processes. This region requires experimentally measured data to validate the prediction of the developed models.

4.2.2. H₂ + H₂O + Salts (NaCl)

The predictive capability of the developed model is to predict H₂ solubility in water (H₂–H₂O– salt systems) as a function of temperature, pressure, and ionic strength. The developed model shows strong agreement with experimental data across various isotherms, as shown in Figure 14, Figure 15 and Figure 16. Figure 14, Figure 15 and Figure 16 represents the isotherm curves of the salt (NaCl) system at ionic strengths of 1.0, 2.0, and 5.0 mol/kg. The isotherm curves presented show the predictions of the developed model against experimental data at the minimum and maximum ionic strength values for the studied system. The developed models capture the general trend of decreasing H₂ solubility in aqueous solutions, which is consistent with observations by Zhu et al. [54]. Similarly, the developed models also follow the inverse relationship between temperature and H₂ solubility.

Figure 17 which describes the 3D surface response (phase diagram) was used to illustrate the impact of salinity variation on the solubility of H₂ in water. The plots depicted in the sections are presented as a function of two varying factors, while the third factor was maintained at three distinct salinity ionic strength levels. The amount of dissolved H₂ in water is depicted in Figure 17 as a function of the aqueous systems’ pressure (MPa) and temperature (K). The salinity is maintained at a constant level by using these four ionic strength values (1.0, 2.0, 3.0 and 4.0 mol/kg).

The salting–out effect, which is the result of the interaction between the H₂ molecules, water, and ions, is responsible for the decrease in H₂ solubility in a saline system as the salinity increases. The majority of water molecules are surrounded by the cations/anions that dissociate from dissolved salts, and only a small number of water molecules are capable of interacting with H₂ [54,59,152]. Consequently, it is hypothesized that the solubility of H₂ is reduced due to the interaction between saline water and H₂ during H₂ during the injection and production processes. In the same vein, the reference [54], reported that the solubility of H₂ increases linearly with pressure. The H₂ molecules are compelled to sink into the dense water phase and become adsorbed as a result of the increased pressure exerted on them.

Furthermore, in Figure 12 through to Figure 17, the developed model’s ability to accurately predict H₂ solubility under various conditions validates its application for underground hydrogen storage and natural hydrogen production scenarios, where such H₂ solubility data is critical. The consistency of the model with empirical data supports its robustness and reliability for simulating behavior in subsurface environments, which is essential for predicting the performance injection in geological formations.

4.3. Salting–Out Effect

The salting–out effect, which is the result of the interaction between the CO₂ molecules, water, and ions, is responsible for the decrease in CO₂ solubility in a saline system as the salinity increases. The majority of water molecules are surrounded by the cations/anions that dissociate from dissolved salts, and only a small number of water molecules are capable of interacting with CO₂ [49,53]. Consequently, it is hypothesized that the solubility of CO₂ is reduced because of the interaction between saline water and CO₂ during the injection procedure. In the same vein, Duan et al. [53] reported that the solubility of CO₂ increases linearly with pressure. The CO₂ molecules are compelled to sink into the dense water phase and become adsorbed due to increased pressure exerted on them. This increase in solubility continues until a pressure point is reached where the solubility value plateaus.

Kaur et al. [217] thermodynamically studied the solubility of CO₂ in different brine systems and reported the salting–out effect of different electrolyte solutions expressed in terms of their ionic strength. Figure 18 shows the salting–out effect of various salts performed in this study. The decreasing trend of CO₂ solubility observed follows the order NaHCO₃ > KCl > CaCl₂ ∼ MgCl₂ > NaCl > Na₂SO₄, and it is consistent with those reported by Zhao et al. [135] on the basis of ionic strength.

Similarly for hydrogen gas solubility in water, Setschenow’s constant [18] or the percentage reduction in solubility, $SO=100\left({\displaystyle \frac{{{mH}_2}^{water}-{{mH}_2}^{brine}}{{{mH}_2}^{water}}}\right)$ under the same thermodynamic factors of temperature, pressure, and salinity [110] are usually used to determine the salting–out (SO) impact on gas solubility for hydrogen. In this work, we employed the SO method to quantify the SO impact on the solubility of H₂ in NaCl solution.

Figure 19 shows the salting–out influence of H₂ on the NaCl solution modeled in the present study. The calculated SO values for H₂ solubility in NaCl brine from this work are shown in

Table 5. The reduction in solubility due to H₂ gas in NaCl brine (Salting–out effect).

	NaCl Molality (mol/kg)
SO (%) at 323.15 K	1	4
Chabab et al. [59]	24.28	60.09
Torín–Ollarves and Trusler r [157]	13.10	42.98
Chabab et al. [152]	16.23	50.70
This study	17.60	53.67

, and the comparison is made with three other published studies. Notably, with the exception of Torín–Ollarves and Trusler’s [157] findings, the results of this study, and two previous research studies [59,152] demonstrate comparable salting–out behavior for H₂. Despite being carried out at low pressures (because pressure only little influences the salting–out effect), the results are consistent with previous work on other gases using Setschenow’s constant [218]. Though solubility is affected by a number of parameters [219], the variance in salting–out behavior is most likely caused by molecule size variations, a trend seen in noble gases (Ballentine et al. [18]). Thus, molecular simulations could offer valuable insight into the distinctive behavior of hydrogen in aqueous solutions in comparison to other gases.

4.4. Evaluation of Developed Solubility Models

The following statistical evaluation metrics were used to assess the performance of the developed models: coefficient of correlation (R²), average percent deviation (APE), and average absolute percent deviation (AAPE). The mathematical expressions for these criteria are provided in Equations (15)–(17).

$$R^2=1-{\displaystyle \frac{{\sum }_{i=0}^n{\left(X_{i, exp}-X_{i, pred}\right)}^2}{{\sum }_{i=0}^n{\left(X_{i, exp}-\overline{X}\right)}^2}}$$

(15)

$$APE={\displaystyle \frac{1}{n}}\sum _{i=0}^n\left({\displaystyle \frac{X_{i, exp}-X_{i, pred}}{X_{i, exp}}}\ast 100\right)$$

(16)

$$AAPE={\displaystyle \frac{1}{n}}\sum _{i=0}^n\left|{\displaystyle \frac{X_{i, exp}-X_{i, pred}}{X_{i, exp}}}\ast 100\right|$$

(17)

In these equations, ${\mathrm{X}}_{\mathrm{i},\mathrm{e}\mathrm{x}\mathrm{p}}$ represent experimentally measured solubility data, ${\mathrm{X}}_{\mathrm{i},\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{d}}$ refers to the solubility values predicted by the model and $\overline{\mathrm{X}}$, denotes the average value of the experimentally obtained solubility dataset. The predictability of the developed correlations was evaluated using statistical analyses. The evaluation of the developed empirical correlations for CO₂ and H₂ solubilities is provided in

Table 6. Summary of the performance of developed empirical correlations against CO₂ and H₂ solubility experimental data.

Correlation	Systems	Parameter Ranges	Correlations Performance Metrics
			No. of Data Points	R2	APE (%)	AAPE (%)
CO2–Water	—		926	0.993	2.34	7.62
CO2–Water– Salt	Mixed Salts (K+, Na+, Ca2+, Mg2+)		391	0.971	−0.7585	8.1914
	KCl		260	0.958	−4.89	12.04
	NaCl		766	0.983	−3.87	10.01
	Na2SO4		226	0.945	0.46	9.13
	NaHCO3		75	0.992	−5.15	8.52
	CaCl2		355	0.989	1.81	6.91
	MgCl2		223	0.988	−0.49	6.81
H2–Water	—		360	0.999	−0.90	4.03
H2–Water– Salt	NaCl		78	0.965	−4.94	9.91

.

Table 6 shows the R², APE, and AAPE metrics for each developed CO₂ solubility correlation. Overall, the correlation for the CO₂–H₂O system is the most reliable, closely followed by the correlation for the CO₂–H₂O–divalent salt system. However, during the development of all CO₂ correlations exhibited good prediction results (R² > 0.945), which means that the proposed models can explain 94.5% of the variability in solubility of CO₂ (mol/kg). The correlation for the CO₂–H₂O system presents the performance compared with experimental data (R² = 0.993 and AAPE = 7.62). The relative closeness of the R² to 1 and low of the AAPE value show that predictions of the developed correlations are closely fitted to the true measured values. Therefore, it is an indication that the mathematical models are good enough to describe the CO₂ experimental measured data.

Similarly, for H₂ solubility data, Table 6 shows the statistical evaluation metrics for each developed H₂ solubility correlation. Overall, the correlation for the H₂–H₂O system is the most reliable, closely followed by the correlation for the H₂–H₂O–salts (NaCl) system. However, the overall performance of the correlations during the development of the H₂ correlations achieved good prediction results (R² > 0.965). which means that the proposed models can explain 96.4% of the variability in solubility of H₂ (mol/kg). The H₂–H₂O system model performs best (R² = 0.999 and AAPE = 4.03). The relative closeness of the R² to 1 and low of the AAPE value show that the predictions of the developed correlations are closely fitted to the experimental values. Therefore, it is an indication that the mathematical models are good enough to describe the H₂ experimental measured data.

5. Conclusions and Recommendations

This paper presents current progress of various techniques for predicting CO₂ solubility and H₂ solubility and experimental databank of CO₂ and H₂ solubility data. The compiled data was used to develop novel simple, non–iterative and reliable models for predicting the solubility factor (mol/kg) in CO₂ and H₂ in pure water and different salt systems. The developed models are a function of pressure, temperature, and salinity in terms of ionic strength. The development and validation of developed CO₂ solubility and H₂ solubility models were conducted using the collected experimental databank from previous studies. The major findings can be summarized as follows:

• Available literature data show that there is sufficient data to evaluate CO₂ solubility in low to medium–pressure regions. However, for CO₂ storage applications in high–pressure and high–temperature regions such as those in deep saline aquifers or gas reservoirs, there is need for more experimental data, especially for mixed salt systems that represent real formation brines.
• For the hydrogen system, there is significant lack of experimental data at higher temperatures (above 400 K) and higher pressures (above 40 MPa). To date, experimental data for mixed salt systems that represent natural formation brine is lacking. These high–temperature and high–pressure region data are critical for understanding H₂ solubility under extreme conditions like those encountered in natural hydrogen production. The limited data in these regions suggests that more experimental studies are needed to understand H₂ solubility under such conditions.
• Reliable and quick empirical models have been developed to accurately predict the solubility of CO in pure water and various salt systems. The pressure range is between 0.1 and 71 MPa, and the temperature range is between 273.15 and 523.15 K. The CO₂ solubility model performs excellently, with an absolute mean error of 7.26 and 8.8% for the pure water and salt systems, respectively, when compared with experimentally measured data.
• Furthermore, the developed simple models can accurately predict the solubility of H₂ in pure water and NaCl salt systems. The pressure range is between 0.1 and 101.3 MPa, and the temperature range is between 273.15 and 636.1 K. Comparison with experimental data shows that the H₂ solubility model performs excellently, with an absolute mean error of 4.03 and 9.1% for the pure water and NaCl salt systems, respectively.
• The salting–out characteristics of various salt systems on CO₂ solubility was accurately captured. Decreasing trend of CO₂ solubility observed based on ionic strength follows the order NaHCO₃ > KCl > CaCl₂ ∼ MgCl₂ > NaCl > Na₂SO₄, and it is consistent with those reported by Zhao et al. [135]. Similarly, the SO characteristics of H₂ gas in NaCl brine are consistent with previous studies by Chabab et al. [152].

It is worth noting that correlations presented here update the CO₂ and H₂ solubility equations presented in the literature and could have a practical application in estimating CO₂ and H₂ solubility in water and saline systems for CO₂–EOR, geologic carbon storage, underground hydrogen storage, and natural hydrogen production processes. However, they may have a limitation: Firstly, these correlations were developed with experimentally measured CO₂ and H₂ solubility data with limited parameter ranges; therefore, these equations can only be used to estimate the CO₂ and H₂ solubility with physical condition ranges presented in Table 1 and Table 2. The application of the developed correlations to calculate CO₂ and H₂ solubility with physical conditions outside these ranges could be associated with the risk of estimations that might be less accurate. Secondly, more experimental data is required to complement the lack of sufficient data, especially the H₂ solubility data at high temperatures, to understand the solubility behavior that occurs in natural hydrogen production. Finally, a comparative study of the performance of equations of state (EoS), empirical correlations, molecular dynamics simulations, and machine learning methods in predicting CO₂ and H₂ solubility is recommended for future studies.