Investigating the Influence of Pore Shape on Shale Gas Recovery with CO₂ Injection Using Molecular Simulation

Abstract

Carbon-dioxide-enhanced shale gas recovery technology has significant potential for large-scale emissions reduction and can help achieve carbon neutrality targets. Previous theoretical studies mainly focused on gas adsorption in one-dimensional pores without considering the influence from the pore geometry. This study evaluates the effects of pore shape on shale gas adsorption. The pure and competitive gas adsorption processes of CO₂ and CH₄ in nanopores were investigated using molecular simulations to improve the prediction of shale gas recovery efficiency. Meanwhile, quantitative analysis was conducted on the effects of the pore shape on the CO₂-EGR efficiency. The results indicate that the density of the adsorption layer in pores is equally distributed in the axial direction when the cone angle is zero; however, when the cone angle is greater than zero, the density of the adsorption layer decreases. Smaller cone-angle pores have stronger gas adsorption affinities, making it challenging to recover the adsorbed CH₄ during the pressure drawdown process. Concurrently, this makes the CO₂ injection method, based on competitive adsorption, efficient. For pores with larger cone angles, the volume occupied by the free gas is larger; thus, the pressure drawdown method displays relatively high recovery efficiency.

1. Introduction

To promote China’s transition to low-carbon development and attain “carbon neutrality”, CO₂ sequestration technology is a necessity. This technology would significantly contribute to accomplishing large-scale, long-term, safe, and stable, deep CO₂ emission reduction. This represents a win–win situation, minimising greenhouse gas emissions whilst boosting the recovery rate of shale gas, since the economic advantages of CO₂ injection to boost oil and gas production could compensate for the costs of CO₂ sequestration. CO₂-enhanced gas recovery (CO₂-EGR) technology is currently considered very promising, owing to the large amount of shale gas resources and the huge potential geological CO₂ storage capacity of shale reservoirs. Based on assessments of world shale oil and gas resources, the recoverable resources from shale gas would measure approximately 7299 trillion cubic feet (tcf) (World Shale Resource Assessments), whilst the potential geological CO₂ storage capacity of the world is estimated at 740 Gt [1]. In CO₂-EGR technology, CO₂ is injected into a shale reservoir, leading to the shale gas in the pore spaces of the shale rock being replaced by CO₂. The injected CO₂ can then be stored in shale gas reservoirs for decades to hundreds of years. Although it has been proven that CO₂-EGR technology can significantly improve shale gas recovery whilst lowering greenhouse gas emissions [2,3,4,5], it is difficult to conduct in situ experiments investigating the recovery mechanisms, given that shale reservoirs have complex conditions, such as high pressure and temperature, and include a large number of nanopores and natural microfractures [6]. At the nanoscale, the interactions among gas molecules, and between pore walls and gas molecules, become particularly significant. Molecular simulation methods, such as grand canonical Monte Carlo (GCMC) simulations, can be useful tools for studying competitive adsorption behaviour and shale gas recovery mechanisms. These methods can both directly reflect the entire process concerning the nanopores and optimise the shale gas recovery path, to achieve increased efficiency of recovery and CO₂ sequestration.

Shale gas is predominantly stored in shale pores, where the proportion of adsorbed gas can be as high as 20 to 85% [7]. Gas adsorption on the shale rock surface is determined by the interactions between the gas and the shale rock. Several variables, such as temperature, pressure, gas composition, shale composition, pore structure, surface properties, and water content, can affect the adsorption properties. Shale is primarily composed of clay minerals and organic matter. Several studies have examined the effects of the mineral composition of shale rocks on CH₄ adsorption, with the results revealing a strong positive correlation between CH₄ adsorption and the total organic carbon (TOC) content [8,9]. Additionally, shales with higher levels of organic matter showed a stronger capacity for CH₄ adsorption [8,9,10,11]. Zhang et al. [11] performed experiments to investigate CH₄ adsorption in different shale samples. Their experimental results showed that the maximum Langmuir adsorption of CH₄ was positively correlated with the TOC content of shale. Two main factors may have accounted for this. Firstly, organic matter in shale reservoirs was the primary source of shale gas and was able to produce hydrocarbons. Secondly, when hydrocarbons were produced, organic matter created many nanoscale pores, which provided the sufficient surface area and pore volume for methane adsorption [12,13]. In general, the TOC content is one of the most important factors affecting shale gas adsorption [11].

Using density functional theory and GCMC simulations, Kurniawan et al. [14] used graphitic slit pores to simulate the organic nanopores in shale reservoirs and studied the adsorption characteristics of methane and its influencing factors. CH₄ density distributions were investigated in graphene slits at 318.15 K, and at pressures ranging from 0.1 to 10 MPa. According to the density profiles of the graphene slit pores, CH₄ molecules can be divided into adsorption and free states. Sun et al. [15] used graphitic cylindrical pores to investigate the impact of kerogen type and pore size on the flow behaviour of CH₄. These simulation results indicated that smaller pores had a stronger affinity for methane and that type III kerogen showed the highest adsorption capacity when compared to type II and type I kerogens.

The CO₂/CH₄ competitive adsorption properties in nanopores have also been widely studied to improve the effectiveness of CO₂-EGR technology. Kurniawan et al. [14] investigated CO₂/CH₄ competitive adsorption in graphene slit pores, using GCMC simulations, and their results showed that the interaction between CO₂ and graphene was stronger than that between CH₄ and graphene. Furthermore, Zeng et al. [16] simulated the CO₂/CH₄ competitive adsorption in organic slit pores and cylindrical pores under pressures of 5–50 MPa and temperatures of 300–360 K. In these cases, the CO₂ adsorption was always stronger than CH₄. Jessen et al. [17] compared the mechanisms of adsorption and desorption of different gas molecules and their mixtures in coal using both experiments and simulations and, subsequently, developed a one-dimensional gas–solid model to explain the experimental results and trends.

Methane adsorption in carbon nanotubes (CNTs) was also investigated using simulations of molecular dynamics, and equations of state for the adsorbed methane molecules were developed [18]. Yuan et al. [19] further studied the effect of the multi-walled CNT radius on methane adsorption and desorption and found a relationship between the pore radius and methane desorption efficiency. Shale rock is dense, heterogeneous, and complex, with pore structures in a range of nanometres and micrometres [20]. Previous research has focused on gas adsorption through one-dimensional pores, such as graphene and CNTs, without considering the influence of pore geometry of other types on the processes of gas adsorption and desorption.

In this study, atomic-level molecular simulations were conducted to examine the pore-shape effects on shale gas adsorption. The effectiveness of the conventional pressure drawdown and CO₂ injection methods for methane desorption was compared, and the impact of the pore shape on this process was further discussed. Corresponding solutions to shale gas recovery were proposed for different application scenarios.

2. Methods

2.1. Simulation Methods

Pores and fractures in shale are intricate and complex, especially considering that shale rocks contain pores of different shapes, such as parallel-plate-shaped, wedge-shaped, or cylindrical pores [21]. Cylindrical pores with two ends open represent a typical pore type in shale [22,23,24]. In this study, cylindrical organic shale nanopores were simulated using CNT structures and heterogeneous pores with one large and one small end, constructed using carbon nanocones (CNCs). The effects of the shape of shale pores on CH₄ adsorption and CH₄ recovery with CO₂ injection were then investigated by altering the cone angle of the CNCs. CO₂ molecules were treated as rigid with the TraPPE-EH force field [25]. CH₄ molecules were modeled with the TraPPE-UA force field [26]. The atom–atom interaction was calculated as:

$$E=E_{\mathrm{ij}}+E_{\mathrm{Coul}}$$

(1)

$$E_{\mathrm{ij}}=4{\epsilon }_{ij}\left[\right(\frac{{\sigma }_{ij}}{r_{ij}}{)}^{12}-(\frac{{\sigma }_{ij}}{r_{ij}}{)}^6]$$

(2)

$$E_{\mathrm{Coul}}=\frac{1}{4\pi {\epsilon }_0}\frac{q_i{q}_j}{r_{ij}}$$

(3)

where i and j denote the atomic numbers; $r_{ij}$ denotes the atomic distance; ${\sigma }_{ij}$ is the zero-potential distance;${\epsilon }_{ij}$ denotes the LJ potential well depth, and ${\epsilon }_0$ is the electrostatic constant. The force field parameters for CH₄, CO₂, and the C atoms of the CNTs and CNCs [27] are listed in

Table 1. Force field parameters for CH₄, CO₂, and C in CNTs or CNCs.

Atom Type	ε/kB 1 (K)	σ (nm)	q(e)
CH4	148	0.373	0
C-CO2	27	0.280	0.70
O-CO2	79	0.305	−0.35
C-CNT/CNC	28.18	0.340	0

¹ k_B is the Boltzman constant.

. Interactions between dissimilar atoms followed the Lorentz–Berthelot (LB) mixing rule [28].

2.2. Simulation Setup

According to Zhu et al. [18], an optimal CNT diameter is present when it maximises adsorption as a result of the competition between the size effect and the curvature effect. The CNT (15,15) was found to be the optimal CNT structure corresponding to the maximum CH₄ adsorption density. CNCs with various cone angles were built while preserving the same volume as the CNT (15,15); the structures of these are shown in Figure 1. In this study, three CNCs with cone angles of 0°, 19.5°, and 30° were chosen to simulate the shale nanopores (CNTs were considered as CNCs with a cone angle of zero), and the effect of pore shape on CH₄ adsorption and CH₄ recovery efficiency was subsequently examined.

Prior to the gas adsorption simulation, force fields were validated by comparing the simulation-derived values with the NIST database for pure gas, and the Peng–Robinson equation of state (PR-EOS) [29] for gas mixtures. Figure 2 shows the pressure–density curves for pure gases and the equimolar CH₄/CO₂ mixture at 338 K for the simulation results, as well as the NIST/PR-EOS values. These results showed that the force field calculations and fluid equation of state correlated very well. The average relative errors between the simulated pressure density values and NIST/PR-EOS values for pure CO₂, pure CH₄, and the equimolar CH₄/CO₂ mixture were 1.58%, 0.45%, and 0.38%, respectively.

The GCMC simulations were conducted to study the CH₄ adsorption and recovery in CNTs and CNCs at 338 K. The pore structure was treated as rigid and connected to gas molecules at both ends in the GCMC simulations. For gas molecules, three types of moves were attempted: translation, insertion, and deletion; however, additionally, rotational moves were included for CO₂ molecules. A total of 1,000,000 cycles were performed per fluid molecule at each experimental pressure. The first half was used for equilibration and the second half was used for sampling. The chemical potentials of the gases were calculated using Widom’s insertion method [30,31]. All the GCMC simulations were conducted using the MCCCS Towhee code [32].

To examine the effects of pore shape on CH₄ adsorption and desorption properties, three carbon nanotubes/cones with consistent volumes but varying cone angles were used in the simulation. The system pressure was adjusted by varying the chemical potential of the gases. Simulated pressure ranged from 10 to 40 MPa during CH₄ adsorption. The adsorption configuration in the equilibrium state of 30 MPa and 338 K was then used as the initial configuration for the CH₄ recovery process. Two CH₄ recovery methods, the conventional pressure drawdown and the CO₂ injection, were compared. The effects of pore shape on the two CH₄ recovery methods were then further studied. At the pressure drawdown stage, the shale reservoir pressure decreased from 30 MPa to 20 MPa, assuming that the gas in the nanopores was in chemical equilibrium with an infinite volume of bulk gas. Further, during the simplified CH₄ recovery process with CO₂ injection, the system pressure was first lowered from 30 MPa to 20 MPa, similar to the pressure reduction during the pressure drawdown process. Subsequently, CO₂ was injected into the system until the bulk mole fraction of CO₂ reached 25%, and the system pressure was increased to 26.1 MPa. After the system reached equilibrium, another pressure drawdown process started, with a pressure decrease from 26.1 MPa to 20 MPa. The pressure and bulk gas composition parameters for the two CH₄ recovery processes are listed in

Table 2. Parameters for the setup of pressure and bulk gas composition during the pressure drawdown and CO₂ injection processes.

	Stage	P (MPa)	Bulk Gas Composition
Pressure drawdown	Initial pressure	30	Pure CH4
	Pressure drawdown	20	Pure CH4
CO2 injection	Initial pressure	30	Pure CH4
	First pressure drawdown	20	Pure CH4
	CO2 injection	26.1	0.75CH4, 0.25CO2
	Second pressure drawdown	20	0.75CH4, 0.25CO2

. To quantify the efficiency of CH₄ recovery and better explain the effects of pore structure on the two processes, the CH₄ recovery efficiency, $\eta$, was measured as the ratio of the number of CH₄ molecules desorbed from the pores throughout the recovery process to the number of CH₄ molecules in the pores at the initial reservoir pressure (30 MPa). This is defined as [33]:

$$\eta =\frac{{\rho }_{30\mathrm{MPa}}-{\rho }_{\mathrm{pore},\mathrm{p}}}{{\rho }_{30\mathrm{MPa}}}$$

(4)

where ${\rho }_{30\mathrm{MPa}}$ is the average CH₄ density under 30 MPa (the initial reservoir pressure), and ${\rho }_{\mathrm{pore},\mathrm{p}}$ is the average CH₄ density in the pore space under pressure p.

3. Results and Discussion

3.1. Effect of Pore Shape on Methane Adsorption Characteristics

To quantitatively characterise the effects of pressure and pore shape on the CH₄ adsorption process, the ${\rho }_{{\mathrm{ave},\mathrm{CH}}_4}$ in CNTs and CNCs at 338 K for different pressures was calculated. The results are displayed in Figure 3.

The pressure effects showed that the ${\rho }_{{\mathrm{ave},\mathrm{CH}}_4}$ within pores increased rapidly with pressure in the low-pressure region, whilst increasing slowly in the comparatively high-pressure region. Since CH₄ adsorption in CNCs increased more in the high-pressure region than that in CNTs, saturation at higher pressures can also be predicted from this trend. In terms of the IUPAC classification of adsorption isotherms, the CH₄ adsorption isotherms of CNCs and CNTs were categorised as type I isotherms, based on this pattern [34]. Therefore, the Langmuir adsorption model was employed to fit the average CH₄ density profile. The Langmuir isotherm model is expressed as [35]:

$${\rho }_{\mathrm{ave}}={\rho }_{\mathrm{L}}\frac{p}{p+p_{\mathrm{L}}}$$

(5)

where ${\rho }_{\mathrm{ave}}$ denotes the average gas density in the pore, p is the bulk pressure, ρ_L is the maximum gas density in the pores, and p_L is the Langmuir pressure. The subsequent results from fitting the Langmuir isotherm model are presented in

Table 3. Langmuir isotherm parameters of methane obtained using nonlinear fitting for CNT/CNC, T= 338 K.

Pore Shape	pL (MPa)	ρL (mmol/cm3)	R2
CNT (15,15)	5.73	14.94	0.991
CNC (19.5°)	11.26	17.42	0.994
CNC (30°)	13.00	18.46	0.988

. The adsorption equilibrium data correlated well using this model, with all correlation coefficients exceeding 0.988. It was consistent with previous studies [33,36]. The Langmuir pressure of CNT (15,15) (p_L = 5.73 MPa) was markedly lower than that of CNC (19.5°) and CNC (30°) (p_L = 11.26 MPa and 13.00 MPa, respectively). The ρ_L of CNT (15,15), CNC (19.5°), and CNC (30°) was 14.94, 17.42, and 18.46 mmol/cm³, respectively. Overall, the maximum gas density and Langmuir pressure both increased with an increase in cone angle. The trends of the CH4 adsorption isotherms in single-walled CNTs are consistent with those in multi-walled CNTs [18], so the effects of CNT walls are not discussed in this work.

To further explore the impact of pore shape on the CH₄ adsorption characteristics, the CH₄ density distributions in CNT (15,15), CNC (19.5°), and CNC (30°) were computed. The results at 338 K and 30 MPa are shown in Figure 4, with the pore wall indicated using the white dashed line. The colour code within the pore space illustrates the gas density from low (blue colour) to high (red colour).

Thus, it is evident that a strong first adsorption layer and comparatively weak second adsorption layer were present adjacent to the pore wall, and the density was significantly higher in this region than in the other regions. In the middle of the pore, known as the free layer [37], the CH₄ distribution was relatively uniform. Compared to previous studies [16,33], clearer distribution of gas adsorption in the pore space is obtained under strict controls on pore roughness. It is also easier to summarize the effects of pore shape on gas adsorption. It is noteworthy that the CH₄ density of the adsorption layers in the CNTs was uniformly distributed in the axial direction, whilst the density of CH₄ in the CNCs decreased with an increase in pore diameter. As shown in Figure 3, the average densities of CH₄ for the three pores were similar at 30 MPa with the same pore volume. Figure 4 shows that the density of the adsorption layer in CNT (15,15) was significantly higher than that of the adsorption layer in CNC (19.5°) and CNC (30°), while the volume of the free gas zone in CNC (30°) was significantly larger than that of the free gas zone in CNC (19.5°) and CNT (15,15). This phenomenon further explains the effects of the cone angle on the trend of ${\rho }_{{\mathrm{ave},\mathrm{CH}}_4}$ with pressure, also shown in Figure 4. At low pressure, the ${\rho }_{{\mathrm{ave},\mathrm{CH}}_4}$ in CNT (15,15) is greater than that in CNC (19.5°) and CNC (30°), owing to stronger adsorption on the pore wall of the CNT (15,15). As the pressure increased, the gas density in the adsorption layer of CNT (15,15) preferentially reached saturation, whereas the gas in the free gas zone of CNC (19.5°) and CNC (30°) could occupy an increased volume and, therefore, gas content further increased as the pressure increased. Generally, the adsorption effect dominated the low-pressure region, and the volume of the gas adsorption zone was larger in pores with smaller cone angles. As the pressure increased, the gas density of the adsorption layer was gradually saturated and the total gas content gradually increased. However, the volume occupied by the free gas zone was larger in the pores that had larger cone angles, and ${\rho }_{{\mathrm{ave},\mathrm{CH}}_4}$ increased more in the pores with larger cone angles than in the pores with smaller cone angles.

3.2. Effect of Pore Shape on the CO₂/CH₄ Competitive Absorption

Shale gas recovery using CO₂ injection is primarily based on the competitive absorption of CH₄/CO₂ in the pore space. In this study, the effects of pore shape on CO₂/CH₄ competitive adsorption characteristics were investigated. Figure 5 shows the equilibrium density distribution of an equimolar CO₂/CH₄ mixture in CNT (15,15) and CNC (19.5°) at 338 K and 20 MPa, with distinct adsorption and free layers formed in the pores. For all three pore shapes, the CO₂ density was significantly higher than CH₄ density, suggesting that CO₂ had a higher adsorption affinity compared to CH₄. The adsorption selectivity of CO₂ over CH₄ in CNT (15,15), CNC (19.5°), and CNC (30°) was 2.09, 1.73, and 1.55, respectively. The CO₂/CH₄ adsorption selectivity decreased with the increase in cone angle but it was always greater than 1, indicating an increased CO₂ density in competitive adsorption. Regarding single-component gas adsorption, the adsorption layer density in CNT (15,15) was uniformly distributed along the axial direction, whereas in CNC (19.5°) and CNC (30°), gas density gradually declined with the increase in pore diameter, whilst the thickness of the adsorption layer also showed a decreasing trend.

3.3. Effect of Pore Shape on Shale Gas Recovery Process

The effects of pore geometry on the CH₄ recovery processes using the CO₂ injection and pressure drawdown methods were further investigated. Following the simulation setup shown in Table 2, the initial pore conditions were assumed to be 30 MPa and 338 K, respectively. Two CH₄ recovery scenarios were compared:

1.

The conventional pressure drawdown method, in which the system pressure was reduced to 20 MPa;

2.

The CO₂ injection method, where the CO₂ huff–puff process was simulated with a combination of the CO₂ injection process and pressure drawdown process.

To quantitatively compare the recovery efficiencies of the two processes and analyse the influence of pore shape on the process, the same initial pressure (30 MPa) and final pressure (20 MPa) were set for both the cases. The setup of the pressure and bulk gas composition during this recovery process is presented in Table 2.

Average densities of CH₄ and CO₂ in the pore space were measured before and after each stage and the evolution of the average gas density in the three pore volumes is shown in Figure 6. The dashed arrows represent the pressure drawdown process whilst the solid arrows represent the CO₂ injection process. At the pressure drawdown stage, ${\rho }_{{\mathrm{ave},\mathrm{CH}}_4}$ gradually decreased, and with the injection of CO₂, CH₄ in the pore space was further recovered. Simultaneously, the injected CO₂ entered the pore space and the CO₂ density increased.

In order to quantify the pore-shape effects on CH₄ recovery processes using the CO₂ injection and pressure drawdown methods, the CH₄ recovery efficiency of the two processes for each of the three pores was calculated, using Equation (3). The results are shown in Figure 7. Recovery efficiencies of the pressure drawdown method in CNT (15,15), CNC (19.5°), and CNC (30°) were 7.53%, 13.17%, and 14.12%, respectively. The recovery efficiency was more than doubled as a result of CO₂ injection. This further verified that CO₂ injection can significantly enhance the recovery efficiency of shale gas. Additionally, the recovery efficiency of the pressure drawdown method increased with an increase in cone angle, whereas the recovery efficiency of the CO₂ injection method decreased with an increase in cone angle. As discussed in Section 3.1, the gas density of the adsorption layer in CNT (15,15) was higher than that of the adsorption layer in CNC (19.5°) and CNC (30°), whilst the volume occupied by the free layer gas was smaller than that of CNC (19.5°) and CNC (30°). The pressure drawdown process mainly affected the gas in the free layer; thus, it is more effective in the case of CNC (30°), in which the volume of the free gas zone is larger than that of the free gas zone in the other pores. However, during CO₂ injection, the CO₂ generally replaced CH₄ in the adsorption layer via competitive adsorption. The adsorption of the CNT (15,15) pore wall was stronger than the CNCs; therefore, the CO₂ injection method could attain a high recovery efficiency. This suggested that for cylindrical pores, particularly those with a small pore diameter, the CO₂ injection method led to a substantial improvement in recovery efficiency.

4. Conclusions

Adsorption properties of pure gases and binary mixtures of CH₄ and CO₂ in CNTs and CNCs with different cone angles were investigated, using GCMC simulation methods. The recovery processes of CH₄ in two cases, that of the conventional pressure drawdown method and the promising CO₂ injection method, were further compared. Effects of pore shape on shale gas adsorption and desorption characteristics were then analysed in detail.

The gas density of the adsorption layer in the CNTs is uniformly distributed along the axial direction, while the density of the adsorption layer in the CNCs decreases with the increase in pore diameter. The gas–wall interaction is stronger in smaller pores than in larger pores, whilst the density of the adsorption layer is higher than that of the adsorption layer in larger pores. Larger pores have a higher ratio of free gas to zone volume. In the low-pressure region, the adsorption effect is dominant, and the average density of gas in pores with a smaller cone angle is higher. As the pressure increases, the gas in the adsorption layer gradually becomes saturated, while in the pores with a larger cone angle, the free gas volume is larger and, therefore, the average density can continue to rise.

For CO₂/CH₄ competitive adsorption, the density of the adsorption layer in the CNC decreases with increasing pore diameter, and the CO₂/CH₄ adsorption selectivity is constantly above 1. For different pore shapes, the CH₄ recovery efficiency using the CO₂ injection method is noticeably higher than that of the CH₄ using the pressure drawdown method. The recovery efficiency of the pressure drawdown method increases with increasing cone angle, whereas that of the CO₂ injection method decreases.

This study focused on the effects of pore shape on shale gas adsorption. The pore surface roughness and element composition are also important factors, which may be considered in further research.